pax_global_header00006660000000000000000000000064123634602020014510gustar00rootroot0000000000000052 comment=4155b611f4e201249c3ebd6b3564e8655029e186 pyGPs-1.2/000077500000000000000000000000001236346020200124145ustar00rootroot00000000000000pyGPs-1.2/CHANGELOG.txt000066400000000000000000000030501236346020200144420ustar00rootroot00000000000000 Changelog pyGPs v1.2 ===================== June 30th 2014 ---------------- structural updates: - input target now can either be in 2-d array with size (n,1) or in 1-d array with size (n,) - setup.py updated - "import pyGPs" instead of "from pyGPs.Core import gp" - rename ".train()" to ".optimize()" - rename "Graph-stuff" to "graphExtension" - rename kernelOnGraph to "nodeKernels" and graphKernel to "graphKernels" - redundancy removed for model.setData(x,y) - rewrite "mean.proceed()" to "getMean()" and "getDerMatrix()" - rewrite "cov.proceed()" to "getCovMatrix()" and "getDerMatrix()" - rename cov.LIN to cov.Linear (to be consistent with mean.Linear) - rename module "valid" to "validation" - add graph dataset Mutag in python file. (.npz and .mat) - add graphUtil.nomalizeKernel() - fix number of iteration problem in graphKernels "PropagationKernel" - add unit testing for covariance, mean functions bug fixes: - derivatives for cov.LINard - derivative of the scalar for cov.covScale - demo_GPR_FITC.py missing pyGPs.mean July 8th 2014 ---------------- structural updates: - add hyperparameter(signal variance s2) for linear covariance - add unit testing for inference,likelihood functions as well as models - NOT show(print) "maximum number of sweep warning in inference EP" any more - documentation updated bug fixes: - typos in lik.Laplace - derivative in lik.Laplace July 14th 2014 ---------------- documentation updates: - online docs updated - API file updated structural updates: - made private for methods that users don't need to call pyGPs-1.2/COPYRIGHT.txt000066400000000000000000000037231236346020200145320ustar00rootroot00000000000000======================================================================================= Marion Neumann [marion dot neumann at uni-bonn dot de] Daniel Marthaler [dan dot marthaler at gmail dot com] Shan Huang [shan dot huang at iais dot fraunhofer dot de] Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] This file is part of pyGPs. The code is released under the BSD 2-Clause (FreeBSD) License. Copyright (c) 2013 Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: - Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. - Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. ======================================================================================= pyGPs-1.2/FUNCTIONALITY.txt000066400000000000000000000065651236346020200152210ustar00rootroot00000000000000 A list of functionality pyGPs =================================== For documentations see documentation and respective modules/functions. ---------------------------- cov functions: ---------------------------- * simple covariance functions: Poly - Polynomial covariance function PiecePoly - Piecewise polynomial kernel with compact support. RBF - Squared Exponential kernel. RBFunit - Squared Exponential kernel with unit magnitude. RBFard - Squared Exponential kernel with Automatic Relevance Determination. Const - Constant kernel. Linear - Linear kernel. LINard - Linear covariance function with Automatic Relevance Detemination. Matern - Matern covariance function. Periodic - Stationary kernel for a smooth periodic function. Noise - Independent covariance function, i.e "white noise". RQ - Rational Quadratic covariance function with isotropic distance measure. RQard - Rational Quadratic covariance function with ARD distance measure. Pre - Precomputed kernel matrix. * composite covariance functions: ProductOfKernel or "*" - product of covariance functions ScaleOfKernel or "*" - scale covariance function (by scalar) SumOfKernel or "+" - sum of (parameterized) covariance functions FITCOfKernel - covariance function to be used together with the FITC approximation ---------------------------- mean functions: ---------------------------- * simple mean functions: Zero - zero mean function One - one mean function Const - constant mean function Linear - linear mean function * composite covariance functions: ProductOfMean or "*" - products of mean functions SumOfMean or "+" - sums of mean functions ScaleOfMean "*" - scaled version of a mean function PowerOfMean "**" - power of a mean function ---------------------------- lik functions: ---------------------------- Erf - Error function, classification, probit regression Gauss - Gaussian likelihood function for regression Laplace - Laplacian likelihood function for regression ---------------------------- inf functions: ---------------------------- Exact - Exact inference (only possible with Gaussian likelihood) EP - Expectation Propagation Laplace - Laplace's Approximation FITC_Exact - Large scale regression with approximate covariance matrix FITC_EP - Large scale inference with approximate covariance matrix FITC_Laplace - Large scale inference with approximate covariance matrix ---------------------------- optimization methods: ---------------------------- Minimize - Minimize by Carl Rasmussen CG - Conjugent gradient BFGS - Quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS) SCG - Scaled conjugent gradient (faster than CG) ---------------------------- evaluation measures: ---------------------------- RMSE - Root mean squared error ACC - Classification accuracy Prec - Precision for class +1 Recall - Recall for class +1 NLPD - Negative log predictive density in transformed observation space pyGPs-1.2/README.md000066400000000000000000000051611236346020200136760ustar00rootroot00000000000000================================================================================ Marion Neumann [marion dot neumann at uni-bonn dot de] Daniel Marthaler [dan dot marthaler at gmail dot com] Shan Huang [shan dot huang at iais dot fraunhofer dot de] Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] This file is part of pyGPs. The software package is released under the BSD 2-Clause (FreeBSD) License. Copyright (c) by Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 ================================================================================ pyGPs is a library containing code for Gaussian Process (GP) Regression and Classification. Here is the online documentation: [ONLINE documentation](http://www-ai.cs.uni-dortmund.de/weblab/static/api_docs/pyGPs/). pyGPs is an object-oriented implementation of GPs. Its functionalities follow roughly the gpml matlab implementaion by Carl Edward Rasmussen and Hannes Nickisch (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2013-01-21). Standard GP regression and (binary) classification as well as FITC (spares GPs) inference is implemented. For a list of implemented covariance, mean, likelihood, and inference functions see list_of_functions.txt. The current implementation is optimized and tested, however, the work on this library is still in progress. We appreciate any feedback. For a comprehensive introduction to functionalities and demonstrations can be found in the *doc* folder; just open /doc/build/html/index.html in your browser to get to the html documentation of the whole package. Further, pyGPs includes implementations of - minimize.py implemented in python by Roland Memisevic 2008, following minimize.m which is copyright (C) 1999 - 2006, Carl Edward Rasmussen - scg.py (Copyright (c) Ian T Nabney (1996-2001)) - brentmin.py (Copyright (c) by Hannes Nickisch 2010-01-10.) Installing pyGPs ------------------ Download the archive and extract it to any local directory. You can either add the local directory to your PYTHONPATH: export PYTHONPATH=$PYTHONPATH:/path/to/local/directory/../parent_folder_of_pyGPs or install the package using setup.py: sudo python setup.py install Requirements -------------- - python 2.6 or 2.7 - scipy, numpy, and matplotlib: open-source packages for scientific computing using the Python programming language. Acknowledgements -------------- The following persons helped to improve this software: Roman Garnett, Maciej Kurek, Hannes Nickisch, Zhao Xu, and Alejandro Molina. This work is partly supported by the Fraunhofer ATTRACT fellowship STREAM. pyGPs-1.2/doc/000077500000000000000000000000001236346020200131615ustar00rootroot00000000000000pyGPs-1.2/doc/Makefile000066400000000000000000000151671236346020200146330ustar00rootroot00000000000000# Makefile for Sphinx documentation # # You can set these variables from the command line. 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These preprocessing implementation are availabe in the source code.hhUubeubeubh)qk}ql(hUhhhh"h$h%h&}qm(h(]h)]h*]h+]qnhah-]qohauh/Kh0hh]qp(h2)qq}qr(hXCross Validationqshhkhh"h$h6h&}qt(h(]h)]h*]h+]h-]uh/Kh0hh]quh9XCross Validationqvqw}qx(hhshhqubaubh=)qy}qz(hX2Now, lets focus on the use of cross-validation. ::q{hhkhh"h$hAh&}q|(h(]h)]h*]h+]h-]uh/Kh0hh]q}h9X/Now, lets focus on the use of cross-validation.q~q}q(hX/Now, lets focus on the use of cross-validation.hhyubaubcdocutils.nodes literal_block q)q}q(hXK = 10 # number of folds for x_train, x_test, y_train, y_test in valid.k_fold_validation(x, y, K): # This is a binary classification problem model = pyGPs.GPC() # Since no prior knowldege, leave everything default model.optimize(x_train, y_train) # Predition ymu, ys2, fmu, fs2, lp = model.predict(x_test, ys=y_test) # ymu for classification is a continuous value over -1 to +1 # If you want predicting result to either one of the classes, take a sign of ymu. ymu_class = np.sign(ymu) # Evluation acc = valid.ACC(ymu_class, y_test) # accuracy rmse = valid.RMSE(ymu_class, y_test) # root-mean-square errorhhkhh"h$U literal_blockqh&}q(U xml:spaceqUpreserveqh+]h*]h(]h)]h-]uh/Kh0hh]qh9XK = 10 # number of folds for x_train, x_test, y_train, y_test in valid.k_fold_validation(x, y, K): # This is a binary classification problem model = pyGPs.GPC() # Since no prior knowldege, leave everything default model.optimize(x_train, y_train) # Predition ymu, ys2, fmu, fs2, lp = model.predict(x_test, ys=y_test) # ymu for classification is a continuous value over -1 to +1 # If you want predicting result to either one of the classes, take a sign of ymu. ymu_class = np.sign(ymu) # Evluation acc = valid.ACC(ymu_class, y_test) # accuracy rmse = valid.RMSE(ymu_class, y_test) # root-mean-square errorqq}q(hUhhubaubeubh)q}q(hUhhhh"h$h%h&}q(h(]h)]h*]h+]qhah-]qhauh/K"h0hh]q(h2)q}q(hXEvaluation measuresqhhhh"h$h6h&}q(h(]h)]h*]h+]h-]uh/K"h0hh]qh9XEvaluation measuresqq}q(hhhhubaubcdocutils.nodes definition_list q)q}q(hUhhhNh$Udefinition_listqh&}q(h(]h)]h*]h+]h-]uh/Nh0hh]qcdocutils.nodes definition_list_item q)q}q(hX)We implemented some classical evaluation measures. - RMSE - root mean squared error - ACC - classification/regression accuracy - Prec - classification precision for class +1 - Recall - classification recall for class +1 - NLPD - negative log predictive density in transformed observation space hhhh"h$Udefinition_list_itemqh&}q(h(]h)]h*]h+]h-]uh/K+h]q(cdocutils.nodes term q)q}q(hX2We implemented some classical evaluation measures.qhhhh"h$Utermqh&}q(h(]h)]h*]h+]h-]uh/K+h]qh9X2We implemented some classical evaluation measures.qq}q(hhhhubaubcdocutils.nodes definition q)q}q(hUh&}q(h(]h)]h*]h+]h-]uhhh]qcdocutils.nodes bullet_list q)q}q(hUh&}q(UbulletqX-h+]h*]h(]h)]h-]uhhh]q(cdocutils.nodes list_item q)q}q(hXRMSE - root mean squared errorqh&}q(h(]h)]h*]h+]h-]uhhh]qh=)q}q(hhhhhh"h$hAh&}q(h(]h)]h*]h+]h-]uh/K$h]qh9XRMSE - root mean squared errorqŅq}q(hhhhubaubah$U list_itemqubh)q}q(hX(ACC - classification/regression accuracyqh&}q(h(]h)]h*]h+]h-]uhhh]qh=)q}q(hhhhhh"h$hAh&}q(h(]h)]h*]h+]h-]uh/K%h]qh9X(ACC - classification/regression accuracyq҅q}q(hhhhubaubah$hubh)q}q(hX,Prec - classification precision for class +1qh&}q(h(]h)]h*]h+]h-]uhhh]qh=)q}q(hhhhhh"h$hAh&}q(h(]h)]h*]h+]h-]uh/K&h]qh9X,Prec - classification precision for class +1qޅq}q(hhhhubaubah$hubh)q}q(hX+Recall - classification recall for class +1qh&}q(h(]h)]h*]h+]h-]uhhh]qh=)q}q(hhhhhh"h$hAh&}q(h(]h)]h*]h+]h-]uh/K'h]qh9X+Recall - classification recall for class +1qꅁq}q(hhhhubaubah$hubh)q}q(hXKNLPD - negative log predictive density in transformed observation space h&}q(h(]h)]h*]h+]h-]uhhh]qh=)q}q(hXGNLPD - negative log predictive density in transformed observation spaceqhhhh"h$hAh&}q(h(]h)]h*]h+]h-]uh/K(h]qh9XGNLPD - negative log predictive density in transformed observation spaceqq}q(hhhhubaubah$hubeh$U bullet_listqubah$U definitionqubeubaubcdocutils.nodes footnote q)q}q(hXSigillito, V. G., Wing, S. P., Hutton, L. V., \& Baker, K. B. (1989). Classification of radar returns from the ionosphere using neural networks. Johns Hopkins APL Technical Digest, 10, 262-266.qh_Khhhh"h$Ufootnoteqh&}r(h(]h)]h*]rhcah+]rhah-]rX1auh/K-h0hh]r(cdocutils.nodes label r)r}r(hUh&}r(h(]h)]h*]h+]h-]uhhh]r h9X1r }r (hUhjubah$Ulabelr ubh=)r }r(hhhhhh"h$hAh&}r(h(]h)]h*]h+]h-]uh/K-h]rh9XSigillito, V. G., Wing, S. P., Hutton, L. V., & Baker, K. B. (1989). Classification of radar returns from the ionosphere using neural networks. Johns Hopkins APL Technical Digest, 10, 262-266.rr}r(hXSigillito, V. G., Wing, S. P., Hutton, L. V., \& Baker, K. B. (1989). Classification of radar returns from the ionosphere using neural networks. Johns Hopkins APL Technical Digest, 10, 262-266.hj ubaubeubeubeubahUU transformerrNU footnote_refsr}rX1]rh]asUrefnamesr}rX1]rh]asUsymbol_footnotesr]rUautofootnote_refsr]rUsymbol_footnote_refsr]r U citationsr!]r"h0hU current_liner#NUtransform_messagesr$]r%Ureporterr&NUid_startr'KU autofootnotesr(]r)U citation_refsr*}r+Uindirect_targetsr,]r-Usettingsr.(cdocutils.frontend Values r/or0}r1(Ufootnote_backlinksr2KUrecord_dependenciesr3NU rfc_base_urlr4Uhttp://tools.ietf.org/html/r5U tracebackr6Upep_referencesr7NUstrip_commentsr8NU toc_backlinksr9Uentryr:U language_coder;Uenr<U datestampr=NU report_levelr>KU _destinationr?NU halt_levelr@KU strip_classesrANh6NUerror_encoding_error_handlerrBUbackslashreplacerCUdebugrDNUembed_stylesheetrEUoutput_encoding_error_handlerrFUstrictrGU sectnum_xformrHKUdump_transformsrINU docinfo_xformrJKUwarning_streamrKNUpep_file_url_templaterLUpep-%04drMUexit_status_levelrNKUconfigrONUstrict_visitorrPNUcloak_email_addressesrQUtrim_footnote_reference_spacerRUenvrSNUdump_pseudo_xmlrTNUexpose_internalsrUNUsectsubtitle_xformrVU source_linkrWNUrfc_referencesrXNUoutput_encodingrYUutf-8rZU source_urlr[NUinput_encodingr\U utf-8-sigr]U_disable_configr^NU id_prefixr_UU tab_widthr`KUerror_encodingraUUTF-8rbU_sourcercU-/Users/hs/Program/git/pyGPs/doc/source/CV.rstrdUgettext_compactreU generatorrfNUdump_internalsrgNU smart_quotesrhU pep_base_urlriUhttp://www.python.org/dev/peps/rjUsyntax_highlightrkUlongrlUinput_encoding_error_handlerrmjGUauto_id_prefixrnUidroUdoctitle_xformrpUstrip_elements_with_classesrqNU _config_filesrr]Ufile_insertion_enabledrsU raw_enabledrtKU dump_settingsruNubUsymbol_footnote_startrvKUidsrw}rx(hhkhhhhhch]hhGhhuUsubstitution_namesry}rzh$h0h&}r{(h(]h+]h*]Usourceh"h)]h-]uU footnotesr|]r}haUrefidsr~}rub.pyGPs-1.2/doc/build/.doctrees/Examples.doctree000066400000000000000000000177671236346020200213150ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(XsourceqXdemosqNXkernels & meansqX optimizersq X pygps/demoq uUsubstitution_defsq }q Uparse_messagesq ]qUcurrent_sourceqNU decorationqNUautofootnote_startqKUnameidsq}q(hUsourceqhUdemosqhU kernels-meansqh U optimizersqh U pygps-demoquUchildrenq]qcdocutils.nodes section q)q}q(U rawsourceqUUparentqhUsourceq cdocutils.nodes reprunicode q!X3/Users/hs/Program/git/pyGPs/doc/source/Examples.rstq"q#}q$bUtagnameq%Usectionq&U attributesq'}q((Udupnamesq)]Uclassesq*]Ubackrefsq+]Uidsq,]q-haUnamesq.]q/hauUlineq0KUdocumentq1hh]q2(cdocutils.nodes title q3)q4}q5(hXDemosq6hhh h#h%Utitleq7h'}q8(h)]h*]h+]h,]h.]uh0Kh1hh]q9cdocutils.nodes Text q:XDemosq;q<}q=(hh6hh4ubaubcdocutils.nodes paragraph q>)q?}q@(hXThere are several demos exemplifying the use of pyGPs for various Gaussian process (:math:`GP`) tasks. 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binary classification (default inference method: EP) model.fit(x, y) # fit default model (mean zero & rbf kernel) with data model.optimize(x, y) # optimize hyperparamters (default optimizer: single run minimize) model.predict(z) # predict test casesh!hh"h%h'hnh)}q(hphqh.]h-]h+]h,]h0]uh2K"h3hh]qhr?}r@(h Uh!j:ubah'UstrongrAubh}q?(h h8h!h6ubaubcdocutils.nodes paragraph q@)qA}qB(h XThe demo in this tutorial can be obtained by running *pyGPs/Demo/demo_GPC_FITC.py*. This demo is more or less a repetition of the demo of `FITC regression`_.h!hh"h%h'U paragraphqCh)}qD(h+]h,]h-]h.]h0]uh2Kh3hh]qE(h}r?(h X( are shown as black circles in the plot.h!j2ubeubeubh)r@}rA(h Uh!hh"h%h'h(h)}rB(h+]h,]h-]h.]rChah0]rDhauh2K1h3hh]rE(h5)rF}rG(h X?Second example :math:`\rightarrow` user-defined inducing pointsrHh!j@h"h%h'h9h)}rI(h+]h,]h-]h.]h0]uh2K1h3hh]rJ(h(hh7h h5ubaubcdocutils.nodes paragraph q?)q@}qA(hXsGPMC is NOT based on multi-class Laplace approximation. It works as a one vs. one classification wrapper. In other words, GPMC trains a GPC model for each pair of two classes, and uses a majority voting scheme over all results to determine the final class. The method only returns the predictive class with highest rating; no other values (such as variance) are returned.qBh hh!h$h&U paragraphqCh(}qD(h*]h+]h,]h-]h/]uh1Kh2hh]qEh;XsGPMC is NOT based on multi-class Laplace approximation. It works as a one vs. one classification wrapper. In other words, GPMC trains a GPC model for each pair of two classes, and uses a majority voting scheme over all results to determine the final class. The method only returns the predictive class with highest rating; no other values (such as variance) are returned.qFqG}qH(hhBh h@ubaubh?)qI}qJ(hXlLets see a practical example to classify the 10 (0,1,2,...9) hand-written digits in the USPS digits dataset.qKh hh!h$h&hCh(}qL(h*]h+]h,]h-]h/]uh1K h2hh]qMh;XlLets see a practical example to classify the 10 (0,1,2,...9) hand-written digits in the USPS digits dataset.qNqO}qP(hhKh hIubaubh)qQ}qR(hUh hh!h$h&h'h(}qS(h*]h+]h,]h-]qThah/]qUhauh1Kh2hh]qV(h4)qW}qX(hX Load dataqYh hQh!h$h&h8h(}qZ(h*]h+]h,]h-]h/]uh1Kh2hh]q[h;X Load dataq\q]}q^(hhYh hWubaubh?)q_}q`(hXThe USPS digits data were gathered at the Center of Excellence in Document Analysis and Recognition (CEDAR) at SUNY Buffalo, as part of a project sponsored by the US Postal Service. The dataset is described in [1]_. ::qah hQh!h$h&hCh(}qb(h*]h+]h,]h-]h/]uh1Kh2hh]qc(h;XThe USPS digits data were gathered at the Center of Excellence in Document Analysis and Recognition (CEDAR) at SUNY Buffalo, as part of a project sponsored by the US Postal Service. The dataset is described in qdqe}qf(hXThe USPS digits data were gathered at the Center of Excellence in Document Analysis and Recognition (CEDAR) at SUNY Buffalo, as part of a project sponsored by the US Postal Service. The dataset is described in h h_ubcdocutils.nodes footnote_reference qg)qh}qi(hX[1]_UresolvedqjKh h_h&Ufootnote_referenceqkh(}ql(h-]qmUid1qnah,]h*]h+]h/]Urefidqohuh]qph;X1qq}qr(hUh hhubaubh;X.qs}qt(hX.h h_ubeubcdocutils.nodes literal_block qu)qv}qw(hXdata = loadmat('data_for_demo/usps_resampled.mat') x = data['train_patterns'].T # train patterns y = data['train_labels'].T # train labels xs = data['test_patterns'].T # test patterns ys = data['test_labels'].T # test labelsh hQh!h$h&U literal_blockqxh(}qy(U xml:spaceqzUpreserveq{h-]h,]h*]h+]h/]uh1Kh2hh]q|h;Xdata = loadmat('data_for_demo/usps_resampled.mat') x = data['train_patterns'].T # train patterns y = data['train_labels'].T # train labels xs = data['test_patterns'].T # test patterns ys = data['test_labels'].T # test labelsq}q~}q(hUh hvubaubh?)q}q(hXLTo be used in GPMC, labels should start from 0 to k (k = number of classes).qh hQh!h$h&hCh(}q(h*]h+]h,]h-]h/]uh1Kh2hh]qh;XLTo be used in GPMC, labels should start from 0 to k (k = number of classes).qq}q(hhh hubaubeubh)q}q(hUh hh!h$h&h'h(}q(h*]h+]h,]h-]qhah/]qh auh1Kh2hh]q(h4)q}q(hX GPMC exampleqh hh!h$h&h8h(}q(h*]h+]h,]h-]h/]uh1Kh2hh]qh;X GPMC exampleqq}q(hhh hubaubh?)q}q(hX4State model with 10-class classification problem: ::qh hh!h$h&hCh(}q(h*]h+]h,]h-]h/]uh1Kh2hh]qh;X1State model with 10-class classification problem:qq}q(hX1State model with 10-class classification problem:h hubaubhu)q}q(hXmodel = pyGPs.GPMC(10)h hh!h$h&hxh(}q(hzh{h-]h,]h*]h+]h/]uh1K h2hh]qh;Xmodel = pyGPs.GPMC(10)qq}q(hUh hubaubh?)q}q(hXPass data to model: ::qh hh!h$h&hCh(}q(h*]h+]h,]h-]h/]uh1K"h2hh]qh;XPass data to model:qq}q(hXPass data to model:h hubaubhu)q}q(hXmodel.setData(x,y)h hh!h$h&hxh(}q(hzh{h-]h,]h*]h+]h/]uh1K$h2hh]qh;Xmodel.setData(x,y)qq}q(hUh hubaubh?)q}q(hXTrain default GPC model for each binary classification problem, and decide label for test patterns of hand-writen digits. The return value *predictive_vote[i,j]* is the probability of being class *j* for test pattern *i*. ::h hh!h$h&hCh(}q(h*]h+]h,]h-]h/]uh1K&h2hh]q(h;XTrain default GPC model for each binary classification problem, and decide label for test patterns of hand-writen digits. The return value qq}q(hXTrain default GPC model for each binary classification problem, and decide label for test patterns of hand-writen digits. The return value h hubcdocutils.nodes emphasis q)q}q(hX*predictive_vote[i,j]*h(}q(h*]h+]h,]h-]h/]uh hh]qh;Xpredictive_vote[i,j]qq}q(hUh hubah&Uemphasisqubh;X# is the probability of being class qąq}q(hX# is the probability of being class h hubh)q}q(hX*j*h(}q(h*]h+]h,]h-]h/]uh hh]qh;Xjq}q(hUh hubah&hubh;X for test pattern qͅq}q(hX for test pattern h hubh)q}q(hX*i*h(}q(h*]h+]h,]h-]h/]uh hh]qh;Xiq}q(hUh hubah&hubh;X.q}q(hX.h hubeubhu)q}q(hXdpredictive_vote = model.optimizeAndPredict(xs) predictive_class = np.argmax(predictive_vote, axis=1)h hh!h$h&hxh(}q(hzh{h-]h,]h*]h+]h/]uh1K*h2hh]qh;Xdpredictive_vote = model.optimizeAndPredict(xs) predictive_class = np.argmax(predictive_vote, axis=1)q܅q}q(hUh hubaubh?)q}q(hXJust like we did for GP classification, you can use specific settings (other than default) for all binary classificiation problem for example by: ::h hh!h$h&hCh(}q(h*]h+]h,]h-]h/]uh1K-h2hh]qh;XJust like we did for GP classification, you can use specific settings (other than default) for all binary classificiation problem for example by:qㅁq}q(hXJust like we did for GP classification, you can use specific settings (other than default) for all binary classificiation problem for example by:h hubaubhu)q}q(hXgm = pyGPs.mean.Zero() k = pyGPs.cov.RBF() model.setPrior(mean=m,kernel=k) model.useInference("Laplace")h hh!h$h&hxh(}q(hzh{h-]h,]h*]h+]h/]uh1K0h2hh]qh;Xgm = pyGPs.mean.Zero() k = pyGPs.cov.RBF() model.setPrior(mean=m,kernel=k) model.useInference("Laplace")qꅁq}q(hUh hubaubh?)q}q(hXXFor more information on how to use non-default settings see `demo_GPC`_ and `demo_GPR`_.qh hh!h$h&hCh(}q(h*]h+]h,]h-]h/]uh1K5h2hh]q(h;X<For more information on how to use non-default settings see qq}q(hX<For more information on how to use non-default settings see h hubcdocutils.nodes reference q)q}q(hX `demo_GPC`_hjKh hh&U referenceqh(}q(UnameXdemo_GPCUrefuriqXGPC.htmlqh-]h,]h*]h+]h/]uh]qh;Xdemo_GPCqq}q(hUh hubaubh;X and rr}r(hX and h hubh)r}r(hX `demo_GPR`_hjKh hh&hh(}r(UnameXdemo_GPRhXGPR.htmlrh-]h,]h*]h+]h/]uh]rh;Xdemo_GPRrr }r (hUh jubaubh;X.r }r (hX.h hubeubcdocutils.nodes target r )r}r(hX.. _demo_GPC: GPC.htmlU referencedrKh hh!h$h&Utargetrh(}r(hhh-]rhah,]h*]h+]h/]rh auh1K8h2hh]ubj )r}r(hX.. _demo_GPR: GPR.htmljKh hh!h$h&jh(}r(hjh-]rhah,]h*]h+]h/]rhauh1K9h2hh]ubh?)r}r(hXvBeside *optimizeAndPredict(xs)*, there is also an option to perform prediction without hyperparameter optimization: ::h hh!h$h&hCh(}r(h*]h+]h,]h-]h/]uh1K(hUh(}r?(h*]h+]h,]h-]h/]uh j3h]r@h;X1rA}rB(hUh j=ubah&UlabelrCubh?)rD}rE(hj5h j3h!h$h&hCh(}rF(h*]h+]h,]h-]h/]uh1KAh]rGh;X`A Database for Handwritten Text Recognition Research, J. J. Hull, IEEE PAMI 16(5) 550-554, 1994.rHrI}rJ(hj5h jDubaubeubeubeubahUU transformerrKNU footnote_refsrL}rMX1]rNhhasUrefnamesrO}rP(X1]rQhhaXdemo_gpc]rRhaXdemo_gpr]rSjauUsymbol_footnotesrT]rUUautofootnote_refsrV]rWUsymbol_footnote_refsrX]rYU citationsrZ]r[h2hU current_liner\NUtransform_messagesr]]r^Ureporterr_NUid_startr`KU autofootnotesra]rbU citation_refsrc}rdUindirect_targetsre]rfUsettingsrg(cdocutils.frontend Values rhori}rj(Ufootnote_backlinksrkKUrecord_dependenciesrlNU rfc_base_urlrmUhttp://tools.ietf.org/html/rnU tracebackroUpep_referencesrpNUstrip_commentsrqNU toc_backlinksrrUentryrsU language_codertUenruU datestamprvNU report_levelrwKU _destinationrxNU halt_levelryKU strip_classesrzNh8NUerror_encoding_error_handlerr{Ubackslashreplacer|Udebugr}NUembed_stylesheetr~Uoutput_encoding_error_handlerrUstrictrU sectnum_xformrKUdump_transformsrNU docinfo_xformrKUwarning_streamrNUpep_file_url_templaterUpep-%04drUexit_status_levelrKUconfigrNUstrict_visitorrNUcloak_email_addressesrUtrim_footnote_reference_spacerUenvrNUdump_pseudo_xmlrNUexpose_internalsrNUsectsubtitle_xformrU source_linkrNUrfc_referencesrNUoutput_encodingrUutf-8rU source_urlrNUinput_encodingrU utf-8-sigrU_disable_configrNU id_prefixrUU tab_widthrKUerror_encodingrUUTF-8rU_sourcerU//Users/hs/Program/git/pyGPs/doc/source/GPMC.rstrUgettext_compactrU generatorrNUdump_internalsrNU smart_quotesrU pep_base_urlrUhttp://www.python.org/dev/peps/rUsyntax_highlightrUlongrUinput_encoding_error_handlerrjUauto_id_prefixrUidrUdoctitle_xformrUstrip_elements_with_classesrNU _config_filesr]Ufile_insertion_enabledrU raw_enabledrKU dump_settingsrNubUsymbol_footnote_startrKUidsr}r(hhhhhjhj3hnhhhhQhjuUsubstitution_namesr}rh&h2h(}r(h*]h-]h,]Usourceh$h+]h/]uU footnotesr]rj3aUrefidsr}rub.pyGPs-1.2/doc/build/.doctrees/GPR.doctree000066400000000000000000000707021236346020200201530ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(Xa bit more things you can doqNXa more complicated exampleqNXbasic regressionqNX load dataq NXgpmlq Ximport packagesq NXa five-line toy exampleq NXkernels & meansq X optimizersquUsubstitution_defsq}qUparse_messagesq]qUcurrent_sourceqNU decorationqNUautofootnote_startqKUnameidsq}q(hUa-bit-more-things-you-can-doqhUa-more-complicated-exampleqhUbasic-regressionqh U load-dataqh Ugpmlqh Uimport-packagesqh Ua-five-line-toy-exampleqh U kernels-meansqhU optimizersq uUchildrenq!]q"cdocutils.nodes section q#)q$}q%(U rawsourceq&UUparentq'hUsourceq(cdocutils.nodes reprunicode q)X./Users/hs/Program/git/pyGPs/doc/source/GPR.rstq*q+}q,bUtagnameq-Usectionq.U attributesq/}q0(Udupnamesq1]Uclassesq2]Ubackrefsq3]Uidsq4]q5haUnamesq6]q7hauUlineq8KUdocumentq9hh!]q:(cdocutils.nodes title q;)q<}q=(h&XBasic Regressionq>h'h$h(h+h-Utitleq?h/}q@(h1]h2]h3]h4]h6]uh8Kh9hh!]qAcdocutils.nodes Text qBXBasic RegressionqCqD}qE(h&h>h'hhBX:You can specify non-default mean and covariance functions:r?r@}rA(h&X:You can specify non-default mean and covariance functions:h'j:ubaubhw)rB}rC(h&Xim = pyGPs.mean.Linear( D=x.shape[1] ) + mean.Const() k = pyGPs.cov.RBF() model.setPrior(mean=m, kernel=k)h'j$h(h+h-hzh/}rD(h|h}h4]h3]h1]h2]h6]uh8K7h9hh!]rEhBXim = pyGPs.mean.Linear( D=x.shape[1] ) + mean.Const() k = pyGPs.cov.RBF() model.setPrior(mean=m, kernel=k)rFrG}rH(h&Uh'jBubaubhF)rI}rJ(h&XiHere, we use a composite mean as the sum of a linear and a constant function, and an rbf kernel. The initial hyperparameters are left to their default values. See `Kernels & Means`_ for a complete documentation of kernel/mean specification and custom kernel/mean construction. Once kernel and mean are specified, they are passed to the prior using *setPrior()*.rKh'j$h(h+h-hJh/}rL(h1]h2]h3]h4]h6]uh8K;h9hh!]rM(hBXHere, we use a composite mean as the sum of a linear and a constant function, and an rbf kernel. The initial hyperparameters are left to their default values. See rNrO}rP(h&XHere, we use a composite mean as the sum of a linear and a constant function, and an rbf kernel. The initial hyperparameters are left to their default values. See h'jIubcdocutils.nodes reference rQ)rR}rS(h&X`Kernels & Means`_UresolvedrTKh'jIh-U referencerUh/}rV(UnameXKernels & MeansUrefurirWX Kernels.htmlrXh4]h3]h1]h2]h6]uh!]rYhBXKernels & MeansrZr[}r\(h&Uh'jRubaubhBX for a complete documentation of kernel/mean specification and custom kernel/mean construction. Once kernel and mean are specified, they are passed to the prior using r]r^}r_(h&X for a complete documentation of kernel/mean specification and custom kernel/mean construction. Once kernel and mean are specified, they are passed to the prior using h'jIubhP)r`}ra(h&X *setPrior()*h/}rb(h1]h2]h3]h4]h6]uh'jIh!]rchBX setPrior()rdre}rf(h&Uh'j`ubah-hXubhBX.rg}rh(h&X.h'jIubeubcdocutils.nodes target ri)rj}rk(h&X!.. _Kernels & Means: Kernels.htmlU referencedrlKh'j$h(h+h-Utargetrmh/}rn(jWjXh4]rohah3]h1]h2]h6]rph auh8K=h9hh!]ubhF)rq}rr(h&X/You can add the traning data to the model explicitly by using *setData()*. So, you avoid passing them into *fit()* or *optimize()* each time used. More importantly, the deafult mean will be adapted to the average value of the trainging labels :math:`y` (if you do not specify mean function by your own).rsh'j$h(h+h-hJh/}rt(h1]h2]h3]h4]h6]uh8K?h9hh!]ru(hBX>You can add the traning data to the model explicitly by using rvrw}rx(h&X>You can add the traning data to the model explicitly by using h'jqubhP)ry}rz(h&X *setData()*h/}r{(h1]h2]h3]h4]h6]uh'jqh!]r|hBX setData()r}r~}r(h&Uh'jyubah-hXubhBX". So, you avoid passing them into rr}r(h&X". So, you avoid passing them into h'jqubhP)r}r(h&X*fit()*h/}r(h1]h2]h3]h4]h6]uh'jqh!]rhBXfit()rr}r(h&Uh'jubah-hXubhBX or rr}r(h&X or h'jqubhP)r}r(h&X *optimize()*h/}r(h1]h2]h3]h4]h6]uh'jqh!]rhBX optimize()rr}r(h&Uh'jubah-hXubhBXq each time used. More importantly, the deafult mean will be adapted to the average value of the trainging labels rr}r(h&Xq each time used. More importantly, the deafult mean will be adapted to the average value of the trainging labels h'jqubh)r}r(h&Uh/}r(UlatexXyh4]h3]h1]h2]h6]uh'jqh!]h-hubhBX3 (if you do not specify mean function by your own).rr}r(h&X3 (if you do not specify mean function by your own).h'jqubeubhF)r}r(h&X2Further, you can plot the data in the 1-d case: ::rh'j$h(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8KAh9hh!]rhBX/Further, you can plot the data in the 1-d case:rr}r(h&X/Further, you can plot the data in the 1-d case:h'jubaubhw)r}r(h&Xmodel.plotData_1d()h'j$h(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8KCh9hh!]rhBXmodel.plotData_1d()rr}r(h&Uh'jubaubj)r}r(h&Uh'j$h(h+h-jh/}r(jXcenterh4]h3]h1]h2]h6]uh8Nh9hh!]rj)r}r(h&Xd.. figure:: _images/d1_2.png :height: 600 px :width: 800 px :align: center :scale: 70 % h/}r(UscaleKFh1]UuriX_images/d1_2.pngrh4]h3]UwidthX800pxh2]j!}rU*jsh6]UheightX600pxuh'jh!]h-j#ubaubhF)r}r(h&XYou can specify a optimization method different from the default, which is a single run of Rasmussen's minimize. For example, you can choose to rerun the optimization method several times with different random initializations: ::h'j$h(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8KKh9hh!]rhBXYou can specify a optimization method different from the default, which is a single run of Rasmussen's minimize. For example, you can choose to rerun the optimization method several times with different random initializations:rr}r(h&XYou can specify a optimization method different from the default, which is a single run of Rasmussen's minimize. For example, you can choose to rerun the optimization method several times with different random initializations:h'jubaubhw)r}r(h&X/model.setOptimizer("Minimize", num_restarts=30)h'j$h(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8KNh9hh!]rhBX/model.setOptimizer("Minimize", num_restarts=30)rr}r(h&Uh'jubaubhF)r}r(h&XThe optimized hyperparameters returned by *optimize()* are then set to be the ones obtained from the run with the best result. The whole functionality for optimization is introduced in detail in the documentation `Optimizers`_.h'j$h(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8KPh9hh!]r(hBX*The optimized hyperparameters returned by rr}r(h&X*The optimized hyperparameters returned by h'jubhP)r}r(h&X *optimize()*h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX optimize()rr}r(h&Uh'jubah-hXubhBX are then set to be the ones obtained from the run with the best result. The whole functionality for optimization is introduced in detail in the documentation rr}r(h&X are then set to be the ones obtained from the run with the best result. The whole functionality for optimization is introduced in detail in the documentation h'jubjQ)r}r(h&X `Optimizers`_jTKh'jh-jUh/}r(UnameX OptimizersjWX Opts.htmlrh4]h3]h1]h2]h6]uh!]rhBX Optimizersrr}r(h&Uh'jubaubhBX.r}r(h&X.h'jubeubji)r}r(h&X.. _Optimizers: Opts.htmljlKh'j$h(h+h-jmh/}r(jWjh4]rh ah3]h1]h2]h6]rhauh8KSh9hh!]ubhF)r}r(h&XInstead of *fit()*, which only fits data using given hyperparameters, *optimize()* will optimize hyperparamters based on marginal likelihood: ::rh'j$h(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8KUh9hh!]r(hBX Instead of rr}r(h&X Instead of h'jubhP)r}r(h&X*fit()*h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXfit()rr}r(h&Uh'jubah-hXubhBX4, which only fits data using given hyperparameters, rr}r(h&X4, which only fits data using given hyperparameters, h'jubhP)r}r(h&X *optimize()*h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX optimize()rr}r(h&Uh'jubah-hXubhBX; will optimize hyperparamters based on marginal likelihood:rr}r(h&X; will optimize hyperparamters based on marginal likelihood:h'jubeubhw)r}r(h&Xmodel.optimize()h'j$h(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8KWh9hh!]rhBXmodel.optimize()rr}r(h&Uh'jubaubhF)r}r(h&X;There are several properties you can get from the model: ::rh'j$h(h+h-hJh/}r (h1]h2]h3]h4]h6]uh8KZh9hh!]r hBX8There are several properties you can get from the model:r r }r (h&X8There are several properties you can get from the model:h'jubaubhw)r}r(h&X-model.nlZ # negative log marginal likelihood model.dnlZ.cov # direvatives of negative log marginal likelihood model.dnlZ.lik model.dnlZ.mean model.posterior.sW # posterior structure model.posterior.alpha model.posterior.L model.covfunc.hyp model.meanfunc.hyp model.likfunc.hyp model.fm # latent mean model.fs2 # latent variance model.ym # predictive mean model.ys2 # predictive variance model.lp # log predictive probabilityh'j$h(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8K\h9hh!]rhBX-model.nlZ # negative log marginal likelihood model.dnlZ.cov # direvatives of negative log marginal likelihood model.dnlZ.lik model.dnlZ.mean model.posterior.sW # posterior structure model.posterior.alpha model.posterior.L model.covfunc.hyp model.meanfunc.hyp model.likfunc.hyp model.fm # latent mean model.fs2 # latent variance model.ym # predictive mean model.ys2 # predictive variance model.lp # log predictive probabilityrr}r(h&Uh'jubaubhF)r}r(h&X7For example, to get the log marginal likelihood use: ::rh'j$h(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Klh9hh!]rhBX4For example, to get the log marginal likelihood use:rr}r(h&X4For example, to get the log marginal likelihood use:h'jubaubhw)r}r(h&XGprint 'Optimized negative log marginal likelihood:', round(model.nlZ,3)h'j$h(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8Knh9hh!]r hBXGprint 'Optimized negative log marginal likelihood:', round(model.nlZ,3)r!r"}r#(h&Uh'jubaubhF)r$}r%(h&XPrediction on the test data will return five values, which are output mean (ymu) resp. variance (ys2), latent mean (fmu) resp. variance (fs2), and log predictive probabilities (lp) ::h'j$h(h+h-hJh/}r&(h1]h2]h3]h4]h6]uh8Kqh9hh!]r'hBXPrediction on the test data will return five values, which are output mean (ymu) resp. variance (ys2), latent mean (fmu) resp. variance (fs2), and log predictive probabilities (lp)r(r)}r*(h&XPrediction on the test data will return five values, which are output mean (ymu) resp. variance (ys2), latent mean (fmu) resp. variance (fs2), and log predictive probabilities (lp)h'j$ubaubhw)r+}r,(h&X'ym, ys2, fm, fs2, lp = model.predict(z)h'j$h(h+h-hzh/}r-(h|h}h4]h3]h1]h2]h6]uh8Kth9hh!]r.hBX'ym, ys2, fm, fs2, lp = model.predict(z)r/r0}r1(h&Uh'j+ubaubhF)r2}r3(h&XPlot data. Note that *GPR.plot()* is a toy method only for visualising 1-d data. Here we got a different posterior by using a different prior other than in the default example. ::r4h'j$h(h+h-hJh/}r5(h1]h2]h3]h4]h6]uh8Kwh9hh!]r6(hBXPlot data. Note that r7r8}r9(h&XPlot data. Note that h'j2ubhP)r:}r;(h&X *GPR.plot()*h/}r<(h1]h2]h3]h4]h6]uh'j2h!]r=hBX GPR.plot()r>r?}r@(h&Uh'j:ubah-hXubhBX is a toy method only for visualising 1-d data. Here we got a different posterior by using a different prior other than in the default example.rArB}rC(h&X is a toy method only for visualising 1-d data. Here we got a different posterior by using a different prior other than in the default example.h'j2ubeubhw)rD}rE(h&X model.plot()h'j$h(h+h-hzh/}rF(h|h}h4]h3]h1]h2]h6]uh8Kyh9hh!]rGhBX model.plot()rHrI}rJ(h&Uh'jDubaubj)rK}rL(h&Uh'j$h(h+h-jh/}rM(jXcenterh4]h3]h1]h2]h6]uh8Nh9hh!]rNj)rO}rP(h&Xf.. figure:: _images/d1_3.png :height: 600 px :width: 800 px :align: center :scale: 70 % h/}rQ(UscaleKFh1]UuriX_images/d1_3.pngrRh4]h3]UwidthX800pxh2]j!}rSU*jRsh6]UheightX600pxuh'jKh!]h-j#ubaubeubh#)rT}rU(h&Uh'h$h(h+h-h.h/}rV(h1]h2]h3]h4]rWhah6]rXhauh8Kh9hh!]rY(h;)rZ}r[(h&XA bit more things you can dor\h'jTh(h+h-h?h/}r](h1]h2]h3]h4]h6]uh8Kh9hh!]r^hBXA bit more things you can dor_r`}ra(h&j\h'jZubaubhF)rb}rc(h&Xg**[For all Models]** Speed up computation time for prediction if you know posterior in advance. Posterior is passed as an object with three fields (attributes) post.alpha, post.sW and post.L. How to use these vectors to represent the posterior can be best seen from Algorithm 2.1 (page 19) in Chapeter 2 of the `GPML`_ book by Rasmussen and Williams, 2006. ::rdh'jTh(h+h-hJh/}re(h1]h2]h3]h4]h6]uh8Kh9hh!]rf(cdocutils.nodes strong rg)rh}ri(h&X**[For all Models]**h/}rj(h1]h2]h3]h4]h6]uh'jbh!]rkhBX[For all Models]rlrm}rn(h&Uh'jhubah-UstrongroubhBX# Speed up computation time for prediction if you know posterior in advance. Posterior is passed as an object with three fields (attributes) post.alpha, post.sW and post.L. How to use these vectors to represent the posterior can be best seen from Algorithm 2.1 (page 19) in Chapeter 2 of the rprq}rr(h&X# Speed up computation time for prediction if you know posterior in advance. Posterior is passed as an object with three fields (attributes) post.alpha, post.sW and post.L. How to use these vectors to represent the posterior can be best seen from Algorithm 2.1 (page 19) in Chapeter 2 of the h'jbubjQ)rs}rt(h&X`GPML`_jTKh'jbh-jUh/}ru(UnameXGPMLjWX4http://www.gaussianprocess.org/gpml/chapters/RW2.pdfrvh4]h3]h1]h2]h6]uh!]rwhBXGPMLrxry}rz(h&Uh'jsubaubhBX& book by Rasmussen and Williams, 2006.r{r|}r}(h&X& book by Rasmussen and Williams, 2006.h'jbubeubhw)r~}r(h&Xlpost = myPosterior() # known in advance ym, ys2, fm, fs2, lp = model.predict_with_posterior( post,z )h'jTh(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8Kh9hh!]rhBXlpost = myPosterior() # known in advance ym, ys2, fm, fs2, lp = model.predict_with_posterior( post,z )rr}r(h&Uh'j~ubaubji)r}r(h&X>.. _GPML: http://www.gaussianprocess.org/gpml/chapters/RW2.pdfjlKh'jTh(h+h-jmh/}r(jWjvh4]rhah3]h1]h2]h6]rh auh8Kh9hh!]ubhF)r}r(h&XX**[Only for Regression]** Specify noise of data (with :math:`\sigma=0.1` by default): ::rh'jTh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Kh9hh!]r(jg)r}r(h&X**[Only for Regression]**h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX[Only for Regression]rr}r(h&Uh'jubah-joubhBX Specify noise of data (with rr}r(h&X Specify noise of data (with h'jubh)r}r(h&Uh/}r(UlatexX \sigma=0.1h4]h3]h1]h2]h6]uh'jh!]h-hubhBX by default):rr}r(h&X by default):h'jubeubhw)r}r(h&X)model.setNoise( log_sigma = np.log(0.1) )h'jTh(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8Kh9hh!]rhBX)model.setNoise( log_sigma = np.log(0.1) )rr}r(h&Uh'jubaubhF)r}r(h&XeYou do not need to specify the noise parameter if you are optimizing the hyperparamters later anyhow.rh'jTh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Kh9hh!]rhBXeYou do not need to specify the noise parameter if you are optimizing the hyperparamters later anyhow.rr}r(h&jh'jubaubhF)r}r(h&XfAll plotting methods have keyword axisvals. You can adjust plotting range if you want. For example: ::rh'jTh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Kh9hh!]rhBXcAll plotting methods have keyword axisvals. You can adjust plotting range if you want. For example:rr}r(h&XcAll plotting methods have keyword axisvals. You can adjust plotting range if you want. For example:h'jubaubhw)r}r(h&X-model.plot(axisvals = [-1.9, 1.9, -0.9, 3.9])h'jTh(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8Kh9hh!]rhBX-model.plot(axisvals = [-1.9, 1.9, -0.9, 3.9])rr}r(h&Uh'jubaubhF)r}r(h&X6Switch to other Inference and Likelihood functions. ::rh'jTh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Kh9hh!]rhBX3Switch to other Inference and Likelihood functions.rr}r(h&X3Switch to other Inference and Likelihood functions.h'jubaubhw)r}r(h&X7model.useInference("EP") model.useLikelihood("Laplace")h'jTh(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8Kh9hh!]rhBX7model.useInference("EP") model.useLikelihood("Laplace")rr}r(h&Uh'jubaubeubeubah&UU transformerrNU footnote_refsr}rUrefnamesr}r(Xgpml]rjsaXkernels & means]rjRaX optimizers]rjauUsymbol_footnotesr]rUautofootnote_refsr]rUsymbol_footnote_refsr]rU citationsr]rh9hU current_linerNUtransform_messagesr]rUreporterrNUid_startrKU autofootnotesr]rU citation_refsr}rUindirect_targetsr]rUsettingsr(cdocutils.frontend Values ror}r(Ufootnote_backlinksrKUrecord_dependenciesrNU 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pep_base_urlr"Uhttp://www.python.org/dev/peps/r#Usyntax_highlightr$Ulongr%Uinput_encoding_error_handlerr&jUauto_id_prefixr'Uidr(Udoctitle_xformr)Ustrip_elements_with_classesr*NU _config_filesr+]Ufile_insertion_enabledr,U raw_enabledr-KU dump_settingsr.NubUsymbol_footnote_startr/KUidsr0}r1(hhhjjhhahjhhhjThj$hh$h juUsubstitution_namesr2}r3h-h9h/}r4(h1]h4]h3]Usourceh+h2]h6]uU footnotesr5]r6Urefidsr7}r8ub.pyGPs-1.2/doc/build/.doctrees/GPR_FITC.doctree000066400000000000000000000425611236346020200207620ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(X%first example default inducing pointsqNXa bit more things you can doqNX+second example user-defined inducing pointsqNXsparse regressionq NXfitc classificationq uUsubstitution_defsq }q Uparse_messagesq ]qUcurrent_sourceqNU decorationqNUautofootnote_startqKUnameidsq}q(hU%first-example-default-inducing-pointsqhUa-bit-more-things-you-can-doqhU+second-example-user-defined-inducing-pointsqh Usparse-regressionqh Ufitc-classificationquUchildrenq]qcdocutils.nodes section q)q}q(U rawsourceqUUparentqhUsourceq cdocutils.nodes reprunicode q!X3/Users/hs/Program/git/pyGPs/doc/source/GPR_FITC.rstq"q#}q$bUtagnameq%Usectionq&U attributesq'}q((Udupnamesq)]Uclassesq*]Ubackrefsq+]Uidsq,]q-haUnamesq.]q/h auUlineq0KUdocumentq1hh]q2(cdocutils.nodes title q3)q4}q5(hXSparse Regressionq6hhh h#h%Utitleq7h'}q8(h)]h*]h+]h,]h.]uh0Kh1hh]q9cdocutils.nodes Text q:XSparse Regressionq;q<}q=(hh6hh4ubaubcdocutils.nodes paragraph q>)q?}q@(hXThe code shown in this tutorial can be obtained by running *pyGPs/Demo/demo_GPR_FITC.py* This demo is more or less similar to the demo of `FITC classification`_.hhh h#h%U paragraphqAh'}qB(h)]h*]h+]h,]h.]uh0Kh1hh]qC(h:X;The code shown in this tutorial can be obtained by running qDqE}qF(hX;The code shown in this tutorial can be obtained by running hh?ubcdocutils.nodes emphasis qG)qH}qI(hX*pyGPs/Demo/demo_GPR_FITC.py*h'}qJ(h)]h*]h+]h,]h.]uhh?h]qKh:XpyGPs/Demo/demo_GPR_FITC.pyqLqM}qN(hUhhHubah%UemphasisqOubh:X2 This demo is more or less similar to the demo of qPqQ}qR(hX2 This demo is more or less similar to the demo of hh?ubcdocutils.nodes reference qS)qT}qU(hX`FITC classification`_UresolvedqVKhh?h%U referenceqWh'}qX(UnameXFITC classificationUrefuriqYX GPC_FITC.htmlqZh,]h+]h)]h*]h.]uh]q[h:XFITC classificationq\q]}q^(hUhhTubaubh:X.q_}q`(hX.hh?ubeubcdocutils.nodes target qa)qb}qc(hX&.. _FITC classification: GPC_FITC.htmlU referencedqdKhhh h#h%Utargetqeh'}qf(hYhZh,]qghah+]h)]h*]h.]qhh auh0Kh1hh]ubh)qi}qj(hUhhh h#h%h&h'}qk(h)]h*]h+]h,]qlhah.]qmhauh0K h1hh]qn(h3)qo}qp(hX9First example :math:`\rightarrow` default inducing pointsqqhhih h#h%h7h'}qr(h)]h*]h+]h,]h.]uh0K h1hh]qs(h:XFirst example qtqu}qv(hXFirst example qwhhoubcsphinx.ext.mathbase math qx)qy}qz(hUh'}q{(UlatexX \rightarrowq|h,]h+]h)]h*]h.]uhhoh]h%Umathq}ubh:X default inducing pointsq~q}q(hX default inducing pointsqhhoubeubh>)q}q(hX,First load the same data as in the GPR demo.qhhih h#h%hAh'}q(h)]h*]h+]h,]h.]uh0K h1hh]qh:X,First load the same data as in the GPR demo.qq}q(hhhhubaubh>)q}q(hX**[Theory]** In case the number of training inputs :math:`x` exceeds a few hundred, approximate inference using Laplace approximation or expectation propagation takes too long. We offer the FITC approximation based on a low-rank plus diagonal approximation to the exact covariance to deal with these cases. The general idea is to use inducing points :math:`u` and to base the computations on cross-covariances between training, test and inducing points only.hhih h#h%hAh'}q(h)]h*]h+]h,]h.]uh0K h1hh]q(cdocutils.nodes strong q)q}q(hX **[Theory]**h'}q(h)]h*]h+]h,]h.]uhhh]qh:X[Theory]qq}q(hUhhubah%Ustrongqubh:X' In case the number of training inputs qq}q(hX' In case the number of training inputs hhubhx)q}q(hUh'}q(UlatexXxh,]h+]h)]h*]h.]uhhh]h%h}ubh:X" exceeds a few hundred, approximate inference using Laplace approximation or expectation propagation takes too long. We offer the FITC approximation based on a low-rank plus diagonal approximation to the exact covariance to deal with these cases. The general idea is to use inducing points qq}q(hX" exceeds a few hundred, approximate inference using Laplace approximation or expectation propagation takes too long. We offer the FITC approximation based on a low-rank plus diagonal approximation to the exact covariance to deal with these cases. The general idea is to use inducing points hhubhx)q}q(hUh'}q(UlatexXuh,]h+]h)]h*]h.]uhhh]h%h}ubh:Xc and to base the computations on cross-covariances between training, test and inducing points only.qq}q(hXc and to base the computations on cross-covariances between training, test and inducing points only.hhubeubh>)q}q(hX(Okay, now the model is FITC regression::qhhih h#h%hAh'}q(h)]h*]h+]h,]h.]uh0Kh1hh]qh:X'Okay, now the model is FITC regression:qq}q(hX'Okay, now the model is FITC regression:hhubaubcdocutils.nodes literal_block q)q}q(hXmodel = pyGPs.GPR_FITC()hhih h#h%U literal_blockqh'}q(U xml:spaceqUpreserveqh,]h+]h)]h*]h.]uh0Kh1hh]qh:Xmodel = pyGPs.GPR_FITC()qq}q(hUhhubaubh>)q}q(hXThe difference between the usage of basic :math:`GP` regression is that we will have to specify inducing points. In the first example here, we will introduce you how to use the default settings.hhih h#h%hAh'}q(h)]h*]h+]h,]h.]uh0Kh1hh]q(h:X*The difference between the usage of basic qq}q(hX*The difference between the usage of basic hhubhx)q}q(hUh'}q(UlatexXGPh,]h+]h)]h*]h.]uhhh]h%h}ubh:X regression is that we will have to specify inducing points. In the first example here, we will introduce you how to use the default settings.qÅq}q(hX regression is that we will have to specify inducing points. In the first example here, we will introduce you how to use the default settings.hhubeubh>)q}q(hXGThe default inducing points are a grid (hypercube for higher dimensions), where each dimension has 5 values in equidistant steps in :math:`[min, max]`, where :math:`min` and :math:`max` are the minimum and maximum values of the input data by default. In order to specify the dimension of input data, we HAVE TO set data first::hhih h#h%hAh'}q(h)]h*]h+]h,]h.]uh0Kh1hh]q(h:XThe default inducing points are a grid (hypercube for higher dimensions), where each dimension has 5 values in equidistant steps in qʅq}q(hXThe default inducing points are a grid (hypercube for higher dimensions), where each dimension has 5 values in equidistant steps in hhubhx)q}q(hUh'}q(UlatexX [min, max]h,]h+]h)]h*]h.]uhhh]h%h}ubh:X, where qЅq}q(hX, where hhubhx)q}q(hUh'}q(UlatexXminh,]h+]h)]h*]h.]uhhh]h%h}ubh:X and qօq}q(hX and hhubhx)q}q(hUh'}q(UlatexXmaxh,]h+]h)]h*]h.]uhhh]h%h}ubh:X are the minimum and maximum values of the input data by default. In order to specify the dimension of input data, we HAVE TO set data first:q܅q}q(hX are the minimum and maximum values of the input data by default. In order to specify the dimension of input data, we HAVE TO set data first:hhubeubh)q}q(hXmodel.setData(x, y)hhih h#h%hh'}q(hhh,]h+]h)]h*]h.]uh0Kh1hh]qh:Xmodel.setData(x, y)qㅁq}q(hUhhubaubh>)q}q(hX@The number of inducing points per axis is :math:`5` per default.qhhih h#h%hAh'}q(h)]h*]h+]h,]h.]uh0Kh1hh]q(h:X*The number of inducing points per axis is q녁q}q(hX*The number of inducing points per axis is hhubhx)q}q(hUh'}q(UlatexX5h,]h+]h)]h*]h.]uhhh]h%h}ubh:X per default.qq}q(hX per default.hhubeubh>)q}q(hX<Now, the regular training and prediction routines follow: ::qhhih h#h%hAh'}q(h)]h*]h+]h,]h.]uh0K!h1hh]qh:X9Now, the regular training and prediction routines follow:qq}q(hX9Now, the regular training and prediction routines follow:hhubaubh)q}q(hX.model.optimize() model.predict(z) model.plot()hhih h#h%hh'}q(hhh,]h+]h)]h*]h.]uh0K#h1hh]qh:X.model.optimize() model.predict(z) model.plot()rr}r(hUhhubaubcdocutils.nodes figure r)r}r(hUhhih h#h%Ufigurerh'}r(UalignrXcenterh,]h+]h)]h*]h.]uh0Nh1hh]r cdocutils.nodes image r )r }r (hXd.. figure:: _images/d3_1.png :height: 600 px :width: 800 px :align: center :scale: 70 % h'}r (UscaleKFh)]UuriX_images/d3_1.pngrh,]h+]UwidthX800pxh*]U candidatesr}rU*jsh.]UheightX600pxuhjh]h%Uimagerubaubh>)r}r(hX\The equidistant default inducing points :math:`u` that are shown in the figure as black x's.rhhih h#h%hAh'}r(h)]h*]h+]h,]h.]uh0K-h1hh]r(h:X(The equidistant default inducing points rr}r(hX(The equidistant default inducing points hjubhx)r}r(hUh'}r(UlatexXuh,]h+]h)]h*]h.]uhjh]h%h}ubh:X+ that are shown in the figure as black x's.rr}r(hX+ that are shown in the figure as black x's.hjubeubh>)r }r!(hXZTo change the number of inducing points per axis just specify a different value per axis::r"hhih h#h%hAh'}r#(h)]h*]h+]h,]h.]uh0K/h1hh]r$h:XYTo change the number of inducing points per axis just specify a different value per axis:r%r&}r'(hXYTo change the number of inducing points per axis just specify a different value per axis:hj ubaubh)r(}r)(hX&model.setData(x, y, value_per_axis=10)hhih h#h%hh'}r*(hhh,]h+]h)]h*]h.]uh0K1h1hh]r+h:X&model.setData(x, y, value_per_axis=10)r,r-}r.(hUhj(ubaubeubh)r/}r0(hUhhh h#h%h&h'}r1(h)]h*]h+]h,]r2hah.]r3hauh0K5h1hh]r4(h3)r5}r6(hX?Second example :math:`\rightarrow` user-defined inducing pointsr7hj/h h#h%h7h'}r8(h)]h*]h+]h,]h.]uh0K5h1hh]r9(h:XSecond example r:r;}r<(hXSecond example r=hj5ubhx)r>}r?(hUh'}r@(UlatexX \rightarrowrAh,]h+]h)]h*]h.]uhj5h]h%h}ubh:X user-defined inducing pointsrBrC}rD(hX user-defined inducing pointsrEhj5ubeubh>)rF}rG(hXAlternatively, a random subset of the training points can be used as inducing points. Note, that there are plenty of methods to set these inducing points. So, in the second example let us use a user-defined set of inducing points.rHhj/h h#h%hAh'}rI(h)]h*]h+]h,]h.]uh0K7h1hh]rJh:XAlternatively, a random subset of the training points can be used as inducing points. Note, that there are plenty of methods to set these inducing points. So, in the second example let us use a user-defined set of inducing points.rKrL}rM(hjHhjFubaubh>)rN}rO(hX7You can pick a set of fixed inducing points by hand: ::rPhj/h h#h%hAh'}rQ(h)]h*]h+]h,]h.]uh0K:h1hh]rRh:X4You can pick a set of fixed inducing points by hand:rSrT}rU(hX4You can pick a set of fixed inducing points by hand:hjNubaubh)rV}rW(hX0u = np.array([[-1], [-0.8], [-0.5], [0.3],[1.]])hj/h h#h%hh'}rX(hhh,]h+]h)]h*]h.]uh0K)r]}r^(hXYou can also use equidistant inducing points :math:`u`, but without the values on the margin of the grid.(i.e. shrinking the range of values) ::r_hj/h h#h%hAh'}r`(h)]h*]h+]h,]h.]uh0K>h1hh]ra(h:X-You can also use equidistant inducing points rbrc}rd(hX-You can also use equidistant inducing points hj]ubhx)re}rf(hUh'}rg(UlatexXuh,]h+]h)]h*]h.]uhj]h]h%h}ubh:XW, but without the values on the margin of the grid.(i.e. shrinking the range of values)rhri}rj(hXW, but without the values on the margin of the grid.(i.e. shrinking the range of values)hj]ubeubh)rk}rl(hXZnum_u = np.fix(x.shape[0]/2) u = np.linspace(-1.3,1.3,num_u).T u = np.reshape(u,(num_u,1))hj/h h#h%hh'}rm(hhh,]h+]h)]h*]h.]uh0K@h1hh]rnh:XZnum_u = np.fix(x.shape[0]/2) u = np.linspace(-1.3,1.3,num_u).T u = np.reshape(u,(num_u,1))rorp}rq(hUhjkubaubh>)rr}rs(hX-Then pass :math:`u` when specifying prior. ::rthj/h h#h%hAh'}ru(h)]h*]h+]h,]h.]uh0KDh1hh]rv(h:X Then pass rwrx}ry(hX Then pass hjrubhx)rz}r{(hUh'}r|(UlatexXuh,]h+]h)]h*]h.]uhjrh]h%h}ubh:X when specifying prior.r}r~}r(hX when specifying prior.hjrubeubh)r}r(hXm = pyGPs.mean.Zero() k = pyGPs.cov.RBFard(log_ell_list=[0.05,0.17], log_sigma=1.) model.setPrior(mean=m, kernel=k, inducing_points=u)hj/h h#h%hh'}r(hhh,]h+]h)]h*]h.]uh0KFh1hh]rh:Xm = pyGPs.mean.Zero() k = pyGPs.cov.RBFard(log_ell_list=[0.05,0.17], log_sigma=1.) model.setPrior(mean=m, kernel=k, inducing_points=u)rr}r(hUhjubaubh>)r}r(hXThe left figure below shows the result of fixed inducing points, and the right figure shows the result for equidistant :math:`u`.rhj/h h#h%hAh'}r(h)]h*]h+]h,]h.]uh0KJh1hh]r(h:XwThe left figure below shows the result of fixed inducing points, and the right figure shows the result for equidistant rr}r(hXwThe left figure below shows the result of fixed inducing points, and the right figure shows the result for equidistant hjubhx)r}r(hUh'}r(UlatexXuh,]h+]h)]h*]h.]uhjh]h%h}ubh:X.r}r(hX.hjubeubj )r}r(hX+.. image:: _images/d3_2.png :width: 45% hj/h h#h%jh'}r(h)]UuriX_images/d3_2.pngrh,]h+]UwidthX45%h*]j}rU*jsh.]uh0Nh1hh]ubj )r}r(hX,.. image:: _images/d3_3.png :width: 45% hj/h h#h%jh'}r(h)]UuriX_images/d3_3.pngrh,]h+]UwidthX45%h*]j}rU*jsh.]uh0Nh1hh]ubh>)r}r(hX**[Theory]** Note that the predictive variance is overestimated outside the support of the inducing inputs. In a multivariate example where densely sampled inducing inputs are infeasible, one can also try to simply use a random subset of the training points.hj/h h#h%hAh'}r(h)]h*]h+]h,]h.]uh0KSh1hh]r(h)r}r(hX **[Theory]**h'}r(h)]h*]h+]h,]h.]uhjh]rh:X[Theory]rr}r(hUhjubah%hubh:X Note that the predictive variance is overestimated outside the support of the inducing inputs. In a multivariate example where densely sampled inducing inputs are infeasible, one can also try to simply use a random subset of the training points.rr}r(hX Note that the predictive variance is overestimated outside the support of the inducing inputs. 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DataqCqD}qE(h&h>h'hqh-Uindexqh/}q(h4]h3]h1]h2]h6]Uentries]q(UsingleqX*pyGPs.GraphExtensions.nodeKernels (module)X(module-pyGPs.GraphExtensions.nodeKernelsUtqauh8Kh9hh!]ubh)q}q(h&Uh'hh(Nh-hh/}q(h4]h3]h1]h2]h6]Uentries]q(hX9VNDKernel() (in module pyGPs.GraphExtensions.nodeKernels)hUtqauh8Nh9hh!]ubcsphinx.addnodes desc q)q}q(h&Uh'hh(Nh-Udescqh/}q(UnoindexqUdomainqXpyh4]h3]h1]h2]h6]UobjtypeqXfunctionqUdesctypeqhuh8Nh9hh!]q(csphinx.addnodes desc_signature q)q}q(h&XVNDKernel(A, alpha=0.5)h'hh(U qh-Udesc_signatureqh/}q(h4]qhaUmoduleqh)X!pyGPs.GraphExtensions.nodeKernelsqq}qbh3]h1]h2]h6]qhaUfullnameqX VNDKernelqUclassqUUfirstquh8Nh9hh!]q(csphinx.addnodes desc_addname q)q}q(h&X"pyGPs.GraphExtensions.nodeKernels.h'hh(hh-U desc_addnameqh/}q(h1]h2]h3]h4]h6]uh8Nh9hh!]qhBX"pyGPs.GraphExtensions.nodeKernels.qq}q(h&Uh'hubaubcsphinx.addnodes desc_name q)q}q(h&hh'hh(hh-U desc_nameqh/}q(h1]h2]h3]h4]h6]uh8Nh9hh!]qhBX VNDKernelqDžq}q(h&Uh'hubaubcsphinx.addnodes desc_parameterlist q)q}q(h&Uh'hh(hh-Udesc_parameterlistqh/}q(h1]h2]h3]h4]h6]uh8Nh9hh!]q(csphinx.addnodes desc_parameter q)q}q(h&XAh/}q(h1]h2]h3]h4]h6]uh'hh!]qhBXAq}q(h&Uh'hubah-Udesc_parameterqubh)q}q(h&X alpha=0.5h/}q(h1]h2]h3]h4]h6]uh'hh!]qhBX alpha=0.5q܅q}q(h&Uh'hubah-hubeubcsphinx.addnodes only q)q}q(h&Uh'hh(Nh-Uonlyqh/}q(UexprUhtmlqh4]h3]h1]h2]h6]uh8Nh9hh!]qcsphinx.addnodes pending_xref q)q}q(h&Uh/}q(UreftypeUviewcodeqUrefdocXGraphqU refdomainUstdqh4]h3]U refexplicith1]h2]h6]U reftargetX*_modules/pyGPs/GraphExtensions/nodeKernelsUrefidhuh'hh!]qcdocutils.nodes inline q)q}q(h&Uh/}q(h1]h2]qU viewcode-linkqah3]h4]h6]uh'hh!]qhBX[source]qq}q(h&Uh'hubah-Uinlinequbah-U pending_xrefqubaubeubcsphinx.addnodes desc_content q)q}q(h&Uh'hh(hh-U desc_contentqh/}q(h1]h2]h3]h4]h6]uh8Nh9hh!]q(hF)r}r(h&XWVon Neumann Diffusion Kernel on graph (Zhou et al., 2004) (also label spreading kernel)rh'hh(Xy/Users/hs/Program/git/pyGPs/pyGPs/GraphExtensions/nodeKernels.py:docstring of pyGPs.GraphExtensions.nodeKernels.VNDKernelrh-hJh/}r(h1]h2]h3]h4]h6]uh8Kh9hh!]rhBXWVon Neumann Diffusion Kernel on graph (Zhou et al., 2004) (also label spreading kernel)rr}r(h&jh'jubaubhF)r }r (h&X/K = (I - alpha*S)^-1, where S = D^-1/2*A*D^-1/2r h'hh(jh-hJh/}r (h1]h2]h3]h4]h6]uh8Kh9hh!]r hBX/K = (I - alpha*S)^-1, where S = D^-1/2*A*D^-1/2rr}r(h&j h'j ubaubcdocutils.nodes field_list r)r}r(h&Uh'hh(Nh-U field_listrh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]r(cdocutils.nodes field r)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(cdocutils.nodes field_name r)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r hBX Parametersr!r"}r#(h&Uh'jubah-U field_namer$ubcdocutils.nodes field_body r%)r&}r'(h&Uh/}r((h1]h2]h3]h4]h6]uh'jh!]r)cdocutils.nodes bullet_list r*)r+}r,(h&Uh/}r-(h1]h2]h3]h4]h6]uh'j&h!]r.(cdocutils.nodes list_item r/)r0}r1(h&Uh/}r2(h1]h2]h3]h4]h6]uh'j+h!]r3hF)r4}r5(h&Uh/}r6(h1]h2]h3]h4]h6]uh'j0h!]r7(cdocutils.nodes strong r8)r9}r:(h&XAh/}r;(h1]h2]h3]h4]h6]uh'j4h!]r<hBXAr=}r>(h&Uh'j9ubah-Ustrongr?ubhBX -- r@rA}rB(h&Uh'j4ubhBXadjacency matrixrCrD}rE(h&Xadjacency matrixh'j4ubeh-hJubah-U list_itemrFubj/)rG}rH(h&Uh/}rI(h1]h2]h3]h4]h6]uh'j+h!]rJhF)rK}rL(h&Uh/}rM(h1]h2]h3]h4]h6]uh'jGh!]rN(j8)rO}rP(h&Xalphah/}rQ(h1]h2]h3]h4]h6]uh'jKh!]rRhBXalpharSrT}rU(h&Uh'jOubah-j?ubhBX -- rVrW}rX(h&Uh'jKubhBXhyperparameter alpharYrZ}r[(h&Xhyperparameter alphah'jKubeh-hJubah-jFubeh-U bullet_listr\ubah-U field_bodyr]ubeh-Ufieldr^ubj)r_}r`(h&Uh/}ra(h1]h2]h3]h4]h6]uh'jh!]rb(j)rc}rd(h&Uh/}re(h1]h2]h3]h4]h6]uh'j_h!]rfhBXReturnsrgrh}ri(h&Uh'jcubah-j$ubj%)rj}rk(h&Uh/}rl(h1]h2]h3]h4]h6]uh'j_h!]rmhF)rn}ro(h&Uh/}rp(h1]h2]h3]h4]h6]uh'jjh!]rqhBX kernel matrixrrrs}rt(h&X kernel matrixh'jnubah-hJubah-j]ubeh-j^ubeubeubeubh)ru}rv(h&Uh'hh(Nh-hh/}rw(h4]h3]h1]h2]h6]Uentries]rx(hX9cosKernel() (in module pyGPs.GraphExtensions.nodeKernels)hUtryauh8Nh9hh!]ubh)rz}r{(h&Uh'hh(Nh-hh/}r|(hhXpyh4]h3]h1]h2]h6]hXfunctionr}hj}uh8Nh9hh!]r~(h)r}r(h&X cosKernel(A)h'jzh(hh-hh/}r(h4]rhahh)X!pyGPs.GraphExtensions.nodeKernelsrr}rbh3]h1]h2]h6]rhahX cosKernelrhUhuh8Nh9hh!]r(h)r}r(h&X"pyGPs.GraphExtensions.nodeKernels.h'jh(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]rhBX"pyGPs.GraphExtensions.nodeKernels.rr}r(h&Uh'jubaubh)r}r(h&jh'jh(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]rhBX cosKernelrr}r(h&Uh'jubaubh)r}r(h&Uh'jh(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]rh)r}r(h&XAh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXAr}r(h&Uh'jubah-hubaubh)r}r(h&Uh'jh(Nh-hh/}r(Uexprhh4]h3]h1]h2]h6]uh8Nh9hh!]rh)r}r(h&Uh/}r(UreftypehUrefdochU refdomainhh4]h3]U refexplicith1]h2]h6]U reftargetX*_modules/pyGPs/GraphExtensions/nodeKernelsUrefidjuh'jh!]rh)r}r(h&Uh/}r(h1]h2]rhah3]h4]h6]uh'jh!]rhBX[source]rr}r(h&Uh'jubah-hubah-hubaubeubh)r}r(h&Uh'jzh(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]r(hF)r}r(h&X*Cosine Kernel (also Inverse Cosine Kernel)rh'jh(Xy/Users/hs/Program/git/pyGPs/pyGPs/GraphExtensions/nodeKernels.py:docstring of 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matrixh'jubah-hJubah-j]ubeh-j^ubeubeubeubh)r}r(h&Uh'hh(Nh-hh/}r(h4]h3]h1]h2]h6]Uentries]r(hX:diffKernel() (in module pyGPs.GraphExtensions.nodeKernels)hUtrauh8Nh9hh!]ubh)r}r(h&Uh'hh(Nh-hh/}r(hhXpyh4]h3]h1]h2]h6]hXfunctionrhjuh8Nh9hh!]r(h)r }r (h&XdiffKernel(A, beta=0.5)h'jh(hh-hh/}r (h4]r hahh)X!pyGPs.GraphExtensions.nodeKernelsr r}rbh3]h1]h2]h6]rhahX diffKernelrhUhuh8Nh9hh!]r(h)r}r(h&X"pyGPs.GraphExtensions.nodeKernels.h'j h(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]rhBX"pyGPs.GraphExtensions.nodeKernels.rr}r(h&Uh'jubaubh)r}r(h&jh'j h(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]rhBX diffKernelrr}r (h&Uh'jubaubh)r!}r"(h&Uh'j h(hh-hh/}r#(h1]h2]h3]h4]h6]uh8Nh9hh!]r$(h)r%}r&(h&XAh/}r'(h1]h2]h3]h4]h6]uh'j!h!]r(hBXAr)}r*(h&Uh'j%ubah-hubh)r+}r,(h&Xbeta=0.5h/}r-(h1]h2]h3]h4]h6]uh'j!h!]r.hBXbeta=0.5r/r0}r1(h&Uh'j+ubah-hubeubh)r2}r3(h&Uh'j h(Nh-hh/}r4(Uexprhh4]h3]h1]h2]h6]uh8Nh9hh!]r5h)r6}r7(h&Uh/}r8(UreftypehUrefdochU refdomainhh4]h3]U refexplicith1]h2]h6]U 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Parametersrkrl}rm(h&Uh'jgubah-j$ubj%)rn}ro(h&Uh/}rp(h1]h2]h3]h4]h6]uh'jch!]rqj*)rr}rs(h&Uh/}rt(h1]h2]h3]h4]h6]uh'jnh!]ru(j/)rv}rw(h&Uh/}rx(h1]h2]h3]h4]h6]uh'jrh!]ryhF)rz}r{(h&Uh/}r|(h1]h2]h3]h4]h6]uh'jvh!]r}(j8)r~}r(h&XAh/}r(h1]h2]h3]h4]h6]uh'jzh!]rhBXAr}r(h&Uh'j~ubah-j?ubhBX -- rr}r(h&Uh'jzubhBXadjacency matrixrr}r(h&Xadjacency matrixh'jzubeh-hJubah-jFubj/)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jrh!]rhF)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j8)r}r(h&Xbetah/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXbetarr}r(h&Uh'jubah-j?ubhBX -- rr}r(h&Uh'jubhBXhyperparameter betarr}r(h&Xhyperparameter betah'jubeh-hJubah-jFubeh-j\ubah-j]ubeh-j^ubj)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'j_h!]r(j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXReturnsrr}r(h&Uh'jubah-j$ubj%)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhF)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX kernel matrixrr}r(h&X kernel matrixh'jubah-hJubah-j]ubeh-j^ubeubeubeubh)r}r(h&Uh'hh(Nh-hh/}r(h4]h3]h1]h2]h6]Uentries]r(hX7normLap() (in module pyGPs.GraphExtensions.nodeKernels)h Utrauh8Nh9hh!]ubh)r}r(h&Uh'hh(Nh-hh/}r(hhXpyh4]h3]h1]h2]h6]hXfunctionrhjuh8Nh9hh!]r(h)r}r(h&X normLap(A)h'jh(hh-hh/}r(h4]rh ahh)X!pyGPs.GraphExtensions.nodeKernelsrr}rbh3]h1]h2]h6]rh ahXnormLaprhUhuh8Nh9hh!]r(h)r}r(h&X"pyGPs.GraphExtensions.nodeKernels.h'jh(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]rhBX"pyGPs.GraphExtensions.nodeKernels.rr}r(h&Uh'jubaubh)r}r(h&jh'jh(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]rhBXnormLaprr}r(h&Uh'jubaubh)r}r(h&Uh'jh(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]rh)r}r(h&XAh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXAr}r(h&Uh'jubah-hubaubh)r}r(h&Uh'jh(Nh-hh/}r(Uexprhh4]h3]h1]h2]h6]uh8Nh9hh!]rh)r}r(h&Uh/}r(UreftypehUrefdochU refdomainhh4]h3]U refexplicith1]h2]h6]U reftargetX*_modules/pyGPs/GraphExtensions/nodeKernelsUrefidjuh'jh!]rh)r}r(h&Uh/}r(h1]h2]rhah3]h4]h6]uh'jh!]rhBX[source]rr}r(h&Uh'jubah-hubah-hubaubeubh)r}r(h&Uh'jh(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]r(hF)r}r(h&XNormalized Laplacianrh'jh(Xw/Users/hs/Program/git/pyGPs/pyGPs/GraphExtensions/nodeKernels.py:docstring of 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pyGPs.GraphExtensions.nodeKernels)hUtr;auh8Nh9hh!]ubh)r<}r=(h&Uh'hh(Nh-hh/}r>(hhXpyh4]h3]h1]h2]h6]hXfunctionr?hj?uh8Nh9hh!]r@(h)rA}rB(h&XpsInvLapKernel(A)h'j<h(hh-hh/}rC(h4]rDhahh)X!pyGPs.GraphExtensions.nodeKernelsrErF}rGbh3]h1]h2]h6]rHhahXpsInvLapKernelrIhUhuh8Nh9hh!]rJ(h)rK}rL(h&X"pyGPs.GraphExtensions.nodeKernels.h'jAh(hh-hh/}rM(h1]h2]h3]h4]h6]uh8Nh9hh!]rNhBX"pyGPs.GraphExtensions.nodeKernels.rOrP}rQ(h&Uh'jKubaubh)rR}rS(h&jIh'jAh(hh-hh/}rT(h1]h2]h3]h4]h6]uh8Nh9hh!]rUhBXpsInvLapKernelrVrW}rX(h&Uh'jRubaubh)rY}rZ(h&Uh'jAh(hh-hh/}r[(h1]h2]h3]h4]h6]uh8Nh9hh!]r\h)r]}r^(h&XAh/}r_(h1]h2]h3]h4]h6]uh'jYh!]r`hBXAra}rb(h&Uh'j]ubah-hubaubh)rc}rd(h&Uh'jAh(Nh-hh/}re(Uexprhh4]h3]h1]h2]h6]uh8Nh9hh!]rfh)rg}rh(h&Uh/}ri(UreftypehUrefdochU refdomainhh4]h3]U refexplicith1]h2]h6]U 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matrixh'jubeh-hJubah-j]ubeh-j^ubj)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXReturnsrr}r(h&Uh'jubah-j$ubj%)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhF)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX kernel matrixrr}r(h&X kernel matrixh'jubah-hJubah-j]ubeh-j^ubeubeubeubh)r}r(h&Uh'hh(Nh-hh/}r(h4]h3]h1]h2]h6]Uentries]r(hX<regLapKernel() (in module pyGPs.GraphExtensions.nodeKernels)hUtrauh8Nh9hh!]ubh)r}r(h&Uh'hh(Nh-hh/}r(hhXpyh4]h3]h1]h2]h6]hXfunctionrhjuh8Nh9hh!]r(h)r}r(h&XregLapKernel(A, sigma=1)h'jh(hh-hh/}r(h4]rhahh)X!pyGPs.GraphExtensions.nodeKernelsrr}rbh3]h1]h2]h6]rhahX regLapKernelrhUhuh8Nh9hh!]r(h)r}r(h&X"pyGPs.GraphExtensions.nodeKernels.h'jh(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]rhBX"pyGPs.GraphExtensions.nodeKernels.rr}r(h&Uh'jubaubh)r}r(h&jh'jh(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]rhBX regLapKernelrr}r(h&Uh'jubaubh)r}r(h&Uh'jh(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]r(h)r}r(h&XAh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXAr}r(h&Uh'jubah-hubh)r}r(h&Xsigma=1h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXsigma=1rr}r(h&Uh'jubah-hubeubh)r}r(h&Uh'jh(Nh-hh/}r(Uexprhh4]h3]h1]h2]h6]uh8Nh9hh!]rh)r}r(h&Uh/}r(UreftypehUrefdochU refdomainhh4]h3]U refexplicith1]h2]h6]U reftargetX*_modules/pyGPs/GraphExtensions/nodeKernelsUrefidjuh'jh!]rh)r}r(h&Uh/}r(h1]h2]rhah3]h4]h6]uh'jh!]rhBX[source]rr}r(h&Uh'jubah-hubah-hubaubeubh)r}r(h&Uh'jh(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]r(hF)r}r(h&XRegularized Laplacian Kernelrh'jh(X|/Users/hs/Program/git/pyGPs/pyGPs/GraphExtensions/nodeKernels.py:docstring of pyGPs.GraphExtensions.nodeKernels.regLapKernelrh-hJh/}r(h1]h2]h3]h4]h6]uh8Kh9hh!]rhBXRegularized Laplacian Kernelrr}r(h&jh'jubaubj)r }r (h&Uh'jh(Nh-jh/}r (h1]h2]h3]h4]h6]uh8Nh9hh!]r (j)r }r(h&Uh/}r(h1]h2]h3]h4]h6]uh'j h!]r(j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'j h!]rhBX Parametersrr}r(h&Uh'jubah-j$ubj%)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'j h!]rj*)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j/)r }r!(h&Uh/}r"(h1]h2]h3]h4]h6]uh'jh!]r#hF)r$}r%(h&Uh/}r&(h1]h2]h3]h4]h6]uh'j h!]r'(j8)r(}r)(h&XAh/}r*(h1]h2]h3]h4]h6]uh'j$h!]r+hBXAr,}r-(h&Uh'j(ubah-j?ubhBX -- r.r/}r0(h&Uh'j$ubhBXadjacency matrixr1r2}r3(h&Xadjacency matrixh'j$ubeh-hJubah-jFubj/)r4}r5(h&Uh/}r6(h1]h2]h3]h4]h6]uh'jh!]r7hF)r8}r9(h&Uh/}r:(h1]h2]h3]h4]h6]uh'j4h!]r;(j8)r<}r=(h&Xsigmah/}r>(h1]h2]h3]h4]h6]uh'j8h!]r?hBXsigmar@rA}rB(h&Uh'j<ubah-j?ubhBX -- rCrD}rE(h&Uh'j8ubhBXhyperparameter sigmarFrG}rH(h&Xhyperparameter sigmah'j8ubeh-hJubah-jFubeh-j\ubah-j]ubeh-j^ubj)rI}rJ(h&Uh/}rK(h1]h2]h3]h4]h6]uh'j h!]rL(j)rM}rN(h&Uh/}rO(h1]h2]h3]h4]h6]uh'jIh!]rPhBXReturnsrQrR}rS(h&Uh'jMubah-j$ubj%)rT}rU(h&Uh/}rV(h1]h2]h3]h4]h6]uh'jIh!]rWhF)rX}rY(h&Uh/}rZ(h1]h2]h3]h4]h6]uh'jTh!]r[hBX kernel matrixr\r]}r^(h&X kernel matrixh'jXubah-hJubah-j]ubeh-j^ubeubeubeubh)r_}r`(h&Uh'hh(Nh-hh/}ra(h4]h3]h1]h2]h6]Uentries]rb(hX8rwKernel() (in module 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Kernelsq:q;}q<(hh5hh3ubaubcdocutils.nodes paragraph q=)q>}q?(hX[The code shown in this tutorial can be executed by running *pyGPs/Demo/demo_GraphKernel.py*q@hhhh"h$U paragraphqAh&}qB(h(]h)]h*]h+]h-]uh/Kh0hh]qC(h9X;The code shown in this tutorial can be executed by running qDqE}qF(hX;The code shown in this tutorial can be executed by running hh>ubcdocutils.nodes emphasis qG)qH}qI(hX *pyGPs/Demo/demo_GraphKernel.py*h&}qJ(h(]h)]h*]h+]h-]uhh>h]qKh9XpyGPs/Demo/demo_GraphKernel.pyqLqM}qN(hUhhHubah$UemphasisqOubeubh)qP}qQ(hUhhhh"h$h%h&}qR(h(]h)]h*]h+]qShah-]qThauh/Kh0hh]qU(h2)qV}qW(hX Load dataqXhhPhh"h$h6h&}qY(h(]h)]h*]h+]h-]uh/Kh0hh]qZh9X Load dataq[q\}q](hhXhhVubaubh=)q^}q_(hXMutag dataset [1]_ was constructed based on data from review of literatures about mutagenicities in Salmonella Typhimurium based on 200 aromatic and heteroaromatic nitro compounds. Mutag dataset is very suitable for machine learning and is widely used as one of the standard dataset. ::hhPhh"h$hAh&}q`(h(]h)]h*]h+]h-]uh/Kh0hh]qa(h9XMutag dataset qbqc}qd(hXMutag dataset hh^ubcdocutils.nodes footnote_reference qe)qf}qg(hX[1]_UresolvedqhKhh^h$Ufootnote_referenceqih&}qj(h+]qkUid1qlah*]h(]h)]h-]Urefidqmhuh]qnh9X1qo}qp(hUhhfubaubh9X  was constructed based on data from review of literatures about mutagenicities in Salmonella Typhimurium based on 200 aromatic and heteroaromatic nitro compounds. Mutag dataset is very suitable for machine learning and is widely used as one of the standard dataset.qqqr}qs(hX  was constructed based on data from review of literatures about mutagenicities in Salmonella Typhimurium based on 200 aromatic and heteroaromatic nitro compounds. Mutag dataset is very suitable for machine learning and is widely used as one of the standard dataset.hh^ubeubcdocutils.nodes literal_block qt)qu}qv(hXdata = np.load('MUTAG.npz') A = csc_matrix( (data['adj_data'], data['adj_indice'], \ data['adj_indptr']), shape=data['adj_shape']) # n x n adjancy array (sparse matrix) gr_id = data['graph_ind'] # n x 1 graph id array node_label = data['responses'] # n x 1 node label array graph_label = data['labels'] # N x 1 graph label arrayhhPhh"h$U literal_blockqwh&}qx(U xml:spaceqyUpreserveqzh+]h*]h(]h)]h-]uh/K h0hh]q{h9Xdata = np.load('MUTAG.npz') A = csc_matrix( (data['adj_data'], data['adj_indice'], \ data['adj_indptr']), shape=data['adj_shape']) # n x n adjancy array (sparse matrix) gr_id = data['graph_ind'] # n x 1 graph id array node_label = data['responses'] # n x 1 node label array graph_label = data['labels'] # N x 1 graph label arrayq|q}}q~(hUhhuubaubh=)q}q(hXNote that adjacency matrix for all graphs is usually too big to fit into memory when using :math:`GP`. Therefore, we generate :math:`A` using sparse matrix *csc_matrix* provided by *scipy*.hhPhh"h$hAh&}q(h(]h)]h*]h+]h-]uh/Kh0hh]q(h9X[Note that adjacency matrix for all graphs is usually too big to fit into memory when using qq}q(hX[Note that adjacency matrix for all graphs is usually too big to fit into memory when using hhubcsphinx.ext.mathbase math q)q}q(hUh&}q(UlatexXGPh+]h*]h(]h)]h-]uhhh]h$Umathqubh9X. Therefore, we generate qq}q(hX. Therefore, we generate hhubh)q}q(hUh&}q(UlatexXAh+]h*]h(]h)]h-]uhhh]h$hubh9X using sparse matrix qq}q(hX using sparse matrix hhubhG)q}q(hX *csc_matrix*h&}q(h(]h)]h*]h+]h-]uhhh]qh9X csc_matrixqq}q(hUhhubah$hOubh9X provided by qq}q(hX provided by hhubhG)q}q(hX*scipy*h&}q(h(]h)]h*]h+]h-]uhhh]qh9Xscipyqq}q(hUhhubah$hOubh9X.q}q(hX.hhubeubeubh)q}q(hUhhhh"h$h%h&}q(h(]h)]h*]h+]qhah-]qhauh/Kh0hh]q(h2)q}q(hX!Compute propagation kernel matrixqhhhh"h$h6h&}q(h(]h)]h*]h+]h-]uh/Kh0hh]qh9X!Compute propagation kernel matrixqq}q(hhhhubaubh=)q}q(hX9Generate prapagation kernel based on given graph data. ::qhhhh"h$hAh&}q(h(]h)]h*]h+]h-]uh/Kh0hh]qh9X6Generate prapagation kernel based on given graph data.qq}q(hX6Generate prapagation kernel based on given graph data.hhubaubht)q}q(hXnum_Iteration = 10 w = 1e-4 dist = 'tv' # possible values: 'tv', 'hellinger' K = graphKernels.propagationKernel(A, node_label, gr_id, num_Iteration, w, dist, 'label_diffusion')hhhh"h$hwh&}q(hyhzh+]h*]h(]h)]h-]uh/Kh0hh]qh9Xnum_Iteration = 10 w = 1e-4 dist = 'tv' # possible values: 'tv', 'hellinger' K = graphKernels.propagationKernel(A, node_label, gr_id, num_Iteration, w, dist, 'label_diffusion')qq}q(hUhhubaubh=)q}q(hX]Adjacency matrix :math:`A` can either be in format of standard numpy matrix or sparse matrix.qhhhh"h$hAh&}q(h(]h)]h*]h+]h-]uh/K h0hh]q(h9XAdjacency matrix qɅq}q(hXAdjacency matrix hhubh)q}q(hUh&}q(UlatexXAh+]h*]h(]h)]h-]uhhh]h$hubh9XC can either be in format of standard numpy matrix or sparse matrix.qυq}q(hXC can either be in format of standard numpy matrix or sparse matrix.hhubeubeubh)q}q(hUhhhh"h$h%h&}q(h(]h)]h*]h+]qhah-]qhauh/K$h0hh]q(h2)q}q(hXStandard GP Classificationqhhhh"h$h6h&}q(h(]h)]h*]h+]h-]uh/K$h0hh]qh9XStandard GP Classificationq݅q}q(hhhhubaubh=)q}q(hX:math:`K` is a big square matrix with all rows and columns of the number of data points. By specifying the indice of training data and test data, we will form two matrix M1 and M2 with the exact format which *pyGPs.Core.cov.Pre* needed. ::hhhh"h$hAh&}q(h(]h)]h*]h+]h-]uh/K%h0hh]q(h)q}q(hUh&}q(UlatexXKh+]h*]h(]h)]h-]uhhh]h$hubh9X is a big square matrix with all rows and columns of the number of data points. By specifying the indice of training data and test data, we will form two matrix M1 and M2 with the exact format which q煁q}q(hX is a big square matrix with all rows and columns of the number of data points. By specifying the indice of training data and test data, we will form two matrix M1 and M2 with the exact format which hhubhG)q}q(hX*pyGPs.Core.cov.Pre*h&}q(h(]h)]h*]h+]h-]uhhh]qh9XpyGPs.Core.cov.Preqq}q(hUhhubah$hOubh9X needed.qq}q(hX needed.hhubeubht)q}q(hX^M1,M2 = graphUtil.formKernelMatrix(Matrix, indice_train, indice_test) k = pyGPs.cov.Pre(M1,M2)hhhh"h$hwh&}q(hyhzh+]h*]h(]h)]h-]uh/K(h0hh]qh9X^M1,M2 = graphUtil.formKernelMatrix(Matrix, indice_train, indice_test) k = pyGPs.cov.Pre(M1,M2)qq}q(hUhhubaubcdocutils.nodes definition_list q)q}q(hUhhhNh$Udefinition_listqh&}q(h(]h)]h*]h+]h-]uh/Nh0hh]r(cdocutils.nodes definition_list_item r)r}r(hXM1 is a matrix with shape **number of training points plus 1** by **number of test points** - cross covariances matrix (train by test) - last row is self covariances (diagonal of test by test)hhhh"h$Udefinition_list_itemrh&}r(h(]h)]h*]h+]h-]uh/K,h]r(cdocutils.nodes term r)r}r (hX[M1 is a matrix with shape **number of training points plus 1** by **number of test points**r hjhh"h$Utermr h&}r (h(]h)]h*]h+]h-]uh/K,h]r (h9XM1 is a matrix with shape rr}r(hXM1 is a matrix with shape hjubcdocutils.nodes strong r)r}r(hX$**number of training points plus 1**h&}r(h(]h)]h*]h+]h-]uhjh]rh9X number of training points plus 1rr}r(hUhjubah$Ustrongrubh9X by rr}r(hX by hjubj)r}r(hX**number of test points**h&}r(h(]h)]h*]h+]h-]uhjh]r h9Xnumber of test pointsr!r"}r#(hUhjubah$jubeubcdocutils.nodes definition r$)r%}r&(hUh&}r'(h(]h)]h*]h+]h-]uhjh]r(cdocutils.nodes bullet_list r))r*}r+(hUh&}r,(Ubulletr-X-h+]h*]h(]h)]h-]uhj%h]r.(cdocutils.nodes list_item r/)r0}r1(hX(cross covariances matrix (train by test)r2h&}r3(h(]h)]h*]h+]h-]uhj*h]r4h=)r5}r6(hj2hj0hh"h$hAh&}r7(h(]h)]h*]h+]h-]uh/K,h]r8h9X(cross covariances matrix (train by test)r9r:}r;(hj2hj5ubaubah$U list_itemr<ubj/)r=}r>(hX7last row is self covariances (diagonal of test by test)r?h&}r@(h(]h)]h*]h+]h-]uhj*h]rAh=)rB}rC(hj?hj=hh"h$hAh&}rD(h(]h)]h*]h+]h-]uh/K-h]rEh9X7last row is self covariances (diagonal of test by test)rFrG}rH(hj?hjBubaubah$j<ubeh$U bullet_listrIubah$U definitionrJubeubj)rK}rL(hX~M2 is a square matrix with **number of training points** for each dimension - training set covariance matrix (train by train) hhhh"h$jh&}rM(h(]h)]h*]h+]h-]uh/K/h0hh]rN(j)rO}rP(hXKM2 is a square matrix with **number of training points** for each dimensionrQhjKhh"h$j h&}rR(h(]h)]h*]h+]h-]uh/K/h]rS(h9XM2 is a square matrix with rTrU}rV(hXM2 is a square matrix with hjOubj)rW}rX(hX**number of training points**h&}rY(h(]h)]h*]h+]h-]uhjOh]rZh9Xnumber of training pointsr[r\}r](hUhjWubah$jubh9X for each dimensionr^r_}r`(hX for each dimensionhjOubeubj$)ra}rb(hUh&}rc(h(]h)]h*]h+]h-]uhjKh]rdj))re}rf(hUh&}rg(j-X-h+]h*]h(]h)]h-]uhjah]rhj/)ri}rj(hX0training set covariance matrix (train by train) h&}rk(h(]h)]h*]h+]h-]uhjeh]rlh=)rm}rn(hX/training set covariance matrix (train by train)rohjihh"h$hAh&}rp(h(]h)]h*]h+]h-]uh/K/h]rqh9X/training set covariance matrix (train by train)rrrs}rt(hjohjmubaubah$j<ubah$jIubah$jJubeubeubh=)ru}rv(hX<The following is the standard way to do GP classification ::rwhhhh"h$hAh&}rx(h(]h)]h*]h+]h-]uh/K1h0hh]ryh9X9The following is the standard way to do GP classificationrzr{}r|(hX9The following is the standard way to do GP classificationhjuubaubht)r}}r~(hX^model = pyGPs.GPC() model.setPrior(kernel=k) model.fit(x_train, y_train) model.predict(x_test)hhhh"h$hwh&}r(hyhzh+]h*]h(]h)]h-]uh/K3h0hh]rh9X^model = pyGPs.GPC() model.setPrior(kernel=k) model.fit(x_train, y_train) model.predict(x_test)rr}r(hUhj}ubaubcdocutils.nodes footnote r)r}r(hXFeragen, A., Kasenburg, N., Petersen, J., de Bruijne, M., Borgwardt, K.M.: Scalable kernels for graphs with continuous attributes. 2013.rhhKhhhh"h$Ufootnoterh&}r(h(]h)]h*]rhlah+]rhah-]rX1auh/KDh0hh]r(cdocutils.nodes label r)r}r(hUh&}r(h(]h)]h*]h+]h-]uhjh]rh9X1r}r(hUhjubah$Ulabelrubh=)r}r(hjhjhh"h$hAh&}r(h(]h)]h*]h+]h-]uh/KDh]rh9XFeragen, A., Kasenburg, N., Petersen, J., de Bruijne, M., Borgwardt, K.M.: Scalable kernels for graphs with continuous attributes. 2013.rr}r(hjhjubaubeubeubeubahUU transformerrNU footnote_refsr}rX1]rhfasUrefnamesr}rX1]rhfasUsymbol_footnotesr]rUautofootnote_refsr]rUsymbol_footnote_refsr]rU citationsr]rh0hU current_linerNUtransform_messagesr]rUreporterrNUid_startrKU autofootnotesr]rU citation_refsr}rUindirect_targetsr]rUsettingsr(cdocutils.frontend Values ror}r(Ufootnote_backlinksrKUrecord_dependenciesrNU rfc_base_urlrUhttp://tools.ietf.org/html/rU tracebackrUpep_referencesrNUstrip_commentsrNU toc_backlinksrUentryrU language_coderUenrU datestamprNU report_levelrKU _destinationrNU halt_levelrKU strip_classesrNh6NUerror_encoding_error_handlerrUbackslashreplacerUdebugrNUembed_stylesheetrUoutput_encoding_error_handlerrUstrictrU sectnum_xformrKUdump_transformsrNU 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footnotesr]rjaUrefidsr}rub.pyGPs-1.2/doc/build/.doctrees/Install.doctree000066400000000000000000000224611236346020200211300ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(X requirementsqNX matplotlibqXscipyqXdownloadq X installationq NXnumpyq Xpython 2.6 or 2.7q uUsubstitution_defsq }qUparse_messagesq]qcdocutils.nodes system_message q)q}q(U rawsourceqUUparentqcdocutils.nodes section q)q}q(hUhhUsourceqcdocutils.nodes reprunicode qX2/Users/hs/Program/git/pyGPs/doc/source/Install.rstqq}qbUtagnameqUsectionqU attributesq }q!(Udupnamesq"]Uclassesq#]Ubackrefsq$]Uidsq%]q&U installationq'aUnamesq(]q)h auUlineq*KUdocumentq+hUchildrenq,]q-(cdocutils.nodes title q.)q/}q0(hX Installationq1hhhhhUtitleq2h }q3(h"]h#]h$]h%]h(]uh*Kh+hh,]q4cdocutils.nodes Text q5X Installationq6q7}q8(hh1hh/ubaubcdocutils.nodes enumerated_list q9)q:}q;(hUhhhhhUenumerated_listqU.h%]h$]h"]Uprefixq?Uh#]h(]Uenumtypeq@UarabicqAuh*Kh+hh,]qBcdocutils.nodes list_item qC)qD}qE(hXPFirst, download_ the archive from github and extract it to any local directory. hh:hhhU list_itemqFh }qG(h"]h#]h$]h%]h(]uh*Nh+hh,]qHcdocutils.nodes paragraph qI)qJ}qK(hXOFirst, download_ the archive from github and extract it to any local directory.hhDhhhU paragraphqLh }qM(h"]h#]h$]h%]h(]uh*Kh,]qN(h5XFirst, qOqP}qQ(hXFirst, hhJubcdocutils.nodes reference qR)qS}qT(hX download_UresolvedqUKhhJhU referenceqVh }qW(UnameXdownloadqXUrefuriqYX5https://github.com/marionmari/pyGPs/releases/tag/v1.1qZh%]h$]h"]h#]h(]uh,]q[h5Xdownloadq\q]}q^(hUhhSubaubh5X? the archive from github and extract it to any local directory.q_q`}qa(hX? the archive from github and extract it to any local directory.hhJubeubaubaubcdocutils.nodes target qb)qc}qd(hXC.. _download: https://github.com/marionmari/pyGPs/releases/tag/v1.1U referencedqeKhhhhhUtargetqfh }qg(hYhZh%]qhUdownloadqiah$]h"]h#]h(]qjh auh*Kh+hh,]ubh9)qk}ql(hUhhhhhhU.UstartqnKh%]h$]h"]h?Uh#]h(]h@hAuh*Kh+hh,]qo(hC)qp}qq(hXYou can either add the local directory to your PYTHONPATH :: export PYTHONPATH=$PYTHONPATH:/path/to/local/directory/../parent_folder_of_pyGPs hhkhhhhFh }qr(h"]h#]h$]h%]h(]uh*Nh+hh,]qs(hI)qt}qu(hX<You can either add the local directory to your PYTHONPATH ::hhphhhhLh }qv(h"]h#]h$]h%]h(]uh*Kh,]qwh5X9You can either add the local directory to your PYTHONPATHqxqy}qz(hX9You can either add the local directory to your PYTHONPATHhhtubaubcdocutils.nodes literal_block q{)q|}q}(hXPexport PYTHONPATH=$PYTHONPATH:/path/to/local/directory/../parent_folder_of_pyGPshhphU literal_blockq~h }q(U xml:spaceqUpreserveqh%]h$]h"]h#]h(]uh*K h,]qh5XPexport PYTHONPATH=$PYTHONPATH:/path/to/local/directory/../parent_folder_of_pyGPsqq}q(hUhh|ubaubeubhC)q}q(hXKOr install the package using setup.py:: sudo python setup.py install hhkhhhhFh }q(h"]h#]h$]h%]h(]uh*Nh+hh,]q(hI)q}q(hX'Or install the package using setup.py::hhhhhhLh }q(h"]h#]h$]h%]h(]uh*K h,]qh5X&Or install the package using setup.py:qq}q(hX&Or install the package using setup.py:hhubaubh{)q}q(hXsudo python setup.py installhhhh~h }q(hhh%]h$]h"]h#]h(]uh*Kh,]qh5Xsudo python setup.py installqq}q(hUhhubaubeubeubh)q}q(hUhhhhhhh }q(h"]h#]h$]h%]qU requirementsqah(]qhauh*Kh+hh,]q(h.)q}q(hX Requirementsqhhhhhh2h }q(h"]h#]h$]h%]h(]uh*Kh+hh,]qh5X Requirementsqq}q(hhhhubaubcdocutils.nodes bullet_list q)q}q(hUhhhhhU bullet_listqh }q(UbulletqX*h%]h$]h"]h#]h(]uh*Kh+hh,]q(hC)q}q(hX`python 2.6 or 2.7`_qhhhhhhFh }q(h"]h#]h$]h%]h(]uh*Nh+hh,]qhI)q}q(hhhhhhhhLh }q(h"]h#]h$]h%]h(]uh*Kh,]qhR)q}q(hhhUKhhhhVh }q(UnameXpython 2.6 or 2.7hYXhttp://www.python.org/qh%]h$]h"]h#]h(]uh,]qh5Xpython 2.6 or 2.7qq}q(hUhhubaubaubaubhC)q}q(hX``scipy`_, `numpy`_, and `matplotlib`_: open-source packages for scientific computing in Python. hhhhhhFh }q(h"]h#]h$]h%]h(]uh*Nh+hh,]qhI)q}q(hX_`scipy`_, `numpy`_, and `matplotlib`_: open-source packages for scientific computing in Python.qhhhhhhLh }q(h"]h#]h$]h%]h(]uh*Kh,]q(hR)q}q(hX`scipy`_hUKhhhhVh }q(UnameXscipyqhYXhttp://www.scipy.org/qh%]h$]h"]h#]h(]uh,]qh5Xscipyq΅q}q(hUhhubaubh5X, qхq}q(hX, hhubhR)q}q(hX`numpy`_hUKhhhhVh }q(UnameXnumpyqhYXhttp://www.numpy.org/qh%]h$]h"]h#]h(]uh,]qh5Xnumpyqڅq}q(hUhhubaubh5X, and q݅q}q(hX, and hhubhR)q}q(hX `matplotlib`_hUKhhhhVh }q(UnameX matplotlibqhYXhttp://matplotlib.org/qh%]h$]h"]h#]h(]uh,]qh5X matplotlibq慁q}q(hUhhubaubh5X:: open-source packages for scientific computing in Python.q酁q}q(hX:: open-source packages for scientific computing in Python.hhubeubaubeubhb)q}q(hX-.. _python 2.6 or 2.7: http://www.python.org/heKhhhhhhfh }q(hYhh%]qUpython-2-6-or-2-7qah$]h"]h#]h(]qh auh*Kh+hh,]ubhb)q}q(hX .. _scipy: http://www.scipy.org/heKhhhhhhfh }q(hYhh%]qUscipyqah$]h"]h#]h(]qhauh*Kh+hh,]ubhb)q}q(hX .. _numpy: http://www.numpy.org/heKhhhhhhfh }q(hYhh%]qUnumpyqah$]h"]h#]h(]qh auh*Kh+hh,]ubhb)q}q(hX&.. _matplotlib: http://matplotlib.org/heKhhhhhhfh }r(hYhh%]rU matplotlibrah$]h"]h#]h(]rhauh*Kh+hh,]ubhI)r}r(hX)Example installation on Ubuntu & Debian::rhhhhhhLh }r(h"]h#]h$]h%]h(]uh*Kh+hh,]rh5X(Example installation on Ubuntu & Debian:r r }r (hX(Example installation on Ubuntu & Debian:hjubaubh{)r }r (hXJsudo apt-get install python2.7 python-numpy python-scipy python-matplotlibhhhhhh~h }r(hhh%]h$]h"]h#]h(]uh*Kh+hh,]rh5XJsudo apt-get install python2.7 python-numpy python-scipy python-matplotlibrr}r(hUhj ubaubhI)r}r(hXHExample installation on Mac via Macports (requires XCode and MacPorts)::rhhhhhhLh }r(h"]h#]h$]h%]h(]uh*K h+hh,]rh5XGExample installation on Mac via Macports (requires XCode and MacPorts):rr}r(hXGExample installation on Mac via Macports (requires XCode and MacPorts):hjubaubh{)r}r(hX@sudo port install python27 py27-numpy py27-scipy py27-matplotlibhhhhhh~h }r(hhh%]h$]h"]h#]h(]uh*K"h+hh,]rh5X@sudo port install python27 py27-numpy py27-scipy py27-matplotlibrr }r!(hUhjubaubhI)r"}r#(hXaFor other systems please check the installation instructions on the respective package web sites.r$hhhhhhLh }r%(h"]h#]h$]h%]h(]uh*K%h+hh,]r&h5XaFor other systems please check the installation instructions on the respective package web sites.r'r(}r)(hj$hj"ubaubeubeubhhhUsystem_messager*h }r+(h"]UlevelKh%]h$]Usourcehh#]h(]UlineKUtypeUINFOr,uh*Kh+hh,]r-hI)r.}r/(hUh }r0(h"]h#]h$]h%]h(]uhhh,]r1h5X:Enumerated list start value not ordinal-1: "2" (ordinal 2)r2r3}r4(hUhj.ubahhLubaubaUcurrent_sourcer5NU decorationr6NUautofootnote_startr7KUnameidsr8}r9(hhhjhhh hih h'h hh huh,]r:hahUU transformerr;NU footnote_refsr<}r=Urefnamesr>}r?(hX]r@hSah]rAhah]rBhaXpython 2.6 or 2.7]rChah]rDhauUsymbol_footnotesrE]rFUautofootnote_refsrG]rHUsymbol_footnote_refsrI]rJU citationsrK]rLh+hU current_linerMNUtransform_messagesrN]rOUreporterrPNUid_startrQKU autofootnotesrR]rSU citation_refsrT}rUUindirect_targetsrV]rWUsettingsrX(cdocutils.frontend Values rYorZ}r[(Ufootnote_backlinksr\KUrecord_dependenciesr]NU rfc_base_urlr^Uhttp://tools.ietf.org/html/r_U tracebackr`Upep_referencesraNUstrip_commentsrbNU toc_backlinksrcUentryrdU language_codereUenrfU datestamprgNU report_levelrhKU _destinationriNU halt_levelrjKU 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dump_settingsrNubUsymbol_footnote_startrKUidsr}r(hhh'hjhhhhihchhhhuUsubstitution_namesr}rhh+h }r(h"]h%]h$]Usourcehh#]h(]uU footnotesr]rUrefidsr}rub.pyGPs-1.2/doc/build/.doctrees/Kernels.doctree000066400000000000000000003774031236346020200211360ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(XpyGPs.Core.cov.RQqX"pyGPs.Core.cov.Kernel.getCovMatrixqXpyGPs.Core.mean.SumOfMeanqX!pyGPs.Core.mean.Mean.getDerMatrixq Xcomposite kernels & meamsq NXpyGPs.Core.mean.Zeroq XpyGPs.Core.mean.Meanq XpyGPs.Core.cov.Periodicq XpyGPs.Core.cov.Kernel.fitcqXpyGPs.Core.cov.ProductOfKernelqXkernels & meansqNXpyGPs.Core.cov.MaternqXpyGPs.Core.cov.PolyqXpyGPs.Core.cov.FITCOfKernelqX"pyGPs.Core.cov.Kernel.getDerMatrixqXsimple kernel & meanqNXpyGPs.Core.cov.ScaleOfKernelqXpyGPs.Core.mean.PowerOfMeanqXpyGPs.Core.cov.RQardqXpyGPs.Core.cov.NoiseqXpyGPs.Core.cov.RBFardqXpyGPs.Core.cov.LinearqXpyGPs.Core.cov.PiecePolyqX$list of means and default parametersqNXpyGPs.Core.cov.RBFunitqXprecomputed kernel matrixqNXpyGPs.Core.cov.LINardq Xsome special casesq!NXpyGPs.Core.mean.Constq"XpyGPs.Core.cov.RBFq#XpyGPs.Core.cov.Kernelq$XpyGPs.Core.mean.ProductOfMeanq%XpyGPs.Core.mean.ScaleOfMeanq&XpyGPs.Core.mean.Linearq'XpyGPs.Core.cov.Preq(XpyGPs.Core.cov.SumOfKernelq)X&list of kernels and default parametersq*NXpyGPs.Core.mean.Oneq+XpyGPs.Core.mean.Mean.getMeanq,XpyGPs.Core.cov.Constq-Xcustomizing kernel & meanq.NuUsubstitution_defsq/}q0Uparse_messagesq1]q2Ucurrent_sourceq3NU decorationq4NUautofootnote_startq5KUnameidsq6}q7(hhhhhhh h h Ucomposite-kernels-meamsq8h h h h h h hhhhhU kernels-meansq9hhhhhhhhhUsimple-kernel-meanq:hhhhhhhhhhhhhhhU$list-of-means-and-default-parametersq;hhhUprecomputed-kernel-matrixqh+h+h,h,h-h-h.Ucustomizing-kernel-meanq?uUchildrenq@]qAcdocutils.nodes section qB)qC}qD(U rawsourceqEUUparentqFhUsourceqGcdocutils.nodes reprunicode qHX2/Users/hs/Program/git/pyGPs/doc/source/Kernels.rstqIqJ}qKbUtagnameqLUsectionqMU attributesqN}qO(UdupnamesqP]UclassesqQ]UbackrefsqR]UidsqS]qTh9aUnamesqU]qVhauUlineqWKUdocumentqXhh@]qY(cdocutils.nodes title qZ)q[}q\(hEXKernels & Meansq]hFhChGhJhLUtitleq^hN}q_(hP]hQ]hR]hS]hU]uhWKhXhh@]q`cdocutils.nodes Text qaXKernels & Meansqbqc}qd(hEh]hFh[ubaubhB)qe}qf(hEUhFhChGhJhLhMhN}qg(hP]hQ]hR]hS]qhh:ahU]qihauhWKhXhh@]qj(hZ)qk}ql(hEXSimple Kernel & MeanqmhFhehGhJhLh^hN}qn(hP]hQ]hR]hS]hU]uhWKhXhh@]qohaXSimple Kernel & Meanqpqq}qr(hEhmhFhkubaubcdocutils.nodes paragraph qs)qt}qu(hEX]You may already seen, we can specify a kernel function like this(same for mean fucntions): ::qvhFhehGhJhLU paragraphqwhN}qx(hP]hQ]hR]hS]hU]uhWKhXhh@]qyhaXZYou may already seen, we can specify a kernel function like this(same for mean fucntions):qzq{}q|(hEXZYou may already seen, we can specify a kernel function like this(same for mean fucntions):hFhtubaubcdocutils.nodes literal_block q})q~}q(hEX.k = pyGPs.cov.RBF( log_ell=-1., log_sigma=0. )hFhehGhJhLU literal_blockqhN}q(U xml:spaceqUpreserveqhS]hR]hP]hQ]hU]uhWKhXhh@]qhaX.k = pyGPs.cov.RBF( log_ell=-1., log_sigma=0. )qq}q(hEUhFh~ubaubhs)q}q(hEX,There are several points need to be noticed:qhFhehGhJhLhwhN}q(hP]hQ]hR]hS]hU]uhWK hXhh@]qhaX,There are several points need to be noticed:qq}q(hEhhFhubaubcdocutils.nodes enumerated_list q)q}q(hEUhFhehGhJhLUenumerated_listqhN}q(UsuffixqU.hS]hR]hP]UprefixqUhQ]hU]UenumtypeqUarabicquhWK hXhh@]q(cdocutils.nodes list_item q)q}q(hEXMost parameters are initilized in their logorithms. This is because we need to make sure they are positive during optimization. e.g. Here length scale and signal variance should always be positive. hFhhGhJhLU list_itemqhN}q(hP]hQ]hR]hS]hU]uhWNhXhh@]qhs)q}q(hEXMost parameters are initilized in their logorithms. This is because we need to make sure they are positive during optimization. e.g. Here length scale and signal variance should always be positive.qhFhhGhJhLhwhN}q(hP]hQ]hR]hS]hU]uhWK h@]qhaXMost parameters are initilized in their logorithms. This is because we need to make sure they are positive during optimization. e.g. Here length scale and signal variance should always be positive.qq}q(hEhhFhubaubaubh)q}q(hEXWMost kernel functions have a scalar in front, namely signal variance(set by log_sigma) hFhhGhJhLhhN}q(hP]hQ]hR]hS]hU]uhWNhXhh@]qhs)q}q(hEXVMost kernel functions have a scalar in front, namely signal variance(set by log_sigma)qhFhhGhJhLhwhN}q(hP]hQ]hR]hS]hU]uhWKh@]qhaXVMost kernel functions have a scalar in front, namely signal variance(set by log_sigma)qq}q(hEhhFhubaubaubh)q}q(hEXWIf you will do optimization later anyway, you can just leave parameters to be default hFhhGhJhLhhN}q(hP]hQ]hR]hS]hU]uhWNhXhh@]qhs)q}q(hEXUIf you will do optimization later anyway, you can just leave parameters to be defaultqhFhhGhJhLhwhN}q(hP]hQ]hR]hS]hU]uhWKh@]qhaXUIf you will do optimization later anyway, you can just leave parameters to be defaultqq}q(hEhhFhubaubaubeubeubhB)q}q(hEUhFhChGhJhLhMhN}q(hP]hQ]hR]hS]qh=ahU]qh!auhWKhXhh@]q(hZ)q}q(hEXSome Special CasesqhFhhGhJhLh^hN}q(hP]hQ]hR]hS]hU]uhWKhXhh@]qhaXSome Special Casesq˅q}q(hEhhFhubaubh)q}q(hEUhFhhGhJhLhhN}q(hU.hS]hR]hP]hUhQ]hU]hhuhWKhXhh@]q(h)q}q(hEXFor some kernels/means, number of hyperparameters depends on the dimension of input data. You can either enter the dimension, which use default values: :: m = pyGPs.mean.Linear( D=x.shape[1] ) or you can initialze with the exact hyperparameters, you should enter as a list, one element for each dimension :: m = pyGPs.mean.Linear( alpha_list=[0.2, 0.4, 0.3] ) All these "hyp-dim-dependent" functions are: * *pyGPs.mean.Linear* * *pyGPs.cov.RBFard* * *pyGPs.cov.LINard* * *pyGPs.cov.RQard* hFhhGNhLhhN}q(hP]hQ]hR]hS]hU]uhWNhXhh@]q(hs)q}q(hEXFor some kernels/means, number of hyperparameters depends on the dimension of input data. You can either enter the dimension, which use default values: ::hFhhGhJhLhwhN}q(hP]hQ]hR]hS]hU]uhWKh@]qhaXFor some kernels/means, number of hyperparameters depends on the dimension of input data. You can either enter the dimension, which use default values:qڅq}q(hEXFor some kernels/means, number of hyperparameters depends on the dimension of input data. You can either enter the dimension, which use default values:hFhubaubh})q}q(hEX%m = pyGPs.mean.Linear( D=x.shape[1] )hFhhLhhN}q(hhhS]hR]hP]hQ]hU]uhWKh@]qhaX%m = pyGPs.mean.Linear( D=x.shape[1] )qᅁq}q(hEUhFhubaubhs)q}q(hEXror you can initialze with the exact hyperparameters, you should enter as a list, one element for each dimension ::hFhhGhJhLhwhN}q(hP]hQ]hR]hS]hU]uhWKh@]qhaXoor you can initialze with the exact hyperparameters, you should enter as a list, one element for each dimensionq腁q}q(hEXoor you can initialze with the exact hyperparameters, you should enter as a list, one element for each dimensionhFhubaubh})q}q(hEX3m = pyGPs.mean.Linear( alpha_list=[0.2, 0.4, 0.3] )hFhhLhhN}q(hhhS]hR]hP]hQ]hU]uhWKh@]qhaX3m = pyGPs.mean.Linear( alpha_list=[0.2, 0.4, 0.3] )qq}q(hEUhFhubaubcdocutils.nodes definition_list q)q}q(hEUhN}q(hP]hQ]hR]hS]hU]uhFhh@]qcdocutils.nodes definition_list_item q)q}q(hEXAll these "hyp-dim-dependent" functions are: * *pyGPs.mean.Linear* * *pyGPs.cov.RBFard* * *pyGPs.cov.LINard* * *pyGPs.cov.RQard* hFhhGhJhLUdefinition_list_itemqhN}q(hP]hQ]hR]hS]hU]uhWK$h@]q(cdocutils.nodes term q)q}q(hEX,All these "hyp-dim-dependent" functions are:rhFhhGhJhLUtermrhN}r(hP]hQ]hR]hS]hU]uhWK$h@]rhaX,All these "hyp-dim-dependent" functions are:rr}r(hEjhFhubaubcdocutils.nodes definition r)r}r (hEUhN}r (hP]hQ]hR]hS]hU]uhFhh@]r cdocutils.nodes bullet_list r )r }r(hEUhN}r(UbulletrX*hS]hR]hP]hQ]hU]uhFjh@]r(h)r}r(hEX*pyGPs.mean.Linear*rhN}r(hP]hQ]hR]hS]hU]uhFj h@]rhs)r}r(hEjhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWK h@]rcdocutils.nodes emphasis r)r}r(hEjhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXpyGPs.mean.Linearr r!}r"(hEUhFjubahLUemphasisr#ubaubahLhubh)r$}r%(hEX*pyGPs.cov.RBFard*r&hN}r'(hP]hQ]hR]hS]hU]uhFj h@]r(hs)r)}r*(hEj&hFj$hGhJhLhwhN}r+(hP]hQ]hR]hS]hU]uhWK!h@]r,j)r-}r.(hEj&hN}r/(hP]hQ]hR]hS]hU]uhFj)h@]r0haXpyGPs.cov.RBFardr1r2}r3(hEUhFj-ubahLj#ubaubahLhubh)r4}r5(hEX*pyGPs.cov.LINard*r6hN}r7(hP]hQ]hR]hS]hU]uhFj h@]r8hs)r9}r:(hEj6hFj4hGhJhLhwhN}r;(hP]hQ]hR]hS]hU]uhWK"h@]r<j)r=}r>(hEj6hN}r?(hP]hQ]hR]hS]hU]uhFj9h@]r@haXpyGPs.cov.LINardrArB}rC(hEUhFj=ubahLj#ubaubahLhubh)rD}rE(hEX*pyGPs.cov.RQard* hN}rF(hP]hQ]hR]hS]hU]uhFj h@]rGhs)rH}rI(hEX*pyGPs.cov.RQard*rJhFjDhGhJhLhwhN}rK(hP]hQ]hR]hS]hU]uhWK#h@]rLj)rM}rN(hEjJhN}rO(hP]hQ]hR]hS]hU]uhFjHh@]rPhaXpyGPs.cov.RQardrQrR}rS(hEUhFjMubahLj#ubaubahLhubehLU bullet_listrTubahLU definitionrUubeubahLUdefinition_listrVubeubh)rW}rX(hEXFor *pyGPs.cov.RBFunit()*, its signal variance is always 1 (because of unit magnitude). Therefore this function do not have a hyperparameter of "signal variance". hFhhGhJhLhhN}rY(hP]hQ]hR]hS]hU]uhWNhXhh@]rZhs)r[}r\(hEXFor *pyGPs.cov.RBFunit()*, its signal variance is always 1 (because of unit magnitude). Therefore this function do not have a hyperparameter of "signal variance".hFjWhGhJhLhwhN}r](hP]hQ]hR]hS]hU]uhWK&h@]r^(haXFor r_r`}ra(hEXFor hFj[ubj)rb}rc(hEX*pyGPs.cov.RBFunit()*hN}rd(hP]hQ]hR]hS]hU]uhFj[h@]rehaXpyGPs.cov.RBFunit()rfrg}rh(hEUhFjbubahLj#ubhaX, its signal variance is always 1 (because of unit magnitude). Therefore this function do not have a hyperparameter of "signal variance".rirj}rk(hEX, its signal variance is always 1 (because of unit magnitude). Therefore this function do not have a hyperparameter of "signal variance".hFj[ubeubaubh)rl}rm(hEX*pyGPs.cov.Poly()* has three parameters, where hyperparameters are: * c -> inhomogeneous offset * sigma -> signal deviation however, * d -> order of polynomial will be treated as normal parameter, i.e. will not be trained hFhhGNhLhhN}rn(hP]hQ]hR]hS]hU]uhWNhXhh@]roh)rp}rq(hEUhN}rr(hP]hQ]hR]hS]hU]uhFjlh@]rs(h)rt}ru(hEX*pyGPs.cov.Poly()* has three parameters, where hyperparameters are: * c -> inhomogeneous offset * sigma -> signal deviation hFjphGhJhLhhN}rv(hP]hQ]hR]hS]hU]uhWK+h@]rw(h)rx}ry(hEXC*pyGPs.cov.Poly()* has three parameters, where hyperparameters are:hFjthGhJhLjhN}rz(hP]hQ]hR]hS]hU]uhWK+h@]r{(j)r|}r}(hEX*pyGPs.cov.Poly()*hN}r~(hP]hQ]hR]hS]hU]uhFjxh@]rhaXpyGPs.cov.Poly()rr}r(hEUhFj|ubahLj#ubhaX1 has three parameters, where hyperparameters are:rr}r(hEX1 has three parameters, where hyperparameters are:hFjxubeubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjth@]rj )r}r(hEUhN}r(jX*hS]hR]hP]hQ]hU]uhFjh@]r(h)r}r(hEXc -> inhomogeneous offsetrhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEjhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWK*h@]rhaXc -> inhomogeneous offsetrr}r(hEjhFjubaubahLhubh)r}r(hEXsigma -> signal deviation hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEXsigma -> signal deviationrhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWK+h@]rhaXsigma -> signal deviationrr}r(hEjhFjubaubahLhubehLjTubahLjUubeubh)r}r(hEXihowever, * d -> order of polynomial will be treated as normal parameter, i.e. will not be trained hFjphGhJhLhhN}r(hP]hQ]hR]hS]hU]uhWK0h@]r(h)r}r(hEXhowever,rhFjhGhJhLjhN}r(hP]hQ]hR]hS]hU]uhWK0h@]rhaXhowever,rr}r(hEjhFjubaubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rj )r}r(hEUhN}r(jX*hS]hR]hP]hQ]hU]uhFjh@]rh)r}r(hEX\d -> order of polynomial will be treated as normal parameter, i.e. will not be trained hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEXZd -> order of polynomial will be treated as normal parameter, i.e. will not be trainedrhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWK.h@]rhaXZd -> order of polynomial will be treated as normal parameter, i.e. will not be trainedrr}r(hEjhFjubaubahLhubahLjTubahLjUubeubehLjVubaubh)r}r(hEXcExplicitly set *pyGPs.cov.Noise* is not necessary, because noise are already added in likelihood. hFhhGhJhLhhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhs)r}r(hEXaExplicitly set *pyGPs.cov.Noise* is not necessary, because noise are already added in likelihood.hFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWK2h@]r(haXExplicitly set rr}r(hEXExplicitly set hFjubj)r}r(hEX*pyGPs.cov.Noise*hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXpyGPs.cov.Noiserr}r(hEUhFjubahLj#ubhaXA is not necessary, because noise are already added in likelihood.rr}r(hEXA is not necessary, because noise are already added in likelihood.hFjubeubaubeubeubhB)r}r(hEUhFhChGhJhLhMhN}r(hP]hQ]hR]hS]rh8ahU]rh auhWK6hXhh@]r(hZ)r}r(hEXComposite Kernels & MeamsrhFjhGhJhLh^hN}r(hP]hQ]hR]hS]hU]uhWK6hXhh@]rhaXComposite Kernels & Meamsrr}r(hEjhFjubaubhs)r}r(hEX9Adding and muliplying Kernels(Means) is really simple: ::rhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWK7hXhh@]rhaX6Adding and muliplying Kernels(Means) is really simple:rr}r(hEX6Adding and muliplying Kernels(Means) is really simple:hFjubaubh})r}r(hEXYk = pyGPs.cov.Periodic() * pyGPs.cov.RBF() k = 0.5*pyGPs.cov.LIN() + pyGPs.cov.Periodic()hFjhGhJhLhhN}r(hhhS]hR]hP]hQ]hU]uhWK9hXhh@]rhaXYk = pyGPs.cov.Periodic() * pyGPs.cov.RBF() k = 0.5*pyGPs.cov.LIN() + pyGPs.cov.Periodic()rr}r(hEUhFjubaubhs)r}r(hEXScalar will also be treated as a hyperparameter. For example, k = s1 * k1 + s2 * k2, then the list of hyperparameters is hyp = [s1, k1.hyp, s2, k2.hyp]. Scalar is passed in logorithm domain such that it will always be positive during optimization.rhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWK(h)r?}r@(hEXM1 is a matrix with shape **number of training points plus 1** by **number of test points** - cross covariances matrix (train by test) - last row is self covariances (diagonal of test by test)hFj;hGhJhLhhN}rA(hP]hQ]hR]hS]hU]uhWKPh@]rB(h)rC}rD(hEX[M1 is a matrix with shape **number of training points plus 1** by **number of test points**rEhFj?hGhJhLjhN}rF(hP]hQ]hR]hS]hU]uhWKPh@]rG(haXM1 is a matrix with shape rHrI}rJ(hEXM1 is a matrix with shape hFjCubcdocutils.nodes strong rK)rL}rM(hEX$**number of training points plus 1**hN}rN(hP]hQ]hR]hS]hU]uhFjCh@]rOhaX number of training points plus 1rPrQ}rR(hEUhFjLubahLUstrongrSubhaX by rTrU}rV(hEX by hFjCubjK)rW}rX(hEX**number of test points**hN}rY(hP]hQ]hR]hS]hU]uhFjCh@]rZhaXnumber of test pointsr[r\}r](hEUhFjWubahLjSubeubj)r^}r_(hEUhN}r`(hP]hQ]hR]hS]hU]uhFj?h@]raj )rb}rc(hEUhN}rd(jX-hS]hR]hP]hQ]hU]uhFj^h@]re(h)rf}rg(hEX(cross covariances matrix (train by test)rhhN}ri(hP]hQ]hR]hS]hU]uhFjbh@]rjhs)rk}rl(hEjhhFjfhGhJhLhwhN}rm(hP]hQ]hR]hS]hU]uhWKPh@]rnhaX(cross covariances matrix (train by test)rorp}rq(hEjhhFjkubaubahLhubh)rr}rs(hEX7last row is self covariances (diagonal of test by test)rthN}ru(hP]hQ]hR]hS]hU]uhFjbh@]rvhs)rw}rx(hEjthFjrhGhJhLhwhN}ry(hP]hQ]hR]hS]hU]uhWKQh@]rzhaX7last row is self covariances (diagonal of test by test)r{r|}r}(hEjthFjwubaubahLhubehLjTubahLjUubeubh)r~}r(hEX~M2 is a square matrix with **number of training points** for each dimension - training set covariance matrix (train by train) hFj;hGhJhLhhN}r(hP]hQ]hR]hS]hU]uhWKShXhh@]r(h)r}r(hEXKM2 is a square matrix with **number of training points** for each dimensionrhFj~hGhJhLjhN}r(hP]hQ]hR]hS]hU]uhWKSh@]r(haXM2 is a square matrix with rr}r(hEXM2 is a square matrix with hFjubjK)r}r(hEX**number of training points**hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXnumber of training pointsrr}r(hEUhFjubahLjSubhaX for each dimensionrr}r(hEX for each dimensionhFjubeubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFj~h@]rj )r}r(hEUhN}r(jX-hS]hR]hP]hQ]hU]uhFjh@]rh)r}r(hEX0training set covariance matrix (train by train) hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEX/training set covariance matrix (train by train)rhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWKSh@]rhaX/training set covariance matrix (train by train)rr}r(hEjhFjubaubahLhubahLjTubahLjUubeubeubhs)r}r(hEXzA precomputed kernel can also be composited with other kernels. You need to explictly add scalar for *pyGPs.cov.Pre()*. ::rhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWKUhXhh@]r(haXeA precomputed kernel can also be composited with other kernels. You need to explictly add scalar for rr}r(hEXeA precomputed kernel can also be composited with other kernels. You need to explictly add scalar for hFjubj)r}r(hEX*pyGPs.cov.Pre()*hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXpyGPs.cov.Pre()rr}r(hEUhFjubahLj#ubhaX.r}r(hEX.hFjubeubh})r}r(hEX/k = 0.5*pyGPs.cov.Pre(M1, M2) + pyGPs.cov.RBF()hFjhGhJhLhhN}r(hhhS]hR]hP]hQ]hU]uhWKWhXhh@]rhaX/k = 0.5*pyGPs.cov.Pre(M1, M2) + pyGPs.cov.RBF()rr}r(hEUhFjubaubeubhB)r}r(hEUhFhChGhJhLhMhN}r(hP]hQ]hR]hS]rh?ahU]rh.auhWK[hXhh@]r(hZ)r}r(hEXCustomizing Kernel & MeanrhFjhGhJhLh^hN}r(hP]hQ]hR]hS]hU]uhWK[hXhh@]rhaXCustomizing Kernel & Meanrr}r(hEjhFjubaubhs)r}r(hEXWe also support you to create your own kernel/mean class, your customized kernel class need to follow the template as below: ::rhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWK\hXhh@]rhaX|We also support you to create your own kernel/mean class, your customized kernel class need to follow the template as below:rr}r(hEX|We also support you to create your own kernel/mean class, your customized kernel class need to follow the template as below:hFjubaubh})r}r(hEXY# Your kernel class needs to inherit base class Kernel, # which is in the module of Core.cov class MyKernel(Kernel): def __init__(self, hyp): ''' Intialize hyperparameters for MyKernel. ''' self.hyp = hyp def getCovMatrix(self,x=None,z=None,mode=None): ''' Return the specific covariance matrix according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self covariance matrix of test data(test by 1). 'train' return training covariance matrix(train by train). 'cross' return cross covariance matrix between x and z(train by test) :return: the corresponding covariance matrix ''' pass def getDerMatrix(self,x=None,z=None,mode=None,der=None): ''' Compute derivatives wrt. hyperparameters according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self derivative matrix of test data(test by 1). 'train' return training derivative matrix(train by train). 'cross' return cross derivative matrix between x and z(train by test) :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' passhFjhGhJhLhhN}r(hhhS]hR]hP]hQ]hU]uhWK^hXhh@]rhaXY# Your kernel class needs to inherit base class Kernel, # which is in the module of Core.cov class MyKernel(Kernel): def __init__(self, hyp): ''' Intialize hyperparameters for MyKernel. ''' self.hyp = hyp def getCovMatrix(self,x=None,z=None,mode=None): ''' Return the specific covariance matrix according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self covariance matrix of test data(test by 1). 'train' return training covariance matrix(train by train). 'cross' return cross covariance matrix between x and z(train by test) :return: the corresponding covariance matrix ''' pass def getDerMatrix(self,x=None,z=None,mode=None,der=None): ''' Compute derivatives wrt. hyperparameters according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self derivative matrix of test data(test by 1). 'train' return training derivative matrix(train by train). 'cross' return cross derivative matrix between x and z(train by test) :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' passrr}r(hEUhFjubaubhs)r}r(hEX!and for customized mean class: ::rhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaXand for customized mean class:rr}r(hEXand for customized mean class:hFjubaubh})r}r(hEXS# Your mean class needs to inherit base class Mean, # which is in the module of Core.mean class MyMean(Mean): def __init__(self, hyp): ''' Intialize hyperparameters for MyMean. ''' self.hyp = hyp def getMean(self, x=None): ''' Get the mean vector. ''' pass def getDerMatrix(self, x=None, der=None): ''' Compute derivatives wrt. hyperparameters. :param x: training data :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' passhFjhGhJhLhhN}r(hhhS]hR]hP]hQ]hU]uhWKhXhh@]rhaXS# Your mean class needs to inherit base class Mean, # which is in the module of Core.mean class MyMean(Mean): def __init__(self, hyp): ''' Intialize hyperparameters for MyMean. ''' self.hyp = hyp def getMean(self, x=None): ''' Get the mean vector. ''' pass def getDerMatrix(self, x=None, der=None): ''' Compute derivatives wrt. hyperparameters. :param x: training data :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' passrr}r(hEUhFjubaubhs)r}r(hEXYou can test your customized mean/kernel function using our framework of unit test. Taking kernel test as an example, you can uncomment method *test_cov_new* in *pyGPs.Testing.unit_test_cov.py* to check the outputs of your kernel function. ::hFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]r(haXYou can test your customized mean/kernel function using our framework of unit test. Taking kernel test as an example, you can uncomment method rr}r(hEXYou can test your customized mean/kernel function using our framework of unit test. Taking kernel test as an example, you can uncomment method hFjubj)r}r(hEX*test_cov_new*hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX test_cov_newrr}r(hEUhFjubahLj#ubhaX in rr}r(hEX in hFjubj)r}r(hEX *pyGPs.Testing.unit_test_cov.py*hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXpyGPs.Testing.unit_test_cov.pyrr}r(hEUhFjubahLj#ubhaX. to check the outputs of your kernel function.rr}r(hEX. to check the outputs of your kernel function.hFjubeubh})r}r(hEX# Test your customized covariance function def test_cov_new(self): k = myKernel() # specify your covariance function self.checkCovariance(k)hFjhGhJhLhhN}r (hhhS]hR]hP]hQ]hU]uhWKhXhh@]r haX# Test your customized covariance function def test_cov_new(self): k = myKernel() # specify your covariance function self.checkCovariance(k)r r }r (hEUhFjubaubhs)r}r(hEXAand testing mean function in *pyGPs.Testing.unit_test_mean.py* ::rhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]r(haXand testing mean function in rr}r(hEXand testing mean function in hFjubj)r}r(hEX!*pyGPs.Testing.unit_test_mean.py*hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXpyGPs.Testing.unit_test_mean.pyrr}r(hEUhFjubahLj#ubeubh})r}r(hEX# Test your customized mean function def test_mean_new(self): m = myMean # specify your mean function self.checkMean(m)hFjhGhJhLhhN}r(hhhS]hR]hP]hQ]hU]uhWKhXhh@]r haX# Test your customized mean function def test_mean_new(self): m = myMean # specify your mean function self.checkMean(m)r!r"}r#(hEUhFjubaubeubhB)r$}r%(hEUhFhChGhJhLhMhN}r&(hP]hQ]hR]hS]r'(Xmodule-pyGPs.Core.covr(h>ehU]r)h*auhWKhXhh@]r*(hZ)r+}r,(hEX&List of Kernels and Default Parametersr-hFj$hGhJhLh^hN}r.(hP]hQ]hR]hS]hU]uhWKhXhh@]r/haX&List of Kernels and Default Parametersr0r1}r2(hEj-hFj+ubaubcsphinx.addnodes index r3)r4}r5(hEUhFj$hGU r6hLUindexr7hN}r8(hS]hR]hP]hQ]hU]Uentries]r9(Usingler:XpyGPs.Core.cov (module)Xmodule-pyGPs.Core.covUtr;auhWKhXhh@]ubj3)r<}r=(hEUhFj$hGNhLj7hN}r>(hS]hR]hP]hQ]hU]Uentries]r?(j:XConst (class in pyGPs.Core.cov)h-Utr@auhWNhXhh@]ubcsphinx.addnodes desc rA)rB}rC(hEUhFj$hGNhLUdescrDhN}rE(UnoindexrFUdomainrGXpyhS]hR]hP]hQ]hU]UobjtyperHXclassrIUdesctyperJjIuhWNhXhh@]rK(csphinx.addnodes desc_signature rL)rM}rN(hEXConst(log_sigma=0.0)hFjBhGU rOhLUdesc_signaturerPhN}rQ(hS]rRh-aUmodulerShHXpyGPs.Core.covrTrU}rVbhR]hP]hQ]hU]rWh-aUfullnamerXXConstrYUclassrZUUfirstr[uhWNhXhh@]r\(csphinx.addnodes desc_annotation r])r^}r_(hEXclass hFjMhGjOhLUdesc_annotationr`hN}ra(hP]hQ]hR]hS]hU]uhWNhXhh@]rbhaXclass rcrd}re(hEUhFj^ubaubcsphinx.addnodes desc_addname rf)rg}rh(hEXpyGPs.Core.cov.hFjMhGjOhLU desc_addnamerihN}rj(hP]hQ]hR]hS]hU]uhWNhXhh@]rkhaXpyGPs.Core.cov.rlrm}rn(hEUhFjgubaubcsphinx.addnodes desc_name ro)rp}rq(hEjYhFjMhGjOhLU desc_namerrhN}rs(hP]hQ]hR]hS]hU]uhWNhXhh@]rthaXConstrurv}rw(hEUhFjpubaubcsphinx.addnodes desc_parameterlist rx)ry}rz(hEUhFjMhGjOhLUdesc_parameterlistr{hN}r|(hP]hQ]hR]hS]hU]uhWNhXhh@]r}csphinx.addnodes desc_parameter r~)r}r(hEX log_sigma=0.0hN}r(hP]hQ]hR]hS]hU]uhFjyh@]rhaX log_sigma=0.0rr}r(hEUhFjubahLUdesc_parameterrubaubcsphinx.addnodes only r)r}r(hEUhFjMhGNhLUonlyrhN}r(UexprUhtmlrhS]hR]hP]hQ]hU]uhWNhXhh@]rcsphinx.addnodes pending_xref r)r}r(hEUhN}r(UreftypeUviewcoderUrefdocXKernelsrU refdomainUstdrhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjYuhFjh@]rcdocutils.nodes inline r)r}r(hEUhN}r(hP]hQ]rU viewcode-linkrahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLUinlinerubahLU pending_xrefrubaubeubcsphinx.addnodes desc_content r)r}r(hEUhFjBhGjOhLU desc_contentrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r}r(hEX$Constant kernel. hyp = [ log_sigma ]rhFjhGXO/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.ConstrhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaX$Constant kernel. hyp = [ log_sigma ]rr}r(hEjhFjubaubcdocutils.nodes field_list r)r}r(hEUhFjhGNhLU field_listrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rcdocutils.nodes field r)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(cdocutils.nodes field_name r)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX Parametersrr}r(hEUhFjubahLU field_namerubcdocutils.nodes field_body r)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r(hEX log_sigmahN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_sigmarr}r(hEUhFjubahLjSubhaX -- rr}r(hEUhFjubhaXsignal deviation.rr}r(hEXsignal deviation.hFjubehLhwubahLU field_bodyrubehLUfieldrubaubeubeubj3)r}r(hEUhFj$hGXV/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.FITCOfKernelrhLj7hN}r(hS]hR]hP]hQ]hU]Uentries]r(j:X&FITCOfKernel (class in pyGPs.Core.cov)hUtrauhWNhXhh@]ubjA)r}r(hEUhFj$hGjhLjDhN}r(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrjJjuhWNhXhh@]r(jL)r}r(hEX FITCOfKernel(cov, inducingInput)hFjhGjOhLjPhN}r(hS]rhajShHXpyGPs.Core.covrr}rbhR]hP]hQ]hU]rhajXX FITCOfKernelrjZUj[uhWNhXhh@]r(j])r}r(hEXclass hFjhGjOhLj`hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXclass rr}r(hEUhFjubaubjf)r}r(hEXpyGPs.Core.cov.hFjhGjOhLjihN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXpyGPs.Core.cov.rr}r(hEUhFjubaubjo)r}r(hEjhFjhGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaX FITCOfKernelrr}r(hEUhFjubaubjx)r}r(hEUhFjhGjOhLj{hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (j~)r }r (hEXcovhN}r (hP]hQ]hR]hS]hU]uhFjh@]rhaXcovrr}r(hEUhFj ubahLjubj~)r}r(hEX inducingInputhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX inducingInputrr}r(hEUhFjubahLjubeubj)r}r(hEUhFjhGNhLjhN}r(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rj)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjuhFjh@]r j)r!}r"(hEUhN}r#(hP]hQ]r$jahR]hS]hU]uhFjh@]r%haX[source]r&r'}r((hEUhFj!ubahLjubahLjubaubeubj)r)}r*(hEUhFjhGjOhLjhN}r+(hP]hQ]hR]hS]hU]uhWNhXhh@]r,hs)r-}r.(hEXNCovariance function to be used together with the FITC approximation. The function allows for more than one output argument and does not respect the interface of a proper covariance function. Instead of outputing the full covariance, it returns cross-covariances between the inputs x, z and the inducing inputs xu as needed by infFITCr/hFj)hGjhLhwhN}r0(hP]hQ]hR]hS]hU]uhWKhXhh@]r1haXNCovariance function to be used together with the FITC approximation. The function allows for more than one output argument and does not respect the interface of a proper covariance function. Instead of outputing the full covariance, it returns cross-covariances between the inputs x, z and the inducing inputs xu as needed by infFITCr2r3}r4(hEj/hFj-ubaubaubeubj3)r5}r6(hEUhFj$hGNhLj7hN}r7(hS]hR]hP]hQ]hU]Uentries]r8(j:X Kernel (class in pyGPs.Core.cov)h$Utr9auhWNhXhh@]ubjA)r:}r;(hEUhFj$hGNhLjDhN}r<(jFjGXpyhS]hR]hP]hQ]hU]jHXclassr=jJj=uhWNhXhh@]r>(jL)r?}r@(hEXKernel()hFj:hGjOhLjPhN}rA(hS]rBh$ajShHXpyGPs.Core.covrCrD}rEbhR]hP]hQ]hU]rFh$ajXXKernelrGjZUj[uhWNhXhh@]rH(j])rI}rJ(hEXclass hFj?hGjOhLj`hN}rK(hP]hQ]hR]hS]hU]uhWNhXhh@]rLhaXclass rMrN}rO(hEUhFjIubaubjf)rP}rQ(hEXpyGPs.Core.cov.hFj?hGjOhLjihN}rR(hP]hQ]hR]hS]hU]uhWNhXhh@]rShaXpyGPs.Core.cov.rTrU}rV(hEUhFjPubaubjo)rW}rX(hEjGhFj?hGjOhLjrhN}rY(hP]hQ]hR]hS]hU]uhWNhXhh@]rZhaXKernelr[r\}r](hEUhFjWubaubj)r^}r_(hEUhFj?hGNhLjhN}r`(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]raj)rb}rc(hEUhN}rd(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjGuhFj^h@]rej)rf}rg(hEUhN}rh(hP]hQ]rijahR]hS]hU]uhFjbh@]rjhaX[source]rkrl}rm(hEUhFjfubahLjubahLjubaubeubj)rn}ro(hEUhFj:hGjOhLjhN}rp(hP]hQ]hR]hS]hU]uhWNhXhh@]rq(hs)rr}rs(hEXThis is a base class of Kernel functions there is no computation in this class, it just defines rules about a kernel class should have each covariance function will inherit it and implement its own behaviourrthFjnhGXP/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.KernelruhLhwhN}rv(hP]hQ]hR]hS]hU]uhWKhXhh@]rwhaXThis is a base class of Kernel functions there is no computation in this class, it just defines rules about a kernel class should have each covariance function will inherit it and implement its own behaviourrxry}rz(hEjthFjrubaubj3)r{}r|(hEUhFjnhGNhLj7hN}r}(hS]hR]hP]hQ]hU]Uentries]r~(j:X%fitc() (pyGPs.Core.cov.Kernel method)hUtrauhWNhXhh@]ubjA)r}r(hEUhFjnhGNhLjDhN}r(jFjGXpyhS]hR]hP]hQ]hU]jHXmethodrjJjuhWNhXhh@]r(jL)r}r(hEXKernel.fitc(inducingInput)hFjhGjOhLjPhN}r(hS]rhajShHXpyGPs.Core.covrr}rbhR]hP]hQ]hU]rhajXX Kernel.fitcrjZjGj[uhWNhXhh@]r(jo)r}r(hEXfitchFjhGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXfitcrr}r(hEUhFjubaubjx)r}r(hEUhFjhGjOhLj{hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rj~)r}r(hEX inducingInputhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX inducingInputrr}r(hEUhFjubahLjubaubj)r}r(hEUhFjhGNhLjhN}r(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rj)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjuhFjh@]rj)r}r(hEUhN}r(hP]hQ]rjahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLjubahLjubaubeubj)r}r(hEUhFjhGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r}r(hEX|Covariance function to be used together with the FITC approximation. Setting FITC gp model will implicitly call this method.rhFjhGXU/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.Kernel.fitcrhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaX|Covariance function to be used together with the FITC approximation. Setting FITC gp model will implicitly call this method.rr}r(hEjhFjubaubj)r}r(hEUhFjhGNhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXReturnsrr}r(hEUhFjubahLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXan instance of FITCOfKernelrr}r(hEXan instance of FITCOfKernelhFjubahLhwubahLjubehLjubaubeubeubj3)r}r(hEUhFjnhGNhLj7hN}r(hS]hR]hP]hQ]hU]Uentries]r(j:X-getCovMatrix() (pyGPs.Core.cov.Kernel method)hUtrauhWNhXhh@]ubjA)r}r(hEUhFjnhGNhLjDhN}r(jFjGXpyrhS]hR]hP]hQ]hU]jHXmethodrjJjuhWNhXhh@]r(jL)r}r(hEX.Kernel.getCovMatrix(x=None, z=None, mode=None)hFjhGjOhLjPhN}r(hS]rhajShHXpyGPs.Core.covrr}rbhR]hP]hQ]hU]rhajXXKernel.getCovMatrixrjZjGj[uhWNhXhh@]r(jo)r}r(hEX getCovMatrixhFjhGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaX getCovMatrixrr}r(hEUhFjubaubjx)r}r(hEUhFjhGjOhLj{hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(j~)r}r(hEXx=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXx=Nonerr}r(hEUhFjubahLjubj~)r}r(hEXz=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXz=Nonerr}r(hEUhFjubahLjubj~)r}r(hEX mode=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]r haX mode=Noner r }r (hEUhFjubahLjubeubj)r }r(hEUhFjhGNhLjhN}r(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rj)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjuhFj h@]rj)r}r(hEUhN}r(hP]hQ]rjahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLjubahLjubaubeubj)r}r(hEUhFjhGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r (hs)r!}r"(hEX=Return the specific covariance matrix according to input moder#hFjhGX]/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.Kernel.getCovMatrixr$hLhwhN}r%(hP]hQ]hR]hS]hU]uhWKhXhh@]r&haX=Return the specific covariance matrix according to input moder'r(}r)(hEj#hFj!ubaubj)r*}r+(hEUhFjhGNhLjhN}r,(hP]hQ]hR]hS]hU]uhWNhXhh@]r-(j)r.}r/(hEUhN}r0(hP]hQ]hR]hS]hU]uhFj*h@]r1(j)r2}r3(hEUhN}r4(hP]hQ]hR]hS]hU]uhFj.h@]r5haX Parametersr6r7}r8(hEUhFj2ubahLjubj)r9}r:(hEUhN}r;(hP]hQ]hR]hS]hU]uhFj.h@]r<j )r=}r>(hEUhN}r?(hP]hQ]hR]hS]hU]uhFj9h@]r@(h)rA}rB(hEUhN}rC(hP]hQ]hR]hS]hU]uhFj=h@]rDhs)rE}rF(hEUhN}rG(hP]hQ]hR]hS]hU]uhFjAh@]rH(jK)rI}rJ(hEXxhN}rK(hP]hQ]hR]hS]hU]uhFjEh@]rLhaXxrM}rN(hEUhFjIubahLjSubhaX -- rOrP}rQ(hEUhFjEubhaX training datarRrS}rT(hEX training datahFjEubehLhwubahLhubh)rU}rV(hEUhN}rW(hP]hQ]hR]hS]hU]uhFj=h@]rXhs)rY}rZ(hEUhN}r[(hP]hQ]hR]hS]hU]uhFjUh@]r\(jK)r]}r^(hEXzhN}r_(hP]hQ]hR]hS]hU]uhFjYh@]r`haXzra}rb(hEUhFj]ubahLjSubhaX -- rcrd}re(hEUhFjYubhaX test datarfrg}rh(hEX test datahFjYubehLhwubahLhubh)ri}rj(hEUhN}rk(hP]hQ]hR]hS]hU]uhFj=h@]rlhs)rm}rn(hEUhN}ro(hP]hQ]hR]hS]hU]uhFjih@]rp(jK)rq}rr(hEXmodehN}rs(hP]hQ]hR]hS]hU]uhFjmh@]rthaXmoderurv}rw(hEUhFjqubahLjSubhaX (rxry}rz(hEUhFjmubj)r{}r|(hEUhN}r}(UreftypeUobjr~U reftargetXstrrU refdomainjhS]hR]U refexplicithP]hQ]hU]uhFjmh@]rj)r}r(hEjhN}r(hP]hQ]hR]hS]hU]uhFj{h@]rhaXstrrr}r(hEUhFjubahLj#ubahLjubhaX)r}r(hEUhFjmubhaX -- rr}r(hEUhFjmubhaX'self_test' return self covariance matrix of test data(test by 1). 'train' return training covariance matrix(train by train). 'cross' return cross covariance matrix between x and z(train by test)rr}r(hEX'self_test' return self covariance matrix of test data(test by 1). 'train' return training covariance matrix(train by train). 'cross' return cross covariance matrix between x and z(train by test)hFjmubehLhwubahLhubehLjTubahLjubehLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFj*h@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXReturnsrr}r(hEUhFjubahLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX#the corresponding covariance matrixrr}r(hEX#the corresponding covariance matrixhFjubahLhwubahLjubehLjubeubeubeubj3)r}r(hEUhFjnhGNhLj7hN}r(hS]hR]hP]hQ]hU]Uentries]r(j:X-getDerMatrix() (pyGPs.Core.cov.Kernel method)hUtrauhWNhXhh@]ubjA)r}r(hEUhFjnhGNhLjDhN}r(jFjGXpyrhS]hR]hP]hQ]hU]jHXmethodrjJjuhWNhXhh@]r(jL)r}r(hEX8Kernel.getDerMatrix(x=None, z=None, mode=None, der=None)hFjhGjOhLjPhN}r(hS]rhajShHXpyGPs.Core.covrr}rbhR]hP]hQ]hU]rhajXXKernel.getDerMatrixrjZjGj[uhWNhXhh@]r(jo)r}r(hEX getDerMatrixhFjhGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaX getDerMatrixrr}r(hEUhFjubaubjx)r}r(hEUhFjhGjOhLj{hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(j~)r}r(hEXx=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXx=Nonerr}r(hEUhFjubahLjubj~)r}r(hEXz=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXz=Nonerr}r(hEUhFjubahLjubj~)r}r(hEX mode=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX mode=Nonerr}r(hEUhFjubahLjubj~)r}r(hEXder=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXder=Nonerr}r(hEUhFjubahLjubeubj)r}r(hEUhFjhGNhLjhN}r(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rj)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjuhFjh@]rj)r}r(hEUhN}r(hP]hQ]rjahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLjubahLjubaubeubj)r}r(hEUhFjhGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r}r(hEX@Compute derivatives wrt. hyperparameters according to input moderhFjhGX]/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.Kernel.getDerMatrixrhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaX@Compute derivatives wrt. hyperparameters according to input moderr}r(hEjhFjubaubj)r}r(hEUhFjhGNhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(j)r}r(hEUhN}r (hP]hQ]hR]hS]hU]uhFjh@]r haX Parametersr r }r (hEUhFjubahLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rj )r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(h)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r(hEXxhN}r (hP]hQ]hR]hS]hU]uhFjh@]r!haXxr"}r#(hEUhFjubahLjSubhaX -- r$r%}r&(hEUhFjubhaX training datar'r(}r)(hEX training datar*hFjubehLhwubahLhubh)r+}r,(hEUhN}r-(hP]hQ]hR]hS]hU]uhFjh@]r.hs)r/}r0(hEUhN}r1(hP]hQ]hR]hS]hU]uhFj+h@]r2(jK)r3}r4(hEXzhN}r5(hP]hQ]hR]hS]hU]uhFj/h@]r6haXzr7}r8(hEUhFj3ubahLjSubhaX -- r9r:}r;(hEUhFj/ubhaX test datar<r=}r>(hEX test datar?hFj/ubehLhwubahLhubh)r@}rA(hEUhN}rB(hP]hQ]hR]hS]hU]uhFjh@]rChs)rD}rE(hEUhN}rF(hP]hQ]hR]hS]hU]uhFj@h@]rG(jK)rH}rI(hEXmodehN}rJ(hP]hQ]hR]hS]hU]uhFjDh@]rKhaXmoderLrM}rN(hEUhFjHubahLjSubhaX (rOrP}rQ(hEUhFjDubj)rR}rS(hEUhN}rT(Ureftypej~U reftargetXstrrUU refdomainjhS]hR]U refexplicithP]hQ]hU]uhFjDh@]rVj)rW}rX(hEjUhN}rY(hP]hQ]hR]hS]hU]uhFjRh@]rZhaXstrr[r\}r](hEUhFjWubahLj#ubahLjubhaX)r^}r_(hEUhFjDubhaX -- r`ra}rb(hEUhFjDubhaX'self_test' return self derivative matrix of test data(test by 1). 'train' return training derivative matrix(train by train). 'cross' return cross derivative matrix between x and z(train by test)rcrd}re(hEX'self_test' return self derivative matrix of test data(test by 1). 'train' return training derivative matrix(train by train). 'cross' return cross derivative matrix between x and z(train by test)rfhFjDubehLhwubahLhubh)rg}rh(hEUhN}ri(hP]hQ]hR]hS]hU]uhFjh@]rjhs)rk}rl(hEUhN}rm(hP]hQ]hR]hS]hU]uhFjgh@]rn(jK)ro}rp(hEXderhN}rq(hP]hQ]hR]hS]hU]uhFjkh@]rrhaXderrsrt}ru(hEUhFjoubahLjSubhaX (rvrw}rx(hEUhFjkubj)ry}rz(hEUhN}r{(Ureftypej~U reftargetXintr|U refdomainjhS]hR]U refexplicithP]hQ]hU]uhFjkh@]r}j)r~}r(hEj|hN}r(hP]hQ]hR]hS]hU]uhFjyh@]rhaXintrr}r(hEUhFj~ubahLj#ubahLjubhaX)r}r(hEUhFjkubhaX -- rr}r(hEUhFjkubhaX7index of hyperparameter whose derivative to be computedrr}r(hEX7index of hyperparameter whose derivative to be computedrhFjkubehLhwubahLhubehLjTubahLjubehLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXReturnsrr}r(hEUhFjubahLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX#the corresponding derivative matrixrr}r(hEX#the corresponding derivative matrixrhFjubahLhwubahLjubehLjubeubeubeubeubeubj3)r}r(hEUhFj$hGNhLj7hN}r(hS]hR]hP]hQ]hU]Uentries]r(j:X LINard (class in pyGPs.Core.cov)h UtrauhWNhXhh@]ubjA)r}r(hEUhFj$hGNhLjDhN}r(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrjJjuhWNhXhh@]r(jL)r}r(hEX!LINard(D=None, log_ell_list=None)hFjhGjOhLjPhN}r(hS]rh ajShHXpyGPs.Core.covrr}rbhR]hP]hQ]hU]rh ajXXLINardrjZUj[uhWNhXhh@]r(j])r}r(hEXclass hFjhGjOhLj`hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXclass rr}r(hEUhFjubaubjf)r}r(hEXpyGPs.Core.cov.hFjhGjOhLjihN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXpyGPs.Core.cov.rr}r(hEUhFjubaubjo)r}r(hEjhFjhGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXLINardrr}r(hEUhFjubaubjx)r}r(hEUhFjhGjOhLj{hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(j~)r}r(hEXD=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXD=Nonerr}r(hEUhFjubahLjubj~)r}r(hEXlog_ell_list=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXlog_ell_list=Nonerr}r(hEUhFjubahLjubeubj)r}r(hEUhFjhGNhLjhN}r(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rj)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjuhFjh@]rj)r}r(hEUhN}r(hP]hQ]rjahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLjubahLjubaubeubj)r}r(hEUhFjhGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r}r(hEXTLinear covariance function with Automatic Relevance Detemination. hyp = log_ell_listrhFjhGXP/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.LINardrhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaXTLinear covariance function with Automatic Relevance Detemination. hyp = log_ell_listrr}r(hEjhFjubaubj)r}r(hEUhFjhGNhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX Parametersr r }r (hEUhFjubahLjubj)r }r (hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rj )r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFj h@]r(h)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r(hEXDhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXDr }r!(hEUhFjubahLjSubhaX -- r"r#}r$(hEUhFjubhaXWdimension of training data. Set if you want default ell, which is 1 for each dimension.r%r&}r'(hEXWdimension of training data. Set if you want default ell, which is 1 for each dimension.r(hFjubehLhwubahLhubh)r)}r*(hEUhN}r+(hP]hQ]hR]hS]hU]uhFjh@]r,hs)r-}r.(hEUhN}r/(hP]hQ]hR]hS]hU]uhFj)h@]r0(jK)r1}r2(hEX log_ell_listhN}r3(hP]hQ]hR]hS]hU]uhFj-h@]r4haX log_ell_listr5r6}r7(hEUhFj1ubahLjSubhaX -- r8r9}r:(hEUhFj-ubhaX/characteristic length scale for each dimension.r;r<}r=(hEX/characteristic length scale for each dimension.r>hFj-ubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)r?}r@(hEUhFj$hGXP/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.LinearrAhLj7hN}rB(hS]hR]hP]hQ]hU]Uentries]rC(j:X Linear (class in pyGPs.Core.cov)hUtrDauhWNhXhh@]ubjA)rE}rF(hEUhFj$hGjAhLjDhN}rG(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrHjJjHuhWNhXhh@]rI(jL)rJ}rK(hEXLinear(log_sigma=0.0)hFjEhGjOhLjPhN}rL(hS]rMhajShHXpyGPs.Core.covrNrO}rPbhR]hP]hQ]hU]rQhajXXLinearrRjZUj[uhWNhXhh@]rS(j])rT}rU(hEXclass hFjJhGjOhLj`hN}rV(hP]hQ]hR]hS]hU]uhWNhXhh@]rWhaXclass rXrY}rZ(hEUhFjTubaubjf)r[}r\(hEXpyGPs.Core.cov.hFjJhGjOhLjihN}r](hP]hQ]hR]hS]hU]uhWNhXhh@]r^haXpyGPs.Core.cov.r_r`}ra(hEUhFj[ubaubjo)rb}rc(hEjRhFjJhGjOhLjrhN}rd(hP]hQ]hR]hS]hU]uhWNhXhh@]rehaXLinearrfrg}rh(hEUhFjbubaubjx)ri}rj(hEUhFjJhGjOhLj{hN}rk(hP]hQ]hR]hS]hU]uhWNhXhh@]rlj~)rm}rn(hEX log_sigma=0.0hN}ro(hP]hQ]hR]hS]hU]uhFjih@]rphaX log_sigma=0.0rqrr}rs(hEUhFjmubahLjubaubj)rt}ru(hEUhFjJhGNhLjhN}rv(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rwj)rx}ry(hEUhN}rz(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjRuhFjth@]r{j)r|}r}(hEUhN}r~(hP]hQ]rjahR]hS]hU]uhFjxh@]rhaX[source]rr}r(hEUhFj|ubahLjubahLjubaubeubj)r}r(hEUhFjEhGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhs)r}r(hEX#Linear kernel. hyp = [ log_sigma ].rhFjhGjAhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaX#Linear kernel. hyp = [ log_sigma ].rr}r(hEjhFjubaubaubeubj3)r}r(hEUhFj$hGNhLj7hN}r(hS]hR]hP]hQ]hU]Uentries]r(j:X Matern (class in pyGPs.Core.cov)hUtrauhWNhXhh@]ubjA)r}r(hEUhFj$hGNhLjDhN}r(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrjJjuhWNhXhh@]r(jL)r}r(hEX'Matern(log_ell=0.0, d=3, log_sigma=0.0)hFjhGjOhLjPhN}r(hS]rhajShHXpyGPs.Core.covrr}rbhR]hP]hQ]hU]rhajXXMaternrjZUj[uhWNhXhh@]r(j])r}r(hEXclass hFjhGjOhLj`hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXclass rr}r(hEUhFjubaubjf)r}r(hEXpyGPs.Core.cov.hFjhGjOhLjihN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXpyGPs.Core.cov.rr}r(hEUhFjubaubjo)r}r(hEjhFjhGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXMaternrr}r(hEUhFjubaubjx)r}r(hEUhFjhGjOhLj{hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(j~)r}r(hEX log_ell=0.0hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_ell=0.0rr}r(hEUhFjubahLjubj~)r}r(hEXd=3hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXd=3rr}r(hEUhFjubahLjubj~)r}r(hEX log_sigma=0.0hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_sigma=0.0rr}r(hEUhFjubahLjubeubj)r}r(hEUhFjhGNhLjhN}r(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rj)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjuhFjh@]rj)r}r(hEUhN}r(hP]hQ]rjahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLjubahLjubaubeubj)r}r(hEUhFjhGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r}r(hEXMatern covariance function with nu = d/2 and isotropic distance measure. For d=1 the function is also known as the exponential covariance function or the Ornstein-Uhlenbeck covariance in 1d. d will be rounded to 1, 3, or 5. hyp = [ log_ell, log_sigma]rhFjhGXP/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.MaternrhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaXMatern covariance function with nu = d/2 and isotropic distance measure. For d=1 the function is also known as the exponential covariance function or the Ornstein-Uhlenbeck covariance in 1d. d will be rounded to 1, 3, or 5. hyp = [ log_ell, log_sigma]rr}r(hEjhFjubaubj)r}r(hEUhFjhGNhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX Parametersrr}r(hEUhFjubahLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rj )r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(h)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFjh@]r (jK)r}r(hEXdhN}r(hP]hQ]hR]hS]hU]uhFj h@]rhaXdr}r(hEUhFjubahLjSubhaX -- rr}r(hEUhFj ubhaX&d is 2 times nu. Can only be 1,3 or 5.rr}r(hEX&d is 2 times nu. Can only be 1,3 or 5.hFj ubehLhwubahLhubh)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r (hP]hQ]hR]hS]hU]uhFjh@]r!(jK)r"}r#(hEXlog_ellhN}r$(hP]hQ]hR]hS]hU]uhFjh@]r%haXlog_ellr&r'}r((hEUhFj"ubahLjSubhaX -- r)r*}r+(hEUhFjubhaXcharacteristic length scale.r,r-}r.(hEXcharacteristic length scale.hFjubehLhwubahLhubh)r/}r0(hEUhN}r1(hP]hQ]hR]hS]hU]uhFjh@]r2hs)r3}r4(hEUhN}r5(hP]hQ]hR]hS]hU]uhFj/h@]r6(jK)r7}r8(hEX log_sigmahN}r9(hP]hQ]hR]hS]hU]uhFj3h@]r:haX log_sigmar;r<}r=(hEUhFj7ubahLjSubhaX -- r>r?}r@(hEUhFj3ubhaXsignal deviation.rArB}rC(hEXsignal deviation.hFj3ubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)rD}rE(hEUhFj$hGNhLj7hN}rF(hS]hR]hP]hQ]hU]Uentries]rG(j:XNoise (class in pyGPs.Core.cov)hUtrHauhWNhXhh@]ubjA)rI}rJ(hEUhFj$hGNhLjDhN}rK(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrLjJjLuhWNhXhh@]rM(jL)rN}rO(hEXNoise(log_sigma=0.0)hFjIhGjOhLjPhN}rP(hS]rQhajShHXpyGPs.Core.covrRrS}rTbhR]hP]hQ]hU]rUhajXXNoiserVjZUj[uhWNhXhh@]rW(j])rX}rY(hEXclass hFjNhGjOhLj`hN}rZ(hP]hQ]hR]hS]hU]uhWNhXhh@]r[haXclass r\r]}r^(hEUhFjXubaubjf)r_}r`(hEXpyGPs.Core.cov.hFjNhGjOhLjihN}ra(hP]hQ]hR]hS]hU]uhWNhXhh@]rbhaXpyGPs.Core.cov.rcrd}re(hEUhFj_ubaubjo)rf}rg(hEjVhFjNhGjOhLjrhN}rh(hP]hQ]hR]hS]hU]uhWNhXhh@]rihaXNoiserjrk}rl(hEUhFjfubaubjx)rm}rn(hEUhFjNhGjOhLj{hN}ro(hP]hQ]hR]hS]hU]uhWNhXhh@]rpj~)rq}rr(hEX log_sigma=0.0hN}rs(hP]hQ]hR]hS]hU]uhFjmh@]rthaX log_sigma=0.0rurv}rw(hEUhFjqubahLjubaubj)rx}ry(hEUhFjNhGNhLjhN}rz(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r{j)r|}r}(hEUhN}r~(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjVuhFjxh@]rj)r}r(hEUhN}r(hP]hQ]rjahR]hS]hU]uhFj|h@]rhaX[source]rr}r(hEUhFjubahLjubahLjubaubeubj)r}r(hEUhFjIhGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r}r(hEXIndependent covariance function, i.e "white noise", with specified variance. Normally NOT used anymore since noise is now added in liklihood. hyp = [ log_sigma ]rhFjhGXO/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.NoiserhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaXIndependent covariance function, i.e "white noise", with specified variance. Normally NOT used anymore since noise is now added in liklihood. hyp = [ log_sigma ]rr}r(hEjhFjubaubj)r}r(hEUhFjhGNhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX Parametersrr}r(hEUhFjubahLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r(hEX log_sigmahN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_sigmarr}r(hEUhFjubahLjSubhaX -- rr}r(hEUhFjubhaXsignal deviation.rr}r(hEXsignal deviation.hFjubehLhwubahLjubehLjubaubeubeubj3)r}r(hEUhFj$hGNhLj7hN}r(hS]hR]hP]hQ]hU]Uentries]r(j:X"Periodic (class in pyGPs.Core.cov)h UtrauhWNhXhh@]ubjA)r}r(hEUhFj$hGNhLjDhN}r(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrjJjuhWNhXhh@]r(jL)r}r(hEX/Periodic(log_ell=0.0, log_p=0.0, log_sigma=0.0)hFjhGjOhLjPhN}r(hS]rh ajShHXpyGPs.Core.covrr}rbhR]hP]hQ]hU]rh ajXXPeriodicrjZUj[uhWNhXhh@]r(j])r}r(hEXclass hFjhGjOhLj`hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXclass rr}r(hEUhFjubaubjf)r}r(hEXpyGPs.Core.cov.hFjhGjOhLjihN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXpyGPs.Core.cov.rr}r(hEUhFjubaubjo)r}r(hEjhFjhGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXPeriodicrr}r(hEUhFjubaubjx)r}r(hEUhFjhGjOhLj{hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(j~)r}r(hEX log_ell=0.0hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_ell=0.0rr}r(hEUhFjubahLjubj~)r}r(hEX log_p=0.0hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_p=0.0rr}r(hEUhFjubahLjubj~)r}r(hEX log_sigma=0.0hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_sigma=0.0rr}r(hEUhFjubahLjubeubj)r}r(hEUhFjhGNhLjhN}r(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rj)r}r (hEUhN}r (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjuhFjh@]r j)r }r (hEUhN}r (hP]hQ]r jahR]hS]hU]uhFjh@]r haX[source]r r }r (hEUhFj ubahLjubahLjubaubeubj)r }r (hEUhFjhGjOhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (hs)r }r (hEXUStationary kernel for a smooth periodic function. hyp = [ log_ell, log_p, log_sigma]r hFj hGXR/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.Periodicr hLhwhN}r (hP]hQ]hR]hS]hU]uhWKhXhh@]r haXUStationary kernel for a smooth periodic function. hyp = [ log_ell, log_p, log_sigma]r r }r (hEj hFj ubaubj)r }r (hEUhFj hGNhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r j)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (j)r }r! (hEUhN}r" (hP]hQ]hR]hS]hU]uhFj h@]r# haX Parametersr$ r% }r& (hEUhFj ubahLjubj)r' }r( (hEUhN}r) (hP]hQ]hR]hS]hU]uhFj h@]r* j )r+ }r, (hEUhN}r- (hP]hQ]hR]hS]hU]uhFj' h@]r. (h)r/ }r0 (hEUhN}r1 (hP]hQ]hR]hS]hU]uhFj+ h@]r2 hs)r3 }r4 (hEUhN}r5 (hP]hQ]hR]hS]hU]uhFj/ h@]r6 (jK)r7 }r8 (hEXlog_phN}r9 (hP]hQ]hR]hS]hU]uhFj3 h@]r: haXlog_pr; r< }r= (hEUhFj7 ubahLjSubhaX -- r> r? }r@ (hEUhFj3 ubhaXperiod.rA rB }rC (hEXperiod.hFj3 ubehLhwubahLhubh)rD }rE (hEUhN}rF (hP]hQ]hR]hS]hU]uhFj+ h@]rG hs)rH }rI (hEUhN}rJ (hP]hQ]hR]hS]hU]uhFjD h@]rK (jK)rL }rM (hEXlog_ellhN}rN (hP]hQ]hR]hS]hU]uhFjH h@]rO haXlog_ellrP rQ }rR (hEUhFjL ubahLjSubhaX -- rS rT }rU (hEUhFjH ubhaXcharacteristic length scale.rV rW }rX (hEXcharacteristic length scale.hFjH ubehLhwubahLhubh)rY }rZ (hEUhN}r[ (hP]hQ]hR]hS]hU]uhFj+ h@]r\ hs)r] }r^ (hEUhN}r_ (hP]hQ]hR]hS]hU]uhFjY h@]r` (jK)ra }rb (hEX log_sigmahN}rc (hP]hQ]hR]hS]hU]uhFj] h@]rd haX log_sigmare rf }rg (hEUhFja ubahLjSubhaX -- rh ri }rj (hEUhFj] ubhaXsignal deviation.rk rl }rm (hEXsignal deviation.hFj] ubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)rn }ro (hEUhFj$hGNhLj7hN}rp (hS]hR]hP]hQ]hU]Uentries]rq (j:X#PiecePoly (class in pyGPs.Core.cov)hUtrr auhWNhXhh@]ubjA)rs }rt (hEUhFj$hGNhLjDhN}ru (jFjGXpyhS]hR]hP]hQ]hU]jHXclassrv jJjv uhWNhXhh@]rw (jL)rx }ry (hEX*PiecePoly(log_ell=0.0, v=2, log_sigma=0.0)hFjs hGjOhLjPhN}rz (hS]r{ hajShHXpyGPs.Core.covr| r} }r~ bhR]hP]hQ]hU]r hajXX PiecePolyr jZUj[uhWNhXhh@]r (j])r }r (hEXclass hFjx hGjOhLj`hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXclass r r }r (hEUhFj ubaubjf)r }r (hEXpyGPs.Core.cov.hFjx hGjOhLjihN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXpyGPs.Core.cov.r r }r (hEUhFj ubaubjo)r }r (hEj hFjx hGjOhLjrhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haX PiecePolyr r }r (hEUhFj ubaubjx)r }r (hEUhFjx hGjOhLj{hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (j~)r }r (hEX log_ell=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_ell=0.0r r }r (hEUhFj ubahLjubj~)r }r (hEXv=2hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXv=2r r }r (hEUhFj ubahLjubj~)r }r (hEX log_sigma=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_sigma=0.0r r }r (hEUhFj ubahLjubeubj)r }r (hEUhFjx hGNhLjhN}r (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r j)r }r (hEUhN}r (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidj uhFj h@]r j)r }r (hEUhN}r (hP]hQ]r jahR]hS]hU]uhFj h@]r haX[source]r r }r (hEUhFj ubahLjubahLjubaubeubj)r }r (hEUhFjs hGjOhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (hs)r }r (hEXMPiecewise polynomial kernel with compact support. hyp = [log_ell, log_sigma]r hFj hGXS/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.PiecePolyr hLhwhN}r (hP]hQ]hR]hS]hU]uhWKhXhh@]r haXMPiecewise polynomial kernel with compact support. hyp = [log_ell, log_sigma]r r }r (hEj hFj ubaubj)r }r (hEUhFj hGNhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r j)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (j)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX Parametersr r }r (hEUhFj ubahLjubj)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r j )r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (h)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (jK)r }r (hEXlog_ellhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXlog_ellr r }r (hEUhFj ubahLjSubhaX -- r r }r (hEUhFj ubhaXcharacteristic length scale.r r }r (hEXcharacteristic length scale.hFj ubehLhwubahLhubh)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (jK)r }r (hEX log_sigmahN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_sigmar r }r (hEUhFj ubahLjSubhaX -- r r }r (hEUhFj ubhaXsignal deviation.r r }r (hEXsignal deviation.hFj ubehLhwubahLhubh)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (jK)r }r (hEXvhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXvr }r (hEUhFj ubahLjSubhaX -- r r }r (hEUhFj ubhaXbdegree v will be rounded to 0,1,2,or 3. (not treated as hyperparameter, i.e. will not be trained).r r }r! (hEXbdegree v will be rounded to 0,1,2,or 3. (not treated as hyperparameter, i.e. will not be trained).hFj ubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)r" }r# (hEUhFj$hGNhLj7hN}r$ (hS]hR]hP]hQ]hU]Uentries]r% (j:XPoly (class in pyGPs.Core.cov)hUtr& auhWNhXhh@]ubjA)r' }r( (hEUhFj$hGNhLjDhN}r) (jFjGXpyhS]hR]hP]hQ]hU]jHXclassr* jJj* uhWNhXhh@]r+ (jL)r, }r- (hEX#Poly(log_c=0.0, d=2, log_sigma=0.0)hFj' hGjOhLjPhN}r. (hS]r/ hajShHXpyGPs.Core.covr0 r1 }r2 bhR]hP]hQ]hU]r3 hajXXPolyr4 jZUj[uhWNhXhh@]r5 (j])r6 }r7 (hEXclass hFj, hGjOhLj`hN}r8 (hP]hQ]hR]hS]hU]uhWNhXhh@]r9 haXclass r: r; }r< (hEUhFj6 ubaubjf)r= }r> (hEXpyGPs.Core.cov.hFj, hGjOhLjihN}r? (hP]hQ]hR]hS]hU]uhWNhXhh@]r@ haXpyGPs.Core.cov.rA rB }rC (hEUhFj= ubaubjo)rD }rE (hEj4 hFj, hGjOhLjrhN}rF (hP]hQ]hR]hS]hU]uhWNhXhh@]rG haXPolyrH rI }rJ (hEUhFjD ubaubjx)rK }rL (hEUhFj, hGjOhLj{hN}rM (hP]hQ]hR]hS]hU]uhWNhXhh@]rN (j~)rO }rP (hEX log_c=0.0hN}rQ (hP]hQ]hR]hS]hU]uhFjK h@]rR haX log_c=0.0rS rT }rU (hEUhFjO ubahLjubj~)rV }rW (hEXd=2hN}rX (hP]hQ]hR]hS]hU]uhFjK h@]rY haXd=2rZ r[ }r\ (hEUhFjV ubahLjubj~)r] }r^ (hEX log_sigma=0.0hN}r_ (hP]hQ]hR]hS]hU]uhFjK h@]r` haX log_sigma=0.0ra rb }rc (hEUhFj] ubahLjubeubj)rd }re (hEUhFj, hGNhLjhN}rf (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rg j)rh }ri (hEUhN}rj (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidj4 uhFjd h@]rk j)rl }rm (hEUhN}rn (hP]hQ]ro jahR]hS]hU]uhFjh h@]rp haX[source]rq rr }rs (hEUhFjl ubahLjubahLjubaubeubj)rt }ru (hEUhFj' hGjOhLjhN}rv (hP]hQ]hR]hS]hU]uhWNhXhh@]rw (hs)rx }ry (hEX:Polynomial covariance function. hyp = [ log_c, log_sigma ]rz hFjt hGXN/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.Polyr{ hLhwhN}r| (hP]hQ]hR]hS]hU]uhWKhXhh@]r} haX:Polynomial covariance function. hyp = [ log_c, log_sigma ]r~ r }r (hEjz hFjx ubaubj)r }r (hEUhFjt hGNhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r j)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (j)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX Parametersr r }r (hEUhFj ubahLjubj)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r j )r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (h)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (jK)r }r (hEXlog_chN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXlog_cr r }r (hEUhFj ubahLjSubhaX -- r r }r (hEUhFj ubhaXinhomogeneous offset.r r }r (hEXinhomogeneous offset.hFj ubehLhwubahLhubh)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (jK)r }r (hEX log_sigmahN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_sigmar r }r (hEUhFj ubahLjSubhaX -- r r }r (hEUhFj ubhaXsignal deviation.r r }r (hEXsignal deviation.hFj ubehLhwubahLhubh)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (jK)r }r (hEXdhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXdr }r (hEUhFj ubahLjSubhaX -- r r }r (hEUhFj ubhaXOdegree of polynomial (not treated as hyperparameter, i.e. will not be trained).r r }r (hEXOdegree of polynomial (not treated as hyperparameter, i.e. will not be trained).hFj ubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)r }r (hEUhFj$hGNhLj7hN}r (hS]hR]hP]hQ]hU]Uentries]r (j:XPre (class in pyGPs.Core.cov)h(Utr auhWNhXhh@]ubjA)r }r (hEUhFj$hGNhLjDhN}r (jFjGXpyhS]hR]hP]hQ]hU]jHXclassr jJj uhWNhXhh@]r (jL)r }r (hEX Pre(M1, M2)hFj hGjOhLjPhN}r (hS]r h(ajShHXpyGPs.Core.covr r }r bhR]hP]hQ]hU]r h(ajXXPrer jZUj[uhWNhXhh@]r (j])r }r (hEXclass hFj hGjOhLj`hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXclass r r }r (hEUhFj ubaubjf)r }r (hEXpyGPs.Core.cov.hFj hGjOhLjihN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXpyGPs.Core.cov.r r }r (hEUhFj ubaubjo)r }r (hEj hFj hGjOhLjrhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXPrer r }r (hEUhFj ubaubjx)r }r (hEUhFj hGjOhLj{hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (j~)r }r (hEXM1hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXM1r r }r (hEUhFj ubahLjubj~)r }r (hEXM2hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXM2r r }r (hEUhFj ubahLjubeubj)r }r (hEUhFj hGNhLjhN}r (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r j)r }r (hEUhN}r (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidj uhFj h@]r j)r }r (hEUhN}r (hP]hQ]r jahR]hS]hU]uhFj h@]r haX[source]r r }r (hEUhFj ubahLjubahLjubaubeubj)r! }r" (hEUhFj hGjOhLjhN}r# (hP]hQ]hR]hS]hU]uhWNhXhh@]r$ (hs)r% }r& (hEXQPrecomputed kernel matrix. No hyperparameters and thus nothing will be optimised.r' hFj! hGXM/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.Prer( hLhwhN}r) (hP]hQ]hR]hS]hU]uhWKhXhh@]r* haXQPrecomputed kernel matrix. No hyperparameters and thus nothing will be optimised.r+ r, }r- (hEj' hFj% ubaubj)r. }r/ (hEUhFj! hGNhLjhN}r0 (hP]hQ]hR]hS]hU]uhWNhXhh@]r1 j)r2 }r3 (hEUhN}r4 (hP]hQ]hR]hS]hU]uhFj. h@]r5 (j)r6 }r7 (hEUhN}r8 (hP]hQ]hR]hS]hU]uhFj2 h@]r9 haX Parametersr: r; }r< (hEUhFj6 ubahLjubj)r= }r> (hEUhN}r? (hP]hQ]hR]hS]hU]uhFj2 h@]r@ j )rA }rB (hEUhN}rC (hP]hQ]hR]hS]hU]uhFj= h@]rD (h)rE }rF (hEUhN}rG (hP]hQ]hR]hS]hU]uhFjA h@]rH hs)rI }rJ (hEUhN}rK (hP]hQ]hR]hS]hU]uhFjE h@]rL (jK)rM }rN (hEXM1hN}rO (hP]hQ]hR]hS]hU]uhFjI h@]rP haXM1rQ rR }rS (hEUhFjM ubahLjSubhaX -- rT rU }rV (hEUhFjI ubhaXbcross covariances matrix(train+1 by test). last row is self covariances (diagonal of test by test)rW rX }rY (hEXbcross covariances matrix(train+1 by test). last row is self covariances (diagonal of test by test)hFjI ubehLhwubahLhubh)rZ }r[ (hEUhN}r\ (hP]hQ]hR]hS]hU]uhFjA h@]r] hs)r^ }r_ (hEUhN}r` (hP]hQ]hR]hS]hU]uhFjZ h@]ra (jK)rb }rc (hEXM2hN}rd (hP]hQ]hR]hS]hU]uhFj^ h@]re haXM2rf rg }rh (hEUhFjb ubahLjSubhaX -- ri rj }rk (hEUhFj^ ubhaX/training set covariance matrix (train by train)rl rm }rn (hEX/training set covariance matrix (train by train)hFj^ ubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)ro }rp (hEUhFj$hGXY/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.ProductOfKernelrq hLj7hN}rr (hS]hR]hP]hQ]hU]Uentries]rs (j:X)ProductOfKernel (class in pyGPs.Core.cov)hUtrt auhWNhXhh@]ubjA)ru }rv (hEUhFj$hGjq hLjDhN}rw (jFjGXpyhS]hR]hP]hQ]hU]jHXclassrx jJjx uhWNhXhh@]ry (jL)rz }r{ (hEXProductOfKernel(cov1, cov2)hFju hGjOhLjPhN}r| (hS]r} hajShHXpyGPs.Core.covr~ r }r bhR]hP]hQ]hU]r hajXXProductOfKernelr jZUj[uhWNhXhh@]r (j])r }r (hEXclass hFjz hGjOhLj`hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXclass r r }r (hEUhFj ubaubjf)r }r (hEXpyGPs.Core.cov.hFjz hGjOhLjihN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXpyGPs.Core.cov.r r }r (hEUhFj ubaubjo)r }r (hEj hFjz hGjOhLjrhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXProductOfKernelr r }r (hEUhFj ubaubjx)r }r (hEUhFjz hGjOhLj{hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (j~)r }r (hEXcov1hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXcov1r r }r (hEUhFj ubahLjubj~)r }r (hEXcov2hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXcov2r r }r (hEUhFj ubahLjubeubj)r }r (hEUhFjz hGNhLjhN}r (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r j)r }r (hEUhN}r (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidj uhFj h@]r j)r }r (hEUhN}r (hP]hQ]r jahR]hS]hU]uhFj h@]r haX[source]r r }r (hEUhFj ubahLjubahLjubaubeubj)r }r (hEUhFju hGjOhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r hs)r }r (hEXProduct of two kernel function.r hFj hGjq hLhwhN}r (hP]hQ]hR]hS]hU]uhWKhXhh@]r haXProduct of two kernel function.r r }r (hEj hFj ubaubaubeubj3)r }r (hEUhFj$hGNhLj7hN}r (hS]hR]hP]hQ]hU]Uentries]r (j:XRBF (class in pyGPs.Core.cov)h#Utr auhWNhXhh@]ubjA)r }r (hEUhFj$hGNhLjDhN}r (jFjGXpyhS]hR]hP]hQ]hU]jHXclassr jJj uhWNhXhh@]r (jL)r }r (hEXRBF(log_ell=0.0, log_sigma=0.0)hFj hGjOhLjPhN}r (hS]r h#ajShHXpyGPs.Core.covr r }r bhR]hP]hQ]hU]r h#ajXXRBFr jZUj[uhWNhXhh@]r (j])r }r (hEXclass hFj hGjOhLj`hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXclass r r }r (hEUhFj ubaubjf)r }r (hEXpyGPs.Core.cov.hFj hGjOhLjihN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXpyGPs.Core.cov.r r }r (hEUhFj ubaubjo)r }r (hEj hFj hGjOhLjrhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXRBFr r }r (hEUhFj ubaubjx)r }r (hEUhFj hGjOhLj{hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (j~)r }r (hEX log_ell=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_ell=0.0r r }r (hEUhFj ubahLjubj~)r }r (hEX log_sigma=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_sigma=0.0r r }r (hEUhFj ubahLjubeubj)r }r (hEUhFj hGNhLjhN}r (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r j)r }r (hEUhN}r (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidj uhFj h@]r j)r }r (hEUhN}r (hP]hQ]r jahR]hS]hU]uhFj h@]r haX[source]r r }r (hEUhFj ubahLjubahLjubaubeubj)r }r (hEUhFj hGjOhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (hs)r }r (hEXVSquared Exponential kernel with isotropic distance measure. hyp = [log_ell, log_sigma]r hFj hGXM/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.RBFr hLhwhN}r (hP]hQ]hR]hS]hU]uhWKhXhh@]r haXVSquared Exponential kernel with isotropic distance measure. hyp = [log_ell, log_sigma]r r }r (hEj hFj ubaubj)r }r (hEUhFj hGNhLjhN}r! (hP]hQ]hR]hS]hU]uhWNhXhh@]r" j)r# }r$ (hEUhN}r% (hP]hQ]hR]hS]hU]uhFj h@]r& (j)r' }r( (hEUhN}r) (hP]hQ]hR]hS]hU]uhFj# h@]r* haX Parametersr+ r, }r- (hEUhFj' ubahLjubj)r. }r/ (hEUhN}r0 (hP]hQ]hR]hS]hU]uhFj# h@]r1 j )r2 }r3 (hEUhN}r4 (hP]hQ]hR]hS]hU]uhFj. h@]r5 (h)r6 }r7 (hEUhN}r8 (hP]hQ]hR]hS]hU]uhFj2 h@]r9 hs)r: }r; (hEUhN}r< (hP]hQ]hR]hS]hU]uhFj6 h@]r= (jK)r> }r? (hEXlog_ellhN}r@ (hP]hQ]hR]hS]hU]uhFj: h@]rA haXlog_ellrB rC }rD (hEUhFj> ubahLjSubhaX -- rE rF }rG (hEUhFj: ubhaXcharacteristic length scale.rH rI }rJ (hEXcharacteristic length scale.hFj: ubehLhwubahLhubh)rK }rL (hEUhN}rM (hP]hQ]hR]hS]hU]uhFj2 h@]rN hs)rO }rP (hEUhN}rQ (hP]hQ]hR]hS]hU]uhFjK h@]rR (jK)rS }rT (hEX log_sigmahN}rU (hP]hQ]hR]hS]hU]uhFjO h@]rV haX log_sigmarW rX }rY (hEUhFjS ubahLjSubhaX -- rZ r[ }r\ (hEUhFjO ubhaXsignal deviation.r] r^ }r_ (hEXsignal deviation.hFjO ubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)r` }ra (hEUhFj$hGNhLj7hN}rb (hS]hR]hP]hQ]hU]Uentries]rc (j:X RBFard (class in pyGPs.Core.cov)hUtrd auhWNhXhh@]ubjA)re }rf (hEUhFj$hGNhLjDhN}rg (jFjGXpyhS]hR]hP]hQ]hU]jHXclassrh jJjh uhWNhXhh@]ri (jL)rj }rk (hEX0RBFard(D=None, log_ell_list=None, log_sigma=0.0)hFje hGjOhLjPhN}rl (hS]rm hajShHXpyGPs.Core.covrn ro }rp bhR]hP]hQ]hU]rq hajXXRBFardrr jZUj[uhWNhXhh@]rs (j])rt }ru (hEXclass hFjj hGjOhLj`hN}rv (hP]hQ]hR]hS]hU]uhWNhXhh@]rw haXclass rx ry }rz (hEUhFjt ubaubjf)r{ }r| (hEXpyGPs.Core.cov.hFjj hGjOhLjihN}r} (hP]hQ]hR]hS]hU]uhWNhXhh@]r~ haXpyGPs.Core.cov.r r }r (hEUhFj{ ubaubjo)r }r (hEjr hFjj hGjOhLjrhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXRBFardr r }r (hEUhFj ubaubjx)r }r (hEUhFjj hGjOhLj{hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (j~)r }r (hEXD=NonehN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXD=Noner r }r (hEUhFj ubahLjubj~)r }r (hEXlog_ell_list=NonehN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXlog_ell_list=Noner r }r (hEUhFj ubahLjubj~)r }r (hEX log_sigma=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_sigma=0.0r r }r (hEUhFj ubahLjubeubj)r }r (hEUhFjj hGNhLjhN}r (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r j)r }r (hEUhN}r (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjr uhFj h@]r j)r }r (hEUhN}r (hP]hQ]r jahR]hS]hU]uhFj h@]r haX[source]r r }r (hEUhFj ubahLjubahLjubaubeubj)r }r (hEUhFje hGjOhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (hs)r }r (hEXcSquared Exponential kernel with Automatic Relevance Determination. hyp = log_ell_list + [log_sigma]r hFj hGXP/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.RBFardr hLhwhN}r (hP]hQ]hR]hS]hU]uhWKhXhh@]r haXcSquared Exponential kernel with Automatic Relevance Determination. hyp = log_ell_list + [log_sigma]r r }r (hEj hFj ubaubj)r }r (hEUhFj hGNhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r j)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (j)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX Parametersr r }r (hEUhFj ubahLjubj)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r j )r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (h)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (jK)r }r (hEXDhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXDr }r (hEUhFj ubahLjSubhaX -- r r }r (hEUhFj ubhaXQdimension of pattern. set if you want default ell, which is 1 for each dimension.r r }r (hEXQdimension of pattern. set if you want default ell, which is 1 for each dimension.hFj ubehLhwubahLhubh)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (jK)r }r (hEX log_ell_listhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_ell_listr r }r (hEUhFj ubahLjSubhaX -- r r }r (hEUhFj ubhaX/characteristic length scale for each dimension.r r }r (hEX/characteristic length scale for each dimension.hFj ubehLhwubahLhubh)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (jK)r }r (hEX log_sigmahN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_sigmar r }r (hEUhFj ubahLjSubhaX -- r r }r (hEUhFj ubhaXsignal deviation.r r }r (hEXsignal deviation.hFj ubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)r }r (hEUhFj$hGNhLj7hN}r (hS]hR]hP]hQ]hU]Uentries]r (j:X!RBFunit (class in pyGPs.Core.cov)hUtr auhWNhXhh@]ubjA)r }r (hEUhFj$hGNhLjDhN}r (jFjGXpyhS]hR]hP]hQ]hU]jHXclassr jJj uhWNhXhh@]r (jL)r }r (hEXRBFunit(log_ell=0.0)hFj hGjOhLjPhN}r (hS]r! hajShHXpyGPs.Core.covr" r# }r$ bhR]hP]hQ]hU]r% hajXXRBFunitr& jZUj[uhWNhXhh@]r' (j])r( }r) (hEXclass hFj hGjOhLj`hN}r* (hP]hQ]hR]hS]hU]uhWNhXhh@]r+ haXclass r, r- }r. (hEUhFj( ubaubjf)r/ }r0 (hEXpyGPs.Core.cov.hFj hGjOhLjihN}r1 (hP]hQ]hR]hS]hU]uhWNhXhh@]r2 haXpyGPs.Core.cov.r3 r4 }r5 (hEUhFj/ ubaubjo)r6 }r7 (hEj& hFj hGjOhLjrhN}r8 (hP]hQ]hR]hS]hU]uhWNhXhh@]r9 haXRBFunitr: r; }r< (hEUhFj6 ubaubjx)r= }r> (hEUhFj hGjOhLj{hN}r? (hP]hQ]hR]hS]hU]uhWNhXhh@]r@ j~)rA }rB (hEX log_ell=0.0hN}rC (hP]hQ]hR]hS]hU]uhFj= h@]rD haX log_ell=0.0rE rF }rG (hEUhFjA ubahLjubaubj)rH }rI (hEUhFj hGNhLjhN}rJ (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rK j)rL }rM (hEUhN}rN (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidj& uhFjH h@]rO j)rP }rQ (hEUhN}rR (hP]hQ]rS jahR]hS]hU]uhFjL h@]rT haX[source]rU rV }rW (hEUhFjP ubahLjubahLjubaubeubj)rX }rY (hEUhFj hGjOhLjhN}rZ (hP]hQ]hR]hS]hU]uhWNhXhh@]r[ (hs)r\ }r] (hEXSquared Exponential kernel with isotropic distance measure with unit magnitude. i.e signal variance is always 1. hyp = [ log_ell ]r^ hFjX hGXQ/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.RBFunitr_ hLhwhN}r` (hP]hQ]hR]hS]hU]uhWKhXhh@]ra haXSquared Exponential kernel with isotropic distance measure with unit magnitude. i.e signal variance is always 1. hyp = [ log_ell ]rb rc }rd (hEj^ hFj\ ubaubj)re }rf (hEUhFjX hGNhLjhN}rg (hP]hQ]hR]hS]hU]uhWNhXhh@]rh j)ri }rj (hEUhN}rk (hP]hQ]hR]hS]hU]uhFje h@]rl (j)rm }rn (hEUhN}ro (hP]hQ]hR]hS]hU]uhFji h@]rp haX Parametersrq rr }rs (hEUhFjm ubahLjubj)rt }ru (hEUhN}rv (hP]hQ]hR]hS]hU]uhFji h@]rw hs)rx }ry (hEUhN}rz (hP]hQ]hR]hS]hU]uhFjt h@]r{ (jK)r| }r} (hEXlog_ellhN}r~ (hP]hQ]hR]hS]hU]uhFjx h@]r haXlog_ellr r }r (hEUhFj| ubahLjSubhaX -- r r }r (hEUhFjx ubhaXcharacteristic length scale.r r }r (hEXcharacteristic length scale.hFjx ubehLhwubahLjubehLjubaubeubeubj3)r }r (hEUhFj$hGNhLj7hN}r (hS]hR]hP]hQ]hU]Uentries]r (j:XRQ (class in pyGPs.Core.cov)hUtr auhWNhXhh@]ubjA)r }r (hEUhFj$hGNhLjDhN}r (jFjGXpyhS]hR]hP]hQ]hU]jHXclassr jJj uhWNhXhh@]r (jL)r }r (hEX-RQ(log_ell=0.0, log_sigma=0.0, log_alpha=0.0)hFj hGjOhLjPhN}r (hS]r hajShHXpyGPs.Core.covr r }r bhR]hP]hQ]hU]r hajXXRQr jZUj[uhWNhXhh@]r (j])r }r (hEXclass hFj hGjOhLj`hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXclass r r }r (hEUhFj ubaubjf)r }r (hEXpyGPs.Core.cov.hFj hGjOhLjihN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXpyGPs.Core.cov.r r }r (hEUhFj ubaubjo)r }r (hEj hFj hGjOhLjrhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXRQr r }r (hEUhFj ubaubjx)r }r (hEUhFj hGjOhLj{hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (j~)r }r (hEX log_ell=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_ell=0.0r r }r (hEUhFj ubahLjubj~)r }r (hEX log_sigma=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_sigma=0.0r r }r (hEUhFj ubahLjubj~)r }r (hEX log_alpha=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_alpha=0.0r r }r (hEUhFj ubahLjubeubj)r }r (hEUhFj hGNhLjhN}r (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r j)r }r (hEUhN}r (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidj uhFj h@]r j)r }r (hEUhN}r (hP]hQ]r jahR]hS]hU]uhFj h@]r haX[source]r r }r (hEUhFj ubahLjubahLjubaubeubj)r }r (hEUhFj hGjOhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (hs)r }r (hEXoRational Quadratic covariance function with isotropic distance measure. hyp = [ log_ell, log_sigma, log_alpha ]r hFj hGXL/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.RQr hLhwhN}r (hP]hQ]hR]hS]hU]uhWKhXhh@]r haXoRational Quadratic covariance function with isotropic distance measure. hyp = [ log_ell, log_sigma, log_alpha ]r r }r (hEj hFj ubaubj)r }r (hEUhFj hGNhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r j)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (j)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX Parametersr r }r (hEUhFj ubahLjubj)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r j )r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (h)r }r(hEUhN}r(hP]hQ]hR]hS]hU]uhFj h@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFj h@]r(jK)r}r(hEXlog_ellhN}r (hP]hQ]hR]hS]hU]uhFjh@]r haXlog_ellr r }r (hEUhFjubahLjSubhaX -- rr}r(hEUhFjubhaXcharacteristic length scale.rr}r(hEXcharacteristic length scale.hFjubehLhwubahLhubh)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFj h@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r(hEX log_sigmahN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_sigmar r!}r"(hEUhFjubahLjSubhaX -- r#r$}r%(hEUhFjubhaXsignal deviation.r&r'}r((hEXsignal deviation.hFjubehLhwubahLhubh)r)}r*(hEUhN}r+(hP]hQ]hR]hS]hU]uhFj h@]r,hs)r-}r.(hEUhN}r/(hP]hQ]hR]hS]hU]uhFj)h@]r0(jK)r1}r2(hEX log_alphahN}r3(hP]hQ]hR]hS]hU]uhFj-h@]r4haX log_alphar5r6}r7(hEUhFj1ubahLjSubhaX -- r8r9}r:(hEUhFj-ubhaX&shape parameter for the RQ covariance.r;r<}r=(hEX&shape parameter for the RQ covariance.hFj-ubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)r>}r?(hEUhFj$hGNhLj7hN}r@(hS]hR]hP]hQ]hU]Uentries]rA(j:XRQard (class in pyGPs.Core.cov)hUtrBauhWNhXhh@]ubjA)rC}rD(hEUhFj$hGNhLjDhN}rE(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrFjJjFuhWNhXhh@]rG(jL)rH}rI(hEX>RQard(D=None, log_ell_list=None, log_sigma=0.0, log_alpha=0.0)hFjChGjOhLjPhN}rJ(hS]rKhajShHXpyGPs.Core.covrLrM}rNbhR]hP]hQ]hU]rOhajXXRQardrPjZUj[uhWNhXhh@]rQ(j])rR}rS(hEXclass hFjHhGjOhLj`hN}rT(hP]hQ]hR]hS]hU]uhWNhXhh@]rUhaXclass rVrW}rX(hEUhFjRubaubjf)rY}rZ(hEXpyGPs.Core.cov.hFjHhGjOhLjihN}r[(hP]hQ]hR]hS]hU]uhWNhXhh@]r\haXpyGPs.Core.cov.r]r^}r_(hEUhFjYubaubjo)r`}ra(hEjPhFjHhGjOhLjrhN}rb(hP]hQ]hR]hS]hU]uhWNhXhh@]rchaXRQardrdre}rf(hEUhFj`ubaubjx)rg}rh(hEUhFjHhGjOhLj{hN}ri(hP]hQ]hR]hS]hU]uhWNhXhh@]rj(j~)rk}rl(hEXD=NonehN}rm(hP]hQ]hR]hS]hU]uhFjgh@]rnhaXD=Nonerorp}rq(hEUhFjkubahLjubj~)rr}rs(hEXlog_ell_list=NonehN}rt(hP]hQ]hR]hS]hU]uhFjgh@]ruhaXlog_ell_list=Nonervrw}rx(hEUhFjrubahLjubj~)ry}rz(hEX log_sigma=0.0hN}r{(hP]hQ]hR]hS]hU]uhFjgh@]r|haX log_sigma=0.0r}r~}r(hEUhFjyubahLjubj~)r}r(hEX log_alpha=0.0hN}r(hP]hQ]hR]hS]hU]uhFjgh@]rhaX log_alpha=0.0rr}r(hEUhFjubahLjubeubj)r}r(hEUhFjHhGNhLjhN}r(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rj)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjPuhFjh@]rj)r}r(hEUhN}r(hP]hQ]rjahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLjubahLjubaubeubj)r}r(hEUhFjChGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r}r(hEXRational Quadratic covariance function with Automatic Relevance Detemination (ARD) distance measure. hyp = log_ell_list + [ log_sigma, log_alpha ]rhFjhGXO/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.RQardrhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaXRational Quadratic covariance function with Automatic Relevance Detemination (ARD) distance measure. hyp = log_ell_list + [ log_sigma, log_alpha ]rr}r(hEjhFjubaubj)r}r(hEUhFjhGNhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX Parametersrr}r(hEUhFjubahLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rj )r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(h)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r(hEXDhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXDr}r(hEUhFjubahLjSubhaX -- rr}r(hEUhFjubhaXSdimension of pattern. set if you want default ell, which is 0.5 for each dimension.rr}r(hEXSdimension of pattern. set if you want default ell, which is 0.5 for each dimension.rhFjubehLhwubahLhubh)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r(hEX log_ell_listhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_ell_listrr}r(hEUhFjubahLjSubhaX -- rr}r(hEUhFjubhaX/characteristic length scale for each dimension.rr}r(hEX/characteristic length scale for each dimension.rhFjubehLhwubahLhubh)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r(hEX log_sigmahN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_sigmarr}r(hEUhFjubahLjSubhaX -- rr}r(hEUhFjubhaXsignal deviation.rr}r(hEXsignal deviation.rhFjubehLhwubahLhubh)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r(hEX log_alphahN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_alpharr }r (hEUhFjubahLjSubhaX -- r r }r (hEUhFjubhaX&shape parameter for the RQ covariance.rr}r(hEX&shape parameter for the RQ covariance.rhFjubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)r}r(hEUhFj$hGXW/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.ScaleOfKernelrhLj7hN}r(hS]hR]hP]hQ]hU]Uentries]r(j:X'ScaleOfKernel (class in pyGPs.Core.cov)hUtrauhWNhXhh@]ubjA)r}r(hEUhFj$hGjhLjDhN}r(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrjJjuhWNhXhh@]r(jL)r}r(hEXScaleOfKernel(cov, scalar)hFjhGjOhLjPhN}r(hS]r hajShHXpyGPs.Core.covr!r"}r#bhR]hP]hQ]hU]r$hajXX ScaleOfKernelr%jZUj[uhWNhXhh@]r&(j])r'}r((hEXclass hFjhGjOhLj`hN}r)(hP]hQ]hR]hS]hU]uhWNhXhh@]r*haXclass r+r,}r-(hEUhFj'ubaubjf)r.}r/(hEXpyGPs.Core.cov.hFjhGjOhLjihN}r0(hP]hQ]hR]hS]hU]uhWNhXhh@]r1haXpyGPs.Core.cov.r2r3}r4(hEUhFj.ubaubjo)r5}r6(hEj%hFjhGjOhLjrhN}r7(hP]hQ]hR]hS]hU]uhWNhXhh@]r8haX ScaleOfKernelr9r:}r;(hEUhFj5ubaubjx)r<}r=(hEUhFjhGjOhLj{hN}r>(hP]hQ]hR]hS]hU]uhWNhXhh@]r?(j~)r@}rA(hEXcovhN}rB(hP]hQ]hR]hS]hU]uhFj<h@]rChaXcovrDrE}rF(hEUhFj@ubahLjubj~)rG}rH(hEXscalarhN}rI(hP]hQ]hR]hS]hU]uhFj<h@]rJhaXscalarrKrL}rM(hEUhFjGubahLjubeubj)rN}rO(hEUhFjhGNhLjhN}rP(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rQj)rR}rS(hEUhN}rT(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidj%uhFjNh@]rUj)rV}rW(hEUhN}rX(hP]hQ]rYjahR]hS]hU]uhFjRh@]rZhaX[source]r[r\}r](hEUhFjVubahLjubahLjubaubeubj)r^}r_(hEUhFjhGjOhLjhN}r`(hP]hQ]hR]hS]hU]uhWNhXhh@]rahs)rb}rc(hEXScale of a kernel function.rdhFj^hGjhLhwhN}re(hP]hQ]hR]hS]hU]uhWKhXhh@]rfhaXScale of a kernel function.rgrh}ri(hEjdhFjbubaubaubeubj3)rj}rk(hEUhFj$hGXU/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.SumOfKernelrlhLj7hN}rm(hS]hR]hP]hQ]hU]Uentries]rn(j:X%SumOfKernel (class in pyGPs.Core.cov)h)UtroauhWNhXhh@]ubjA)rp}rq(hEUhFj$hGjlhLjDhN}rr(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrsjJjsuhWNhXhh@]rt(jL)ru}rv(hEXSumOfKernel(cov1, cov2)hFjphGjOhLjPhN}rw(hS]rxh)ajShHXpyGPs.Core.covryrz}r{bhR]hP]hQ]hU]r|h)ajXX SumOfKernelr}jZUj[uhWNhXhh@]r~(j])r}r(hEXclass hFjuhGjOhLj`hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXclass rr}r(hEUhFjubaubjf)r}r(hEXpyGPs.Core.cov.hFjuhGjOhLjihN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXpyGPs.Core.cov.rr}r(hEUhFjubaubjo)r}r(hEj}hFjuhGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaX SumOfKernelrr}r(hEUhFjubaubjx)r}r(hEUhFjuhGjOhLj{hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(j~)r}r(hEXcov1hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXcov1rr}r(hEUhFjubahLjubj~)r}r(hEXcov2hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXcov2rr}r(hEUhFjubahLjubeubj)r}r(hEUhFjuhGNhLjhN}r(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rj)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidj}uhFjh@]rj)r}r(hEUhN}r(hP]hQ]rjahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLjubahLjubaubeubj)r}r(hEUhFjphGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhs)r}r(hEXSum of two kernel function.rhFjhGjlhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaXSum of two kernel function.rr}r(hEjhFjubaubaubeubeubhB)r}r(hEUhFhChGhJhLhMhN}r(hP]hQ]hR]hS]r(Xmodule-pyGPs.Core.meanrh;ehU]rhauhWKhXhh@]r(hZ)r}r(hEX$List of Means and Default ParametersrhFjhGhJhLh^hN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaX$List of Means and Default Parametersrr}r(hEjhFjubaubj3)r}r(hEUhFjhGj6hLj7hN}r(hS]hR]hP]hQ]hU]Uentries]r(j:XpyGPs.Core.mean (module)Xmodule-pyGPs.Core.meanUtrauhWKhXhh@]ubj3)r}r(hEUhFjhGNhLj7hN}r(hS]hR]hP]hQ]hU]Uentries]r(j:X Const (class in pyGPs.Core.mean)h"UtrauhWNhXhh@]ubjA)r}r(hEUhFjhGNhLjDhN}r(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrjJjuhWNhXhh@]r(jL)r}r(hEX Const(c=5.0)hFjhGjOhLjPhN}r(hS]rh"ajShHXpyGPs.Core.meanrr}rbhR]hP]hQ]hU]rh"ajXXConstrjZUj[uhWNhXhh@]r(j])r}r(hEXclass hFjhGjOhLj`hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXclass rr}r(hEUhFjubaubjf)r}r(hEXpyGPs.Core.mean.hFjhGjOhLjihN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXpyGPs.Core.mean.rr}r(hEUhFjubaubjo)r}r(hEjhFjhGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXConstrr}r(hEUhFjubaubjx)r}r(hEUhFjhGjOhLj{hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rj~)r}r(hEXc=5.0hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXc=5.0rr}r (hEUhFjubahLjubaubj)r }r (hEUhFjhGNhLjhN}r (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r j)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/meanUrefidjuhFj h@]rj)r}r(hEUhN}r(hP]hQ]rjahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLjubahLjubaubeubj)r}r(hEUhFjhGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r}r(hEX!Constant mean function. hyp = [c]r hFjhGXQ/Users/hs/Program/git/pyGPs/pyGPs/Core/mean.py:docstring of pyGPs.Core.mean.Constr!hLhwhN}r"(hP]hQ]hR]hS]hU]uhWKhXhh@]r#haX!Constant mean function. hyp = [c]r$r%}r&(hEj hFjubaubj)r'}r((hEUhFjhGNhLjhN}r)(hP]hQ]hR]hS]hU]uhWNhXhh@]r*j)r+}r,(hEUhN}r-(hP]hQ]hR]hS]hU]uhFj'h@]r.(j)r/}r0(hEUhN}r1(hP]hQ]hR]hS]hU]uhFj+h@]r2haX Parametersr3r4}r5(hEUhFj/ubahLjubj)r6}r7(hEUhN}r8(hP]hQ]hR]hS]hU]uhFj+h@]r9hs)r:}r;(hEUhN}r<(hP]hQ]hR]hS]hU]uhFj6h@]r=(jK)r>}r?(hEXchN}r@(hP]hQ]hR]hS]hU]uhFj:h@]rAhaXcrB}rC(hEUhFj>ubahLjSubhaX -- rDrE}rF(hEUhFj:ubhaXconstant value for meanrGrH}rI(hEXconstant value for meanhFj:ubehLhwubahLjubehLjubaubeubeubj3)rJ}rK(hEUhFjhGNhLj7hN}rL(hS]hR]hP]hQ]hU]Uentries]rM(j:X!Linear (class in pyGPs.Core.mean)h'UtrNauhWNhXhh@]ubjA)rO}rP(hEUhFjhGNhLjDhN}rQ(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrRjJjRuhWNhXhh@]rS(jL)rT}rU(hEXLinear(D=None, alpha_list=None)hFjOhGjOhLjPhN}rV(hS]rWh'ajShHXpyGPs.Core.meanrXrY}rZbhR]hP]hQ]hU]r[h'ajXXLinearr\jZUj[uhWNhXhh@]r](j])r^}r_(hEXclass hFjThGjOhLj`hN}r`(hP]hQ]hR]hS]hU]uhWNhXhh@]rahaXclass rbrc}rd(hEUhFj^ubaubjf)re}rf(hEXpyGPs.Core.mean.hFjThGjOhLjihN}rg(hP]hQ]hR]hS]hU]uhWNhXhh@]rhhaXpyGPs.Core.mean.rirj}rk(hEUhFjeubaubjo)rl}rm(hEj\hFjThGjOhLjrhN}rn(hP]hQ]hR]hS]hU]uhWNhXhh@]rohaXLinearrprq}rr(hEUhFjlubaubjx)rs}rt(hEUhFjThGjOhLj{hN}ru(hP]hQ]hR]hS]hU]uhWNhXhh@]rv(j~)rw}rx(hEXD=NonehN}ry(hP]hQ]hR]hS]hU]uhFjsh@]rzhaXD=Noner{r|}r}(hEUhFjwubahLjubj~)r~}r(hEXalpha_list=NonehN}r(hP]hQ]hR]hS]hU]uhFjsh@]rhaXalpha_list=Nonerr}r(hEUhFj~ubahLjubeubj)r}r(hEUhFjThGNhLjhN}r(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rj)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/meanUrefidj\uhFjh@]rj)r}r(hEUhN}r(hP]hQ]rjahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLjubahLjubaubeubj)r}r(hEUhFjOhGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r}r(hEX+Linear mean function. self.hyp = alpha_listrhFjhGXR/Users/hs/Program/git/pyGPs/pyGPs/Core/mean.py:docstring of pyGPs.Core.mean.LinearrhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaX+Linear mean function. self.hyp = alpha_listrr}r(hEjhFjubaubj)r}r(hEUhFjhGNhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX Parametersrr}r(hEUhFjubahLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r(hEXDhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXDr}r(hEUhFjubahLjSubhaX -- rr}r(hEUhFjubhaX[dimension of training data. Set if you want default alpha, which is 0.5 for each dimension.rr}r(hEX[dimension of training data. 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Usually, the approximate posterior to be returned admits the form: N(m=K*alpha, V=inv(inv(K)+W)), where alpha is a vector and W is diagonal; if not, then L contains instead -inv(K+inv(W)), and sW is unused. For more information on the individual approximation methods and their implementations, see the respective inference function below. See also gp.py :param meanfunc: mean function :param covfunc: covariance function :param likfunc: likelihood function :param x: training data :param y: training labels :param nargout: specify the number of output(1,2 or 3) :return: posterior, negative-log-marginal-likelihood, derivative for negative-log-marginal-likelihood-likelihood ''' passh(jch)h,h.hh0}r{(hhh5]h4]h2]h3]h7]uh9K!h:hh"]r|hCX# Your inference class needs to inherit base class Inference, # which is in the module of Core.inf class MyKernel(Kernel): def __init__(self): pass def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): ''' Inference computation based on inputs. post, nlZ, dnlZ = inf.proceed(mean, cov, lik, x, y) INPUT: cov: name of the covariance function (see covFunctions.m) lik: name of the likelihood function (see likFunctions.m) x: n by D matrix of training inputs y: 1d array (of size n) of targets OUTPUT: post(instance of postStruct): struct representation of the (approximate) posterior containing: nlZ: returned value of the negative log marginal likelihood dnlZ(instance of dnlZStruct): struct representation for derivatives of the negative log marginal likelihood w.r.t. each hyperparameter. Usually, the approximate posterior to be returned admits the form: N(m=K*alpha, V=inv(inv(K)+W)), where alpha is a vector and W is diagonal; if not, then L contains instead -inv(K+inv(W)), and sW is unused. For more information on the individual approximation methods and their implementations, see the respective inference function below. See also gp.py :param meanfunc: mean function :param covfunc: covariance function :param likfunc: likelihood function :param x: training data :param y: training labels :param nargout: specify the number of output(1,2 or 3) :return: posterior, negative-log-marginal-likelihood, derivative for negative-log-marginal-likelihood-likelihood ''' passr}r~}r(h'Uh(jyubaubh|)r}r(h'XIwhere **postStruct** and **dnlZStruct** is also defined in *Core.inf*. ::rh(jch)h,h.hh0}r(h2]h3]h4]h5]h7]uh9KJh:hh"]r(hCXwhere rr}r(h'Xwhere h(jubh)r}r(h'X**postStruct**h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCX postStructrr}r(h'Uh(jubah.hubhCX and rr}r(h'X and h(jubh)r}r(h'X**dnlZStruct**h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCX dnlZStructrr}r(h'Uh(jubah.hubhCX is also defined in rr}r(h'X is also defined in h(jubh)r}r(h'X *Core.inf*h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCXCore.infrr}r(h'Uh(jubah.hubhCX.r}r(h'X.h(jubeubh)r}r(h'Xclass postStruct(object): ''' Data structure for posterior post.alpha -> 1d array containing inv(K)*m, where K is the prior covariance matrix and m the approx posterior mean post.sW: -> 1d array containing diagonal of sqrt(W) the approximate posterior covariance matrix is inv(inv(K)+W) post.L -> 2d array, L = chol(sW*K*sW+identity(n)) ''' class dnlZStruct(object): ''' Data structure for the derivatives of mean, cov and lik functions. dnlZ.mean -> list of derivatives for each hyperparameters in mean function dnlZ.cov -> list of derivatives for each hyperparameters in covariance function dnlZ.lik -> list of derivatives for each hyperparameters in likelihood function '''h(jch)h,h.hh0}r(hhh5]h4]h2]h3]h7]uh9KLh:hh"]rhCXclass postStruct(object): ''' Data structure for posterior post.alpha -> 1d array containing inv(K)*m, where K is the prior covariance matrix and m the approx posterior mean post.sW: -> 1d array containing diagonal of sqrt(W) the approximate posterior covariance matrix is inv(inv(K)+W) post.L -> 2d array, L = chol(sW*K*sW+identity(n)) ''' class dnlZStruct(object): ''' Data structure for the derivatives of mean, cov and lik functions. dnlZ.mean -> list of derivatives for each hyperparameters in mean function dnlZ.cov -> list of derivatives for each hyperparameters in covariance function dnlZ.lik -> list of derivatives for each hyperparameters in likelihood function '''rr}r(h'Uh(jubaubh|)r}r(h'X!and for customized mean class: ::rh(jch)h,h.hh0}r(h2]h3]h4]h5]h7]uh9K`h:hh"]rhCXand for customized mean class:rr}r(h'Xand for customized mean class:h(jubaubh)r}r(h'XS# Your mean class needs to inherit base class Mean, # which is in the module of Core.mean class MyMean(Mean): def __init__(self, hyp): ''' Intialize hyperparameters for MyMean. ''' self.hyp = hyp def getMean(self, x=None): ''' Get the mean vector. ''' pass def getDerMatrix(self, x=None, der=None): ''' Compute derivatives wrt. hyperparameters. :param x: training data :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' passh(jch)h,h.hh0}r(hhh5]h4]h2]h3]h7]uh9Kbh:hh"]rhCXS# Your mean class needs to inherit base class Mean, # which is in the module of Core.mean class MyMean(Mean): def __init__(self, hyp): ''' Intialize hyperparameters for MyMean. ''' self.hyp = hyp def getMean(self, x=None): ''' Get the mean vector. ''' pass def getDerMatrix(self, x=None, der=None): ''' Compute derivatives wrt. hyperparameters. :param x: training data :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' passrr}r(h'Uh(jubaubh|)r}r(h'XCustomizing likelihood function is more complicated. We will omit it here to keep this this page not too long. However, you can find detailed explaination either in the **source code** *Core.lik* or in coresponding section of **manual**.h(jch)h,h.hh0}r(h2]h3]h4]h5]h7]uh9K~h:hh"]r(hCXCustomizing likelihood function is more complicated. We will omit it here to keep this this page not too long. However, you can find detailed explaination either in the rr}r(h'XCustomizing likelihood function is more complicated. We will omit it here to keep this this page not too long. However, you can find detailed explaination either in the h(jubh)r}r(h'X**source code**h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCX source coderr}r(h'Uh(jubah.hubhCX r}r(h'X h(jubh)r}r(h'X *Core.lik*h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCXCore.likrr}r(h'Uh(jubah.hubhCX or in coresponding section of rr}r(h'X or in coresponding section of h(jubh)r}r(h'X **manual**h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCXmanualrr}r(h'Uh(jubah.hubhCX.r}r(h'X.h(jubeubh|)r}r(h'XJust like testing kernel/mean fucntions, you can also find unit test module for likelihood and inference functions. To test your customized inference function, uncomment the following method in *pyGPs.Testing.unit_test_inf.py*. ::h(jch)h,h.hh0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]r(hCXJust like testing kernel/mean fucntions, you can also find unit test module for likelihood and inference functions. To test your customized inference function, uncomment the following method in rr}r(h'XJust like testing kernel/mean fucntions, you can also find unit test module for likelihood and inference functions. To test your customized inference function, uncomment the following method in h(jubh)r}r(h'X *pyGPs.Testing.unit_test_inf.py*h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCXpyGPs.Testing.unit_test_inf.pyrr}r(h'Uh(jubah.hubhCX.r}r(h'X.h(jubeubh)r}r(h'X # Test your customized inference function def test_inf_new(self): # specify your inf function # set mean/cov/lik functions post, nlZ, dnlZ = inffunc.proceed(meanfunc, covfunc, likfunc, self.x, self.y, nargout=3) self.checkFITCOutput(post, nlZ, dnlZ)h(jch)h,h.hh0}r(hhh5]h4]h2]h3]h7]uh9Kh:hh"]rhCX # Test your customized inference function def test_inf_new(self): # specify your inf function # set mean/cov/lik functions post, nlZ, dnlZ = inffunc.proceed(meanfunc, covfunc, likfunc, self.x, self.y, nargout=3) self.checkFITCOutput(post, nlZ, dnlZ)rr}r(h'Uh(jubaubh|)r}r(h'XNand test customized likelihood function in *pyGPs.Testing.unit_test_lik.py* ::rh(jch)h,h.hh0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]r(hCX+and test customized likelihood function in rr}r(h'X+and test customized likelihood function in h(jubh)r}r(h'X *pyGPs.Testing.unit_test_lik.py*h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCXpyGPs.Testing.unit_test_lik.pyrr}r(h'Uh(jubah.hubeubh)r}r(h'X# Test your customized likelihood function def test_cov_new(self): likelihood = myLikelihood() # specify your likelihood function self.checkLikelihood(likelihood)h(jch)h,h.hh0}r(hhh5]h4]h2]h3]h7]uh9Kh:hh"]rhCX# Test your customized likelihood function def test_cov_new(self): likelihood = myLikelihood() # specify your likelihood function self.checkLikelihood(likelihood)rr }r (h'Uh(jubaubeubh$)r }r (h'Uh(h%h)h,h.h/h0}r (h2]h3]h4]h5]rh!ah7]rhauh9Kh:hh"]r(h<)r}r(h'XList of Likelihoodsrh(j h)h,h.h@h0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]rhCXList of Likelihoodsrr}r(h'jh(jubaubcsphinx.addnodes index r)r}r(h'Uh(j h)XM/Users/hs/Program/git/pyGPs/pyGPs/Core/lik.py:docstring of pyGPs.Core.lik.Erfrh.Uindexrh0}r(h5]h4]h2]h3]h7]Uentries]r(Usingler XErf (class in pyGPs.Core.lik)hUtr!auh9Nh:hh"]ubcsphinx.addnodes desc r")r#}r$(h'Uh(j h)jh.Udescr%h0}r&(Unoindexr'Udomainr(Xpyh5]h4]h2]h3]h7]Uobjtyper)Xclassr*Udesctyper+j*uh9Nh:hh"]r,(csphinx.addnodes desc_signature r-)r.}r/(h'XErf()h(j#h)U r0h.Udesc_signaturer1h0}r2(h5]r3haUmoduler4h*XpyGPs.Core.likr5r6}r7bh4]h2]h3]h7]r8haUfullnamer9XErfr:Uclassr;UUfirstr<uh9Nh:hh"]r=(csphinx.addnodes desc_annotation r>)r?}r@(h'Xclass h(j.h)j0h.Udesc_annotationrAh0}rB(h2]h3]h4]h5]h7]uh9Nh:hh"]rChCXclass rDrE}rF(h'Uh(j?ubaubcsphinx.addnodes desc_addname rG)rH}rI(h'XpyGPs.Core.lik.h(j.h)j0h.U desc_addnamerJh0}rK(h2]h3]h4]h5]h7]uh9Nh:hh"]rLhCXpyGPs.Core.lik.rMrN}rO(h'Uh(jHubaubcsphinx.addnodes desc_name rP)rQ}rR(h'j:h(j.h)j0h.U desc_namerSh0}rT(h2]h3]h4]h5]h7]uh9Nh:hh"]rUhCXErfrVrW}rX(h'Uh(jQubaubcsphinx.addnodes only rY)rZ}r[(h'Uh(j.h)Nh.Uonlyr\h0}r](UexprUhtmlr^h5]h4]h2]h3]h7]uh9Nh:hh"]r_csphinx.addnodes pending_xref r`)ra}rb(h'Uh0}rc(UreftypeUviewcoderdUrefdocX LikelihoodsreU refdomainUstdrfh5]h4]U refexplicith2]h3]h7]U reftargetX_modules/pyGPs/Core/likUrefidj:uh(jZh"]rgcdocutils.nodes inline rh)ri}rj(h'Uh0}rk(h2]h3]rlU viewcode-linkrmah4]h5]h7]uh(jah"]rnhCX[source]rorp}rq(h'Uh(jiubah.Uinlinerrubah.U pending_xrefrsubaubeubcsphinx.addnodes desc_content rt)ru}rv(h'Uh(j#h)j0h.U desc_contentrwh0}rx(h2]h3]h4]h5]h7]uh9Nh:hh"]ry(h|)rz}r{(h'XiError function or cumulative Gaussian likelihood function for binary classification or probit regression.r|h(juh)jh.hh0}r}(h2]h3]h4]h5]h7]uh9Kh:hh"]r~hCXiError function or cumulative Gaussian likelihood function for binary classification or probit regression.rr}r(h'j|h(jzubaubh|)r}r(h'X@:math:`Erf(t)=\frac{1}{2}(1+erf(\frac{t}{\sqrt{2}}))=normcdf(t)`h(juh)jh.hh0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]rcsphinx.ext.mathbase math r)r}r(h'Uh0}r(UlatexX8Erf(t)=\frac{1}{2}(1+erf(\frac{t}{\sqrt{2}}))=normcdf(t)h5]h4]h2]h3]h7]uh(jh"]h.Umathrubaubeubeubj)r}r(h'Uh(j h)XO/Users/hs/Program/git/pyGPs/pyGPs/Core/lik.py:docstring of pyGPs.Core.lik.Gaussrh.jh0}r(h5]h4]h2]h3]h7]Uentries]r(j XGauss (class in pyGPs.Core.lik)hUtrauh9Nh:hh"]ubj")r}r(h'Uh(j h)jh.j%h0}r(j'j(Xpyh5]h4]h2]h3]h7]j)Xclassrj+juh9Nh:hh"]r(j-)r}r(h'X$Gauss(log_sigma=-2.3025850929940455)h(jh)j0h.j1h0}r(h5]rhaj4h*XpyGPs.Core.likrr}rbh4]h2]h3]h7]rhaj9XGaussrj;Uj<uh9Nh:hh"]r(j>)r}r(h'Xclass h(jh)j0h.jAh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXclass rr}r(h'Uh(jubaubjG)r}r(h'XpyGPs.Core.lik.h(jh)j0h.jJh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXpyGPs.Core.lik.rr}r(h'Uh(jubaubjP)r}r(h'jh(jh)j0h.jSh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXGaussrr}r(h'Uh(jubaubcsphinx.addnodes desc_parameterlist r)r}r(h'Uh(jh)j0h.Udesc_parameterlistrh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rcsphinx.addnodes desc_parameter r)r}r(h'Xlog_sigma=-2.3025850929940455h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCXlog_sigma=-2.3025850929940455rr}r(h'Uh(jubah.Udesc_parameterrubaubjY)r}r(h'Uh(jh)Nh.j\h0}r(Uexprj^h5]h4]h2]h3]h7]uh9Nh:hh"]rj`)r}r(h'Uh0}r(UreftypejdUrefdocjeU refdomainjfh5]h4]U refexplicith2]h3]h7]U reftargetX_modules/pyGPs/Core/likUrefidjuh(jh"]rjh)r}r(h'Uh0}r(h2]h3]rjmah4]h5]h7]uh(jh"]rhCX[source]rr}r(h'Uh(jubah.jrubah.jsubaubeubjt)r}r(h'Uh(jh)j0h.jwh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]r(h|)r}r(h'X,Gaussian likelihood function for regression.rh(jh)jh.hh0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]rhCX,Gaussian likelihood function for regression.rr}r(h'jh(jubaubh|)r}r(h'X:math:`Gauss(t)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(t-y)^2}{2\sigma^2}}`, where :math:`y` is the mean and :math:`\sigma` is the standard deviation.h(jh)jh.hh0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]r(j)r}r(h'Uh0}r(UlatexXDGauss(t)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(t-y)^2}{2\sigma^2}}h5]h4]h2]h3]h7]uh(jh"]h.jubhCX, where rr}r(h'X, where h(jubj)r}r(h'Uh0}r(UlatexXyh5]h4]h2]h3]h7]uh(jh"]h.jubhCX is the mean and rr}r(h'X is the mean and h(jubj)r}r(h'Uh0}r(UlatexX\sigmah5]h4]h2]h3]h7]uh(jh"]h.jubhCX is the standard deviation.rr}r(h'X is the standard deviation.h(jubeubh|)r}r(h'Xhyp = [ log_sigma ]rh(jh)jh.hh0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]rhCXhyp = [ log_sigma ]rr}r(h'jh(jubaubeubeubj)r}r(h'Uh(j h)XQ/Users/hs/Program/git/pyGPs/pyGPs/Core/lik.py:docstring of pyGPs.Core.lik.Laplacerh.jh0}r(h5]h4]h2]h3]h7]Uentries]r(j X!Laplace (class in pyGPs.Core.lik)hUtrauh9Nh:hh"]ubj")r}r(h'Uh(j h)jh.j%h0}r(j'j(Xpyh5]h4]h2]h3]h7]j)Xclassrj+juh9Nh:hh"]r(j-)r }r (h'X&Laplace(log_sigma=-2.3025850929940455)h(jh)j0h.j1h0}r (h5]r haj4h*XpyGPs.Core.likr r}rbh4]h2]h3]h7]rhaj9XLaplacerj;Uj<uh9Nh:hh"]r(j>)r}r(h'Xclass h(j h)j0h.jAh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXclass rr}r(h'Uh(jubaubjG)r}r(h'XpyGPs.Core.lik.h(j h)j0h.jJh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXpyGPs.Core.lik.rr}r (h'Uh(jubaubjP)r!}r"(h'jh(j h)j0h.jSh0}r#(h2]h3]h4]h5]h7]uh9Nh:hh"]r$hCXLaplacer%r&}r'(h'Uh(j!ubaubj)r(}r)(h'Uh(j h)j0h.jh0}r*(h2]h3]h4]h5]h7]uh9Nh:hh"]r+j)r,}r-(h'Xlog_sigma=-2.3025850929940455h0}r.(h2]h3]h4]h5]h7]uh(j(h"]r/hCXlog_sigma=-2.3025850929940455r0r1}r2(h'Uh(j,ubah.jubaubjY)r3}r4(h'Uh(j h)Nh.j\h0}r5(Uexprj^h5]h4]h2]h3]h7]uh9Nh:hh"]r6j`)r7}r8(h'Uh0}r9(UreftypejdUrefdocjeU refdomainjfh5]h4]U refexplicith2]h3]h7]U reftargetX_modules/pyGPs/Core/likUrefidjuh(j3h"]r:jh)r;}r<(h'Uh0}r=(h2]h3]r>jmah4]h5]h7]uh(j7h"]r?hCX[source]r@rA}rB(h'Uh(j;ubah.jrubah.jsubaubeubjt)rC}rD(h'Uh(jh)j0h.jwh0}rE(h2]h3]h4]h5]h7]uh9Nh:hh"]rF(h|)rG}rH(h'XKLaplacian likelihood function for regression. ONLY works with EP inference!rIh(jCh)jh.hh0}rJ(h2]h3]h4]h5]h7]uh9Kh:hh"]rKhCXKLaplacian likelihood function for regression. ONLY works with EP inference!rLrM}rN(h'jIh(jGubaubh|)rO}rP(h'X:math:`Laplace(t) = \frac{1}{2b}e^{-\frac{|t-y|}{b}}` where :math:`b=\frac{\sigma}{\sqrt{2}}`, :math:`y` is the mean and :math:`\sigma` is the standard deviation.h(jCh)jh.hh0}rQ(h2]h3]h4]h5]h7]uh9Kh:hh"]rR(j)rS}rT(h'Uh0}rU(UlatexX-Laplace(t) = \frac{1}{2b}e^{-\frac{|t-y|}{b}}h5]h4]h2]h3]h7]uh(jOh"]h.jubhCX where rVrW}rX(h'X where h(jOubj)rY}rZ(h'Uh0}r[(UlatexXb=\frac{\sigma}{\sqrt{2}}h5]h4]h2]h3]h7]uh(jOh"]h.jubhCX, r\r]}r^(h'X, h(jOubj)r_}r`(h'Uh0}ra(UlatexXyh5]h4]h2]h3]h7]uh(jOh"]h.jubhCX is the mean and rbrc}rd(h'X is the mean and h(jOubj)re}rf(h'Uh0}rg(UlatexX\sigmah5]h4]h2]h3]h7]uh(jOh"]h.jubhCX is the standard deviation.rhri}rj(h'X is the standard deviation.h(jOubeubh|)rk}rl(h'Xhyp = [ log_sigma ]rmh(jCh)jh.hh0}rn(h2]h3]h4]h5]h7]uh9Kh:hh"]rohCXhyp = [ log_sigma ]rprq}rr(h'jmh(jkubaubeubeubeubh$)rs}rt(h'Uh(h%h)h,h.h/h0}ru(h2]h3]h4]h5]rvh ah7]rwhauh9Kh:hh"]rx(h<)ry}rz(h'XList of Inferencer{h(jsh)h,h.h@h0}r|(h2]h3]h4]h5]h7]uh9Kh:hh"]r}hCXList of Inferencer~r}r(h'j{h(jyubaubj)r}r(h'Uh(jsh)XO/Users/hs/Program/git/pyGPs/pyGPs/Core/inf.py:docstring of pyGPs.Core.inf.Exactrh.jh0}r(h5]h4]h2]h3]h7]Uentries]r(j XExact (class in pyGPs.Core.inf)hUtrauh9Nh:hh"]ubj")r}r(h'Uh(jsh)jh.j%h0}r(j'j(Xpyh5]h4]h2]h3]h7]j)Xclassrj+juh9Nh:hh"]r(j-)r}r(h'XExact()h(jh)j0h.j1h0}r(h5]rhaj4h*XpyGPs.Core.infrr}rbh4]h2]h3]h7]rhaj9XExactrj;Uj<uh9Nh:hh"]r(j>)r}r(h'Xclass h(jh)j0h.jAh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXclass rr}r(h'Uh(jubaubjG)r}r(h'XpyGPs.Core.inf.h(jh)j0h.jJh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXpyGPs.Core.inf.rr}r(h'Uh(jubaubjP)r}r(h'jh(jh)j0h.jSh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXExactrr}r(h'Uh(jubaubjY)r}r(h'Uh(jh)Nh.j\h0}r(Uexprj^h5]h4]h2]h3]h7]uh9Nh:hh"]rj`)r}r(h'Uh0}r(UreftypejdUrefdocjeU refdomainjfh5]h4]U refexplicith2]h3]h7]U reftargetX_modules/pyGPs/Core/infUrefidjuh(jh"]rjh)r}r(h'Uh0}r(h2]h3]rjmah4]h5]h7]uh(jh"]rhCX[source]rr}r(h'Uh(jubah.jrubah.jsubaubeubjt)r}r(h'Uh(jh)j0h.jwh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rh|)r}r(h'XExact inference for a GP with Gaussian likelihood. Compute a parametrization of the posterior, the negative log marginal likelihood and its derivatives w.r.t. the hyperparameters.rh(jh)jh.hh0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]rhCXExact inference for a GP with Gaussian likelihood. 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The function is equivalent to infExact with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Quu) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu, snu2 = sn2/1e6 is the noise of the inducing inputs and Quu = Kuu + snu2*eye(nu).rh(jh)jUh.hh0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]rhCXMFITC approximation to the posterior Gaussian process. The function is equivalent to infExact with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Quu) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu, snu2 = sn2/1e6 is the noise of the inducing inputs and Quu = Kuu + snu2*eye(nu).rr}r(h'jh(jubaubaubeubj)r}r(h'Uh(jsh)XQ/Users/hs/Program/git/pyGPs/pyGPs/Core/inf.py:docstring of pyGPs.Core.inf.FITC_EPrh.jh0}r(h5]h4]h2]h3]h7]Uentries]r(j X!FITC_EP (class in pyGPs.Core.inf)h Utrauh9Nh:hh"]ubj")r}r(h'Uh(jsh)jh.j%h0}r(j'j(Xpyh5]h4]h2]h3]h7]j)Xclassrj+juh9Nh:hh"]r(j-)r}r(h'X FITC_EP()h(jh)j0h.j1h0}r(h5]rh aj4h*XpyGPs.Core.infrr}rbh4]h2]h3]h7]rh aj9XFITC_EPrj;Uj<uh9Nh:hh"]r(j>)r}r(h'Xclass h(jh)j0h.jAh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXclass rr}r(h'Uh(jubaubjG)r}r(h'XpyGPs.Core.inf.h(jh)j0h.jJh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXpyGPs.Core.inf.rr}r(h'Uh(jubaubjP)r}r(h'jh(jh)j0h.jSh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXFITC_EPrr}r(h'Uh(jubaubjY)r}r(h'Uh(jh)Nh.j\h0}r(Uexprj^h5]h4]h2]h3]h7]uh9Nh:hh"]rj`)r}r(h'Uh0}r(UreftypejdUrefdocjeU refdomainjfh5]h4]U refexplicith2]h3]h7]U reftargetX_modules/pyGPs/Core/infUrefidjuh(jh"]rjh)r}r(h'Uh0}r(h2]h3]rjmah4]h5]h7]uh(jh"]rhCX[source]rr}r(h'Uh(jubah.jrubah.jsubaubeubjt)r}r(h'Uh(jh)j0h.jwh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rh|)r}r(h'X{FITC-EP approximation to the posterior Gaussian process. The function is equivalent to infEP with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Kuu + snu2 * eye(nu)) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu and snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard deviation of the inducing inputs snu to be a one per mil of the measurement noise's standard deviation sn. In case of a likelihood without noise parameter sn2, we simply use snu2 = 1e-6. For details, see The Generalized FITC Approximation, Andrew Naish-Guzman and Sean Holden, NIPS, 2007.rh(jh)jh.hh0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]rhCX{FITC-EP approximation to the posterior Gaussian process. The function is equivalent to infEP with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Kuu + snu2 * eye(nu)) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu and snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard deviation of the inducing inputs snu to be a one per mil of the measurement noise's standard deviation sn. In case of a likelihood without noise parameter sn2, we simply use snu2 = 1e-6. For details, see The Generalized FITC Approximation, Andrew Naish-Guzman and Sean Holden, NIPS, 2007.rr}r(h'jh(jubaubaubeubj)r}r(h'Uh(jsh)XV/Users/hs/Program/git/pyGPs/pyGPs/Core/inf.py:docstring of pyGPs.Core.inf.FITC_Laplacerh.jh0}r(h5]h4]h2]h3]h7]Uentries]r(j X&FITC_Laplace (class in pyGPs.Core.inf)hUtrauh9Nh:hh"]ubj")r}r(h'Uh(jsh)jh.j%h0}r(j'j(Xpyh5]h4]h2]h3]h7]j)Xclassrj+juh9Nh:hh"]r(j-)r}r(h'XFITC_Laplace()rh(jh)j0h.j1h0}r(h5]rhaj4h*XpyGPs.Core.infrr}rbh4]h2]h3]h7]rhaj9X FITC_Laplacerj;Uj<uh9Nh:hh"]r(j>)r}r(h'Xclass h(jh)j0h.jAh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXclass rr}r(h'Uh(jubaubjG)r}r(h'XpyGPs.Core.inf.h(jh)j0h.jJh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXpyGPs.Core.inf.rr}r(h'Uh(jubaubjP)r}r(h'jh(jh)j0h.jSh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCX FITC_Laplacerr}r (h'Uh(jubaubjY)r }r (h'Uh(jh)Nh.j\h0}r (Uexprj^h5]h4]h2]h3]h7]uh9Nh:hh"]r j`)r}r(h'Uh0}r(UreftypejdUrefdocjeU refdomainjfh5]h4]U refexplicith2]h3]h7]U reftargetX_modules/pyGPs/Core/infUrefidjuh(j h"]rjh)r}r(h'Uh0}r(h2]h3]rjmah4]h5]h7]uh(jh"]rhCX[source]rr}r(h'Uh(jubah.jrubah.jsubaubeubjt)r}r(h'Uh(jh)j0h.jwh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rh|)r}r(h'XFITC-Laplace approximation to the posterior Gaussian process. The function is equivalent to infLaplace with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Kuu + snu2 * eye(nu)) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu and snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard deviation of the inducing inputs snu to be a one per mil of the measurement noise's standard deviation sn. In case of a likelihood without noise parameter sn2, we simply use snu2 = 1e-6.r h(jh)jh.hh0}r!(h2]h3]h4]h5]h7]uh9Kh:hh"]r"hCXFITC-Laplace approximation to the posterior Gaussian process. The function is equivalent to infLaplace with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Kuu + snu2 * eye(nu)) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu and snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard deviation of the inducing inputs snu to be a one per mil of the measurement noise's standard deviation sn. 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MethodsqDqE}qF(hhAhh?ubaubcdocutils.nodes paragraph qG)qH}qI(hX]As you may have already seen in the demos, the optimizer is initialized in the following way:qJhh9hhh U paragraphqKh"}qL(h$]h%]h&]h']h)]uh+Kh,hh]qMh5X]As you may have already seen in the demos, the optimizer is initialized in the following way:qNqO}qP(hhJhhHubaubcsphinx.addnodes index qQ)qR}qS(hUhh9hNh UindexqTh"}qU(h']h&]h$]h%]h)]Uentries]qV(UsingleqWX(setOptimizer() (pyGPs.Core.gp.GP method)hUtqXauh+Nh,hh]ubcsphinx.addnodes desc qY)qZ}q[(hUhh9hNh Udescq\h"}q](Unoindexq^Udomainq_Xpyh']h&]h$]h%]h)]Uobjtypeq`XmethodqaUdesctypeqbhauh+Nh,hh]qc(csphinx.addnodes desc_signature qd)qe}qf(hXlGP.setOptimizer(method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None)hhZhU qgh Udesc_signatureqhh"}qi(h']qjhaUmoduleqkhX pyGPs.Core.gpqlqm}qnbh&]h$]h%]h)]qohaUfullnameqpXGP.setOptimizerqqUclassqrXGPUfirstqsuh+Nh,hh]qt(csphinx.addnodes desc_addname qu)qv}qw(hXGP.hhehhgh U desc_addnameqxh"}qy(h$]h%]h&]h']h)]uh+Nh,hh]qzh5XGP.q{q|}q}(hUhhvubaubcsphinx.addnodes desc_name q~)q}q(hX setOptimizerhhehhgh U desc_nameqh"}q(h$]h%]h&]h']h)]uh+Nh,hh]qh5X setOptimizerqq}q(hUhhubaubcsphinx.addnodes desc_parameterlist q)q}q(hUhhehhgh Udesc_parameterlistqh"}q(h$]h%]h&]h']h)]uh+Nh,hh]q(csphinx.addnodes desc_parameter q)q}q(hXmethodh"}q(h$]h%]h&]h']h)]uhhh]qh5Xmethodqq}q(hUhhubah Udesc_parameterqubh)q}q(hXnum_restarts=Noneh"}q(h$]h%]h&]h']h)]uhhh]qh5Xnum_restarts=Noneqq}q(hUhhubah hubh)q}q(hXmin_threshold=Noneh"}q(h$]h%]h&]h']h)]uhhh]qh5Xmin_threshold=Noneqq}q(hUhhubah hubh)q}q(hXmeanRange=Noneh"}q(h$]h%]h&]h']h)]uhhh]qh5XmeanRange=Noneqq}q(hUhhubah hubh)q}q(hX covRange=Noneh"}q(h$]h%]h&]h']h)]uhhh]qh5X covRange=Noneqq}q(hUhhubah hubh)q}q(hX likRange=Noneh"}q(h$]h%]h&]h']h)]uhhh]qh5X likRange=Noneqq}q(hUhhubah hubeubcsphinx.addnodes only q)q}q(hUhhehNh Uonlyqh"}q(UexprUhtmlqh']h&]h$]h%]h)]uh+Nh,hh]qcsphinx.addnodes pending_xref q)q}q(hUh"}q(UreftypeUviewcodeqUrefdocXOptsqU refdomainUstdqh']h&]U refexplicith$]h%]h)]U reftargetX_modules/pyGPs/Core/gpUrefidhquhhh]qcdocutils.nodes inline q)q}q(hUh"}q(h$]h%]qU viewcode-linkqah&]h']h)]uhhh]qh5X[source]qυq}q(hUhhubah Uinlinequbah U pending_xrefqubaubeubcsphinx.addnodes desc_content q)q}q(hUhhZhhgh U desc_contentqh"}q(h$]h%]h&]h']h)]uh+Nh,hh]q(hG)q}q(hXhThis method is used to sepecify optimization configuration. By default, gp uses a single run "minimize".qhhhXW/Users/hs/Program/git/pyGPs/pyGPs/Core/gp.py:docstring of pyGPs.Core.gp.GP.setOptimizerqh hKh"}q(h$]h%]h&]h']h)]uh+Kh,hh]qh5XhThis method is used to sepecify optimization configuration. By default, gp uses a single run "minimize".qq}q(hhhhubaubcdocutils.nodes field_list q)q}q(hUhhhNh U field_listqh"}q(h$]h%]h&]h']h)]uh+Nh,hh]qcdocutils.nodes field q)q}q(hUh"}q(h$]h%]h&]h']h)]uhhh]q(cdocutils.nodes field_name q)q}q(hUh"}q(h$]h%]h&]h']h)]uhhh]qh5X Parametersqq}q(hUhhubah U field_namequbcdocutils.nodes field_body q)q}q(hUh"}q(h$]h%]h&]h']h)]uhhh]qcdocutils.nodes bullet_list q)q}q(hUh"}q(h$]h%]h&]h']h)]uhhh]r(cdocutils.nodes list_item r)r}r(hUh"}r(h$]h%]h&]h']h)]uhhh]rhG)r}r(hUh"}r(h$]h%]h&]h']h)]uhjh]r (cdocutils.nodes strong r )r }r (hXmethodh"}r (h$]h%]h&]h']h)]uhjh]rh5Xmethodrr}r(hUhj ubah Ustrongrubh5X -- rr}r(hUhjubhG)r}r(hX*Optimization methods. Possible values are:rhjhhh hKh"}r(h$]h%]h&]h']h)]uh+Kh]rh5X*Optimization methods. Possible values are:rr}r(hjhjubaubhG)r}r(hXX"Minimize" -> minimize by Carl Rasmussen (python implementation of "minimize" in GPML)r hjhhh hKh"}r!(h$]h%]h&]h']h)]uh+Kh]r"h5XX"Minimize" -> minimize by Carl Rasmussen (python implementation of "minimize" in GPML)r#r$}r%(hj hjubaubhG)r&}r'(hX""CG" -> conjugent gradientr(hjhhh hKh"}r)(h$]h%]h&]h']h)]uh+Kh]r*h5X""CG" -> conjugent gradientr+r,}r-(hj(hj&ubaubhG)r.}r/(hXU"BFGS" -> quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS)r0hjhhh hKh"}r1(h$]h%]h&]h']h)]uh+K h]r2h5XU"BFGS" -> quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS)r3r4}r5(hj0hj.ubaubhG)r6}r7(hX:"SCG" -> scaled conjugent gradient (faster than CG)r8hjhhh hKh"}r9(h$]h%]h&]h']h)]uh+K h]r:h5X:"SCG" -> scaled conjugent gradient (faster than CG)r;r<}r=(hj8hj6ubaubeh hKubah U list_itemr>ubj)r?}r@(hUh"}rA(h$]h%]h&]h']h)]uhhh]rBhG)rC}rD(hUh"}rE(h$]h%]h&]h']h)]uhj?h]rF(j )rG}rH(hX num_restartsh"}rI(h$]h%]h&]h']h)]uhjCh]rJh5X num_restartsrKrL}rM(hUhjGubah jubh5X -- rNrO}rP(hUhjCubh5XSet if you want to run mulitiple times of optimization with different initial guess. It specifys the maximum number of runs/restarts/trials.rQrR}rS(hXSet if you want to run mulitiple times of optimization with different initial guess. It specifys the maximum number of runs/restarts/trials.rThjCubeh hKubah j>ubj)rU}rV(hUh"}rW(h$]h%]h&]h']h)]uhhh]rXhG)rY}rZ(hUh"}r[(h$]h%]h&]h']h)]uhjUh]r\(j )r]}r^(hX min_thresholdh"}r_(h$]h%]h&]h']h)]uhjYh]r`h5X min_thresholdrarb}rc(hUhj]ubah jubh5X -- rdre}rf(hUhjYubh5XSet if you want to run mulitiple times of optimization with different initial guess. It specifys the threshold of objective function value. Stop optimization when this value is reached.rgrh}ri(hXSet if you want to run mulitiple times of optimization with different initial guess. It specifys the threshold of objective function value. Stop optimization when this value is reached.rjhjYubeh hKubah j>ubj)rk}rl(hUh"}rm(h$]h%]h&]h']h)]uhhh]rnhG)ro}rp(hUh"}rq(h$]h%]h&]h']h)]uhjkh]rr(j )rs}rt(hX meanRangeh"}ru(h$]h%]h&]h']h)]uhjoh]rvh5X meanRangerwrx}ry(hUhjsubah jubh5X -- rzr{}r|(hUhjoubhG)r}}r~(hXThe range of initial guess for mean hyperparameters. e.g. meanRange = [(-2,2), (-5,5), (0,1)]. Each tuple specifys the range (low, high) of this hyperparameter, This is only the range of initial guess, during optimization process, optimal hyperparameters may go out of this range.rhjohhh hKh"}r(h$]h%]h&]h']h)]uh+Kh]rh5XThe range of initial guess for mean hyperparameters. e.g. meanRange = [(-2,2), (-5,5), (0,1)]. Each tuple specifys the range (low, high) of this hyperparameter, This is only the range of initial guess, during optimization process, optimal hyperparameters may go out of this range.rr}r(hjhj}ubaubhG)r}r(hX*(-5,5) for each hyperparameter by default.rhjohhh hKh"}r(h$]h%]h&]h']h)]uh+Kh]rh5X*(-5,5) for each hyperparameter by default.rr}r(hjhjubaubeh hKubah j>ubj)r}r(hUh"}r(h$]h%]h&]h']h)]uhhh]rhG)r}r(hUh"}r(h$]h%]h&]h']h)]uhjh]r(j )r}r(hXcovRangeh"}r(h$]h%]h&]h']h)]uhjh]rh5XcovRangerr}r(hUhjubah jubh5X -- rr}r(hUhjubh5XJThe range of initial guess for kernel hyperparameters. Usage see meanRangerr}r(hXJThe range of initial guess for kernel hyperparameters. Usage see meanRangerhjubeh hKubah j>ubj)r}r(hUh"}r(h$]h%]h&]h']h)]uhhh]rhG)r}r(hUh"}r(h$]h%]h&]h']h)]uhjh]r(j )r}r(hXlikRangeh"}r(h$]h%]h&]h']h)]uhjh]rh5XlikRangerr}r(hUhjubah jubh5X -- rr}r(hUhjubh5XNThe range of initial guess for likelihood hyperparameters. Usage see meanRangerr}r(hXNThe range of initial guess for likelihood hyperparameters. Usage see meanRangerhjubeh hKubah j>ubeh U bullet_listrubah U field_bodyrubeh UfieldrubaubeubeubeubeubahUU transformerrNU footnote_refsr}rUrefnamesr}rUsymbol_footnotesr]rUautofootnote_refsr]rUsymbol_footnote_refsr]rU citationsr]rh,hU current_linerNUtransform_messagesr]rUreporterrNUid_startrKU autofootnotesr]rU citation_refsr}rUindirect_targetsr]rUsettingsr(cdocutils.frontend Values ror}r(Ufootnote_backlinksrKUrecord_dependenciesrNU rfc_base_urlrUhttp://tools.ietf.org/html/rU tracebackrUpep_referencesrNUstrip_commentsrNU toc_backlinksrUentryrU language_coderUenrU datestamprNU report_levelrKU _destinationrNU halt_levelrKU strip_classesrNh2NUerror_encoding_error_handlerrUbackslashreplacerUdebugrNUembed_stylesheetrUoutput_encoding_error_handlerrUstrictrU sectnum_xformrKUdump_transformsrNU docinfo_xformrKUwarning_streamrNUpep_file_url_templaterUpep-%04drUexit_status_levelrKUconfigrNUstrict_visitorrNUcloak_email_addressesrUtrim_footnote_reference_spacerUenvrNUdump_pseudo_xmlrNUexpose_internalsrNUsectsubtitle_xformrU source_linkrNUrfc_referencesrNUoutput_encodingrUutf-8rU source_urlrNUinput_encodingrU utf-8-sigrU_disable_configrNU id_prefixrUU tab_widthrKUerror_encodingrUUTF-8rU_sourcer U//Users/hs/Program/git/pyGPs/doc/source/Opts.rstr Ugettext_compactr U generatorr NUdump_internalsr NU smart_quotesrU pep_base_urlrUhttp://www.python.org/dev/peps/rUsyntax_highlightrUlongrUinput_encoding_error_handlerrjUauto_id_prefixrUidrUdoctitle_xformrUstrip_elements_with_classesrNU _config_filesr]Ufile_insertion_enabledrU raw_enabledrKU dump_settingsrNubUsymbol_footnote_startrKUidsr}r(hhehh9hhuUsubstitution_namesr}r h h,h"}r!(h$]h']h&]Usourcehh%]h)]uU footnotesr"]r#Urefidsr$}r%ub.pyGPs-1.2/doc/build/.doctrees/SemiSupervised.doctree000066400000000000000000000641041236346020200224710ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(X$semi-supervised learning with graphsqNXstructured kernelsqX load dataqNX1Xform a nearest neighbour graphq NX2Xresultq NXgp classificationq NXimportq NXkernel on graphq NuUsubstitution_defsq}qUparse_messagesq]qUcurrent_sourceqNU decorationqNUautofootnote_startqKUnameidsq}q(hU$semi-supervised-learning-with-graphsqhUstructured-kernelsqhU load-dataqX1Uid3qh Uform-a-nearest-neighbour-graphqX2Uid4qh Uresultqh Ugp-classificationqh Uimportqh Ukernel-on-graphq uUchildrenq!]q"cdocutils.nodes section q#)q$}q%(U rawsourceq&UUparentq'hUsourceq(cdocutils.nodes reprunicode q)X9/Users/hs/Program/git/pyGPs/doc/source/SemiSupervised.rstq*q+}q,bUtagnameq-Usectionq.U attributesq/}q0(Udupnamesq1]Uclassesq2]Ubackrefsq3]Uidsq4]q5haUnamesq6]q7hauUlineq8KUdocumentq9hh!]q:(cdocutils.nodes title q;)q<}q=(h&X$Semi-supervised Learning with Graphsq>h'h$h(h+h-Utitleq?h/}q@(h1]h2]h3]h4]h6]uh8Kh9hh!]qAcdocutils.nodes Text qBX$Semi-supervised Learning with GraphsqCqD}qE(h&h>h'hr?}r@(h&X graph.h'j3ubeubhF)rA}rB(h&XBelow shows an example of such symmetrized Euclidean :math:`2-NN` graph on some 1s and 2s taking from Xiaojin Zhu's doctoral thesis [2]_.rCh'hh(h+h-hJh/}rD(h1]h2]h3]h4]h6]uh8K-h9hh!]rE(hBX5Below shows an example of such symmetrized Euclidean rFrG}rH(h&X5Below shows an example of such symmetrized Euclidean h'jAubh)rI}rJ(h&Uh/}rK(UlatexX2-NNh4]h3]h1]h2]h6]uh'jAh!]h-hubhBXC graph on some 1s and 2s taking from Xiaojin Zhu's doctoral thesis rLrM}rN(h&XC graph on some 1s and 2s taking from Xiaojin Zhu's doctoral thesis h'jAubh)rO}rP(h&X[2]_hKh'jAh-hh/}rQ(h4]rRUid2rSah3]h1]h2]h6]hhuh!]rThBX2rU}rV(h&Uh'jOubaubhBX.rW}rX(h&X.h'jAubeubcdocutils.nodes figure rY)rZ}r[(h&Uh'hh(h+h-Ufigurer\h/}r](Ualignr^Xcenterh4]h3]h1]h2]h6]uh8Nh9hh!]r_h)r`}ra(h&X4.. figure:: _images/2nnGraph.png :align: center h/}rb(UuriX_images/2nnGraph.pngrch4]h3]h1]h2]h}rdU*jcsh6]uh'jZh!]h-hubaubeubh#)re}rf(h&Uh'h$h(h+h-h.h/}rg(h1]h2]h3]h4]rhh ah6]rih auh8K4h9hh!]rj(h;)rk}rl(h&XKernel on graphrmh'jeh(h+h-h?h/}rn(h1]h2]h3]h4]h6]uh8K4h9hh!]rohBXKernel on graphrprq}rr(h&jmh'jkubaubhF)rs}rt(h&XSeveral classical kernels on graph described in `Structured Kernels`_ can be built from adjacency matrix :math:`A`. We use diffusion kernel for this example to get the precomputed kernel matrix. ::ruh'jeh(h+h-hJh/}rv(h1]h2]h3]h4]h6]uh8K5h9hh!]rw(hBX0Several classical kernels on graph described in rxry}rz(h&X0Several classical kernels on graph described in h'jsubcdocutils.nodes reference r{)r|}r}(h&X`Structured Kernels`_hKh'jsh-U referencer~h/}r(UnameXStructured KernelsUrefurirX Graph.htmlrh4]h3]h1]h2]h6]uh!]rhBXStructured Kernelsrr}r(h&Uh'j|ubaubhBX$ can be built from adjacency matrix rr}r(h&X$ can be built from adjacency matrix h'jsubh)r}r(h&Uh/}r(UlatexXAh4]h3]h1]h2]h6]uh'jsh!]h-hubhBXP. We use diffusion kernel for this example to get the precomputed kernel matrix.rr}r(h&XP. We use diffusion kernel for this example to get the precomputed kernel matrix.h'jsubeubho)r}r(h&X"Matrix = nodeKernels.diffKernel(A)h'jeh(h+h-hrh/}r(hthuh4]h3]h1]h2]h6]uh8K7h9hh!]rhBX"Matrix = nodeKernels.diffKernel(A)rr}r(h&Uh'jubaubcdocutils.nodes target r)r}r(h&X".. _Structured Kernels: Graph.htmlU referencedrKh'jeh(h+h-Utargetrh/}r(jjh4]rhah3]h1]h2]h6]rhauh8K9h9hh!]ubhF)r}r(h&XThis a big square matrix with all rows and columns of the number of data points. By specifying the indice of training data and test data, we will form two matrix M1 and M2 with the exact format which *pyGPs.Core.cov.Pre* needed. ::h'jeh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8K;h9hh!]r(hBXThis a big square matrix with all rows and columns of the number of data points. By specifying the indice of training data and test data, we will form two matrix M1 and M2 with the exact format which rr}r(h&XThis a big square matrix with all rows and columns of the number of data points. By specifying the indice of training data and test data, we will form two matrix M1 and M2 with the exact format which h'jubhP)r}r(h&X*pyGPs.Core.cov.Pre*h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXpyGPs.Core.cov.Prerr}r(h&Uh'jubah-hXubhBX needed.rr}r(h&X needed.h'jubeubho)r}r(h&XEM1,M2 = graphUtil.formKernelMatrix(Matrix, indice_train, indice_test)h'jeh(h+h-hrh/}r(hthuh4]h3]h1]h2]h6]uh8K>h9hh!]rhBXEM1,M2 = graphUtil.formKernelMatrix(Matrix, indice_train, indice_test)rr}r(h&Uh'jubaubcdocutils.nodes definition_list r)r}r(h&Uh'jeh(Nh-Udefinition_listrh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]r(cdocutils.nodes definition_list_item r)r}r(h&XM1 is a matrix with shape **number of training points plus 1** by **number of test points** - cross covariances matrix (train by test) - last row is self covariances (diagonal of test by test)h'jh(h+h-Udefinition_list_itemrh/}r(h1]h2]h3]h4]h6]uh8KAh!]r(cdocutils.nodes term r)r}r(h&X[M1 is a matrix with shape **number of training points plus 1** by **number of test points**rh'jh(h+h-Utermrh/}r(h1]h2]h3]h4]h6]uh8KAh!]r(hBXM1 is a matrix with shape rr}r(h&XM1 is a matrix with shape h'jubcdocutils.nodes strong r)r}r(h&X$**number of training points plus 1**h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX number of training points plus 1rr}r(h&Uh'jubah-UstrongrubhBX by rr}r(h&X by h'jubj)r}r(h&X**number of test points**h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXnumber of test pointsrr}r(h&Uh'jubah-jubeubcdocutils.nodes definition r)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rcdocutils.nodes bullet_list r)r}r(h&Uh/}r(UbulletrX-h4]h3]h1]h2]h6]uh'jh!]r(cdocutils.nodes list_item r)r}r(h&X(cross covariances matrix (train by test)rh/}r(h1]h2]h3]h4]h6]uh'jh!]rhF)r}r(h&jh'jh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8KAh!]rhBX(cross covariances matrix (train by test)rr}r(h&jh'jubaubah-U list_itemrubj)r}r(h&X7last row is self covariances (diagonal of test by test)rh/}r(h1]h2]h3]h4]h6]uh'jh!]rhF)r}r(h&jh'jh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8KBh!]rhBX7last row is self covariances (diagonal of test by test)rr}r(h&jh'jubaubah-jubeh-U bullet_listrubah-U definitionrubeubj)r}r(h&XM2 is a square matrix with **number of training points** for each dimension - training set covariance matrix (train by train) h'jh(h+h-jh/}r(h1]h2]h3]h4]h6]uh8KEh9hh!]r (j)r }r (h&XKM2 is a square matrix with **number of training points** for each dimensionr h'jh(h+h-jh/}r (h1]h2]h3]h4]h6]uh8KEh!]r(hBXM2 is a square matrix with rr}r(h&XM2 is a square matrix with h'j ubj)r}r(h&X**number of training points**h/}r(h1]h2]h3]h4]h6]uh'j h!]rhBXnumber of training pointsrr}r(h&Uh'jubah-jubhBX for each dimensionrr}r(h&X for each dimensionh'j ubeubj)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rj)r }r!(h&Uh/}r"(jX-h4]h3]h1]h2]h6]uh'jh!]r#j)r$}r%(h&X1training set covariance matrix (train by train) h/}r&(h1]h2]h3]h4]h6]uh'j h!]r'hF)r(}r)(h&X/training set covariance matrix (train by train)r*h'j$h(h+h-hJh/}r+(h1]h2]h3]h4]h6]uh8KDh!]r,hBX/training set covariance matrix (train by train)r-r.}r/(h&j*h'j(ubaubah-jubah-jubah-jubeubeubeubh#)r0}r1(h&Uh'h$h(h+h-h.h/}r2(h1]h2]h3]h4]r3hah6]r4h auh8KHh9hh!]r5(h;)r6}r7(h&XGP classificationr8h'j0h(h+h-h?h/}r9(h1]h2]h3]h4]h6]uh8KHh9hh!]r:hBXGP classificationr;r<}r=(h&j8h'j6ubaubhF)r>}r?(h&XEvery ingredients for a basic semi-supervised learning is prepared now. Lets see how to proceed for :math:`GP` classification. First, the normal way with rbf kernel we have seen several times ::r@h'j0h(h+h-hJh/}rA(h1]h2]h3]h4]h6]uh8KIh9hh!]rB(hBXeEvery ingredients for a basic semi-supervised learning is prepared now. Lets see how to proceed for rCrD}rE(h&XeEvery ingredients for a basic semi-supervised learning is prepared now. Lets see how to proceed for h'j>ubh)rF}rG(h&Uh/}rH(UlatexXGPh4]h3]h1]h2]h6]uh'j>h!]h-hubhBXQ classification. First, the normal way with rbf kernel we have seen several timesrIrJ}rK(h&XQ classification. First, the normal way with rbf kernel we have seen several timesh'j>ubeubho)rL}rM(h&X@model = pyGPs.GPC() k = pyGPs.cov.RBF() model.setPrior(kernel=k)h'j0h(h+h-hrh/}rN(hthuh4]h3]h1]h2]h6]uh8KKh9hh!]rOhBX@model = pyGPs.GPC() k = pyGPs.cov.RBF() model.setPrior(kernel=k)rPrQ}rR(h&Uh'jLubaubhF)rS}rT(h&XThen lets use our kernel precomputed matrix. If you only use precomputed kernel matrix, there is no training data. However you still need to specify :math:`x` just to fit in the usage of pyGPs for generality reason. You can create any :math:`x` as long as the dimension is correct. ::h'j0h(h+h-hJh/}rU(h1]h2]h3]h4]h6]uh8KOh9hh!]rV(hBXThen lets use our kernel precomputed matrix. If you only use precomputed kernel matrix, there is no training data. However you still need to specify rWrX}rY(h&XThen lets use our kernel precomputed matrix. If you only use precomputed kernel matrix, there is no training data. However you still need to specify h'jSubh)rZ}r[(h&Uh/}r\(UlatexXxh4]h3]h1]h2]h6]uh'jSh!]h-hubhBXM just to fit in the usage of pyGPs for generality reason. You can create any r]r^}r_(h&XM just to fit in the usage of pyGPs for generality reason. You can create any h'jSubh)r`}ra(h&Uh/}rb(UlatexXxh4]h3]h1]h2]h6]uh'jSh!]h-hubhBX% as long as the dimension is correct.rcrd}re(h&X% as long as the dimension is correct.h'jSubeubho)rf}rg(h&XWx = np.zeros((n,1)) k = pyGPs.cov.Pre(M1,M2) + pyGPs.cov.RBF() model.setPrior(kernel=k)h'j0h(h+h-hrh/}rh(hthuh4]h3]h1]h2]h6]uh8KSh9hh!]rihBXWx = np.zeros((n,1)) k = pyGPs.cov.Pre(M1,M2) + pyGPs.cov.RBF() model.setPrior(kernel=k)rjrk}rl(h&Uh'jfubaubhF)rm}rn(h&XmMoreover, you can composite a kernel for both precomputed matrix and regular kernel function if necessary. ::roh'j0h(h+h-hJh/}rp(h1]h2]h3]h4]h6]uh8KWh9hh!]rqhBXjMoreover, you can composite a kernel for both precomputed matrix and regular kernel function if necessary.rrrs}rt(h&XjMoreover, you can composite a kernel for both precomputed matrix and regular kernel function if necessary.h'jmubaubho)ru}rv(h&XGk = pyGPs.cov.Pre(M1,M2) + pyGPs.cov.RBFunit() model.setPrior(kernel=k)h'j0h(h+h-hrh/}rw(hthuh4]h3]h1]h2]h6]uh8KYh9hh!]rxhBXGk = pyGPs.cov.Pre(M1,M2) + pyGPs.cov.RBFunit() model.setPrior(kernel=k)ryrz}r{(h&Uh'juubaubhF)r|}r}(h&XQThe rest way of using pyGPs is exactly the same as the demo of GP classification.r~h'j0h(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8K\h9hh!]rhBXQThe rest way of using pyGPs is exactly the same as the demo of GP classification.rr}r(h&j~h'j|ubaubeubh#)r}r(h&Uh'h$h(h+h-h.h/}r(h1]h2]h3]h4]rhah6]rh auh8K`h9hh!]r(h;)r}r(h&XResultrh'jh(h+h-h?h/}r(h1]h2]h3]h4]h6]uh8K`h9hh!]rhBXResultrr}r(h&jh'jubaubhF)r}r(h&XFor our manually created graph data, an rbf kernel works better than a diffusion kernel on the graph (higher accuracy). The performance in general should depend on the application as well as features of data.rh'jh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Kah9hh!]rhBXFor our manually created graph data, an rbf kernel works better than a diffusion kernel on the graph (higher accuracy). The performance in general should depend on the application as well as features of data.rr}r(h&jh'jubaubhF)r}r(h&XCThe left image shows the digit that using diffusion kernel will predict the wrong result (should be :math:`2`), but rbf kernel does the job fine. The right image shows the digit that rbf kernel predicts the wrong class, diffusion kernel on the other hand, predicts correctly due to graph information! (should be :math:`1`).h'jh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Kch9hh!]r(hBXdThe left image shows the digit that using diffusion kernel will predict the wrong result (should be rr}r(h&XdThe left image shows the digit that using diffusion kernel will predict the wrong result (should be h'jubh)r}r(h&Uh/}r(UlatexX2h4]h3]h1]h2]h6]uh'jh!]h-hubhBX), but rbf kernel does the job fine. The right image shows the digit that rbf kernel predicts the wrong class, diffusion kernel on the other hand, predicts correctly due to graph information! (should be rr}r(h&X), but rbf kernel does the job fine. The right image shows the digit that rbf kernel predicts the wrong class, diffusion kernel on the other hand, predicts correctly due to graph information! (should be h'jubh)r}r(h&Uh/}r(UlatexX1h4]h3]h1]h2]h6]uh'jh!]h-hubhBX).rr}r(h&X).h'jubeubhF)r}r(h&XInterestingly, using a composite kernel with diffusion kernel on graph and an rbf kernel together. All test cases including the following are predicted correctly.rh'jh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Kfh9hh!]rhBXInterestingly, using a composite kernel with diffusion kernel on graph and an rbf kernel together. All test cases including the following are predicted correctly.rr}r(h&jh'jubaubh)r}r(h&X5.. image:: _images/digitDiffwrong.png :width: 50% h'jh(h+h-hh/}r(h1]UuriX_images/digitDiffwrong.pngrh4]h3]UwidthX50%h2]h}rU*jsh6]uh8Nh9hh!]ubh)r}r(h&X5.. image:: _images/digitRBFwrong.png :width: 50% h'jh(h+h-hh/}r(h1]UuriX_images/digitRBFwrong.pngrh4]h3]UwidthX50%h2]h}rU*jsh6]uh8Nh9hh!]ubcdocutils.nodes footnote r)r}r(h&X`A Database for Handwritten Text Recognition Research, J. J. Hull, IEEE PAMI 16(5) 550-554, 1994.rhKh'jh(h+h-Ufootnoterh/}r(h1]h2]h3]rhah4]rhah6]rX1auh8Koh9hh!]r(cdocutils.nodes label r)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX1r}r(h&Uh'jubah-UlabelrubhF)r}r(h&jh'jh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Koh!]rhBX`A Database for Handwritten Text Recognition Research, J. J. Hull, IEEE PAMI 16(5) 550-554, 1994.rr}r(h&jh'jubaubeubj)r}r(h&XGSemi-Supervised Learning with Graphs, Xiaojin Zhu, CMU-LTI-05-192, 2005rhKh'jh(h+h-jh/}r(h1]h2]h3]rjSah4]rhah6]rX2auh8Kph9hh!]r(j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX2r}r(h&Uh'jubah-jubhF)r}r(h&jh'jh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Kph!]rhBXGSemi-Supervised Learning with Graphs, Xiaojin Zhu, CMU-LTI-05-192, 2005rr}r(h&jh'jubaubeubeubeubah&UU transformerrNU footnote_refsr}r(X1]rhaX2]rjOauUrefnamesr}r(X1]rhaX2]rjOaXstructured kernels]rj|auUsymbol_footnotesr]rUautofootnote_refsr]rUsymbol_footnote_refsr]rU citationsr]rh9hU current_linerNUtransform_messagesr]rUreporterrNUid_startrKU autofootnotesr]rU citation_refsr}rUindirect_targetsr]r Usettingsr (cdocutils.frontend Values r or }r (Ufootnote_backlinksrKUrecord_dependenciesrNU rfc_base_urlrUhttp://tools.ietf.org/html/rU tracebackrUpep_referencesrNUstrip_commentsrNU toc_backlinksrUentryrU language_coderUenrU datestamprNU report_levelrKU _destinationrNU halt_levelrKU strip_classesrNh?NUerror_encoding_error_handlerrUbackslashreplacerUdebugr NUembed_stylesheetr!Uoutput_encoding_error_handlerr"Ustrictr#U sectnum_xformr$KUdump_transformsr%NU docinfo_xformr&KUwarning_streamr'NUpep_file_url_templater(Upep-%04dr)Uexit_status_levelr*KUconfigr+NUstrict_visitorr,NUcloak_email_addressesr-Utrim_footnote_reference_spacer.Uenvr/NUdump_pseudo_xmlr0NUexpose_internalsr1NUsectsubtitle_xformr2U source_linkr3NUrfc_referencesr4NUoutput_encodingr5Uutf-8r6U source_urlr7NUinput_encodingr8U utf-8-sigr9U_disable_configr:NU id_prefixr;UU tab_widthr<KUerror_encodingr=UUTF-8r>U_sourcer?U9/Users/hs/Program/git/pyGPs/doc/source/SemiSupervised.rstr@Ugettext_compactrAU generatorrBNUdump_internalsrCNU smart_quotesrDU pep_base_urlrEUhttp://www.python.org/dev/peps/rFUsyntax_highlightrGUlongrHUinput_encoding_error_handlerrIj#Uauto_id_prefixrJUidrKUdoctitle_xformrLUstrip_elements_with_classesrMNU _config_filesrN]Ufile_insertion_enabledrOU raw_enabledrPKU dump_settingsrQNubUsymbol_footnote_startrRKUidsrS}rT(hjhh$hjjSjOhjhhhhzhjh jehhYhhhj0uUsubstitution_namesrU}rVh-h9h/}rW(h1]h4]h3]Usourceh+h2]h6]uU footnotesrX]rY(jjeUrefidsrZ}r[ub.pyGPs-1.2/doc/build/.doctrees/Theory.doctree000066400000000000000000000520071236346020200207730ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(Xgps & functionalityqNXgpmlqXlist of functionalityqNuUsubstitution_defsq }q Uparse_messagesq ]q Ucurrent_sourceq NU decorationqNUautofootnote_startqKUnameidsq}q(hUgps-functionalityqhUgpmlqhUlist-of-functionalityquUchildrenq]qcdocutils.nodes section q)q}q(U rawsourceqUUparentqhUsourceqcdocutils.nodes reprunicode qX1/Users/hs/Program/git/pyGPs/doc/source/Theory.rstqq}q bUtagnameq!Usectionq"U attributesq#}q$(Udupnamesq%]Uclassesq&]Ubackrefsq']Uidsq(]q)haUnamesq*]q+hauUlineq,KUdocumentq-hh]q.(cdocutils.nodes title q/)q0}q1(hXGPs & Functionalityq2hhhhh!Utitleq3h#}q4(h%]h&]h']h(]h*]uh,Kh-hh]q5cdocutils.nodes Text q6XGPs & Functionalityq7q8}q9(hh2hh0ubaubcdocutils.nodes paragraph q:)q;}q<(hXGaussian Processes (GPs) can conveniently be used for Bayesian supervised learning, such as regression and classification. In its simplest form, GP inference can be implemented in a few lines of code. However, in practice, things typically get a little more complicated: you might want to use expressive covariance and mean functions, learn good values for hyperparameters, use non-Gaussian likelihood functions (rendering exact inference intractable), use approximate inference algorithms, or combinations of many or all of the above.q=hhhhh!U paragraphq>h#}q?(h%]h&]h']h(]h*]uh,Kh-hh]q@h6XGaussian Processes (GPs) can conveniently be used for Bayesian supervised learning, such as regression and classification. In its simplest form, GP inference can be implemented in a few lines of code. However, in practice, things typically get a little more complicated: you might want to use expressive covariance and mean functions, learn good values for hyperparameters, use non-Gaussian likelihood functions (rendering exact inference intractable), use approximate inference algorithms, or combinations of many or all of the above.qAqB}qC(hh=hh;ubaubh:)qD}qE(hXA comprehensive introduction to Gaussian Processes for Machine Learning is provided in the `GPML`_ book by Rasmussen and Williams, 2006.qFhhhhh!h>h#}qG(h%]h&]h']h(]h*]uh,K h-hh]qH(h6X[A comprehensive introduction to Gaussian Processes for Machine Learning is provided in the qIqJ}qK(hX[A comprehensive introduction to Gaussian Processes for Machine Learning is provided in the hhDubcdocutils.nodes reference qL)qM}qN(hX`GPML`_UresolvedqOKhhDh!U referenceqPh#}qQ(UnameXGPMLUrefuriqRX#http://www.gaussianprocess.org/gpmlqSh(]h']h%]h&]h*]uh]qTh6XGPMLqUqV}qW(hUhhMubaubh6X& book by Rasmussen and Williams, 2006.qXqY}qZ(hX& book by Rasmussen and Williams, 2006.hhDubeubh)q[}q\(hUhhhhh!h"h#}q](h%]h&]h']h(]q^hah*]q_hauh,Kh-hh]q`(h/)qa}qb(hXList of Functionalityqchh[hhh!h3h#}qd(h%]h&]h']h(]h*]uh,Kh-hh]qeh6XList of Functionalityqfqg}qh(hhchhaubaubh:)qi}qj(hX8This table lists the functionality implemented in pyGPs.qkhh[hhh!h>h#}ql(h%]h&]h']h(]h*]uh,Kh-hh]qmh6X8This table lists the functionality implemented in pyGPs.qnqo}qp(hhkhhiubaubcdocutils.nodes table qq)qr}qs(hUhh[hhh!Utableqth#}qu(h%]h&]h']h(]h*]uh,Nh-hh]qvcdocutils.nodes tgroup qw)qx}qy(hUh#}qz(h(]h']h%]h&]h*]UcolsKuhhrh]q{(cdocutils.nodes colspec q|)q}}q~(hUh#}q(h(]h']h%]h&]h*]UcolwidthK uhhxh]h!Ucolspecqubh|)q}q(hUh#}q(h(]h']h%]h&]h*]UcolwidthKuhhxh]h!hubh|)q}q(hUh#}q(h(]h']h%]h&]h*]UcolwidthK uhhxh]h!hubh|)q}q(hUh#}q(h(]h']h%]h&]h*]UcolwidthKuhhxh]h!hubh|)q}q(hUh#}q(h(]h']h%]h&]h*]UcolwidthKuhhxh]h!hubh|)q}q(hUh#}q(h(]h']h%]h&]h*]UcolwidthKuhhxh]h!hubcdocutils.nodes thead q)q}q(hUh#}q(h%]h&]h']h(]h*]uhhxh]qcdocutils.nodes row 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Suttorp and C. Igel, Approximation of Gaussian process regression models after training. In M. Verleysen (Hrsg.), Proceedings of the 16th European Symposium on Artificial Neural Networks (ESANN 2008) , pp. 427–432 (2008).q Uresolvedq!Khcdocutils.nodes section q")q#}q$(hUhhUsourceq%hUtagnameq&Usectionq'h}q((h]h]h]h]q)Uregression-on-uci-housing-dataq*ah]q+hauUlineq,KUdocumentq-hUchildrenq.]q/(cdocutils.nodes title q0)q1}q2(hXRegression on UCI Housing dataq3hh#h%hh&Utitleq4h}q5(h]h]h]h]h]uh,Kh-hh.]q6cdocutils.nodes Text q7XRegression on UCI Housing dataq8q9}q:(hh3hh1ubaubcdocutils.nodes paragraph q;)q<}q=(hX5Boston Housing is a fairly standard dataset used for testing regression problems. It contains 506 data points with 12 numeric attributes, and one binary categorical attribute. The goal is to determine median home values, based on various census attributes. This dataset is available at the `UCI Repository`_.hh#h%hh&U paragraphq>h}q?(h]h]h]h]h]uh,Kh-hh.]q@(h7X#Boston Housing is a fairly standard dataset used for testing regression problems. It contains 506 data points with 12 numeric attributes, and one binary categorical attribute. The goal is to determine median home values, based on various census attributes. This dataset is available at the qAqB}qC(hX#Boston Housing is a fairly standard dataset used for testing regression problems. It contains 506 data points with 12 numeric attributes, and one binary categorical attribute. The goal is to determine median home values, based on various census attributes. This dataset is available at the hhh}qS(h]h]h]h]h]uh,Kh-hh.]qT(h7XThe demo follows that in qUqV}qW(hXThe demo follows that in hhQubcdocutils.nodes footnote_reference qX)qY}qZ(hX[1]_h!KhhQh&Ufootnote_referenceq[h}q\(h]q]Uid1q^ah]h]h]h]Urefidq_Uid2q`uh.]qah7X1qb}qc(hUhhYubaubh7X. The data set was preprocessed as follows: each continuous feature was transformed to zero mean and unit variance (The categorical variable was dropped). The data was partitioned into qdqe}qf(hX. The data set was preprocessed as follows: each continuous feature was transformed to zero mean and unit variance (The categorical variable was dropped). The data was partitioned into hhQubcsphinx.ext.mathbase math qg)qh}qi(hUh}qj(UlatexX481h]h]h]h]h]uhhQh.]h&Umathqkubh7X points for training and qlqm}qn(hX points for training and hhQubhg)qo}qp(hUh}qq(UlatexX25h]h]h]h]h]uhhQh.]h&hkubh7X points for testing.qrqs}qt(hX points for testing.hhQubeubh;)qu}qv(hXyThe mean function used was :func:`src.Core.means.meanZero` and the covariance (using the :func:`src.Core.kernels.covSum` function) was a composite of :func:`src.Core.kernels.covSEiso` and :func:`src.Core.kernels.covNoise`. The initial values of the hyperparameters were selected randomly from a zero-mean, unit-variance normal distribtion. The actual values were: :math:`[ -0.75, \; 0.59, \; -0.45 ]`. The initial likelihood hyperparameter was :math:`-2.30`. The regression started with initial negative log marginal likelihood of :math:` 752.46`. Note the initial zero mean and the variance that is uniform over the test set. ::hh#h%hh&h>h}qw(h]h]h]h]h]uh,K h-hh.]qx(h7XThe mean function used was qyqz}q{(hXThe mean function used was hhuubcsphinx.addnodes pending_xref q|)q}}q~(hX:func:`src.Core.means.meanZero`qhhuh%hh&U pending_xrefqh}q(UreftypeXfuncUrefwarnqU reftargetqXsrc.Core.means.meanZeroU refdomainXpyqh]h]U refexplicith]h]h]UrefdocqX demoHousingqUpy:classqNU py:moduleqNuh,K h.]qcdocutils.nodes literal q)q}q(hhh}q(h]h]q(UxrefqhXpy-funcqeh]h]h]uhh}h.]qh7Xsrc.Core.means.meanZero()qq}q(hUhhubah&Uliteralqubaubh7X and the covariance (using the qq}q(hX and the covariance (using the hhuubh|)q}q(hX:func:`src.Core.kernels.covSum`qhhuh%hh&hh}q(UreftypeXfunchhXsrc.Core.kernels.covSumU refdomainXpyqh]h]U refexplicith]h]h]hhhNhNuh,K h.]qh)q}q(hhh}q(h]h]q(hhXpy-funcqeh]h]h]uhhh.]qh7Xsrc.Core.kernels.covSum()qq}q(hUhhubah&hubaubh7X function) was a composite of qq}q(hX function) was a composite of hhuubh|)q}q(hX!:func:`src.Core.kernels.covSEiso`qhhuh%hh&hh}q(UreftypeXfunchhXsrc.Core.kernels.covSEisoU refdomainXpyqh]h]U refexplicith]h]h]hhhNhNuh,K h.]qh)q}q(hhh}q(h]h]q(hhXpy-funcqeh]h]h]uhhh.]qh7Xsrc.Core.kernels.covSEiso()qq}q(hUhhubah&hubaubh7X and qq}q(hX and hhuubh|)q}q(hX!:func:`src.Core.kernels.covNoise`qhhuh%hh&hh}q(UreftypeXfunchhXsrc.Core.kernels.covNoiseU refdomainXpyqh]h]U refexplicith]h]h]hhhNhNuh,K h.]qh)q}q(hhh}q(h]h]q(hhXpy-funcqeh]h]h]uhhh.]qh7Xsrc.Core.kernels.covNoise()qɅq}q(hUhhubah&hubaubh7X. The initial values of the hyperparameters were selected randomly from a zero-mean, unit-variance normal distribtion. The actual values were: q̅q}q(hX. The initial values of the hyperparameters were selected randomly from a zero-mean, unit-variance normal distribtion. The actual values were: hhuubhg)q}q(hUh}q(UlatexX[ -0.75, \; 0.59, \; -0.45 ]h]h]h]h]h]uhhuh.]h&hkubh7X,. The initial likelihood hyperparameter was q҅q}q(hX,. The initial likelihood hyperparameter was hhuubhg)q}q(hUh}q(UlatexX-2.30h]h]h]h]h]uhhuh.]h&hkubh7X. The regression started with initial negative log marginal likelihood of :math:` 752.46`. Note the initial zero mean and the variance that is uniform over the test set.q؅q}q(hX. The regression started with initial negative log marginal likelihood of :math:` 752.46`. Note the initial zero mean and the variance that is uniform over the test set.hhuubeubcdocutils.nodes literal_block q)q}q(hXmodel = pyGPs.GPR() model.optimize(x,y) ym, ys2, fm, fs2, lp = model.predict(xs) xa = np.concatenate((data[:,:4],data[:,5:-1]),axis=1) xa = (xa - np.mean(xa,axis=0))/(np.std(xa,axis=0)+1.e-16) ya, ys2a, fma, fs2a, lpa = model.predict(xa)hh#h%hh&U literal_blockqh}q(U xml:spaceqUpreserveqh]h]h]h]h]uh,Kh-hh.]qh7Xmodel = pyGPs.GPR() model.optimize(x,y) ym, ys2, fm, fs2, lp = model.predict(xs) xa = np.concatenate((data[:,:4],data[:,5:-1]),axis=1) xa = (xa - np.mean(xa,axis=0))/(np.std(xa,axis=0)+1.e-16) ya, ys2a, fma, fs2a, lpa = model.predict(xa)qㅁq}q(hUhhubaubcdocutils.nodes figure q)q}q(hUhh#h%hh&Ufigureqh}q(UalignqXcenterh]h]h]h]h]uh,Nh-hh.]qcdocutils.nodes image q)q}q(hXT.. figure:: _images/demoH1.png :align: center :width: 600pt :height: 300pt h}q(h]UuriX_images/demoH1.pngqh]h]UwidthX600pth]U candidatesq}qU*hsh]UheightX300ptuhhh.]h&Uimagequbaubh;)q}q(hXAfter hyperparameter optimization, the covariance hyperparameters were :math:`[ 1.17, \; 0.45, \; -1.41 ]` and the likelihood hyperparameter was :math:`-2.27`. The final negative log marginal likelihood (optimized) was :math:`214.46`.hh#h%hh&h>h}q(h]h]h]h]h]uh,Kh-hh.]q(h7XGAfter hyperparameter optimization, the covariance hyperparameters were qq}q(hXGAfter hyperparameter optimization, the covariance hyperparameters were hhubhg)q}q(hUh}q(UlatexX[ 1.17, \; 0.45, \; -1.41 ]h]h]h]h]h]uhhh.]h&hkubh7X' and the likelihood hyperparameter was qr}r(hX' and the likelihood hyperparameter was hhubhg)r}r(hUh}r(UlatexX-2.27h]h]h]h]h]uhhh.]h&hkubh7X?. The final negative log marginal likelihood (optimized) was rr}r(hX?. The final negative log marginal likelihood (optimized) was hhubhg)r}r (hUh}r (UlatexX214.46h]h]h]h]h]uhhh.]h&hkubh7X.r }r (hX.hhubeubh)r }r(hUhh#h%hh&hh}r(hXcenterh]h]h]h]h]uh,Nh-hh.]rh)r}r(hXT.. figure:: _images/demoH2.png :align: center :width: 600pt :height: 300pt h}r(h]UuriX_images/demoH2.pngrh]h]UwidthX600pth]h}rU*jsh]UheightX300ptuhj h.]h&hubaubcdocutils.nodes target r)r}r(hXB.. _UCI Repository: http://archive.ics.uci.edu/ml/datasets/HousingU referencedrKhh#h%hh&Utargetrh}r(hIhJh]rUuci-repositoryrah]h]h]h]rhauh,K%h-hh.]ubheubh%hh&Ufootnoterh}r (h]h]h]r!h^ah]r"h`ah]r#X1auh,K'h-hh.]r$(cdocutils.nodes label r%)r&}r'(hUh}r((h]h]h]h]h]uhhh.]r)h7X1r*}r+(hUhj&ubah&Ulabelr,ubcdocutils.nodes enumerated_list r-)r.}r/(hUh}r0(Usuffixr1U.Ustartr2Kh]h]h]Uprefixr3Uh]h]Uenumtyper4U upperalphar5uhhh.]r6cdocutils.nodes list_item r7)r8}r9(hXSuttorp and C. Igel, Approximation of Gaussian process regression models after training. In M. Verleysen (Hrsg.), Proceedings of the 16th European Symposium on Artificial Neural Networks (ESANN 2008) , pp. 427–432 (2008).r:h}r;(h]h]h]h]h]uhj.h.]r<h;)r=}r>(hj:hj8h%hh&h>h}r?(h]h]h]h]h]uh,K'h.]r@h7XSuttorp and C. Igel, Approximation of Gaussian process regression models after training. In M. Verleysen (Hrsg.), Proceedings of the 16th European Symposium on Artificial Neural Networks (ESANN 2008) , pp. 427–432 (2008).rArB}rC(hj:hj=ubaubah&U list_itemrDubah&Uenumerated_listrEubeubh.]rFh;)rG}rH(hUh}rI(h]h]h]h]h]uhh h.]rJh7X;Enumerated list start value not ordinal-1: "T" (ordinal 20)rKrL}rM(hUhjGubah&h>ubah&Usystem_messagerNubaUcurrent_sourcerONU decorationrPNUautofootnote_startrQKUnameidsrR}rS(X1h`hjhh*uh.]rTh#ahUU transformerrUNU footnote_refsrV}rWX1]rXhYasUrefnamesrY}rZ(X1]r[hYaXuci repository]r\hEauUsymbol_footnotesr]]r^Uautofootnote_refsr_]r`Usymbol_footnote_refsra]rbU citationsrc]rdh-hU current_linereNUtransform_messagesrf]rgUreporterrhNUid_startriKU autofootnotesrj]rkU citation_refsrl}rmUindirect_targetsrn]roUsettingsrp(cdocutils.frontend Values rqorr}rs(Ufootnote_backlinksrtKUrecord_dependenciesruNU rfc_base_urlrvUhttp://tools.ietf.org/html/rwU tracebackrxUpep_referencesryNUstrip_commentsrzNU toc_backlinksr{Uentryr|U language_coder}Uenr~U datestamprNU report_levelrKU _destinationrNU halt_levelrKU strip_classesrNh4NUerror_encoding_error_handlerrUbackslashreplacerUdebugrNUembed_stylesheetrUoutput_encoding_error_handlerrUstrictrU sectnum_xformrKUdump_transformsrNU docinfo_xformrKUwarning_streamrNUpep_file_url_templaterUpep-%04drUexit_status_levelrKUconfigrNUstrict_visitorrNUcloak_email_addressesrUtrim_footnote_reference_spacerUenvrNUdump_pseudo_xmlrNUexpose_internalsrNUsectsubtitle_xformrU source_linkrNUrfc_referencesrNUoutput_encodingrUutf-8rU source_urlrNUinput_encodingrU utf-8-sigrU_disable_configrNU id_prefixrUU tab_widthrKUerror_encodingrUUTF-8rU_sourcerU6/Users/hs/Program/git/pyGPs/doc/source/demoHousing.rstrUgettext_compactrU generatorrNUdump_internalsrNU smart_quotesrU pep_base_urlrUhttp://www.python.org/dev/peps/rUsyntax_highlightrUlongrUinput_encoding_error_handlerrjUauto_id_prefixrUidrUdoctitle_xformrUstrip_elements_with_classesrNU _config_filesr]Ufile_insertion_enabledrU raw_enabledrKU dump_settingsrNubUsymbol_footnote_startrKUidsr}r(h`hjjh*h#h^hYuUsubstitution_namesr}rh&h-h}r(h]h]h]Usourcehh]h]uU footnotesr]rhaUrefidsr}rub.pyGPs-1.2/doc/build/.doctrees/demoMaunaLoa.doctree000066400000000000000000000642571236346020200220750ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(X1XgpmlqXregression on mauna loa dataqNX[2]quUsubstitution_defsq }q Uparse_messagesq ]q Ucurrent_sourceq NU decorationqNUautofootnote_startqKUnameidsq}q(X1Uid2qhUgpmlqhUregression-on-mauna-loa-dataqhUid3quUchildrenq]qcdocutils.nodes section q)q}q(U rawsourceqUUparentqhUsourceqcdocutils.nodes reprunicode qX7/Users/hs/Program/git/pyGPs/doc/source/demoMaunaLoa.rstqq }q!bUtagnameq"Usectionq#U attributesq$}q%(Udupnamesq&]Uclassesq']Ubackrefsq(]Uidsq)]q*haUnamesq+]q,hauUlineq-KUdocumentq.hh]q/(cdocutils.nodes title q0)q1}q2(hXRegression on Mauna Loa dataq3hhhh h"Utitleq4h$}q5(h&]h']h(]h)]h+]uh-Kh.hh]q6cdocutils.nodes Text q7XRegression on Mauna Loa dataq8q9}q:(hh3hh1ubaubcdocutils.nodes paragraph q;)q<}q=(hXThis example does regression on the Hawaiian Mauna Loa data (example taken from chapter :math:`5` of the `GPML`_ book by Rasmussen and Williams, 2006)q>hhhh h"U paragraphq?h$}q@(h&]h']h(]h)]h+]uh-Kh.hh]qA(h7XXThis example does regression on the Hawaiian Mauna Loa data (example taken from chapter qBqC}qD(hXXThis example does regression on the Hawaiian Mauna Loa data (example taken from chapter hhThe data is shown in the above plot. Our goal is to model the qq}q(hX>The data is shown in the above plot. Our goal is to model the hhubhE)q}q(hUh$}q(UlatexXCO_2h)]h(]h&]h']h+]uhhh]h"hIubh7X% concentration as a function of time qƅq}q(hX% concentration as a function of time hhubhE)q}q(hUh$}q(UlatexXth)]h(]h&]h']h+]uhhh]h"hIubh7X. Several features are immediately apparent: a long term rising trend, a pronounced seasonal variation and some smaller irregularities. In the following, contributions to a combined covariance function which takes care of these individual properties are suggessted. This is meant primarily to illustrate the power and flexibility of the Gaussian process framework—it is possible that other choices would be more appropriate for this data set.q̅q}q(hX. Several features are immediately apparent: a long term rising trend, a pronounced seasonal variation and some smaller irregularities. In the following, contributions to a combined covariance function which takes care of these individual properties are suggessted. This is meant primarily to illustrate the power and flexibility of the Gaussian process framework—it is possible that other choices would be more appropriate for this data set.hhubeubh;)q}q(hXTo model the long term smooth rising trend, a squared exponential (SE) covariance term with two hyperparameters controlling the amplitude :math:`\theta_1` and characteristic length-scale :math:`\theta_2` is used:hhhh h"h?h$}q(h&]h']h(]h)]h+]uh-K#h.hh]q(h7XTo model the long term smooth rising trend, a squared exponential (SE) covariance term with two hyperparameters controlling the amplitude qӅq}q(hXTo model the long term smooth rising trend, a squared exponential (SE) covariance term with two hyperparameters controlling the amplitude hhubhE)q}q(hUh$}q(UlatexX\theta_1h)]h(]h&]h']h+]uhhh]h"hIubh7X! and characteristic length-scale qمq}q(hX! and characteristic length-scale hhubhE)q}q(hUh$}q(UlatexX\theta_2h)]h(]h&]h']h+]uhhh]h"hIubh7X is used:q߅q}q(hX is used:hhubeubcsphinx.ext.mathbase displaymath q)q}q(hUhhhh h"U displaymathqh$}q(UlatexqXIk_1(x,x') = \theta_1^2 \exp \left(-\frac{(x-x')^2}{2\theta_2^2}\right). Unowrapqh)]h(]h&]h']h+]UdocnameqX demoMaunaLoaqUlabelqNuh-K'h.hh]ubh;)q}q(hXNote that we just use a smooth trend; actually enforcing the trend *a priori* to be increasing is probably not so simple and (hopefully) not desirable. We can use the periodic covariance function with a period of one year to model the seasonal variation. However, it is not clear that the seasonal trend is exactly periodic, so we modify it by taking the product with a squared exponential component to allow a decay away from exact periodicity:hhhh h"h?h$}q(h&]h']h(]h)]h+]uh-K)h.hh]q(h7XCNote that we just use a smooth trend; actually enforcing the trend qq}q(hXCNote that we just use a smooth trend; actually enforcing the trend hhubh)q}q(hX *a priori*h$}q(h&]h']h(]h)]h+]uhhh]qh7Xa prioriqq}q(hUhhubah"hubh7Xp to be increasing is probably not so simple and (hopefully) not desirable. We can use the periodic covariance function with a period of one year to model the seasonal variation. However, it is not clear that the seasonal trend is exactly periodic, so we modify it by taking the product with a squared exponential component to allow a decay away from exact periodicity:qq}q(hXp to be increasing is probably not so simple and (hopefully) not desirable. We can use the periodic covariance function with a period of one year to model the seasonal variation. However, it is not clear that the seasonal trend is exactly periodic, so we modify it by taking the product with a squared exponential component to allow a decay away from exact periodicity:hhubeubh)q}q(hUhhhh h"hh$}q(hXmk_2(x,x') = \theta_3^2 \exp\left(-\frac{(x-x')^2}{2\theta_4^2} \frac{2\sin^2(\pi(x-x'))}{\theta_5^2}\right).hh)]h(]h&]h']h+]hhhNuh-K/h.hh]ubh;)r}r(hXwhere :math:`\theta_3` gives the magnitude, :math:`\theta_4` the decay-time for the periodic component, and :math:`\theta_5` the smoothness of the periodic component; the period has been fixed to one (year). The seasonal component in the data is caused primarily by different rates of :math:`CO_2` uptake for plants depending on the season, and it is probably reasonable to assume that this pattern may itself change slowly over time, partially due to the elevation of the :math:`CO_2` level itself; if this effect turns out not to be relevant, then it can be effectively removed at the fitting stage by allowing :math:`\theta_4` to become very large.hhhh h"h?h$}r(h&]h']h(]h)]h+]uh-K3h.hh]r(h7Xwhere rr}r(hXwhere hjubhE)r}r(hUh$}r (UlatexX\theta_3h)]h(]h&]h']h+]uhjh]h"hIubh7X gives the magnitude, r r }r (hX gives the magnitude, hjubhE)r }r(hUh$}r(UlatexX\theta_4h)]h(]h&]h']h+]uhjh]h"hIubh7X0 the decay-time for the periodic component, and rr}r(hX0 the decay-time for the periodic component, and hjubhE)r}r(hUh$}r(UlatexX\theta_5h)]h(]h&]h']h+]uhjh]h"hIubh7X the smoothness of the periodic component; the period has been fixed to one (year). The seasonal component in the data is caused primarily by different rates of rr}r(hX the smoothness of the periodic component; the period has been fixed to one (year). The seasonal component in the data is caused primarily by different rates of hjubhE)r}r(hUh$}r(UlatexXCO_2h)]h(]h&]h']h+]uhjh]h"hIubh7X uptake for plants depending on the season, and it is probably reasonable to assume that this pattern may itself change slowly over time, partially due to the elevation of the rr}r(hX uptake for plants depending on the season, and it is probably reasonable to assume that this pattern may itself change slowly over time, partially due to the elevation of the hjubhE)r}r (hUh$}r!(UlatexXCO_2h)]h(]h&]h']h+]uhjh]h"hIubh7X level itself; if this effect turns out not to be relevant, then it can be effectively removed at the fitting stage by allowing r"r#}r$(hX level itself; if this effect turns out not to be relevant, then it can be effectively removed at the fitting stage by allowing hjubhE)r%}r&(hUh$}r'(UlatexX\theta_4h)]h(]h&]h']h+]uhjh]h"hIubh7X to become very large.r(r)}r*(hX to become very large.hjubeubh;)r+}r,(hXSTo model the (small) medium term irregularities, a rational quadratic term is used:r-hhhh h"h?h$}r.(h&]h']h(]h)]h+]uh-Kh.hh]ubh;)r6}r7(hXwhere :math:`\theta_6` is the magnitude, :math:`\theta_7` is the typical length-scale and :math:`\theta_8` is the shape parameter determining diffuseness of the length-scales.hhhh h"h?h$}r8(h&]h']h(]h)]h+]uh-KBh.hh]r9(h7Xwhere r:r;}r<(hXwhere hj6ubhE)r=}r>(hUh$}r?(UlatexX\theta_6h)]h(]h&]h']h+]uhj6h]h"hIubh7X is the magnitude, r@rA}rB(hX is the magnitude, hj6ubhE)rC}rD(hUh$}rE(UlatexX\theta_7h)]h(]h&]h']h+]uhj6h]h"hIubh7X! is the typical length-scale and rFrG}rH(hX! is the typical length-scale and hj6ubhE)rI}rJ(hUh$}rK(UlatexX\theta_8h)]h(]h&]h']h+]uhj6h]h"hIubh7XE is the shape parameter determining diffuseness of the length-scales.rLrM}rN(hXE is the shape parameter determining diffuseness of the length-scales.hj6ubeubh;)rO}rP(hXOne could also have used a squared exponential form for this component, but it turns out that the rational quadratic works better (gives higher marginal likelihood), probably because it can accommodate several length-scales simultaneously.rQhhhh h"h?h$}rR(h&]h']h(]h)]h+]uh-KEh.hh]rSh7XOne could also have used a squared exponential form for this component, but it turns out that the rational quadratic works better (gives higher marginal likelihood), probably because it can accommodate several length-scales simultaneously.rTrU}rV(hjQhjOubaubh;)rW}rX(hXoFinally we specify a noise model as the sum of a squared exponential contrubition and an independent component:rYhhhh h"h?h$}rZ(h&]h']h(]h)]h+]uh-KIh.hh]r[h7XoFinally we specify a noise model as the sum of a squared exponential contrubition and an independent component:r\r]}r^(hjYhjWubaubh)r_}r`(hUhhhh h"hh$}ra(hXkk_4(x_p,x_q) = \theta_9^2\exp\left(-\frac{(x_p - x_q)^2}{2\theta_{10}^2}\right) + \theta_{11}^2\delta_{pq}.hh)]h(]h&]h']h+]hhhNuh-KKh.hh]ubh;)rb}rc(hXwhere :math:`\theta_9` is the magnitude of the correlated noise component, :math:`\theta_{10}` is its length scale and :math:`\theta_{11}` is the magnitude of the independent noise component. Noise in the series could be caused by measurement inaccuracies, and by local short-term weather phenomena, so it is probably reasonable to assume at least a modest amount of correlation in time. Notice that the correlated noise component, the first term has an identical expression to the long term component in the trend covariance. When optimizing the hyperparameters, we will see that one of these components becomes large with a long length-scale (the long term trend), while the other remains small with a short length-scale (noise). The fact that we have chosen to call one of these components ‘signal’ and the other one ‘noise’ is only a question of interpretation. Presumably, we are less interested in very short-term effect, and thus call it noise; if on the other hand we were interested in this effect, we would call it signal.hhhh h"h?h$}rd(h&]h']h(]h)]h+]uh-KOh.hh]re(h7Xwhere rfrg}rh(hXwhere hjbubhE)ri}rj(hUh$}rk(UlatexX\theta_9h)]h(]h&]h']h+]uhjbh]h"hIubh7X5 is the magnitude of the correlated noise component, rlrm}rn(hX5 is the magnitude of the correlated noise component, hjbubhE)ro}rp(hUh$}rq(UlatexX \theta_{10}h)]h(]h&]h']h+]uhjbh]h"hIubh7X is its length scale and rrrs}rt(hX is its length scale and hjbubhE)ru}rv(hUh$}rw(UlatexX \theta_{11}h)]h(]h&]h']h+]uhjbh]h"hIubh7X is the magnitude of the independent noise component. Noise in the series could be caused by measurement inaccuracies, and by local short-term weather phenomena, so it is probably reasonable to assume at least a modest amount of correlation in time. Notice that the correlated noise component, the first term has an identical expression to the long term component in the trend covariance. When optimizing the hyperparameters, we will see that one of these components becomes large with a long length-scale (the long term trend), while the other remains small with a short length-scale (noise). The fact that we have chosen to call one of these components ‘signal’ and the other one ‘noise’ is only a question of interpretation. Presumably, we are less interested in very short-term effect, and thus call it noise; if on the other hand we were interested in this effect, we would call it signal.rxry}rz(hX is the magnitude of the independent noise component. Noise in the series could be caused by measurement inaccuracies, and by local short-term weather phenomena, so it is probably reasonable to assume at least a modest amount of correlation in time. Notice that the correlated noise component, the first term has an identical expression to the long term component in the trend covariance. When optimizing the hyperparameters, we will see that one of these components becomes large with a long length-scale (the long term trend), while the other remains small with a short length-scale (noise). The fact that we have chosen to call one of these components ‘signal’ and the other one ‘noise’ is only a question of interpretation. Presumably, we are less interested in very short-term effect, and thus call it noise; if on the other hand we were interested in this effect, we would call it signal.hjbubeubh;)r{}r|(hX!The final covariance function is:r}hhhh h"h?h$}r~(h&]h']h(]h)]h+]uh-K]h.hh]rh7X!The final covariance function is:rr}r(hj}hj{ubaubh)r}r(hUhhhh h"hh$}r(hX7k(x,x') = k_1(x,x') + k_2(x,x') + k_3(x,x') + k_4(x,x')hh)]h(]h&]h']h+]hhhNuh-K_h.hh]ubh;)r}r(hXFwith hyperparameters :math:`\theta = (\theta_1,\ldots,\theta_{11})` ::rhhhh h"h?h$}r(h&]h']h(]h)]h+]uh-Kch.hh]r(h7Xwith hyperparameters rr}r(hXwith hyperparameters hjubhE)r}r(hUh$}r(UlatexX&\theta = (\theta_1,\ldots,\theta_{11})h)]h(]h&]h']h+]uhjh]h"hIubeubcdocutils.nodes literal_block r)r}r(hXX# DEFINE parameterized covariance function k1 = pyGPs.cov.RBF(np.log(67.), np.log(66.)) k2 = pyGPs.cov.Periodic(np.log(1.3), np.log(1.0), np.log(2.4)) * cov.RBF(np.log(90.), np.log(2.4)) k3 = pyGPs.cov.RQ(np.log(1.2), np.log(0.66), np.log(0.78)) k4 = pyGPs.cov.RBF(np.log(1.6/12.), np.log(0.18)) + cov.Noise(np.log(0.19)) k = k1 + k2 + k3 + k4hhhh h"U literal_blockrh$}r(U xml:spacerUpreserverh)]h(]h&]h']h+]uh-Keh.hh]rh7XX# DEFINE parameterized covariance function k1 = pyGPs.cov.RBF(np.log(67.), np.log(66.)) k2 = pyGPs.cov.Periodic(np.log(1.3), np.log(1.0), np.log(2.4)) * cov.RBF(np.log(90.), np.log(2.4)) k3 = pyGPs.cov.RQ(np.log(1.2), np.log(0.66), np.log(0.78)) k4 = pyGPs.cov.RBF(np.log(1.6/12.), np.log(0.18)) + cov.Noise(np.log(0.19)) k = k1 + k2 + k3 + k4rr}r(hUhjubaubh;)r}r(hX"After running the minimization, ::rhhhh h"h?h$}r(h&]h']h(]h)]h+]uh-Kmh.hh]rh7XAfter running the minimization,rr}r(hXAfter running the minimization,hjubaubj)r}r(hX?t0 = clock() model.optimize(x,y) t1 = clock() model.predict(xs)hhhh h"jh$}r(jjh)]h(]h&]h']h+]uh-Koh.hh]rh7X?t0 = clock() model.optimize(x,y) t1 = clock() model.predict(xs)rr}r(hUhjubaubh;)r}r(hX!The extrapolated data looks like:rhhhh h"h?h$}r(h&]h']h(]h)]h+]uh-Kth.hh]rh7X!The extrapolated data looks like:rr}r(hjhjubaubh)r}r(hUhhhh h"hh$}r(hXcenterh)]h(]h&]h']h+]uh-Nh.hh]rh)r}r(hXg.. figure:: _images/demoML2.png :height: 600 px :width: 800 px :align: center :scale: 70 % h$}r(UscaleKFh&]UuriX_images/demoML2.pngrh)]h(]UwidthX800pxh']h}rU*jsh+]UheightX600pxuhjh]h"hubaubh;)r}r(hXzand the optimized values of the hyperparameters allow for a principled analysis of different components driving the model.rhhhh h"h?h$}r(h&]h']h(]h)]h+]uh-K|h.hh]rh7Xzand the optimized values of the hyperparameters allow for a principled analysis of different components driving the model.rr}r(hjhjubaubcdocutils.nodes footnote r)r}r(hX Keeling, C. D. and Whorf, T. P. (2004). Atmospheric :math:`CO_2` Records from Sites in the SIO Air Sampling Network. In Trends: A Compendium of Data on Global Change. Carbon Dioxide Information Analysis Center, Oak Ridge National Laboratory, Oak Ridge, Tenn., U.S.A. hPKhhhh h"Ufootnoterh$}r(h&]h']h(]rh|ah)]rhah+]rX1auh-K~h.hh]r(cdocutils.nodes label r)r}r(hUh$}r(h&]h']h(]h)]h+]uhjh]rh7X1r}r(hUhjubah"hubh;)r}r(hX Keeling, C. D. and Whorf, T. P. (2004). Atmospheric :math:`CO_2` Records from Sites in the SIO Air Sampling Network. In Trends: A Compendium of Data on Global Change. Carbon Dioxide Information Analysis Center, Oak Ridge National Laboratory, Oak Ridge, Tenn., U.S.A.rhjhh h"h?h$}r(h&]h']h(]h)]h+]uh-K~h]r(h7X4Keeling, C. D. and Whorf, T. P. (2004). Atmospheric rr}r(hX4Keeling, C. D. and Whorf, T. P. (2004). Atmospheric hjubhE)r}r(hUh$}r(UlatexXCO_2h)]h(]h&]h']h+]uhjh]h"hIubh7X Records from Sites in the SIO Air Sampling Network. In Trends: A Compendium of Data on Global Change. Carbon Dioxide Information Analysis Center, Oak Ridge National Laboratory, Oak Ridge, Tenn., U.S.A.rr}r(hX Records from Sites in the SIO Air Sampling Network. In Trends: A Compendium of Data on Global Change. 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Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ # likelihood functions are provided to be used by the gp.py function: # # Erf (Error function, classification, probit regression) # Logistic [NOT IMPLEMENTED!] (Logistic, classification, logit regression) # Uni [NOT IMPLEMENTED!] (Uniform likelihood, classification) # # Gauss (Gaussian, regression) # Laplace (Laplacian or double exponential, regression) # Sech2 [NOT IMPLEMENTED!] (Sech-square, regression) # T [NOT IMPLEMENTED!] (Student's t, regression) # # Poisson [NOT IMPLEMENTED!] (Poisson regression, count data) # # Mix [NOT IMPLEMENTED!] (Mixture of individual covariance functions) # # The likelihood functions have two possible modes, the mode being selected # as follows (where "lik" stands for "proceed" method for any likelihood function): # # # 1) With two or three input arguments: [PREDICTION MODE] # # lp = lik(y, mu) OR lp, ymu, ys2 = lik(y, mu, s2) # # This allows to evaluate the predictive distribution. Let p(y_*|f_*) be the # likelihood of a test point and N(f_*|mu,s2) an approximation to the posterior # marginal p(f_*|x_*,x,y) as returned by an inference method. The predictive # distribution p(y_*|x_*,x,y) is approximated by. # q(y_*) = \int N(f_*|mu,s2) p(y_*|f_*) df_* # # lp = log( q(y) ) for a particular value of y, if s2 is [] or 0, this # corresponds to log( p(y|mu) ) # ymu and ys2 the mean and variance of the predictive marginal q(y) # note that these two numbers do not depend on a particular # value of y # All vectors have the same size. # # # 2) With four or five input arguments, the fouth being an object of class "Inference" [INFERENCE MODE] # # lik(y, mu, s2, inf) OR lik(y, mu, s2, inf, i) # # There are two cases for inf, namely a) infLaplace, b) infEP # The last input i, refers to derivatives w.r.t. the ith hyperparameter. # # a1) lp,dlp,d2lp,d3lp = lik(y, f, [], 'infLaplace') # lp, dlp, d2lp and d3lp correspond to derivatives of the log likelihood # log(p(y|f)) w.r.t. to the latent location f. # lp = log( p(y|f) ) # dlp = d log( p(y|f) ) / df # d2lp = d^2 log( p(y|f) ) / df^2 # d3lp = d^3 log( p(y|f) ) / df^3 # # a2) lp_dhyp,dlp_dhyp,d2lp_dhyp = lik(y, f, [], 'infLaplace', i) # returns derivatives w.r.t. to the ith hyperparameter # lp_dhyp = d log( p(y|f) ) / ( dhyp_i) # dlp_dhyp = d^2 log( p(y|f) ) / (df dhyp_i) # d2lp_dhyp = d^3 log( p(y|f) ) / (df^2 dhyp_i) # # # b1) lZ,dlZ,d2lZ = lik(y, mu, s2, 'infEP') # let Z = \int p(y|f) N(f|mu,s2) df then # lZ = log(Z) # dlZ = d log(Z) / dmu # d2lZ = d^2 log(Z) / dmu^2 # # b2) dlZhyp = lik(y, mu, s2, 'infEP', i) # returns derivatives w.r.t. to the ith hyperparameter # dlZhyp = d log(Z) / dhyp_i # # # Cumulative likelihoods are designed for binary classification. Therefore, they # only look at the sign of the targets y; zero values are treated as +1. # # Some examples for valid likelihood functions: # lik = likGauss([0.1]) # lik = likErf() # # See the documentation for the individual likelihood for the computations specific # to each likelihood function. # # # This is a object-oriented python implementation of gpml functionality # (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18). # based on the functional-version of python implementation # (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013) # # Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013 import numpy as np from scipy.special import erf import inf class Likelihood(object): """Base function for Likelihood function""" def __init__(self): self.hyp = [] def proceed(self): ''' The likelihood functions have two possible modes based on inputs. See lik.py for detail documentation. ''' pass class Gauss(Likelihood): ''' Gaussian likelihood function for regression. :math:`Gauss(t)=\\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-\\frac{(t-y)^2}{2\\sigma^2}}`, where :math:`y` is the mean and :math:`\\sigma` is the standard deviation. hyp = [ log_sigma ] ''' def __init__(self, log_sigma=np.log(0.1) ): self.hyp = [log_sigma] def proceed(self, y=None, mu=None, s2=None, inffunc=None, der=None, nargout=1): sn2 = np.exp(2. * self.hyp[0]) if inffunc == None: # prediction mode if y == None: y = np.zeros_like(mu) s2zero = True if (not s2==None) and np.linalg.norm(s2) > 0: s2zero = False if s2zero: # log probability lp = -(y-mu)**2 /sn2/2 - np.log(2.*np.pi*sn2)/2. s2 = np.zeros_like(s2) else: inf_func = inf.EP() # prediction lp = self.proceed(y, mu, s2, inf_func) if nargout>1: ymu = mu # first y moment if nargout>2: ys2 = s2 + sn2 # second y moment return lp,ymu,ys2 else: return lp,ymu else: return lp else: if isinstance(inffunc, inf.EP): if der == None: # no derivative mode lZ = -(y-mu)**2/(sn2+s2)/2. - np.log(2*np.pi*(sn2+s2))/2. # log part function if nargout>1: dlZ = (y-mu)/(sn2+s2) # 1st derivative w.r.t. mean if nargout>2: d2lZ = -1/(sn2+s2) # 2nd derivative w.r.t. mean return lZ,dlZ,d2lZ else: return lZ,dlZ else: return lZ else: # derivative mode dlZhyp = ((y-mu)**2/(sn2+s2)-1) / (1+s2/sn2) # deriv. w.r.t. hyp.lik return dlZhyp elif isinstance(inffunc, inf.Laplace): if der == None: # no derivative mode if y == None: y=0 ymmu = y-mu lp = -ymmu**2/(2*sn2) - np.log(2*np.pi*sn2)/2. if nargout>1: dlp = ymmu/sn2 # dlp, derivative of log likelihood if nargout>2: # d2lp, 2nd derivative of log likelihood d2lp = -np.ones_like(ymmu)/sn2 if nargout>3: # d3lp, 3rd derivative of log likelihood d3lp = np.zeros_like(ymmu) return lp,dlp,d2lp,d3lp else: return lp,dlp,d2lp else: return lp,dlp else: return lp else: # derivative mode lp_dhyp = (y-mu)**2/sn2 - 1 # derivative of log likelihood w.r.t. hypers dlp_dhyp = 2*(mu-y)/sn2 # first derivative, d2lp_dhyp = 2*np.ones_like(mu)/sn2 # and also of the second mu derivative return lp_dhyp,dlp_dhyp,d2lp_dhyp ''' elif isinstance(inffunc, infVB): if der == None: # variational lower site bound # t(s) = exp(-(y-s)^2/2sn2)/sqrt(2*pi*sn2) # the bound has the form: b*s - s.^2/(2*ga) - h(ga)/2 with b=y/ga ga = s2 n = len(ga) b = y/ga y = y*np.ones((n,1)) db = -y/ga**2 d2b = 2*y/ga**3 h = np.zeros((n,1)) dh = h d2h = h # allocate memory for return args id = (ga <= sn2 + 1e-8) # OK below noise variance h[id] = y[id]**2/ga[id] + np.log(2*np.pi*sn2) h[np.logical_not(id)] = np.inf dh[id] = -y[id]**2/ga[id]**2 d2h[id] = 2*y[id]**2/ga[id]**3 id = ga < 0 h[id] = np.inf dh[id] = 0 d2h[id] = 0 # neg. var. treatment varargout = [h,b,dh,db,d2h,d2b] else: ga = s2 n = len(ga) dhhyp = np.zeros((n,1)) dhhyp[ga<=sn2] = 2 dhhyp[ga<0] = 0 # negative variances get a special treatment varargout = dhhyp # deriv. w.r.t. hyp.lik else: raise Exception('Incorrect inference in lik.Gauss\n') ''' class Erf(Likelihood): ''' Error function or cumulative Gaussian likelihood function for binary classification or probit regression. :math:`Erf(t)=\\frac{1}{2}(1+erf(\\frac{t}{\\sqrt{2}}))=normcdf(t)` ''' def __init__(self): self.hyp = [] def proceed(self, y=None, mu=None, s2=None, inffunc=None, der=None, nargout=1): if not y == None: y = np.sign(y) y[y==0] = 1 else: y = 1 # allow only +/- 1 values if inffunc == None: # prediction mode if inf is not present y = y*np.ones_like(mu) # make y a vector s2zero = True; if not s2 == None: if np.linalg.norm(s2)>0: s2zero = False # s2==0? if s2zero: # log probability evaluation p,lp = self.cumGauss(y,mu,2) else: # prediction lp = self.proceed(y, mu, s2, inf.EP()) p = np.exp(lp) if nargout>1: ymu = 2*p-1 # first y moment if nargout>2: ys2 = 4*p*(1-p) # second y moment return lp,ymu,ys2 else: return lp,ymu else: return lp else: # inference mode if isinstance(inffunc, inf.Laplace): if der == None: # no derivative mode f = mu; yf = y*f # product latents and labels p,lp = self.cumGauss(y,f,2) if nargout>1: # derivative of log likelihood n_p = self.gauOverCumGauss(yf,p) dlp = y*n_p # derivative of log likelihood if nargout>2: # 2nd derivative of log likelihood d2lp = -n_p**2 - yf*n_p if nargout>3: # 3rd derivative of log likelihood d3lp = 2*y*n_p**3 + 3*f*n_p**2 + y*(f**2-1)*n_p return lp,dlp,d2lp,d3lp else: return lp,dlp,d2lp else: return lp,dlp else: return lp else: # derivative mode return [] # derivative w.r.t. hypers elif isinstance(inffunc, inf.EP): if der == None: # no derivative mode z = mu/np.sqrt(1+s2) junk,lZ = self.cumGauss(y,z,2) # log part function if not y == None: z = z*y if nargout>1: if y == None: y = 1 n_p = self.gauOverCumGauss(z,np.exp(lZ)) dlZ = y*n_p/np.sqrt(1.+s2) # 1st derivative wrt mean if nargout>2: d2lZ = -n_p*(z+n_p)/(1.+s2) # 2nd derivative wrt mean return lZ,dlZ,d2lZ else: return lZ,dlZ else: return lZ else: # derivative mode return [] # deriv. wrt hyp.lik ''' if inffunc == 'inf.infVB': if der == None: # no derivative mode # naive variational lower bound based on asymptotical properties of lik # normcdf(t) -> -(t*A_hat^2-2dt+c)/2 for t->-np.inf (tight lower bound) d = 0.158482605320942; c = -1.785873318175113; ga = s2; n = len(ga); b = d*y*np.ones((n,1)); db = np.zeros((n,1)); d2b = db h = -2.*c*np.ones((n,1)); h[ga>1] = np.inf; dh = np.zeros((n,1)); d2h = dh varargout = [h,b,dh,db,d2h,d2b] else: # derivative mode varargout = [] # deriv. wrt hyp.lik ''' def cumGauss(self, y=None, f=None, nargout=1): # return [p,lp] = cumGauss(y,f) if not y == None: yf = y*f else: yf = f p = (1. + erf(yf/np.sqrt(2.)))/2. # likelihood if nargout>1: lp = self.logphi(yf,p) return p,lp else: return p def gauOverCumGauss(self,f,p): # return n_p = gauOverCumGauss(f,p) n_p = np.zeros_like(f) # safely compute Gaussian over cumulative Gaussian ok = f>-5 # naive evaluation for large values of f n_p[ok] = (np.exp(-f[ok]**2/2)/np.sqrt(2*np.pi)) / p[ok] bd = f<-6 # tight upper bound evaluation n_p[bd] = np.sqrt(f[bd]**2/4+1)-f[bd]/2 interp = np.logical_and(np.logical_not(ok),np.logical_not(bd)) # linearly interpolate between both of them tmp = f[interp] lam = -5. - f[interp] n_p[interp] = (1-lam)*(np.exp(-tmp**2/2)/np.sqrt(2*np.pi))/p[interp] + lam *(np.sqrt(tmp**2/4+1)-tmp/2); return n_p def logphi(self,z,p): # return lp = logphi(z,p) lp = np.zeros_like(z) # allocate memory zmin = -6.2; zmax = -5.5; ok = z>zmax # safe evaluation for large values bd = z0: s2zero = False # s2==0? if s2zero: # log probability evaluation lp = -np.abs(y-mu)/b -np.log(2*b); s2 = 0 else: # prediction lp = self.proceed(y, mu, s2, inf.EP()) if nargout>1: ymu = mu # first y moment if nargout>2: ys2 = s2 + sn**2 # second y moment return lp,ymu,ys2 else: return lp,ymu else: return lp else: # inference mode if isinstance(inffunc, inf.Laplace): if der == None: # no derivative mode if y == None: y = np.zeros_like(mu) ymmu = y-mu lp = np.abs(ymmu)/b - np.log(2*b) if nargout>1: # derivative of log likelihood dlp = np.sign(ymmu)/b if nargout>2: # 2nd derivative of log likelihood d2lp = np.zeros_like(ymmu) if nargout>3: # 3rd derivative of log likelihood d3lp = np.zeros_like(ymmu) return lp,dlp,d2lp,d3lp else: return lp,dlp,d2lp else: return lp,dlp else: return lp else: # derivative mode return [] # derivative w.r.t. hypers elif isinstance(inffunc, inf.EP): n = np.max([len(y.flatten()),len(mu.flatten()),len(s2.flatten()),len(sn.flatten())]) on = np.ones((n,1)) y = y*on; mu = mu*on; s2 = s2*on; sn = sn*on; fac = 1e3; # factor between the widths of the two distributions ... # ... from when one considered a delta peak, we use 3 orders of magnitude #idlik = np.reshape( (fac*sn) < np.sqrt(s2) , (sn.shape[0],) ) # Likelihood is a delta peak #idgau = np.reshape( (fac*np.sqrt(s2)) < sn , (sn.shape[0],) ) # Gaussian is a delta peak idlik = (fac*sn) < np.sqrt(s2) idgau = (fac*np.sqrt(s2)) < sn id = np.logical_and(np.logical_not(idgau),np.logical_not(idlik)) # interesting case in between if der == None: # no derivative mode lZ = np.zeros((n,1)) dlZ = np.zeros((n,1)) d2lZ = np.zeros((n,1)) if np.any(idlik): l = Gauss(log_sigma=np.log(s2[idlik])/2) a = l.proceed(mu[idlik], y[idlik]) lZ[idlik] = a[0]; dlZ[idlik] = a[1]; d2lZ[idlik] = a[2] if np.any(idgau): l = Laplace(log_hyp=np.log(sn[idgau])) a = l.proceed(mu=mu[idgau], y=y[idgau]) lZ[idgau] = a[0]; dlZ[idgau] = a[1]; d2lZ[idgau] = a[2] if np.any(id): # substitution to obtain unit variance, zero mean Laplacian tvar = s2[id]/(sn[id]**2+1e-16) tmu = (mu[id]-y[id])/(sn[id]+1e-16) # an implementation based on logphi(t) = log(normcdf(t)) zp = (tmu+np.sqrt(2)*tvar)/np.sqrt(tvar) zm = (tmu-np.sqrt(2)*tvar)/np.sqrt(tvar) ap = self._logphi(-zp)+np.sqrt(2)*tmu am = self._logphi( zm)-np.sqrt(2)*tmu apam = np.vstack((ap,am)).T lZ[id] = self._logsum2exp(apam) + tvar - np.log(sn[id]*np.sqrt(2.)) if nargout>1: lqp = -0.5*zp**2 - 0.5*np.log(2*np.pi) - self._logphi(-zp); # log( N(z)/Phi(z) ) lqm = -0.5*zm**2 - 0.5*np.log(2*np.pi) - self._logphi( zm); dap = -np.exp(lqp-0.5*np.log(s2[id])) + np.sqrt(2)/sn[id] dam = np.exp(lqm-0.5*np.log(s2[id])) - np.sqrt(2)/sn[id] _z1 = np.vstack((ap,am)).T _z2 = np.vstack((dap,dam)).T _x = np.array([[1],[1]]) dlZ[id] = self._expABz_expAx(_z1, _x, _z2, _x) if nargout>2: a = np.sqrt(8.)/sn[id]/np.sqrt(s2[id]); bp = 2./sn[id]**2 - (a - zp/s2[id])*np.exp(lqp) bm = 2./sn[id]**2 - (a + zm/s2[id])*np.exp(lqm) _x = np.reshape(np.array([1,1]),(2,1)) _z1 = np.reshape(np.array([ap,am]),(1,2)) _z2 = np.reshape(np.array([bp,bm]),(1,2)) d2lZ[id] = self._expABz_expAx(_z1, _x, _z2, _x) - dlZ[id]**2 return lZ,dlZ,d2lZ else: return lZ,dlZ else: return lZ else: # derivative mode dlZhyp = np.zeros((n,1)) if np.any(idlik): dlZhyp[idlik] = 0 if np.any(idgau): l = Laplace(log_hyp=np.log(sn[idgau])) a = l.proceed(mu=mu[idgau], y=y[idgau], inffunc='inf.Laplace', nargout=1) dlZhyp[idgau] = a[0] if np.any(id): # substitution to obtain unit variance, zero mean Laplacian tmu = (mu[id]-y[id])/(sn[id]+1e-16); tvar = s2[id]/(sn[id]**2+1e-16) zp = (tvar+tmu/np.sqrt(2))/np.sqrt(tvar); vp = tvar+np.sqrt(2)*tmu zm = (tvar-tmu/np.sqrt(2))/np.sqrt(tvar); vm = tvar-np.sqrt(2)*tmu dzp = (-s2[id]/sn[id]+tmu*sn[id]/np.sqrt(2)) / np.sqrt(s2[id]) dvp = -2*tvar - np.sqrt(2)*tmu dzm = (-s2[id]/sn[id]-tmu*sn[id]/np.sqrt(2)) / np.sqrt(s2[id]) dvm = -2*tvar + np.sqrt(2)*tmu lezp = self._lerfc(zp); # ap = exp(vp).*ezp lezm = self._lerfc(zm); # am = exp(vm).*ezm vmax = np.max(np.array([vp+lezp,vm+lezm]),axis=0); # subtract max to avoid numerical pb ep = np.exp(vp+lezp-vmax) em = np.exp(vm+lezm-vmax) dap = ep*(dvp - 2/np.sqrt(np.pi)*np.exp(-zp**2-lezp)*dzp) dam = em*(dvm - 2/np.sqrt(np.pi)*np.exp(-zm**2-lezm)*dzm) dlZhyp[id] = (dap+dam)/(ep+em) - 1; return dlZhyp # deriv. wrt hyp.lik elif isinstance(inffunc, inf.VB): n = len(s2.flatten()); b = np.zeros((n,1)); y = y*np.ones((n,1)); z = y return b,z def _lerfc(self,t): ''' numerically safe implementation of f(t) = log(1-erf(t)) = log(erfc(t))''' from scipy.special import erfc f = np.zeros_like(t) tmin = 20; tmax = 25 ok = ttmax # evaluate tight bound nok = np.logical_not(ok) interp = np.logical_and(nok,np.logical_not(bd)) # interpolate between both of them f[nok] = np.log(2/np.sqrt(np.pi)) -t[nok]**2 -np.log(t[nok]+np.sqrt( t[nok]**2+4/np.pi )) lam = 1/(1+np.exp( 12*(0.5-(t[interp]-tmin)/(tmax-tmin)) )) # interp. weights f[interp] = lam*f[interp] + (1-lam)*np.log(erfc( t[interp] )) f[ok] += np.log(erfc( t[ok] )) # safe eval return f def _expABz_expAx(self,A,x,B,z): ''' Computes y = ( (exp(A).*B)*z ) ./ ( exp(A)*x ) in a numerically safe way The function is not general in the sense that it yields correct values for all types of inputs. We assume that the values are close together. ''' N = A.shape[1] maxA = np.max(A,axis=1) # number of columns, max over columns maxA = np.array([maxA]).T A = A - np.dot(maxA, np.ones((1,N))) # subtract maximum value y = ( np.dot((np.exp(A)*B),z) ) / ( np.dot(np.exp(A),x) ) return y[0] def _logphi(self,z): ''' Safe implementation of the log of phi(x) = \int_{-\infty}^x N(f|0,1) df returns lp = log(normcdf(z)) ''' lp = np.zeros_like(z) # allocate memory zmin = -6.2; zmax = -5.5; ok = z>zmax # safe evaluation for large values bd = z self.searchConfig.num_restarts/2: print "[CG] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter) raise Exception("Over half of the trails failed for conjugate gradient") if self.searchConfig.num_restarts and self.trailsCounter > self.searchConfig.num_restarts-1: # if exceed num_restarts print "[CG] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter) return optimalHyp, funcValue if self.searchConfig.min_threshold and funcValue <= self.searchConfig.min_threshold: # reach provided mininal print "[CG] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter) return optimalHyp, funcValue return optimalHyp, funcValue class BFGS(Optimizer): '''quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS)''' def __init__(self, model, searchConfig = None): super(BFGS, self).__init__() self.model = model self.searchConfig = searchConfig self.trailsCounter = 0 self.errorCounter = 0 def findMin(self, x, y): meanfunc = self.model.meanfunc covfunc = self.model.covfunc likfunc = self.model.likfunc inffunc = self.model.inffunc hypInArray = self.convert_to_array() try: opt = bfgs(self.nlml, hypInArray, self.dnlml, maxiter=100, disp=False, full_output=True) optimalHyp = deepcopy(opt[0]) funcValue = opt[1] warnFlag = opt[6] if warnFlag == 1: print "Maximum number of iterations exceeded." elif warnFlag == 2: print "Gradient and/or function calls not changing." except: self.errorCounter += 1 if not self.searchConfig: raise Exception("Can not use BFGS. Try other hyparameters") self.trailsCounter += 1 if self.searchConfig: searchRange = self.searchConfig.meanRange + self.searchConfig.covRange + self.searchConfig.likRange if not (self.searchConfig.num_restarts or self.searchConfig.min_threshold): raise Exception('Specify at least one of the stop conditions') while True: self.trailsCounter += 1 # increase counter for i in xrange(hypInArray.shape[0]): # random init of hyp hypInArray[i]= np.random.uniform(low=searchRange[i][0], high=searchRange[i][1]) # value this time is better than optiaml min value try: thisopt = bfgs(self.nlml, hypInArray, self.dnlml, maxiter=100, disp=False, full_output=True) if thisopt[1] < funcValue: funcValue = thisopt[1] optimalHyp = thisopt[0] except: self.errorCounter += 1 if self.searchConfig.num_restarts and self.errorCounter > self.searchConfig.num_restarts/2: print "[BFGS] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter) raise Exception("Over half of the trails failed for BFGS") if self.searchConfig.num_restarts and self.trailsCounter > self.searchConfig.num_restarts-1: # if exceed num_restarts print "[BFGS] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter) return optimalHyp, funcValue if self.searchConfig.min_threshold and funcValue <= self.searchConfig.min_threshold: # reach provided mininal print "[BFGS] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter) return optimalHyp, funcValue return optimalHyp, funcValue class Minimize(Optimizer): '''minimize by Carl Rasmussen (python implementation of "minimize" in GPML)''' def __init__(self, model, searchConfig = None): super(Minimize, self).__init__() self.model = model self.searchConfig = searchConfig self.trailsCounter = 0 self.errorCounter = 0 def findMin(self, x, y): meanfunc = self.model.meanfunc covfunc = self.model.covfunc likfunc = self.model.likfunc inffunc = self.model.inffunc hypInArray = self.convert_to_array() try: opt = minimize.run(self.nlzAnddnlz, hypInArray, length=-40) optimalHyp = deepcopy(opt[0]) funcValue = opt[1][-1] except: self.errorCounter += 1 if not self.searchConfig: raise Exception("Can not use minimize. Try other hyparameters") self.trailsCounter += 1 if self.searchConfig: searchRange = self.searchConfig.meanRange + self.searchConfig.covRange + self.searchConfig.likRange if not (self.searchConfig.num_restarts or self.searchConfig.min_threshold): raise Exception('Specify at least one of the stop conditions') while True: self.trailsCounter += 1 # increase counter for i in xrange(hypInArray.shape[0]): # random init of hyp hypInArray[i]= np.random.uniform(low=searchRange[i][0], high=searchRange[i][1]) # value this time is better than optiaml min value try: thisopt = minimize.run(self.nlzAnddnlz, hypInArray, length=-40) if thisopt[1][-1] < funcValue: funcValue = thisopt[1][-1] optimalHyp = thisopt[0] except: self.errorCounter += 1 if self.searchConfig.num_restarts and self.errorCounter > self.searchConfig.num_restarts/2: print "[Minimize] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter) raise Exception("Over half of the trails failed for minimize") if self.searchConfig.num_restarts and self.trailsCounter > self.searchConfig.num_restarts-1: # if exceed num_restarts print "[Minimize] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter) return optimalHyp, funcValue if self.searchConfig.min_threshold and funcValue <= self.searchConfig.min_threshold: # reach provided mininal print "[Minimize] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter) return optimalHyp, funcValue return optimalHyp, funcValue class SCG(Optimizer): '''Scaled conjugent gradient (faster than CG)''' def __init__(self, model, searchConfig = None): super(SCG, self).__init__() self.model = model self.searchConfig = searchConfig self.trailsCounter = 0 self.errorCounter = 0 def findMin(self, x, y): meanfunc = self.model.meanfunc covfunc = self.model.covfunc likfunc = self.model.likfunc inffunc = self.model.inffunc hypInArray = self.convert_to_array() try: opt = scg.run(self.nlzAnddnlz, hypInArray) optimalHyp = deepcopy(opt[0]) funcValue = opt[1][-1] except: self.errorCounter += 1 if not self.searchConfig: raise Exception("Can not use Scaled conjugate gradient. Try other hyparameters") self.trailsCounter += 1 if self.searchConfig: searchRange = self.searchConfig.meanRange + self.searchConfig.covRange + self.searchConfig.likRange if not (self.searchConfig.num_restarts or self.searchConfig.min_threshold): raise Exception('Specify at least one of the stop conditions') while True: self.trailsCounter += 1 # increase counter for i in xrange(hypInArray.shape[0]): # random init of hyp hypInArray[i]= np.random.uniform(low=searchRange[i][0], high=searchRange[i][1]) # value this time is better than optiaml min value try: thisopt = scg.run(self.nlzAnddnlz, hypInArray) if thisopt[1][-1] < funcValue: funcValue = thisopt[1][-1] optimalHyp = thisopt[0] except: self.errorCounter += 1 if self.searchConfig.num_restarts and self.errorCounter > self.searchConfig.num_restarts/2: print "[SCG] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter) raise Exception("Over half of the trails failed for Scaled conjugate gradient") if self.searchConfig.num_restarts and self.trailsCounter > self.searchConfig.num_restarts-1: # if exceed num_restarts print "[SCG] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter) return optimalHyp, funcValue if self.searchConfig.min_threshold and funcValue <= self.searchConfig.min_threshold: # reach provided mininal print "[SCG] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter) return optimalHyp, funcValue return optimalHyp, funcValue }rgM(X CG.__init__XdefKFKMXBFGSXclassKKX SCG.findMinXdefKM*X OptimizerXclassKKDXOptimizer.__init__XdefKK XMinimizeXclassKKX CG.findMinXdefKMKXOptimizer.convert_to_arrayXdefK4K9XCGXclassKDKXOptimizer.apply_in_objectsXdefK9KDX SCG.__init__XdefKKXOptimizer.dnlmlXdefK&K-XOptimizer.nlzAnddnlzXdefK-K4X BFGS.findMinXdefKKXMinimize.__init__XdefKKƇX BFGS.__init__XdefKKXSCGXclassKM*XOptimizer.nlmlXdefK K&XMinimize.findMinXdefKKu}rhM(XMinimizejcMXBFGSjcMXCGjcMXSCGjcMuj]MXpyGPs.Core.meanriMrjM}rkMbX$#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ # mean functions to be use by Gaussian process functions. There are two # different kinds of mean functions: simple and composite: # # simple mean functions: # # Zero - zero mean function # One - one mean function # Const - constant mean function # Linear - linear mean function # # composite mean functions: # # ScaleOfMean - scaled version of a mean function # PowerOfMean - power of a mean function # ProductOfMean - products of mean functions # SumOfMean - sums of mean functions # # # This is a object-oriented python implementation of gpml functionality # (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18). # based on the functional-version of python implementation # (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013) # # # Copyright (c) by Marion Neumann and Shan Huang, Sep.2013 import numpy as np import math class Mean(object): """ The base function for mean function """ def __init__(self): super(Mean, self).__init__() self.hyp = [] self.para = [] # overloading def __add__(self,mean): ''' Overloading + operator. :param mean: mean function :return: an instance of SumOfMean ''' return SumOfMean(self,mean) # overloading def __mul__(self,other): ''' Overloading * operator. Using * for both multiplication with scalar and product of means depending on the type of the two objects. :param other: mean function as product or int/float as scalar :return: an instance of ScaleOfMean or ProductOfMean ''' if isinstance(other, int) or isinstance(other, float): return ScaleOfMean(self,other) elif isinstance(other, Mean): return ProductOfMean(self,other) else: print "only numbers and Means are allowed for *" # overloading __rmul__ = __mul__ # overloading def __pow__(self,number): ''' Overloading ** operator. :param int number: power of the mean function :return: an instance of PowerOfMean ''' if isinstance(number, int) and number > 0: return PowerOfMean(self,number) else: print "only non-zero integers are supported for **" class ProductOfMean(Mean): '''Product of two mean fucntions.''' def __init__(self,mean1,mean2): self.mean1 = mean1 self.mean2 = mean2 if mean1.hyp and mean2.hyp: self._hyp = mean1.hyp + mean2.hyp elif not mean1.hyp: self._hyp = mean2.hyp elif not mean2.hyp: self._hyp = mean1.hyp def sethyp(self,hyp): assert len(hyp) == len(self._hyp) len1 = len(self.mean1.hyp) self._hyp = hyp self.mean1.hyp = self._hyp[:len1] self.mean2.hyp = self._hyp[len1:] def gethyp(self): return self._hyp hyp = property(gethyp,sethyp) def getMean(self, x=None): A = self.mean1.getMean(x) * self.mean2.getMean(x) return A def getDerMatrix(self, x=None, der=None): if der < len(self.mean1.hyp): A = self.mean1.getDerMatrix(x, der) * self.mean2.getMean(x) elif der < len(self.hyp): der2 = der - len(self.mean1.hyp) A = self.mean2.getDerMatrix(x, der2) * self.mean1.getMean(x) else: raise Exception("Error: der out of range for meanProduct") return A class SumOfMean(Mean): '''Sum of two mean functions.''' def __init__(self,mean1,mean2): self.mean1 = mean1 self.mean2 = mean2 if mean1.hyp and mean2.hyp: self._hyp = mean1.hyp + mean2.hyp elif not mean1.hyp: self._hyp = mean2.hyp elif not mean2.hyp: self._hyp = mean1.hyp def sethyp(self,hyp): assert len(hyp) == len(self._hyp) len1 = len(self.mean1.hyp) self._hyp = hyp self.mean1.hyp = self._hyp[:len1] self.mean2.hyp = self._hyp[len1:] def gethyp(self): return self._hyp hyp = property(gethyp,sethyp) def getMean(self, x=None): A = self.mean1.getMean(x) + self.mean2.getMean(x) return A def getDerMatrix(self, x=None, der=None): if der < len(self.mean1.hyp): A = self.mean1.getDerMatrix(x, der) elif der < len(self.hyp): der2 = der - len(self.mean1.hyp) A = self.mean2.getDerMatrix(x, der2) else: raise Exception("Error: der out of range for meanSum") return A class ScaleOfMean(Mean): '''Scale of a mean function.''' def __init__(self,mean,scalar): self.mean = mean if mean.hyp: self._hyp = [scalar] + mean.hyp else: self._hyp = [scalar] def sethyp(self,hyp): assert len(hyp) == len(self._hyp) self._hyp = hyp self.mean.hyp = self._hyp[1:] def gethyp(self): return self._hyp hyp = property(gethyp,sethyp) def getMean(self, x=None): c = self.hyp[0] # scale parameter A = c * self.mean.getMean(x) # accumulate means return A def getDerMatrix(self, x=None, der=None): c = self.hyp[0] # scale parameter if der == 0: # compute derivative w.r.t. c A = self.mean.getMean(x) else: A = c * self.mean.getDerMatrix(x,der-1) return A class PowerOfMean(Mean): '''Power of a mean fucntion.''' def __init__(self, mean, d): self.mean = mean if mean.hyp: self._hyp = [d] + mean.hyp else: self._hyp = [d] def sethyp(self,hyp): assert len(hyp) == len(self._hyp) self._hyp = hyp self.mean.hyp = self._hyp[1:] def gethyp(self): return self._hyp hyp = property(gethyp,sethyp) def getMean(self, x=None): d = np.abs(np.floor(self.hyp[0])) d = max(d,1) A = self.mean.getMean(x) **d # accumulate means return A def getDerMatrix(self, x=None, der=None): d = np.abs(np.floor(self.hyp[0])) d = max(d,1) if der == 0: # compute derivative w.r.t. c a = self.mean.getMean(x) A = a**d * np.log(a) else: A = d * self.mean.getMean(x) ** (d-1) * self.mean.getDerMatrix(x, der-1) return A class Zero(Mean): '''Zero mean.''' def __init__(self): self.hyp = [] self.name = '0' def getMean(self, x=None): n, D = x.shape A = np.zeros((n,1)) return A def getDerMatrix(self, x=None, der=None): n, D = x.shape A = np.zeros((n,1)) return A class One(Mean): ''' One mean.''' def __init__(self): self.hyp = [] self.name = '1' def getMean(self, x=None): n, D = x.shape A = np.ones((n,1)) return A def getDerMatrix(self, x=None, der=None): n, D = x.shape A = np.zeros((n,1)) return A class Const(Mean): ''' Constant mean function. hyp = [c] :param c: constant value for mean ''' def __init__(self, c=5.): self.hyp = [c] def getMean(self, x=None): n,D = x.shape A = self.hyp[0] * np.ones((n,1)) return A def getDerMatrix(self, x=None, der=None): n,D = x.shape if der == 0: # compute derivative vector wrt c A = np.ones((n,1)) else: A = np.zeros((n,1)) return A class Linear(Mean): ''' Linear mean function. self.hyp = alpha_list :param D: dimension of training data. Set if you want default alpha, which is 0.5 for each dimension. :alpha_list: scalar alpha for each dimension ''' def __init__(self, D=None, alpha_list=None): if alpha_list == None: self.hyp = [0.5 for i in xrange(D)] else: self.hyp = alpha_list def getMean(self, x=None): n, D = x.shape c = np.array(self.hyp) c = np.reshape(c,(len(c),1)) A = np.dot(x,c) return A def getDerMatrix(self, x=None, der=None): n, D = x.shape c = np.array(self.hyp) c = np.reshape(c,(len(c),1)) if isinstance(der, int) and der < D: # compute derivative vector wrt meanparameters A = np.reshape(x[:,der], (len(x[:,der]),1) ) else: A = np.zeros((n,1)) return A if __name__ == '__main__': pass }rlM(XScaleOfMean.__init__XdefKKXScaleOfMean.getMeanXdefKKX One.getMeanXdefMM X Zero.getMeanXdefKKX One.__init__XdefMMXSumOfMean.getMeanXdefKKXOne.getDerMatrixXdefM MXScaleOfMean.getDerMatrixXdefKKʇX Mean.__add__XdefK5K?XProductOfMean.getDerMatrixXdefKzKXSumOfMean.sethypXdefKKXConstXclassMM(X Mean.__mul__XdefK?KPXScaleOfMean.gethypXdefKKXProductOfMean.getMeanXdefKvKzXLinearXclassM(MIXSumOfMean.getDerMatrixXdefKKXPowerOfMean.gethypXdefKK؇XZero.getDerMatrixXdefKKXProductOfMean.sethypXdefKlKrX ScaleOfMeanXclassKKʇXSumOfMean.gethypXdefKKX ProductOfMeanXclassKaKXPowerOfMean.getMeanXdefKKX Mean.__pow__XdefKSKaXLinear.getDerMatrixXdefM<MIX SumOfMeanXclassKKXPowerOfMean.getDerMatrixXdefKKX Mean.__init__XdefK/K5XPowerOfMean.sethypXdefKKևX Zero.__init__XdefKKXLinear.getMeanXdefM5M<X Const.getMeanXdefMMXProductOfMean.gethypXdefKrKtXProductOfMean.__init__XdefKcKlXMeanXclassK+KaXScaleOfMean.sethypXdefKKX PowerOfMeanXclassKKXConst.getDerMatrixXdefMM(XOneXclassKMXZeroXclassKKXConst.__init__XdefMMXLinear.__init__XdefM/M5XSumOfMean.__init__XdefKKXPowerOfMean.__init__XdefKK҇u}rmM(XConsthXLinearhX SumOfMeanhX PowerOfMeanhXOnehX ScaleOfMeanhXZerohX ProductOfMeanhXMeanhuj]MXpyGPs.Core.infrnMroM}rpMbX#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ # Inference methods: Compute the (approximate) posterior for a Gaussian process. # Methods currently implemented include: # # infExact Exact inference (only possible with Gaussian likelihood) # infLaplace Laplace's Approximation # infEP Expectation Propagation # infVB [NOT IMPLEMENTED!] Variational Bayes Approximation # # infFITC Large scale regression with approximate covariance matrix # infFITC_Laplace Large scale inference with approximate covariance matrix # infFITC_EP Large scale inference with approximate covariance matrix # # infMCMC [NOT IMPLEMENTED!] # Markov Chain Monte Carlo and Annealed Importance Sampling # We offer two samplers. # - hmc: Hybrid Monte Carlo # - ess: Elliptical Slice Sampling # No derivatives w.r.t. to hyperparameters are provided. # # infLOO [NOT IMPLEMENTED!] # Leave-One-Out predictive probability and Least-Squares Approxim. # # The interface to the approximation methods is the following: # # post nlZ dnlZ = inf.proceed(mean, cov, lik, x, y) # # where: # INPUT: # cov name of the covariance function (see covFunctions.m) # lik name of the likelihood function (see likFunctions.m) # x n by D matrix of training inputs # y 1d array (of size n) of targets # # OUTPUT: # post struct representation of the (approximate) posterior containing: # alpha 1d array containing inv(K)*m, # where K is the prior covariance matrix and m the approx posterior mean # sW 1d array containing diagonal of sqrt(W) # the approximate posterior covariance matrix is inv(inv(K)+W) # L 2d array, L = chol(sW*K*sW+identity(n)) # nlZ returned value of the negative log marginal likelihood # dnlZ 1d array of partial derivatives of the negative log marginal likelihood # w.r.t. each hyperparameter # # Usually, the approximate posterior to be returned admits the form # N(m=K*alpha, V=inv(inv(K)+W)), where alpha is a vector and W is diagonal; # if not, then L contains instead -inv(K+inv(W)), and sW is unused. # # For more information on the individual approximation methods and their # implementations, see the respective inf* function below. See also gp.py # # # This is a object-oriented python implementation of gpml functionality # (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18). # based on the functional-version of python implementation # (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013) # # Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013 import numpy as np import lik, cov from copy import copy,deepcopy from tools import solve_chol,brentmin,cholupdate np.seterr(all='ignore') class postStruct(object): ''' Data structure for posterior ''' def __init__(self): self.alpha = np.array([]) self.L = np.array([]) self.sW = np.array([]) class dnlZStruct(object): ''' Data structure for the derivatives of mean, cov and lik functions ''' def __init__(self, m, c, l): self.mean = [] self.cov = [] self.lik = [] if m.hyp: self.mean = [0 for i in xrange(len(m.hyp))] if c.hyp: self.cov = [0 for i in xrange(len(c.hyp))] if l.hyp: self.lik = [0 for i in xrange(len(l.hyp))] class Inference(object): ''' Base class for inference. Defined several tool methods in it. ''' def __init__(self): pass def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): ''' Inference computation based on inputs. post, nlZ, dnlZ = inf.proceed(mean, cov, lik, x, y) :param meanfunc: mean function :param covfunc: covariance function :param likfunc: likelihood function :param x: training data :param y: training labels :param nargout: specify the number of output(1,2 or 3) :return: posterior, negative-log-marginal-likelihood, derivative for negative-log-marginal-likelihood-likelihood ''' pass def epComputeParams(self, K, y, ttau, tnu, likfunc, m, inffunc): n = len(y) # number of training cases ssi = np.sqrt(ttau) # compute Sigma and mu L = np.linalg.cholesky(np.eye(n)+np.dot(ssi,ssi.T)*K).T # L'*L=B=eye(n)+sW*K*sW V = np.linalg.solve(L.T,np.tile(ssi,(1,n))*K) Sigma = K - np.dot(V.T,V) mu = np.dot(Sigma,tnu) Dsigma = np.reshape(np.diag(Sigma),(np.diag(Sigma).shape[0],1)) tau_n = 1/Dsigma - ttau # compute the log marginal likelihood nu_n = mu/Dsigma-tnu + m*tau_n # vectors of cavity parameters lZ = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc) nlZ = np.log(np.diag(L)).sum() - lZ.sum() - np.dot(tnu.T,np.dot(Sigma,tnu))/2 \ - np.dot((nu_n-m*tau_n).T,((ttau/tau_n*(nu_n-m*tau_n)-2*tnu) / (ttau+tau_n)))/2 \ + (tnu**2/(tau_n+ttau)).sum()/2.- np.log(1.+ttau/tau_n).sum()/2. return Sigma, mu, nlZ[0], L def logdetA(self,K,w,nargout): ''' Compute the log determinant ldA and the inverse iA of a square nxn matrix A = eye(n) + K*diag(w) from its LU decomposition; for negative definite A, we return ldA = Inf. We also return mwiA = -diag(w)*inv(A). [ldA,iA,mwiA] = logdetA(K,w)''' n = K.shape[0] assert(K.shape[0] == K.shape[1]) A = np.eye(n) + K*np.tile(w.T,(n,1)) [L,U,P] = np.linalg.lu(A) # compute LU decomposition, A = P'*L*U u = np.diag(U) signU = np.prod(np.sign(u)) # sign of U detP = 1 # compute sign (and det) of the permutation matrix P p = np.dot(P,np.array(range(n)).T) for ii in range(n): if ii != p[ii]: detP = -detP j= [jj for jj,val in enumerate(p) if val == ii] p[ii,j[0]] = p[j[0],ii] if signU != detP: # log becomes complex for negative values, encoded by infinity ldA = np.inf else: # det(L) = 1 and U triangular => det(A) = det(P)*prod(diag(U)) ldA = np.log(np.abs(u)).sum() if nargout>1: iA = inv(U)*inv(L)*P; if nargout>2: mwiA = -np.tile(w,(1,n))*iA return ldA,iA,mwiA else: return ldA,iA else: return ldA def Psi_line(self,s,dalpha,alpha,K,m,likfunc,y,inffunc): '''Criterion Psi at alpha + s*dalpha for line search [Psi,alpha,f,dlp,W] = Psi_line(s,dalpha,alpha,hyp,K,m,lik,y,inf)''' alpha = alpha + s*dalpha f = np.dot(K,alpha) + m [lp,dlp,d2lp] = likfunc.proceed(y,f,None,inffunc,None,3) W = -d2lp Psi = np.dot(alpha.T,(f-m))/2. - lp.sum() return Psi[0],alpha,f,dlp,W def epfitcZ(self,d,P,R,nn,gg,ttau,tnu,d0,R0,P0,y,likfunc,m,inffunc): '''Compute the marginal likelihood approximation effort is O(n*nu^2) provided that nu0: R = cholupdate(R,np.sqrt(r)*np.dot(R.T,v),'-') else: R = cholupdate(R,np.sqrt(-r)*np.dot(R.T,v),'+') ttemp = np.dot(R0.T,np.dot(R.T,np.dot(R,np.dot(R0,P_i)))) gg = gg + ((dbi-dwi*(hi-m[ii]))/t) * np.reshape(ttemp,(ttemp.shape[0],1)) # O(nu^2) w[ii] = w_i; b[ii] = b_i; # update site parameters O(1) P_i = P_i/t # O(nu) return d,P_i,R,nn,gg,w,b def mvmZ(self,x,RVdd,t): '''Matrix vector multiplication with Z=inv(K+inv(W))''' Zx = t*x - np.dot(RVdd.T,np.dot(RVdd,x)) return Zx def mvmK(self,al,V,d0): '''Matrix vector multiplication with approximate covariance matrix''' Kal = np.dot(V.T,np.dot(V,al)) + d0*al return Kal def Psi_lineFITC(self,s,dalpha,alpha,V,d0,m,likfunc,y,inffunc): '''Criterion Psi at alpha + s*dalpha for line search''' alpha = alpha + s*dalpha f = self.mvmK(alpha,V,d0) + m vargout = likfunc.proceed(y,f,None,inffunc,None,3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W = -d2lp Psi = np.dot(alpha.T,(f-m))/2. - lp.sum() return Psi[0],alpha,f,dlp,W def fitcRefresh(self,d0,P0,R0,R0P0, w): '''Refresh the representation of the posterior from initial and site parameters to prevent possible loss of numerical precision after many epfitcUpdates effort is O(n*nu^2) provided that nu1: # do we want the marginal likelihood? nlZ = np.dot((y-m).T,alpha)/2. + np.log(np.diag(L)).sum() + n*np.log(2*np.pi*sn2)/2. # -log marg lik if nargout>2: # do we want derivatives? dnlZ = dnlZStruct(meanfunc, covfunc, likfunc) # allocate space for derivatives Q = solve_chol(L,np.eye(n))/sn2 - np.dot(alpha,alpha.T) # precompute for convenience dnlZ.lik = [sn2*np.trace(Q)] if covfunc.hyp: for ii in range(len(covfunc.hyp)): dnlZ.cov[ii] = (Q*covfunc.getDerMatrix(x=x, mode='train', der=ii)).sum()/2. if meanfunc.hyp: for ii in range(len(meanfunc.hyp)): dnlZ.mean[ii] = np.dot(-meanfunc.getDerMatrix(x, ii).T,alpha) dnlZ.mean[ii] = dnlZ.mean[ii][0,0] return post, nlZ[0,0], dnlZ return post, nlZ[0,0] return post class FITC_Exact(Inference): ''' FITC approximation to the posterior Gaussian process. The function is equivalent to infExact with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Quu) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu, snu2 = sn2/1e6 is the noise of the inducing inputs and Quu = Kuu + snu2*eye(nu). ''' def __init__(self): self.name = 'FICT exact inference' def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): if not isinstance(likfunc, lik.Gauss): # NOTE: no explicit call to likGauss raise Exception ('Exact inference only possible with Gaussian likelihood') if not isinstance(covfunc, cov.FITCOfKernel): raise Exception('Only covFITC supported.') # check cov diagK,Kuu,Ku = covfunc.getCovMatrix(x=x, mode='train') # evaluate covariance matrix m = meanfunc.getMean(x) # evaluate mean vector n, D = x.shape nu = Kuu.shape[0] sn2 = np.exp(2*likfunc.hyp[0]) # noise variance of likGauss snu2 = 1.e-6*sn2 # hard coded inducing inputs noise Luu = np.linalg.cholesky(Kuu+snu2*np.eye(nu)).T # Kuu + snu2*I = Luu'*Luu V = np.linalg.solve(Luu.T,Ku) # V = inv(Luu')*Ku => V'*V = Q g_sn2 = diagK + sn2 - np.array([(V*V).sum(axis=0)]).T # g + sn2 = diag(K) + sn2 - diag(Q) Lu = np.linalg.cholesky(np.eye(nu) + np.dot(V/np.tile(g_sn2.T,(nu,1)),V.T)).T # Lu'*Lu=I+V*diag(1/g_sn2)*V' r = (y-m)/np.sqrt(g_sn2) be = np.linalg.solve(Lu.T,np.dot(V,r/np.sqrt(g_sn2))) iKuu = solve_chol(Luu,np.eye(nu)) # inv(Kuu + snu2*I) = iKuu post = postStruct() post.alpha = np.linalg.solve(Luu,np.linalg.solve(Lu,be))# return the posterior parameters post.L = solve_chol(np.dot(Lu,Luu),np.eye(nu)) - iKuu # Sigma-inv(Kuu) post.sW = np.ones((n,1))/np.sqrt(sn2) # unused for FITC prediction with gp.m if nargout>1: # do we want the marginal likelihood nlZ = np.log(np.diag(Lu)).sum() + (np.log(g_sn2).sum() + n*np.log(2*np.pi) + np.dot(r.T,r) - np.dot(be.T,be))/2. if nargout>2: # do we want derivatives? dnlZ = dnlZStruct(meanfunc, covfunc, likfunc) # allocate space for derivatives al = r/np.sqrt(g_sn2) - np.dot(V.T,np.linalg.solve(Lu,be))/g_sn2 # al = (Kt+sn2*eye(n))\y B = np.dot(iKuu,Ku) w = np.dot(B,al) W = np.linalg.solve(Lu.T,V/np.tile(g_sn2.T,(nu,1))) for ii in range(len(covfunc.hyp)): [ddiagKi,dKuui,dKui] = covfunc.getDerMatrix(x=x, mode='train', der=ii) # eval cov deriv R = 2.*dKui-np.dot(dKuui,B) v = ddiagKi - np.array([(R*B).sum(axis=0)]).T # diag part of cov deriv dnlZ.cov[ii] = ( np.dot(ddiagKi.T,1./g_sn2) + np.dot(w.T,(np.dot(dKuui,w)-2.*np.dot(dKui,al))) \ - np.dot(al.T,(v*al)) - np.dot(np.array([(W*W).sum(axis=0)]),v) - (np.dot(R,W.T)*np.dot(B,W.T)).sum() )/2. dnlZ.cov[ii] = dnlZ.cov[ii][0,0] dnlZ.lik = sn2*((1./g_sn2).sum() - (np.array([(W*W).sum(axis=0)])).sum() - np.dot(al.T,al)) dKuui = 2*snu2 R = -dKuui*B v = -np.array([(R*B).sum(axis=0)]).T # diag part of cov deriv dnlZ.lik += (np.dot(w.T,np.dot(dKuui,w)) -np.dot(al.T,(v*al)) \ - np.dot(np.array([(W*W).sum(axis=0)]),v) - (np.dot(R,W.T)*np.dot(B,W.T)).sum() )/2. dnlZ.lik = list(dnlZ.lik[0]) for ii in range(len(meanfunc.hyp)): dnlZ.mean[ii] = np.dot(-meanfunc.getDerMatrix(x, ii).T, al) dnlZ.mean[ii] = dnlZ.mean[ii][0,0] return post, nlZ[0,0], dnlZ return post, nlZ[0,0] return post class Laplace(Inference): ''' Laplace's Approximation to the posterior Gaussian process. ''' def __init__(self): self.last_alpha = None def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): tol = 1e-6 # tolerance for when to stop the Newton iterations smax = 2; Nline = 20; thr = 1e-4 # line search parameters maxit = 20 # max number of Newton steps in f inffunc = self K = covfunc.getCovMatrix(x=x, mode='train') # evaluate the covariance matrix m = meanfunc.getMean(x) # evaluate the mean vector n, D = x.shape Psi_old = np.inf # make sure while loop starts by the largest old objective val if self.last_alpha == None: # find a good starting point for alpha and f alpha = np.zeros((n,1)) f = np.dot(K,alpha) + m # start at mean if sizes not match vargout = likfunc.proceed(y, f, None, inffunc, None, 3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W= -d2lp Psi_new = -lp.sum() else: alpha = self.last_alpha f = np.dot(K,alpha) + m # try last one vargout = likfunc.proceed(y, f, None, inffunc, None, 3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W= -d2lp Psi_new = np.dot(alpha.T,(f-m))/2. - lp.sum() # objective for last alpha vargout = - likfunc.proceed(y, m, None, inffunc, None, 1) Psi_def = vargout[0] # objective for default init f==m if Psi_def < Psi_new: # if default is better, we use it alpha = np.zeros((n,1)) f = np.dot(K,alpha) + m vargout = likfunc.proceed(y, f, None, inffunc, None, 3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W=-d2lp; Psi_new = -lp.sum() isWneg = np.any(W<0) # flag indicating whether we found negative values of W it = 0 # this happens for the Student's t likelihood while (Psi_old - Psi_new > tol) and it2: # do we want derivatives? dnlZ = dnlZStruct(meanfunc, covfunc, likfunc) # allocate space for derivatives if isWneg: # switch between Cholesky and LU decomposition mode Z = -post.L # inv(K+inv(W)) g = np.atleast_2d((iA*K).sum(axis=1)).T /2 # deriv. of ln|B| wrt W; g = diag(inv(inv(K)+diag(W)))/2 else: Z = np.tile(sW,(1,n))*solve_chol(post.L,np.diag(np.reshape(sW,(sW.shape[0],)))) #sW*inv(B)*sW=inv(K+inv(W)) C = np.linalg.solve(post.L.T,np.tile(sW,(1,n))*K) # deriv. of ln|B| wrt W g = np.atleast_2d((np.diag(K)-(C**2).sum(axis=0).T)).T /2. # g = diag(inv(inv(K)+W))/2 dfhat = g* d3lp # deriv. of nlZ wrt. fhat: dfhat=diag(inv(inv(K)+W)).*d3lp/2 for ii in range(len(covfunc.hyp)): # covariance hypers dK = covfunc.getDerMatrix(x=x, mode='train', der=ii) dnlZ.cov[ii] = (Z*dK).sum()/2. - np.dot(alpha.T,np.dot(dK,alpha))/2. # explicit part b = np.dot(dK,dlp) # b-K*(Z*b) = inv(eye(n)+K*diag(W))*b dnlZ.cov[ii] -= np.dot(dfhat.T,b-np.dot(K,np.dot(Z,b))) # implicit part dnlZ.cov[ii] = dnlZ.cov[ii][0,0] for ii in range(len(likfunc.hyp)): # likelihood hypers [lp_dhyp,dlp_dhyp,d2lp_dhyp] = likfunc.proceed(y,f,None,inffunc,ii,3) dnlZ.lik[ii] = -np.dot(g.T,d2lp_dhyp) - lp_dhyp.sum() # explicit part b = np.dot(K,dlp_dhyp) # b-K*(Z*b) = inv(eye(n)+K*diag(W))*b dnlZ.lik[ii] -= np.dot(dfhat.T,b-np.dot(K,np.dot(Z,b))) # implicit part dnlZ.lik[ii] = dnlZ.lik[ii][0,0] for ii in range(len(meanfunc.hyp)): # mean hypers dm = meanfunc.getDerMatrix(x, ii) dnlZ.mean[ii] = -np.dot(alpha.T,dm) # explicit part dnlZ.mean[ii] -= np.dot(dfhat.T,dm-np.dot(K,np.dot(Z,dm))) # implicit part dnlZ.mean[ii] = dnlZ.mean[ii][0,0] return post,nlZ[0],dnlZ else: return post, nlZ[0] class FITC_Laplace(Inference): ''' FITC-Laplace approximation to the posterior Gaussian process. The function is equivalent to infLaplace with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Kuu + snu2 * eye(nu)) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu and snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard deviation of the inducing inputs snu to be a one per mil of the measurement noise's standard deviation sn. In case of a likelihood without noise parameter sn2, we simply use snu2 = 1e-6. ''' def __init__(self): self.last_alpha = None def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): if not isinstance(covfunc, cov.FITCOfKernel): raise Exception('Only covFITC supported.') tol = 1e-6 # tolerance for when to stop the Newton iterations smax = 2; Nline = 100; thr = 1e-4 # line search parameters maxit = 20 # max number of Newton steps in f inffunc = Laplace() diagK,Kuu,Ku = covfunc.getCovMatrix(x=x, mode='train') # evaluate the covariance matrix m = meanfunc.getMean(x) # evaluate the mean vector if likfunc.hyp: # hard coded inducing inputs noise sn2 = np.exp(2.*likfunc.hyp[-1]) snu2 = 1.e-6*sn2 # similar to infFITC else: snu2 = 1.e-6 n, D = x.shape nu = Kuu.shape[0] rot180 = lambda A: np.rot90(np.rot90(A)) # little helper functions chol_inv = lambda A: np.linalg.solve( rot180( np.linalg.cholesky(rot180(A)) ),np.eye(nu)) # chol(inv(A)) R0 = chol_inv(Kuu+snu2*np.eye(nu)) # initial R, used for refresh O(nu^3) V = np.dot(R0,Ku); d0 = diagK - np.array([(V*V).sum(axis=0)]).T # initial d, needed Psi_old = np.inf # make sure while loop starts by the largest old objective val if self.last_alpha == None: # find a good starting point for alpha and f alpha = np.zeros((n,1)) f = self.mvmK(alpha,V,d0) + m # start at mean if sizes not match vargout = likfunc.proceed(y, f, None, inffunc, None, 3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W=-d2lp; Psi_new = -lp.sum() else: alpha = self.last_alpha f = self.mvmK(alpha,V,d0) + m # try last one vargout = likfunc.proceed(y, f, None, inffunc, None, 3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W=-d2lp Psi_new = np.dot(alpha.T,(f-m))/2. - lp.sum() # objective for last alpha vargout = - likfunc.proceed(y, m, None, inffunc, None, 1) Psi_def = vargout[0] # objective for default init f==m if Psi_def < Psi_new: # if default is better, we use it alpha = np.zeros((n,1)) f = self.mvmK(alpha,V,d0) + m vargout = likfunc.proceed(y, f, None, inffunc, None, 3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W=-d2lp; Psi_new = -lp.sum() isWneg = np.any(W<0) # flag indicating whether we found negative values of W it = 0 # this happens for the Student's t likelihood while (Psi_old - Psi_new > tol) and it2: # do we want derivatives? dnlZ = dnlZStruct(meanfunc, covfunc, likfunc) # allocate space for derivatives [d,P,R] = self.fitcRefresh(d0,Ku,R0,V,W) # g = diag(inv(inv(K)+W))/2 g = d/2 + 0.5*np.atleast_2d((np.dot(np.dot(R,R0),P)**2).sum(axis=0)).T t = W/(1+W*d0) dfhat = g*d3lp # deriv. of nlZ wrt. fhat: dfhat=diag(inv(inv(K)+W)).*d3lp/2 for ii in range(len(covfunc.hyp)): # covariance hypers ddiagK,dKuu,dKu = covfunc.getDerMatrix(x=x, mode='train', der=ii) # eval cov derivatives dA = 2.*dKu.T-np.dot(R0tV.T,dKuu) # dQ = dA*R0tV w = np.atleast_2d((dA*R0tV.T).sum(axis=1)).T # w = diag(dQ) v = ddiagK-w # v = diag(dK)-diag(dQ); dnlZ.cov[ii] = np.dot(ddiagK.T,t) - np.dot((RVdd*RVdd).sum(axis=0),v) # explicit part dnlZ.cov[ii] -= (np.dot(RVdd,dA)*np.dot(RVdd,R0tV.T)).sum() # explicit part dnlZ.cov[ii] = 0.5*dnlZ.cov[ii] - np.dot(alpha.T,np.dot(dA,np.dot(R0tV,alpha))+v*alpha)/2. # explicit b = np.dot(dA,np.dot(R0tV,dlp)) + v*dlp # b-K*(Z*b) = inv(eye(n)+K*diag(W))*b KZb = self.mvmK(self.mvmZ(b,RVdd,t),V,d0) dnlZ.cov[ii] -= np.dot(dfhat.T,(b-KZb)) # implicit part dnlZ.cov[ii] = dnlZ.cov[ii][0,0] for ii in range(len(likfunc.hyp)): # likelihood hypers vargout = likfunc.proceed(y,f,None,inffunc,ii,3) lp_dhyp = vargout[0]; dlp_dhyp = vargout[1]; d2lp_dhyp = vargout[2] dnlZ.lik[ii] = -np.dot(g.T,d2lp_dhyp) - lp_dhyp.sum() # explicit part b = self.mvmK(dlp_dhyp,V,d0) # implicit part dnlZ.lik[ii] -= np.dot(dfhat.T,b-self.mvmK(self.mvmZ(b,RVdd,t),V,d0)) if ii == len(likfunc.hyp)-1: # since snu2 is a fixed fraction of sn2, there is a covariance-like term # in the derivative as well snu = np.sqrt(snu2); T = chol_inv(Kuu + snu2*np.eye(nu)); T = np.dot(T.T,np.dot(T,snu*Ku)); t = np.array([(T*T).sum(axis=0)]).T z = np.dot(alpha.T,np.dot(T.T,np.dot(T,alpha))-t*alpha) - np.dot(np.array([(RVdd*RVdd).sum(axis=0)]),t) z += (np.dot(RVdd,T.T)**2).sum() b = (t*dlp-np.dot(T.T,np.dot(T,dlp)))/2. KZb = self.mvmK(self.mvmZ(b,RVdd,t),V,d0) z -= np.dot(dfhat.T,b-KZb) dnlZ.lik[ii] += z dnlZ.lik[ii] = dnlZ.lik[ii][0,0] for ii in range(len(meanfunc.hyp)): # mean hypers dm = meanfunc.getDerMatrix(x, ii) dnlZ.mean[ii] = -np.dot(alpha.T,dm) # explicit part Zdm = self.mvmZ(dm,RVdd,t) dnlZ.mean[ii] -= np.dot(dfhat.T,(dm-self.mvmK(Zdm,V,d0))) # implicit part dnlZ.mean[ii] = dnlZ.mean[ii][0,0] return post,nlZ[0,0],dnlZ else: return post, nlZ[0,0] class EP(Inference): ''' Expectation Propagation approximation to the posterior Gaussian Process. ''' def __init__(self): self.name = 'Expectation Propagation' self.last_ttau = None self.last_tnu = None def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): tol = 1e-4; max_sweep = 10; min_sweep = 2 # tolerance to stop EP iterations n = x.shape[0] inffunc = self K = covfunc.getCovMatrix(x=x, mode='train') # evaluate the covariance matrix m = meanfunc.getMean(x) # evaluate the mean vector nlZ0 = -likfunc.proceed(y, m, np.reshape(np.diag(K),(np.diag(K).shape[0],1)), inffunc).sum() if self.last_ttau == None: # find starting point for tilde parameters ttau = np.zeros((n,1)) # initialize to zero if we have no better guess tnu = np.zeros((n,1)) Sigma = K # initialize Sigma and mu, the parameters of .. mu = np.zeros((n,1)) # .. the Gaussian posterior approximation nlZ = nlZ0 else: ttau = self.last_ttau # try the tilde values from previous call tnu = self.last_tnu Sigma, mu, nlZ, L = self.epComputeParams(K, y, ttau, tnu, likfunc, m, inffunc) if nlZ > nlZ0: # if zero is better .. ttau = np.zeros((n,1)) # .. then initialize with zero instead tnu = np.zeros((n,1)) Sigma = K # initialize Sigma and mu, the parameters of .. mu = np.zeros((n,1)) # .. the Gaussian posterior approximation nlZ = nlZ0 nlZ_old = np.inf; sweep = 0 # converged, max. sweeps or min. sweeps? while (np.abs(nlZ-nlZ_old) > tol and sweep < max_sweep) or (sweep < min_sweep): nlZ_old = nlZ; sweep += 1 rperm = xrange(n) # randperm(n) for ii in rperm: # iterate EP updates (in random order) over examples tau_ni = 1/Sigma[ii,ii] - ttau[ii]# first find the cavity distribution .. nu_ni = mu[ii]/Sigma[ii,ii] + m[ii]*tau_ni - tnu[ii] # .. params tau_ni and nu_ni # compute the desired derivatives of the indivdual log partition function lZ,dlZ,d2lZ = likfunc.proceed(y[ii], nu_ni/tau_ni, 1/tau_ni, inffunc, None, 3) ttau_old = copy(ttau[ii]) # then find the new tilde parameters, keep copy of old ttau[ii] = -d2lZ /(1.+d2lZ/tau_ni) ttau[ii] = max(ttau[ii],0) # enforce positivity i.e. lower bound ttau by zero tnu[ii] = ( dlZ + (m[ii]-nu_ni/tau_ni)*d2lZ )/(1.+d2lZ/tau_ni) ds2 = ttau[ii] - ttau_old # finally rank-1 update Sigma .. si = np.reshape(Sigma[:,ii],(Sigma.shape[0],1)) Sigma = Sigma - ds2/(1.+ds2*si[ii])*np.dot(si,si.T) # takes 70# of total time mu = np.dot(Sigma,tnu) # .. and recompute mu # recompute since repeated rank-one updates can destroy numerical precision Sigma, mu, nlZ, L = self.epComputeParams(K, y, ttau, tnu, likfunc, m, inffunc) if sweep == max_sweep: pass # print '[warning] maximum number of sweeps reached in function infEP' self.last_ttau = ttau; self.last_tnu = tnu # remember for next call sW = np.sqrt(ttau); alpha = tnu-sW*solve_chol(L,sW*np.dot(K,tnu)) post = postStruct() post.alpha = alpha # return the posterior params post.sW = sW post.L = L if nargout>2: # do we want derivatives? dnlZ = dnlZStruct(meanfunc, covfunc, likfunc) # allocate space for derivatives ssi = np.sqrt(ttau) V = np.linalg.solve(L.T,np.tile(ssi,(1,n))*K) Sigma = K - np.dot(V.T,V) mu = np.dot(Sigma,tnu) Dsigma = np.reshape(np.diag(Sigma),(np.diag(Sigma).shape[0],1)) tau_n = 1/Dsigma-ttau # compute the log marginal likelihood nu_n = mu/Dsigma-tnu # vectors of cavity parameters F = np.dot(alpha,alpha.T) - np.tile(sW,(1,n))* \ solve_chol(L,np.diag(np.reshape(sW,(sW.shape[0],)))) # covariance hypers for jj in range(len(covfunc.hyp)): dK = covfunc.getDerMatrix(x=x, mode='train', der=jj) dnlZ.cov[jj] = -(F*dK).sum()/2. for ii in range(len(likfunc.hyp)): dlik = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc, ii) dnlZ.lik[ii] = -dlik.sum() junk,dlZ = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc, None, 2) # mean hyps for ii in range(len(meanfunc.hyp)): dm = meanfunc.getDerMatrix(x, ii) dnlZ.mean[ii] = -np.dot(dlZ.T,dm) dnlZ.mean[ii] = dnlZ.mean[ii][0,0] return post, nlZ[0], dnlZ else: return post, nlZ[0] class FITC_EP(Inference): ''' FITC-EP approximation to the posterior Gaussian process. The function is equivalent to infEP with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Kuu + snu2 * eye(nu)) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu and snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard deviation of the inducing inputs snu to be a one per mil of the measurement noise's standard deviation sn. In case of a likelihood without noise parameter sn2, we simply use snu2 = 1e-6. For details, see The Generalized FITC Approximation, Andrew Naish-Guzman and Sean Holden, NIPS, 2007. ''' def __init__(self): self.name = 'FITC Expectation Propagation' self.last_ttau = None self.last_tnu = None def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): if not isinstance(covfunc, cov.FITCOfKernel): raise Exception('Only covFITC supported.') # check cov tol = 1e-4; max_sweep = 10; min_sweep = 2 # tolerance to stop EP iterations inffunc = EP() diagK,Kuu,Ku = covfunc.getCovMatrix(x=x, mode='train') # evaluate the covariance matrix m = meanfunc.getMean(x) # evaluate the mean vector if likfunc.hyp: # hard coded inducing inputs noise sn2 = np.exp(2.*likfunc.hyp[-1]) snu2 = 1.e-6*sn2 # similar to infFITC else: snu2 = 1.e-6 n, D = x.shape; nu = Kuu.shape[0] rot180 = lambda A: np.rot90(np.rot90(A)) # little helper functions chol_inv = lambda A: np.linalg.solve( rot180( np.linalg.cholesky(rot180(A)) ),np.eye(nu)) # chol(inv(A)) R0 = chol_inv(Kuu+snu2*np.eye(nu)) # initial R, used for refresh O(nu^3) V = np.dot(R0,Ku); d0 = diagK - np.array([(V*V).sum(axis=0)]).T # initial d, needed for refresh O(n*nu^2) # A note on naming: variables are given short but descriptive names in # accordance with Rasmussen & Williams "GPs for Machine Learning" (2006): mu # and s2 are mean and variance, nu and tau are natural parameters. A leading t # means tilde, a subscript _ni means "not i" (for cavity parameters), or _n # for a vector of cavity parameters. # marginal likelihood for ttau = tnu = zeros(n,1); equals n*log(2) for likCum* nlZ0 = -1.* likfunc.proceed(y, m, np.reshape(diagK,(diagK.shape[0],1)), inffunc).sum() if self.last_ttau == None: # find starting point for tilde parameters ttau = np.zeros((n,1)) # initialize to zero if we have no better guess tnu = np.zeros((n,1)) [d,P,R,nn,gg] = self.epfitcRefresh(d0,Ku,R0,V,ttau,tnu) # compute initial repres. nlZ = nlZ0 else: ttau = self.last_ttau # try the tilde values from previous call tnu = self.last_tnu [d,P,R,nn,gg] = self.epfitcRefresh(d0,Ku,R0,V,ttau,tnu) # compute initial repres. nlZ = self.epfitcZ(d,P,R,nn,gg,ttau,tnu,d0,R0,Ku,y,likfunc,m,inffunc)[0] if nlZ > nlZ0: # if zero is better .. ttau = np.zeros((n,1)) # .. then initialize with zero instead tnu = np.zeros((n,1)) [d,P,R,nn,gg] = self.epfitcRefresh(d0,Ku,R0,V,ttau,tnu) # initial repres. nlZ = nlZ0 nlZ_old = np.inf; sweep = 0 # converged, max. sweeps or min. sweeps? while (np.abs(nlZ-nlZ_old) > tol and sweep < max_sweep) or (sweep < min_sweep): nlZ_old = nlZ sweep += 1 rperm = range(n) # randperm(n) for ii in rperm: # iterate EP updates (in random order) over examples p_i = np.reshape(P[:,ii],(P.shape[0],1)) t = np.dot(R,np.dot(R0,p_i)) # temporary variables sigma_i = d[ii] + np.dot(t.T,t); mu_i = nn[ii] + np.dot(p_i.T,gg) # post moments O(nu^2) tau_ni = 1/sigma_i - ttau[ii] # first find the cavity distribution .. nu_ni = mu_i/sigma_i + m[ii]*tau_ni - tnu[ii] # .. params tau_ni and nu_ni # compute the desired derivatives of the indivdual log partition function vargout = likfunc.proceed(y[ii], nu_ni/tau_ni, 1/tau_ni, inffunc, None, 3) lZ = vargout[0]; dlZ = vargout[1]; d2lZ = vargout[2] ttau_i = -d2lZ /(1.+d2lZ/tau_ni) ttau_i = max(ttau_i,0) # enforce positivity i.e. lower bound ttau by zero tnu_i = ( dlZ + (m[ii]-nu_ni/tau_ni)*d2lZ )/(1.+d2lZ/tau_ni) [d,P[:,ii],R,nn,gg,ttau,tnu] = self.epfitcUpdate(d,P[:,ii],R,nn,gg,ttau,tnu,ii,ttau_i,tnu_i,m,d0,Ku,R0)# update representation # recompute since repeated rank-one updates can destroy numerical precision [d,P,R,nn,gg] = self.epfitcRefresh(d0,Ku,R0,V,ttau,tnu) [nlZ,nu_n,tau_n] = self.epfitcZ(d,P,R,nn,gg,ttau,tnu,d0,R0,Ku,y,likfunc,m,inffunc) if sweep == max_sweep: pass # print '[warning] maximum number of sweeps reached in function infEP' self.last_ttau = ttau self.last_tnu = tnu # remember for next call post = postStruct() post.sW = np.sqrt(ttau) # unused for FITC_EP prediction with gp.m dd = 1/(d0+1/ttau) alpha = tnu/ttau*dd RV = np.dot(R,V) R0tV = np.dot(R0.T,V) alpha = alpha - np.dot(RV.T,np.dot(RV,alpha))*dd # long alpha vector for ordinary infEP post.alpha = np.dot(R0tV,alpha) # alpha = R0'*V*inv(Kt+diag(1./ttau))*(tnu./ttau) B = R0tV*np.tile(dd.T,(nu,1)); L = np.dot(B,R0tV.T); B = np.dot(B,RV.T) post.L = np.dot(B,B.T) - L # L = -R0'*V*inv(Kt+diag(1./ttau))*V'*R0 if nargout>2: # do we want derivatives? dnlZ = dnlZStruct(meanfunc, covfunc, likfunc) # allocate space for derivatives RVdd = RV*np.tile(dd.T,(nu,1)) for ii in range(len(covfunc.hyp)): ddiagK,dKuu,dKu = covfunc.getDerMatrix(x=x, mode='train', der=ii) dA = 2*dKu.T - np.dot(R0tV.T,dKuu) # dQ = dA*R0tV w = np.atleast_2d((dA*R0tV.T).sum(axis=1)).T # w = diag(dQ) v = ddiagK - w # v = diag(dK)-diag(dQ) z = np.dot(dd.T,(v+w)) - np.dot(np.atleast_2d((RVdd*RVdd).sum(axis=0)), v) \ - (np.dot(RVdd,dA).T * np.dot(R0tV,RVdd.T)).sum() dnlZ.cov[ii] = (z - np.dot(alpha.T,(alpha*v)) - np.dot(np.dot(alpha.T,dA),np.dot(R0tV,alpha)))/2. dnlZ.cov[ii] = dnlZ.cov[ii][0,0] for ii in range(len(likfunc.hyp)): # likelihood hypers dlik = likfunc.proceed(y, nu_n/tau_n+m, 1/tau_n, inffunc, ii, 1) dnlZ.lik[ii] = -dlik.sum() if ii == len(likfunc.hyp)-1: # since snu2 is a fixed fraction of sn2, there is a covariance-like term # in the derivative as well v = np.atleast_2d((R0tV*R0tV).sum(axis=0)).T z = (np.dot(RVdd,R0tV.T)**2).sum() - np.dot(np.atleast_2d((RVdd*RVdd).sum(axis=0)),v) z = z + np.dot(post.alpha.T,post.alpha) - np.dot(alpha.T,(v*alpha)) dnlZ.lik[ii] += snu2*z dnlZ.lik[ii] = dnlZ.lik[ii][0,0] [junk,dlZ] = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc, None, 2) # mean hyps for ii in range(len(meanfunc.hyp)): dm = meanfunc.getDerMatrix(x, ii) dnlZ.mean[ii] = -np.dot(dlZ.T,dm) dnlZ.mean[ii] = dnlZ.mean[ii][0,0] return post, nlZ[0,0], dnlZ else: return post, nlZ[0,0] if __name__ == '__main__': pass }rqM(XInference.mvmKXdefKKXInference.mvmZXdefKKXpostStruct.__init__XdefKUKZXExact.__init__XdefMM X postStructXclassKQKZXdnlZStruct.__init__XdefK^KiX Exact.proceedXdefM MAXFITC_Exact.proceedXdefMLMXInference.__init__XdefKmKpXInference.Psi_lineFITCXdefKMXInference.epComputeParamsXdefKKXFITC_Laplace.proceedXdefMMXLaplaceXclassMMXFITC_Laplace.__init__XdefMMX FITC_ExactXclassMAMXInference.logdetAXdefKKXFITC_EP.proceedXdefMMjXExactXclassMMAX FITC_LaplaceXclassMMXFITC_EPXclassMMjXLaplace.proceedXdefMMXInference.fitcRefreshXdefMMXInference.epfitcUpdateXdefKKXInference.proceedXdefKpKXInference.Psi_lineXdefKKXInference.epfitcRefreshXdefKKއXEPXclassMMX dnlZStructXclassKZKiXFITC_Exact.__init__XdefMIMLX InferenceXclassKiMXLaplace.__init__XdefMMX EP.proceedXdefMMXFITC_EP.__init__XdefMMXInference.epfitcZXdefKK͇X EP.__init__XdefMMu}rrM(XInference.epfitcRefreshjcMXEPhX FITC_LaplacehX InferencejcMX dnlZStructjcMXInference.Psi_lineFITCjcMXInference.mvmZjcMXInference.Psi_linejcMXFITC_EPhX postStructjcMXInference.proceedjcMX FITC_ExacthXInference.logdetAjcMXInference.mvmKjcMXInference.epfitcZjcMXExacthXLaplacehXInference.fitcRefreshjcMuj]MX pyGPs.Core.gprsMrtM}ruMbX#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ # MEANING OF NOTATION: # # inffunc function specifying the inference method # covfunc prior covariance function (see below) # meanfunc prior mean function # likfunc likelihood function # x n by D matrix of training inputs # y column vector of length n of training targets # xs n by D matrix of test inputs # ys column vector of length nn of true test targets (optional) # nlZ returned value of the negative log marginal likelihood # dnlZ column vector of partial derivatives of the negative # log marginal likelihood w.r.t. each hyperparameter # ym column vector (of length ns) of predictive output means # ys2 column vector (of length ns) of predictive output variances # fm column vector (of length ns) of predictive latent means # fs2 column vector (of length ns) of predictive latent variances # lp column vector (of length ns) of log predictive probabilities # post struct representation of the (approximate) posterior # post consists of post.alpha, post.L, post.sW # # This is a object-oriented python implementation of gpml functionality # (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18). # based on the functional-version of python implementation # (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013) # # Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013 import itertools import numpy as np import matplotlib.pyplot as plt import inf, mean, lik, cov, opt from tools import unique from copy import deepcopy import pyGPs SHADEDCOLOR = [0.7539, 0.89453125, 0.62890625, 1.0] MEANCOLOR = [ 0.2109375, 0.63385, 0.1796875, 1.0] DATACOLOR = [0.12109375, 0.46875, 1., 1.0] class GP(object): '''Base class for GP model''' def __init__(self): super(GP, self).__init__() self.usingDefaultMean = True # was using default mean function now? self.meanfunc = None self.covfunc = None self.likfunc = None self.inffunc = None self.optimizer = None self.nlZ = None # negative log marginal likelihood self.dnlZ = None # column vector of partial derivatives of the negative # log marginal likelihood w.r.t. each hyperparameter self.posterior = None # struct representation of the (approximate) posterior self.x = None # n by D matrix of training inputs self.y = None # column vector of length n of training targets self.xs = None # n by D matrix of test inputs self.ys = None # column vector of length nn of true test targets (optional) self.ym = None # column vector (of length ns) of predictive output means self.ys2 = None # column vector (of length ns) of predictive output variances self.fm = None # column vector (of length ns) of predictive latent means self.fs2 = None # column vector (of length ns) of predictive latent variances self.lp = None # column vector (of length ns) of log predictive probabilities def setData(self, x, y): '''Pass training data and traning labels to model.''' if x.ndim == 1: x = np.reshape(x, (x.shape[0],1)) if y.ndim == 1: y = np.reshape(y, (y.shape[0],1)) self.x = x self.y = y if self.usingDefaultMean: c = np.mean(y) self.meanfunc = mean.Const(c) # adapt default prior mean wrt. training labels def plotData_1d(self, axisvals=None): '''Plot 1d (toy) data.''' plt.figure() plt.plot(self.x, self.y, ls='None', marker='+', color=DATACOLOR, ms=12, mew=2) if axisvals: plt.axis(axisvals) plt.grid() plt.xlabel('input x') plt.ylabel('target y') plt.show() def plotData_2d(self,x1,x2,t1,t2,p1,p2,axisvals=None): '''Plot 2d (toy) data.''' fig = plt.figure() plt.plot(x1[:,0], x1[:,1], 'b+', markersize = 12) plt.plot(x2[:,0], x2[:,1], 'r+', markersize = 12) pc = plt.contour(t1, t2, np.reshape(p2/(p1+p2), (t1.shape[0],t1.shape[1]) )) fig.colorbar(pc) plt.grid() if axisvals: plt.axis(axisvals) plt.show() def setPrior(self, mean=None, kernel=None): '''Set prior mean and cov''' if mean != None: self.meanfunc = mean self.usingDefaultMean = False if kernel != None: self.covfunc = kernel if type(kernel) is cov.Pre: self.usingDefaultMean = False def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None): '''This method is used to sepecify optimization configuration. By default, gp uses a single run "minimize". :param method: Optimization methods. Possible values are: "Minimize" -> minimize by Carl Rasmussen (python implementation of "minimize" in GPML) "CG" -> conjugent gradient "BFGS" -> quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS) "SCG" -> scaled conjugent gradient (faster than CG) :param num_restarts: Set if you want to run mulitiple times of optimization with different initial guess. It specifys the maximum number of runs/restarts/trials. :param min_threshold: Set if you want to run mulitiple times of optimization with different initial guess. It specifys the threshold of objective function value. Stop optimization when this value is reached. :param meanRange: The range of initial guess for mean hyperparameters. e.g. meanRange = [(-2,2), (-5,5), (0,1)]. Each tuple specifys the range (low, high) of this hyperparameter, This is only the range of initial guess, during optimization process, optimal hyperparameters may go out of this range. (-5,5) for each hyperparameter by default. :param covRange: The range of initial guess for kernel hyperparameters. Usage see meanRange :param likRange: The range of initial guess for likelihood hyperparameters. Usage see meanRange ''' pass def optimize(self, x=None, y=None): ''' Train optimal hyperparameters based on training data, adjust new hyperparameters to all mean/cov/lik functions ''' if x != None: if x.ndim == 1: x = np.reshape(x, (x.shape[0],1)) self.x = x if y != None: if y.ndim == 1: y = np.reshape(y, (y.shape[0],1)) self.y = y if self.usingDefaultMean and self.meanfunc == None: c = np.mean(y) self.meanfunc = mean.Const(c) # adapt default prior mean wrt. training labels # optimize optimalHyp, optimalNlZ = self.optimizer.findMin(self.x, self.y) self.nlZ= optimalNlZ # apply optimal hyp to all mean/cov/lik functions here self.optimizer.apply_in_objects(optimalHyp) self.fit() def fit(self, x=None, y=None, der=True): ''' Fit the training data. Update negative log marginal likelihood(nlZ), partial derivatives of nlZ w.r.t. each hyperparameter(dnlZ), and struct representation of the (approximate) posterior(post), which consists of post.alpha, post.L, post.sW. ''' if x != None: if x.ndim == 1: x = np.reshape(x, (x.shape[0],1)) self.x = x if y != None: if y.ndim == 1: y = np.reshape(y, (y.shape[0],1)) self.y = y if self.usingDefaultMean and self.meanfunc == None: c = np.mean(y) self.meanfunc = mean.Const(c) # adapt default prior mean wrt. training labels # call inference method if isinstance(self.likfunc, lik.Erf): #or is instance(self.likfunc, lik.Logistic): uy = unique(self.y) ind = ( uy != 1 ) if any( uy[ind] != -1): raise Exception('You attempt classification using labels different from {+1,-1}') if not der: post, nlZ = self.inffunc.proceed(self.meanfunc, self.covfunc, self.likfunc, self.x, self.y, 2) self.nlZ = nlZ self.posterior = deepcopy(post) return nlZ, post else: post, nlZ, dnlZ = self.inffunc.proceed(self.meanfunc, self.covfunc, self.likfunc, self.x, self.y, 3) self.nlZ = nlZ self.dnlZ = deepcopy(dnlZ) self.posterior = deepcopy(post) return nlZ, dnlZ, post def predict(self, xs, ys=None): ''' Prediction according to given inputs. Get predictive output means(ym), predictive output variances(ys2), predictive latent means(fm), predictive latent variances(fs2), log predictive probabilities(lp). :param xs: test input :param ys: test target(optional) :return: ym, ys2, fm, fs2, lp ''' meanfunc = self.meanfunc covfunc = self.covfunc likfunc = self.likfunc inffunc = self.inffunc x = self.x y = self.y if xs.ndim == 1: xs = np.reshape(xs, (xs.shape[0],1)) self.xs = xs if ys != None: if ys.ndim == 1: ys = np.reshape(ys, (ys.shape[0],1)) self.ys = ys if self.posterior == None: self.fit() alpha = self.posterior.alpha L = self.posterior.L sW = self.posterior.sW nz = range(len(alpha[:,0])) # non-sparse representation if L == []: # in case L is not provided, we compute it K = covfunc.getCovMatrix(x=x[nz,:], mode='train') L = np.linalg.cholesky( (np.eye(nz) + np.dot(sW,sW.T)*K).T ) Ltril = np.all( np.tril(L,-1) == 0 ) # is L an upper triangular matrix? ns = xs.shape[0] # number of data points nperbatch = 1000 # number of data points per mini batch nact = 0 # number of already processed test data points ymu = np.zeros((ns,1)) ys2 = np.zeros((ns,1)) fmu = np.zeros((ns,1)) fs2 = np.zeros((ns,1)) lp = np.zeros((ns,1)) while nact<=ns-1: # process minibatches of test cases to save memory id = range(nact,min(nact+nperbatch,ns)) # data points to process kss = covfunc.getCovMatrix(z=xs[id,:], mode='self_test') # self-variances Ks = covfunc.getCovMatrix(x=x[nz,:], z=xs[id,:], mode='cross') # cross-covariances ms = meanfunc.getMean(xs[id,:]) N = (alpha.shape)[1] # number of alphas (usually 1; more in case of sampling) Fmu = np.tile(ms,(1,N)) + np.dot(Ks.T,alpha[nz]) # conditional mean fs|f fmu[id] = np.reshape(Fmu.sum(axis=1)/N,(len(id),1)) # predictive means if Ltril: # L is triangular => use Cholesky parameters (alpha,sW,L) V = np.linalg.solve(L.T,np.tile(sW,(1,len(id)))*Ks) fs2[id] = kss - np.array([(V*V).sum(axis=0)]).T # predictive variances else: # L is not triangular => use alternative parametrization fs2[id] = kss + np.array([(Ks*np.dot(L,Ks)).sum(axis=0)]).T # predictive variances fs2[id] = np.maximum(fs2[id],0) # remove numerical noise i.e. negative variances Fs2 = np.tile(fs2[id],(1,N)) # we have multiple values in case of sampling if ys == None: [Lp, Ymu, Ys2] = likfunc.proceed(None,Fmu[:],Fs2[:],None,None,3) else: [Lp, Ymu, Ys2] = likfunc.proceed(np.tile(ys[id],(1,N)), Fmu[:], Fs2[:],None,None,3) lp[id] = np.reshape( np.reshape(Lp,(np.prod(Lp.shape),N)).sum(axis=1)/N , (len(id),1) ) # log probability; sample averaging ymu[id] = np.reshape( np.reshape(Ymu,(np.prod(Ymu.shape),N)).sum(axis=1)/N ,(len(id),1) ) # predictive mean ys|y and ... ys2[id] = np.reshape( np.reshape(Ys2,(np.prod(Ys2.shape),N)).sum(axis=1)/N , (len(id),1) ) # .. variance nact = id[-1]+1 # set counter to index of next data point self.ym = ymu self.ys2 = ys2 self.lp = lp self.fm = fmu self.fs2 = fs2 if ys == None: return ymu, ys2, fmu, fs2, None else: return ymu, ys2, fmu, fs2, lp def predict_with_posterior(self, post, xs, ys=None): ''' Prediction with provided posterior (i.e. you already have the posterior and thus don't need fitting/training phases) Get predictive output means(ym), predictive output variances(ys2), predictive latent means(fm), predictive latent variances(fs2), log predictive probabilities(lp). :param post: struct representation of posterior :param xs: test input :param ys: test target(optional) :return: ym, ys2, fm, fs2, lp ''' meanfunc = self.meanfunc covfunc = self.covfunc likfunc = self.likfunc inffunc = self.inffunc x = self.x y = self.y if xs.ndim == 1: xs = np.reshape(xs, (xs.shape[0],1)) self.xs = xs if ys != None: if ys.ndim == 1: ys = np.reshape(ys, (ys.shape[0],1)) self.ys = ys self.posterior = deepcopy(post) alpha = post.alpha L = post.L sW = post.sW nz = range(len(alpha[:,0])) # non-sparse representation if L == []: # in case L is not provided, we compute it K = covfunc.getCovMatrix(x=x[nz,:], mode='train') L = np.linalg.cholesky( (np.eye(nz) + np.dot(sW,sW.T)*K).T ) Ltril = np.all( np.tril(L,-1) == 0 ) # is L an upper triangular matrix? ns = xs.shape[0] # number of data points nperbatch = 1000 # number of data points per mini batch nact = 0 # number of already processed test data points ymu = np.zeros((ns,1)) ys2 = np.zeros((ns,1)) fmu = np.zeros((ns,1)) fs2 = np.zeros((ns,1)) lp = np.zeros((ns,1)) while nact<=ns-1: # process minibatches of test cases to save memory id = range(nact,min(nact+nperbatch,ns)) # data points to process kss = covfunc.getCovMatrix(z=xs[id,:], mode='self_test') # self-variances Ks = covfunc.getCovMatrix(x=x[nz,:], z=xs[id,:], mode='cross') # cross-covariances ms = meanfunc.getMean(xs[id,:]) N = (alpha.shape)[1] # number of alphas (usually 1; more in case of sampling) Fmu = np.tile(ms,(1,N)) + np.dot(Ks.T,alpha[nz]) # conditional mean fs|f fmu[id] = np.reshape(Fmu.sum(axis=1)/N,(len(id),1)) # predictive means if Ltril: # L is triangular => use Cholesky parameters (alpha,sW,L) V = np.linalg.solve(L.T,np.tile(sW,(1,len(id)))*Ks) fs2[id] = kss - np.array([(V*V).sum(axis=0)]).T # predictive variances else: # L is not triangular => use alternative parametrization fs2[id] = kss + np.array([(Ks*np.dot(L,Ks)).sum(axis=0)]).T # predictive variances fs2[id] = np.maximum(fs2[id],0) # remove numerical noise i.e. negative variances Fs2 = np.tile(fs2[id],(1,N)) # we have multiple values in case of sampling if ys == None: [Lp, Ymu, Ys2] = likfunc.proceed(None,Fmu[:],Fs2[:],None,None,3) else: [Lp, Ymu, Ys2] = likfunc.proceed(np.tile(ys[id],(1,N)), Fmu[:], Fs2[:],None,None,3) lp[id] = np.reshape( np.reshape(Lp,(np.prod(Lp.shape),N)).sum(axis=1)/N , (len(id),1) ) # log probability; sample averaging ymu[id] = np.reshape( np.reshape(Ymu,(np.prod(Ymu.shape),N)).sum(axis=1)/N ,(len(id),1) ) # predictive mean ys|y and ... ys2[id] = np.reshape( np.reshape(Ys2,(np.prod(Ys2.shape),N)).sum(axis=1)/N , (len(id),1) ) # .. variance nact = id[-1]+1 # set counter to index of next data point self.ym = ymu self.ys2 = ys2 self.lp = lp self.fm = fmu self.fs2 = fs2 if ys == None: return ymu, ys2, fmu, fs2, None else: return ymu, ys2, fmu, fs2, lp class GPR(GP): '''Gaussian Process Regression''' def __init__(self): super(GPR, self).__init__() self.meanfunc = mean.Zero() # default prior mean self.covfunc = cov.RBF() # default prior covariance self.likfunc = lik.Gauss() # likihood with default noise variance 0.1 self.inffunc = inf.Exact() # inference method self.optimizer = opt.Minimize(self) # default optimizer def setNoise(self,log_sigma): '''Set noise variance other than default''' self.likfunc = lik.Gauss(log_sigma) def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None): '''Set Optimizer. See base class.''' conf = None if (num_restarts!=None) or (min_threshold!=None): conf = pyGPs.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc) conf.num_restarts = num_restarts conf.min_threshold = min_threshold if meanRange != None: conf.meanRange = meanRange if covRange != None: conf.covRange = covRange if likRange != None: conf.likRange = likRange if method == "Minimize": self.optimizer = opt.Minimize(self,conf) elif method == "SCG": self.optimizer = opt.SCG(self,conf) elif method == "CG": self.optimizer = opt.CG(self,conf) elif method == "BFGS": self.optimizer = opt.BFGS(self,conf) def plot(self,axisvals=None): '''Plot 1d GP regression.''' xs = self.xs x = self.x y = self.y ym = self.ym ys2 = self.ys2 plt.figure() xss = np.reshape(xs,(xs.shape[0],)) ymm = np.reshape(ym,(ym.shape[0],)) ys22 = np.reshape(ys2,(ys2.shape[0],)) plt.plot(x, y, color=DATACOLOR, ls='None', marker='+',ms=12, mew=2) plt.plot(xs, ym, color=MEANCOLOR, ls='-', lw=3.) plt.fill_between(xss,ymm + 2.*np.sqrt(ys22), ymm - 2.*np.sqrt(ys22), facecolor=SHADEDCOLOR,linewidths=0.0) plt.grid() if axisvals != None: plt.axis(axisvals) plt.xlabel('input x') plt.ylabel('target y') plt.show() def useInference(self, newInf): ''' Use another inference techinique other than default exact inference. :param str newInf: 'Laplace' or 'EP' ''' if newInf == "Laplace": self.inffunc = inf.Laplace() elif newInf == "EP": self.inffunc = inf.EP() else: raise Exception('Possible inf values are "Laplace", "EP".') def useLikelihood(self,newLik): if newLik == "Laplace": self.likfunc = lik.Laplace() self.inffunc = inf.EP() else: raise Exception('Possible lik values are "Laplace".') class GPC(GP): '''Gaussian Process Classification''' def __init__(self): super(GPC, self).__init__() self.meanfunc = mean.Zero() # default prior mean self.covfunc = cov.RBF() # default prior covariance self.likfunc = lik.Erf() # erf likihood self.inffunc = inf.EP() # default inference method self.optimizer = opt.Minimize(self) # default optimizer def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None): '''Set Optimizer. See base class.''' conf = None if (num_restarts!=None) or (min_threshold!=None): conf = pyGPs.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc) conf.num_restarts = num_restarts conf.min_threshold = min_threshold if meanRange != None: conf.meanRange = meanRange if covRange != None: conf.covRange = covRange if likRange != None: conf.likRange = likRange if method == "Minimize": self.optimizer = opt.Minimize(self,conf) elif method == "SCG": self.optimizer = opt.SCG(self,conf) elif method == "CG": self.optimizer = opt.CG(self,conf) elif method == "BFGS": self.optimizer = opt.BFGS(self,conf) def plot(self,x1,x2,t1,t2,axisvals=None): '''Plot 2d gp classification.''' fig = plt.figure() plt.plot(x1[:,0], x1[:,1], 'b+', markersize = 12) plt.plot(x2[:,0], x2[:,1], 'r+', markersize = 12) pc = plt.contour(t1, t2, np.reshape(np.exp(self.lp), (t1.shape[0],t1.shape[1]) )) fig.colorbar(pc) plt.grid() if axisvals != None: plt.axis(axisvals) plt.show() def useInference(self, newInf): ''' Use another inference techinique other than default EP inference. :param str newInf: 'Laplace' ''' if newInf == "Laplace": self.inffunc = inf.Laplace() else: raise Exception('Possible inf values are "Laplace".') def useLikelihood(self,newLik): ''' Use another likelihood function other than default Erf inference. (Not used in this version) :param str newLik: 'Logistic' ''' if newLik == "Logistic": pass #self.likfunc = lik.Logistic() else: raise Exception('Possible lik values are "Logistic".') class GPMC(object): '''This is a one vs. one classification wrapper for GPC''' def __init__(self, n_class): self.meanfunc = mean.Zero() # default prior mean self.covfunc = cov.RBF() # default prior covariance self.n_class = n_class # number of different classes self.x_all = None self.y_all = None self.newInf = None # new inference? -> call useInference self.newLik = None # new likelihood? -> call useLikelihood self.newPrior = False def setPrior(self, mean=None, kernel=None): '''set prior mean function and covariance function''' if mean != None: self.meanfunc = mean if kernel != None: self.covfunc = kernel if type(kernel) is cov.Pre: self.usingDefaultMean = False self.newPrior = True def useInference(self, newInf): '''Use another inference techinique other than default EP inference.''' self.newInf = newInf def useLikelihood(self,newLik): '''Use another likelihood function other than default Erf inference.''' self.newLik = newLik def setData(self,x,y): '''Set data for multi-class model''' if x.ndim == 1: x = np.reshape(x, (x.shape[0],1)) if y.ndim == 1: y = np.reshape(y, (y.shape[0],1)) self.x_all = x self.y_all = y def fitAndPredict(self, xs): ''' Fitting model and prediction. predictive_vote is a matrix where row i is each test point i column j is the probability for being class j :return: predictive_vote ''' if xs.ndim == 1: xs = np.reshape(xs, (xs.shape[0],1)) predictive_vote = np.zeros((xs.shape[0],self.n_class)) for i in xrange(self.n_class): # classifier for class i... for j in xrange(i+1,self.n_class): # ...and class j x,y = self.createBinaryClass(i,j) model = GPC() if self.newPrior: model.setPrior(mean=self.meanfunc, kernel=self.covfunc) if self.newInf: model.useInference(self.newInf) if self.newLik: model.useLikelihood(self.newLik) model.fit(x,y) # fitting ym = model.predict(xs)[0] ym += 1 # now scale into 0 to 2, ym=0 is class j, ym=2 is class i vote_i = np.zeros((xs.shape[0],self.n_class)) vote_j = np.zeros((xs.shape[0],self.n_class)) vote_i[:,i:i+1] = ym vote_j[:,j:j+1] = 2-ym predictive_vote += vote_i predictive_vote += vote_j predictive_vote /= predictive_vote.sum(axis=1)[:,np.newaxis] return predictive_vote def optimizeAndPredict(self, xs): ''' Optimizing model and prediction. predictive_vote is a matrix where row i is each test point i column j is the probability for being class j :return: predictive_vote ''' if xs.ndim == 1: xs = np.reshape(xs, (xs.shape[0],1)) predictive_vote = np.zeros((xs.shape[0],self.n_class)) for i in xrange(self.n_class): # classifier for class i... for j in xrange(i+1,self.n_class): # ...and class j x,y = self.createBinaryClass(i,j) model = GPC() if self.newPrior: model.setPrior(mean=self.meanfunc, kernel=self.covfunc) if self.newInf: model.useInference(self.newInf) if self.newLik: model.useLikelihood(self.newLik) model.optimize(x,y) # training ym = model.predict(xs)[0] ym += 1 # now scale into 0 to 2, ym=0 is class j, ym=2 is class i vote_i = np.zeros((xs.shape[0],self.n_class)) vote_j = np.zeros((xs.shape[0],self.n_class)) vote_i[:,i:i+1] = ym vote_j[:,j:j+1] = 2-ym predictive_vote += vote_i predictive_vote += vote_j predictive_vote /= predictive_vote.sum(axis=1)[:,np.newaxis] return predictive_vote def createBinaryClass(self, i,j): ''' Create dataset x(data) and y(label) which only contains class i and j. Relabel class i to +1 and class j to -1''' class_i = [] class_j = [] for index in xrange(len(self.y_all)): # check all classes target = self.y_all[index] if target == i: class_i.append(index) elif target == j: class_j.append(index) n1 = len(class_i) n2 = len(class_j) class_i.extend(class_j) x = self.x_all[class_i,:] y = np.concatenate((np.ones((1,n1)),-np.ones((1,n2))),axis=1).T return x,y class GP_FITC(GP): '''FITC GP base class''' def __init__(self): super(GP_FITC, self).__init__() self.u = None # overriding def setData(self, x, y, value_per_axis=5): '''Set training data and derive deault inducing_points. :param x: training data :param y: training target :param int value_per_axis: number of value in each dimension when using a uni-distant default inducing points''' if x.ndim == 1: x = np.reshape(x, (x.shape[0],1)) if y.ndim == 1: y = np.reshape(y, (y.shape[0],1)) self.x = x self.y = y if self.usingDefaultMean: c = np.mean(y) self.meanfunc = mean.Const(c) # adapt default prior mean wrt. training labels # get range of x in each dimension # 5 uniformally selected value for each dimension gridAxis=[] for d in xrange(x.shape[1]): column = x[:,d] mini = np.min(column) maxi = np.max(column) axis = np.linspace(mini,maxi,value_per_axis) gridAxis.append(axis) # default inducing points-> a grid if self.u == None: self.u = np.array(list(itertools.product(*gridAxis))) self.covfunc = self.covfunc.fitc(self.u) # overriding def setPrior(self, mean=None, kernel=None, inducing_points=None): '''Set GP prior and inducing points''' if kernel != None: if inducing_points != None: self.covfunc = kernel.fitc(inducing_points) self.u = inducing_points else: if self.u != None: self.covfunc = kernel.fitc(self.u) else: raise error("To use default inducing points, please call setData() first!") if type(kernel) is cov.Pre: self.usingDefaultMean = False if mean != None: self.meanfunc = mean self.usingDefaultMean = False class GPR_FITC(GP_FITC): '''Gaussian Process Regression FITC''' def __init__(self): super(GPR_FITC, self).__init__() self.meanfunc = mean.Zero() # default prior mean self.covfunc = cov.RBF() # default prior covariance self.likfunc = lik.Gauss() # likihood with default noise variance 0.1 self.inffunc = inf.FITC_Exact() # inference method self.optimizer = opt.Minimize(self) # default optimizer self.u = None # no default inducing points def setNoise(self,log_sigma): '''Set noise variance other than default value''' self.likfunc = lik.Gauss(log_sigma) def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None): '''Set Optimizer. See base class.''' conf = None if (num_restarts!=None) or (min_threshold!=None): conf = pyGPs.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc) conf.num_restarts = num_restarts conf.min_threshold = min_threshold if meanRange != None: conf.meanRange = meanRange if covRange != None: conf.covRange = covRange if likRange != None: conf.likRange = likRange if method == "Minimize": self.optimizer = opt.Minimize(self,conf) elif method == "SCG": self.optimizer = opt.SCG(self,conf) elif method == "CG": self.optimizer = opt.CG(self,conf) elif method == "BFGS": self.optimizer = opt.BFGS(self,conf) def plot(self,axisvals=None): '''Plot 1d GP FITC regression.''' plt.figure() xss = np.reshape(self.xs,(self.xs.shape[0],)) ymm = np.reshape(self.ym,(self.ym.shape[0],)) ys22 = np.reshape(self.ys2,(self.ys2.shape[0],)) plt.plot(self.x, self.y, color=DATACOLOR, ls='None', marker='+',ms=12, mew=2) plt.plot(self.xs, self.ym, color=MEANCOLOR, ls='-', lw=3.) plt.fill_between(xss,ymm + 2.*np.sqrt(ys22), ymm - 2.*np.sqrt(ys22), facecolor=SHADEDCOLOR,linewidths=0.0) plt.grid() if axisvals != None: plt.axis(axisvals) plt.xlabel('input x') plt.ylabel('output y') plt.plot(self.u,np.ones_like(self.u), ls='None', color='k',marker='x',markersize=12,mew=2) plt.show() def useInference(self, newInf): ''' Use another inference techinique other than default exact inference. :param str newInf: 'Laplace' or 'EP' ''' if newInf == "Laplace": self.inffunc = inf.FITC_Laplace() elif newInf == "EP": self.inffunc = inf.FITC_EP() else: raise Exception('Possible inf values are "Laplace", "EP".') def useLikelihood(self,newLik): ''' Use another inference techinique other than default Gaussian likelihood. :param str newLik: 'Laplace' ''' if newLik == "Laplace": self.likfunc = lik.Laplace() self.inffunc = inf.FITC_EP() else: raise Exception('Possible lik values are "Laplace".') class GPC_FITC(GP_FITC): '''Gaussian Process Classification FITC''' def __init__(self): super(GPC_FITC, self).__init__() self.meanfunc = mean.Zero() # default prior mean self.covfunc = cov.RBF() # default prior covariance self.likfunc = lik.Erf() # erf liklihood self.inffunc = inf.FITC_EP() # default inference method self.optimizer = opt.Minimize(self) # default optimizer self.u = None # no default inducing points def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None): '''Set optimizer. See base class.''' conf = None if (num_restarts!=None) or (min_threshold!=None): conf = pyGPs.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc) conf.num_restarts = num_restarts conf.min_threshold = min_threshold if meanRange != None: conf.meanRange = meanRange if covRange != None: conf.covRange = covRange if likRange != None: conf.likRange = likRange if method == "Minimize": self.optimizer = opt.Minimize(self,conf) elif method == "SCG": self.optimizer = opt.SCG(self,conf) elif method == "CG": self.optimizer = opt.CG(self,conf) elif method == "BFGS": self.optimizer = opt.BFGS(self,conf) def plot(self,x1,x2,t1,t2,axisvals=None): '''Plot 2d GP FITC classification.''' fig = plt.figure() plt.plot(x1[:,0], x1[:,1], 'b+', markersize = 12) plt.plot(x2[:,0], x2[:,1], 'r+', markersize = 12) plt.plot(self.u[:,0],self.u[:,1],'ko', markersize=12) pc = plt.contour(t1, t2, np.reshape(np.exp(self.lp), (t1.shape[0],t1.shape[1]) )) fig.colorbar(pc) plt.grid() if axisvals != None: plt.axis(axisvals) plt.show() def useInference(self, newInf): ''' Use another inference techinique other than default exact inference. :param str newInf: 'Laplace' or 'EP' ''' if newInf == "Laplace": self.inffunc = inf.FITC_Laplace() else: raise Exception('Possible inf values are "Laplace".') def useLikelihood(self,newLik): ''' Use another inference techinique other than default Erf likelihood. 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A = A.astype(float) ## INITIALIZE kernel matrix K = np.ndarray(shape=(num_graphs,num_graphs,h_max+1), dtype=float) ## INITIALIZE label probabilities of labeled nodes (num_nodes x num_labels) if l.shape[1]==1: lab_prob = np.zeros((num_nodes,num_labels),dtype=np.float64) for i in xrange(num_nodes): if l[i,0] > 0: lab_prob[i,l[i,0]-1] = 1 else: lab_prob = l if spsp.issparse(lab_prob): lab_prob = lab_prob.todense() lab_orig = lab_prob.copy() # lab_orig is the orignal label information ## GET indices of labeled and unlabeled nodes idx_unlab = np.where(np.max(lab_prob, axis=1)!=1)[0] idx_all = np.array(np.arange(0,A.shape[0]), ndmin=2) idx = np.array(np.setdiff1d(idx_all,idx_unlab), ndmin=2) ## INITIALIZE unlabeled/uninitialized nodes UNIFORMLY idx_unif = np.where(np.sum(lab_prob, axis=1)==0)[0] if idx_unif.shape[0] != 0: lab_prob[idx_unif,:] = 1./lab_prob.shape[1] ## row normalize A -> transition matrix T if (ktype=='label_propagation') or (ktype=='label_diffusion'): row_sums = np.array(A.sum(axis=1))[:,0] T = A.copy() T.data /= row_sums[A.nonzero()[0]] # so A and T are sparse matrix #=========================================================================== # ## PROPAGATION KERNEL ITERATIONS #=========================================================================== for h in xrange(h_max+1): print 'ITERATION: ', h if h > 0: ## LABEL UPDATE if showEachStep: print '...computing LABEL UPDATE' if ktype == 'label_propagation': lab_prob[idx,:] = lab_orig[idx,:] # PUSH BACK original LABELS lab_prob = T*lab_prob elif ktype == 'label_diffusion': lab_prob = T*lab_prob elif ktype == 'label_spreading': # S = D^(-1/2)*W*D^(-1/2) # y(t+1) = alpha*S*y(t)+(1-alpha)*y(0) alpha = 0.8 # compute S diag = np.array(A.sum(axis=1)).T**(-1/2) D = spsp.lil_matrix((A.shape[0], A.shape[1]), dtype=float) # D.setdiag(diag[0,:]) D = D.tocsr() S = D*A*D lab_prob = np.dot((alpha*S.todense()),lab_prob) + (1-alpha)*lab_orig ## COMPUTE hashvalues if showEachStep: print '...computing hashvalues' # determine path to take depending on chosen distance use_cauchy = (p =='L1') or (p =='tv') take_sqrt = (p =='hellinger') if take_sqrt: lab_prob = np.sqrt(lab_prob) # generate random projection vector v = np.random.normal(size=(lab_prob.shape[1], 1)) if use_cauchy: # if X, Y are N(0, 1), then X / Y has a standard Cauchy distribution v /= np.random.normal(size=(lab_prob.shape[1], 1)) # random offset b = w * np.random.rand() # compute hashes # hashLabels is a Vector with length: number of nodes (hashvalue for respective node) hashLabels = (np.dot(lab_prob,v) + b) / w hashLabels = np.floor(hashLabels) # take floor uniqueHash, hashLabels = np.unique(hashLabels, return_inverse=True) # map to consecutive integer from 0 ## COMPUTE kernel contribution if showEachStep: print '...computing KERNEL contribution' # aggregate counts on graphs # counts is a matrix: number of graphs x number of hashlabels num_bins = len(uniqueHash) counts = np.zeros((num_graphs, num_bins)) # init counts matrix # accumulate counts of hash labels for i in xrange(num_nodes): counts[ (gr_id[i,0]-1), hashLabels[i] ] +=1 # compute base kernel (here: LINEAR kernel) K_h = np.dot(counts,counts.T) ## SUM kernel contributions if SUM: if h == 0: K[:,:,h] = K_h else: K[:,:,h] = K[:,:,h-1] + K_h else: K[:,:,h] = K_h if showEachStep: print K[:,:,h] ## VISUALIZE KERNELS if VIS: for h in range(h_max+1): K_h = K[:,:,h] plt.figure() plt.title('height:'+str(h)) imgplot = plt.imshow(K_h) imgplot.set_interpolation('nearest') plt.colorbar() plt.show() return K }r{MXpropagationKernelXdefKKs}r|MXpropagationKernelhsj]MXpyGPs.Core.covr}Mr~M}rMbXE#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ # covariance functions to be use by Gaussian process functions. There are two # different kinds of covariance functions: simple and composite: # # simple cov functions: # # Poly - Polynomial covariance function # PiecePoly - Piecewise polynomial kernel with compact support. # RBF - Squared Exponential kernel. # RBFunit - Squared Exponential kernel with unit magnitude. # RBFard - Squared Exponential kernel with Automatic Relevance Determination. # Const - Constant kernel. # Linear - Linear kernel. # LINard - Linear covariance function with Automatic Relevance Detemination. # Matern - Matern covariance function. # Periodic - Stationary kernel for a smooth periodic function. # Noise - Independent covariance function, i.e "white noise". # RQ - Rational Quadratic covariance function with isotropic distance measure. # RQard - Rational Quadratic covariance function with ARD distance measure. # Pre - Precomputed kernel matrix. # # composite covariance functions: # # ScaleOfKernel - scaled version of a covariance function # ProductOfKernel - products of covariance functions # SumOfKernel - sums of covariance functions # FITCOfKernel - Covariance function to be used together with the FITC approximation # # # This is a object-oriented python implementation of gpml functionality # (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18). # based on the functional-version of python implementation # (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013) # # Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013 import numpy as np import math import scipy.spatial.distance as spdist class Kernel(object): """ This is a base class of Kernel functions there is no computation in this class, it just defines rules about a kernel class should have each covariance function will inherit it and implement its own behaviour """ def __init__(self): self.hyp = [] self.para = [] def getCovMatrix(self,x=None,z=None,mode=None): ''' Return the specific covariance matrix according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self covariance matrix of test data(test by 1). 'train' return training covariance matrix(train by train). 'cross' return cross covariance matrix between x and z(train by test) :return: the corresponding covariance matrix ''' pass def getDerMatrix(self,x=None,z=None,mode=None,der=None): ''' Compute derivatives wrt. hyperparameters according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self derivative matrix of test data(test by 1). 'train' return training derivative matrix(train by train). 'cross' return cross derivative matrix between x and z(train by test) :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' pass # overloading def __add__(self,cov): ''' Overloading + operator. :param cov: covariance function :return: an instance of SumOfKernel ''' return SumOfKernel(self,cov) # overloading def __mul__(self,other): ''' Overloading * operator. Using * for both multiplication with scalar and product of kernels depending on the type of the two objects. :param other: covariance function as product or int/float as scalar :return: an instance of ScaleOfKernel or ProductOfKernel ''' if isinstance(other, int) or isinstance(other, float): return ScaleOfKernel(self,other) elif isinstance(other, Kernel): return ProductOfKernel(self,other) else: print "only numbers and Kernels are supported operand types for *" # overloading __rmul__ = __mul__ def fitc(self,inducingInput): ''' Covariance function to be used together with the FITC approximation. Setting FITC gp model will implicitly call this method. :return: an instance of FITCOfKernel ''' return FITCOfKernel(self,inducingInput) # can be replaced by spdist from scipy def sq_dist(self, a, b=None): '''Compute a matrix of all pairwise squared distances between two sets of vectors, stored in the row of the two matrices: a (of size n by D) and b (of size m by D).''' n = a.shape[0] D = a.shape[1] m = n if b == None: b = a.transpose() else: m = b.shape[0] b = b.transpose() C = np.zeros((n,m)) for d in range(0,D): tt = a[:,d] tt = tt.reshape(n,1) tem = np.kron(np.ones((1,m)), tt) tem = tem - np.kron(np.ones((n,1)), b[d,:]) C = C + tem * tem return C class ProductOfKernel(Kernel): '''Product of two kernel function.''' def __init__(self,cov1,cov2): self.cov1 = cov1 self.cov2 = cov2 self._hyp = cov1.hyp + cov2.hyp def sethyp(self,hyp): assert len(hyp) == len(self._hyp) len1 = len(self.cov1.hyp) self._hyp = hyp self.cov1.hyp = self._hyp[:len1] self.cov2.hyp = self._hyp[len1:] def gethyp(self): return self._hyp hyp = property(gethyp,sethyp) def getCovMatrix(self,x=None,z=None,mode=None): A = self.cov1.getCovMatrix(x,z,mode) * self.cov2.getCovMatrix(x,z,mode) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): if der < len(self.cov1.hyp): A = self.cov1.getDerMatrix(x,z,mode,der) * self.cov2.getCovMatrix(x,z,mode) elif der < len(self.hyp): der2 = der - len(self.cov1.hyp) A = self.cov2.getDerMatrix(x,z,mode,der2) * self.cov1.getCovMatrix(x,z,mode) else: raise Exception("Error: der out of range for covProduct") return A class SumOfKernel(Kernel): '''Sum of two kernel function.''' def __init__(self,cov1,cov2): self.cov1 = cov1 self.cov2 = cov2 self._hyp = cov1.hyp + cov2.hyp def sethyp(self,hyp): assert len(hyp) == len(self._hyp) len1 = len(self.cov1.hyp) self._hyp = hyp self.cov1.hyp = self._hyp[:len1] self.cov2.hyp = self._hyp[len1:] def gethyp(self): return self._hyp hyp = property(gethyp,sethyp) def getCovMatrix(self,x=None,z=None,mode=None): A = self.cov1.getCovMatrix(x,z,mode) + self.cov2.getCovMatrix(x,z,mode) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): if der < len(self.cov1.hyp): A = self.cov1.getDerMatrix(x,z,mode,der) elif der < len(self.hyp): der2 = der - len(self.cov1.hyp) A = self.cov2.getDerMatrix(x,z,mode,der2) else: raise Exception("Error: der out of range for covSum") return A class ScaleOfKernel(Kernel): '''Scale of a kernel function.''' def __init__(self,cov,scalar): self.cov = cov if cov.hyp: self._hyp = [scalar] + cov.hyp else: self._hyp = [scalar] def sethyp(self,hyp): assert len(hyp) == len(self._hyp) self._hyp = hyp self.cov.hyp = self._hyp[1:] def gethyp(self): return self._hyp hyp = property(gethyp,sethyp) def getCovMatrix(self,x=None,z=None,mode=None): sf2 = np.exp(self.hyp[0]) # scale parameter A = sf2 * self.cov.getCovMatrix(x,z,mode) # accumulate cov return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): sf2 = np.exp(self.hyp[0]) # scale parameter if der == 0: # compute derivative w.r.t. sf2 A = 2. * sf2 * self.cov.getCovMatrix(x,z,mode) else: A = sf2 * self.cov.getDerMatrix(x,z,mode,der-1) return A class FITCOfKernel(Kernel): ''' Covariance function to be used together with the FITC approximation. The function allows for more than one output argument and does not respect the interface of a proper covariance function. Instead of outputing the full covariance, it returns cross-covariances between the inputs x, z and the inducing inputs xu as needed by infFITC ''' def __init__(self,cov,inducingInput): self.inducingInput = inducingInput self.covfunc = cov self._hyp = cov.hyp def getHyp(self): return self._hyp def setHyp(self, hyp): self._hyp = hyp self.covfunc.hyp = hyp hyp = property(getHyp,setHyp) def getCovMatrix(self,x=None,z=None,mode=None): xu = self.inducingInput if x != None: try: assert(xu.shape[1] == x.shape[1]) except AssertionError: raise Exception('Dimensionality of inducing inputs must match training inputs') if mode == 'self_test': # self covariances for the test cases K = self.covfunc.getCovMatrix(z=z,mode='self_test') return K elif mode == 'train': # compute covariance matix for training set K = self.covfunc.getCovMatrix(z=x,mode='self_test') Kuu = self.covfunc.getCovMatrix(x=xu,mode='train') Ku = self.covfunc.getCovMatrix(x=xu,z=x,mode='cross') return K, Kuu, Ku elif mode == 'cross': # compute covariance between data sets x and z K = self.covfunc.getCovMatrix(x=xu,z=z,mode='cross') return K def getDerMatrix(self,x=None,z=None,mode=None,der=None): xu = self.inducingInput if x!= None: try: assert(xu.shape[1] == x.shape[1]) except AssertionError: raise Exception('Dimensionality of inducing inputs must match training inputs') if mode == 'self_test': # self covariances for the test cases K = self.covfunc.getDerMatrix(z=z,mode='self_test',der=der) return K elif mode == 'train': # compute covariance matix for training set K = self.covfunc.getDerMatrix(z=x,mode='self_test',der=der) Kuu = self.covfunc.getDerMatrix(x=xu,mode='train',der=der) Ku = self.covfunc.getDerMatrix(x=xu,z=x,mode='cross',der=der) return K, Kuu, Ku elif mode == 'cross': # compute covariance between data sets x and z K = self.covfunc.getDerMatrix(x=xu,z=z,mode='cross',der=der) return K class Poly(Kernel): ''' Polynomial covariance function. hyp = [ log_c, log_sigma ] :param log_c: inhomogeneous offset. :param log_sigma: signal deviation. :param d: degree of polynomial (not treated as hyperparameter, i.e. will not be trained). ''' def __init__(self, log_c=0., d=2, log_sigma=0. ): self.hyp = [log_c, log_sigma] self.para = [d] def getCovMatrix(self,x=None,z=None,mode=None): c = np.exp(self.hyp[0]) # inhomogeneous offset sf2 = np.exp(2.*self.hyp[1]) # signal variance ord = self.para[0] # order of polynomial if np.abs(ord-np.round(ord)) < 1e-8: # remove numerical error from format of parameter ord = int(round(ord)) assert(ord >= 1.) # only nonzero integers for ord ord = int(ord) if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.reshape(np.sum(z*z,1), (nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.dot(x,x.T) elif mode == 'cross': # compute covariance between data sets x and z A = np.dot(x,z.T) A = sf2 * (c + A)**ord return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): c = np.exp(self.hyp[0]) # inhomogeneous offset sf2 = np.exp(2.*self.hyp[1]) # signal variance ord = self.para[0] # order of polynomial if np.abs(ord-np.round(ord)) < 1e-8: # remove numerical error from format of parameter ord = int(round(ord)) assert(ord >= 1.) # only nonzero integers for ord ord = int(ord) if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.reshape(np.sum(z*z,1), (nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.dot(x,x.T) elif mode == 'cross': # compute covariance between data sets x and z A = np.dot(x,z.T) if der == 0: # compute derivative matrix wrt 1st parameter A = c * ord * sf2 * (c+A)**(ord-1) elif der == 1: # compute derivative matrix wrt 2nd parameter A = 2. * sf2 * (c + A)**ord elif der == 2: # no derivative wrt 3rd parameter A = np.zeros_like(A) # do nothing (d is not learned) else: raise Exception("Wrong derivative entry in Poly") return A class PiecePoly(Kernel): ''' Piecewise polynomial kernel with compact support. hyp = [log_ell, log_sigma] :param log_ell: characteristic length scale. :param log_sigma: signal deviation. :param v: degree v will be rounded to 0,1,2,or 3. (not treated as hyperparameter, i.e. will not be trained). ''' def __init__(self, log_ell=0., v=2, log_sigma=0. ): self.hyp = [log_ell, log_sigma] self.para = [v] def ppmax(self,A,B): return np.maximum(A,B*np.ones_like(A)) def func(self,v,r,j): if v == 0: return 1 elif v == 1: return ( 1. + (j+1) * r ) elif v == 2: return ( 1. + (j+2)*r + (j*j + 4.*j+ 3)/3.*r*r ) elif v == 3: return ( 1. + (j+3)*r + (6.*j*j+36.*j+45.)/15.*r*r + (j*j*j+9.*j*j+23.*j+15.)/15.*r*r*r ) else: raise Exception (["Wrong degree in PiecePoly. Should be 0,1,2 or 3, is " + str(v)]) def dfunc(self,v,r,j): if v == 0: return 0 elif v == 1: return ( j+1 ) elif v == 2: return ( (j+2) + 2.*(j*j+ 4.*j+ 3.)/3.*r ) elif v == 3: return ( (j+3) + 2.*(6.*j*j+36.*j+45.)/15.*r + (j*j*j+9.*j*j+23.*j+15.)/5.*r*r ) else: raise Exception (["Wrong degree in PiecePoly. Should be 0,1,2 or 3, is " + str(v)]) def pp(self,r,j,v,func): return func(v,r,j)*(self.ppmax(1-r,0)**(j+v)) def dpp(self,r,j,v,func,dfunc): return self.ppmax(1-r,0)**(j+v-1) * r * ( (j+v)*self.func(v,r,j) - self.ppmax(1-r,0) * self.dfunc(v,r,j) ) def getCovMatrix(self,x=None,z=None,mode=None): if x != None: n, D = x.shape if z != None: nn, D = z.shape ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.*self.hyp[1]) # signal variance v = self.para[0] # degree (v = 0,1,2 or 3 only) if np.abs(v-np.round(v)) < 1e-8: # remove numerical error from format of parameter v = int(round(v)) assert(int(v) in range(4)) # Only allowed degrees: 0,1,2 or 3 v = int(v) j = np.floor(0.5*D) + v + 1 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.sqrt( spdist.cdist(x/ell, x/ell, 'sqeuclidean') ) elif mode == 'cross': # compute covariance between data sets x and z A = np.sqrt( spdist.cdist(x/ell, z/ell, 'sqeuclidean') ) A = sf2 * self.pp(A,j,v,self.func) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): if x != None: n, D = x.shape if z != None: nn, D = z.shape ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.*self.hyp[1]) # signal variance v = self.para[0] # degree (v = 0,1,2 or 3 only) if np.abs(v-np.round(v)) < 1e-8: # remove numerical error from format of parameter v = int(round(v)) assert(int(v) in range(4)) # Only allowed degrees: 0,1,2 or 3 v = int(v) j = np.floor(0.5*D) + v + 1 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.sqrt( spdist.cdist(x/ell, x/ell, 'sqeuclidean') ) elif mode == 'cross': # compute covariance between data sets x and z A = np.sqrt( spdist.cdist(x/ell, z/ell, 'sqeuclidean') ) if der == 0: # compute derivative matrix wrt 1st parameter A = sf2 * self.dpp(A,j,v,self.func,self.dfunc) elif der == 1: # compute derivative matrix wrt 2nd parameter A = 2. * sf2 * self.pp(A,j,v,self.func) elif der == 2: # wants to compute derivative wrt order A = np.zeros_like(A) else: raise Exception("Wrong derivative entry in PiecePoly") return A class RBF(Kernel): ''' Squared Exponential kernel with isotropic distance measure. hyp = [log_ell, log_sigma] :param log_ell: characteristic length scale. :param log_sigma: signal deviation. ''' def __init__(self, log_ell=0., log_sigma=0.): self.hyp = [log_ell, log_sigma] def getCovMatrix(self,x=None,z=None,mode=None): ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.*self.hyp[1]) # signal variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for training set A = spdist.cdist(x/ell,x/ell,'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z A = spdist.cdist(x/ell,z/ell,'sqeuclidean') A = sf2 * np.exp(-0.5*A) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.*self.hyp[1]) # signal variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = spdist.cdist(x/ell,x/ell,'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z A = spdist.cdist(x/ell,z/ell,'sqeuclidean') if der == 0: # compute derivative matrix wrt 1st parameter A = sf2 * np.exp(-0.5*A) * A elif der == 1: # compute derivative matrix wrt 2nd parameter A = 2. * sf2 * np.exp(-0.5*A) else: raise Exception("Calling for a derivative in RBF that does not exist") return A class RBFunit(Kernel): ''' Squared Exponential kernel with isotropic distance measure with unit magnitude. i.e signal variance is always 1. hyp = [ log_ell ] :param log_ell: characteristic length scale. ''' def __init__(self, log_ell=0.): self.hyp = [log_ell] def getCovMatrix(self,x=None,z=None,mode=None): ell = np.exp(self.hyp[0]) # characteristic length scale if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = spdist.cdist(x/ell,x/ell,'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z A = spdist.cdist(x/ell,z/ell,'sqeuclidean') A = np.exp(-0.5*A) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): ell = np.exp(self.hyp[0]) # characteristic length scale if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = spdist.cdist(x/ell,x/ell,'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z A = spdist.cdist(x/ell,z/ell,'sqeuclidean') if der == 0: # compute derivative matrix wrt 1st parameter A = np.exp(-0.5*A) * A else: raise Exception("Wrong derivative index in RDFunit") return A class RBFard(Kernel): ''' Squared Exponential kernel with Automatic Relevance Determination. hyp = log_ell_list + [log_sigma] :param D: dimension of pattern. set if you want default ell, which is 1 for each dimension. :param log_ell_list: characteristic length scale for each dimension. :param log_sigma: signal deviation. ''' def __init__(self, D=None, log_ell_list=None, log_sigma=0.): if log_ell_list == None: self.hyp = [0. for i in xrange(D)] + [log_sigma] else: self.hyp = log_ell_list + [log_sigma] def getCovMatrix(self,x=None,z=None,mode=None): if x != None: n, D = x.shape if z != None: nn, D = z.shape ell = 1./np.exp(self.hyp[0:D]) # characteristic length scale sf2 = np.exp(2.*self.hyp[D]) # signal variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x tem = np.dot(np.diag(ell),x.T).T A = spdist.cdist(tem,tem,'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z A = spdist.cdist(np.dot(np.diag(ell),x.T).T,np.dot(np.diag(ell),z.T).T,'sqeuclidean') A = sf2*np.exp(-0.5*A) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): if x != None: n, D = x.shape if z != None: nn, D = z.shape ell = 1./np.exp(self.hyp[0:D]) # characteristic length scale sf2 = np.exp(2.*self.hyp[D]) # signal variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x tem = np.dot(np.diag(ell),x.T).T A = spdist.cdist(tem,tem,'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z A = spdist.cdist(np.dot(np.diag(ell),x.T).T,np.dot(np.diag(ell),z.T).T,'sqeuclidean') A = sf2*np.exp(-0.5*A) if der < D: # compute derivative matrix wrt length scale parameters if mode == 'self_test': A = A*0 elif mode == 'train': tem = np.atleast_2d(x[:,der])/ell[der] A *= spdist.cdist(tem,tem,'sqeuclidean') elif mode == 'cross': A *= spdist.cdist(np.atleast_2d(x[:,der]).T/ell[der],np.atleast_2d(z[:,der]).T/ell[der],'sqeuclidean') elif der == D: # compute derivative matrix wrt magnitude parameter A = 2.*A else: raise Exception("Wrong derivative index in RDFard") return A class Const(Kernel): ''' Constant kernel. hyp = [ log_sigma ] :param log_sigma: signal deviation. ''' def __init__(self, log_sigma=0.): self.hyp = [log_sigma] def getCovMatrix(self,x=None,z=None,mode=None): sf2 = np.exp(self.hyp[0]) # s2 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = sf2 * np.ones((nn,1)) elif mode == 'train': # compute covariance matix for dataset x n,D = x.shape A = sf2 * np.ones((n,n)) elif mode == 'cross': # compute covariance between data sets x and z n,D = x.shape nn,D = z.shape A = sf2 * np.ones((n,nn)) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): sf2 = np.exp(self.hyp[0]) # s2 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = sf2 * np.ones((nn,1)) elif mode == 'train': # compute covariance matix for dataset x n,D = x.shape A = sf2 * np.ones((n,n)) elif mode == 'cross': # compute covariance between data sets x and z n,D = x.shape nn,D = z.shape A = sf2 * np.ones((n,nn)) if der == 0: # compute derivative matrix wrt sf2 A = 2. * A else: raise Exception("Wrong derivative entry in covConst") return A class Linear(Kernel): ''' Linear kernel. hyp = [ log_sigma ]. ''' def __init__(self, log_sigma=0.): self.hyp = [ log_sigma ] def getCovMatrix(self,x=None,z=None,mode=None): sf2 = np.exp(self.hyp[0]) # s2 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.reshape(np.sum(z*z,1), (nn,1)) elif mode == 'train': # compute covariance matix for dataset x n,D = x.shape A = np.dot(x,x.T) + np.eye(n)*1e-16 # required for numerical accuracy elif mode == 'cross': # compute covariance between data sets x and z A = np.dot(x,z.T) A = sf2 * A return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): sf2 = np.exp(self.hyp[0]) # s2 if der == 0: if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.reshape(np.sum(z*z,1), (nn,1)) elif mode == 'train': # compute covariance matix for dataset x n,D = x.shape A = np.dot(x,x.T) + np.eye(n)*1e-16 # required for numerical accuracy elif mode == 'cross': # compute covariance between data sets x and z A = np.dot(x,z.T) A = 2 * sf2 * A else: raise Exception("Wrong derivative index in covLinear") return A class LINard(Kernel): ''' Linear covariance function with Automatic Relevance Detemination. hyp = log_ell_list :param D: dimension of training data. Set if you want default ell, which is 1 for each dimension. :param log_ell_list: characteristic length scale for each dimension. ''' def __init__(self, D=None, log_ell_list=None): if log_ell_list == None: self.hyp = [0. for i in xrange(D)] else: self.hyp = log_ell_list def getCovMatrix(self,x=None,z=None,mode=None): ell = np.exp(self.hyp) # ARD parameters if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.reshape(np.sum(z*z,1), (nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.dot(x,x.T) elif mode == 'cross': # compute covariance between data sets x and z z = np.dot(z,np.diag(1./ell)) A = np.dot(x,z.T) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): ell = np.exp(self.hyp) # ARD parameters if x != None: n, D = x.shape if z != None: nn, D = z.shape if der < D: if mode == 'self_test': tem = np.atleast_2d(z[:,der]).T A = -2.* tem * tem elif mode == 'train': A = -2.*np.dot(np.atleast_2d(x[:,der]).T,np.atleast_2d(x[:,der])) elif mode == 'cross': z = np.dot(z,np.diag(1./ell)) A = -2.*np.dot(np.atleast_2d(x[:,der]).T, np.atleast_2d(z[:,der])) # cross covariances else: raise Exception("Wrong derivative index in covLINard") return A class Matern(Kernel): ''' Matern covariance function with nu = d/2 and isotropic distance measure. For d=1 the function is also known as the exponential covariance function or the Ornstein-Uhlenbeck covariance in 1d. d will be rounded to 1, 3, or 5. hyp = [ log_ell, log_sigma] :param d: d is 2 times nu. Can only be 1,3 or 5. :param log_ell: characteristic length scale. :param log_sigma: signal deviation. ''' def __init__(self, log_ell=0., d=3, log_sigma=0. ): self.hyp = [ log_ell, log_sigma ] self.para = [d] def func(self,d,t): if d == 1: return 1 elif d == 3: return 1 + t elif d == 5: return 1 + t*(1+t/3.) else: raise Exception("Wrong value for d in Matern") def dfunc(self,d,t): if d == 1: return 1 elif d == 3: return t elif d == 5: return t*(1+t/3.) else: raise Exception("Wrong value for d in Matern") def mfunc(self,d,t): return self.func(d,t)*np.exp(-1.*t) def dmfunc(self,d,t): return self.dfunc(d,t)*t*np.exp(-1.*t) def getCovMatrix(self,x=None,z=None,mode=None): ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.* self.hyp[1]) # signal variance d = self.para[0] # 2 times nu if np.abs(d-np.round(d)) < 1e-8: # remove numerical error from format of parameter d = int(round(d)) d = int(d) try: assert(d in [1,3,5]) # check for valid values of d except AssertionError: print "Warning: You specified d to be neither 1,3 nor 5. We set to d=3. " d = 3 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x x = np.sqrt(d)*x/ell A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean')) elif mode == 'cross': # compute covariance between data sets x and z x = np.sqrt(d)*x/ell z = np.sqrt(d)*z/ell A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean')) A = sf2 * self.mfunc(d,A) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.* self.hyp[1]) # signal variance d = self.para[0] # 2 times nu if np.abs(d-np.round(d)) < 1e-8: # remove numerical error from format of parameter d = int(round(d)) d = int(d) try: assert(d in [1,3,5]) # check for valid values of d except AssertionError: print "Warning: You specified d to be neither 1,3 nor 5. We set to d=3. " d = 3 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x x = np.sqrt(d)*x/ell A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean')) elif mode == 'cross': # compute covariance between data sets x and z x = np.sqrt(d)*x/ell z = np.sqrt(d)*z/ell A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean')) A = sf2 * self.mfunc(d,A) if der == 0: # compute derivative matrix wrt 1st parameter A = sf2 * self.dmfunc(d,A) elif der == 1: # compute derivative matrix wrt 2nd parameter A = 2 * sf2 * self.mfunc(d,A) elif der == 2: # no derivative wrt 3rd parameter A = np.zeros_like(A) # do nothing (d is not learned) else: raise Exception("Wrong derivative value in Matern") return A class Periodic(Kernel): ''' Stationary kernel for a smooth periodic function. hyp = [ log_ell, log_p, log_sigma] :param log_p: period. :param log_ell: characteristic length scale. :param log_sigma: signal deviation. ''' def __init__(self, log_ell=0., log_p=0., log_sigma=0. ): self.hyp = [ log_ell, log_p, log_sigma] def getCovMatrix(self,x=None,z=None,mode=None): ell = np.exp(self.hyp[0]) # characteristic length scale p = np.exp(self.hyp[1]) # period sf2 = np.exp(2.*self.hyp[2]) # signal variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean')) elif mode == 'cross': # compute covariance between data sets x and z A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean')) A = np.pi*A/p A = np.sin(A)/ell A = A * A A = sf2 *np.exp(-2.*A) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): ell = np.exp(self.hyp[0]) # characteristic length scale p = np.exp(self.hyp[1]) # period sf2 = np.exp(2.*self.hyp[2]) # signal variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean')) elif mode == 'cross': # compute covariance between data sets x and z A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean')) A = np.pi*A/p if der == 0: # compute derivative matrix wrt 1st parameter A = np.sin(A)/ell A = A * A A = 4. *sf2 *np.exp(-2.*A) * A elif der == 1: # compute derivative matrix wrt 2nd parameter R = np.sin(A)/ell A = 4 * sf2/ell * np.exp(-2.*R*R)*R*np.cos(A)*A elif der == 2: # compute derivative matrix wrt 3rd parameter A = np.sin(A)/ell A = A * A A = 2. * sf2 * np.exp(-2.*A) else: raise Exception("Wrong derivative index in covPeriodic") return A class Noise(Kernel): ''' Independent covariance function, i.e "white noise", with specified variance. Normally NOT used anymore since noise is now added in liklihood. hyp = [ log_sigma ] :param log_sigma: signal deviation. ''' def __init__(self, log_sigma=0.): self.hyp = [log_sigma] def getCovMatrix(self,x=None,z=None,mode=None): tol = 1.e-9 # Tolerance for declaring two vectors "equal" s2 = np.exp(2.*self.hyp[0]) # noise variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x n,D = x.shape A = np.eye(n) elif mode == 'cross': # compute covariance between data sets x and z M = spdist.cdist(x, z, 'sqeuclidean') A = np.zeros_like(M,dtype=np.float) A[M < tol] = 1. A = s2*A return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): tol = 1.e-9 # Tolerance for declaring two vectors "equal" s2 = np.exp(2.*self.hyp[0]) # noise variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x n,D = x.shape A = np.eye(n) elif mode == 'cross': # compute covariance between data sets x and z M = spdist.cdist(x, z, 'sqeuclidean') A = np.zeros_like(M,dtype=np.float) A[M < tol] = 1. if der == 0: A = 2.*s2*A else: raise Exception("Wrong derivative index in covNoise") return A class RQ(Kernel): ''' Rational Quadratic covariance function with isotropic distance measure. hyp = [ log_ell, log_sigma, log_alpha ] :param log_ell: characteristic length scale. :param log_sigma: signal deviation. :param log_alpha: shape parameter for the RQ covariance. ''' def __init__(self, log_ell=0., log_sigma=0., log_alpha=0.): self.hyp = [ log_ell, log_sigma, log_alpha ] def getCovMatrix(self,x=None,z=None,mode=None): ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.*self.hyp[1]) # signal variance alpha = np.exp(self.hyp[2]) if mode == 'self_test': # self covariances for the test cases nn,D = z.shape D2 = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x D2 = spdist.cdist(x/ell, x/ell, 'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z D2 = spdist.cdist(x/ell, z/ell, 'sqeuclidean') A = sf2 * ( ( 1.0 + 0.5*D2/alpha )**(-alpha) ) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.*self.hyp[1]) # signal variance alpha = np.exp(self.hyp[2]) if mode == 'self_test': # self covariances for the test cases nn,D = z.shape D2 = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x D2 = spdist.cdist(x/ell, x/ell, 'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z D2 = spdist.cdist(x/ell, z/ell, 'sqeuclidean') if der == 0: # compute derivative matrix wrt 1st parameter A = sf2 * ( 1.0 + 0.5*D2/alpha )**(-alpha-1) * D2 elif der == 1: # compute derivative matrix wrt 2nd parameter A = 2.* sf2 * ( ( 1.0 + 0.5*D2/alpha )**(-alpha) ) elif der == 2: # compute derivative matrix wrt 3rd parameter K = ( 1.0 + 0.5*D2/alpha ) A = sf2 * K**(-alpha) * (0.5*D2/K - alpha*np.log(K) ) else: raise Exception("Wrong derivative index in covRQ") return A class RQard(Kernel): ''' Rational Quadratic covariance function with Automatic Relevance Detemination (ARD) distance measure. hyp = log_ell_list + [ log_sigma, log_alpha ] :param D: dimension of pattern. set if you want default ell, which is 0.5 for each dimension. :param log_ell_list: characteristic length scale for each dimension. :param log_sigma: signal deviation. :param log_alpha: shape parameter for the RQ covariance. ''' def __init__(self, D=None, log_ell_list=None, log_sigma=0., log_alpha=0.): if log_ell_list == None: self.hyp = [0. for i in xrange(D)] + [ log_sigma, log_alpha ] else: self.hyp = log_ell_list + [ log_sigma, log_alpha ] def getCovMatrix(self,x=None,z=None,mode=None): if x != None: n, D = x.shape if z != None: nn, D = z.shape ell = 1./np.exp(self.hyp[0:D]) # characteristic length scale sf2 = np.exp(2.*self.hyp[D]) # signal variance alpha = np.exp(self.hyp[D+1]) if mode == 'self_test': # self covariances for the test cases nn,D = z.shape D2 = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x tmp = np.dot(np.diag(ell),x.T).T D2 = spdist.cdist(tmp, tmp, 'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z D2 = spdist.cdist(np.dot(np.diag(ell),x.T).T, np.dot(np.diag(ell),z.T).T, 'sqeuclidean') A = sf2 * ( ( 1.0 + 0.5*D2/alpha )**(-alpha) ) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): if x != None: n, D = x.shape if z != None: nn, D = z.shape ell = 1./np.exp(self.hyp[0:D]) # characteristic length scale sf2 = np.exp(2.*self.hyp[D]) # signal variance alpha = np.exp(self.hyp[D+1]) if mode == 'self_test': # self covariances for the test cases nn,D = z.shape D2 = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x tmp = np.dot(np.diag(ell),x.T).T D2 = spdist.cdist(tmp, tmp, 'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z D2 = spdist.cdist(np.dot(np.diag(ell),x.T).T, np.dot(np.diag(ell),z.T).T, 'sqeuclidean') if der < D: if mode == 'self_test': A = D2*0 elif mode == 'train': tmp = np.atleast_2d(x[:,der])/ell[der] A = sf2 * ( 1.0 + 0.5*D2/alpha )**(-alpha-1) * spdist.cdist(tmp, tmp, 'sqeuclidean') elif mode == 'cross': A = sf2 * ( 1.0 + 0.5*D2/alpha )**(-alpha-1) * spdist.cdist(np.atleast_2d(x[:,der]).T/ell[der], np.atleast_2d(z[:,der]).T/ell[der], 'sqeuclidean') elif der==D: # compute derivative matrix wrt magnitude parameter A = 2. * sf2 * ( ( 1.0 + 0.5*D2/alpha )**(-alpha) ) elif der==(D+1): # compute derivative matrix wrt magnitude parameter K = ( 1.0 + 0.5*D2/alpha ) A = sf2 * K**(-alpha) * ( 0.5*D2/K - alpha*np.log(K) ) else: raise Exception("Wrong derivative index in covRQard") return A class Pre(Kernel): ''' Precomputed kernel matrix. No hyperparameters and thus nothing will be optimised. :param M1: cross covariances matrix(train+1 by test). last row is self covariances (diagonal of test by test) :param M2: training set covariance matrix (train by train) ''' def __init__(self,M1,M2): self.M1 = M1 self.M2 = M2 self.hyp = [] def getCovMatrix(self,x=None,z=None,mode=None): if mode == 'self_test': # diagonal covariance between test_test A = self.M1[-1,:] # self covariances for the test cases (last row) A = np.reshape(A, (A.shape[0],1)) elif mode == 'train': # covariance between train_train A = self.M2 elif mode == 'cross': # covariance between train_test A = self.M1[:-1,:] return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): if der != None: raise Exception("Error: NO optimization in precomputed kernel matrix") return 0 if __name__ == '__main__': pass }rM(XConstXclassMjMXPre.getDerMatrixXdefMJMQXPeriodic.getDerMatrixXdefMmMX PiecePolyXclassMqMXLinear.__init__XdefMMX 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hj;Pubj+P)rBP}rCP(hUh}rDP(UlatexX \rightarrowh]h]h]h]h]uhj;Ph]hj/PubhX user-defined inducing pointsrEPrFP}rGP(hX user-defined inducing pointshj;PubehjMubahjMubahj NubjM)rHP}rIP(hUh}rJP(h]h]h]h]h]uhjPh]rKPjM)rLP}rMP(hUh}rNP(h]h]h]h]h]uhjHPh]rOPjM)rPP}rQP(hUh}rRP(U anchornameU#a-bit-more-things-you-can-doUrefurih h]h]h]h]h]UinternaluhjLPh]rSPhXA bit more things you can dorTPrUP}rVP(hXA bit more things you can dohjPPubahjMubahjMubahj Nubehj(Nubehj Nubahj(Nubh6jM)rWP}rXP(hUh}rYP(h]h]h]h]h]uh]rZPjM)r[P}r\P(hUh}r]P(h]h]h]h]h]uhjWPh]r^PjM)r_P}r`P(hUh}raP(h]h]h]h]h]uhj[Ph]rbPjM)rcP}rdP(hUh}reP(U anchornameUUrefurih6h]h]h]h]h]Uinternaluhj_Ph]rfPhXRegression on UCI Housing datargPrhP}riP(hhhjcPubahjMubahjMubahj Nubahj(NubhjM)rjP}rkP(hUh}rlP(h]h]h]h]h]uh]rmPjM)rnP}roP(hUh}rpP(h]h]h]h]h]uhjjPh]rqP(jM)rrP}rsP(hUh}rtP(h]h]h]h]h]uhjnPh]ruPjM)rvP}rwP(hUh}rxP(U anchornameUUrefurihh]h]h]h]h]UinternaluhjrPh]ryPhXLikelihoods & InferencerzPr{P}r|P(hjhjvPubahU 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anchornameU#list-of-inferenceUrefurihh]h]h]h]h]UinternaluhjPh]rPhXList of InferencerPrP}rP(hXList of InferencehjPubahj}Pubahj~PubahjPubehU bullet_listrPubehjPubahjPubhAjM)rP}rP(hUh}rP(h]h]h]h]h]uh]rPjM)rP}rP(hUh}rP(h]h]h]h]h]uhjPh]rP(jM)rP}rP(hUh}rP(h]h]h]h]h]uhjPh]rPjM)rP}rP(hUh}rP(U anchornameUUrefurihAh]h]h]h]h]UinternaluhjPh]rPhXBasic ClassificationrPrP}rP(hjhjPubahjMubahjMubjM)rP}rP(hUh}rP(h]h]h]h]h]uhjPh]rP(jM)rP}rP(hUh}rP(h]h]h]h]h]uhjPh]rPjM)rP}rP(hUh}rP(h]h]h]h]h]uhjPh]rPjM)rP}rP(hUh}rP(U anchornameU #load-dataUrefurihAh]h]h]h]h]UinternaluhjPh]rPhX Load datarPrP}rP(hX Load datahjPubahjMubahjMubahj NubjM)rP}rP(hUh}rP(h]h]h]h]h]uhjPh]rPjM)rP}rP(hUh}rP(h]h]h]h]h]uhjPh]rPjM)rP}rP(hUh}rP(U anchornameU##first-example-state-default-valuesUrefurihAh]h]h]h]h]UinternaluhjPh]rP(hXFirst example rPrP}rP(hXFirst example hjPubj+P)rP}rP(hUh}rP(UlatexX \rightarrowh]h]h]h]h]uhjPh]hj/PubhX state default valuesrPrP}rP(hX state default valueshjPubehjMubahjMubahj 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anchornameU#about-the-packageUrefurij h]h]h]h]h]Uinternaluhj;Qh]rBQhXAbout the packagerCQrDQ}rEQ(hXAbout the packagehj?QubahjMubahjMubahj NubjM)rFQ}rGQ(hUh}rHQ(h]h]h]h]h]uhj3Qh]rIQ(jM)rJQ}rKQ(hUh}rLQ(h]h]h]h]h]uhjFQh]rMQjM)rNQ}rOQ(hUh}rPQ(U anchornameU#getting-startedUrefurij h]h]h]h]h]UinternaluhjJQh]rQQhXGetting startedrRQrSQ}rTQ(hXGetting startedhjNQubahjMubahjMubjM)rUQ}rVQ(hUh}rWQ(h]h]h]h]h]uhjFQh]rXQcsphinx.addnodes toctree rYQ)rZQ}r[Q(hUh}r\Q(UnumberedKUparentj U titlesonlyUglobh]h]h]h]U includehiddenUentries]r]Q(NjMr^QNjMr_QNjMr`QNjMraQNjMrbQNjMrcQNjMrdQeUhiddenUmaxdepthKU includefiles]reQ(jMjMjMjMjMjMjMeh]uhjUQh]hUtoctreeubahj(Nubehj Nubehj(Nubehj Nubahj(NubjjM)rfQ}rgQ(hUh}rhQ(h]h]h]h]h]uh]riQjM)rjQ}rkQ(hUh}rlQ(h]h]h]h]h]uhjfQh]rmQ(jM)rnQ}roQ(hUh}rpQ(h]h]h]h]h]uhjjQh]rqQjM)rrQ}rsQ(hUh}rtQ(U anchornameUh]h]h]Urefurijh]h]UinternaluhjnQh]ruQhXDemosrvQrwQ}rxQ(hjhjrQubahjNubahjNubjM)ryQ}rzQ(hUh}r{Q(h]h]h]h]h]uhjjQh]r|Q(jYQ)r}Q}r~Q(hUh}rQ(UnumberedKU includehiddenUparentjh]Uglobh]h]U 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Nubahj(Nubj$jM)rQ}rQ(hUh}rQ(h]h]h]h]h]uh]rQjM)rQ}rQ(hUh}rQ(h]h]h]h]h]uhjQh]rQ(jM)rQ}rQ(hUh}rQ(h]h]h]h]h]uhjQh]rQjM)rQ}rQ(hUh}rQ(U anchornameUh]h]h]Urefurij$h]h]UinternaluhjQh]rQhX Graph KernelsrQrQ}rQ(hj,hjQubahjNubahjNubjM)rQ}rQ(hUh}rQ(h]h]h]h]h]uhjQh]rQ(jM)rQ}rQ(hUh}rQ(h]h]h]h]h]uhjQh]rQjM)rQ}rQ(hUh}rQ(h]h]h]h]h]uhjQh]rQjM)rQ}rQ(hUh}rQ(U anchornameU #load-datah]h]h]Urefurij$h]h]UinternaluhjQh]rQhX Load datarQrQ}rQ(hX Load datahjQubahjNubahjNubahjNubjM)rQ}rQ(hUh}rQ(h]h]h]h]h]uhjQh]rQjM)rQ}rQ(hUh}rQ(h]h]h]h]h]uhjQh]rQjM)rQ}rQ(hUh}rQ(U anchornameU"#compute-propagation-kernel-matrixh]h]h]Urefurij$h]h]UinternaluhjQh]rQhX!Compute propagation kernel matrixrQrQ}rQ(hX!Compute propagation kernel matrixhjQubahjNubahjNubahjNubjM)rQ}rQ(hUh}rQ(h]h]h]h]h]uhjQh]rQjM)rQ}rQ(hUh}rQ(h]h]h]h]h]uhjQh]rQjM)rQ}rQ(hUh}rQ(U anchornameU#standard-gp-classificationh]h]h]Urefurij$h]h]UinternaluhjQh]rQhXStandard GP ClassificationrQrQ}rR(hXStandard GP ClassificationhjQubahjNubahjNubahjNubehj@OubehjNubahj@Oubj-jM)rR}rR(hUh}rR(h]h]h]h]h]uh]rRjM)rR}rR(hUh}rR(h]h]h]h]h]uhjRh]rR(jM)r R}r R(hUh}r R(h]h]h]h]h]uhjRh]r RjM)r R}rR(hUh}rR(U anchornameUUrefurij-h]h]h]h]h]Uinternaluhj Rh]rRhX CopyrightrRrR}rR(hj5hj RubahjMubahjMubjM)rR}rR(hUh}rR(h]h]h]h]h]uhjRh]rR(jM)rR}rR(hUh}rR(h]h]h]h]h]uhjRh]rRjM)rR}rR(hUh}rR(h]h]h]h]h]uhjRh]rRjM)r R}r!R(hUh}r"R(U anchornameU#authorsUrefurij-h]h]h]h]h]UinternaluhjRh]r#RhXAuthorsr$Rr%R}r&R(hXAuthorshj RubahjMubahjMubahj NubjM)r'R}r(R(hUh}r)R(h]h]h]h]h]uhjRh]r*RjM)r+R}r,R(hUh}r-R(h]h]h]h]h]uhj'Rh]r.RjM)r/R}r0R(hUh}r1R(U anchornameU#licenseUrefurij-h]h]h]h]h]Uinternaluhj+Rh]r2RhXLicenser3Rr4R}r5R(hXLicensehj/RubahjMubahjMubahj Nubehj(Nubehj Nubahj(NubhjM)r6R}r7R(hUh}r8R(h]h]h]h]h]uh]r9RjM)r:R}r;R(hUh}rR}r?R(hUh}r@R(h]h]h]h]h]uhj:Rh]rARjM)rBR}rCR(hUh}rDR(U anchornameUUrefurihh]h]h]h]h]Uinternaluhj>Rh]rERhXSparse 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anchornameU#a-bit-more-things-you-can-doUrefurihh]h]h]h]h]Uinternaluhj{Rh]rRhXA bit more things you can dorRrR}rR(hXA bit more things you can dohjRubahjMubahjMubahj Nubehj(Nubehj Nubahj(NubhGjM)rR}rR(hUh}rR(h]h]h]h]h]uh]rRjM)rR}rR(hUh}rR(h]h]h]h]h]uhjRh]rR(jM)rR}rR(hUh}rR(h]h]h]h]h]uhjRh]rRjM)rR}rR(hUh}rR(U anchornameUUrefurihGh]h]h]h]h]UinternaluhjRh]rRhXBasic RegressionrRrR}rR(hjEhjRubahjMubahjMubjM)rR}rR(hUh}rR(h]h]h]h]h]uhjRh]rR(jM)rR}rR(hUh}rR(h]h]h]h]h]uhjRh]rRjM)rR}rR(hUh}rR(h]h]h]h]h]uhjRh]rRjM)rR}rR(hUh}rR(U anchornameU#import-packagesUrefurihGh]h]h]h]h]UinternaluhjRh]rRhXImport packagesrRrR}rR(hXImport packageshjRubahjMubahjMubahj NubjM)rR}rR(hUh}rR(h]h]h]h]h]uhjRh]rRjM)rR}rR(hUh}rR(h]h]h]h]h]uhjRh]rRjM)rR}rR(hUh}rR(U anchornameU #load-dataUrefurihGh]h]h]h]h]UinternaluhjRh]rRhX Load datarRrR}rR(hX Load datahjRubahjMubahjMubahj NubjM)rR}rR(hUh}rR(h]h]h]h]h]uhjRh]rRjM)rR}rR(hUh}rR(h]h]h]h]h]uhjRh]rRjM)rR}rR(hUh}rR(U 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2013.UparentqhUsourceqcdocutils.nodes reprunicode qX0/Users/hs/Program/git/pyGPs/doc/source/index.rstq q!}q"bUtagnameq#Ucommentq$U attributesq%}q&(U xml:spaceq'Upreserveq(Uidsq)]Ubackrefsq*]Udupnamesq+]Uclassesq,]Unamesq-]uUlineq.KUdocumentq/hh]q0cdocutils.nodes Text q1XwObject-Oriented Gaussian Processes documentation master file, created by sphinx-quickstart on Thu Nov 14 19:54:52 2013.q2q3}q4(hUhhubaubcdocutils.nodes section q5)q6}q7(hUhhhh!h#Usectionq8h%}q9(h+]h,]h*]h)]q:hah-]q;h auh.Kh/hh]q<(cdocutils.nodes title q=)q>}q?(hX(pyGPs - A Package for Gaussian Processesq@hh6hh!h#UtitleqAh%}qB(h+]h,]h*]h)]h-]uh.Kh/hh]qCh1X(pyGPs - A Package for Gaussian ProcessesqDqE}qF(hh@hh>ubaubh5)qG}qH(hUhh6hh!h#h8h%}qI(h+]h,]h*]h)]qJhah-]qKhauh.Kh/hh]qL(h=)qM}qN(hXAbout the packageqOhhGhh!h#hAh%}qP(h+]h,]h*]h)]h-]uh.Kh/hh]qQh1XAbout the packageqRqS}qT(hhOhhMubaubcdocutils.nodes paragraph qU)qV}qW(hXpyGPs is a library hosting Python implementations of Gaussian processes (GPs) for machine learning. pyGPs bridges the gap between systems designed primarily for users, who mainly want to apply GPs and need basic machine learning routines for model training, evaluation, and visualiztion, and expressive systems for developers, who focus on extending the core functionalities as covariance and likelihood functions, as well as inference techniques.qXhhGhh!h#U paragraphqYh%}qZ(h+]h,]h*]h)]h-]uh.Kh/hh]q[h1XpyGPs is a library hosting Python implementations of Gaussian processes (GPs) for machine learning. pyGPs bridges the gap between systems designed primarily for users, who mainly want to apply GPs and need basic machine learning routines for model training, evaluation, and visualiztion, and expressive systems for developers, who focus on extending the core functionalities as covariance and likelihood functions, as well as inference techniques.q\q]}q^(hhXhhVubaubhU)q_}q`(hXThe software package is released under the BSD 2-Clause (FreeBSD) License. :doc:`Copyright <_static/copyright>` (c) by Marion Neumann, Shan Huang, Daniel Marthaler, & Kristian Kersting, Feb.2014hhGhh!h#hYh%}qa(h+]h,]h*]h)]h-]uh.Kh/hh]qb(h1XKThe software package is released under the BSD 2-Clause (FreeBSD) License. qcqd}qe(hXKThe software package is released under the BSD 2-Clause (FreeBSD) License. hh_ubcsphinx.addnodes pending_xref qf)qg}qh(hX$:doc:`Copyright <_static/copyright>`qihh_hh!h#U pending_xrefqjh%}qk(UreftypeXdocqlUrefwarnqmU reftargetqnX_static/copyrightU refdomainUh)]h*]U refexplicith+]h,]h-]UrefdocqoXindexqpuh.Kh]qqcdocutils.nodes literal qr)qs}qt(hhih%}qu(h+]h,]qv(Uxrefqwhleh*]h)]h-]uhhgh]qxh1X Copyrightqyqz}q{(hUhhsubah#Uliteralq|ubaubh1XS (c) by Marion Neumann, Shan Huang, Daniel Marthaler, & Kristian Kersting, Feb.2014q}q~}q(hXS (c) by Marion Neumann, Shan Huang, Daniel Marthaler, & Kristian Kersting, Feb.2014hh_ubeubhU)q}q(hX'Further, it includes implementations ofqhhGhh!h#hYh%}q(h+]h,]h*]h)]h-]uh.Kh/hh]qh1X'Further, it includes implementations ofqq}q(hhhhubaubcdocutils.nodes bullet_list q)q}q(hUhhGhh!h#U bullet_listqh%}q(UbulletqX*h)]h*]h+]h,]h-]uh.Kh/hh]q(cdocutils.nodes list_item q)q}q(hXminimize.py implemented in python by Roland Memisevic 2008, following minimize.m (Copyright (c) Carl Edward Rasmussen (1999-2006)) hhhh!h#U list_itemqh%}q(h+]h,]h*]h)]h-]uh.Nh/hh]qhU)q}q(hXminimize.py implemented in python by Roland Memisevic 2008, following minimize.m (Copyright (c) Carl Edward Rasmussen (1999-2006))qhhhh!h#hYh%}q(h+]h,]h*]h)]h-]uh.Kh]qh1Xminimize.py implemented in python by Roland Memisevic 2008, following minimize.m (Copyright (c) Carl Edward Rasmussen (1999-2006))qq}q(hhhhubaubaubh)q}q(hX0scg.py (Copyright (c) Ian T Nabney (1996-2001)) hhhh!h#hh%}q(h+]h,]h*]h)]h-]uh.Nh/hh]qhU)q}q(hX/scg.py (Copyright (c) Ian T Nabney (1996-2001))qhhhh!h#hYh%}q(h+]h,]h*]h)]h-]uh.Kh]qh1X/scg.py (Copyright (c) Ian T Nabney (1996-2001))qq}q(hhhhubaubaubh)q}q(hX7brentmin.py (Copyright (c) Hannes Nickisch 2010-01-10) hhhh!h#hh%}q(h+]h,]h*]h)]h-]uh.Nh/hh]qhU)q}q(hX6brentmin.py (Copyright (c) Hannes Nickisch 2010-01-10)qhhhh!h#hYh%}q(h+]h,]h*]h)]h-]uh.Kh]qh1X6brentmin.py (Copyright (c) Hannes Nickisch 2010-01-10)qq}q(hhhhubaubaubh)q}q(hXFITC functionality (following matlab implementations under Copyright (c) by Ed Snelson, Carl Edward Rasmussen and Hannes Nickisch, 2011-11-02) hhhh!h#hh%}q(h+]h,]h*]h)]h-]uh.Nh/hh]qhU)q}q(hXFITC functionality (following matlab implementations under Copyright (c) by Ed Snelson, Carl Edward Rasmussen and Hannes Nickisch, 2011-11-02)qhhhh!h#hYh%}q(h+]h,]h*]h)]h-]uh.Kh]qh1XFITC functionality (following matlab implementations under Copyright (c) by Ed Snelson, Carl Edward Rasmussen and Hannes Nickisch, 2011-11-02)qq}q(hhhhubaubaubeubhU)q}q(hXThe most recent stable release is pyGPs v1.2. See :doc:`changelog <_static/changelog>` for a list of changes to the previous release. A Developer's Guide containing api is found in :doc:`api.pdf <_static/api.pdf>`, but for the casual user, the guide is below. If you observe problems or bugs, please let us know. You can also download a `procedual implementation`_ of GP functionality from Github. However, the procedual version will not be supported in future.hhGhh!h#hYh%}q(h+]h,]h*]h)]h-]uh.Kh/hh]q(h1X2The most recent stable release is pyGPs v1.2. See qŅq}q(hX2The most recent stable release is pyGPs v1.2. See hhubhf)q}q(hX$:doc:`changelog <_static/changelog>`qhhhh!h#hjh%}q(UreftypeXdocqhmhnX_static/changelogU refdomainUh)]h*]U refexplicith+]h,]h-]hohpuh.Kh]qhr)q}q(hhh%}q(h+]h,]q(hwheh*]h)]h-]uhhh]qh1X changelogqӅq}q(hUhhubah#h|ubaubh1X_ for a list of changes to the previous release. A Developer's Guide containing api is found in qօq}q(hX_ for a list of changes to the previous release. A Developer's Guide containing api is found in hhubhf)q}q(hX :doc:`api.pdf <_static/api.pdf>`qhhhh!h#hjh%}q(UreftypeXdocqhmhnX_static/api.pdfU refdomainUh)]h*]U refexplicith+]h,]h-]hohpuh.Kh]qhr)q}q(hhh%}q(h+]h,]q(hwheh*]h)]h-]uhhh]qh1Xapi.pdfq䅁q}q(hUhhubah#h|ubaubh1X|, but for the casual user, the guide is below. If you observe problems or bugs, please let us know. You can also download a q煁q}q(hX|, but for the casual user, the guide is below. If you observe problems or bugs, please let us know. You can also download a hhubcdocutils.nodes reference q)q}q(hX`procedual implementation`_UresolvedqKhhh#U referenceqh%}q(UnameXprocedual implementationUrefuriqX&https://github.com/marionmari/pyGP_PR/qh)]h*]h+]h,]h-]uh]qh1Xprocedual implementationqq}q(hUhhubaubh1Xa of GP functionality from Github. However, the procedual version will not be supported in future.qq}q(hXa of GP functionality from Github. However, the procedual version will not be supported in future.hhubeubcdocutils.nodes target q)q}q(hXD.. _procedual implementation: https://github.com/marionmari/pyGP_PR/U referencedqKhhGhh!h#Utargetqh%}q(hhh)]qhah*]h+]h,]h-]rhauh.K"h/hh]ubcdocutils.nodes definition_list r)r}r(hUhhGhNh#Udefinition_listrh%}r(h+]h,]h*]h)]h-]uh.Nh/hh]rcdocutils.nodes definition_list_item r)r}r (hXAuthors: - Marion Neumann [marion dot neumann at uni-bonn dot de] - Shan Huang [shan dot huang at iais dot fraunhofer dot de] - Daniel Marthaler [marthaler at ge dot com] - Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] hjhh!h#Udefinition_list_itemr h%}r (h+]h,]h*]h)]h-]uh.K*h]r (cdocutils.nodes term r )r}r(hXAuthors:rhjhh!h#Utermrh%}r(h+]h,]h*]h)]h-]uh.K*h]rh1XAuthors:rr}r(hjhjubaubcdocutils.nodes definition r)r}r(hUh%}r(h+]h,]h*]h)]h-]uhjh]rh)r}r(hUh%}r(hX-h)]h*]h+]h,]h-]uhjh]r(h)r }r!(hX6Marion Neumann [marion dot neumann at uni-bonn dot de]r"h%}r#(h+]h,]h*]h)]h-]uhjh]r$hU)r%}r&(hj"hj hh!h#hYh%}r'(h+]h,]h*]h)]h-]uh.K'h]r(h1X6Marion Neumann [marion dot neumann at uni-bonn dot de]r)r*}r+(hj"hj%ubaubah#hubh)r,}r-(hX9Shan Huang [shan dot huang at iais dot fraunhofer dot de]r.h%}r/(h+]h,]h*]h)]h-]uhjh]r0hU)r1}r2(hj.hj,hh!h#hYh%}r3(h+]h,]h*]h)]h-]uh.K(h]r4h1X9Shan Huang [shan dot huang at iais dot fraunhofer dot de]r5r6}r7(hj.hj1ubaubah#hubh)r8}r9(hX*Daniel Marthaler [marthaler at ge dot com]r:h%}r;(h+]h,]h*]h)]h-]uhjh]r<hU)r=}r>(hj:hj8hh!h#hYh%}r?(h+]h,]h*]h)]h-]uh.K)h]r@h1X*Daniel Marthaler [marthaler at ge dot com]rArB}rC(hj:hj=ubaubah#hubh)rD}rE(hXGKristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] h%}rF(h+]h,]h*]h)]h-]uhjh]rGhU)rH}rI(hXFKristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]rJhjDhh!h#hYh%}rK(h+]h,]h*]h)]h-]uh.K*h]rLh1XFKristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]rMrN}rO(hjJhjHubaubah#hubeh#hubah#U definitionrPubeubaubhU)rQ}rR(hXThe following persons helped to improve this software: Roman 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IENDB`pyGPs-1.2/doc/build/_modules/000077500000000000000000000000001236346020200160675ustar00rootroot00000000000000pyGPs-1.2/doc/build/_modules/index.html000066400000000000000000000071471236346020200200750ustar00rootroot00000000000000 Overview: module code — pyGPs v1.2 documentation

All modules for which code is available

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© Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting. Created using Sphinx 1.2.
pyGPs-1.2/doc/build/_modules/pyGPs/000077500000000000000000000000001236346020200171315ustar00rootroot00000000000000pyGPs-1.2/doc/build/_modules/pyGPs/Core.html000066400000000000000000000100641236346020200207100ustar00rootroot00000000000000 pyGPs.Core — pyGPs v1.2 documentation

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Source code for pyGPs.Core

import inf
import cov
import mean
import lik
import gp
import opt



__all__ = ['inf', 'cov', 'mean', 'lik','gp','opt']

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© Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting. Created using Sphinx 1.2.
pyGPs-1.2/doc/build/_modules/pyGPs/Core/000077500000000000000000000000001236346020200200215ustar00rootroot00000000000000pyGPs-1.2/doc/build/_modules/pyGPs/Core/cov.html000066400000000000000000007142321236346020200215070ustar00rootroot00000000000000 pyGPs.Core.cov — pyGPs v1.2 documentation

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Source code for pyGPs.Core.cov

#================================================================================
#    Marion Neumann [marion dot neumann at uni-bonn dot de]
#    Daniel Marthaler [marthaler at ge dot com]
#    Shan Huang [shan dot huang at iais dot fraunhofer dot de]
#    Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]
#
#    This file is part of pyGPs.
#    The software package is released under the BSD 2-Clause (FreeBSD) License.
#
#    Copyright (c) by
#    Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014
#================================================================================
# covariance functions to be use by Gaussian process functions. There are two
# different kinds of covariance functions: simple and composite:
#
# simple cov functions:
#
# Poly          - Polynomial covariance function
# PiecePoly     - Piecewise polynomial kernel with compact support.
# RBF           - Squared Exponential kernel.
# RBFunit       - Squared Exponential kernel with unit magnitude.
# RBFard        - Squared Exponential kernel with Automatic Relevance Determination.
# Const         - Constant kernel.
# Linear        - Linear kernel.
# LINard        - Linear covariance function with Automatic Relevance Detemination.
# Matern        - Matern covariance function.
# Periodic      - Stationary kernel for a smooth periodic function.
# Noise         - Independent covariance function, i.e "white noise".
# RQ            - Rational Quadratic covariance function with isotropic distance measure.
# RQard         - Rational Quadratic covariance function with ARD distance measure.
# Pre           - Precomputed kernel matrix.
# 
# composite covariance functions:
#
#   ScaleOfKernel     - scaled version of a covariance function
#   ProductOfKernel   - products of covariance functions
#   SumOfKernel       - sums of covariance functions
#   FITCOfKernel      - Covariance function to be used together with the FITC approximation
#
#
# This is a object-oriented python implementation of gpml functionality 
# (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18).
# based on the functional-version of python implementation
# (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013)
# 
# Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013


import numpy as np
import math
import scipy.spatial.distance as spdist



[docs]class Kernel(object):
    """
    This is a base class of Kernel functions
    there is no computation in this class, it just defines rules about a kernel class should have
    each covariance function will inherit it and implement its own behaviour
    """
    def __init__(self):
        self.hyp = []
        self.para = []


[docs]    def getCovMatrix(self,x=None,z=None,mode=None):
        '''
        Return the specific covariance matrix according to input mode

        :param x: training data
        :param z: test data
        :param str mode: 'self_test' return self covariance matrix of test data(test by 1). 
        'train' return training covariance matrix(train by train).
        'cross' return cross covariance matrix between x and z(train by test)

        :return: the corresponding covariance matrix
        '''
        pass



[docs]    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        '''
        Compute derivatives wrt. hyperparameters according to input mode

        :param x: training data
        :param z: test data
        :param str mode: 'self_test' return self derivative matrix of test data(test by 1). 
        'train' return training derivative matrix(train by train).
        'cross' return cross derivative matrix between x and z(train by test)
        :param int der: index of hyperparameter whose derivative to be computed

        :return: the corresponding derivative matrix
        '''
        pass

    # overloading 

    def __add__(self,cov):
        '''
        Overloading + operator.
        
        :param cov: covariance function
        :return: an instance of SumOfKernel
        '''
        return SumOfKernel(self,cov)

    # overloading 
    def __mul__(self,other):
        '''
        Overloading * operator.
        Using * for both multiplication with scalar and product of kernels
        depending on the type of the two objects.
        
        :param other: covariance function as product or int/float as scalar
        :return: an instance of ScaleOfKernel or ProductOfKernel
        '''
        if isinstance(other, int) or isinstance(other, float):
            return ScaleOfKernel(self,other)
        elif isinstance(other, Kernel):
            return ProductOfKernel(self,other)
        else:
            print "only numbers and Kernels are supported operand types for *"

    # overloading 
    __rmul__ = __mul__


[docs]    def fitc(self,inducingInput):
        '''
        Covariance function to be used together with the FITC approximation.
        Setting FITC gp model will implicitly call this method.

        :return: an instance of FITCOfKernel
        '''
        return FITCOfKernel(self,inducingInput)
    
    # can be replaced by spdist from scipy


[docs]    def sq_dist(self, a, b=None):
        '''Compute a matrix of all pairwise squared distances
        between two sets of vectors, stored in the row of the two matrices:
        a (of size n by D) and b (of size m by D).'''
        n = a.shape[0]
        D = a.shape[1]
        m = n    
        if b == None:
            b = a.transpose()
        else:
            m = b.shape[0]
            b = b.transpose()
        C = np.zeros((n,m))
        for d in range(0,D):
            tt  = a[:,d]
            tt  = tt.reshape(n,1)
            tem = np.kron(np.ones((1,m)), tt)
            tem = tem - np.kron(np.ones((n,1)), b[d,:])
            C   = C + tem * tem 
        return C

              
        


[docs]class ProductOfKernel(Kernel):
    '''Product of two kernel function.'''
    def __init__(self,cov1,cov2):
        self.cov1 = cov1
        self.cov2 = cov2
        self._hyp = cov1.hyp + cov2.hyp

    def sethyp(self,hyp):
        assert len(hyp) == len(self._hyp)
        len1 = len(self.cov1.hyp)
        self._hyp = hyp 
        self.cov1.hyp = self._hyp[:len1]
        self.cov2.hyp = self._hyp[len1:]
    def gethyp(self):
        return self._hyp
    hyp = property(gethyp,sethyp)

    def getCovMatrix(self,x=None,z=None,mode=None):
        A = self.cov1.getCovMatrix(x,z,mode) * self.cov2.getCovMatrix(x,z,mode)
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        if der < len(self.cov1.hyp):
            A = self.cov1.getDerMatrix(x,z,mode,der) * self.cov2.getCovMatrix(x,z,mode)
        elif der < len(self.hyp):
            der2 = der - len(self.cov1.hyp)
            A = self.cov2.getDerMatrix(x,z,mode,der2) * self.cov1.getCovMatrix(x,z,mode)
        else:
            raise Exception("Error: der out of range for covProduct") 
        return A





[docs]class SumOfKernel(Kernel):
    '''Sum of two kernel function.'''
    def __init__(self,cov1,cov2):
        self.cov1 = cov1
        self.cov2 = cov2
        self._hyp = cov1.hyp + cov2.hyp
    def sethyp(self,hyp):
        assert len(hyp) == len(self._hyp)
        len1 = len(self.cov1.hyp)
        self._hyp = hyp 
        self.cov1.hyp = self._hyp[:len1]
        self.cov2.hyp = self._hyp[len1:]
    def gethyp(self):
        return self._hyp
    hyp = property(gethyp,sethyp)

    def getCovMatrix(self,x=None,z=None,mode=None):
        A = self.cov1.getCovMatrix(x,z,mode) + self.cov2.getCovMatrix(x,z,mode)
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        if der < len(self.cov1.hyp):
            A = self.cov1.getDerMatrix(x,z,mode,der) 
        elif der < len(self.hyp):
            der2 = der - len(self.cov1.hyp)
            A = self.cov2.getDerMatrix(x,z,mode,der2) 
        else:
            raise Exception("Error: der out of range for covSum") 
        return A





[docs]class ScaleOfKernel(Kernel):
    '''Scale of a kernel function.'''
    def __init__(self,cov,scalar):
        self.cov = cov
        if cov.hyp:
            self._hyp = [scalar] + cov.hyp 
        else:
            self._hyp = [scalar]
    def sethyp(self,hyp):
        assert len(hyp) == len(self._hyp)
        self._hyp = hyp 
        self.cov.hyp = self._hyp[1:]
    def gethyp(self):
        return self._hyp
    hyp = property(gethyp,sethyp)

    def getCovMatrix(self,x=None,z=None,mode=None):
        sf2 = np.exp(self.hyp[0])                     # scale parameter  
        A = sf2 * self.cov.getCovMatrix(x,z,mode)     # accumulate cov 
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        sf2 = np.exp(self.hyp[0])                     # scale parameter  
        if der == 0:                                  # compute derivative w.r.t. sf2
            A = 2. * sf2 * self.cov.getCovMatrix(x,z,mode)
        else:                                 
            A = sf2 * self.cov.getDerMatrix(x,z,mode,der-1) 
        return A
     




[docs]class FITCOfKernel(Kernel):
    '''
    Covariance function to be used together with the FITC approximation.
    The function allows for more than one output argument and does not respect the
    interface of a proper covariance function. 
    Instead of outputing the full covariance, it returns cross-covariances between
    the inputs x, z and the inducing inputs xu as needed by infFITC
    '''
    def __init__(self,cov,inducingInput):
        self.inducingInput = inducingInput
        self.covfunc = cov
        self._hyp = cov.hyp

    def getHyp(self):
        return self._hyp
    def setHyp(self, hyp):
        self._hyp = hyp
        self.covfunc.hyp = hyp
    hyp = property(getHyp,setHyp)

    def getCovMatrix(self,x=None,z=None,mode=None):
        xu = self.inducingInput
        if x != None:
            try:
                assert(xu.shape[1] == x.shape[1])
            except AssertionError:
                raise Exception('Dimensionality of inducing inputs must match training inputs') 
        if mode == 'self_test':           # self covariances for the test cases
            K = self.covfunc.getCovMatrix(z=z,mode='self_test')
            return K
        elif mode == 'train':             # compute covariance matix for training set
            K   = self.covfunc.getCovMatrix(z=x,mode='self_test')
            Kuu = self.covfunc.getCovMatrix(x=xu,mode='train')
            Ku  = self.covfunc.getCovMatrix(x=xu,z=x,mode='cross')
            return K, Kuu, Ku
        elif mode == 'cross':             # compute covariance between data sets x and z 
            K = self.covfunc.getCovMatrix(x=xu,z=z,mode='cross')
            return K 

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        xu = self.inducingInput
        if x!= None:
            try:
                assert(xu.shape[1] == x.shape[1])
            except AssertionError:
                raise Exception('Dimensionality of inducing inputs must match training inputs') 
        if mode == 'self_test':           # self covariances for the test cases
            K = self.covfunc.getDerMatrix(z=z,mode='self_test',der=der)
            return K
        elif mode == 'train':             # compute covariance matix for training set
            K   = self.covfunc.getDerMatrix(z=x,mode='self_test',der=der)
            Kuu = self.covfunc.getDerMatrix(x=xu,mode='train',der=der)
            Ku  = self.covfunc.getDerMatrix(x=xu,z=x,mode='cross',der=der)
            return K, Kuu, Ku
        elif mode == 'cross':             # compute covariance between data sets x and z 
            K = self.covfunc.getDerMatrix(x=xu,z=z,mode='cross',der=der)
            return K 
        




[docs]class Poly(Kernel):
    '''
    Polynomial covariance function. hyp = [ log_c, log_sigma ]

    :param log_c: inhomogeneous offset. 
    :param log_sigma: signal deviation. 
    :param d: degree of polynomial (not treated as hyperparameter, i.e. will not be trained). 
    '''
    def __init__(self, log_c=0., d=2, log_sigma=0. ):
        self.hyp = [log_c, log_sigma]
        self.para = [d] 

    def getCovMatrix(self,x=None,z=None,mode=None):
        c   = np.exp(self.hyp[0])             # inhomogeneous offset
        sf2 = np.exp(2.*self.hyp[1])          # signal variance
        ord = self.para[0]                    # order of polynomial
        if np.abs(ord-np.round(ord)) < 1e-8:  # remove numerical error from format of parameter
            ord = int(round(ord))
        assert(ord >= 1.)                     # only nonzero integers for ord
        ord = int(ord) 
        if mode == 'self_test':               # self covariances for the test cases
            nn,D = z.shape
            A = np.reshape(np.sum(z*z,1), (nn,1))
        elif mode == 'train':                 # compute covariance matix for dataset x
            A = np.dot(x,x.T)
        elif mode == 'cross':                 # compute covariance between data sets x and z 
            A = np.dot(x,z.T) 
        A = sf2 * (c + A)**ord
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        c   = np.exp(self.hyp[0])             # inhomogeneous offset
        sf2 = np.exp(2.*self.hyp[1])          # signal variance
        ord = self.para[0]                    # order of polynomial
        if np.abs(ord-np.round(ord)) < 1e-8:  # remove numerical error from format of parameter
            ord = int(round(ord))
        assert(ord >= 1.)                     # only nonzero integers for ord
        ord = int(ord) 
        if mode == 'self_test':               # self covariances for the test cases
            nn,D = z.shape
            A = np.reshape(np.sum(z*z,1), (nn,1))
        elif mode == 'train':                 # compute covariance matix for dataset x
            A = np.dot(x,x.T)
        elif mode == 'cross':                 # compute covariance between data sets x and z 
            A = np.dot(x,z.T) 
        if der == 0:                          # compute derivative matrix wrt 1st parameter             
            A = c * ord * sf2 * (c+A)**(ord-1)
        elif der == 1:                        # compute derivative matrix wrt 2nd parameter
            A = 2. * sf2 * (c + A)**ord
        elif der == 2:                        # no derivative wrt 3rd parameter
            A = np.zeros_like(A)              # do nothing (d is not learned)
        else:
            raise Exception("Wrong derivative entry in Poly")
        return A
        




[docs]class PiecePoly(Kernel):
    '''
    Piecewise polynomial kernel with compact support. 
    hyp = [log_ell, log_sigma]

    :param log_ell: characteristic length scale. 
    :param log_sigma: signal deviation. 
    :param v: degree v will be rounded to 0,1,2,or 3. (not treated as hyperparameter, i.e. will not be trained). 
    '''
    def __init__(self, log_ell=0., v=2, log_sigma=0. ):
        self.hyp = [log_ell, log_sigma]
        self.para = [v]

    def ppmax(self,A,B):
        return np.maximum(A,B*np.ones_like(A))

    def func(self,v,r,j):
        if v == 0:
            return 1
        elif v == 1:
            return ( 1. + (j+1) * r )
        elif v == 2:
            return ( 1. + (j+2)*r + (j*j + 4.*j+ 3)/3.*r*r )
        elif v == 3:
            return ( 1. + (j+3)*r + (6.*j*j+36.*j+45.)/15.*r*r + (j*j*j+9.*j*j+23.*j+15.)/15.*r*r*r )
        else:
             raise Exception (["Wrong degree in PiecePoly.  Should be 0,1,2 or 3, is " + str(v)])

    def dfunc(self,v,r,j):
        if v == 0:
            return 0
        elif v == 1:
            return ( j+1 )
        elif v == 2:
            return ( (j+2) + 2.*(j*j+ 4.*j+ 3.)/3.*r )
        elif v == 3:
            return ( (j+3) + 2.*(6.*j*j+36.*j+45.)/15.*r + (j*j*j+9.*j*j+23.*j+15.)/5.*r*r )
        else:
            raise Exception (["Wrong degree in PiecePoly.  Should be 0,1,2 or 3, is " + str(v)])

    def pp(self,r,j,v,func):
        return func(v,r,j)*(self.ppmax(1-r,0)**(j+v))

    def dpp(self,r,j,v,func,dfunc):
        return self.ppmax(1-r,0)**(j+v-1) * r * ( (j+v)*self.func(v,r,j) - self.ppmax(1-r,0) * self.dfunc(v,r,j) )

    def getCovMatrix(self,x=None,z=None,mode=None):
        if x != None:
            n, D = x.shape
        if z != None:
            nn, D = z.shape
        ell = np.exp(self.hyp[0])            # characteristic length scale
        sf2 = np.exp(2.*self.hyp[1])         # signal variance
        v   = self.para[0]                   # degree (v = 0,1,2 or 3 only)
        if np.abs(v-np.round(v)) < 1e-8:     # remove numerical error from format of parameter
            v = int(round(v))
        assert(int(v) in range(4))           # Only allowed degrees: 0,1,2 or 3
        v = int(v)
        j = np.floor(0.5*D) + v + 1
        if mode == 'self_test':              # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':                # compute covariance matix for dataset x
            A = np.sqrt( spdist.cdist(x/ell, x/ell, 'sqeuclidean') )
        elif mode == 'cross':                # compute covariance between data sets x and z 
            A = np.sqrt( spdist.cdist(x/ell, z/ell, 'sqeuclidean') )
        A = sf2 * self.pp(A,j,v,self.func)
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        if x != None:
            n, D = x.shape
        if z != None:
            nn, D = z.shape
        ell = np.exp(self.hyp[0])            # characteristic length scale
        sf2 = np.exp(2.*self.hyp[1])         # signal variance
        v   = self.para[0]                   # degree (v = 0,1,2 or 3 only)
        if np.abs(v-np.round(v)) < 1e-8:     # remove numerical error from format of parameter
            v = int(round(v))
        assert(int(v) in range(4))           # Only allowed degrees: 0,1,2 or 3
        v = int(v)
        j = np.floor(0.5*D) + v + 1
        if mode == 'self_test':              # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':                # compute covariance matix for dataset x
            A = np.sqrt( spdist.cdist(x/ell, x/ell, 'sqeuclidean') )
        elif mode == 'cross':                # compute covariance between data sets x and z 
            A = np.sqrt( spdist.cdist(x/ell, z/ell, 'sqeuclidean') )
        if der == 0:                            # compute derivative matrix wrt 1st parameter
            A = sf2 * self.dpp(A,j,v,self.func,self.dfunc)
        elif der == 1:                          # compute derivative matrix wrt 2nd parameter
            A = 2. * sf2 * self.pp(A,j,v,self.func)
        elif der == 2:                          # wants to compute derivative wrt order
            A = np.zeros_like(A)
        else:
            raise Exception("Wrong derivative entry in PiecePoly")
        return A





[docs]class RBF(Kernel):
    '''
    Squared Exponential kernel with isotropic distance measure. hyp = [log_ell, log_sigma]

    :param log_ell: characteristic length scale. 
    :param log_sigma: signal deviation. 
    '''
    def __init__(self, log_ell=0., log_sigma=0.):
        self.hyp = [log_ell, log_sigma]

    def getCovMatrix(self,x=None,z=None,mode=None):
        ell = np.exp(self.hyp[0])         # characteristic length scale
        sf2 = np.exp(2.*self.hyp[1])      # signal variance
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for training set
            A = spdist.cdist(x/ell,x/ell,'sqeuclidean')
        elif mode == 'cross':             # compute covariance between data sets x and z 
            A = spdist.cdist(x/ell,z/ell,'sqeuclidean') 
        A = sf2 * np.exp(-0.5*A)
        return A


    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        ell = np.exp(self.hyp[0])         # characteristic length scale
        sf2 = np.exp(2.*self.hyp[1])      # signal variance
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            A = spdist.cdist(x/ell,x/ell,'sqeuclidean')
        elif mode == 'cross':             # compute covariance between data sets x and z 
            A = spdist.cdist(x/ell,z/ell,'sqeuclidean') 
        if der == 0:    # compute derivative matrix wrt 1st parameter
            A = sf2 * np.exp(-0.5*A) * A
        elif der == 1:  # compute derivative matrix wrt 2nd parameter
            A = 2. * sf2 * np.exp(-0.5*A)
        else:
            raise Exception("Calling for a derivative in RBF that does not exist")
        return A
        




[docs]class RBFunit(Kernel):
    '''
    Squared Exponential kernel with isotropic distance measure with unit magnitude.
    i.e signal variance is always 1. hyp = [ log_ell ]

    :param log_ell: characteristic length scale. 
    '''
    def __init__(self, log_ell=0.):
        self.hyp = [log_ell]

    def getCovMatrix(self,x=None,z=None,mode=None):
        ell = np.exp(self.hyp[0])         # characteristic length scale
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            A = spdist.cdist(x/ell,x/ell,'sqeuclidean')
        elif mode == 'cross':             # compute covariance between data sets x and z 
            A = spdist.cdist(x/ell,z/ell,'sqeuclidean') 
        A = np.exp(-0.5*A)
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        ell = np.exp(self.hyp[0])         # characteristic length scale
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            A = spdist.cdist(x/ell,x/ell,'sqeuclidean')
        elif mode == 'cross':             # compute covariance between data sets x and z 
            A = spdist.cdist(x/ell,z/ell,'sqeuclidean') 
        if der == 0:           # compute derivative matrix wrt 1st parameter
            A = np.exp(-0.5*A) * A
        else:
            raise Exception("Wrong derivative index in RDFunit")
        return A





[docs]class RBFard(Kernel):
    '''
    Squared Exponential kernel with Automatic Relevance Determination.
    hyp = log_ell_list + [log_sigma]

    :param D: dimension of pattern. set if you want default ell, which is 1 for each dimension.
    :param log_ell_list: characteristic length scale for each dimension.
    :param log_sigma: signal deviation. 
    '''
    def __init__(self, D=None, log_ell_list=None, log_sigma=0.):
        if log_ell_list == None:
            self.hyp = [0. for i in xrange(D)] + [log_sigma]
        else:
            self.hyp = log_ell_list + [log_sigma]

    def getCovMatrix(self,x=None,z=None,mode=None):
        if x != None:
            n, D = x.shape
        if z != None:
            nn, D = z.shape
        ell = 1./np.exp(self.hyp[0:D])    # characteristic length scale
        sf2 = np.exp(2.*self.hyp[D])      # signal variance
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            tem = np.dot(np.diag(ell),x.T).T
            A = spdist.cdist(tem,tem,'sqeuclidean')
        elif mode == 'cross':             # compute covariance between data sets x and z 
            A = spdist.cdist(np.dot(np.diag(ell),x.T).T,np.dot(np.diag(ell),z.T).T,'sqeuclidean')
        A = sf2*np.exp(-0.5*A)
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        if x != None:
            n, D = x.shape
        if z != None:
            nn, D = z.shape
        ell = 1./np.exp(self.hyp[0:D])    # characteristic length scale
        sf2 = np.exp(2.*self.hyp[D])      # signal variance
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            tem = np.dot(np.diag(ell),x.T).T
            A = spdist.cdist(tem,tem,'sqeuclidean')
        elif mode == 'cross':             # compute covariance between data sets x and z 
            A = spdist.cdist(np.dot(np.diag(ell),x.T).T,np.dot(np.diag(ell),z.T).T,'sqeuclidean')
        A = sf2*np.exp(-0.5*A)
        if der < D:                       # compute derivative matrix wrt length scale parameters
            if mode == 'self_test':
                A = A*0
            elif mode == 'train':
                tem = np.atleast_2d(x[:,der])/ell[der]
                A *= spdist.cdist(tem,tem,'sqeuclidean')
            elif mode == 'cross':
                A *= spdist.cdist(np.atleast_2d(x[:,der]).T/ell[der],np.atleast_2d(z[:,der]).T/ell[der],'sqeuclidean')
        elif der == D:                    # compute derivative matrix wrt magnitude parameter
            A = 2.*A
        else:
            raise Exception("Wrong derivative index in RDFard") 
        return A





[docs]class Const(Kernel):
    '''
    Constant kernel. hyp = [ log_sigma ]

    :param log_sigma: signal deviation. 
    '''
    def __init__(self, log_sigma=0.):
        self.hyp = [log_sigma]

    def getCovMatrix(self,x=None,z=None,mode=None):
        sf2 = np.exp(self.hyp[0])         # s2
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = sf2 * np.ones((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            n,D = x.shape
            A = sf2 * np.ones((n,n))
        elif mode == 'cross':             # compute covariance between data sets x and z 
            n,D  = x.shape
            nn,D = z.shape
            A = sf2 * np.ones((n,nn))
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        sf2 = np.exp(self.hyp[0])         # s2
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = sf2 * np.ones((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            n,D = x.shape
            A = sf2 * np.ones((n,n))
        elif mode == 'cross':             # compute covariance between data sets x and z 
            n,D  = x.shape
            nn,D = z.shape
            A = sf2 * np.ones((n,nn))
        if der == 0:                      # compute derivative matrix wrt sf2
            A = 2. * A
        else:
            raise Exception("Wrong derivative entry in covConst")
        return A
        




[docs]class Linear(Kernel):
    '''
    Linear kernel. hyp = [ log_sigma ].
    '''
    def __init__(self, log_sigma=0.):
        self.hyp = [ log_sigma ]

    def getCovMatrix(self,x=None,z=None,mode=None):
        sf2 = np.exp(self.hyp[0])         # s2
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.reshape(np.sum(z*z,1), (nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            n,D = x.shape
            A = np.dot(x,x.T) + np.eye(n)*1e-16    # required for numerical accuracy
        elif mode == 'cross':             # compute covariance between data sets x and z 
            A = np.dot(x,z.T)
        A = sf2 * A
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        sf2 = np.exp(self.hyp[0])         # s2
        if der == 0:
            if mode == 'self_test':           # self covariances for the test cases
                nn,D = z.shape
                A = np.reshape(np.sum(z*z,1), (nn,1))
            elif mode == 'train':             # compute covariance matix for dataset x
                n,D = x.shape
                A = np.dot(x,x.T) + np.eye(n)*1e-16    # required for numerical accuracy
            elif mode == 'cross':             # compute covariance between data sets x and z 
                A = np.dot(x,z.T)
            A = 2 * sf2 * A
        else:
            raise Exception("Wrong derivative index in covLinear")
        return A





[docs]class LINard(Kernel):
    '''
    Linear covariance function with Automatic Relevance Detemination.
    hyp = log_ell_list 

    :param D: dimension of training data. Set if you want default ell, which is 1 for each dimension.
    :param log_ell_list: characteristic length scale for each dimension.
    '''
    def __init__(self, D=None, log_ell_list=None):
        if log_ell_list == None:
            self.hyp = [0. for i in xrange(D)]
        else:
            self.hyp = log_ell_list

    def getCovMatrix(self,x=None,z=None,mode=None):
        ell = np.exp(self.hyp)            # ARD parameters
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.reshape(np.sum(z*z,1), (nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            A = np.dot(x,x.T) 
        elif mode == 'cross':             # compute covariance between data sets x and z
            z = np.dot(z,np.diag(1./ell))
            A = np.dot(x,z.T) 
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        ell = np.exp(self.hyp)            # ARD parameters
        if x != None:
            n, D = x.shape
        if z != None:
            nn, D = z.shape
        if der < D:
            if mode == 'self_test':
                tem = np.atleast_2d(z[:,der]).T
                A = -2.* tem * tem
            elif mode == 'train':
                A = -2.*np.dot(np.atleast_2d(x[:,der]).T,np.atleast_2d(x[:,der]))
            elif mode == 'cross':
                z = np.dot(z,np.diag(1./ell))
                A = -2.*np.dot(np.atleast_2d(x[:,der]).T, np.atleast_2d(z[:,der])) # cross covariances
        else:
            raise Exception("Wrong derivative index in covLINard")
        return A





[docs]class Matern(Kernel):
    '''
    Matern covariance function with nu = d/2 and isotropic distance measure. 
    For d=1 the function is also known as the exponential covariance function 
    or the Ornstein-Uhlenbeck covariance in 1d.
    d will be rounded to 1, 3, or 5.
    hyp = [ log_ell, log_sigma]
    
    :param d: d is 2 times nu. Can only be 1,3 or 5.
    :param log_ell: characteristic length scale. 
    :param log_sigma: signal deviation. 
    '''
    def __init__(self, log_ell=0., d=3, log_sigma=0. ):
        self.hyp = [ log_ell, log_sigma ]
        self.para = [d]

    def func(self,d,t):
        if d == 1:
            return 1
        elif d == 3:
            return 1 + t
        elif d == 5:
            return 1 + t*(1+t/3.)
        else:
            raise Exception("Wrong value for d in Matern")

    def dfunc(self,d,t):
        if d == 1:
            return 1
        elif d == 3:
            return t
        elif d == 5:
            return t*(1+t/3.)
        else:
            raise Exception("Wrong value for d in Matern")

    def mfunc(self,d,t):
        return self.func(d,t)*np.exp(-1.*t)

    def dmfunc(self,d,t):
        return self.dfunc(d,t)*t*np.exp(-1.*t)

    def getCovMatrix(self,x=None,z=None,mode=None):
        ell = np.exp(self.hyp[0])        # characteristic length scale
        sf2 = np.exp(2.* self.hyp[1])    # signal variance
        d   = self.para[0]               # 2 times nu
        if np.abs(d-np.round(d)) < 1e-8: # remove numerical error from format of parameter
            d = int(round(d))
        d = int(d)
        try:
            assert(d in [1,3,5])         # check for valid values of d
        except AssertionError:
            print "Warning: You specified d to be neither 1,3 nor 5. We set to d=3. "
            d = 3
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            x = np.sqrt(d)*x/ell   
            A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean'))
        elif mode == 'cross':             # compute covariance between data sets x and z
            x = np.sqrt(d)*x/ell
            z = np.sqrt(d)*z/ell
            A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean'))
        A = sf2 * self.mfunc(d,A)
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        ell = np.exp(self.hyp[0])        # characteristic length scale
        sf2 = np.exp(2.* self.hyp[1])    # signal variance
        d   = self.para[0]               # 2 times nu
        if np.abs(d-np.round(d)) < 1e-8: # remove numerical error from format of parameter
            d = int(round(d))
        d = int(d)
        try:
            assert(d in [1,3,5])         # check for valid values of d
        except AssertionError:
            print "Warning: You specified d to be neither 1,3 nor 5. We set to d=3. "
            d = 3
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            x = np.sqrt(d)*x/ell   
            A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean'))
        elif mode == 'cross':             # compute covariance between data sets x and z
            x = np.sqrt(d)*x/ell
            z = np.sqrt(d)*z/ell
            A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean'))
        A = sf2 * self.mfunc(d,A)
        if der == 0:                    # compute derivative matrix wrt 1st parameter
            A = sf2 * self.dmfunc(d,A)
        elif der == 1:                  # compute derivative matrix wrt 2nd parameter
            A = 2 * sf2 * self.mfunc(d,A)
        elif der == 2:                  # no derivative wrt 3rd parameter
            A = np.zeros_like(A)        # do nothing (d is not learned)
        else:
            raise Exception("Wrong derivative value in Matern")
        return A
        




[docs]class Periodic(Kernel):
    '''
    Stationary kernel for a smooth periodic function. 
    hyp = [ log_ell, log_p, log_sigma]

    :param log_p: period.
    :param log_ell: characteristic length scale. 
    :param log_sigma: signal deviation. 
    '''
    def __init__(self, log_ell=0., log_p=0., log_sigma=0. ): 
        self.hyp = [ log_ell, log_p, log_sigma]

    def getCovMatrix(self,x=None,z=None,mode=None):
        ell = np.exp(self.hyp[0])        # characteristic length scale
        p   = np.exp(self.hyp[1])        # period
        sf2 = np.exp(2.*self.hyp[2])     # signal variance
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean'))
        elif mode == 'cross':             # compute covariance between data sets x and z
            A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean'))
        A = np.pi*A/p
        A = np.sin(A)/ell
        A = A * A
        A = sf2 *np.exp(-2.*A)
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        ell = np.exp(self.hyp[0])        # characteristic length scale
        p   = np.exp(self.hyp[1])        # period
        sf2 = np.exp(2.*self.hyp[2])     # signal variance
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean'))
        elif mode == 'cross':             # compute covariance between data sets x and z
            A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean'))
        A = np.pi*A/p
        if der == 0:            # compute derivative matrix wrt 1st parameter
            A = np.sin(A)/ell
            A = A * A
            A = 4. *sf2 *np.exp(-2.*A) * A
        elif der == 1:          # compute derivative matrix wrt 2nd parameter
            R = np.sin(A)/ell
            A = 4 * sf2/ell * np.exp(-2.*R*R)*R*np.cos(A)*A
        elif der == 2:          # compute derivative matrix wrt 3rd parameter
            A = np.sin(A)/ell
            A = A * A
            A = 2. * sf2 * np.exp(-2.*A)
        else:
            raise Exception("Wrong derivative index in covPeriodic")
        return A
        




[docs]class Noise(Kernel):
    '''
    Independent covariance function, i.e "white noise", with specified variance.
    Normally NOT used anymore since noise is now added in liklihood.
    hyp = [ log_sigma ]

    :param log_sigma: signal deviation. 
    '''
    def __init__(self, log_sigma=0.):
        self.hyp = [log_sigma]

    def getCovMatrix(self,x=None,z=None,mode=None):
        tol = 1.e-9                       # Tolerance for declaring two vectors "equal"
        s2 = np.exp(2.*self.hyp[0])       # noise variance
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            n,D = x.shape
            A = np.eye(n)
        elif mode == 'cross':             # compute covariance between data sets x and z
            M = spdist.cdist(x, z, 'sqeuclidean')
            A = np.zeros_like(M,dtype=np.float)
            A[M < tol] = 1.
        A = s2*A
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        tol = 1.e-9                       # Tolerance for declaring two vectors "equal"
        s2 = np.exp(2.*self.hyp[0])       # noise variance
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            n,D = x.shape
            A = np.eye(n)
        elif mode == 'cross':             # compute covariance between data sets x and z
            M = spdist.cdist(x, z, 'sqeuclidean')
            A = np.zeros_like(M,dtype=np.float)
            A[M < tol] = 1.
        if der == 0:
            A = 2.*s2*A
        else:
            raise Exception("Wrong derivative index in covNoise")
        return A
        




[docs]class RQ(Kernel):
    ''' 
    Rational Quadratic covariance function with isotropic distance measure.
    hyp = [ log_ell, log_sigma, log_alpha ]
    
    :param log_ell: characteristic length scale. 
    :param log_sigma: signal deviation.
    :param log_alpha: shape parameter for the RQ covariance.
    '''
    def __init__(self, log_ell=0., log_sigma=0., log_alpha=0.):
        self.hyp = [ log_ell, log_sigma, log_alpha ]

    def getCovMatrix(self,x=None,z=None,mode=None):
        ell   = np.exp(self.hyp[0])       # characteristic length scale
        sf2   = np.exp(2.*self.hyp[1])    # signal variance
        alpha = np.exp(self.hyp[2])
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            D2 = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            D2 = spdist.cdist(x/ell, x/ell, 'sqeuclidean')
        elif mode == 'cross':             # compute covariance between data sets x and z
            D2 = spdist.cdist(x/ell, z/ell, 'sqeuclidean')
        A = sf2 * ( ( 1.0 + 0.5*D2/alpha )**(-alpha) )
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        ell   = np.exp(self.hyp[0])       # characteristic length scale
        sf2   = np.exp(2.*self.hyp[1])    # signal variance
        alpha = np.exp(self.hyp[2])
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            D2 = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            D2 = spdist.cdist(x/ell, x/ell, 'sqeuclidean')
        elif mode == 'cross':             # compute covariance between data sets x and z
            D2 = spdist.cdist(x/ell, z/ell, 'sqeuclidean')
        if der == 0:                # compute derivative matrix wrt 1st parameter
            A = sf2 * ( 1.0 + 0.5*D2/alpha )**(-alpha-1) * D2
        elif der == 1:              # compute derivative matrix wrt 2nd parameter
            A = 2.* sf2 * ( ( 1.0 + 0.5*D2/alpha )**(-alpha) )
        elif der == 2:              # compute derivative matrix wrt 3rd parameter
            K = ( 1.0 + 0.5*D2/alpha )
            A = sf2 * K**(-alpha) * (0.5*D2/K - alpha*np.log(K) )
        else:
            raise Exception("Wrong derivative index in covRQ")
        return A





[docs]class RQard(Kernel):
    '''
    Rational Quadratic covariance function with Automatic Relevance Detemination
    (ARD) distance measure.
    hyp = log_ell_list + [ log_sigma, log_alpha ]

    :param D: dimension of pattern. set if you want default ell, which is 0.5 for each dimension.
    :param log_ell_list: characteristic length scale for each dimension.
    :param log_sigma: signal deviation. 
    :param log_alpha: shape parameter for the RQ covariance.
    '''
    def __init__(self, D=None, log_ell_list=None, log_sigma=0., log_alpha=0.):
        if log_ell_list == None:
            self.hyp = [0. for i in xrange(D)] + [ log_sigma, log_alpha ]
        else:
            self.hyp = log_ell_list + [ log_sigma, log_alpha ]

    def getCovMatrix(self,x=None,z=None,mode=None):
        if x != None:
            n, D = x.shape
        if z != None:
            nn, D = z.shape
        ell = 1./np.exp(self.hyp[0:D])    # characteristic length scale
        sf2 = np.exp(2.*self.hyp[D])      # signal variance
        alpha = np.exp(self.hyp[D+1])
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            D2 = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            tmp = np.dot(np.diag(ell),x.T).T
            D2 = spdist.cdist(tmp, tmp, 'sqeuclidean')
        elif mode == 'cross':             # compute covariance between data sets x and z
            D2 = spdist.cdist(np.dot(np.diag(ell),x.T).T, np.dot(np.diag(ell),z.T).T, 'sqeuclidean')
        A = sf2 * ( ( 1.0 + 0.5*D2/alpha )**(-alpha) )
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        if x != None:
            n, D = x.shape
        if z != None:
            nn, D = z.shape
        ell = 1./np.exp(self.hyp[0:D])    # characteristic length scale
        sf2 = np.exp(2.*self.hyp[D])      # signal variance
        alpha = np.exp(self.hyp[D+1])
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            D2 = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            tmp = np.dot(np.diag(ell),x.T).T
            D2 = spdist.cdist(tmp, tmp, 'sqeuclidean')
        elif mode == 'cross':             # compute covariance between data sets x and z
            D2 = spdist.cdist(np.dot(np.diag(ell),x.T).T, np.dot(np.diag(ell),z.T).T, 'sqeuclidean')
        if der < D:
            if mode == 'self_test':
                A = D2*0
            elif mode == 'train':
                tmp = np.atleast_2d(x[:,der])/ell[der]
                A = sf2 * ( 1.0 + 0.5*D2/alpha )**(-alpha-1) * spdist.cdist(tmp, tmp, 'sqeuclidean')
            elif mode == 'cross':
                A = sf2 * ( 1.0 + 0.5*D2/alpha )**(-alpha-1) * spdist.cdist(np.atleast_2d(x[:,der]).T/ell[der], np.atleast_2d(z[:,der]).T/ell[der], 'sqeuclidean')
        elif der==D:                # compute derivative matrix wrt magnitude parameter
            A = 2. * sf2 * ( ( 1.0 + 0.5*D2/alpha )**(-alpha) )
        elif der==(D+1):            # compute derivative matrix wrt magnitude parameter
            K = ( 1.0 + 0.5*D2/alpha )
            A = sf2 * K**(-alpha) * ( 0.5*D2/K - alpha*np.log(K) )
        else:
            raise Exception("Wrong derivative index in covRQard") 
        return A
        




[docs]class Pre(Kernel):
    '''
    Precomputed kernel matrix. No hyperparameters and thus nothing will be optimised. 

    :param M1: cross covariances matrix(train+1 by test).
               last row is self covariances (diagonal of test by test)
    :param M2: training set covariance matrix (train by train)   
    '''
    def __init__(self,M1,M2):
        self.M1 = M1
        self.M2 = M2
        self.hyp = []

    def getCovMatrix(self,x=None,z=None,mode=None):
        if mode == 'self_test':           # diagonal covariance between test_test
            A = self.M1[-1,:]             # self covariances for the test cases (last row) 
            A = np.reshape(A, (A.shape[0],1))
        elif mode == 'train':             # covariance between train_train
            A = self.M2
        elif mode == 'cross':             # covariance between train_test
            A = self.M1[:-1,:]
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        if der != None:
            raise Exception("Error: NO optimization in precomputed kernel matrix")
        return 0
        
    


if __name__ == '__main__':
    pass

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© Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting. Created using Sphinx 1.2.
pyGPs-1.2/doc/build/_modules/pyGPs/Core/gp.html000066400000000000000000005443401236346020200213270ustar00rootroot00000000000000 pyGPs.Core.gp — pyGPs v1.2 documentation

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Source code for pyGPs.Core.gp

#================================================================================
#    Marion Neumann [marion dot neumann at uni-bonn dot de]
#    Daniel Marthaler [marthaler at ge dot com]
#    Shan Huang [shan dot huang at iais dot fraunhofer dot de]
#    Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]
#
#    This file is part of pyGPs.
#    The software package is released under the BSD 2-Clause (FreeBSD) License.
#
#    Copyright (c) by
#    Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014
#================================================================================

#   MEANING OF NOTATION:
#  
#   inffunc      function specifying the inference method 
#   covfunc      prior covariance function (see below)
#   meanfunc     prior mean function
#   likfunc      likelihood function
#   x            n by D matrix of training inputs
#   y            column vector of length n of training targets
#   xs           n by D matrix of test inputs
#   ys           column vector of length nn of true test targets (optional)
#   nlZ          returned value of the negative log marginal likelihood
#   dnlZ         column vector of partial derivatives of the negative
#                    log marginal likelihood w.r.t. each hyperparameter
#   ym           column vector (of length ns) of predictive output means
#   ys2          column vector (of length ns) of predictive output variances
#   fm           column vector (of length ns) of predictive latent means
#   fs2          column vector (of length ns) of predictive latent variances
#   lp           column vector (of length ns) of log predictive probabilities
#   post         struct representation of the (approximate) posterior
#                post consists of post.alpha, post.L, post.sW
#
# This is a object-oriented python implementation of gpml functionality 
# (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18).
# based on the functional-version of python implementation
# (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013)
# 
# Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013

import itertools
import numpy as np
import matplotlib.pyplot as plt
import inf, mean, lik, cov, opt
from tools import unique
from copy import deepcopy
import pyGPs

SHADEDCOLOR = [0.7539, 0.89453125, 0.62890625, 1.0]
MEANCOLOR = [ 0.2109375, 0.63385, 0.1796875, 1.0]
DATACOLOR = [0.12109375, 0.46875, 1., 1.0]


[docs]class GP(object):
    '''Base class for GP model'''
    def __init__(self):
        super(GP, self).__init__()
        self.usingDefaultMean = True  # was using default mean function now?
        self.meanfunc = None
        self.covfunc = None
        self.likfunc = None
        self.inffunc = None
        self.optimizer = None 
        self.nlZ = None           # negative log marginal likelihood
        self.dnlZ = None          # column vector of partial derivatives of the negative
                                  # log marginal likelihood w.r.t. each hyperparameter
        self.posterior = None     # struct representation of the (approximate) posterior 
        self.x = None             # n by D matrix of training inputs
        self.y = None             # column vector of length n of training targets
        self.xs = None            # n by D matrix of test inputs
        self.ys = None            # column vector of length nn of true test targets (optional)
        self.ym = None            # column vector (of length ns) of predictive output means
        self.ys2 = None           # column vector (of length ns) of predictive output variances
        self.fm = None            # column vector (of length ns) of predictive latent means
        self.fs2 = None           # column vector (of length ns) of predictive latent variances
        self.lp = None            # column vector (of length ns) of log predictive probabilities



[docs]    def setData(self, x, y):
        '''Pass training data and traning labels to model.''' 
        if x.ndim == 1:
            x = np.reshape(x, (x.shape[0],1))
        if y.ndim == 1:
            y = np.reshape(y, (y.shape[0],1))  
        self.x = x
        self.y = y
        if self.usingDefaultMean:
            c = np.mean(y)
            self.meanfunc = mean.Const(c)    # adapt default prior mean wrt. training labels



[docs]    def plotData_1d(self, axisvals=None):
        '''Plot 1d (toy) data.'''
        plt.figure()
        plt.plot(self.x, self.y, ls='None', marker='+', color=DATACOLOR, ms=12, mew=2)
        if axisvals:
            plt.axis(axisvals)
        plt.grid()
        plt.xlabel('input x')
        plt.ylabel('target y')
        plt.show()
    


[docs]    def plotData_2d(self,x1,x2,t1,t2,p1,p2,axisvals=None):
        '''Plot 2d (toy) data.'''
        fig = plt.figure()
        plt.plot(x1[:,0], x1[:,1], 'b+', markersize = 12)
        plt.plot(x2[:,0], x2[:,1], 'r+', markersize = 12)
        pc = plt.contour(t1, t2, np.reshape(p2/(p1+p2), (t1.shape[0],t1.shape[1]) ))
        fig.colorbar(pc)
        plt.grid()
        if axisvals:
            plt.axis(axisvals)
        plt.show()



[docs]    def setPrior(self, mean=None, kernel=None):
        '''Set prior mean and cov'''
        if mean != None:
            self.meanfunc = mean
            self.usingDefaultMean = False
        if kernel != None:
            self.covfunc = kernel
            if type(kernel) is cov.Pre:
                self.usingDefaultMean = False
    


[docs]    def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None):
        '''This method is used to sepecify optimization configuration. By default, gp uses a single run "minimize".

        :param method: Optimization methods. Possible values are:

                        "Minimize"   -> minimize by Carl Rasmussen (python implementation of "minimize" in GPML)

                        "CG"         -> conjugent gradient

                        "BFGS"       -> quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS)

                        "SCG"        -> scaled conjugent gradient (faster than CG) 
        :param num_restarts: Set if you want to run mulitiple times of optimization with different initial guess. 
                             It specifys the maximum number of runs/restarts/trials.
        :param min_threshold: Set if you want to run mulitiple times of optimization with different initial guess. 
                              It specifys the threshold of objective function value. Stop optimization when this value is reached.
        :param meanRange: The range of initial guess for mean hyperparameters. 
                          e.g. meanRange = [(-2,2), (-5,5), (0,1)].
                          Each tuple specifys the range (low, high) of this hyperparameter,
                          This is only the range of initial guess, during optimization process, optimal hyperparameters may go out of this range.

                          (-5,5) for each hyperparameter by default.
        :param covRange: The range of initial guess for kernel hyperparameters. Usage see meanRange
        :param likRange: The range of initial guess for likelihood hyperparameters. Usage see meanRange

        '''
        pass




[docs]    def optimize(self, x=None, y=None):
        '''
        Train optimal hyperparameters based on training data,
        adjust new hyperparameters to all mean/cov/lik functions
        '''
        if x != None:
            if x.ndim == 1:
                x = np.reshape(x, (x.shape[0],1))   
            self.x = x 

        if y != None:
            if y.ndim == 1:
                y = np.reshape(y, (y.shape[0],1))  
            self.y = y

        if self.usingDefaultMean and self.meanfunc == None:
            c = np.mean(y)
            self.meanfunc = mean.Const(c)    # adapt default prior mean wrt. training labels

        # optimize 
        optimalHyp, optimalNlZ = self.optimizer.findMin(self.x, self.y)
        self.nlZ= optimalNlZ

        # apply optimal hyp to all mean/cov/lik functions here
        self.optimizer.apply_in_objects(optimalHyp)
        self.fit()




[docs]    def fit(self, x=None, y=None, der=True):
        '''
        Fit the training data. Update negative log marginal likelihood(nlZ), 
        partial derivatives of nlZ w.r.t. each hyperparameter(dnlZ),
        and struct representation of the (approximate) posterior(post), 
        which consists of post.alpha, post.L, post.sW.
        '''
        if x != None:
            if x.ndim == 1:
                x = np.reshape(x, (x.shape[0],1))   
            self.x = x 
        if y != None:
            if y.ndim == 1:
                y = np.reshape(y, (y.shape[0],1))  
            self.y = y
        if self.usingDefaultMean and self.meanfunc == None:
            c = np.mean(y)
            self.meanfunc = mean.Const(c)    # adapt default prior mean wrt. training labels
        # call inference method
        if isinstance(self.likfunc, lik.Erf): #or is instance(self.likfunc, lik.Logistic):  
            uy = unique(self.y)        
            ind = ( uy != 1 )
            if any( uy[ind] != -1):
                raise Exception('You attempt classification using labels different from {+1,-1}')
        if not der:
            post, nlZ = self.inffunc.proceed(self.meanfunc, self.covfunc, self.likfunc, self.x, self.y, 2)
            self.nlZ = nlZ
            self.posterior = deepcopy(post)
            return nlZ, post          
        else:
            post, nlZ, dnlZ = self.inffunc.proceed(self.meanfunc, self.covfunc, self.likfunc, self.x, self.y, 3) 
            self.nlZ = nlZ 
            self.dnlZ = deepcopy(dnlZ)
            self.posterior = deepcopy(post)
            return nlZ, dnlZ, post    




[docs]    def predict(self, xs, ys=None):
        '''
        Prediction according to given inputs.
        Get predictive output means(ym), 
        predictive output variances(ys2),
        predictive latent means(fm),
        predictive latent variances(fs2),
        log predictive probabilities(lp).

        :param xs: test input
        :param ys: test target(optional)

        :return: ym, ys2, fm, fs2, lp
        '''
        meanfunc = self.meanfunc
        covfunc = self.covfunc
        likfunc = self.likfunc
        inffunc = self.inffunc
        x = self.x
        y = self.y
        if xs.ndim == 1:
            xs = np.reshape(xs, (xs.shape[0],1))   
        self.xs = xs
        if ys != None:
            if ys.ndim == 1:
                ys = np.reshape(ys, (ys.shape[0],1)) 
            self.ys = ys
        
        if self.posterior == None:   
            self.fit()        
        alpha = self.posterior.alpha
        L     = self.posterior.L
        sW    = self.posterior.sW
        
        nz = range(len(alpha[:,0]))         # non-sparse representation 
        if L == []:                         # in case L is not provided, we compute it
            K = covfunc.getCovMatrix(x=x[nz,:], mode='train')
            L = np.linalg.cholesky( (np.eye(nz) + np.dot(sW,sW.T)*K).T )
        Ltril     = np.all( np.tril(L,-1) == 0 ) # is L an upper triangular matrix?
        ns        = xs.shape[0]                  # number of data points
        nperbatch = 1000                         # number of data points per mini batch
        nact      = 0                            # number of already processed test data points
        ymu = np.zeros((ns,1))
        ys2 = np.zeros((ns,1))
        fmu = np.zeros((ns,1))
        fs2 = np.zeros((ns,1))
        lp  = np.zeros((ns,1))
        while nact<=ns-1:                              # process minibatches of test cases to save memory
            id  = range(nact,min(nact+nperbatch,ns))   # data points to process
            kss = covfunc.getCovMatrix(z=xs[id,:], mode='self_test')    # self-variances
            Ks  = covfunc.getCovMatrix(x=x[nz,:], z=xs[id,:], mode='cross')   # cross-covariances
            ms  = meanfunc.getMean(xs[id,:])         
            N   = (alpha.shape)[1]                     # number of alphas (usually 1; more in case of sampling)
            Fmu = np.tile(ms,(1,N)) + np.dot(Ks.T,alpha[nz])          # conditional mean fs|f
            fmu[id] = np.reshape(Fmu.sum(axis=1)/N,(len(id),1))       # predictive means
            if Ltril: # L is triangular => use Cholesky parameters (alpha,sW,L)
                V       = np.linalg.solve(L.T,np.tile(sW,(1,len(id)))*Ks)
                fs2[id] = kss - np.array([(V*V).sum(axis=0)]).T             # predictive variances
            else:     # L is not triangular => use alternative parametrization
                fs2[id] = kss + np.array([(Ks*np.dot(L,Ks)).sum(axis=0)]).T # predictive variances
            fs2[id] = np.maximum(fs2[id],0)            # remove numerical noise i.e. negative variances
            Fs2 = np.tile(fs2[id],(1,N))               # we have multiple values in case of sampling
            if ys == None:
                [Lp, Ymu, Ys2] = likfunc.proceed(None,Fmu[:],Fs2[:],None,None,3)
            else:
                [Lp, Ymu, Ys2] = likfunc.proceed(np.tile(ys[id],(1,N)), Fmu[:], Fs2[:],None,None,3)
            lp[id]  = np.reshape( np.reshape(Lp,(np.prod(Lp.shape),N)).sum(axis=1)/N , (len(id),1) )   # log probability; sample averaging
            ymu[id] = np.reshape( np.reshape(Ymu,(np.prod(Ymu.shape),N)).sum(axis=1)/N ,(len(id),1) )  # predictive mean ys|y and ...
            ys2[id] = np.reshape( np.reshape(Ys2,(np.prod(Ys2.shape),N)).sum(axis=1)/N , (len(id),1) ) # .. variance
            nact = id[-1]+1                  # set counter to index of next data point
        self.ym = ymu
        self.ys2 = ys2
        self.lp = lp
        self.fm = fmu
        self.fs2 = fs2
        if ys == None:
            return ymu, ys2, fmu, fs2, None
        else:
            return ymu, ys2, fmu, fs2, lp 




[docs]    def predict_with_posterior(self, post, xs, ys=None):
        '''
        Prediction with provided posterior
        (i.e. you already have the posterior and thus don't need fitting/training phases)
        Get predictive output means(ym), 
        predictive output variances(ys2),
        predictive latent means(fm),
        predictive latent variances(fs2),
        log predictive probabilities(lp).

        :param post: struct representation of posterior
        :param xs: test input
        :param ys: test target(optional)

        :return: ym, ys2, fm, fs2, lp
        '''
        meanfunc = self.meanfunc
        covfunc = self.covfunc
        likfunc = self.likfunc
        inffunc = self.inffunc
        x = self.x
        y = self.y
        if xs.ndim == 1:
            xs = np.reshape(xs, (xs.shape[0],1))   
        self.xs = xs
        if ys != None:
            if ys.ndim == 1:
                ys = np.reshape(ys, (ys.shape[0],1)) 
            self.ys = ys

        self.posterior = deepcopy(post)
        alpha = post.alpha
        L     = post.L
        sW    = post.sW

        nz = range(len(alpha[:,0]))         # non-sparse representation 
        if L == []:                         # in case L is not provided, we compute it
            K = covfunc.getCovMatrix(x=x[nz,:], mode='train')
            L = np.linalg.cholesky( (np.eye(nz) + np.dot(sW,sW.T)*K).T )
        Ltril     = np.all( np.tril(L,-1) == 0 ) # is L an upper triangular matrix?
        ns        = xs.shape[0]                  # number of data points
        nperbatch = 1000                         # number of data points per mini batch
        nact      = 0                            # number of already processed test data points
        ymu = np.zeros((ns,1))
        ys2 = np.zeros((ns,1))
        fmu = np.zeros((ns,1))
        fs2 = np.zeros((ns,1))
        lp  = np.zeros((ns,1))
        while nact<=ns-1:                              # process minibatches of test cases to save memory
            id  = range(nact,min(nact+nperbatch,ns))   # data points to process
            kss = covfunc.getCovMatrix(z=xs[id,:], mode='self_test')    # self-variances
            Ks  = covfunc.getCovMatrix(x=x[nz,:], z=xs[id,:], mode='cross')   # cross-covariances
            ms  = meanfunc.getMean(xs[id,:])         
            N   = (alpha.shape)[1]                     # number of alphas (usually 1; more in case of sampling)
            Fmu = np.tile(ms,(1,N)) + np.dot(Ks.T,alpha[nz])          # conditional mean fs|f
            fmu[id] = np.reshape(Fmu.sum(axis=1)/N,(len(id),1))       # predictive means
            if Ltril: # L is triangular => use Cholesky parameters (alpha,sW,L)
                V       = np.linalg.solve(L.T,np.tile(sW,(1,len(id)))*Ks)
                fs2[id] = kss - np.array([(V*V).sum(axis=0)]).T             # predictive variances
            else:     # L is not triangular => use alternative parametrization
                fs2[id] = kss + np.array([(Ks*np.dot(L,Ks)).sum(axis=0)]).T # predictive variances
            fs2[id] = np.maximum(fs2[id],0)            # remove numerical noise i.e. negative variances
            Fs2 = np.tile(fs2[id],(1,N))               # we have multiple values in case of sampling
            if ys == None:
                [Lp, Ymu, Ys2] = likfunc.proceed(None,Fmu[:],Fs2[:],None,None,3)
            else:
                [Lp, Ymu, Ys2] = likfunc.proceed(np.tile(ys[id],(1,N)), Fmu[:], Fs2[:],None,None,3)
            lp[id]  = np.reshape( np.reshape(Lp,(np.prod(Lp.shape),N)).sum(axis=1)/N , (len(id),1) )   # log probability; sample averaging
            ymu[id] = np.reshape( np.reshape(Ymu,(np.prod(Ymu.shape),N)).sum(axis=1)/N ,(len(id),1) )  # predictive mean ys|y and ...
            ys2[id] = np.reshape( np.reshape(Ys2,(np.prod(Ys2.shape),N)).sum(axis=1)/N , (len(id),1) ) # .. variance
            nact = id[-1]+1                  # set counter to index of next data point
        self.ym = ymu
        self.ys2 = ys2
        self.lp = lp
        self.fm = fmu
        self.fs2 = fs2
        if ys == None:
            return ymu, ys2, fmu, fs2, None
        else:
            return ymu, ys2, fmu, fs2, lp 








[docs]class GPR(GP):
    '''Gaussian Process Regression'''
    def __init__(self):
        super(GPR, self).__init__()
        self.meanfunc = mean.Zero()                        # default prior mean 
        self.covfunc = cov.RBF()                           # default prior covariance
        self.likfunc = lik.Gauss()                         # likihood with default noise variance 0.1
        self.inffunc = inf.Exact()                         # inference method
        self.optimizer = opt.Minimize(self)                # default optimizer       


[docs]    def setNoise(self,log_sigma):
        '''Set noise variance other than default'''
        self.likfunc = lik.Gauss(log_sigma)



[docs]    def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None):
        '''Set Optimizer. See base class.'''
        conf = None
        if (num_restarts!=None) or (min_threshold!=None):
            conf = pyGPs.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc)
            conf.num_restarts = num_restarts
            conf.min_threshold = min_threshold
            if meanRange != None:
                conf.meanRange = meanRange
            if covRange != None:
                conf.covRange = covRange
            if likRange != None:
                conf.likRange = likRange   
        if method == "Minimize":
            self.optimizer = opt.Minimize(self,conf)            
        elif method == "SCG":
            self.optimizer = opt.SCG(self,conf)  
        elif method == "CG":
            self.optimizer = opt.CG(self,conf)  
        elif method == "BFGS":
            self.optimizer = opt.BFGS(self,conf)  
                       


[docs]    def plot(self,axisvals=None):
        '''Plot 1d GP regression.'''
        xs = self.xs
        x = self.x
        y = self.y
        ym = self.ym
        ys2 = self.ys2
        plt.figure()
        xss  = np.reshape(xs,(xs.shape[0],))
        ymm  = np.reshape(ym,(ym.shape[0],))
        ys22 = np.reshape(ys2,(ys2.shape[0],))
        plt.plot(x, y, color=DATACOLOR, ls='None', marker='+',ms=12, mew=2)
        plt.plot(xs, ym, color=MEANCOLOR, ls='-', lw=3.)
        plt.fill_between(xss,ymm + 2.*np.sqrt(ys22), ymm - 2.*np.sqrt(ys22), facecolor=SHADEDCOLOR,linewidths=0.0)
        plt.grid()
        if axisvals != None:
            plt.axis(axisvals)
        plt.xlabel('input x')
        plt.ylabel('target y')
        plt.show()



[docs]    def useInference(self, newInf):
        '''
        Use another inference techinique other than default exact inference.
        
        :param str newInf: 'Laplace' or 'EP'
        '''

        if newInf == "Laplace":
            self.inffunc = inf.Laplace()
        elif newInf == "EP":
            self.inffunc = inf.EP()
        else:
            raise Exception('Possible inf values are "Laplace", "EP".')


    def useLikelihood(self,newLik):
        if newLik == "Laplace":
            self.likfunc = lik.Laplace()
            self.inffunc = inf.EP()
        else:
            raise Exception('Possible lik values are "Laplace".')





[docs]class GPC(GP):
    '''Gaussian Process Classification'''
    def __init__(self):
        super(GPC, self).__init__()
        self.meanfunc = mean.Zero()                        # default prior mean 
        self.covfunc = cov.RBF()                           # default prior covariance
        self.likfunc = lik.Erf()                           # erf likihood 
        self.inffunc = inf.EP()                            # default inference method
        self.optimizer = opt.Minimize(self)                # default optimizer       


[docs]    def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None):
        '''Set Optimizer. See base class.'''
        conf = None
        if (num_restarts!=None) or (min_threshold!=None):
            conf = pyGPs.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc)
            conf.num_restarts = num_restarts
            conf.min_threshold = min_threshold
            if meanRange != None:
                conf.meanRange = meanRange
            if covRange != None:
                conf.covRange = covRange
            if likRange != None:
                conf.likRange = likRange   
        if method == "Minimize":
            self.optimizer = opt.Minimize(self,conf)            
        elif method == "SCG":
            self.optimizer = opt.SCG(self,conf)  
        elif method == "CG":
            self.optimizer = opt.CG(self,conf)  
        elif method == "BFGS":
            self.optimizer = opt.BFGS(self,conf)  
                       


[docs]    def plot(self,x1,x2,t1,t2,axisvals=None): 
        '''Plot 2d gp classification.'''
        fig = plt.figure()
        plt.plot(x1[:,0], x1[:,1], 'b+', markersize = 12)
        plt.plot(x2[:,0], x2[:,1], 'r+', markersize = 12)
        pc = plt.contour(t1, t2, np.reshape(np.exp(self.lp), (t1.shape[0],t1.shape[1]) ))
        fig.colorbar(pc)
        plt.grid()
        if axisvals != None:
            plt.axis(axisvals)
        plt.show()



[docs]    def useInference(self, newInf):
        '''
        Use another inference techinique other than default EP inference.

        :param str newInf: 'Laplace'
        '''
        if newInf == "Laplace":
            self.inffunc = inf.Laplace()
        else:
            raise Exception('Possible inf values are "Laplace".')



[docs]    def useLikelihood(self,newLik):
        '''
        Use another likelihood function other than default Erf inference.
        (Not used in this version)

        :param str newLik: 'Logistic'
        '''
        if newLik == "Logistic":
            pass
            #self.likfunc = lik.Logistic()
        else:
            raise Exception('Possible lik values are "Logistic".')






[docs]class GPMC(object):
    '''This is a one vs. one classification wrapper for GPC'''
    def __init__(self, n_class):
        self.meanfunc = mean.Zero()                # default prior mean 
        self.covfunc = cov.RBF()                   # default prior covariance
        self.n_class = n_class                     # number of different classes
        self.x_all = None
        self.y_all = None
        self.newInf = None                         # new inference? -> call useInference 
        self.newLik = None                         # new likelihood? -> call useLikelihood
        self.newPrior = False


[docs]    def setPrior(self, mean=None, kernel=None):
        '''set prior mean function and covariance function'''
        if mean != None:
            self.meanfunc = mean
        if kernel != None:
            self.covfunc = kernel
            if type(kernel) is cov.Pre:
                self.usingDefaultMean = False
        self.newPrior = True



[docs]    def useInference(self, newInf):
        '''Use another inference techinique other than default EP inference.'''
        self.newInf = newInf



[docs]    def useLikelihood(self,newLik):
        '''Use another likelihood function other than default Erf inference.'''
        self.newLik = newLik    
    


[docs]    def setData(self,x,y):
        '''Set data for multi-class model'''
        if x.ndim == 1:
            x = np.reshape(x, (x.shape[0],1))  
        if y.ndim == 1:
            y = np.reshape(y, (y.shape[0],1))  
        self.x_all = x
        self.y_all = y



[docs]    def fitAndPredict(self, xs):
        '''
        Fitting model and prediction.
        predictive_vote is a matrix where
        row i is each test point i
        column j is the probability for being class j

        :return: predictive_vote
        '''
        if xs.ndim == 1:
            xs = np.reshape(xs, (xs.shape[0],1)) 
        predictive_vote = np.zeros((xs.shape[0],self.n_class))
        for i in xrange(self.n_class):         # classifier for class i...
            for j in xrange(i+1,self.n_class): # ...and class j
                x,y = self.createBinaryClass(i,j)
                model = GPC()
                if self.newPrior:
                    model.setPrior(mean=self.meanfunc, kernel=self.covfunc)
                if self.newInf:
                    model.useInference(self.newInf)
                if self.newLik:
                    model.useLikelihood(self.newLik)
                model.fit(x,y)               # fitting
                ym = model.predict(xs)[0]
                ym += 1     # now scale into 0 to 2,  ym=0 is class j, ym=2 is class i 
                vote_i = np.zeros((xs.shape[0],self.n_class))
                vote_j = np.zeros((xs.shape[0],self.n_class))
                vote_i[:,i:i+1] = ym
                vote_j[:,j:j+1] = 2-ym
                predictive_vote += vote_i
                predictive_vote += vote_j
        predictive_vote /=  predictive_vote.sum(axis=1)[:,np.newaxis]
        return predictive_vote




[docs]    def optimizeAndPredict(self, xs):
        '''
        Optimizing model and prediction.
        predictive_vote is a matrix where
        row i is each test point i
        column j is the probability for being class j

        :return: predictive_vote
        '''
        if xs.ndim == 1:
            xs = np.reshape(xs, (xs.shape[0],1)) 
        predictive_vote = np.zeros((xs.shape[0],self.n_class))
        for i in xrange(self.n_class):         # classifier for class i...
            for j in xrange(i+1,self.n_class): # ...and class j
                x,y = self.createBinaryClass(i,j)
                model = GPC()
                if self.newPrior:
                    model.setPrior(mean=self.meanfunc, kernel=self.covfunc)
                if self.newInf:
                    model.useInference(self.newInf)
                if self.newLik:
                    model.useLikelihood(self.newLik)
                model.optimize(x,y)               # training
                ym = model.predict(xs)[0]
                ym += 1     # now scale into 0 to 2,  ym=0 is class j, ym=2 is class i 
                vote_i = np.zeros((xs.shape[0],self.n_class))
                vote_j = np.zeros((xs.shape[0],self.n_class))
                vote_i[:,i:i+1] = ym
                vote_j[:,j:j+1] = 2-ym
                predictive_vote += vote_i
                predictive_vote += vote_j
        predictive_vote /=  predictive_vote.sum(axis=1)[:,np.newaxis]
        return predictive_vote



[docs]    def createBinaryClass(self, i,j):
        ''' Create dataset x(data) and y(label) which only contains class i and j.
        Relabel class i to +1 and class j to -1'''
        class_i = []
        class_j = []
        for index in xrange(len(self.y_all)):       # check all classes
            target = self.y_all[index]
            if target == i:
                class_i.append(index)
            elif target == j:
                class_j.append(index)
        n1 = len(class_i)
        n2 = len(class_j)
        class_i.extend(class_j)
        x = self.x_all[class_i,:]
        y = np.concatenate((np.ones((1,n1)),-np.ones((1,n2))),axis=1).T
        return x,y







[docs]class GP_FITC(GP):
    '''FITC GP base class'''
    def __init__(self):
        super(GP_FITC, self).__init__()
        self.u = None

    # overriding

[docs]    def setData(self, x, y, value_per_axis=5):
        '''Set training data and derive deault inducing_points.

        :param x: training data
        :param y: training target
        :param int value_per_axis: number of value in each dimension 
        when using a uni-distant default inducing points'''
        if x.ndim == 1:
            x = np.reshape(x, (x.shape[0],1))  
        if y.ndim == 1:
            y = np.reshape(y, (y.shape[0],1))  
        self.x = x
        self.y = y
        if self.usingDefaultMean:
            c = np.mean(y)
            self.meanfunc = mean.Const(c)    # adapt default prior mean wrt. training labels

        # get range of x in each dimension
        # 5 uniformally selected value for each dimension
        gridAxis=[]
        for d in xrange(x.shape[1]):
            column = x[:,d]
            mini = np.min(column)
            maxi = np.max(column)
            axis = np.linspace(mini,maxi,value_per_axis)
            gridAxis.append(axis)
        # default inducing points-> a grid
        if self.u == None:
            self.u = np.array(list(itertools.product(*gridAxis)))
            self.covfunc = self.covfunc.fitc(self.u)

    # overriding


[docs]    def setPrior(self, mean=None, kernel=None, inducing_points=None):
        '''Set GP prior and inducing points'''
        if kernel != None:
            if inducing_points != None:
                self.covfunc = kernel.fitc(inducing_points)
                self.u = inducing_points
            else:
                if self.u != None:
                    self.covfunc = kernel.fitc(self.u)
                else:
                    raise error("To use default inducing points, please call setData() first!")
            if type(kernel) is cov.Pre:
                self.usingDefaultMean = False
        if mean != None:
            self.meanfunc = mean
            self.usingDefaultMean = False






[docs]class GPR_FITC(GP_FITC):
    '''Gaussian Process Regression FITC'''
    def __init__(self):
        super(GPR_FITC, self).__init__()
        self.meanfunc = mean.Zero()                        # default prior mean 
        self.covfunc = cov.RBF()                           # default prior covariance
        self.likfunc = lik.Gauss()                         # likihood with default noise variance 0.1
        self.inffunc = inf.FITC_Exact()                    # inference method
        self.optimizer = opt.Minimize(self)                # default optimizer 
        self.u = None                                      # no default inducing points


[docs]    def setNoise(self,log_sigma):
        '''Set noise variance other than default value'''
        self.likfunc = lik.Gauss(log_sigma)



[docs]    def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None):
        '''Set Optimizer. See base class.'''
        conf = None
        if (num_restarts!=None) or (min_threshold!=None):
            conf = pyGPs.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc)
            conf.num_restarts = num_restarts
            conf.min_threshold = min_threshold
            if meanRange != None:
                conf.meanRange = meanRange
            if covRange != None:
                conf.covRange = covRange
            if likRange != None:
                conf.likRange = likRange   
        if method == "Minimize":
            self.optimizer = opt.Minimize(self,conf)            
        elif method == "SCG":
            self.optimizer = opt.SCG(self,conf)  
        elif method == "CG":
            self.optimizer = opt.CG(self,conf)  
        elif method == "BFGS":
            self.optimizer = opt.BFGS(self,conf)  
                       


[docs]    def plot(self,axisvals=None):
        '''Plot 1d GP FITC regression.'''
        plt.figure()
        xss  = np.reshape(self.xs,(self.xs.shape[0],))
        ymm  = np.reshape(self.ym,(self.ym.shape[0],))
        ys22 = np.reshape(self.ys2,(self.ys2.shape[0],))
        plt.plot(self.x, self.y, color=DATACOLOR, ls='None', marker='+',ms=12, mew=2)
        plt.plot(self.xs, self.ym, color=MEANCOLOR, ls='-', lw=3.)
        plt.fill_between(xss,ymm + 2.*np.sqrt(ys22), ymm - 2.*np.sqrt(ys22), facecolor=SHADEDCOLOR,linewidths=0.0)
        plt.grid()
        if axisvals != None:                
            plt.axis(axisvals)
        plt.xlabel('input x')
        plt.ylabel('output y')
        plt.plot(self.u,np.ones_like(self.u), ls='None', color='k',marker='x',markersize=12,mew=2)
        plt.show()



[docs]    def useInference(self, newInf):
        '''
        Use another inference techinique other than default exact inference.

        :param str newInf: 'Laplace' or 'EP'
        '''
        if newInf == "Laplace":
            self.inffunc = inf.FITC_Laplace()
        elif newInf == "EP":
            self.inffunc = inf.FITC_EP()
        else:
            raise Exception('Possible inf values are "Laplace", "EP".')



[docs]    def useLikelihood(self,newLik):
        '''
        Use another inference techinique other than default Gaussian likelihood.

        :param str newLik: 'Laplace'
        '''
        if newLik == "Laplace":
            self.likfunc = lik.Laplace()
            self.inffunc = inf.FITC_EP()
        else:
            raise Exception('Possible lik values are "Laplace".')   



   


[docs]class GPC_FITC(GP_FITC):
    '''Gaussian Process Classification FITC'''
    def __init__(self):
        super(GPC_FITC, self).__init__()
        self.meanfunc = mean.Zero()                        # default prior mean 
        self.covfunc = cov.RBF()                           # default prior covariance
        self.likfunc = lik.Erf()                           # erf liklihood
        self.inffunc = inf.FITC_EP()                       # default inference method
        self.optimizer = opt.Minimize(self)                # default optimizer 
        self.u = None                                      # no default inducing points


[docs]    def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None):
        '''Set optimizer. See base class.'''
        conf = None
        if (num_restarts!=None) or (min_threshold!=None):
            conf = pyGPs.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc)
            conf.num_restarts = num_restarts
            conf.min_threshold = min_threshold
            if meanRange != None:
                conf.meanRange = meanRange
            if covRange != None:
                conf.covRange = covRange
            if likRange != None:
                conf.likRange = likRange   
        if method == "Minimize":
            self.optimizer = opt.Minimize(self,conf)            
        elif method == "SCG":
            self.optimizer = opt.SCG(self,conf)  
        elif method == "CG":
            self.optimizer = opt.CG(self,conf)  
        elif method == "BFGS":
            self.optimizer = opt.BFGS(self,conf)  
                       


[docs]    def plot(self,x1,x2,t1,t2,axisvals=None): 
        '''Plot 2d GP FITC classification.'''
        fig = plt.figure()
        plt.plot(x1[:,0], x1[:,1], 'b+', markersize = 12)
        plt.plot(x2[:,0], x2[:,1], 'r+', markersize = 12)
        plt.plot(self.u[:,0],self.u[:,1],'ko', markersize=12)
        pc = plt.contour(t1, t2, np.reshape(np.exp(self.lp), (t1.shape[0],t1.shape[1]) ))
        fig.colorbar(pc)
        plt.grid()
        if axisvals != None:    
            plt.axis(axisvals)
        plt.show()  



[docs]    def useInference(self, newInf):
        '''
        Use another inference techinique other than default exact inference.

        :param str newInf: 'Laplace' or 'EP'
        '''
        if newInf == "Laplace":
            self.inffunc = inf.FITC_Laplace()
        else:
            raise Exception('Possible inf values are "Laplace".')



[docs]    def useLikelihood(self,newLik):
        '''
        Use another inference techinique other than default Erf likelihood.
        (Not used in this version)

        :param str newLik: 'Logistic'
        '''
        if newLik == "Logistic":
            pass
            #self.likfunc = lik.Logistic()
        else:
            raise Exception('Possible lik values are "Logistic".')

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© Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting. Created using Sphinx 1.2.
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Source code for pyGPs.Core.inf

#================================================================================
#    Marion Neumann [marion dot neumann at uni-bonn dot de]
#    Daniel Marthaler [marthaler at ge dot com]
#    Shan Huang [shan dot huang at iais dot fraunhofer dot de]
#    Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]
#
#    This file is part of pyGPs.
#    The software package is released under the BSD 2-Clause (FreeBSD) License.
#
#    Copyright (c) by
#    Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014
#================================================================================

# Inference methods: Compute the (approximate) posterior for a Gaussian process.
# Methods currently implemented include:
#
#   infExact         Exact inference (only possible with Gaussian likelihood)
#   infLaplace       Laplace's Approximation
#   infEP            Expectation Propagation
#   infVB            [NOT IMPLEMENTED!] Variational Bayes Approximation 
#
#   infFITC          Large scale regression with approximate covariance matrix
#   infFITC_Laplace  Large scale inference  with approximate covariance matrix
#   infFITC_EP       Large scale inference  with approximate covariance matrix
#
#   infMCMC     [NOT IMPLEMENTED!]
#               Markov Chain Monte Carlo and Annealed Importance Sampling
#               We offer two samplers.
#                 - hmc: Hybrid Monte Carlo
#                 - ess: Elliptical Slice Sampling
#               No derivatives w.r.t. to hyperparameters are provided.
#
#   infLOO      [NOT IMPLEMENTED!]
#               Leave-One-Out predictive probability and Least-Squares Approxim.
#
# The interface to the approximation methods is the following:
#
# post nlZ dnlZ = inf.proceed(mean, cov, lik, x, y)
#
# where:
#   INPUT:
#   cov     name of the covariance function (see covFunctions.m)
#   lik     name of the likelihood function (see likFunctions.m)
#   x       n by D matrix of training inputs 
#   y       1d array (of size n) of targets
#
#   OUTPUT:
#   post    struct representation of the (approximate) posterior containing: 
#           alpha   1d array containing inv(K)*m, 
#                   where K is the prior covariance matrix and m the approx posterior mean
#           sW      1d array containing diagonal of sqrt(W)
#                   the approximate posterior covariance matrix is inv(inv(K)+W)
#           L       2d array, L = chol(sW*K*sW+identity(n))
#   nlZ     returned value of the negative log marginal likelihood
#   dnlZ    1d array of partial derivatives of the negative log marginal likelihood
#           w.r.t. each hyperparameter
#
# Usually, the approximate posterior to be returned admits the form
# N(m=K*alpha, V=inv(inv(K)+W)), where alpha is a vector and W is diagonal;
# if not, then L contains instead -inv(K+inv(W)), and sW is unused.
#
# For more information on the individual approximation methods and their
# implementations, see the respective inf* function below. See also gp.py
#
#
# This is a object-oriented python implementation of gpml functionality 
# (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18).
# based on the functional-version of python implementation
# (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013)
# 
# Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013


import numpy as np
import lik, cov
from copy import copy,deepcopy
from tools import solve_chol,brentmin,cholupdate

np.seterr(all='ignore')


[docs]class postStruct(object):
    '''
    Data structure for posterior
    '''
    def __init__(self):
        self.alpha = np.array([])
        self.L     = np.array([])
        self.sW    = np.array([])



[docs]class dnlZStruct(object):
    '''
    Data structure for the derivatives of mean, cov and lik functions
    '''
    def __init__(self, m, c, l):
        self.mean = []
        self.cov = []
        self.lik = []
        if m.hyp:
            self.mean = [0 for i in xrange(len(m.hyp))]
        if c.hyp:
            self.cov  = [0 for i in xrange(len(c.hyp))]
        if l.hyp:
            self.lik  = [0 for i in xrange(len(l.hyp))]        



[docs]class Inference(object):
    '''
    Base class for inference. Defined several tool methods in it. 
    '''
    def __init__(self):
        pass


[docs]    def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1):
        '''
        Inference computation based on inputs.
        post, nlZ, dnlZ = inf.proceed(mean, cov, lik, x, y)

        :param meanfunc: mean function
        :param covfunc: covariance function
        :param likfunc: likelihood function
        :param x: training data
        :param y: training labels
        :param nargout: specify the number of output(1,2 or 3)
        :return: posterior, negative-log-marginal-likelihood, derivative for negative-log-marginal-likelihood-likelihood
        '''
        pass


    def epComputeParams(self, K, y, ttau, tnu, likfunc, m, inffunc):
        n     = len(y)                                                # number of training cases
        ssi   = np.sqrt(ttau)                                         # compute Sigma and mu
        L     = np.linalg.cholesky(np.eye(n)+np.dot(ssi,ssi.T)*K).T   # L'*L=B=eye(n)+sW*K*sW
        V     = np.linalg.solve(L.T,np.tile(ssi,(1,n))*K)
        Sigma = K - np.dot(V.T,V)
        mu    = np.dot(Sigma,tnu)
        Dsigma = np.reshape(np.diag(Sigma),(np.diag(Sigma).shape[0],1)) 
        tau_n = 1/Dsigma - ttau               # compute the log marginal likelihood
        nu_n  = mu/Dsigma-tnu + m*tau_n       # vectors of cavity parameters
        lZ    = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc)
        nlZ   = np.log(np.diag(L)).sum() - lZ.sum() - np.dot(tnu.T,np.dot(Sigma,tnu))/2  \
                - np.dot((nu_n-m*tau_n).T,((ttau/tau_n*(nu_n-m*tau_n)-2*tnu) / (ttau+tau_n)))/2 \
                + (tnu**2/(tau_n+ttau)).sum()/2.- np.log(1.+ttau/tau_n).sum()/2.
        return Sigma, mu, nlZ[0], L
    

[docs]    def logdetA(self,K,w,nargout):
        '''
        Compute the log determinant ldA and the inverse iA of a square nxn matrix
        A = eye(n) + K*diag(w) from its LU decomposition; for negative definite A, we 
        return ldA = Inf. We also return mwiA = -diag(w)*inv(A).
        [ldA,iA,mwiA] = logdetA(K,w)'''
        n = K.shape[0]
        assert(K.shape[0] == K.shape[1])
        A = np.eye(n) + K*np.tile(w.T,(n,1))
        [L,U,P] = np.linalg.lu(A)    # compute LU decomposition, A = P'*L*U
        u = np.diag(U)           
        signU = np.prod(np.sign(u))  # sign of U
        detP = 1                     # compute sign (and det) of the permutation matrix P
        p = np.dot(P,np.array(range(n)).T)
        for ii in range(n):
            if ii != p[ii]: 
                detP = -detP
                j= [jj for jj,val in enumerate(p) if val == ii]
                p[ii,j[0]] = p[j[0],ii] 
        if signU != detP:            # log becomes complex for negative values, encoded by infinity
            ldA = np.inf
        else:                        # det(L) = 1 and U triangular => det(A) = det(P)*prod(diag(U))
            ldA = np.log(np.abs(u)).sum()
        if nargout>1:
            iA = inv(U)*inv(L)*P; 
            if nargout>2: 
                mwiA = -np.tile(w,(1,n))*iA
                return ldA,iA,mwiA
            else:
                return ldA,iA
        else:
            return ldA  
    


[docs]    def Psi_line(self,s,dalpha,alpha,K,m,likfunc,y,inffunc):
        '''Criterion Psi at alpha + s*dalpha for line search
        [Psi,alpha,f,dlp,W] = Psi_line(s,dalpha,alpha,hyp,K,m,lik,y,inf)'''
        alpha = alpha + s*dalpha
        f = np.dot(K,alpha) + m
        [lp,dlp,d2lp] = likfunc.proceed(y,f,None,inffunc,None,3) 
        W = -d2lp
        Psi = np.dot(alpha.T,(f-m))/2. - lp.sum()
        return Psi[0],alpha,f,dlp,W  
    


[docs]    def epfitcZ(self,d,P,R,nn,gg,ttau,tnu,d0,R0,P0,y,likfunc,m,inffunc):
        '''Compute the marginal likelihood approximation
        effort is O(n*nu^2) provided that nu<n'''
        T = np.dot(np.dot(R,R0),P)              # temporary variable
        diag_sigma = d + np.array([(T*T).sum(axis=0)]).T 
        mu = nn + np.dot(P.T,gg)                # post moments O(n*nu^2)
        tau_n = 1./diag_sigma-ttau              # compute the log marginal likelihood
        nu_n  = mu/diag_sigma - tnu + m*tau_n   # vectors of cavity parameters
        lZ = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc, None, 1)
        nu = gg.shape[0]
        U = np.dot(R0,P0).T*np.tile(1/np.sqrt(d0+1/ttau),(1,nu))
        L = np.linalg.cholesky(np.eye(nu)+np.dot(U.T,U)).T
        ld = 2.*np.log(np.diag(L)).sum() + (np.log(d0+1/ttau)).sum() + (np.log(ttau)).sum()
        t = np.dot(T,tnu); tnu_Sigma_tnu = np.dot(tnu.T,(d*tnu)) + np.dot(t.T,t)
        nlZ = ld/2. - lZ.sum() -tnu_Sigma_tnu/2. \
            -np.dot((nu_n-m*tau_n).T,((ttau/tau_n*(nu_n-m*tau_n)-2.*tnu)/(ttau+tau_n)))/2. \
            + (tnu**2/(tau_n+ttau)).sum()/2. - (np.log(1+ttau/tau_n)).sum()/2.
        return nlZ,nu_n,tau_n
    


[docs]    def epfitcRefresh(self,d0,P0,R0,R0P0,w,b):
        '''Refresh the representation of the posterior from initial and site parameters
        to prevent possible loss of numerical precision after many epfitcUpdates
        effort is O(n*nu^2) provided that nu<n
        Sigma = inv(inv(K)+diag(W)) = diag(d) + P'*R0'*R'*R*R0*P.'''
        nu = R0.shape[0]                                 # number of inducing points
        rot180   = lambda A: np.rot90(np.rot90(A))       # little helper functions
        chol_inv = lambda A: np.linalg.solve( rot180( np.linalg.cholesky(rot180(A)) ),np.eye(nu)) # chol(inv(A))
        t  = 1/(1+d0*w)                                  # temporary variable O(n)
        d  = d0*t                                        # O(n)
        P  = np.tile(t.T,(nu,1))*P0                      # O(n*nu)
        T  = np.tile((w*t).T,(nu,1))*R0P0                # temporary variable O(n*nu^2)
        R  = chol_inv(np.eye(nu)+np.dot(R0P0,T.T))       # O(n*nu^3)
        nn = d*b                                                  # O(n)
        gg = np.dot(R0.T,np.dot(R.T,np.dot(R,np.dot(R0P0,t*b))))  # O(n*nu)
        return d,P,R,nn,gg
    

    def epfitcUpdate(self,d,P_i,R,nn,gg,w,b,ii,w_i,b_i,m,d0,P0,R0):
        dwi = w_i-w[ii]
        dbi = b_i-b[ii]
        hi = nn[ii] + m[ii] + np.dot(P_i.T,gg)           # posterior mean of site i O(nu)
        t = 1+dwi*d[ii]
        d[ii] = d[ii]/t                                  # O(1)
        nn[ii] = d[ii]*b_i                               # O(1)
        r = 1+d0[ii]*w[ii]
        r = (r*r)/dwi + r*d0[ii]
        v = np.dot(R,np.dot(R0,P0[:,ii]))
        v = np.reshape(v,(v.shape[0],1))
        r = 1/(r+np.dot(v.T,v))
        if r>0:
            R = cholupdate(R,np.sqrt(r)*np.dot(R.T,v),'-')
        else:
            R = cholupdate(R,np.sqrt(-r)*np.dot(R.T,v),'+')
        ttemp = np.dot(R0.T,np.dot(R.T,np.dot(R,np.dot(R0,P_i))))
        gg = gg + ((dbi-dwi*(hi-m[ii]))/t) * np.reshape(ttemp,(ttemp.shape[0],1)) # O(nu^2)
        w[ii] = w_i; b[ii] = b_i;                          # update site parameters O(1)
        P_i = P_i/t                                        # O(nu)
        return d,P_i,R,nn,gg,w,b
    

[docs]    def mvmZ(self,x,RVdd,t):
        '''Matrix vector multiplication with Z=inv(K+inv(W))'''
        Zx = t*x - np.dot(RVdd.T,np.dot(RVdd,x))
        return Zx



[docs]    def mvmK(self,al,V,d0):
        '''Matrix vector multiplication with approximate covariance matrix'''
        Kal = np.dot(V.T,np.dot(V,al)) + d0*al
        return Kal



[docs]    def Psi_lineFITC(self,s,dalpha,alpha,V,d0,m,likfunc,y,inffunc):
        '''Criterion Psi at alpha + s*dalpha for line search'''
        alpha = alpha + s*dalpha
        f = self.mvmK(alpha,V,d0) + m
        vargout = likfunc.proceed(y,f,None,inffunc,None,3) 
        lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] 
        W = -d2lp
        Psi = np.dot(alpha.T,(f-m))/2. - lp.sum()
        return Psi[0],alpha,f,dlp,W



[docs]    def fitcRefresh(self,d0,P0,R0,R0P0, w):
        '''Refresh the representation of the posterior from initial and site parameters
        to prevent possible loss of numerical precision after many epfitcUpdates
        effort is O(n*nu^2) provided that nu<n
        Sigma = inv(inv(K)+diag(W)) = diag(d) + P'*R0'*R'*R*R0*P.'''
        nu = R0.shape[0]                                  # number of inducing points
        rot180   = lambda A: np.rot90(np.rot90(A))        # little helper functions
        chol_inv = lambda A: np.linalg.solve( rot180( np.linalg.cholesky(rot180(A)) ),np.eye(nu)) # chol(inv(A))
        t  = 1/(1+d0*w)                                   # temporary variable O(n)
        d  = d0*t                                         # O(n)
        P  = np.tile(t.T,(nu,1))*P0;                      # O(n*nu)
        T  = np.tile((w*t).T,(nu,1))*R0P0;                # temporary variable O(n*nu^2)
        R  = chol_inv(np.eye(nu)+np.dot(R0P0,T.T))        # O(n*nu^3)
        return d,P,R




[docs]class Exact(Inference):
    '''
    Exact inference for a GP with Gaussian likelihood. Compute a parametrization
    of the posterior, the negative log marginal likelihood and its derivatives
    w.r.t. the hyperparameters.
    '''
    def __init__(self):
        self.name = "Exact inference"
    def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1):
        if not isinstance(likfunc, lik.Gauss):
            raise Exception ('Exact inference only possible with Gaussian likelihood')
        n, D = x.shape
        K = covfunc.getCovMatrix(x=x, mode='train')            # evaluate covariance matrix
        m = meanfunc.getMean(x)                                # evaluate mean vector
        
        sn2   = np.exp(2*likfunc.hyp[0])                       # noise variance of likGauss
        L     = np.linalg.cholesky(K/sn2+np.eye(n)).T          # Cholesky factor of covariance with noise
        alpha = solve_chol(L,y-m)/sn2
        post = postStruct()
        post.alpha = alpha                                     # return the posterior parameters
        post.sW    = np.ones((n,1))/np.sqrt(sn2)               # sqrt of noise precision vector
        post.L     = L                                         # L = chol(eye(n)+sW*sW'.*K)

        if nargout>1:                                          # do we want the marginal likelihood?
            nlZ = np.dot((y-m).T,alpha)/2. + np.log(np.diag(L)).sum() + n*np.log(2*np.pi*sn2)/2. # -log marg lik
            if nargout>2:                                      # do we want derivatives?
                dnlZ = dnlZStruct(meanfunc, covfunc, likfunc)  # allocate space for derivatives
                Q = solve_chol(L,np.eye(n))/sn2 - np.dot(alpha,alpha.T) # precompute for convenience
                dnlZ.lik = [sn2*np.trace(Q)]
                if covfunc.hyp:
                    for ii in range(len(covfunc.hyp)):
                        dnlZ.cov[ii] = (Q*covfunc.getDerMatrix(x=x, mode='train', der=ii)).sum()/2.
                if meanfunc.hyp:
                    for ii in range(len(meanfunc.hyp)): 
                        dnlZ.mean[ii] = np.dot(-meanfunc.getDerMatrix(x, ii).T,alpha)
                        dnlZ.mean[ii] = dnlZ.mean[ii][0,0]
                return post, nlZ[0,0], dnlZ
            return post, nlZ[0,0]
        return post




[docs]class FITC_Exact(Inference):
    '''
    FITC approximation to the posterior Gaussian process. The function is
    equivalent to infExact with the covariance function:
    Kt = Q + G; G = diag(g); g = diag(K-Q);  Q = Ku' * inv(Quu) * Ku; 
    where Ku and Kuu are covariances w.r.t. to inducing inputs xu, snu2 = sn2/1e6
    is the noise of the inducing inputs and Quu = Kuu + snu2*eye(nu).
    '''
    def __init__(self):
        self.name = 'FICT exact inference'

    def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1):
        if not isinstance(likfunc, lik.Gauss):                  # NOTE: no explicit call to likGauss
            raise Exception ('Exact inference only possible with Gaussian likelihood')
        if not isinstance(covfunc, cov.FITCOfKernel):
            raise Exception('Only covFITC supported.')          # check cov

        diagK,Kuu,Ku = covfunc.getCovMatrix(x=x, mode='train')  # evaluate covariance matrix
        m  = meanfunc.getMean(x)                                # evaluate mean vector
        n, D = x.shape
        nu = Kuu.shape[0]

        sn2   = np.exp(2*likfunc.hyp[0])                        # noise variance of likGauss
        snu2  = 1.e-6*sn2                                       # hard coded inducing inputs noise
        Luu   = np.linalg.cholesky(Kuu+snu2*np.eye(nu)).T       # Kuu + snu2*I = Luu'*Luu
        V     = np.linalg.solve(Luu.T,Ku)                       # V = inv(Luu')*Ku => V'*V = Q
        g_sn2 = diagK + sn2 - np.array([(V*V).sum(axis=0)]).T   # g + sn2 = diag(K) + sn2 - diag(Q)
        Lu    = np.linalg.cholesky(np.eye(nu) + np.dot(V/np.tile(g_sn2.T,(nu,1)),V.T)).T  # Lu'*Lu=I+V*diag(1/g_sn2)*V'
        r     = (y-m)/np.sqrt(g_sn2)
        be    = np.linalg.solve(Lu.T,np.dot(V,r/np.sqrt(g_sn2)))
        iKuu  = solve_chol(Luu,np.eye(nu))                      # inv(Kuu + snu2*I) = iKuu
        
        post = postStruct()
        post.alpha = np.linalg.solve(Luu,np.linalg.solve(Lu,be))# return the posterior parameters
        post.L  = solve_chol(np.dot(Lu,Luu),np.eye(nu)) - iKuu  # Sigma-inv(Kuu)
        post.sW = np.ones((n,1))/np.sqrt(sn2)                   # unused for FITC prediction  with gp.m

        if nargout>1:                                           # do we want the marginal likelihood
            nlZ = np.log(np.diag(Lu)).sum() + (np.log(g_sn2).sum() + n*np.log(2*np.pi) + np.dot(r.T,r) - np.dot(be.T,be))/2. 
            if nargout>2:                                       # do we want derivatives?
                dnlZ = dnlZStruct(meanfunc, covfunc, likfunc)   # allocate space for derivatives
                al = r/np.sqrt(g_sn2) - np.dot(V.T,np.linalg.solve(Lu,be))/g_sn2 # al = (Kt+sn2*eye(n))\y
                B = np.dot(iKuu,Ku)
                w = np.dot(B,al)
                W = np.linalg.solve(Lu.T,V/np.tile(g_sn2.T,(nu,1)))
                for ii in range(len(covfunc.hyp)):
                    [ddiagKi,dKuui,dKui] = covfunc.getDerMatrix(x=x, mode='train', der=ii)    # eval cov deriv
                    R = 2.*dKui-np.dot(dKuui,B)
                    v = ddiagKi - np.array([(R*B).sum(axis=0)]).T          # diag part of cov deriv
                    dnlZ.cov[ii] = ( np.dot(ddiagKi.T,1./g_sn2) + np.dot(w.T,(np.dot(dKuui,w)-2.*np.dot(dKui,al))) \
                                   - np.dot(al.T,(v*al)) - np.dot(np.array([(W*W).sum(axis=0)]),v) - (np.dot(R,W.T)*np.dot(B,W.T)).sum() )/2.
                    dnlZ.cov[ii] = dnlZ.cov[ii][0,0]
                dnlZ.lik = sn2*((1./g_sn2).sum() - (np.array([(W*W).sum(axis=0)])).sum() - np.dot(al.T,al))
                dKuui = 2*snu2
                R = -dKuui*B
                v = -np.array([(R*B).sum(axis=0)]).T     # diag part of cov deriv
                dnlZ.lik += (np.dot(w.T,np.dot(dKuui,w)) -np.dot(al.T,(v*al)) \
                                 - np.dot(np.array([(W*W).sum(axis=0)]),v) - (np.dot(R,W.T)*np.dot(B,W.T)).sum() )/2. 
                dnlZ.lik = list(dnlZ.lik[0])
                for ii in range(len(meanfunc.hyp)):
                    dnlZ.mean[ii] = np.dot(-meanfunc.getDerMatrix(x, ii).T, al) 
                    dnlZ.mean[ii] = dnlZ.mean[ii][0,0]

                return post, nlZ[0,0], dnlZ
            return post, nlZ[0,0]
        return post





[docs]class Laplace(Inference): 
    '''
    Laplace's Approximation to the posterior Gaussian process.
    '''
    def __init__(self):
        self.last_alpha = None

    def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1):
        tol = 1e-6                           # tolerance for when to stop the Newton iterations
        smax = 2; Nline = 20; thr = 1e-4     # line search parameters
        maxit = 20                           # max number of Newton steps in f
        inffunc = self
        K = covfunc.getCovMatrix(x=x, mode='train')    # evaluate the covariance matrix
        m = meanfunc.getMean(x)              # evaluate the mean vector
        n, D = x.shape
        Psi_old = np.inf                     # make sure while loop starts by the largest old objective val
        if self.last_alpha == None:          # find a good starting point for alpha and f
            alpha = np.zeros((n,1))
            f = np.dot(K,alpha) + m          # start at mean if sizes not match 
            vargout = likfunc.proceed(y, f, None, inffunc, None, 3)
            lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2]
            W= -d2lp
            Psi_new = -lp.sum()
        else:
            alpha = self.last_alpha
            f = np.dot(K,alpha) + m                       # try last one
            vargout = likfunc.proceed(y, f, None, inffunc, None, 3)
            lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2]
            W= -d2lp
            Psi_new = np.dot(alpha.T,(f-m))/2. - lp.sum() # objective for last alpha
            vargout = - likfunc.proceed(y, m, None, inffunc, None, 1)
            Psi_def =  vargout[0]                         # objective for default init f==m
            if Psi_def < Psi_new:                         # if default is better, we use it
                alpha = np.zeros((n,1))
                f = np.dot(K,alpha) + m 
                vargout = likfunc.proceed(y, f, None, inffunc, None, 3)
                lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2]
                W=-d2lp; Psi_new = -lp.sum()
        isWneg = np.any(W<0)       # flag indicating whether we found negative values of W
        it = 0                     # this happens for the Student's t likelihood

        while (Psi_old - Psi_new > tol) and it<maxit:      # begin Newton
            Psi_old = Psi_new; it += 1
            if isWneg:                   # stabilise the Newton direction in case W has negative values
                W = np.maximum(W,0)      # stabilise the Hessian to guarantee postive definiteness
                tol = 1e-10              # increase accuracy to also get the derivatives right
            sW = np.sqrt(W); L = np.linalg.cholesky(np.eye(n) + np.dot(sW,sW.T)*K).T
            b = W*(f-m) + dlp; 
            dalpha = b - sW*solve_chol(L,sW*np.dot(K,b)) - alpha
            vargout = brentmin(0,smax,Nline,thr,self.Psi_line,4,dalpha,alpha,K,m,likfunc,y,inffunc)
            s = vargout[0]
            Psi_new = vargout[1]
            Nfun = vargout[2]
            alpha = vargout[3]
            f = vargout[4]
            dlp = vargout[5]
            W = vargout[6]
            isWneg = np.any(W<0)
        self.last_alpha = alpha                 # remember for next call
        vargout = likfunc.proceed(y,f,None,inffunc,None,4) 
        lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2]; d3lp = vargout[3] 
        W = -d2lp; isWneg = np.any(W<0)
        post = postStruct()
        post.alpha = alpha                      # return the posterior parameters
        post.sW = np.sqrt(np.abs(W))*np.sign(W) # preserve sign in case of negative
        if isWneg:
            [ldA,iA,post.L] = self.logdetA(K,W,3)
            nlZ = np.dot(alpha.T,(f-m))/2. - lp.sum() + ldA/2.
            nlZ = nlZ[0] 
        else:
            sW = post.sW
            post.L = np.linalg.cholesky(np.eye(n)+np.dot(sW,sW.T)*K).T 
            nlZ = np.dot(alpha.T,(f-m))/2. + (np.log(np.diag(post.L))-np.reshape(lp,(lp.shape[0],))).sum()
            nlZ = nlZ[0]
        if nargout>2:                                           # do we want derivatives?
            dnlZ = dnlZStruct(meanfunc, covfunc, likfunc)       # allocate space for derivatives
            if isWneg:                                          # switch between Cholesky and LU decomposition mode
                Z = -post.L                                     # inv(K+inv(W))
                g = np.atleast_2d((iA*K).sum(axis=1)).T /2      # deriv. of ln|B| wrt W; g = diag(inv(inv(K)+diag(W)))/2
            else:
                Z = np.tile(sW,(1,n))*solve_chol(post.L,np.diag(np.reshape(sW,(sW.shape[0],)))) #sW*inv(B)*sW=inv(K+inv(W))
                C = np.linalg.solve(post.L.T,np.tile(sW,(1,n))*K)            # deriv. of ln|B| wrt W
                g = np.atleast_2d((np.diag(K)-(C**2).sum(axis=0).T)).T /2.   # g = diag(inv(inv(K)+W))/2
            dfhat = g* d3lp                 # deriv. of nlZ wrt. fhat: dfhat=diag(inv(inv(K)+W)).*d3lp/2
            for ii in range(len(covfunc.hyp)):                               # covariance hypers
                dK = covfunc.getDerMatrix(x=x, mode='train', der=ii)
                dnlZ.cov[ii] = (Z*dK).sum()/2. - np.dot(alpha.T,np.dot(dK,alpha))/2.   # explicit part
                b = np.dot(dK,dlp)                                           # b-K*(Z*b) = inv(eye(n)+K*diag(W))*b
                dnlZ.cov[ii] -= np.dot(dfhat.T,b-np.dot(K,np.dot(Z,b)))      # implicit part
                dnlZ.cov[ii] = dnlZ.cov[ii][0,0]
            for ii in range(len(likfunc.hyp)):                               # likelihood hypers
                [lp_dhyp,dlp_dhyp,d2lp_dhyp] = likfunc.proceed(y,f,None,inffunc,ii,3)
                dnlZ.lik[ii] = -np.dot(g.T,d2lp_dhyp) - lp_dhyp.sum()        # explicit part
                b = np.dot(K,dlp_dhyp)                                       # b-K*(Z*b) = inv(eye(n)+K*diag(W))*b
                dnlZ.lik[ii] -= np.dot(dfhat.T,b-np.dot(K,np.dot(Z,b)))      # implicit part
                dnlZ.lik[ii] = dnlZ.lik[ii][0,0]
            for ii in range(len(meanfunc.hyp)):                              # mean hypers
                dm = meanfunc.getDerMatrix(x, ii)
                dnlZ.mean[ii] = -np.dot(alpha.T,dm)                          # explicit part
                dnlZ.mean[ii] -= np.dot(dfhat.T,dm-np.dot(K,np.dot(Z,dm)))   # implicit part
                dnlZ.mean[ii] = dnlZ.mean[ii][0,0]
            return post,nlZ[0],dnlZ
        else:
            return post, nlZ[0]





[docs]class FITC_Laplace(Inference):
    '''
    FITC-Laplace approximation to the posterior Gaussian process. The function is
    equivalent to infLaplace with the covariance function:
    Kt = Q + G; G = diag(g); g = diag(K-Q);  Q = Ku' * inv(Kuu + snu2 * eye(nu)) * Ku;
    where Ku and Kuu are covariances w.r.t. to inducing inputs xu and
    snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard
    deviation of the inducing inputs snu to be a one per mil of the measurement 
    noise's standard deviation sn. In case of a likelihood without noise
    parameter sn2, we simply use snu2 = 1e-6.
    '''
    def __init__(self):
        self.last_alpha = None

    def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1):
        if not isinstance(covfunc, cov.FITCOfKernel):
            raise Exception('Only covFITC supported.')            
        tol = 1e-6                             # tolerance for when to stop the Newton iterations
        smax = 2; Nline = 100; thr = 1e-4      # line search parameters
        maxit = 20                             # max number of Newton steps in f
        inffunc = Laplace()
        diagK,Kuu,Ku = covfunc.getCovMatrix(x=x, mode='train')      # evaluate the covariance matrix
        m = meanfunc.getMean(x)                # evaluate the mean vector
        if likfunc.hyp:                        # hard coded inducing inputs noise
            sn2  = np.exp(2.*likfunc.hyp[-1]) 
            snu2 = 1.e-6*sn2                   # similar to infFITC
        else:
            snu2 = 1.e-6        
        
        n, D = x.shape
        nu = Kuu.shape[0]
        rot180   = lambda A: np.rot90(np.rot90(A))      # little helper functions
        chol_inv = lambda A: np.linalg.solve( rot180( np.linalg.cholesky(rot180(A)) ),np.eye(nu)) # chol(inv(A))
        R0 = chol_inv(Kuu+snu2*np.eye(nu))              # initial R, used for refresh O(nu^3)
        V  = np.dot(R0,Ku); d0 = diagK - np.array([(V*V).sum(axis=0)]).T     # initial d, needed
    
        Psi_old = np.inf                    # make sure while loop starts by the largest old objective val
        if self.last_alpha == None:         # find a good starting point for alpha and f
            alpha = np.zeros((n,1))
            f = self.mvmK(alpha,V,d0) + m   # start at mean if sizes not match 
            vargout = likfunc.proceed(y, f, None, inffunc, None, 3)
            lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2]
            W=-d2lp; Psi_new = -lp.sum()
        else:
            alpha = self.last_alpha
            f = self.mvmK(alpha,V,d0) + m                           # try last one
            vargout = likfunc.proceed(y, f, None, inffunc, None, 3)
            lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2]
            W=-d2lp
            Psi_new = np.dot(alpha.T,(f-m))/2. - lp.sum()           # objective for last alpha
            vargout = - likfunc.proceed(y, m, None, inffunc, None, 1)
            Psi_def =  vargout[0]                                   # objective for default init f==m
            if Psi_def < Psi_new:                                   # if default is better, we use it
                alpha = np.zeros((n,1))
                f = self.mvmK(alpha,V,d0) + m
                vargout = likfunc.proceed(y, f, None, inffunc, None, 3)
                lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2]
                W=-d2lp; Psi_new = -lp.sum()

        isWneg = np.any(W<0)                # flag indicating whether we found negative values of W
        it = 0                              # this happens for the Student's t likelihood

        while (Psi_old - Psi_new > tol) and it<maxit:               # begin Newton
            Psi_old = Psi_new
            it += 1
            if isWneg:                      # stabilise the Newton direction in case W has negative values
                W = np.maximum(W,0)         # stabilise the Hessian to guarantee postive definiteness
                tol = 1e-8                  # increase accuracy to also get the derivatives right
            b = W*(f-m) + dlp; dd = 1/(1+W*d0)
            RV = np.dot( chol_inv( np.eye(nu) + np.dot(V*np.tile((W*dd).T,(nu,1)),V.T)),V ) 
            dalpha = dd*b - (W*dd)*np.dot(RV.T,np.dot(RV,(dd*b))) - alpha # Newt dir + line search
            vargout = brentmin(0,smax,Nline,thr,self.Psi_lineFITC,4,dalpha,alpha,V,d0,m,likfunc,y,inffunc)
            s = vargout[0]; Psi_new = vargout[1]; Nfun = vargout[2]; alpha = vargout[3]
            f = vargout[4]; dlp = vargout[5]; W = vargout[6]
            isWneg = np.any(W<0)

        self.last_alpha = alpha                                     # remember for next call
        vargout = likfunc.proceed(y,f,None,inffunc,None,4) 
        lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2]; d3lp = vargout[3]  

        W=-d2lp; isWneg = np.any(W<0)
        post = postStruct()
        post.alpha = np.dot(R0.T,np.dot(V,alpha))                   # return the posterior parameters
        post.sW = np.sqrt(np.abs(W))*np.sign(W)                     # preserve sign in case of negative
        dd = 1/(1+d0*W)                                             # temporary variable O(n)
        A = np.eye(nu) + np.dot(V*np.tile((W*dd).T,(nu,1)),V.T)     # temporary variable O(n*nu^2)
        R0tV = np.dot(R0.T,V); B = R0tV*np.tile((W*dd).T,(nu,1))    # temporary variables O(n*nu^2)
        post.L = -np.dot(B,R0tV.T)                                  # L = -R0'*V*inv(Kt+diag(1./ttau))*V'*R0, first part
        if np.any(1+d0*W<0):
            raise Exception('W is too negative; nlZ and dnlZ cannot be computed.')
        nlZ = np.dot(alpha.T,(f-m))/2. - lp.sum() - np.log(dd).sum()/2. + \
            np.log(np.diag(np.linalg.cholesky(A).T)).sum()
        RV = np.dot(chol_inv(A),V)
        RVdd = RV * np.tile((W*dd).T,(nu,1))                        # RVdd needed for dnlZ
        B = np.dot(B,RV.T)
        post.L += np.dot(B,B.T)

        if nargout>2:                                               # do we want derivatives?
            dnlZ = dnlZStruct(meanfunc, covfunc, likfunc)           # allocate space for derivatives
            [d,P,R] = self.fitcRefresh(d0,Ku,R0,V,W)                # g = diag(inv(inv(K)+W))/2
            g = d/2 + 0.5*np.atleast_2d((np.dot(np.dot(R,R0),P)**2).sum(axis=0)).T
            t = W/(1+W*d0)
            
            dfhat = g*d3lp  # deriv. of nlZ wrt. fhat: dfhat=diag(inv(inv(K)+W)).*d3lp/2
            for ii in range(len(covfunc.hyp)):                      # covariance hypers
                ddiagK,dKuu,dKu = covfunc.getDerMatrix(x=x, mode='train', der=ii)      # eval cov derivatives
                dA = 2.*dKu.T-np.dot(R0tV.T,dKuu)                   # dQ = dA*R0tV
                w = np.atleast_2d((dA*R0tV.T).sum(axis=1)).T        # w = diag(dQ)
                v = ddiagK-w                                        # v = diag(dK)-diag(dQ);
                dnlZ.cov[ii] = np.dot(ddiagK.T,t) - np.dot((RVdd*RVdd).sum(axis=0),v)   # explicit part
                dnlZ.cov[ii] -= (np.dot(RVdd,dA)*np.dot(RVdd,R0tV.T)).sum()             # explicit part
                dnlZ.cov[ii] = 0.5*dnlZ.cov[ii] - np.dot(alpha.T,np.dot(dA,np.dot(R0tV,alpha))+v*alpha)/2. # explicit
                b = np.dot(dA,np.dot(R0tV,dlp)) + v*dlp             # b-K*(Z*b) = inv(eye(n)+K*diag(W))*b
                KZb = self.mvmK(self.mvmZ(b,RVdd,t),V,d0)
                dnlZ.cov[ii] -= np.dot(dfhat.T,(b-KZb))             # implicit part
                dnlZ.cov[ii] = dnlZ.cov[ii][0,0]
                
            for ii in range(len(likfunc.hyp)):                      # likelihood hypers
                vargout = likfunc.proceed(y,f,None,inffunc,ii,3)
                lp_dhyp = vargout[0]; dlp_dhyp = vargout[1]; d2lp_dhyp = vargout[2] 
                dnlZ.lik[ii] = -np.dot(g.T,d2lp_dhyp) - lp_dhyp.sum() # explicit part
                b = self.mvmK(dlp_dhyp,V,d0)                          # implicit part
                dnlZ.lik[ii] -= np.dot(dfhat.T,b-self.mvmK(self.mvmZ(b,RVdd,t),V,d0))
                if ii == len(likfunc.hyp)-1:
                    # since snu2 is a fixed fraction of sn2, there is a covariance-like term
                    # in the derivative as well
                    snu = np.sqrt(snu2);
                    T = chol_inv(Kuu + snu2*np.eye(nu)); 
                    T = np.dot(T.T,np.dot(T,snu*Ku)); 
                    t = np.array([(T*T).sum(axis=0)]).T 
                    z = np.dot(alpha.T,np.dot(T.T,np.dot(T,alpha))-t*alpha) - np.dot(np.array([(RVdd*RVdd).sum(axis=0)]),t)
                    z += (np.dot(RVdd,T.T)**2).sum()
                    b = (t*dlp-np.dot(T.T,np.dot(T,dlp)))/2.
                    KZb = self.mvmK(self.mvmZ(b,RVdd,t),V,d0)
                    z -= np.dot(dfhat.T,b-KZb)
                    dnlZ.lik[ii] += z
                    dnlZ.lik[ii] = dnlZ.lik[ii][0,0]

            for ii in range(len(meanfunc.hyp)):                           # mean hypers
                dm = meanfunc.getDerMatrix(x, ii)
                dnlZ.mean[ii] = -np.dot(alpha.T,dm)                       # explicit part
                Zdm = self.mvmZ(dm,RVdd,t)
                dnlZ.mean[ii] -= np.dot(dfhat.T,(dm-self.mvmK(Zdm,V,d0))) # implicit part
                dnlZ.mean[ii] = dnlZ.mean[ii][0,0]

            return post,nlZ[0,0],dnlZ
        else:
            return post, nlZ[0,0]





[docs]class EP(Inference):
    '''
    Expectation Propagation approximation to the posterior Gaussian Process.
    '''
    def __init__(self):
        self.name = 'Expectation Propagation'
        self.last_ttau = None
        self.last_tnu = None
    def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1):
        tol = 1e-4; max_sweep = 10; min_sweep = 2 # tolerance to stop EP iterations
        n = x.shape[0]
        inffunc = self
        K = covfunc.getCovMatrix(x=x, mode='train') # evaluate the covariance matrix
        m = meanfunc.getMean(x)                   # evaluate the mean vector
        nlZ0 = -likfunc.proceed(y, m, np.reshape(np.diag(K),(np.diag(K).shape[0],1)), inffunc).sum()
        if self.last_ttau == None:                # find starting point for tilde parameters
            ttau  = np.zeros((n,1))               # initialize to zero if we have no better guess
            tnu   = np.zeros((n,1))
            Sigma = K                             # initialize Sigma and mu, the parameters of ..
            mu    = np.zeros((n,1))               # .. the Gaussian posterior approximation
            nlZ   = nlZ0
        else:
            ttau = self.last_ttau                 # try the tilde values from previous call
            tnu  = self.last_tnu
            Sigma, mu, nlZ, L = self.epComputeParams(K, y, ttau, tnu, likfunc, m, inffunc)
            if nlZ > nlZ0:                        # if zero is better ..
                ttau = np.zeros((n,1))            # .. then initialize with zero instead
                tnu  = np.zeros((n,1)) 
                Sigma = K                         # initialize Sigma and mu, the parameters of ..
                mu = np.zeros((n,1))              # .. the Gaussian posterior approximation
                nlZ = nlZ0
        nlZ_old = np.inf; sweep = 0               # converged, max. sweeps or min. sweeps?
        while (np.abs(nlZ-nlZ_old) > tol and sweep < max_sweep) or (sweep < min_sweep):
            nlZ_old = nlZ; sweep += 1
            rperm = xrange(n)                     # randperm(n)
            for ii in rperm:                      # iterate EP updates (in random order) over examples
                tau_ni = 1/Sigma[ii,ii] - ttau[ii]#  first find the cavity distribution ..
                nu_ni  = mu[ii]/Sigma[ii,ii] + m[ii]*tau_ni - tnu[ii]    # .. params tau_ni and nu_ni
                # compute the desired derivatives of the indivdual log partition function
                lZ,dlZ,d2lZ = likfunc.proceed(y[ii], nu_ni/tau_ni, 1/tau_ni, inffunc, None, 3)
                ttau_old = copy(ttau[ii])         # then find the new tilde parameters, keep copy of old
                ttau[ii] = -d2lZ  /(1.+d2lZ/tau_ni)
                ttau[ii] = max(ttau[ii],0)        # enforce positivity i.e. lower bound ttau by zero
                tnu[ii]  = ( dlZ + (m[ii]-nu_ni/tau_ni)*d2lZ )/(1.+d2lZ/tau_ni)
                ds2 = ttau[ii] - ttau_old         # finally rank-1 update Sigma ..
                si  = np.reshape(Sigma[:,ii],(Sigma.shape[0],1))
                Sigma = Sigma - ds2/(1.+ds2*si[ii])*np.dot(si,si.T)   # takes 70# of total time
                mu = np.dot(Sigma,tnu)                                # .. and recompute mu
            # recompute since repeated rank-one updates can destroy numerical precision
            Sigma, mu, nlZ, L = self.epComputeParams(K, y, ttau, tnu, likfunc, m, inffunc)
        if sweep == max_sweep:
            pass
            # print '[warning] maximum number of sweeps reached in function infEP'
        self.last_ttau = ttau; self.last_tnu = tnu          # remember for next call
        sW = np.sqrt(ttau); alpha = tnu-sW*solve_chol(L,sW*np.dot(K,tnu))
        post = postStruct()
        post.alpha = alpha                                  # return the posterior params
        post.sW    = sW
        post.L     = L
        if nargout>2:                                       # do we want derivatives?
            dnlZ = dnlZStruct(meanfunc, covfunc, likfunc)   # allocate space for derivatives
            ssi  = np.sqrt(ttau)
            V = np.linalg.solve(L.T,np.tile(ssi,(1,n))*K)
            Sigma = K - np.dot(V.T,V)
            mu = np.dot(Sigma,tnu)
            Dsigma = np.reshape(np.diag(Sigma),(np.diag(Sigma).shape[0],1))
            tau_n = 1/Dsigma-ttau                           # compute the log marginal likelihood
            nu_n  = mu/Dsigma-tnu                           # vectors of cavity parameters
            F = np.dot(alpha,alpha.T) - np.tile(sW,(1,n))* \
                solve_chol(L,np.diag(np.reshape(sW,(sW.shape[0],))))   # covariance hypers
            for jj in range(len(covfunc.hyp)):
                dK = covfunc.getDerMatrix(x=x, mode='train', der=jj)
                dnlZ.cov[jj] = -(F*dK).sum()/2.
            for ii in range(len(likfunc.hyp)):
                dlik = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc, ii)
                dnlZ.lik[ii] = -dlik.sum()    
            junk,dlZ = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc, None, 2) # mean hyps
            for ii in range(len(meanfunc.hyp)):
                dm = meanfunc.getDerMatrix(x, ii)
                dnlZ.mean[ii] = -np.dot(dlZ.T,dm)
                dnlZ.mean[ii] = dnlZ.mean[ii][0,0]
            return post, nlZ[0], dnlZ
        else:
            return post, nlZ[0]





[docs]class FITC_EP(Inference):
    '''
    FITC-EP approximation to the posterior Gaussian process. The function is
    equivalent to infEP with the covariance function:
    Kt = Q + G; G = diag(g); g = diag(K-Q);  Q = Ku' * inv(Kuu + snu2 * eye(nu)) * Ku;
    where Ku and Kuu are covariances w.r.t. to inducing inputs xu and
    snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard
    deviation of the inducing inputs snu to be a one per mil of the measurement 
    noise's standard deviation sn. In case of a likelihood without noise
    parameter sn2, we simply use snu2 = 1e-6.
    For details, see The Generalized FITC Approximation, Andrew Naish-Guzman and
    Sean Holden, NIPS, 2007.
    '''
    def __init__(self):
        self.name = 'FITC Expectation Propagation'
        self.last_ttau = None
        self.last_tnu = None

    def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1):
        if not isinstance(covfunc, cov.FITCOfKernel):
            raise Exception('Only covFITC supported.')  # check cov
        tol = 1e-4; max_sweep = 10; min_sweep = 2       # tolerance to stop EP iterations
        inffunc = EP()

        diagK,Kuu,Ku = covfunc.getCovMatrix(x=x, mode='train')  # evaluate the covariance matrix
        m = meanfunc.getMean(x)                         # evaluate the mean vector

        if likfunc.hyp:                                 # hard coded inducing inputs noise
            sn2  = np.exp(2.*likfunc.hyp[-1]) 
            snu2 = 1.e-6*sn2                            # similar to infFITC
        else:
            snu2 = 1.e-6
    
        n, D = x.shape; nu = Kuu.shape[0]
        rot180   = lambda A: np.rot90(np.rot90(A))      # little helper functions
        chol_inv = lambda A: np.linalg.solve( rot180( np.linalg.cholesky(rot180(A)) ),np.eye(nu)) # chol(inv(A))

        R0 = chol_inv(Kuu+snu2*np.eye(nu))              # initial R, used for refresh O(nu^3)
        V  = np.dot(R0,Ku); d0 = diagK - np.array([(V*V).sum(axis=0)]).T # initial d, needed for refresh O(n*nu^2)

        # A note on naming: variables are given short but descriptive names in 
        # accordance with Rasmussen & Williams "GPs for Machine Learning" (2006): mu
        # and s2 are mean and variance, nu and tau are natural parameters. A leading t
        # means tilde, a subscript _ni means "not i" (for cavity parameters), or _n
        # for a vector of cavity parameters.

        # marginal likelihood for ttau = tnu = zeros(n,1); equals n*log(2) for likCum*
        nlZ0 = -1.* likfunc.proceed(y, m, np.reshape(diagK,(diagK.shape[0],1)), inffunc).sum()
        if self.last_ttau == None:                      # find starting point for tilde parameters
            ttau  = np.zeros((n,1))                     # initialize to zero if we have no better guess
            tnu   = np.zeros((n,1))
            [d,P,R,nn,gg] = self.epfitcRefresh(d0,Ku,R0,V,ttau,tnu)   # compute initial repres.
            nlZ = nlZ0
        else:
            ttau = self.last_ttau                       # try the tilde values from previous call
            tnu  = self.last_tnu
            [d,P,R,nn,gg] = self.epfitcRefresh(d0,Ku,R0,V,ttau,tnu) # compute initial repres.
            nlZ = self.epfitcZ(d,P,R,nn,gg,ttau,tnu,d0,R0,Ku,y,likfunc,m,inffunc)[0]
            if nlZ > nlZ0:                              # if zero is better ..
                ttau = np.zeros((n,1))                  # .. then initialize with zero instead
                tnu  = np.zeros((n,1))
                [d,P,R,nn,gg] = self.epfitcRefresh(d0,Ku,R0,V,ttau,tnu) # initial repres.
                nlZ = nlZ0

        nlZ_old = np.inf; sweep = 0                     # converged, max. sweeps or min. sweeps?
        while (np.abs(nlZ-nlZ_old) > tol and sweep < max_sweep) or (sweep < min_sweep):
            nlZ_old = nlZ
            sweep += 1
            rperm = range(n)                            # randperm(n)
            for ii in rperm:                            # iterate EP updates (in random order) over examples
                p_i = np.reshape(P[:,ii],(P.shape[0],1))
                t = np.dot(R,np.dot(R0,p_i))            # temporary variables
                sigma_i = d[ii] + np.dot(t.T,t); mu_i = nn[ii] + np.dot(p_i.T,gg) # post moments O(nu^2)    
                tau_ni = 1/sigma_i - ttau[ii]                   #  first find the cavity distribution ..
                nu_ni  = mu_i/sigma_i + m[ii]*tau_ni - tnu[ii]  # .. params tau_ni and nu_ni
                # compute the desired derivatives of the indivdual log partition function
                vargout = likfunc.proceed(y[ii], nu_ni/tau_ni, 1/tau_ni, inffunc, None, 3)
                lZ = vargout[0]; dlZ = vargout[1]; d2lZ = vargout[2]
                ttau_i = -d2lZ  /(1.+d2lZ/tau_ni)
                ttau_i = max(ttau_i,0)                  # enforce positivity i.e. lower bound ttau by zero
                tnu_i  = ( dlZ + (m[ii]-nu_ni/tau_ni)*d2lZ )/(1.+d2lZ/tau_ni)
                [d,P[:,ii],R,nn,gg,ttau,tnu] = self.epfitcUpdate(d,P[:,ii],R,nn,gg,ttau,tnu,ii,ttau_i,tnu_i,m,d0,Ku,R0)# update representation
      
            # recompute since repeated rank-one updates can destroy numerical precision
            [d,P,R,nn,gg] = self.epfitcRefresh(d0,Ku,R0,V,ttau,tnu)
            [nlZ,nu_n,tau_n] = self.epfitcZ(d,P,R,nn,gg,ttau,tnu,d0,R0,Ku,y,likfunc,m,inffunc)
        if sweep == max_sweep:
            pass
            # print '[warning] maximum number of sweeps reached in function infEP'
        
        self.last_ttau = ttau
        self.last_tnu = tnu       # remember for next call
        post = postStruct()
        post.sW = np.sqrt(ttau)   # unused for FITC_EP prediction with gp.m
        dd = 1/(d0+1/ttau)
        alpha = tnu/ttau*dd
        RV = np.dot(R,V)
        R0tV = np.dot(R0.T,V)
        alpha = alpha - np.dot(RV.T,np.dot(RV,alpha))*dd      # long alpha vector for ordinary infEP
        post.alpha = np.dot(R0tV,alpha)                       # alpha = R0'*V*inv(Kt+diag(1./ttau))*(tnu./ttau)
        B = R0tV*np.tile(dd.T,(nu,1)); L = np.dot(B,R0tV.T); B = np.dot(B,RV.T)
        post.L = np.dot(B,B.T) - L                            # L = -R0'*V*inv(Kt+diag(1./ttau))*V'*R0

        if nargout>2:                                         # do we want derivatives?
            dnlZ = dnlZStruct(meanfunc, covfunc, likfunc)     # allocate space for derivatives
            RVdd = RV*np.tile(dd.T,(nu,1))
            for ii in range(len(covfunc.hyp)):
                ddiagK,dKuu,dKu = covfunc.getDerMatrix(x=x, mode='train', der=ii)
                dA = 2*dKu.T - np.dot(R0tV.T,dKuu)            # dQ = dA*R0tV
                w = np.atleast_2d((dA*R0tV.T).sum(axis=1)).T  # w = diag(dQ)
                v = ddiagK - w                                # v = diag(dK)-diag(dQ)
                z = np.dot(dd.T,(v+w)) - np.dot(np.atleast_2d((RVdd*RVdd).sum(axis=0)), v) \
                       - (np.dot(RVdd,dA).T * np.dot(R0tV,RVdd.T)).sum()
                dnlZ.cov[ii] = (z - np.dot(alpha.T,(alpha*v)) - np.dot(np.dot(alpha.T,dA),np.dot(R0tV,alpha)))/2.
                dnlZ.cov[ii] = dnlZ.cov[ii][0,0]
            for ii in range(len(likfunc.hyp)):                # likelihood hypers
                dlik = likfunc.proceed(y, nu_n/tau_n+m, 1/tau_n, inffunc, ii, 1)
                dnlZ.lik[ii] = -dlik.sum()                                 
                if ii == len(likfunc.hyp)-1:
                    # since snu2 is a fixed fraction of sn2, there is a covariance-like term
                    # in the derivative as well
                    v = np.atleast_2d((R0tV*R0tV).sum(axis=0)).T
                    z = (np.dot(RVdd,R0tV.T)**2).sum()  - np.dot(np.atleast_2d((RVdd*RVdd).sum(axis=0)),v)
                    z = z + np.dot(post.alpha.T,post.alpha) - np.dot(alpha.T,(v*alpha))
                    dnlZ.lik[ii] += snu2*z
                    dnlZ.lik[ii] = dnlZ.lik[ii][0,0]
            [junk,dlZ] = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc, None, 2) # mean hyps
            for ii in range(len(meanfunc.hyp)):
                dm = meanfunc.getDerMatrix(x, ii)
                dnlZ.mean[ii] = -np.dot(dlZ.T,dm)
                dnlZ.mean[ii] = dnlZ.mean[ii][0,0]
        
            return post, nlZ[0,0], dnlZ
        else:
            return post, nlZ[0,0]
    




if __name__ == '__main__':
    pass

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© Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting. Created using Sphinx 1.2.
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Source code for pyGPs.Core.lik

#    Marion Neumann [marion dot neumann at uni-bonn dot de]
#    Daniel Marthaler [marthaler at ge dot com]
#    Shan Huang [shan dot huang at iais dot fraunhofer dot de]
#    Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]
#
#    This file is part of pyGPs.
#    The software package is released under the BSD 2-Clause (FreeBSD) License.
#
#    Copyright (c) by
#    Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014
#================================================================================

# likelihood functions are provided to be used by the gp.py function:
#
#   Erf         (Error function, classification, probit regression)
#   Logistic    [NOT IMPLEMENTED!] (Logistic, classification, logit regression)
#   Uni         [NOT IMPLEMENTED!] (Uniform likelihood, classification)
#
#   Gauss       (Gaussian, regression)
#   Laplace     (Laplacian or double exponential, regression)
#   Sech2       [NOT IMPLEMENTED!] (Sech-square, regression)
#   T           [NOT IMPLEMENTED!] (Student's t, regression)
#
#   Poisson     [NOT IMPLEMENTED!] (Poisson regression, count data)
#
#   Mix         [NOT IMPLEMENTED!] (Mixture of individual covariance functions)
#
# The likelihood functions have two possible modes, the mode being selected
# as follows (where "lik" stands for "proceed" method for any likelihood function):
#
#
# 1) With two or three input arguments:                       [PREDICTION MODE]
#
#    lp = lik(y, mu) OR lp, ymu, ys2 = lik(y, mu, s2)
#
# This allows to evaluate the predictive distribution. Let p(y_*|f_*) be the
# likelihood of a test point and N(f_*|mu,s2) an approximation to the posterior
# marginal p(f_*|x_*,x,y) as returned by an inference method. The predictive
# distribution p(y_*|x_*,x,y) is approximated by.
#   q(y_*) = \int N(f_*|mu,s2) p(y_*|f_*) df_*
#
#   lp = log( q(y) ) for a particular value of y, if s2 is [] or 0, this
#                    corresponds to log( p(y|mu) )
#   ymu and ys2      the mean and variance of the predictive marginal q(y)
#                    note that these two numbers do not depend on a particular 
#                    value of y 
# All vectors have the same size.
#
#
# 2) With four or five input arguments, the fouth being an object of class "Inference" [INFERENCE MODE]
#
#  lik(y, mu, s2, inf) OR lik(y, mu, s2, inf, i)
#
# There are two cases for inf, namely a) infLaplace, b) infEP 
# The last input i, refers to derivatives w.r.t. the ith hyperparameter. 
#
# a1)   lp,dlp,d2lp,d3lp = lik(y, f, [], 'infLaplace')
#       lp, dlp, d2lp and d3lp correspond to derivatives of the log likelihood 
#       log(p(y|f)) w.r.t. to the latent location f.
#           lp = log( p(y|f) )
#           dlp = d   log( p(y|f) ) / df
#           d2lp = d^2 log( p(y|f) ) / df^2
#           d3lp = d^3 log( p(y|f) ) / df^3
#
# a2)   lp_dhyp,dlp_dhyp,d2lp_dhyp = lik(y, f, [], 'infLaplace', i)
#       returns derivatives w.r.t. to the ith hyperparameter
#           lp_dhyp = d   log( p(y|f) ) / (     dhyp_i)
#           dlp_dhyp = d^2 log( p(y|f) ) / (df   dhyp_i)
#           d2lp_dhyp = d^3 log( p(y|f) ) / (df^2 dhyp_i)
#
#
# b1)   lZ,dlZ,d2lZ = lik(y, mu, s2, 'infEP')
#       let Z = \int p(y|f) N(f|mu,s2) df then
#           lZ =     log(Z)
#           dlZ = d   log(Z) / dmu
#           d2lZ = d^2 log(Z) / dmu^2
#
# b2)   dlZhyp = lik(y, mu, s2, 'infEP', i)
#       returns derivatives w.r.t. to the ith hyperparameter
#           dlZhyp = d log(Z) / dhyp_i
#
#
# Cumulative likelihoods are designed for binary classification. Therefore, they
# only look at the sign of the targets y; zero values are treated as +1.
#
# Some examples for valid likelihood functions:
#      lik = likGauss([0.1])
#      lik = likErf()

#
# See the documentation for the individual likelihood for the computations specific 
# to each likelihood function.
#
#
# This is a object-oriented python implementation of gpml functionality 
# (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18).
# based on the functional-version of python implementation
# (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013)
# 
# Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013


import numpy as np
from scipy.special import erf
import inf


[docs]class Likelihood(object):
    """Base function for Likelihood function"""
    def __init__(self):
        self.hyp = []

[docs]    def proceed(self):
        '''
        The likelihood functions have two possible modes based on inputs. 
        See lik.py for detail documentation.
        '''
        pass




[docs]class Gauss(Likelihood):
    '''
    Gaussian likelihood function for regression. 

    :math:`Gauss(t)=\\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-\\frac{(t-y)^2}{2\\sigma^2}}`,
    where :math:`y` is the mean and :math:`\\sigma` is the standard deviation.

    hyp = [ log_sigma ]
    '''
    def __init__(self, log_sigma=np.log(0.1) ):
        self.hyp = [log_sigma]

    def proceed(self, y=None, mu=None, s2=None, inffunc=None, der=None, nargout=1):
        sn2 = np.exp(2. * self.hyp[0])
        if inffunc == None:              # prediction mode 
            if y == None:
                y = np.zeros_like(mu)
            s2zero = True
            if (not s2==None) and np.linalg.norm(s2) > 0:
                s2zero = False     
            if s2zero:                   # log probability
                lp = -(y-mu)**2 /sn2/2 - np.log(2.*np.pi*sn2)/2. 
                s2 = np.zeros_like(s2)
            else:
                inf_func = inf.EP()   # prediction
                lp = self.proceed(y, mu, s2, inf_func)
            if nargout>1:
                ymu = mu                 # first y moment
                if nargout>2:
                    ys2 = s2 + sn2       # second y moment
                    return lp,ymu,ys2
                else:
                    return lp,ymu
            else:
                return lp  
        else:
            if isinstance(inffunc, inf.EP):
                if der == None:                                  # no derivative mode
                    lZ = -(y-mu)**2/(sn2+s2)/2. - np.log(2*np.pi*(sn2+s2))/2. # log part function
                    if nargout>1:
                        dlZ  = (y-mu)/(sn2+s2)                   # 1st derivative w.r.t. mean
                        if nargout>2:
                            d2lZ = -1/(sn2+s2)                   # 2nd derivative w.r.t. mean
                            return lZ,dlZ,d2lZ
                        else:
                           return lZ,dlZ
                    else:
                        return lZ
                else:                                            # derivative mode
                    dlZhyp = ((y-mu)**2/(sn2+s2)-1) / (1+s2/sn2) # deriv. w.r.t. hyp.lik
                    return dlZhyp
            elif isinstance(inffunc, inf.Laplace):
                if der == None:                                  # no derivative mode
                    if y == None: 
                        y=0 
                    ymmu = y-mu
                    lp = -ymmu**2/(2*sn2) - np.log(2*np.pi*sn2)/2. 
                    if nargout>1:
                        dlp = ymmu/sn2                           # dlp, derivative of log likelihood
                        if nargout>2:                            # d2lp, 2nd derivative of log likelihood
                            d2lp = -np.ones_like(ymmu)/sn2
                            if nargout>3:                        # d3lp, 3rd derivative of log likelihood
                                d3lp = np.zeros_like(ymmu)
                                return lp,dlp,d2lp,d3lp
                            else:
                                return lp,dlp,d2lp
                        else:
                            return lp,dlp
                    else:
                        return lp
                else:                                            # derivative mode
                    lp_dhyp   = (y-mu)**2/sn2 - 1                # derivative of log likelihood w.r.t. hypers
                    dlp_dhyp  = 2*(mu-y)/sn2                     # first derivative,
                    d2lp_dhyp = 2*np.ones_like(mu)/sn2           # and also of the second mu derivative
                    return lp_dhyp,dlp_dhyp,d2lp_dhyp
            '''
            elif isinstance(inffunc, infVB):
                if der == None:
                    # variational lower site bound
                    # t(s) = exp(-(y-s)^2/2sn2)/sqrt(2*pi*sn2)
                    # the bound has the form: b*s - s.^2/(2*ga) - h(ga)/2 with b=y/ga
                    ga  = s2
                    n   = len(ga)
                    b   = y/ga
                    y   = y*np.ones((n,1))
                    db  = -y/ga**2 
                    d2b = 2*y/ga**3
                    h   = np.zeros((n,1))
                    dh  = h
                    d2h = h                           # allocate memory for return args
                    id  = (ga <= sn2 + 1e-8)          # OK below noise variance
                    h[id]   = y[id]**2/ga[id] + np.log(2*np.pi*sn2)
                    h[np.logical_not(id)] = np.inf
                    dh[id]  = -y[id]**2/ga[id]**2
                    d2h[id] = 2*y[id]**2/ga[id]**3
                    id = ga < 0
                    h[id] = np.inf
                    dh[id] = 0
                    d2h[id] = 0                       # neg. var. treatment
                    varargout = [h,b,dh,db,d2h,d2b]
                else:
                    ga = s2 
                    n  = len(ga)
                    dhhyp = np.zeros((n,1))
                    dhhyp[ga<=sn2] = 2
                    dhhyp[ga<0] = 0                   # negative variances get a special treatment
                    varargout = dhhyp                 # deriv. w.r.t. hyp.lik
            else:
                raise Exception('Incorrect inference in lik.Gauss\n')
        '''




[docs]class Erf(Likelihood):
    '''
    Error function or cumulative Gaussian likelihood function for binary
    classification or probit regression. 

    :math:`Erf(t)=\\frac{1}{2}(1+erf(\\frac{t}{\\sqrt{2}}))=normcdf(t)`
    '''
    def __init__(self):
        self.hyp = []

    def proceed(self, y=None, mu=None, s2=None, inffunc=None, der=None, nargout=1):
        if not y == None:
            y = np.sign(y)
            y[y==0] = 1
        else:
            y = 1                                        # allow only +/- 1 values
        if inffunc == None:                              # prediction mode if inf is not present
            y = y*np.ones_like(mu)                       # make y a vector
            s2zero = True; 
            if not s2 == None: 
                if np.linalg.norm(s2)>0:
                    s2zero = False                       # s2==0?       
            if s2zero:                                   # log probability evaluation
                p,lp = self.cumGauss(y,mu,2)
            else:                                        # prediction
                lp = self.proceed(y, mu, s2, inf.EP())
                p = np.exp(lp)
            if nargout>1:
                ymu = 2*p-1                              # first y moment
                if nargout>2:
                    ys2 = 4*p*(1-p)                      # second y moment
                    return lp,ymu,ys2
                else:
                    return lp,ymu
            else:
                return lp
        else:                                            # inference mode
            if isinstance(inffunc, inf.Laplace):
                if der == None:                          # no derivative mode
                    f = mu; yf = y*f                     # product latents and labels
                    p,lp = self.cumGauss(y,f,2)
                    if nargout>1:                        # derivative of log likelihood
                        n_p = self.gauOverCumGauss(yf,p)
                        dlp = y*n_p                      # derivative of log likelihood
                        if nargout>2:                    # 2nd derivative of log likelihood
                            d2lp = -n_p**2 - yf*n_p
                            if nargout>3:                # 3rd derivative of log likelihood
                                d3lp = 2*y*n_p**3 + 3*f*n_p**2 + y*(f**2-1)*n_p 
                                return lp,dlp,d2lp,d3lp
                            else:
                                return lp,dlp,d2lp
                        else:
                            return lp,dlp
                    else:
                        return lp
                else:                                    # derivative mode
                    return []                            # derivative w.r.t. hypers

            elif isinstance(inffunc, inf.EP):
                if der == None:                          # no derivative mode
                    z = mu/np.sqrt(1+s2) 
                    junk,lZ = self.cumGauss(y,z,2)       # log part function
                    if not y == None:
                         z = z*y
                    if nargout>1:
                        if y == None: y = 1
                        n_p = self.gauOverCumGauss(z,np.exp(lZ))
                        dlZ = y*n_p/np.sqrt(1.+s2)       # 1st derivative wrt mean
                        if nargout>2:
                            d2lZ = -n_p*(z+n_p)/(1.+s2)  # 2nd derivative wrt mean
                            return lZ,dlZ,d2lZ
                        else:
                            return lZ,dlZ
                    else:
                        return lZ
                else:                                    # derivative mode
                    return []                       # deriv. wrt hyp.lik
        '''
        if inffunc == 'inf.infVB':
            if der == None:                              # no derivative mode
                # naive variational lower bound based on asymptotical properties of lik
                # normcdf(t) -> -(t*A_hat^2-2dt+c)/2 for t->-np.inf (tight lower bound)
                d =  0.158482605320942;
                c = -1.785873318175113;
                ga = s2; n = len(ga); b = d*y*np.ones((n,1)); db = np.zeros((n,1)); d2b = db
                h = -2.*c*np.ones((n,1)); h[ga>1] = np.inf; dh = np.zeros((n,1)); d2h = dh   
                varargout = [h,b,dh,db,d2h,d2b]
            else:                                        # derivative mode
                varargout = []                           # deriv. wrt hyp.lik
        '''

    def cumGauss(self, y=None, f=None, nargout=1):
        # return [p,lp] = cumGauss(y,f)
        if not y == None: 
            yf = y*f 
        else:
            yf = f 
        p = (1. + erf(yf/np.sqrt(2.)))/2. # likelihood
        if nargout>1: 
            lp = self.logphi(yf,p)
            return p,lp 
        else:
            return p

    def gauOverCumGauss(self,f,p):
        # return n_p = gauOverCumGauss(f,p)
        n_p = np.zeros_like(f)       # safely compute Gaussian over cumulative Gaussian
        ok = f>-5                    # naive evaluation for large values of f
        n_p[ok] = (np.exp(-f[ok]**2/2)/np.sqrt(2*np.pi)) / p[ok] 
        bd = f<-6                    # tight upper bound evaluation
        n_p[bd] = np.sqrt(f[bd]**2/4+1)-f[bd]/2
        interp = np.logical_and(np.logical_not(ok),np.logical_not(bd)) # linearly interpolate between both of them
        tmp = f[interp]
        lam = -5. - f[interp]
        n_p[interp] = (1-lam)*(np.exp(-tmp**2/2)/np.sqrt(2*np.pi))/p[interp] + lam *(np.sqrt(tmp**2/4+1)-tmp/2);
        return n_p

    def logphi(self,z,p):
        # return lp = logphi(z,p)
        lp = np.zeros_like(z)                       # allocate memory
        zmin = -6.2; zmax = -5.5;
        ok = z>zmax                                 # safe evaluation for large values
        bd = z<zmin                                 # use asymptotics
        nok = np.logical_not(ok)
        ip = np.logical_and(nok,np.logical_not(bd)) # interpolate between both of them
        lam = 1/(1.+np.exp( 25.*(0.5-(z[ip]-zmin)/(zmax-zmin)) ))  # interp. weights
        lp[ok] = np.log(p[ok])
        lp[nok] = -np.log(np.pi)/2. -z[nok]**2/2. - np.log( np.sqrt(z[nok]**2/2.+2.) - z[nok]/np.sqrt(2.) )
        lp[ip] = (1-lam)*lp[ip] + lam*np.log( p[ip] )
        return lp





[docs]class Laplace(Likelihood):
    ''' 
    Laplacian likelihood function for regression. ONLY works with EP inference!

    :math:`Laplace(t) = \\frac{1}{2b}e^{-\\frac{|t-y|}{b}}` where :math:`b=\\frac{\\sigma}{\\sqrt{2}}`,
    :math:`y` is the mean and :math:`\\sigma` is the standard deviation.

    hyp = [ log_sigma ]
    '''
    def __init__(self, log_sigma=np.log(0.1) ):
        self.hyp = [ log_sigma ]

    def proceed(self, y=None, mu=None, s2=None, inffunc=None, der=None, nargout=1):
        sn = np.exp(self.hyp); b = sn/np.sqrt(2);
        if y == None:
            y = np.zeros_like(mu) 
        if inffunc == None:                              # prediction mode if inf is not present
            if y == None:
                y = np.zeros_like(mu)
            s2zero = True; 
            if not s2 == None: 
                if np.linalg.norm(s2)>0:
                    s2zero = False                       # s2==0?       
            if s2zero:                                   # log probability evaluation
                lp = -np.abs(y-mu)/b -np.log(2*b); s2 = 0
            else:                                        # prediction
                lp = self.proceed(y, mu, s2, inf.EP())
            if nargout>1:
                ymu = mu                              # first y moment
                if nargout>2:
                    ys2 = s2 + sn**2                  # second y moment
                    return lp,ymu,ys2
                else:
                    return lp,ymu
            else:
                return lp
        else:                                            # inference mode
            if isinstance(inffunc, inf.Laplace):
                if der == None:                          # no derivative mode
                    if y == None:
                        y = np.zeros_like(mu) 
                    ymmu = y-mu
                    lp = np.abs(ymmu)/b - np.log(2*b)
                    if nargout>1:                        # derivative of log likelihood
                        dlp = np.sign(ymmu)/b
                        if nargout>2:                    # 2nd derivative of log likelihood
                            d2lp = np.zeros_like(ymmu)
                            if nargout>3:                # 3rd derivative of log likelihood
                                d3lp = np.zeros_like(ymmu)
                                return lp,dlp,d2lp,d3lp
                            else:
                                return lp,dlp,d2lp
                        else:
                            return lp,dlp
                    else:
                        return lp
                else:                                    # derivative mode
                    return []                            # derivative w.r.t. hypers
            elif isinstance(inffunc, inf.EP):
                n = np.max([len(y.flatten()),len(mu.flatten()),len(s2.flatten()),len(sn.flatten())])
                on = np.ones((n,1))
                y = y*on; mu = mu*on; s2 = s2*on; sn = sn*on; 
                fac = 1e3;          # factor between the widths of the two distributions ...
                                    # ... from when one considered a delta peak, we use 3 orders of magnitude
                #idlik = np.reshape( (fac*sn) < np.sqrt(s2) , (sn.shape[0],) ) # Likelihood is a delta peak
                #idgau = np.reshape( (fac*np.sqrt(s2)) < sn , (sn.shape[0],) ) # Gaussian is a delta peak
                idlik = (fac*sn) < np.sqrt(s2) 
                idgau = (fac*np.sqrt(s2)) < sn 
                id    = np.logical_and(np.logical_not(idgau),np.logical_not(idlik)) # interesting case in between

                if der == None:                          # no derivative mode
                    lZ = np.zeros((n,1))
                    dlZ = np.zeros((n,1))
                    d2lZ = np.zeros((n,1))
                    if np.any(idlik):
                        l = Gauss(log_sigma=np.log(s2[idlik])/2)
                        a = l.proceed(mu[idlik], y[idlik])
                        lZ[idlik] = a[0]; dlZ[idlik] = a[1]; d2lZ[idlik] = a[2]
                    if np.any(idgau):
                        l = Laplace(log_hyp=np.log(sn[idgau]))
                        a = l.proceed(mu=mu[idgau], y=y[idgau])
                        lZ[idgau] = a[0]; dlZ[idgau] = a[1]; d2lZ[idgau] = a[2] 
                    if np.any(id):
                        # substitution to obtain unit variance, zero mean Laplacian
                        tvar = s2[id]/(sn[id]**2+1e-16)
                        tmu = (mu[id]-y[id])/(sn[id]+1e-16)
                        # an implementation based on logphi(t) = log(normcdf(t))
                        zp = (tmu+np.sqrt(2)*tvar)/np.sqrt(tvar)
                        zm = (tmu-np.sqrt(2)*tvar)/np.sqrt(tvar)
                        ap =  self._logphi(-zp)+np.sqrt(2)*tmu
                        am =  self._logphi( zm)-np.sqrt(2)*tmu
                        apam = np.vstack((ap,am)).T
                        lZ[id] = self._logsum2exp(apam) + tvar - np.log(sn[id]*np.sqrt(2.))

                    if nargout>1:
                        lqp = -0.5*zp**2 - 0.5*np.log(2*np.pi) - self._logphi(-zp);       # log( N(z)/Phi(z) )
                        lqm = -0.5*zm**2 - 0.5*np.log(2*np.pi) - self._logphi( zm);
                        dap = -np.exp(lqp-0.5*np.log(s2[id])) + np.sqrt(2)/sn[id]
                        dam =  np.exp(lqm-0.5*np.log(s2[id])) - np.sqrt(2)/sn[id]
                        _z1 = np.vstack((ap,am)).T
                        _z2 = np.vstack((dap,dam)).T
                        _x = np.array([[1],[1]])
                        dlZ[id] = self._expABz_expAx(_z1, _x, _z2, _x)
                        if nargout>2:
                            a = np.sqrt(8.)/sn[id]/np.sqrt(s2[id]);
                            bp = 2./sn[id]**2 - (a - zp/s2[id])*np.exp(lqp)
                            bm = 2./sn[id]**2 - (a + zm/s2[id])*np.exp(lqm)
                            _x = np.reshape(np.array([1,1]),(2,1))
                            _z1 = np.reshape(np.array([ap,am]),(1,2))
                            _z2 = np.reshape(np.array([bp,bm]),(1,2))
                            d2lZ[id] = self._expABz_expAx(_z1, _x, _z2, _x) - dlZ[id]**2
                            return lZ,dlZ,d2lZ
                        else:
                            return lZ,dlZ
                    else:
                        return lZ
                else:                                    # derivative mode
                    dlZhyp = np.zeros((n,1))
                    if np.any(idlik):
                        dlZhyp[idlik] = 0
                    if np.any(idgau):
                        l = Laplace(log_hyp=np.log(sn[idgau]))
                        a =  l.proceed(mu=mu[idgau], y=y[idgau], inffunc='inf.Laplace', nargout=1)
                        dlZhyp[idgau] = a[0]

                    if np.any(id):
                        # substitution to obtain unit variance, zero mean Laplacian
                        tmu = (mu[id]-y[id])/(sn[id]+1e-16);        tvar = s2[id]/(sn[id]**2+1e-16)
                        zp  = (tvar+tmu/np.sqrt(2))/np.sqrt(tvar);  vp = tvar+np.sqrt(2)*tmu
                        zm  = (tvar-tmu/np.sqrt(2))/np.sqrt(tvar);  vm = tvar-np.sqrt(2)*tmu
                        dzp = (-s2[id]/sn[id]+tmu*sn[id]/np.sqrt(2)) / np.sqrt(s2[id])
                        dvp = -2*tvar - np.sqrt(2)*tmu
                        dzm = (-s2[id]/sn[id]-tmu*sn[id]/np.sqrt(2)) / np.sqrt(s2[id])
                        dvm = -2*tvar + np.sqrt(2)*tmu
                        lezp = self._lerfc(zp); # ap = exp(vp).*ezp
                        lezm = self._lerfc(zm); # am = exp(vm).*ezm
                        vmax = np.max(np.array([vp+lezp,vm+lezm]),axis=0); # subtract max to avoid numerical pb
                        ep  = np.exp(vp+lezp-vmax)
                        em  = np.exp(vm+lezm-vmax)
                        dap = ep*(dvp - 2/np.sqrt(np.pi)*np.exp(-zp**2-lezp)*dzp)
                        dam = em*(dvm - 2/np.sqrt(np.pi)*np.exp(-zm**2-lezm)*dzm)        
                        dlZhyp[id] = (dap+dam)/(ep+em) - 1;       
                    return dlZhyp               # deriv. wrt hyp.lik
            elif isinstance(inffunc, inf.VB):
                n = len(s2.flatten()); b = np.zeros((n,1)); y = y*np.ones((n,1)); z = y
                return b,z

    def _lerfc(self,t):
        ''' numerically safe implementation of f(t) = log(1-erf(t)) = log(erfc(t))'''
        from scipy.special import erfc
        f  = np.zeros_like(t)
        tmin = 20; tmax = 25
        ok = t<tmin                              # log(1-erf(t)) is safe to evaluate
        bd = t>tmax                              # evaluate tight bound
        nok = np.logical_not(ok)
        interp = np.logical_and(nok,np.logical_not(bd)) # interpolate between both of them
        f[nok] = np.log(2/np.sqrt(np.pi)) -t[nok]**2 -np.log(t[nok]+np.sqrt( t[nok]**2+4/np.pi ))
        lam = 1/(1+np.exp( 12*(0.5-(t[interp]-tmin)/(tmax-tmin)) ))   # interp. weights
        f[interp] = lam*f[interp] + (1-lam)*np.log(erfc( t[interp] ))
        f[ok] += np.log(erfc( t[ok] ))             # safe eval
        return f

    def _expABz_expAx(self,A,x,B,z):
        ''' 
        Computes y = ( (exp(A).*B)*z ) ./ ( exp(A)*x ) in a numerically safe way
        The function is not general in the sense that it yields correct values for
        all types of inputs. We assume that the values are close together.
        '''
        N = A.shape[1]
        maxA = np.max(A,axis=1)                    # number of columns, max over columns
        maxA = np.array([maxA]).T
        A = A - np.dot(maxA, np.ones((1,N)))       # subtract maximum value
        y = ( np.dot((np.exp(A)*B),z) ) / ( np.dot(np.exp(A),x) )
        return y[0]

    def _logphi(self,z):
        ''' Safe implementation of the log of phi(x) = \int_{-\infty}^x N(f|0,1) df
         returns lp = log(normcdf(z))
        '''
        lp = np.zeros_like(z)                       # allocate memory
        zmin = -6.2; zmax = -5.5;
        ok = z>zmax                                 # safe evaluation for large values
        bd = z<zmin                                 # use asymptotics
        nok = np.logical_not(ok)
        ip = np.logical_and(nok,np.logical_not(bd)) # interpolate between both of them
        lam = 1./(1.+np.exp( 25.*(0.5-(z[ip]-zmin)/(zmax-zmin)) ))  # interp. weights
        lp[ok] = np.log( 0.5*( 1.+erf(z[ok]/np.sqrt(2.)) ) )
        lp[nok] = -0.5*(np.log(np.pi) + z[nok]**2) - np.log( np.sqrt(2.+0.5*(z[nok]**2)) - z[nok]/np.sqrt(2)) 
        lp[ip] = (1-lam)*lp[ip] + lam*np.log( 0.5*( 1.+erf(z[ip]/np.sqrt(2.)) ) )
        return lp

    def _logsum2exp(self,logx):
        '''computes y = log( sum(exp(x),2) ) in a numerically safe way 
        by subtracting the row maximum to avoid cancelation after taking 
        the exp the sum is done along the rows'''
        N = logx.shape[1]
        max_logx = logx.max(1)
        max_logx = np.array([max_logx]).T
        # we have all values in the log domain, and want to calculate a sum
        x = np.exp(logx - np.dot(max_logx,np.ones((1,N))))
        y = np.log(np.array([np.sum(x,1)]).T) + max_logx
        return list(y.flatten())


 


if __name__ == '__main__':
    pass

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© Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting. Created using Sphinx 1.2.
pyGPs-1.2/doc/build/_modules/pyGPs/Core/mean.html000066400000000000000000001457321236346020200216430ustar00rootroot00000000000000 pyGPs.Core.mean — pyGPs v1.2 documentation

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Source code for pyGPs.Core.mean

#================================================================================
#    Marion Neumann [marion dot neumann at uni-bonn dot de]
#    Daniel Marthaler [marthaler at ge dot com]
#    Shan Huang [shan dot huang at iais dot fraunhofer dot de]
#    Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]
#
#    This file is part of pyGPs.
#    The software package is released under the BSD 2-Clause (FreeBSD) License.
#
#    Copyright (c) by
#    Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014
#================================================================================

# mean functions to be use by Gaussian process functions. There are two
# different kinds of mean functions: simple and composite:
#
# simple mean functions:
#
#   Zero      - zero mean function
#   One       - one mean function
#   Const     - constant mean function
#   Linear    - linear mean function
# 
# composite mean functions:
#
#   ScaleOfMean     - scaled version of a mean function
#   PowerOfMean     - power of a mean function
#   ProductOfMean   - products of mean functions
#   SumOfMean       - sums of mean functions
#
#
# This is a object-oriented python implementation of gpml functionality 
# (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18).
# based on the functional-version of python implementation
# (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013)
# 
#
# Copyright (c) by Marion Neumann and Shan Huang, Sep.2013

import numpy as np
import math


[docs]class Mean(object):
    """
    The base function for mean function
    """
    def __init__(self):
        super(Mean, self).__init__()
        self.hyp = []
        self.para = []

    # overloading 
    def __add__(self,mean):
        '''
        Overloading + operator.
        
        :param mean: mean function
        :return: an instance of SumOfMean
        '''
        return SumOfMean(self,mean)

    # overloading
    def __mul__(self,other):
        '''
        Overloading * operator.
        Using * for both multiplication with scalar and product of means
        depending on the type of the two objects.
        
        :param other: mean function as product or int/float as scalar
        :return: an instance of ScaleOfMean or ProductOfMean
        '''
        if isinstance(other, int) or isinstance(other, float):
            return ScaleOfMean(self,other)
        elif isinstance(other, Mean):
            return ProductOfMean(self,other)
        else:
            print "only numbers and Means are allowed for *"

    # overloading
    __rmul__ = __mul__

    # overloading
    def __pow__(self,number):
        '''
        Overloading ** operator.
        
        :param int number: power of the mean function
        :return: an instance of PowerOfMean
        '''
        if isinstance(number, int) and number > 0:
            return PowerOfMean(self,number)
        else:
            print "only non-zero integers are supported for **"
    




[docs]class ProductOfMean(Mean):
    '''Product of two mean fucntions.'''
    def __init__(self,mean1,mean2):
        self.mean1 = mean1
        self.mean2 = mean2
        if mean1.hyp and mean2.hyp:
            self._hyp = mean1.hyp + mean2.hyp
        elif not mean1.hyp:
            self._hyp = mean2.hyp
        elif not mean2.hyp:
            self._hyp = mean1.hyp
    def sethyp(self,hyp):
        assert len(hyp) == len(self._hyp)
        len1 = len(self.mean1.hyp)
        self._hyp = hyp 
        self.mean1.hyp = self._hyp[:len1]
        self.mean2.hyp = self._hyp[len1:]
    def gethyp(self):
        return self._hyp
    hyp = property(gethyp,sethyp)
    
    def getMean(self, x=None):
        A = self.mean1.getMean(x) * self.mean2.getMean(x)
        return A

    def getDerMatrix(self, x=None, der=None):
        if der < len(self.mean1.hyp):
            A = self.mean1.getDerMatrix(x, der) * self.mean2.getMean(x)
        elif der < len(self.hyp):
            der2 = der - len(self.mean1.hyp)
            A = self.mean2.getDerMatrix(x, der2) * self.mean1.getMean(x) 
        else:
            raise Exception("Error: der out of range for meanProduct")
        return A





[docs]class SumOfMean(Mean):
    '''Sum of two mean functions.'''
    def __init__(self,mean1,mean2):
        self.mean1 = mean1
        self.mean2 = mean2
        if mean1.hyp and mean2.hyp:
            self._hyp = mean1.hyp + mean2.hyp
        elif not mean1.hyp:
            self._hyp = mean2.hyp
        elif not mean2.hyp:
            self._hyp = mean1.hyp
    def sethyp(self,hyp):
        assert len(hyp) == len(self._hyp)
        len1 = len(self.mean1.hyp)
        self._hyp = hyp 
        self.mean1.hyp = self._hyp[:len1]
        self.mean2.hyp = self._hyp[len1:]
    def gethyp(self):
        return self._hyp
    hyp = property(gethyp,sethyp)

    def getMean(self, x=None):
        A = self.mean1.getMean(x) + self.mean2.getMean(x)
        return A

    def getDerMatrix(self, x=None, der=None):
        if der < len(self.mean1.hyp):
            A = self.mean1.getDerMatrix(x, der)
        elif der < len(self.hyp):
            der2 = der - len(self.mean1.hyp)
            A = self.mean2.getDerMatrix(x, der2) 
        else:
            raise Exception("Error: der out of range for meanSum")
        return A





[docs]class ScaleOfMean(Mean):
    '''Scale of a mean function.'''
    def __init__(self,mean,scalar):
        self.mean = mean
        if mean.hyp:
            self._hyp = [scalar] + mean.hyp 
        else:
            self._hyp = [scalar]
    def sethyp(self,hyp):
        assert len(hyp) == len(self._hyp)
        self._hyp = hyp 
        self.mean.hyp = self._hyp[1:]
    def gethyp(self):
        return self._hyp
    hyp = property(gethyp,sethyp)

    def getMean(self, x=None):
        c = self.hyp[0]                          # scale parameter
        A = c * self.mean.getMean(x)             # accumulate means
        return A

    def getDerMatrix(self, x=None, der=None):
        c = self.hyp[0]                          # scale parameter
        if der == 0:                             # compute derivative w.r.t. c
            A = self.mean.getMean(x)
        else:
            A = c * self.mean.getDerMatrix(x,der-1)
        return A





[docs]class PowerOfMean(Mean):
    '''Power of a mean fucntion.'''
    def __init__(self, mean, d):
        self.mean = mean
        if mean.hyp:
            self._hyp = [d] + mean.hyp 
        else:
            self._hyp = [d]
    def sethyp(self,hyp):
        assert len(hyp) == len(self._hyp)
        self._hyp = hyp 
        self.mean.hyp = self._hyp[1:]
    def gethyp(self):
        return self._hyp
    hyp = property(gethyp,sethyp)

    def getMean(self, x=None):
        d = np.abs(np.floor(self.hyp[0])) 
        d = max(d,1)
        A = self.mean.getMean(x) **d              # accumulate means
        return A

    def getDerMatrix(self, x=None, der=None):
        d = np.abs(np.floor(self.hyp[0])) 
        d = max(d,1)
        if der == 0:                             # compute derivative w.r.t. c
            a = self.mean.getMean(x)
            A = a**d * np.log(a) 
        else:
            A = d * self.mean.getMean(x) ** (d-1) * self.mean.getDerMatrix(x, der-1)
        return A





[docs]class Zero(Mean):
    '''Zero mean.'''
    def __init__(self):
        self.hyp = []
        self.name = '0'

    def getMean(self, x=None):
        n, D = x.shape
        A = np.zeros((n,1))
        return A

    def getDerMatrix(self, x=None, der=None):
        n, D = x.shape
        A = np.zeros((n,1))
        return A





[docs]class One(Mean):
    ''' One mean.'''
    def __init__(self):
        self.hyp = []
        self.name = '1'

    def getMean(self, x=None):
        n, D = x.shape
        A = np.ones((n,1))
        return A

    def getDerMatrix(self, x=None, der=None):
        n, D = x.shape
        A = np.zeros((n,1))
        return A





[docs]class Const(Mean):
    '''
    Constant mean function. hyp = [c]

    :param c: constant value for mean
    '''
    def __init__(self, c=5.):
        self.hyp = [c]

    def getMean(self, x=None):
        n,D = x.shape
        A = self.hyp[0] * np.ones((n,1))
        return A

    def getDerMatrix(self, x=None, der=None):
        n,D = x.shape
        if der == 0:                  # compute derivative vector wrt c
            A = np.ones((n,1))
        else:
            A = np.zeros((n,1)) 
        return A          





[docs]class Linear(Mean):
    '''
    Linear mean function. self.hyp = alpha_list

    :param D: dimension of training data. Set if you want default alpha, which is 0.5 for each dimension.
    :alpha_list: scalar alpha for each dimension
    '''
    def __init__(self, D=None, alpha_list=None):
        if alpha_list == None:
            self.hyp = [0.5 for i in xrange(D)]
        else:
            self.hyp = alpha_list

    def getMean(self, x=None):
        n, D = x.shape
        c = np.array(self.hyp)
        c = np.reshape(c,(len(c),1))
        A = np.dot(x,c)
        return A

    def getDerMatrix(self, x=None, der=None):
        n, D = x.shape
        c = np.array(self.hyp)
        c = np.reshape(c,(len(c),1))
        if isinstance(der, int) and der < D:     # compute derivative vector wrt meanparameters
            A = np.reshape(x[:,der], (len(x[:,der]),1) ) 
        else:   
            A = np.zeros((n,1)) 
        return A





if __name__ == '__main__':
    pass

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© Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting. Created using Sphinx 1.2.
pyGPs-1.2/doc/build/_modules/pyGPs/Core/opt.html000066400000000000000000002002071236346020200215120ustar00rootroot00000000000000 pyGPs.Core.opt — pyGPs v1.2 documentation

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Source code for pyGPs.Core.opt

#================================================================================
#    Marion Neumann [marion dot neumann at uni-bonn dot de]
#    Daniel Marthaler [marthaler at ge dot com]
#    Shan Huang [shan dot huang at iais dot fraunhofer dot de]
#    Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]
#
#    This file is part of pyGPs.
#    The software package is released under the BSD 2-Clause (FreeBSD) License.
#
#    Copyright (c) by
#    Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014
#================================================================================

# This is a object-oriented python implementation of gpml functionality 
# (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18).
# based on the functional-version of python implementation
# (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013)
# 
# Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013

import numpy as np
import gp
from scipy.optimize import fmin_bfgs as bfgs
from scipy.optimize import fmin_cg as cg
from ..Optimization import minimize, scg
from copy import deepcopy

class Optimizer(object):
    def __init__(self):
        self.model = None

    def nlml(self, hypInArray):
        '''Find negative-log-marginal-likelihood'''
        self.apply_in_objects(hypInArray)
        nlZ, dnlZ = self.model.fit(der=False)
        return nlZ

    def dnlml(self, hypInArray):
        '''Find derivatives wrt. negative-log-marginal-likelihood'''
        self.apply_in_objects(hypInArray)
        nlZ, dnlZ, post = self.model.fit()
        dnlml_List = dnlZ.mean + dnlZ.cov + dnlZ.lik
        return np.array(dnlml_List)

    def nlzAnddnlz(self, hypInArray):
        '''Find negative-log-marginal-likelihood and derivatives in one pass(faster)'''
        self.apply_in_objects(hypInArray)
        nlZ, dnlZ, post = self.model.fit()
        dnlml_List = dnlZ.mean + dnlZ.cov + dnlZ.lik
        return nlZ, np.array(dnlml_List)

    def convert_to_array(self):
        '''Convert all hyparameters in the model to an array'''
        hyplist = self.model.meanfunc.hyp + self.model.covfunc.hyp + self.model.likfunc.hyp
        return np.array(hyplist)

    def apply_in_objects(self, hypInArray):
        '''Apply the values in the input array to hyparameters of model.'''
        Lm = len(self.model.meanfunc.hyp)
        Lc = len(self.model.covfunc.hyp)
        hypInList = hypInArray.tolist()
        self.model.meanfunc.hyp  = hypInList[:Lm]
        self.model.covfunc.hyp   = hypInList[Lm:(Lm+Lc)]
        self.model.likfunc.hyp   = hypInList[(Lm+Lc):]

       


[docs]class CG(Optimizer):
    '''Conjugent gradient'''
    def __init__(self, model, searchConfig = None):
        super(CG, self).__init__()
        self.model = model
        self.searchConfig = searchConfig
        self.trailsCounter = 0
        self.errorCounter = 0

    def findMin(self, x, y):
        meanfunc = self.model.meanfunc
        covfunc = self.model.covfunc
        likfunc = self.model.likfunc
        inffunc = self.model.inffunc
        hypInArray = self.convert_to_array()
        try:
            opt = cg(self.nlml, hypInArray, self.dnlml, maxiter=100, disp=False, full_output=True)
            optimalHyp = deepcopy(opt[0])
            funcValue  = opt[1]
            warnFlag   = opt[4]
            if warnFlag == 1:
                print "Maximum number of iterations exceeded."
            elif warnFlag ==  2:
                print "Gradient and/or function calls not changing."
        except:
            self.errorCounter += 1
            if not self.searchConfig:         
                raise Exception("Can not use conjugate gradient. Try other hyparameters")
        self.trailsCounter += 1

        if self.searchConfig:
            searchRange = self.searchConfig.meanRange + self.searchConfig.covRange + self.searchConfig.likRange 
            if not (self.searchConfig.num_restarts or self.searchConfig.min_threshold):
                raise Exception('Specify at least one of the stop conditions')
            while True:
                self.trailsCounter += 1                 # increase counter
                for i in xrange(hypInArray.shape[0]):   # random init of hyp
                    hypInArray[i]= np.random.uniform(low=searchRange[i][0], high=searchRange[i][1])
                # value this time is better than optiaml min value
                try:
                    thisopt = cg(self.nlml, hypInArray, self.dnlml, maxiter=100, disp=False, full_output=True)
                    if thisopt[1] < funcValue:
                        funcValue  = thisopt[1]
                        optimalHyp = thisopt[0]
                except:
                    self.errorCounter += 1
                if self.searchConfig.num_restarts and self.errorCounter > self.searchConfig.num_restarts/2:
                    print "[CG] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter)
                    raise Exception("Over half of the trails failed for conjugate gradient")
                if self.searchConfig.num_restarts and self.trailsCounter > self.searchConfig.num_restarts-1:         # if exceed num_restarts
                    print "[CG] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter)
                    return optimalHyp, funcValue
                if self.searchConfig.min_threshold and funcValue <= self.searchConfig.min_threshold:           # reach provided mininal
                    print "[CG] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter)
                    return optimalHyp, funcValue 
        return optimalHyp, funcValue





[docs]class BFGS(Optimizer):
    '''quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS)'''
    def __init__(self, model, searchConfig = None):
        super(BFGS, self).__init__()
        self.model = model
        self.searchConfig = searchConfig
        self.trailsCounter = 0
        self.errorCounter = 0

    def findMin(self, x, y):
        meanfunc = self.model.meanfunc
        covfunc = self.model.covfunc
        likfunc = self.model.likfunc
        inffunc = self.model.inffunc
        hypInArray = self.convert_to_array()

        try:
            opt = bfgs(self.nlml, hypInArray, self.dnlml, maxiter=100, disp=False, full_output=True)
            optimalHyp = deepcopy(opt[0])
            funcValue  = opt[1]
            warnFlag   = opt[6]
            if warnFlag == 1:
                print "Maximum number of iterations exceeded."
            elif warnFlag ==  2:
                print "Gradient and/or function calls not changing."
        except:
            self.errorCounter += 1
            if not self.searchConfig:         
                raise Exception("Can not use BFGS. Try other hyparameters")
        self.trailsCounter += 1


        if self.searchConfig:
            searchRange = self.searchConfig.meanRange + self.searchConfig.covRange + self.searchConfig.likRange 
            if not (self.searchConfig.num_restarts or self.searchConfig.min_threshold):
                raise Exception('Specify at least one of the stop conditions')
            while True:
                self.trailsCounter += 1                 # increase counter
                for i in xrange(hypInArray.shape[0]):   # random init of hyp
                    hypInArray[i]= np.random.uniform(low=searchRange[i][0], high=searchRange[i][1])
                # value this time is better than optiaml min value
                try:
                    thisopt = bfgs(self.nlml, hypInArray, self.dnlml, maxiter=100, disp=False, full_output=True)
                    if thisopt[1] < funcValue:
                        funcValue  = thisopt[1]
                        optimalHyp = thisopt[0]
                except:
                    self.errorCounter += 1
                if self.searchConfig.num_restarts and self.errorCounter > self.searchConfig.num_restarts/2:
                    print "[BFGS] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter)
                    raise Exception("Over half of the trails failed for BFGS")
                if self.searchConfig.num_restarts and self.trailsCounter > self.searchConfig.num_restarts-1:         # if exceed num_restarts
                    print "[BFGS] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter)
                    return optimalHyp, funcValue
                if self.searchConfig.min_threshold and funcValue <= self.searchConfig.min_threshold:           # reach provided mininal
                    print "[BFGS] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter)
                    return optimalHyp, funcValue 

        return optimalHyp, funcValue





[docs]class Minimize(Optimizer):
    '''minimize by Carl Rasmussen (python implementation of "minimize" in GPML)'''
    def __init__(self, model, searchConfig = None):
        super(Minimize, self).__init__()
        self.model = model
        self.searchConfig = searchConfig
        self.trailsCounter = 0
        self.errorCounter = 0

    def findMin(self, x, y):
        meanfunc = self.model.meanfunc
        covfunc = self.model.covfunc
        likfunc = self.model.likfunc
        inffunc = self.model.inffunc
        hypInArray = self.convert_to_array()

        try: 
            opt = minimize.run(self.nlzAnddnlz, hypInArray, length=-40)
            optimalHyp = deepcopy(opt[0])
            funcValue  = opt[1][-1]  
        except:
            self.errorCounter += 1
            if not self.searchConfig:         
                raise Exception("Can not use minimize. Try other hyparameters")
        self.trailsCounter += 1

        if self.searchConfig:
            searchRange = self.searchConfig.meanRange + self.searchConfig.covRange + self.searchConfig.likRange 
            if not (self.searchConfig.num_restarts or self.searchConfig.min_threshold):
                raise Exception('Specify at least one of the stop conditions')
            while True:
                self.trailsCounter += 1                 # increase counter
                for i in xrange(hypInArray.shape[0]):   # random init of hyp
                    hypInArray[i]= np.random.uniform(low=searchRange[i][0], high=searchRange[i][1])
                # value this time is better than optiaml min value
                try:
                    thisopt = minimize.run(self.nlzAnddnlz, hypInArray, length=-40)
                    if thisopt[1][-1] < funcValue:
                        funcValue  = thisopt[1][-1]
                        optimalHyp = thisopt[0]
                except:
                    self.errorCounter += 1
                if self.searchConfig.num_restarts and self.errorCounter > self.searchConfig.num_restarts/2:
                    print "[Minimize] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter)
                    raise Exception("Over half of the trails failed for minimize")
                if self.searchConfig.num_restarts and self.trailsCounter > self.searchConfig.num_restarts-1:         # if exceed num_restarts
                    print "[Minimize] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter)
                    return optimalHyp, funcValue
                if self.searchConfig.min_threshold and funcValue <= self.searchConfig.min_threshold:           # reach provided mininal
                    print "[Minimize] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter)
                    return optimalHyp, funcValue                   
        return optimalHyp, funcValue





[docs]class SCG(Optimizer):
    '''Scaled conjugent gradient (faster than CG)'''
    def __init__(self, model, searchConfig = None):
        super(SCG, self).__init__()
        self.model = model
        self.searchConfig = searchConfig
        self.trailsCounter = 0
        self.errorCounter = 0

    def findMin(self, x, y):
        meanfunc = self.model.meanfunc
        covfunc = self.model.covfunc
        likfunc = self.model.likfunc
        inffunc = self.model.inffunc
        hypInArray = self.convert_to_array()
        try:
            opt = scg.run(self.nlzAnddnlz, hypInArray)
            optimalHyp = deepcopy(opt[0])
            funcValue  = opt[1][-1]
        except:
            self.errorCounter += 1
            if not self.searchConfig:         
                raise Exception("Can not use Scaled conjugate gradient. Try other hyparameters")
        self.trailsCounter += 1

        if self.searchConfig:
            searchRange = self.searchConfig.meanRange + self.searchConfig.covRange + self.searchConfig.likRange 
            if not (self.searchConfig.num_restarts or self.searchConfig.min_threshold):
                raise Exception('Specify at least one of the stop conditions')
            while True:
                self.trailsCounter += 1                 # increase counter
                for i in xrange(hypInArray.shape[0]):   # random init of hyp
                    hypInArray[i]= np.random.uniform(low=searchRange[i][0], high=searchRange[i][1])
                # value this time is better than optiaml min value
                try:
                    thisopt = scg.run(self.nlzAnddnlz, hypInArray)
                    if thisopt[1][-1] < funcValue:
                        funcValue  = thisopt[1][-1]
                        optimalHyp = thisopt[0]
                except:
                    self.errorCounter += 1
                if self.searchConfig.num_restarts and self.errorCounter > self.searchConfig.num_restarts/2:
                    print "[SCG] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter)
                    raise Exception("Over half of the trails failed for Scaled conjugate gradient")
                if self.searchConfig.num_restarts and self.trailsCounter > self.searchConfig.num_restarts-1:         # if exceed num_restarts
                    print "[SCG] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter)
                    return optimalHyp, funcValue
                if self.searchConfig.min_threshold and funcValue <= self.searchConfig.min_threshold:           # reach provided mininal
                    print "[SCG] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter)
                    return optimalHyp, funcValue 

        return optimalHyp, funcValue

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© Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting. Created using Sphinx 1.2.
pyGPs-1.2/doc/build/_modules/pyGPs/GraphExtensions/000077500000000000000000000000001236346020200222525ustar00rootroot00000000000000pyGPs-1.2/doc/build/_modules/pyGPs/GraphExtensions/graphKernels.html000066400000000000000000000757471236346020200256110ustar00rootroot00000000000000 pyGPs.GraphExtensions.graphKernels — pyGPs v1.2 documentation

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Source code for pyGPs.GraphExtensions.graphKernels

#================================================================================
#    Marion Neumann [marion dot neumann at uni-bonn dot de]
#    Daniel Marthaler [marthaler at ge dot com]
#    Shan Huang [shan dot huang at iais dot fraunhofer dot de]
#    Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]
#
#    This file is part of pyGPs.
#    The software package is released under the BSD 2-Clause (FreeBSD) License.
#
#    Copyright (c) by
#    Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014
#================================================================================

'''
Created on April, 2014

@author: marion, shan
'''
import scipy.sparse as spsp
import numpy as np
import matplotlib.pyplot as plt
    

[docs]def propagationKernel(A, l, gr_id, h_max, w, p, ktype=None, SUM=True, VIS=False, showEachStep=False):
    '''
    Propagation kernel for graphs.

    :param A: adjacency matrix (num_nodes x num_nodes)
    :param l: label array (num_nodes x 1); values [1,...,k] or -1 for unlabeled nodes 
              OR label array (num_nodes x num_labels); values [0,1], unlabeled nodes have only 0 entries
    :param gr_id: graph indicator array (num_nodes x 1); values [0,..,n]
    :param h_max: number of iterations
    :param w: bin widths parameter
    :param p: distance ('tv', 'hellinger', 'L1', 'L2')
    :param ktype: type of propagation kernel ['diffusion', 'label_propagation', 'label_spreading', 'belief_propagation']
    
    :return: kernel matrix 
    '''
    # ToDO: SPLIT Varible label array and label probability
        
    # CAUTION: number of labels (k) > 1 !!! 
    
    #===========================================================================
    # ## Propagation Kernel COMPUTATION
    #===========================================================================
    num_graphs = gr_id.max().astype(int)
    num_nodes = A.shape[0]
    if l.shape[1]==1:
        num_labels = l.max().astype(int)
    else:
        num_labels = l.shape[1]
 
    #===========================================================================
    # ## INITIALIZATION
    #===========================================================================
    
    ## CHECK and ADJUST sparseness
    if not spsp.issparse(A): 
        A = spsp.csr_matrix(A)
    else:
        if not spsp.isspmatrix_csr(A):     
            A = A.tocsr()
    # MAKE sure A is float!        
    A = A.astype(float)                  
    ## INITIALIZE kernel matrix
    K = np.ndarray(shape=(num_graphs,num_graphs,h_max+1), dtype=float)
    ## INITIALIZE label probabilities of labeled nodes (num_nodes x num_labels)
    if l.shape[1]==1:
        lab_prob = np.zeros((num_nodes,num_labels),dtype=np.float64)            
        for i in xrange(num_nodes):
            if l[i,0] > 0:
                lab_prob[i,l[i,0]-1] = 1
    else: 
        lab_prob = l
        if spsp.issparse(lab_prob):
            lab_prob = lab_prob.todense()
    lab_orig  = lab_prob.copy()      # lab_orig is the orignal label information       
    ## GET indices of labeled and unlabeled nodes
    idx_unlab = np.where(np.max(lab_prob, axis=1)!=1)[0]
    idx_all   = np.array(np.arange(0,A.shape[0]), ndmin=2)
    idx       = np.array(np.setdiff1d(idx_all,idx_unlab), ndmin=2)     
    ## INITIALIZE unlabeled/uninitialized nodes UNIFORMLY
    idx_unif = np.where(np.sum(lab_prob, axis=1)==0)[0]         
    if idx_unif.shape[0] != 0:
        lab_prob[idx_unif,:] = 1./lab_prob.shape[1] 
    ## row normalize A -> transition matrix T
    if (ktype=='label_propagation') or (ktype=='label_diffusion'):
        row_sums = np.array(A.sum(axis=1))[:,0] 
        T = A.copy()
        T.data /= row_sums[A.nonzero()[0]]    # so A and T are sparse matrix

    #===========================================================================
    # ## PROPAGATION KERNEL ITERATIONS
    #===========================================================================
    for h in xrange(h_max+1):  
        print 'ITERATION: ', h
        if h > 0:
            ## LABEL UPDATE 
            if showEachStep:
                print '...computing LABEL UPDATE'
            
            if ktype == 'label_propagation':
                lab_prob[idx,:] = lab_orig[idx,:]   # PUSH BACK original LABELS
                lab_prob = T*lab_prob
                
            elif ktype == 'label_diffusion': 
                lab_prob = T*lab_prob
            
            elif ktype == 'label_spreading':
                # S = D^(-1/2)*W*D^(-1/2)
                # y(t+1) = alpha*S*y(t)+(1-alpha)*y(0)
                alpha = 0.8
                # compute S
                diag = np.array(A.sum(axis=1)).T**(-1/2)
                D = spsp.lil_matrix((A.shape[0], A.shape[1]), dtype=float) #
                D.setdiag(diag[0,:])
                D = D.tocsr()
                S = D*A*D   
                lab_prob = np.dot((alpha*S.todense()),lab_prob) + (1-alpha)*lab_orig
        
        ## COMPUTE hashvalues 
        if showEachStep:
            print '...computing hashvalues'
        # determine path to take depending on chosen distance
        use_cauchy = (p =='L1') or (p =='tv')
        take_sqrt  = (p =='hellinger') 
        if take_sqrt:
            lab_prob = np.sqrt(lab_prob)      
        # generate random projection vector
        v = np.random.normal(size=(lab_prob.shape[1], 1))
        if use_cauchy:
            # if X, Y are N(0, 1), then X / Y has a standard Cauchy distribution
            v /= np.random.normal(size=(lab_prob.shape[1], 1))           
        # random offset
        b = w * np.random.rand()         
        # compute hashes
        # hashLabels is a Vector with length: number of nodes (hashvalue for respective node)
        hashLabels = (np.dot(lab_prob,v) + b) / w
        hashLabels = np.floor(hashLabels)                           # take floor
        uniqueHash, hashLabels = np.unique(hashLabels, return_inverse=True)  # map to consecutive integer from 0        
        ## COMPUTE kernel contribution 
        if showEachStep:
            print '...computing KERNEL contribution'                     
        # aggregate counts on graphs
        # counts is a matrix: number of graphs x number of hashlabels
        num_bins = len(uniqueHash)
        counts = np.zeros((num_graphs, num_bins))      # init counts matrix 
        # accumulate counts of hash labels                              
        for i in xrange(num_nodes):
            counts[ (gr_id[i,0]-1), hashLabels[i] ] +=1
            
        # compute base kernel (here: LINEAR kernel)
        K_h = np.dot(counts,counts.T)      
        
        ## SUM kernel contributions
        if SUM:
            if h == 0:
                K[:,:,h] = K_h 
            else:
                K[:,:,h] = K[:,:,h-1] + K_h 
        else:
            K[:,:,h] = K_h
        
        if showEachStep:
            print K[:,:,h] 

    ## VISUALIZE KERNELS
    if VIS:  
        for h in range(h_max+1): 
            K_h = K[:,:,h]
            plt.figure()
            plt.title('height:'+str(h))
            imgplot = plt.imshow(K_h)
            imgplot.set_interpolation('nearest')
            plt.colorbar()
        plt.show()
    
    return K

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© Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting. Created using Sphinx 1.2.
pyGPs-1.2/doc/build/_modules/pyGPs/GraphExtensions/nodeKernels.html000066400000000000000000000455021236346020200254170ustar00rootroot00000000000000 pyGPs.GraphExtensions.nodeKernels — pyGPs v1.2 documentation

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Source code for pyGPs.GraphExtensions.nodeKernels

#================================================================================
#    Marion Neumann [marion dot neumann at uni-bonn dot de]
#    Daniel Marthaler [marthaler at ge dot com]
#    Shan Huang [shan dot huang at iais dot fraunhofer dot de]
#    Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]
#
#    This file is part of pyGPs.
#    The software package is released under the BSD 2-Clause (FreeBSD) License.
#
#    Copyright (c) by
#    Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014
#================================================================================


import numpy as np
import scipy.linalg as spla
import scipy.sparse as spsp



[docs]def normLap(A):
    '''
    Normalized Laplacian

    :param A: adjacency matrix
    :return: kernel matrix
    '''
    I = np.identity(A.shape[0])
    d = A.sum(axis=0)
    d = np.sqrt(1./d)    
    D = np.diag(d)
    L = I - np.dot( np.dot(D,A),D )
    return L




[docs]def regLapKernel(A, sigma=1):
    '''
    Regularized Laplacian Kernel 
        
    :param A: adjacency matrix
    :param sigma: hyperparameter sigma
    :return: kernel matrix
    '''
    I = np.identity(A.shape[0])
    L = normLap(A)
    K = np.linalg.inv( I+(sigma**2)*L )
    return K




[docs]def psInvLapKernel(A):
    '''
    Pseudo inverse of the normalized Laplacian.

    :param A: adjacency matrix
    :return: kernel matrix
    '''
    L = normLap(A)
    K = np.linalg.pinv(L)
    return K




[docs]def diffKernel(A, beta=0.5):
    '''
    Diffusion Process Kernel 
        
    K = exp(beta * H), where H = -L = A-D
    
    K = Q exp(beta * Lambda) Q.T
        
    :param A: adjacency matrix
    :param beta: hyperparameter beta
    :return: kernel matrix
    '''
    A = np.array(A) # make sure that A is a numpy array!!
    H = A - np.diag(np.sum(A, axis=1))
    w, Q = np.linalg.eigh(H)
    Lambda = np.diag(np.exp(beta*w))
    K = np.dot(np.dot(Q, Lambda), Q.T)
    return K




[docs]def VNDKernel(A, alpha=0.5):
    '''
    Von Neumann Diffusion Kernel on graph (Zhou et al., 2004)
    (also label spreading kernel)
        
    K = (I - alpha*S)^-1, where S = D^-1/2*A*D^-1/2
        
    :param A: adjacency matrix
    :param alpha: hyperparameter alpha
    :return: kernel matrix
    ''' 
    I = np.identity(A.shape[0])
    d = A.sum(axis=0)
    d = np.sqrt(1./d)    
    D = np.diag(d)
    S = np.dot( np.dot(D,A),D )
    K = np.linalg.inv( I - alpha*S )
    return K




[docs]def rwKernel(A, p=1, a=2):
    '''
    p-step Random Walk Kernel with a>1
       
    K = (aI-L)^p, p>1 and L is the normalized Laplacian 
     
    :param A: adjacency matrix
    :param p: step parameter
    :param a: hyperparameter a
    :return: kernel matrix
    '''
    if type(p) != int:
	p = int(p)
    if p < 1:
	raise Exception('Step parameter p needs to be larger than 0.') 	
    if a <= 1:	
	a=1.0001
    I = np.identity(A.shape[0])
    L = normLap(A)
    K = np.linalg.matrix_power( a*I - L, p)	
    return K





[docs]def cosKernel(A):
    '''
    Cosine Kernel (also Inverse Cosine Kernel)
    
    K = cos (L*pi/4), where L is the normalized Laplacian
                
    :param A: adjacency matrix
    :return: kernel matrix
    '''
    L = normLap(A)
    K = np.cos(L*np.pi/4)   
    return K

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© Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting. Created using Sphinx 1.2.
pyGPs-1.2/doc/build/_sources/000077500000000000000000000000001236346020200161025ustar00rootroot00000000000000pyGPs-1.2/doc/build/_sources/CV.txt000066400000000000000000000036401236346020200171560ustar00rootroot00000000000000K-fold Cross-Validation ================================== In this demo, we'll show you the typical process of using GP for machine learning from loading data, preprocessing, training, predicting to validation and evaluation. Load data -------------------- We use the ionosphere dataset [1]_ from Johns Hopkins University Ionosphere database. It is available in UCI machine learning repository. Then we need to do some data cleaning. Here we deal with label in ionosphere data, change "b" to"-1", and "g" to "+1". These preprocessing implementation are availabe in the source code. Cross Validation ---------------------------- Now, lets focus on the use of cross-validation. :: K = 10 # number of folds for x_train, x_test, y_train, y_test in valid.k_fold_validation(x, y, K): # This is a binary classification problem model = pyGPs.GPC() # Since no prior knowldege, leave everything default model.optimize(x_train, y_train) # Predition ymu, ys2, fmu, fs2, lp = model.predict(x_test, ys=y_test) # ymu for classification is a continuous value over -1 to +1 # If you want predicting result to either one of the classes, take a sign of ymu. ymu_class = np.sign(ymu) # Evluation acc = valid.ACC(ymu_class, y_test) # accuracy rmse = valid.RMSE(ymu_class, y_test) # root-mean-square error Evaluation measures ----------------------------- We implemented some classical evaluation measures. - RMSE - root mean squared error - ACC - classification/regression accuracy - Prec - classification precision for class +1 - Recall - classification recall for class +1 - NLPD - negative log predictive density in transformed observation space .. [1] Sigillito, V. G., Wing, S. P., Hutton, L. V., \& Baker, K. B. (1989). Classification of radar returns from the ionosphere using neural networks. Johns Hopkins APL Technical Digest, 10, 262-266. pyGPs-1.2/doc/build/_sources/Examples.txt000066400000000000000000000021701236346020200204210ustar00rootroot00000000000000Demos ========================= There are several demos exemplifying the use of pyGPs for various Gaussian process (:math:`GP`) tasks. We recommend to first go through *Basic GP Regression* which introduces the :math:`GP` regression model. Basic regression is the most intuitive and simplest learning task feasable with :math:`GPs`. The other demos will then provide a general insight into more advanced functionalities of the package. You will also find the implementation of the demos in the source_ folder under `pyGPs/Demo`_. .. _pyGPs/Demo: https://github.com/marionmari/pyGPs/tree/master/pyGPs/Demo The Demos give some theoretical explanations. Further, it is useful to have a look at our documentation on `Kernels & Means`_ and `Optimizers`_. .. _Kernels & Means: Kernels.html .. _Optimizers: Opts.html .. _source: https://github.com/marionmari/pyGPs Regression .. toctree:: :maxdepth: 1 GPR GPR_FITC Classification .. toctree:: :maxdepth: 1 GPC GPC_FITC GPMC Some examples for real-world data .. toctree:: :maxdepth: 1 CV demoMaunaLoa demoHousing SemiSupervised GraphKernel pyGPs-1.2/doc/build/_sources/GPC.txt000066400000000000000000000074671236346020200172720ustar00rootroot00000000000000Basic Classification ========================= The demo shown in this tutorial can be obtained by running *pyGPs/Demo/demo_GPC.py*. Load data -------------------- First, we import the data:: # GPC target class are +1 and -1 demoData = np.load('data_for_demo/classification_data.npz') x = demoData['x'] # training data y = demoData['y'] # training target z = demoData['xstar'] # test data The :math:`120` data points were generated from two Gaussians with different means and covariances. One Gaussian is isotropic and contains :math:`2/3` of the data (blue), the other is highly correlated and contains :math:`1/3` of the points (red). Note, that the labels for the targets are specified to be :math:`\pm 1` (and not :math:`0/1`). In the plot, we superimpose the data points with the posterior equi-probability contour lines for the probability of the second class given complete information about the generating mechanism. .. figure:: _images/d2_1.png :height: 600 px :width: 800 px :align: center :scale: 70 % First example :math:`\rightarrow` state default values ---------------------------------------------------------------- Again, lets see the simplest use of gp classification at first :: model = pyGPs.gp.GPC() # binary classification (default inference method: EP) model.fit(x, y) # fit default model (mean zero & rbf kernel) with data model.optimize(x, y) # optimize hyperparamters (default optimizer: single run minimize) model.predict(z) # predict test cases Note, that inference is done via expectation propagation (EP) approximation by deault. How to set inference to Laplace approximation, see :ref:`more_on_GPC`. Second example :math:`\rightarrow` GP classification ------------------------------------------------------------ So we first state the model to be :math:`GP` classification now:: model = pyGPs.GPC() The rest is similar to GPR:: k = pyGPs.cov.RBFard(log_ell_list=[0.05,0.17], log_sigma=1.) model.setPrior(kernel=k) model.setData(x, y) model.plotData_2d(x1,x2,t1,t2,p1,p2) model.fit() model.optimize() model.predict(z, ys=np.ones((z.shape[0],1))) **[Theory]** In this example, we used an RBF kernel (squared exponential covariance function) with automatic relevance determination (ARD). This covariance function has one characteristic length-scale parameter for each dimension of the input space (here :math:`2` in total), and a signal magnitude parameter, resulting in a total of :math:`3` hyperparameters. ARD with separate length-scales for each input dimension is a very powerful tool to learn which inputs are important for the predictions: if length-scales are short, input dimensions are very important, and when they grow very large (compared to the spread of the data), the corresponding input dimensions will be mostly ignored. Note, *pyGPs.GPC().plot()* is a toy method for 2-d data:: model.plot(x1,x2,t1,t2) The contour plot for the predictive distribution is shown below. Note, that the predictive probability is fairly close to the probabilities of the generating process in regions of high data density. Note also, that as you move away from the data, the probability approaches :math:`1/3`, the overall class probability. .. figure:: _images/d2_2.png :height: 600 px :width: 800 px :align: center :scale: 70 % Examining the two ARD characteristic length-scale parameters after learning, you will find that they are fairly similar, reflecting the fact that for this data set, both input dimensions are important. .. _more_on_GPC: A bit more things you can do ---------------------------------------------------- GPC uses expectation propagation (EP) inference and Error function likelihood by default, you can explictly change to other methods: :: model.useInference("Laplace") pyGPs-1.2/doc/build/_sources/GPC_FITC.txt000066400000000000000000000063401236346020200200640ustar00rootroot00000000000000Sparse Classification ======================== The demo in this tutorial can be obtained by running *pyGPs/Demo/demo_GPC_FITC.py*. This demo is more or less a repetition of the demo of `FITC regression`_. .. _FITC regression: GPR_FITC.html First example :math:`\rightarrow` default inducing points ------------------------------------------------------------- First load the same data as in the GPC demo. **[Theory]** In case the number of training inputs :math:`x` exceeds a few hundred, approximate inference using Laplacian Approximation or Expectation Propagation takes too long. As in regression, we offer the FITC approximation based on a low-rank plus diagonal approximation to the exact covariance to deal with these cases. The general idea is to use inducing points :math:`u` and to base the computations on cross-covariances between training, test and inducing points only. Okay, now the model is FITC classificiation:: model = pyGPs.GPC_FITC() The difference between the usage of basic :math:`GP` is that we will have to specify inducing points. In our first example, we will introduce how to perform sparse GPC with the default settings. The default inducing points form a grid (hypercube in higher dimension), where each dimension has :math:`5` values in equidistant steps in :math:`[min, max]`, where :math:`min` and :math:`max` are the minimum and maximum values of the input data by default. In order to specify the dimension of input data, we HAVE TO set data first:: model.setData(x, y) The number of inducing points per axis is :math:`5` per default. How to change this, see :ref:`more_on_GPC_FITC`. Then, the regular process follows:: model.optimize() model.predict(z, ys=np.ones((z.shape[0],1))) model.plot(x1,x2,t1,t2) .. figure:: _images/d4_1.png :height: 600 px :width: 800 px :align: center :scale: 70 % The equispaced default inducing points :math:`u` are shown as black circles in the plot. Second example :math:`\rightarrow` user-defined inducing points -------------------------------------------------------------------- Alternatively, a random subset of the training points can be used as inducing points. Note, that there are various different ways of how to set the inducing points. So, in the second example let us use a user-defined set of inducing points:: u1,u2 = np.meshgrid(np.linspace(-2,2,5),np.linspace(-2,2,5)) u = np.array(zip(np.reshape(u2,(np.prod(u2.shape),)),np.reshape(u1,(np.prod(u1.shape),)))) Here, we also use a grid euqually spaced, but without the values on the margin of the grid.(i.e. shrinking the grid) Then, we can just pass :math:`u` when specifying prior:: m = pyGPs.mean.Zero() k = pyGPs.cov.RBFard(log_ell_list=[0.05,0.17], log_sigma=1.) model.setPrior(mean=m, kernel=k, inducing_points=u) The prediction results for this set of inducing points are shown below: .. figure:: _images/d4_2.png :height: 600 px :width: 800 px :align: center :scale: 70 % .. _more_on_GPC_FITC: A bit more things you can do ------------------------------ As in standard GPC, it is possible to use other inference/likelihood in the FITC method:: model.useInference("Laplace") Change the number of inducing points per axis:: model.setData(x, y, value_per_axis=10) pyGPs-1.2/doc/build/_sources/GPMC.txt000066400000000000000000000043321236346020200173730ustar00rootroot00000000000000Multi-class Classification =========================== GPMC is NOT based on multi-class Laplace approximation. It works as a one vs. one classification wrapper. In other words, GPMC trains a GPC model for each pair of two classes, and uses a majority voting scheme over all results to determine the final class. The method only returns the predictive class with highest rating; no other values (such as variance) are returned. Lets see a practical example to classify the 10 (0,1,2,...9) hand-written digits in the USPS digits dataset. Load data -------------------- The USPS digits data were gathered at the Center of Excellence in Document Analysis and Recognition (CEDAR) at SUNY Buffalo, as part of a project sponsored by the US Postal Service. The dataset is described in [1]_. :: data = loadmat('data_for_demo/usps_resampled.mat') x = data['train_patterns'].T # train patterns y = data['train_labels'].T # train labels xs = data['test_patterns'].T # test patterns ys = data['test_labels'].T # test labels To be used in GPMC, labels should start from 0 to k (k = number of classes). GPMC example --------------------- State model with 10-class classification problem: :: model = pyGPs.GPMC(10) Pass data to model: :: model.setData(x,y) Train default GPC model for each binary classification problem, and decide label for test patterns of hand-writen digits. The return value *predictive_vote[i,j]* is the probability of being class *j* for test pattern *i*. :: predictive_vote = model.optimizeAndPredict(xs) predictive_class = np.argmax(predictive_vote, axis=1) Just like we did for GP classification, you can use specific settings (other than default) for all binary classificiation problem for example by: :: m = pyGPs.mean.Zero() k = pyGPs.cov.RBF() model.setPrior(mean=m,kernel=k) model.useInference("Laplace") For more information on how to use non-default settings see `demo_GPC`_ and `demo_GPR`_. .. _demo_GPC: GPC.html .. _demo_GPR: GPR.html Beside *optimizeAndPredict(xs)*, there is also an option to perform prediction without hyperparameter optimization: :: model.fitAndPredict(xs) .. [1] A Database for Handwritten Text Recognition Research, J. J. Hull, IEEE PAMI 16(5) 550-554, 1994. pyGPs-1.2/doc/build/_sources/GPR.txt000066400000000000000000000150431236346020200172760ustar00rootroot00000000000000Basic Regression ========================= The code shown in this tutorial can be executed by running *pyGPs/Demo/demo_GPR.py* This demo will not only introduce the regression model, it also provides the general insight of how to use the package. This general information will not be repeated in the other demos. Import packages -------------------- Once you installed pyGPs, the typical way to import it is: :: import pyGPs import numpy as np Load data -------------------- First, load the data for this demo. The data consists of :math:`n=20` 1-d data points drawn from a unit Gaussian. This is the same data used in the GPML example (it is hardcoded in *data/regression_data.npz*). Note that target vector :math:`y` can be in 2d matrix with shape (nn,1) or 1d vector with shape (nn,) where :math:`nn` is number of test inputs. pyGPs can work with either format. :: demoData = np.load('data_for_demo/regression_data.npz') x = demoData['x'] # training data y = demoData['y'] # training target z = demoData['xstar'] # test data A five-line toy example --------------------------- Now lets do regression with Gaussian processes. Using pyGPs for regression is really simple; here is the most basic example: :: model = pyGPs.GPR() # specify model (GP regression) model.fit(x, y) # fit default model (mean zero & rbf kernel) with data model.optimize(x, y) # optimize hyperparamters (default optimizer: single run minimize) model.predict(z) # predict test cases model.plot() # and plot result By default, GPR uses a zero mean, the rbf kernel and a Gaussian likelihood. Default optimizer is a single run of Rasmussen's minimize. You will see below how to set non-default values in another example. *pyGPs.GPR().plot()* will plot the result, where the dark line is the posterior mean and the green-shaded area is the posterior variance. Note, that *plot()* is not a general method as it is not trivial to visualize high dimensional data. Here, *pyGPs.GPR().plot()* works for 1-d data only, while *pyGPs.GPC().plot()* is a toy method visualising 2-d input data in a classification scenario. .. figure:: _images/d1_1.png :height: 600 px :width: 800 px :align: center :scale: 70 % A more complicated example --------------------------- Now lets do another example to get insight into more advanced features of the toolbox. You can specify non-default mean and covariance functions: :: m = pyGPs.mean.Linear( D=x.shape[1] ) + mean.Const() k = pyGPs.cov.RBF() model.setPrior(mean=m, kernel=k) Here, we use a composite mean as the sum of a linear and a constant function, and an rbf kernel. The initial hyperparameters are left to their default values. See `Kernels & Means`_ for a complete documentation of kernel/mean specification and custom kernel/mean construction. Once kernel and mean are specified, they are passed to the prior using *setPrior()*. .. _Kernels & Means: Kernels.html You can add the traning data to the model explicitly by using *setData()*. So, you avoid passing them into *fit()* or *optimize()* each time used. More importantly, the deafult mean will be adapted to the average value of the trainging labels :math:`y` (if you do not specify mean function by your own). Further, you can plot the data in the 1-d case: :: model.plotData_1d() .. figure:: _images/d1_2.png :height: 600 px :width: 800 px :align: center :scale: 70 % You can specify a optimization method different from the default, which is a single run of Rasmussen's minimize. For example, you can choose to rerun the optimization method several times with different random initializations: :: model.setOptimizer("Minimize", num_restarts=30) The optimized hyperparameters returned by *optimize()* are then set to be the ones obtained from the run with the best result. The whole functionality for optimization is introduced in detail in the documentation `Optimizers`_. .. _Optimizers: Opts.html Instead of *fit()*, which only fits data using given hyperparameters, *optimize()* will optimize hyperparamters based on marginal likelihood: :: model.optimize() There are several properties you can get from the model: :: model.nlZ # negative log marginal likelihood model.dnlZ.cov # direvatives of negative log marginal likelihood model.dnlZ.lik model.dnlZ.mean model.posterior.sW # posterior structure model.posterior.alpha model.posterior.L model.covfunc.hyp model.meanfunc.hyp model.likfunc.hyp model.fm # latent mean model.fs2 # latent variance model.ym # predictive mean model.ys2 # predictive variance model.lp # log predictive probability For example, to get the log marginal likelihood use: :: print 'Optimized negative log marginal likelihood:', round(model.nlZ,3) Prediction on the test data will return five values, which are output mean (ymu) resp. variance (ys2), latent mean (fmu) resp. variance (fs2), and log predictive probabilities (lp) :: ym, ys2, fm, fs2, lp = model.predict(z) Plot data. Note that *GPR.plot()* is a toy method only for visualising 1-d data. Here we got a different posterior by using a different prior other than in the default example. :: model.plot() .. figure:: _images/d1_3.png :height: 600 px :width: 800 px :align: center :scale: 70 % A bit more things you can do ------------------------------- **[For all Models]** Speed up computation time for prediction if you know posterior in advance. Posterior is passed as an object with three fields (attributes) post.alpha, post.sW and post.L. How to use these vectors to represent the posterior can be best seen from Algorithm 2.1 (page 19) in Chapeter 2 of the `GPML`_ book by Rasmussen and Williams, 2006. :: post = myPosterior() # known in advance ym, ys2, fm, fs2, lp = model.predict_with_posterior( post,z ) .. _GPML: http://www.gaussianprocess.org/gpml/chapters/RW2.pdf **[Only for Regression]** Specify noise of data (with :math:`\sigma=0.1` by default): :: model.setNoise( log_sigma = np.log(0.1) ) You do not need to specify the noise parameter if you are optimizing the hyperparamters later anyhow. All plotting methods have keyword axisvals. You can adjust plotting range if you want. For example: :: model.plot(axisvals = [-1.9, 1.9, -0.9, 3.9]) Switch to other Inference and Likelihood functions. :: model.useInference("EP") model.useLikelihood("Laplace") pyGPs-1.2/doc/build/_sources/GPR_FITC.txt000066400000000000000000000070011236346020200200760ustar00rootroot00000000000000Sparse Regression ========================= The code shown in this tutorial can be obtained by running *pyGPs/Demo/demo_GPR_FITC.py* This demo is more or less similar to the demo of `FITC classification`_. .. _FITC classification: GPC_FITC.html First example :math:`\rightarrow` default inducing points -------------------------------------------------------------- First load the same data as in the GPR demo. **[Theory]** In case the number of training inputs :math:`x` exceeds a few hundred, approximate inference using Laplace approximation or expectation propagation takes too long. We offer the FITC approximation based on a low-rank plus diagonal approximation to the exact covariance to deal with these cases. The general idea is to use inducing points :math:`u` and to base the computations on cross-covariances between training, test and inducing points only. Okay, now the model is FITC regression:: model = pyGPs.GPR_FITC() The difference between the usage of basic :math:`GP` regression is that we will have to specify inducing points. In the first example here, we will introduce you how to use the default settings. The default inducing points are a grid (hypercube for higher dimensions), where each dimension has 5 values in equidistant steps in :math:`[min, max]`, where :math:`min` and :math:`max` are the minimum and maximum values of the input data by default. In order to specify the dimension of input data, we HAVE TO set data first:: model.setData(x, y) The number of inducing points per axis is :math:`5` per default. Now, the regular training and prediction routines follow: :: model.optimize() model.predict(z) model.plot() .. figure:: _images/d3_1.png :height: 600 px :width: 800 px :align: center :scale: 70 % The equidistant default inducing points :math:`u` that are shown in the figure as black x's. To change the number of inducing points per axis just specify a different value per axis:: model.setData(x, y, value_per_axis=10) Second example :math:`\rightarrow` user-defined inducing points ------------------------------------------------------------------ Alternatively, a random subset of the training points can be used as inducing points. Note, that there are plenty of methods to set these inducing points. So, in the second example let us use a user-defined set of inducing points. You can pick a set of fixed inducing points by hand: :: u = np.array([[-1], [-0.8], [-0.5], [0.3],[1.]]) You can also use equidistant inducing points :math:`u`, but without the values on the margin of the grid.(i.e. shrinking the range of values) :: num_u = np.fix(x.shape[0]/2) u = np.linspace(-1.3,1.3,num_u).T u = np.reshape(u,(num_u,1)) Then pass :math:`u` when specifying prior. :: m = pyGPs.mean.Zero() k = pyGPs.cov.RBFard(log_ell_list=[0.05,0.17], log_sigma=1.) model.setPrior(mean=m, kernel=k, inducing_points=u) The left figure below shows the result of fixed inducing points, and the right figure shows the result for equidistant :math:`u`. .. image:: _images/d3_2.png :width: 45% .. image:: _images/d3_3.png :width: 45% **[Theory]** Note that the predictive variance is overestimated outside the support of the inducing inputs. In a multivariate example where densely sampled inducing inputs are infeasible, one can also try to simply use a random subset of the training points. A bit more things you can do ----------------------------- Switch to other Inference and Likelihood functions. :: model.useInference("EP") model.useLikelihood("Laplace") pyGPs-1.2/doc/build/_sources/Graph.txt000066400000000000000000000010761236346020200177100ustar00rootroot00000000000000Kernels for Graph Data ============================ You can refer to our demo for `semi-supervised learning`_ for a simple usage of kernels for graph data. And there is also a demo for `graph kernel`_ using an example of propagation kernel. .. _`semi-supervised learning`: SemiSupervised.html .. _`graph kernel`: GraphKernel.html Kernels on Graph --------------------------------------- .. automodule:: pyGPs.GraphExtensions.nodeKernels :members: Graph Kernels --------------------------------------- .. automodule:: pyGPs.GraphExtensions.graphKernels :members: pyGPs-1.2/doc/build/_sources/GraphKernel.txt000066400000000000000000000050571236346020200210540ustar00rootroot00000000000000Graph Kernels ========================= The code shown in this tutorial can be executed by running *pyGPs/Demo/demo_GraphKernel.py* Load data -------------------- Mutag dataset [1]_ was constructed based on data from review of literatures about mutagenicities in Salmonella Typhimurium based on 200 aromatic and heteroaromatic nitro compounds. Mutag dataset is very suitable for machine learning and is widely used as one of the standard dataset. :: data = np.load('MUTAG.npz') A = csc_matrix( (data['adj_data'], data['adj_indice'], \ data['adj_indptr']), shape=data['adj_shape']) # n x n adjancy array (sparse matrix) gr_id = data['graph_ind'] # n x 1 graph id array node_label = data['responses'] # n x 1 node label array graph_label = data['labels'] # N x 1 graph label array Note that adjacency matrix for all graphs is usually too big to fit into memory when using :math:`GP`. Therefore, we generate :math:`A` using sparse matrix *csc_matrix* provided by *scipy*. Compute propagation kernel matrix ---------------------------------------- Generate prapagation kernel based on given graph data. :: num_Iteration = 10 w = 1e-4 dist = 'tv' # possible values: 'tv', 'hellinger' K = graphKernels.propagationKernel(A, node_label, gr_id, num_Iteration, w, dist, 'label_diffusion') Adjacency matrix :math:`A` can either be in format of standard numpy matrix or sparse matrix. Standard GP Classification ---------------------------------------- :math:`K` is a big square matrix with all rows and columns of the number of data points. By specifying the indice of training data and test data, we will form two matrix M1 and M2 with the exact format which *pyGPs.Core.cov.Pre* needed. :: M1,M2 = graphUtil.formKernelMatrix(Matrix, indice_train, indice_test) k = pyGPs.cov.Pre(M1,M2) M1 is a matrix with shape **number of training points plus 1** by **number of test points** - cross covariances matrix (train by test) - last row is self covariances (diagonal of test by test) M2 is a square matrix with **number of training points** for each dimension - training set covariance matrix (train by train) The following is the standard way to do GP classification :: model = pyGPs.GPC() model.setPrior(kernel=k) model.fit(x_train, y_train) model.predict(x_test) .. [1] Feragen, A., Kasenburg, N., Petersen, J., de Bruijne, M., Borgwardt, K.M.: Scalable kernels for graphs with continuous attributes. 2013.pyGPs-1.2/doc/build/_sources/Install.txt000066400000000000000000000021111236346020200202440ustar00rootroot00000000000000Installation ============================ 1. First, download_ the archive from github and extract it to any local directory. .. _download: https://github.com/marionmari/pyGPs/releases/tag/v1.1 2. You can either add the local directory to your PYTHONPATH :: export PYTHONPATH=$PYTHONPATH:/path/to/local/directory/../parent_folder_of_pyGPs 3. Or install the package using setup.py:: sudo python setup.py install Requirements ------------ * `python 2.6 or 2.7`_ * `scipy`_, `numpy`_, and `matplotlib`_: open-source packages for scientific computing in Python. .. _python 2.6 or 2.7: http://www.python.org/ .. _scipy: http://www.scipy.org/ .. _numpy: http://www.numpy.org/ .. _matplotlib: http://matplotlib.org/ Example installation on Ubuntu & Debian:: sudo apt-get install python2.7 python-numpy python-scipy python-matplotlib Example installation on Mac via Macports (requires XCode and MacPorts):: sudo port install python27 py27-numpy py27-scipy py27-matplotlib For other systems please check the installation instructions on the respective package web sites. pyGPs-1.2/doc/build/_sources/Kernels.txt000066400000000000000000000150061236346020200202500ustar00rootroot00000000000000Kernels & Means ============================ Simple Kernel & Mean --------------------- You may already seen, we can specify a kernel function like this(same for mean fucntions): :: k = pyGPs.cov.RBF( log_ell=-1., log_sigma=0. ) There are several points need to be noticed: 1. Most parameters are initilized in their logorithms. This is because we need to make sure they are positive during optimization. e.g. Here length scale and signal variance should always be positive. 2. Most kernel functions have a scalar in front, namely signal variance(set by log_sigma) 3. If you will do optimization later anyway, you can just leave parameters to be default Some Special Cases --------------------- 1. For some kernels/means, number of hyperparameters depends on the dimension of input data. You can either enter the dimension, which use default values: :: m = pyGPs.mean.Linear( D=x.shape[1] ) or you can initialze with the exact hyperparameters, you should enter as a list, one element for each dimension :: m = pyGPs.mean.Linear( alpha_list=[0.2, 0.4, 0.3] ) All these "hyp-dim-dependent" functions are: * *pyGPs.mean.Linear* * *pyGPs.cov.RBFard* * *pyGPs.cov.LINard* * *pyGPs.cov.RQard* 2. For *pyGPs.cov.RBFunit()*, its signal variance is always 1 (because of unit magnitude). Therefore this function do not have a hyperparameter of "signal variance". 3. *pyGPs.cov.Poly()* has three parameters, where hyperparameters are: * c -> inhomogeneous offset * sigma -> signal deviation however, * d -> order of polynomial will be treated as normal parameter, i.e. will not be trained 4. Explicitly set *pyGPs.cov.Noise* is not necessary, because noise are already added in likelihood. Composite Kernels & Meams ---------------------------- Adding and muliplying Kernels(Means) is really simple: :: k = pyGPs.cov.Periodic() * pyGPs.cov.RBF() k = 0.5*pyGPs.cov.LIN() + pyGPs.cov.Periodic() Scalar will also be treated as a hyperparameter. For example, k = s1 * k1 + s2 * k2, then the list of hyperparameters is hyp = [s1, k1.hyp, s2, k2.hyp]. Scalar is passed in logorithm domain such that it will always be positive during optimization. Beside + / * , there is also a power operator for mean functions: :: m = ( pyGPs.mean.One() + pyGPs.mean.Linear(alpha_list=[0.2]) )**2 Precomputed Kernel Matrix ----------------------------- In certain cases, you may have a precomputed kernel matrix, but its non-trivial to write down the exact formula of kernel functions. Then you can specify your kernel in the following way. A precomputed kernel also fits with other kernels. In other words, it can also be composited as the way other kernels functions do. :: k = pyGPs.cov.Pre(M1, M2) M1 and M2 are your precomputed kernel matrix, where, M1 is a matrix with shape **number of training points plus 1** by **number of test points** - cross covariances matrix (train by test) - last row is self covariances (diagonal of test by test) M2 is a square matrix with **number of training points** for each dimension - training set covariance matrix (train by train) A precomputed kernel can also be composited with other kernels. You need to explictly add scalar for *pyGPs.cov.Pre()*. :: k = 0.5*pyGPs.cov.Pre(M1, M2) + pyGPs.cov.RBF() Customizing Kernel & Mean ------------------------------ We also support you to create your own kernel/mean class, your customized kernel class need to follow the template as below: :: # Your kernel class needs to inherit base class Kernel, # which is in the module of Core.cov class MyKernel(Kernel): def __init__(self, hyp): ''' Intialize hyperparameters for MyKernel. ''' self.hyp = hyp def getCovMatrix(self,x=None,z=None,mode=None): ''' Return the specific covariance matrix according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self covariance matrix of test data(test by 1). 'train' return training covariance matrix(train by train). 'cross' return cross covariance matrix between x and z(train by test) :return: the corresponding covariance matrix ''' pass def getDerMatrix(self,x=None,z=None,mode=None,der=None): ''' Compute derivatives wrt. hyperparameters according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self derivative matrix of test data(test by 1). 'train' return training derivative matrix(train by train). 'cross' return cross derivative matrix between x and z(train by test) :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' pass and for customized mean class: :: # Your mean class needs to inherit base class Mean, # which is in the module of Core.mean class MyMean(Mean): def __init__(self, hyp): ''' Intialize hyperparameters for MyMean. ''' self.hyp = hyp def getMean(self, x=None): ''' Get the mean vector. ''' pass def getDerMatrix(self, x=None, der=None): ''' Compute derivatives wrt. hyperparameters. :param x: training data :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' pass You can test your customized mean/kernel function using our framework of unit test. Taking kernel test as an example, you can uncomment method *test_cov_new* in *pyGPs.Testing.unit_test_cov.py* to check the outputs of your kernel function. :: # Test your customized covariance function def test_cov_new(self): k = myKernel() # specify your covariance function self.checkCovariance(k) and testing mean function in *pyGPs.Testing.unit_test_mean.py* :: # Test your customized mean function def test_mean_new(self): m = myMean # specify your mean function self.checkMean(m) List of Kernels and Default Parameters --------------------------------------- .. automodule:: pyGPs.Core.cov :members: List of Means and Default Parameters ----------------------------------------- .. automodule:: pyGPs.Core.mean :members: pyGPs-1.2/doc/build/_sources/Likelihoods.txt000066400000000000000000000137021236346020200211140ustar00rootroot00000000000000Likelihoods & Inference ============================ Changing Likelihood & Inference ------------------------------------ By default, * GPR uses Gaussian likelihood and exact inference. * GPC uses Error functionlikelihood and EP inference. * FITC model uses same default with corresponding FITC inference. * GPMC calls GPC and thus uses the default setting of GPC You can change to other likelihood or inference methods using: :: model.useLikelihood(newLik) model.useInference(newInf) *newLik* and *newInf* are **Strings**. Currently the options are: 1. Regression model * newLik: **"Laplace"**. Note this will force inference method to be EP. * newInf: **"EP"**, **"Laplace"**. 2. Classification model (including GPMC) * newInf: **"Laplace"** Customizing Likelihood & Inference ------------------------------------ We also support you to create your own likelihood/inference class, your customized inference class need to follow the template as below: :: # Your inference class needs to inherit base class Inference, # which is in the module of Core.inf class MyKernel(Kernel): def __init__(self): pass def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): ''' Inference computation based on inputs. post, nlZ, dnlZ = inf.proceed(mean, cov, lik, x, y) INPUT: cov: name of the covariance function (see covFunctions.m) lik: name of the likelihood function (see likFunctions.m) x: n by D matrix of training inputs y: 1d array (of size n) of targets OUTPUT: post(instance of postStruct): struct representation of the (approximate) posterior containing: nlZ: returned value of the negative log marginal likelihood dnlZ(instance of dnlZStruct): struct representation for derivatives of the negative log marginal likelihood w.r.t. each hyperparameter. Usually, the approximate posterior to be returned admits the form: N(m=K*alpha, V=inv(inv(K)+W)), where alpha is a vector and W is diagonal; if not, then L contains instead -inv(K+inv(W)), and sW is unused. For more information on the individual approximation methods and their implementations, see the respective inference function below. See also gp.py :param meanfunc: mean function :param covfunc: covariance function :param likfunc: likelihood function :param x: training data :param y: training labels :param nargout: specify the number of output(1,2 or 3) :return: posterior, negative-log-marginal-likelihood, derivative for negative-log-marginal-likelihood-likelihood ''' pass where **postStruct** and **dnlZStruct** is also defined in *Core.inf*. :: class postStruct(object): ''' Data structure for posterior post.alpha -> 1d array containing inv(K)*m, where K is the prior covariance matrix and m the approx posterior mean post.sW: -> 1d array containing diagonal of sqrt(W) the approximate posterior covariance matrix is inv(inv(K)+W) post.L -> 2d array, L = chol(sW*K*sW+identity(n)) ''' class dnlZStruct(object): ''' Data structure for the derivatives of mean, cov and lik functions. dnlZ.mean -> list of derivatives for each hyperparameters in mean function dnlZ.cov -> list of derivatives for each hyperparameters in covariance function dnlZ.lik -> list of derivatives for each hyperparameters in likelihood function ''' and for customized mean class: :: # Your mean class needs to inherit base class Mean, # which is in the module of Core.mean class MyMean(Mean): def __init__(self, hyp): ''' Intialize hyperparameters for MyMean. ''' self.hyp = hyp def getMean(self, x=None): ''' Get the mean vector. ''' pass def getDerMatrix(self, x=None, der=None): ''' Compute derivatives wrt. hyperparameters. :param x: training data :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' pass Customizing likelihood function is more complicated. We will omit it here to keep this this page not too long. However, you can find detailed explaination either in the **source code** *Core.lik* or in coresponding section of **manual**. Just like testing kernel/mean fucntions, you can also find unit test module for likelihood and inference functions. To test your customized inference function, uncomment the following method in *pyGPs.Testing.unit_test_inf.py*. :: # Test your customized inference function def test_inf_new(self): # specify your inf function # set mean/cov/lik functions post, nlZ, dnlZ = inffunc.proceed(meanfunc, covfunc, likfunc, self.x, self.y, nargout=3) self.checkFITCOutput(post, nlZ, dnlZ) and test customized likelihood function in *pyGPs.Testing.unit_test_lik.py* :: # Test your customized likelihood function def test_cov_new(self): likelihood = myLikelihood() # specify your likelihood function self.checkLikelihood(likelihood) List of Likelihoods --------------------------------------- .. autoclass:: pyGPs.Core.lik.Erf :members: .. autoclass:: pyGPs.Core.lik.Gauss :members: .. autoclass:: pyGPs.Core.lik.Laplace :members: List of Inference ----------------------------------------- .. autoclass:: pyGPs.Core.inf.Exact :members: .. autoclass:: pyGPs.Core.inf.EP :members: .. autoclass:: pyGPs.Core.inf.Laplace :members: .. autoclass:: pyGPs.Core.inf.FITC_Exact :members: .. autoclass:: pyGPs.Core.inf.FITC_EP :members: .. autoclass:: pyGPs.Core.inf.FITC_Laplace :members: pyGPs-1.2/doc/build/_sources/Opts.txt000066400000000000000000000003541236346020200175720ustar00rootroot00000000000000Optimizers ============================ Optimization Methods ---------------------------- As you may have already seen in the demos, the optimizer is initialized in the following way: .. automethod:: pyGPs.Core.gp.GP.setOptimizer pyGPs-1.2/doc/build/_sources/SemiSupervised.txt000066400000000000000000000116771236346020200216260ustar00rootroot00000000000000Semi-supervised Learning with Graphs ======================================= The code shown in this tutorial can be executed by running *pyGPs/Demo/demo_NodeKernel.py* Import ------------------- You may want to import some extensions we provide as follows: :: from pyGPs.GraphExtension import graphUtil, nodeKernels from pyGPs.Validation import valid Load data -------------------- We used the same dataset from GPMC example. i.e. The USPS digits dataset [1]_. Each digit of :math:`16*16` pixels is flattened into a :math:`256` dimension vector. For the simplicity of demo, we only selected digits :math:`1` s and :math:`2` s such that we have a binary classification problem where digit :math:`1` for class +1 and digit :math:`2` for class -1. We also reduced the dataset into :math:`100` samples per digit, where the original dataset consist of thousands of samples for each digit. Here are samples for two digits for :math:`1` .. image:: _images/digit1_1.png :width: 30% .. image:: _images/digit1_2.png :width: 30% and samples for two digits for :math:`2` .. image:: _images/digit2_1.png :width: 30% .. image:: _images/digit2_2.png :width: 30% Form a nearest neighbour graph -------------------------------- We form a nearest-neighbor graph based on Euclidean distance of the vector representation of digits. Neighboring images have small Euclidean distance. Each digit is a node in the graph. There is an edge if digit :math:`i` is the k-nearest neighbour of digit :math:`j`. We form a symmetrized graph such that we connect nodes :math:`j`, :math:`i` if i is in j’s kNN and vice versa, and therefore a node can have more than k edges. You should import the corresponding module from *pyGPs.GraphStuff* :: x,y = load_binary(1,2,reduce=True) A = graphUtil.formKnnGraph(x,2) A is the adjacency matrix of this :math:`2-NN` graph. Below shows an example of such symmetrized Euclidean :math:`2-NN` graph on some 1s and 2s taking from Xiaojin Zhu's doctoral thesis [2]_. .. figure:: _images/2nnGraph.png :align: center Kernel on graph ------------------ Several classical kernels on graph described in `Structured Kernels`_ can be built from adjacency matrix :math:`A`. We use diffusion kernel for this example to get the precomputed kernel matrix. :: Matrix = nodeKernels.diffKernel(A) .. _Structured Kernels: Graph.html This a big square matrix with all rows and columns of the number of data points. By specifying the indice of training data and test data, we will form two matrix M1 and M2 with the exact format which *pyGPs.Core.cov.Pre* needed. :: M1,M2 = graphUtil.formKernelMatrix(Matrix, indice_train, indice_test) M1 is a matrix with shape **number of training points plus 1** by **number of test points** - cross covariances matrix (train by test) - last row is self covariances (diagonal of test by test) M2 is a square matrix with **number of training points** for each dimension - training set covariance matrix (train by train) GP classification ----------------------- Every ingredients for a basic semi-supervised learning is prepared now. Lets see how to proceed for :math:`GP` classification. First, the normal way with rbf kernel we have seen several times :: model = pyGPs.GPC() k = pyGPs.cov.RBF() model.setPrior(kernel=k) Then lets use our kernel precomputed matrix. If you only use precomputed kernel matrix, there is no training data. However you still need to specify :math:`x` just to fit in the usage of pyGPs for generality reason. You can create any :math:`x` as long as the dimension is correct. :: x = np.zeros((n,1)) k = pyGPs.cov.Pre(M1,M2) + pyGPs.cov.RBF() model.setPrior(kernel=k) Moreover, you can composite a kernel for both precomputed matrix and regular kernel function if necessary. :: k = pyGPs.cov.Pre(M1,M2) + pyGPs.cov.RBFunit() model.setPrior(kernel=k) The rest way of using pyGPs is exactly the same as the demo of GP classification. Result ----------------------- For our manually created graph data, an rbf kernel works better than a diffusion kernel on the graph (higher accuracy). The performance in general should depend on the application as well as features of data. The left image shows the digit that using diffusion kernel will predict the wrong result (should be :math:`2`), but rbf kernel does the job fine. The right image shows the digit that rbf kernel predicts the wrong class, diffusion kernel on the other hand, predicts correctly due to graph information! (should be :math:`1`). Interestingly, using a composite kernel with diffusion kernel on graph and an rbf kernel together. All test cases including the following are predicted correctly. .. image:: _images/digitDiffwrong.png :width: 50% .. image:: _images/digitRBFwrong.png :width: 50% .. [1] A Database for Handwritten Text Recognition Research, J. J. Hull, IEEE PAMI 16(5) 550-554, 1994. .. [2] Semi-Supervised Learning with Graphs, Xiaojin Zhu, CMU-LTI-05-192, 2005 pyGPs-1.2/doc/build/_sources/Theory.txt000066400000000000000000000164221236346020200201220ustar00rootroot00000000000000GPs & Functionality ======================== Gaussian Processes (GPs) can conveniently be used for Bayesian supervised learning, such as regression and classification. In its simplest form, GP inference can be implemented in a few lines of code. However, in practice, things typically get a little more complicated: you might want to use expressive covariance and mean functions, learn good values for hyperparameters, use non-Gaussian likelihood functions (rendering exact inference intractable), use approximate inference algorithms, or combinations of many or all of the above. A comprehensive introduction to Gaussian Processes for Machine Learning is provided in the `GPML`_ book by Rasmussen and Williams, 2006. List of Functionality ------------------------ This table lists the functionality implemented in pyGPs. +-------------+-------------------+------------+-------------------------+---------------+------------------+ |Functionality| Kernel | Mean | Likelihood | Inference | Optimizer | +=============+===================+============+=========================+===============+==================+ | Simple | Constant | Constant | Gaussian | Exact | Minimize | | +-------------------+------------+-------------------------+---------------+------------------+ | | Linear | Linear |Cumulative Gaussian (Erf)| EP | CG | | +-------------------+------------+-------------------------+---------------+------------------+ | | Linear ard | One | Laplace | Laplace | SCG | | +-------------------+------------+-------------------------+---------------+------------------+ | | Matérn (1,3,5) | Zero | | | BFGS | | +-------------------+------------+ | +------------------+ | | Periodic | | | | | | +-------------------+ | | | | | | Polynomial | | | | | | +-------------------+ | | | | | | Piecewise Poly | | | | | | +-------------------+ | | | | | | RBF iso | | | | | | +-------------------+ | | | | | | RBF ard | | | | | | +-------------------+ | | | | | | RBF unit | | | | | | +-------------------+ | | | | | | RQ iso | | | | | | +-------------------+ | | | | | | RQ ard | | | | | | +-------------------+ | | | | | | Noise | | | | | +-------------+-------------------+------------+-------------------------+---------------+------------------+ | Composite | Sum (+) | Sum (+) | | | | | +-------------------+------------+ | | | | | Product (*) | Product (*)| | | | | +-------------------+------------+ | | | | | Scale (*) | Scale (*) | | | | | +-------------------+------------+ | | | | | | Power (**) | | | | +-------------+-------------------+------------+-------------------------+---------------+------------------+ | Sparse GP | | | | FITC Exact | | | | | | +---------------+ | | | | | | FITC EP | | | | | | +---------------+ | | | | | | FITC Laplace | | +-------------+-------------------+------------+-------------------------+---------------+------------------+ | Graphs | Diffusion | | | | | | +-------------------+ | | | | | | VN Diffusion | | | | | | +-------------------+ | | | | | |Pseudo Inv Laplace | | | | | | +-------------------+ | | | | | |Regularized Laplace| | | | | | +-------------------+ | | | | | |p-step Random Walk | | | | | | +-------------------+ | | | | | |Inverse Cosine | | | | | +-------------+-------------------+------------+-------------------------+---------------+------------------+ | Other | Customized | Customized | | | | | +-------------------+------------+ | | | | |Precomputed Kernel | | | | | +-------------+-------------------+------------+-------------------------+---------------+------------------+ pyGPs also provide cross-validation and some built-in evalation methods. .. _GPML: http://www.gaussianprocess.org/gpml pyGPs-1.2/doc/build/_sources/_static/000077500000000000000000000000001236346020200175305ustar00rootroot00000000000000pyGPs-1.2/doc/build/_sources/_static/changelog.txt000066400000000000000000000030461236346020200222230ustar00rootroot00000000000000 Changelog pyGPs v1.2 ===================== June 30th 2014 ---------------- structural updates: - input target now can either be in 2-d array with size (n,1) or in 1-d array with size (n,) - setup.py updated - "import pyGPs" instead of "from pyGPs.Core import gp" - rename ".train()" to ".optimize()" - rename "Graph-stuff" to "graphExtension" - rename kernelOnGraph to "nodeKernels" and graphKernel to "graphKernels" - redundancy removed for model.setData(x,y) - rewrite "mean.proceed()" to "getMean()" and "getDerMatrix()" - rewrite "cov.proceed()" to "getCovMatrix()" and "getDerMatrix()" - rename cov.LIN to cov.Linear (to be consistent with mean.Linear) - rename module "valid" to "validation" - add graph dataset Mutag in python file. (.npz and .mat) - add graphUtil.nomalizeKernel() - fix number of iteration problem in graphKernels "PropagationKernel" - add unit testing for covariance, mean functions bug fixes: - derivatives for cov.LINard - derivative of the scalar for cov.covScale - demo_GPR_FITC.py missing pyGPs.mean July 8th 2014 ---------------- structural updates: - add hyperparameter(signal variance s2) for linear covariance - add unit testing for inference,likelihood functions as well as models - NOT show(print) "maximum number of sweep warning in inference EP" any more - documentation updated bug fixes: - typos in lik.Laplace - derivative in lik.Laplace July 14th 2014 ---------------- documentation updates: - online docs updated - API file updated structural updates: - made private for methods that users don't need to call pyGPs-1.2/doc/build/_sources/_static/copyright.txt000066400000000000000000000035241236346020200223050ustar00rootroot00000000000000Copyright ========= Authors ------- * Marion Neumann [marion dot neumann at uni-bonn dot de] * Shan Huang [shan dot huang at iais dot fraunhofer dot de] * Daniel Marthaler [marthaler at ge dot com] * Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] This file is part of pyGPs. License ------- The code is released under the BSD 2-Clause (FreeBSD) License. Copyright (c) 2014 Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: - Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. - Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. pyGPs-1.2/doc/build/_sources/demoHousing.txt000066400000000000000000000043421236346020200211270ustar00rootroot00000000000000Regression on UCI Housing data ------------------------------ Boston Housing is a fairly standard dataset used for testing regression problems. It contains 506 data points with 12 numeric attributes, and one binary categorical attribute. The goal is to determine median home values, based on various census attributes. This dataset is available at the `UCI Repository`_. The demo follows that in [1]_. The data set was preprocessed as follows: each continuous feature was transformed to zero mean and unit variance (The categorical variable was dropped). The data was partitioned into :math:`481` points for training and :math:`25` points for testing. The mean function used was :func:`src.Core.means.meanZero` and the covariance (using the :func:`src.Core.kernels.covSum` function) was a composite of :func:`src.Core.kernels.covSEiso` and :func:`src.Core.kernels.covNoise`. The initial values of the hyperparameters were selected randomly from a zero-mean, unit-variance normal distribtion. The actual values were: :math:`[ -0.75, \; 0.59, \; -0.45 ]`. The initial likelihood hyperparameter was :math:`-2.30`. The regression started with initial negative log marginal likelihood of :math:` 752.46`. Note the initial zero mean and the variance that is uniform over the test set. :: model = pyGPs.GPR() model.optimize(x,y) ym, ys2, fm, fs2, lp = model.predict(xs) xa = np.concatenate((data[:,:4],data[:,5:-1]),axis=1) xa = (xa - np.mean(xa,axis=0))/(np.std(xa,axis=0)+1.e-16) ya, ys2a, fma, fs2a, lpa = model.predict(xa) .. figure:: _images/demoH1.png :align: center :width: 600pt :height: 300pt After hyperparameter optimization, the covariance hyperparameters were :math:`[ 1.17, \; 0.45, \; -1.41 ]` and the likelihood hyperparameter was :math:`-2.27`. The final negative log marginal likelihood (optimized) was :math:`214.46`. .. figure:: _images/demoH2.png :align: center :width: 600pt :height: 300pt .. _UCI Repository: http://archive.ics.uci.edu/ml/datasets/Housing .. [1] T. Suttorp and C. Igel, Approximation of Gaussian process regression models after training. In M. Verleysen (Hrsg.), Proceedings of the 16th European Symposium on Artificial Neural Networks (ESANN 2008) , pp. 427–432 (2008). pyGPs-1.2/doc/build/_sources/demoMaunaLoa.txt000066400000000000000000000147501236346020200212140ustar00rootroot00000000000000Regression on Mauna Loa data ---------------------------- This example does regression on the Hawaiian Mauna Loa data (example taken from chapter :math:`5` of the `GPML`_ book by Rasmussen and Williams, 2006) We will use a modelling problem concerning the concentration of :math:`CO_2` in the atmosphere to illustrate how the marginal likelihood can be used to set multiple hyperparameters in hierarchical Gaussian process models. A complex covariance function is derived by combining several different kinds of simple covariance functions, and the resulting model provides an excellent fit to the data as well as insight into its properties by interpretation of the adapted hyperparameters. Although the data is one-dimensional, and therefore easy to visualize, a total of :math:`11` hyperparameters are used, which in practice rules out the use of cross-validation for setting parameters, except for the gradient-based LOO-CV procedure. The data [1]_ consists of monthly average atmospheric :math:`CO_2` concentrations (in parts per million by volume (ppmv)) derived from *in-situ* air samples collected at the Mauna Loa Observatory, Hawaii, between :math:`1958` and :math:`2003` (with some missing values) `[2]`_. .. figure:: _images/demoML1.png :height: 600 px :width: 800 px :align: center :scale: 70 % The data is shown in the above plot. Our goal is to model the :math:`CO_2` concentration as a function of time :math:`t`. Several features are immediately apparent: a long term rising trend, a pronounced seasonal variation and some smaller irregularities. In the following, contributions to a combined covariance function which takes care of these individual properties are suggessted. This is meant primarily to illustrate the power and flexibility of the Gaussian process framework—it is possible that other choices would be more appropriate for this data set. To model the long term smooth rising trend, a squared exponential (SE) covariance term with two hyperparameters controlling the amplitude :math:`\theta_1` and characteristic length-scale :math:`\theta_2` is used: .. math:: k_1(x,x') = \theta_1^2 \exp \left(-\frac{(x-x')^2}{2\theta_2^2}\right). Note that we just use a smooth trend; actually enforcing the trend *a priori* to be increasing is probably not so simple and (hopefully) not desirable. We can use the periodic covariance function with a period of one year to model the seasonal variation. However, it is not clear that the seasonal trend is exactly periodic, so we modify it by taking the product with a squared exponential component to allow a decay away from exact periodicity: .. math:: k_2(x,x') = \theta_3^2 \exp\left(-\frac{(x-x')^2}{2\theta_4^2} \frac{2\sin^2(\pi(x-x'))}{\theta_5^2}\right). where :math:`\theta_3` gives the magnitude, :math:`\theta_4` the decay-time for the periodic component, and :math:`\theta_5` the smoothness of the periodic component; the period has been fixed to one (year). The seasonal component in the data is caused primarily by different rates of :math:`CO_2` uptake for plants depending on the season, and it is probably reasonable to assume that this pattern may itself change slowly over time, partially due to the elevation of the :math:`CO_2` level itself; if this effect turns out not to be relevant, then it can be effectively removed at the fitting stage by allowing :math:`\theta_4` to become very large. To model the (small) medium term irregularities, a rational quadratic term is used: .. math:: k_3(x,x') = \theta_6^2\left(1+\frac{(x-x')^2}{2\theta_8\theta_7^2}\right)^{\theta_8}. where :math:`\theta_6` is the magnitude, :math:`\theta_7` is the typical length-scale and :math:`\theta_8` is the shape parameter determining diffuseness of the length-scales. One could also have used a squared exponential form for this component, but it turns out that the rational quadratic works better (gives higher marginal likelihood), probably because it can accommodate several length-scales simultaneously. Finally we specify a noise model as the sum of a squared exponential contrubition and an independent component: .. math:: k_4(x_p,x_q) = \theta_9^2\exp\left(-\frac{(x_p - x_q)^2}{2\theta_{10}^2}\right) + \theta_{11}^2\delta_{pq}. where :math:`\theta_9` is the magnitude of the correlated noise component, :math:`\theta_{10}` is its length scale and :math:`\theta_{11}` is the magnitude of the independent noise component. Noise in the series could be caused by measurement inaccuracies, and by local short-term weather phenomena, so it is probably reasonable to assume at least a modest amount of correlation in time. Notice that the correlated noise component, the first term has an identical expression to the long term component in the trend covariance. When optimizing the hyperparameters, we will see that one of these components becomes large with a long length-scale (the long term trend), while the other remains small with a short length-scale (noise). The fact that we have chosen to call one of these components ‘signal’ and the other one ‘noise’ is only a question of interpretation. Presumably, we are less interested in very short-term effect, and thus call it noise; if on the other hand we were interested in this effect, we would call it signal. The final covariance function is: .. math:: k(x,x') = k_1(x,x') + k_2(x,x') + k_3(x,x') + k_4(x,x') with hyperparameters :math:`\theta = (\theta_1,\ldots,\theta_{11})` :: # DEFINE parameterized covariance function k1 = pyGPs.cov.RBF(np.log(67.), np.log(66.)) k2 = pyGPs.cov.Periodic(np.log(1.3), np.log(1.0), np.log(2.4)) * cov.RBF(np.log(90.), np.log(2.4)) k3 = pyGPs.cov.RQ(np.log(1.2), np.log(0.66), np.log(0.78)) k4 = pyGPs.cov.RBF(np.log(1.6/12.), np.log(0.18)) + cov.Noise(np.log(0.19)) k = k1 + k2 + k3 + k4 After running the minimization, :: t0 = clock() model.optimize(x,y) t1 = clock() model.predict(xs) The extrapolated data looks like: .. figure:: _images/demoML2.png :height: 600 px :width: 800 px :align: center :scale: 70 % and the optimized values of the hyperparameters allow for a principled analysis of different components driving the model. .. [1] Keeling, C. D. and Whorf, T. P. (2004). Atmospheric :math:`CO_2` Records from Sites in the SIO Air Sampling Network. In Trends: A Compendium of Data on Global Change. Carbon Dioxide Information Analysis Center, Oak Ridge National Laboratory, Oak Ridge, Tenn., U.S.A. .. _[2]: http://cdiac.esd.ornl.gov/ftp/trends/co2/maunaloa.co2 .. _GPML: http://www.gaussianprocess.org/gpml/chapters/RW5.pdf pyGPs-1.2/doc/build/_sources/index.txt000066400000000000000000000047151236346020200177610ustar00rootroot00000000000000.. Object-Oriented Gaussian Processes documentation master file, created by sphinx-quickstart on Thu Nov 14 19:54:52 2013. pyGPs - A Package for Gaussian Processes =========================================== About the package --------------------- pyGPs is a library hosting Python implementations of Gaussian processes (GPs) for machine learning. pyGPs bridges the gap between systems designed primarily for users, who mainly want to apply GPs and need basic machine learning routines for model training, evaluation, and visualiztion, and expressive systems for developers, who focus on extending the core functionalities as covariance and likelihood functions, as well as inference techniques. The software package is released under the BSD 2-Clause (FreeBSD) License. :doc:`Copyright <_static/copyright>` (c) by Marion Neumann, Shan Huang, Daniel Marthaler, & Kristian Kersting, Feb.2014 Further, it includes implementations of * minimize.py implemented in python by Roland Memisevic 2008, following minimize.m (Copyright (c) Carl Edward Rasmussen (1999-2006)) * scg.py (Copyright (c) Ian T Nabney (1996-2001)) * brentmin.py (Copyright (c) Hannes Nickisch 2010-01-10) * FITC functionality (following matlab implementations under Copyright (c) by Ed Snelson, Carl Edward Rasmussen and Hannes Nickisch, 2011-11-02) The most recent stable release is pyGPs v1.2. See :doc:`changelog <_static/changelog>` for a list of changes to the previous release. A Developer's Guide containing api is found in :doc:`api.pdf <_static/api.pdf>`, but for the casual user, the guide is below. If you observe problems or bugs, please let us know. You can also download a `procedual implementation`_ of GP functionality from Github. However, the procedual version will not be supported in future. .. _procedual implementation: https://github.com/marionmari/pyGP_PR/ Authors: - Marion Neumann [marion dot neumann at uni-bonn dot de] - Shan Huang [shan dot huang at iais dot fraunhofer dot de] - Daniel Marthaler [marthaler at ge dot com] - Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] The following persons helped to improve this software: Roman Garnett, Maciej Kurek, Hannes Nickisch, Zhao Xu, and Alejandro Molina. This work is partly supported by the Fraunhofer ATTRACT fellowship STREAM. Getting started ------------------------------ .. toctree:: :maxdepth: 1 Install Theory Examples Kernels Likelihoods Opts Graph pyGPs-1.2/doc/build/_static/000077500000000000000000000000001236346020200157065ustar00rootroot00000000000000pyGPs-1.2/doc/build/_static/ajax-loader.gif000066400000000000000000000012411236346020200205620ustar00rootroot00000000000000GIF89aU|NU|l!Created with ajaxload.info! ! NETSCAPE2.0,30Ikc:Nf E1º.`q-[9ݦ9 JkH! ,4N!  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Changelog pyGPs v1.2

June 30th 2014

structural updates:

  • input target now can either be in 2-d array with size (n,1) or in 1-d array with size (n,)
  • setup.py updated
  • “import pyGPs” instead of “from pyGPs.Core import gp”
  • rename ”.train()” to ”.optimize()”
  • rename “Graph-stuff” to “graphExtension”
  • rename kernelOnGraph to “nodeKernels” and graphKernel to “graphKernels”
  • redundancy removed for model.setData(x,y)
  • rewrite “mean.proceed()” to “getMean()” and “getDerMatrix()”
  • rewrite “cov.proceed()” to “getCovMatrix()” and “getDerMatrix()”
  • rename cov.LIN to cov.Linear (to be consistent with mean.Linear)
  • rename module “valid” to “validation”
  • add graph dataset Mutag in python file. (.npz and .mat)
  • add graphUtil.nomalizeKernel()
  • fix number of iteration problem in graphKernels “PropagationKernel”
  • add unit testing for covariance, mean functions

bug fixes:

  • derivatives for cov.LINard
  • derivative of the scalar for cov.covScale
  • demo_GPR_FITC.py missing pyGPs.mean

July 8th 2014

structural updates:

  • add hyperparameter(signal variance s2) for linear covariance
  • add unit testing for inference,likelihood functions as well as models
  • NOT show(print) “maximum number of sweep warning in inference EP” any more
  • documentation updated

bug fixes:

  • typos in lik.Laplace
  • derivative in lik.Laplace

July 14th 2014

documentation updates:

  • online docs updated
  • API file updated

structural updates:

  • made private for methods that users don’t need to call

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© Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting. Created using Sphinx 1.2.
pyGPs-1.2/doc/build/_static/changelog.rst000066400000000000000000000030461236346020200203720ustar00rootroot00000000000000 Changelog pyGPs v1.2 ===================== June 30th 2014 ---------------- structural updates: - input target now can either be in 2-d array with size (n,1) or in 1-d array with size (n,) - setup.py updated - "import pyGPs" instead of "from pyGPs.Core import gp" - rename ".train()" to ".optimize()" - rename "Graph-stuff" to "graphExtension" - rename kernelOnGraph to "nodeKernels" and graphKernel to "graphKernels" - redundancy removed for model.setData(x,y) - rewrite "mean.proceed()" to "getMean()" and "getDerMatrix()" - rewrite "cov.proceed()" to "getCovMatrix()" and "getDerMatrix()" - rename cov.LIN to cov.Linear (to be consistent with mean.Linear) - rename module "valid" to "validation" - add graph dataset Mutag in python file. (.npz and .mat) - add graphUtil.nomalizeKernel() - fix number of iteration problem in graphKernels "PropagationKernel" - add unit testing for covariance, mean functions bug fixes: - derivatives for cov.LINard - derivative of the scalar for cov.covScale - demo_GPR_FITC.py missing pyGPs.mean July 8th 2014 ---------------- structural updates: - add hyperparameter(signal variance s2) for linear covariance - add unit testing for inference,likelihood functions as well as models - NOT show(print) "maximum number of sweep warning in inference EP" any more - documentation updated bug fixes: - typos in lik.Laplace - derivative in lik.Laplace July 14th 2014 ---------------- documentation updates: - online docs updated - API file updated structural updates: - made private for methods that users don't need to call pyGPs-1.2/doc/build/_static/comment-bright.png000066400000000000000000000066541236346020200213460ustar00rootroot00000000000000PNG  IHDRa OiCCPPhotoshop ICC profilexڝSgTS=BKKoR RB&*! 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pyGPs-1.2/doc/build/_static/copyright.rst000066400000000000000000000035241236346020200204540ustar00rootroot00000000000000Copyright ========= Authors ------- * Marion Neumann [marion dot neumann at uni-bonn dot de] * Shan Huang [shan dot huang at iais dot fraunhofer dot de] * Daniel Marthaler [marthaler at ge dot com] * Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] This file is part of pyGPs. License ------- The code is released under the BSD 2-Clause (FreeBSD) License. Copyright (c) 2014 Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: - Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. - Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. pyGPs-1.2/doc/build/_static/doctools.js000066400000000000000000000147341236346020200201030ustar00rootroot00000000000000/* * doctools.js * ~~~~~~~~~~~ * * Sphinx JavaScript utilities for all documentation. * * :copyright: Copyright 2007-2013 by the Sphinx team, see AUTHORS. * :license: BSD, see LICENSE for details. * */ /** * select a different prefix for underscore */ $u = _.noConflict(); /** * make the code below compatible with browsers without * an installed firebug like debugger if (!window.console || !console.firebug) { var names = ["log", "debug", "info", "warn", "error", "assert", "dir", "dirxml", "group", "groupEnd", "time", "timeEnd", "count", "trace", "profile", "profileEnd"]; window.console = {}; for (var i = 0; i < names.length; ++i) window.console[names[i]] = function() {}; } */ /** * small helper function to urldecode strings */ jQuery.urldecode = function(x) { return decodeURIComponent(x).replace(/\+/g, ' '); }; /** * small helper function to urlencode strings */ jQuery.urlencode = encodeURIComponent; /** * This function returns the parsed url parameters of the * current request. 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      • '; $(document).ready(function() { init(); }); })(jQuery); $(document).ready(function() { // add comment anchors for all paragraphs that are commentable $('.sphinx-has-comment').comment(); // highlight search words in search results $("div.context").each(function() { var params = $.getQueryParameters(); var terms = (params.q) ? params.q[0].split(/\s+/) : []; var result = $(this); $.each(terms, function() { result.highlightText(this.toLowerCase(), 'highlighted'); }); }); // directly open comment window if requested var anchor = document.location.hash; if (anchor.substring(0, 9) == '#comment-') { $('#ao' + anchor.substring(9)).click(); document.location.hash = '#s' + anchor.substring(9); } }); pyGPs-1.2/doc/build/doctrees/000077500000000000000000000000001236346020200160705ustar00rootroot00000000000000pyGPs-1.2/doc/build/doctrees/CV.doctree000066400000000000000000000241701236346020200177530ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(Xk-fold cross-validationqNXcross validationqNXevaluation measuresqNX1X load dataq NuUsubstitution_defsq }q Uparse_messagesq ]q Ucurrent_sourceqNU decorationqNUautofootnote_startqKUnameidsq}q(hUk-fold-cross-validationqhUcross-validationqhUevaluation-measuresqX1Uid2qh U load-dataquUchildrenq]qcdocutils.nodes section q)q}q(U rawsourceqUUparentqhUsourceqcdocutils.nodes reprunicode q X=/Users/marion/workspace/workspace_GPs/pyGPs/doc/source/CV.rstq!q"}q#bUtagnameq$Usectionq%U attributesq&}q'(Udupnamesq(]Uclassesq)]Ubackrefsq*]Uidsq+]q,haUnamesq-]q.hauUlineq/KUdocumentq0hh]q1(cdocutils.nodes title q2)q3}q4(hXK-fold Cross-Validationq5hhhh"h$Utitleq6h&}q7(h(]h)]h*]h+]h-]uh/Kh0hh]q8cdocutils.nodes Text q9XK-fold Cross-Validationq:q;}q<(hh5hh3ubaubcdocutils.nodes paragraph q=)q>}q?(hXIn this demo, we'll show you the typical process of using GP for machine learning from loading data, preprocessing, training, predicting to validation and evaluation.q@hhhh"h$U paragraphqAh&}qB(h(]h)]h*]h+]h-]uh/Kh0hh]qCh9XIn this demo, we'll show you the typical process of using GP for machine learning from loading data, preprocessing, training, predicting to validation and evaluation.qDqE}qF(hh@hh>ubaubh)qG}qH(hUhhhh"h$h%h&}qI(h(]h)]h*]h+]qJhah-]qKh auh/Kh0hh]qL(h2)qM}qN(hX Load dataqOhhGhh"h$h6h&}qP(h(]h)]h*]h+]h-]uh/Kh0hh]qQh9X Load dataqRqS}qT(hhOhhMubaubh=)qU}qV(hXDWe use the ionosphere dataset [1]_ from Johns Hopkins University Ionosphere database. It is available in UCI machine learning repository. Then we need to do some data cleaning. Here we deal with label in ionosphere data, change "b" to"-1", and "g" to "+1". These preprocessing implementation are availabe in the source code.hhGhh"h$hAh&}qW(h(]h)]h*]h+]h-]uh/Kh0hh]qX(h9XWe use the ionosphere dataset qYqZ}q[(hXWe use the ionosphere dataset hhUubcdocutils.nodes footnote_reference q\)q]}q^(hX[1]_Uresolvedq_KhhUh$Ufootnote_referenceq`h&}qa(h+]qbUid1qcah*]h(]h)]h-]Urefidqdhuh]qeh9X1qf}qg(hUhh]ubaubh9X" from Johns Hopkins University Ionosphere database. It is available in UCI machine learning repository. Then we need to do some data cleaning. Here we deal with label in ionosphere data, change "b" to"-1", and "g" to "+1". These preprocessing implementation are availabe in the source code.qhqi}qj(hX" from Johns Hopkins University Ionosphere database. It is available in UCI machine learning repository. Then we need to do some data cleaning. Here we deal with label in ionosphere data, change "b" to"-1", and "g" to "+1". These preprocessing implementation are availabe in the source code.hhUubeubeubh)qk}ql(hUhhhh"h$h%h&}qm(h(]h)]h*]h+]qnhah-]qohauh/Kh0hh]qp(h2)qq}qr(hXCross Validationqshhkhh"h$h6h&}qt(h(]h)]h*]h+]h-]uh/Kh0hh]quh9XCross Validationqvqw}qx(hhshhqubaubh=)qy}qz(hX2Now, lets focus on the use of cross-validation. ::q{hhkhh"h$hAh&}q|(h(]h)]h*]h+]h-]uh/Kh0hh]q}h9X/Now, lets focus on the use of cross-validation.q~q}q(hX/Now, lets focus on the use of cross-validation.hhyubaubcdocutils.nodes literal_block q)q}q(hXK = 10 # number of folds for x_train, x_test, y_train, y_test in valid.k_fold_validation(x, y, K): # This is a binary classification problem model = pyGPs.GPC() # Since no prior knowldege, leave everything default model.optimize(x_train, y_train) # Predition ymu, ys2, fmu, fs2, lp = model.predict(x_test, ys=y_test) # ymu for classification is a continuous value over -1 to +1 # If you want predicting result to either one of the classes, take a sign of ymu. ymu_class = np.sign(ymu) # Evluation acc = valid.ACC(ymu_class, y_test) # accuracy rmse = valid.RMSE(ymu_class, y_test) # root-mean-square errorhhkhh"h$U literal_blockqh&}q(U xml:spaceqUpreserveqh+]h*]h(]h)]h-]uh/Kh0hh]qh9XK = 10 # number of folds for x_train, x_test, y_train, y_test in valid.k_fold_validation(x, y, K): # This is a binary classification problem model = pyGPs.GPC() # Since no prior knowldege, leave everything default model.optimize(x_train, y_train) # Predition ymu, ys2, fmu, fs2, lp = model.predict(x_test, ys=y_test) # ymu for classification is a continuous value over -1 to +1 # If you want predicting result to either one of the classes, take a sign of ymu. ymu_class = np.sign(ymu) # Evluation acc = valid.ACC(ymu_class, y_test) # accuracy rmse = valid.RMSE(ymu_class, y_test) # root-mean-square errorqq}q(hUhhubaubeubh)q}q(hUhhhh"h$h%h&}q(h(]h)]h*]h+]qhah-]qhauh/K"h0hh]q(h2)q}q(hXEvaluation measuresqhhhh"h$h6h&}q(h(]h)]h*]h+]h-]uh/K"h0hh]qh9XEvaluation measuresqq}q(hhhhubaubcdocutils.nodes definition_list q)q}q(hUhhhNh$Udefinition_listqh&}q(h(]h)]h*]h+]h-]uh/Nh0hh]qcdocutils.nodes definition_list_item q)q}q(hX)We implemented some classical evaluation measures. - RMSE - root mean squared error - ACC - classification/regression accuracy - Prec - classification precision for class +1 - Recall - classification recall for class +1 - NLPD - negative log predictive density in transformed observation space hhhh"h$Udefinition_list_itemqh&}q(h(]h)]h*]h+]h-]uh/K+h]q(cdocutils.nodes term q)q}q(hX2We implemented some classical evaluation measures.qhhhh"h$Utermqh&}q(h(]h)]h*]h+]h-]uh/K+h]qh9X2We implemented some classical evaluation measures.qq}q(hhhhubaubcdocutils.nodes definition q)q}q(hUh&}q(h(]h)]h*]h+]h-]uhhh]qcdocutils.nodes bullet_list q)q}q(hUh&}q(UbulletqX-h+]h*]h(]h)]h-]uhhh]q(cdocutils.nodes list_item q)q}q(hXRMSE - root mean squared errorqh&}q(h(]h)]h*]h+]h-]uhhh]qh=)q}q(hhhhhh"h$hAh&}q(h(]h)]h*]h+]h-]uh/K$h]qh9XRMSE - root mean squared errorqŅq}q(hhhhubaubah$U list_itemqubh)q}q(hX(ACC - classification/regression accuracyqh&}q(h(]h)]h*]h+]h-]uhhh]qh=)q}q(hhhhhh"h$hAh&}q(h(]h)]h*]h+]h-]uh/K%h]qh9X(ACC - classification/regression accuracyq҅q}q(hhhhubaubah$hubh)q}q(hX,Prec - classification precision for class +1qh&}q(h(]h)]h*]h+]h-]uhhh]qh=)q}q(hhhhhh"h$hAh&}q(h(]h)]h*]h+]h-]uh/K&h]qh9X,Prec - classification precision for class +1qޅq}q(hhhhubaubah$hubh)q}q(hX+Recall - classification recall for class +1qh&}q(h(]h)]h*]h+]h-]uhhh]qh=)q}q(hhhhhh"h$hAh&}q(h(]h)]h*]h+]h-]uh/K'h]qh9X+Recall - classification recall for class +1qꅁq}q(hhhhubaubah$hubh)q}q(hXKNLPD - negative log predictive density in transformed observation space h&}q(h(]h)]h*]h+]h-]uhhh]qh=)q}q(hXGNLPD - negative log predictive density in transformed observation spaceqhhhh"h$hAh&}q(h(]h)]h*]h+]h-]uh/K(h]qh9XGNLPD - negative log predictive density in transformed observation spaceqq}q(hhhhubaubah$hubeh$U bullet_listqubah$U definitionqubeubaubcdocutils.nodes footnote q)q}q(hXSigillito, V. G., Wing, S. P., Hutton, L. V., \& Baker, K. B. (1989). Classification of radar returns from the ionosphere using neural networks. Johns Hopkins APL Technical Digest, 10, 262-266.qh_Khhhh"h$Ufootnoteqh&}r(h(]h)]h*]rhcah+]rhah-]rX1auh/K-h0hh]r(cdocutils.nodes label r)r}r(hUh&}r(h(]h)]h*]h+]h-]uhhh]r h9X1r }r (hUhjubah$Ulabelr ubh=)r }r(hhhhhh"h$hAh&}r(h(]h)]h*]h+]h-]uh/K-h]rh9XSigillito, V. G., Wing, S. P., Hutton, L. V., & Baker, K. B. (1989). Classification of radar returns from the ionosphere using neural networks. Johns Hopkins APL Technical Digest, 10, 262-266.rr}r(hXSigillito, V. G., Wing, S. P., Hutton, L. V., \& Baker, K. B. (1989). Classification of radar returns from the ionosphere using neural networks. Johns Hopkins APL Technical Digest, 10, 262-266.hj ubaubeubeubeubahUU transformerrNU footnote_refsr}rX1]rh]asUrefnamesr}rX1]rh]asUsymbol_footnotesr]rUautofootnote_refsr]rUsymbol_footnote_refsr]r U citationsr!]r"h0hU current_liner#NUtransform_messagesr$]r%Ureporterr&NUid_startr'KU autofootnotesr(]r)U citation_refsr*}r+Uindirect_targetsr,]r-Usettingsr.(cdocutils.frontend Values r/or0}r1(Ufootnote_backlinksr2KUrecord_dependenciesr3NU rfc_base_urlr4Uhttp://tools.ietf.org/html/r5U tracebackr6Upep_referencesr7NUstrip_commentsr8NU toc_backlinksr9Uentryr:U language_coder;Uenr<U datestampr=NU report_levelr>KU _destinationr?NU halt_levelr@KU strip_classesrANh6NUerror_encoding_error_handlerrBUbackslashreplacerCUdebugrDNUembed_stylesheetrEUoutput_encoding_error_handlerrFUstrictrGU sectnum_xformrHKUdump_transformsrINU docinfo_xformrJKUwarning_streamrKNUpep_file_url_templaterLUpep-%04drMUexit_status_levelrNKUconfigrONUstrict_visitorrPNUcloak_email_addressesrQUtrim_footnote_reference_spacerRUenvrSNUdump_pseudo_xmlrTNUexpose_internalsrUNUsectsubtitle_xformrVU source_linkrWNUrfc_referencesrXNUoutput_encodingrYUutf-8rZU source_urlr[NUinput_encodingr\U utf-8-sigr]U_disable_configr^NU id_prefixr_UU tab_widthr`KUerror_encodingraUUTF-8rbU_sourcercU=/Users/marion/workspace/workspace_GPs/pyGPs/doc/source/CV.rstrdUgettext_compactreU generatorrfNUdump_internalsrgNU smart_quotesrhU pep_base_urlriUhttp://www.python.org/dev/peps/rjUsyntax_highlightrkUlongrlUinput_encoding_error_handlerrmjGUauto_id_prefixrnUidroUdoctitle_xformrpUstrip_elements_with_classesrqNU _config_filesrr]Ufile_insertion_enabledrsU raw_enabledrtKU dump_settingsruNubUsymbol_footnote_startrvKUidsrw}rx(hhkhhhhhch]hhGhhuUsubstitution_namesry}rzh$h0h&}r{(h(]h+]h*]Usourceh"h)]h-]uU footnotesr|]r}haUrefidsr~}rub.pyGPs-1.2/doc/build/doctrees/Examples.doctree000066400000000000000000000200341236346020200212140ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(XsourceqXdemosqNXkernels & meansqX optimizersq X pygps/demoq uUsubstitution_defsq }q Uparse_messagesq ]qUcurrent_sourceqNU decorationqNUautofootnote_startqKUnameidsq}q(hUsourceqhUdemosqhU kernels-meansqh U optimizersqh U pygps-demoquUchildrenq]qcdocutils.nodes section q)q}q(U rawsourceqUUparentqhUsourceq cdocutils.nodes reprunicode q!XC/Users/marion/workspace/workspace_GPs/pyGPs/doc/source/Examples.rstq"q#}q$bUtagnameq%Usectionq&U attributesq'}q((Udupnamesq)]Uclassesq*]Ubackrefsq+]Uidsq,]q-haUnamesq.]q/hauUlineq0KUdocumentq1hh]q2(cdocutils.nodes title q3)q4}q5(hXDemosq6hhh h#h%Utitleq7h'}q8(h)]h*]h+]h,]h.]uh0Kh1hh]q9cdocutils.nodes Text q:XDemosq;q<}q=(hh6hh4ubaubcdocutils.nodes paragraph q>)q?}q@(hXThere are several demos exemplifying the use of pyGPs for various Gaussian process (:math:`GP`) tasks. 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You will also find the implementation of the demos in the hh?ubcdocutils.nodes reference qg)qh}qi(hXsource_UresolvedqjKhh?h%U referenceqkh'}ql(UnameXsourceqmUrefuriqnX#https://github.com/marionmari/pyGPsqoh,]h+]h)]h*]h.]uh]qph:Xsourceqqqr}qs(hUhhhubaubh:X folder under qtqu}qv(hX folder under hh?ubhg)qw}qx(hX `pyGPs/Demo`_hjKhh?h%hkh'}qy(UnameX pyGPs/DemohnX:https://github.com/marionmari/pyGPs/tree/master/pyGPs/Demoqzh,]h+]h)]h*]h.]uh]q{h:X pyGPs/Demoq|q}}q~(hUhhwubaubh:X.q}q(hX.hh?ubeubcdocutils.nodes target q)q}q(hXJ.. _pyGPs/Demo: https://github.com/marionmari/pyGPs/tree/master/pyGPs/DemoU referencedqKhhh h#h%Utargetqh'}q(hnhzh,]qhah+]h)]h*]h.]qh auh0K h1hh]ubh>)q}q(hXThe Demos give some theoretical explanations. Further, it is useful to have a look at our documentation on `Kernels & Means`_ and `Optimizers`_.qhhh h#h%hAh'}q(h)]h*]h+]h,]h.]uh0K h1hh]q(h:XkThe Demos give some theoretical explanations. 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Further, it is useful to have a look at our documentation on hhubhg)q}q(hX`Kernels & Means`_hjKhhh%hkh'}q(UnameXKernels & MeanshnX Kernels.htmlqh,]h+]h)]h*]h.]uh]qh:XKernels & Meansqq}q(hUhhubaubh:X and qq}q(hX and hhubhg)q}q(hX `Optimizers`_hjKhhh%hkh'}q(UnameX OptimizershnX Opts.htmlqh,]h+]h)]h*]h.]uh]qh:X Optimizersqq}q(hUhhubaubh:X.q}q(hX.hhubeubh)q}q(hX!.. _Kernels & Means: Kernels.htmlhKhhh h#h%hh'}q(hnhh,]qhah+]h)]h*]h.]qhauh0K h1hh]ubh)q}q(hX.. _Optimizers: Opts.htmlhKhhh h#h%hh'}q(hnhh,]qhah+]h)]h*]h.]qh auh0Kh1hh]ubh)q}q(hX/.. _source: https://github.com/marionmari/pyGPshKhhh h#h%hh'}q(hnhoh,]qhah+]h)]h*]h.]qhauh0Kh1hh]ubh>)q}q(hX Regressionqhhh h#h%hAh'}q(h)]h*]h+]h,]h.]uh0Kh1hh]qh:X Regressionqq}q(hhhhubaubcdocutils.nodes compound q)q}q(hUhhh h#h%Ucompoundqh'}q(h)]h*]qUtoctree-wrapperqah+]h,]h.]uh0Nh1hh]qcsphinx.addnodes toctree q)q}q(hUhhh h#h%Utoctreeqh'}q(UnumberedqKU includehiddenqˉhXExamplesqU titlesonlyq͉UglobqΉh,]h+]h)]h*]h.]Uentriesq]q(NXGPRqцqNXGPR_FITCqӆqeUhiddenqՉU includefilesq]q(hheUmaxdepthqKuh0Kh]ubaubh>)q}q(hXClassificationqhhh h#h%hAh'}q(h)]h*]h+]h,]h.]uh0Kh1hh]qh:XClassificationqޅq}q(hhhhubaubh)q}q(hUhhh h#h%hh'}q(h)]h*]qhah+]h,]h.]uh0Nh1hh]qh)q}q(hUhhh h#h%hh'}q(hKhˉhhh͉hΉh,]h+]h)]h*]h.]h]q(NXGPCqqNXGPC_FITCqqNXGPMCqqehՉh]q(hhhehKuh0Kh]ubaubh>)q}q(hX!Some examples for real-world dataqhhh h#h%hAh'}q(h)]h*]h+]h,]h.]uh0K#h1hh]qh:X!Some examples for real-world dataqq}q(hhhhubaubh)q}q(hUhhh h#h%hh'}q(h)]h*]qhah+]h,]h.]uh0Nh1hh]qh)q}q(hUhhh h#h%hh'}r(hKhˉhhh͉hΉh,]h+]h)]h*]h.]h]r(NXCVrrNX demoMaunaLoarrNX demoHousingrrNXSemiSupervisedrr NX GraphKernelr r ehՉh]r (jjjjj ehKuh0K%h]ubaubeubahUU transformerr NU footnote_refsr}rUrefnamesr}r(hm]rhhaXkernels & means]rhaX optimizers]rhaX pygps/demo]rhwauUsymbol_footnotesr]rUautofootnote_refsr]rUsymbol_footnote_refsr]rU citationsr]rh1hU current_linerNUtransform_messagesr]r Ureporterr!NUid_startr"KU autofootnotesr#]r$U citation_refsr%}r&Uindirect_targetsr']r(Usettingsr)(cdocutils.frontend Values r*or+}r,(Ufootnote_backlinksr-KUrecord_dependenciesr.NU rfc_base_urlr/Uhttp://tools.ietf.org/html/r0U tracebackr1Upep_referencesr2NUstrip_commentsr3NU toc_backlinksr4Uentryr5U language_coder6Uenr7U datestampr8NU report_levelr9KU _destinationr:NU halt_levelr;KU strip_classesr<Nh7NUerror_encoding_error_handlerr=Ubackslashreplacer>Udebugr?NUembed_stylesheetr@Uoutput_encoding_error_handlerrAUstrictrBU sectnum_xformrCKUdump_transformsrDNU docinfo_xformrEKUwarning_streamrFNUpep_file_url_templaterGUpep-%04drHUexit_status_levelrIKUconfigrJNUstrict_visitorrKNUcloak_email_addressesrLUtrim_footnote_reference_spacerMUenvrNNUdump_pseudo_xmlrONUexpose_internalsrPNUsectsubtitle_xformrQU source_linkrRNUrfc_referencesrSNUoutput_encodingrTUutf-8rUU source_urlrVNUinput_encodingrWU utf-8-sigrXU_disable_configrYNU id_prefixrZUU tab_widthr[KUerror_encodingr\UUTF-8r]U_sourcer^UC/Users/marion/workspace/workspace_GPs/pyGPs/doc/source/Examples.rstr_Ugettext_compactr`U generatorraNUdump_internalsrbNU smart_quotesrcU pep_base_urlrdUhttp://www.python.org/dev/peps/reUsyntax_highlightrfUlongrgUinput_encoding_error_handlerrhjBUauto_id_prefixriUidrjUdoctitle_xformrkUstrip_elements_with_classesrlNU _config_filesrm]rnUfile_insertion_enabledroU raw_enabledrpKU dump_settingsrqNubUsymbol_footnote_startrrKUidsrs}rt(hhhhhhhhhhuUsubstitution_namesru}rvh%h1h'}rw(h)]h,]h+]Usourceh#h*]h.]uU footnotesrx]ryUrefidsrz}r{ub.pyGPs-1.2/doc/build/doctrees/GPC.doctree000066400000000000000000000430651236346020200200600ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(Xa bit more things you can doqNX"first example state default valuesqNX load dataqNX more_on_gpcq X second example gp classificationq NXbasic classificationq NuUsubstitution_defsq }q Uparse_messagesq]qUcurrent_sourceqNU decorationqNUautofootnote_startqKUnameidsq}q(hUa-bit-more-things-you-can-doqhU"first-example-state-default-valuesqhU load-dataqh U more-on-gpcqh U second-example-gp-classificationqh Ubasic-classificationquUchildrenq]qcdocutils.nodes section q)q}q(U rawsourceq UUparentq!hUsourceq"cdocutils.nodes reprunicode q#X>/Users/marion/workspace/workspace_GPs/pyGPs/doc/source/GPC.rstq$q%}q&bUtagnameq'Usectionq(U attributesq)}q*(Udupnamesq+]Uclassesq,]Ubackrefsq-]Uidsq.]q/haUnamesq0]q1h auUlineq2KUdocumentq3hh]q4(cdocutils.nodes title q5)q6}q7(h XBasic Classificationq8h!hh"h%h'Utitleq9h)}q:(h+]h,]h-]h.]h0]uh2Kh3hh]q;cdocutils.nodes Text q}q?(h h8h!h6ubaubcdocutils.nodes paragraph q@)qA}qB(h XTThe demo shown in this tutorial can be obtained by running *pyGPs/Demo/demo_GPC.py*.qCh!hh"h%h'U paragraphqDh)}qE(h+]h,]h-]h.]h0]uh2Kh3hh]qF(hAgain, lets see the simplest use of gp classification at firstqЅq}q(h X>Again, lets see the simplest use of gp classification at firsth!hubaubhk)q}q(h XBmodel = pyGPs.gp.GPC() # binary classification (default inference method: EP) model.fit(x, y) # fit default model (mean zero & rbf kernel) with data model.optimize(x, y) # optimize hyperparamters (default optimizer: single run minimize) model.predict(z) # predict test casesh!hh"h%h'hnh)}q(hphqh.]h-]h+]h,]h0]uh2K"h3hh]qhr?}r@(h Uh!j:ubah'UstrongrAubh/Users/marion/workspace/workspace_GPs/pyGPs/doc/source/GPC.rstr Ugettext_compactr U generatorr NUdump_internalsrNU smart_quotesrU pep_base_urlrUhttp://www.python.org/dev/peps/rUsyntax_highlightrUlongrUinput_encoding_error_handlerrjUauto_id_prefixrUidrUdoctitle_xformrUstrip_elements_with_classesrNU _config_filesr]rUfile_insertion_enabledrU raw_enabledrKU dump_settingsrNubUsymbol_footnote_startrKUidsr}r (hhhhhhUhjhhhjuUsubstitution_namesr!}r"h'h3h)}r#(h+]h.]h-]Usourceh%h,]h0]uU footnotesr$]r%Urefidsr&}r'h]r(jasub.pyGPs-1.2/doc/build/doctrees/GPC_FITC.doctree000066400000000000000000000416751236346020200206720ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(Xa bit more things you can doqNX+second example user-defined inducing pointsqNXsparse classificationqNXmore_on_gpc_fitcq X%first example default inducing pointsq NXfitc regressionq uUsubstitution_defsq }q Uparse_messagesq]qUcurrent_sourceqNU decorationqNUautofootnote_startqKUnameidsq}q(hUa-bit-more-things-you-can-doqhU+second-example-user-defined-inducing-pointsqhUsparse-classificationqh Umore-on-gpc-fitcqh U%first-example-default-inducing-pointsqh Ufitc-regressionquUchildrenq]qcdocutils.nodes section q)q}q(U rawsourceq UUparentq!hUsourceq"cdocutils.nodes reprunicode q#XC/Users/marion/workspace/workspace_GPs/pyGPs/doc/source/GPC_FITC.rstq$q%}q&bUtagnameq'Usectionq(U attributesq)}q*(Udupnamesq+]Uclassesq,]Ubackrefsq-]Uidsq.]q/haUnamesq0]q1hauUlineq2KUdocumentq3hh]q4(cdocutils.nodes title q5)q6}q7(h XSparse Classificationq8h!hh"h%h'Utitleq9h)}q:(h+]h,]h-]h.]h0]uh2Kh3hh]q;cdocutils.nodes Text q}q?(h h8h!h6ubaubcdocutils.nodes paragraph q@)qA}qB(h XThe demo in this tutorial can be obtained by running *pyGPs/Demo/demo_GPC_FITC.py*. This demo is more or less a repetition of the demo of `FITC regression`_.h!hh"h%h'U paragraphqCh)}qD(h+]h,]h-]h.]h0]uh2Kh3hh]qE(h}r?(h X( are shown as black circles in the plot.h!j2ubeubeubh)r@}rA(h Uh!hh"h%h'h(h)}rB(h+]h,]h-]h.]rChah0]rDhauh2K1h3hh]rE(h5)rF}rG(h X?Second example :math:`\rightarrow` user-defined inducing pointsrHh!j@h"h%h'h9h)}rI(h+]h,]h-]h.]h0]uh2K1h3hh]rJ(h(hh7h h5ubaubcdocutils.nodes paragraph q?)q@}qA(hXsGPMC is NOT based on multi-class Laplace approximation. It works as a one vs. one classification wrapper. In other words, GPMC trains a GPC model for each pair of two classes, and uses a majority voting scheme over all results to determine the final class. The method only returns the predictive class with highest rating; no other values (such as variance) are returned.qBh hh!h$h&U paragraphqCh(}qD(h*]h+]h,]h-]h/]uh1Kh2hh]qEh;XsGPMC is NOT based on multi-class Laplace approximation. It works as a one vs. one classification wrapper. In other words, GPMC trains a GPC model for each pair of two classes, and uses a majority voting scheme over all results to determine the final class. The method only returns the predictive class with highest rating; no other values (such as variance) are returned.qFqG}qH(hhBh h@ubaubh?)qI}qJ(hXlLets see a practical example to classify the 10 (0,1,2,...9) hand-written digits in the USPS digits dataset.qKh hh!h$h&hCh(}qL(h*]h+]h,]h-]h/]uh1K h2hh]qMh;XlLets see a practical example to classify the 10 (0,1,2,...9) hand-written digits in the USPS digits dataset.qNqO}qP(hhKh hIubaubh)qQ}qR(hUh hh!h$h&h'h(}qS(h*]h+]h,]h-]qThah/]qUhauh1Kh2hh]qV(h4)qW}qX(hX Load dataqYh hQh!h$h&h8h(}qZ(h*]h+]h,]h-]h/]uh1Kh2hh]q[h;X Load dataq\q]}q^(hhYh hWubaubh?)q_}q`(hXThe USPS digits data were gathered at the Center of Excellence in Document Analysis and Recognition (CEDAR) at SUNY Buffalo, as part of a project sponsored by the US Postal Service. The dataset is described in [1]_. ::qah hQh!h$h&hCh(}qb(h*]h+]h,]h-]h/]uh1Kh2hh]qc(h;XThe USPS digits data were gathered at the Center of Excellence in Document Analysis and Recognition (CEDAR) at SUNY Buffalo, as part of a project sponsored by the US Postal Service. The dataset is described in qdqe}qf(hXThe USPS digits data were gathered at the Center of Excellence in Document Analysis and Recognition (CEDAR) at SUNY Buffalo, as part of a project sponsored by the US Postal Service. The dataset is described in h h_ubcdocutils.nodes footnote_reference qg)qh}qi(hX[1]_UresolvedqjKh h_h&Ufootnote_referenceqkh(}ql(h-]qmUid1qnah,]h*]h+]h/]Urefidqohuh]qph;X1qq}qr(hUh hhubaubh;X.qs}qt(hX.h h_ubeubcdocutils.nodes literal_block qu)qv}qw(hXdata = loadmat('data_for_demo/usps_resampled.mat') x = data['train_patterns'].T # train patterns y = data['train_labels'].T # train labels xs = data['test_patterns'].T # test patterns ys = data['test_labels'].T # test labelsh hQh!h$h&U literal_blockqxh(}qy(U xml:spaceqzUpreserveq{h-]h,]h*]h+]h/]uh1Kh2hh]q|h;Xdata = loadmat('data_for_demo/usps_resampled.mat') x = data['train_patterns'].T # train patterns y = data['train_labels'].T # train labels xs = data['test_patterns'].T # test patterns ys = data['test_labels'].T # test labelsq}q~}q(hUh hvubaubh?)q}q(hXLTo be used in GPMC, labels should start from 0 to k (k = number of classes).qh hQh!h$h&hCh(}q(h*]h+]h,]h-]h/]uh1Kh2hh]qh;XLTo be used in GPMC, labels should start from 0 to k (k = number of classes).qq}q(hhh hubaubeubh)q}q(hUh hh!h$h&h'h(}q(h*]h+]h,]h-]qhah/]qh auh1Kh2hh]q(h4)q}q(hX GPMC exampleqh hh!h$h&h8h(}q(h*]h+]h,]h-]h/]uh1Kh2hh]qh;X GPMC exampleqq}q(hhh hubaubh?)q}q(hX4State model with 10-class classification problem: ::qh hh!h$h&hCh(}q(h*]h+]h,]h-]h/]uh1Kh2hh]qh;X1State model with 10-class classification problem:qq}q(hX1State model with 10-class classification problem:h hubaubhu)q}q(hXmodel = pyGPs.GPMC(10)h hh!h$h&hxh(}q(hzh{h-]h,]h*]h+]h/]uh1K h2hh]qh;Xmodel = pyGPs.GPMC(10)qq}q(hUh hubaubh?)q}q(hXPass data to model: ::qh hh!h$h&hCh(}q(h*]h+]h,]h-]h/]uh1K"h2hh]qh;XPass data to model:qq}q(hXPass data to model:h hubaubhu)q}q(hXmodel.setData(x,y)h hh!h$h&hxh(}q(hzh{h-]h,]h*]h+]h/]uh1K$h2hh]qh;Xmodel.setData(x,y)qq}q(hUh hubaubh?)q}q(hXTrain default GPC model for each binary classification problem, and decide label for test patterns of hand-writen digits. The return value *predictive_vote[i,j]* is the probability of being class *j* for test pattern *i*. ::h hh!h$h&hCh(}q(h*]h+]h,]h-]h/]uh1K&h2hh]q(h;XTrain default GPC model for each binary classification problem, and decide label for test patterns of hand-writen digits. The return value qq}q(hXTrain default GPC model for each binary classification problem, and decide label for test patterns of hand-writen digits. The return value h hubcdocutils.nodes emphasis q)q}q(hX*predictive_vote[i,j]*h(}q(h*]h+]h,]h-]h/]uh hh]qh;Xpredictive_vote[i,j]qq}q(hUh hubah&Uemphasisqubh;X# is the probability of being class qąq}q(hX# is the probability of being class h hubh)q}q(hX*j*h(}q(h*]h+]h,]h-]h/]uh hh]qh;Xjq}q(hUh hubah&hubh;X for test pattern qͅq}q(hX for test pattern h hubh)q}q(hX*i*h(}q(h*]h+]h,]h-]h/]uh hh]qh;Xiq}q(hUh hubah&hubh;X.q}q(hX.h hubeubhu)q}q(hXdpredictive_vote = model.optimizeAndPredict(xs) predictive_class = np.argmax(predictive_vote, axis=1)h hh!h$h&hxh(}q(hzh{h-]h,]h*]h+]h/]uh1K*h2hh]qh;Xdpredictive_vote = model.optimizeAndPredict(xs) predictive_class = np.argmax(predictive_vote, axis=1)q܅q}q(hUh hubaubh?)q}q(hXJust like we did for GP classification, you can use specific settings (other than default) for all binary classificiation problem for example by: ::h hh!h$h&hCh(}q(h*]h+]h,]h-]h/]uh1K-h2hh]qh;XJust like we did for GP classification, you can use specific settings (other than default) for all binary classificiation problem for example by:qㅁq}q(hXJust like we did for GP classification, you can use specific settings (other than default) for all binary classificiation problem for example by:h hubaubhu)q}q(hXgm = pyGPs.mean.Zero() k = pyGPs.cov.RBF() model.setPrior(mean=m,kernel=k) model.useInference("Laplace")h hh!h$h&hxh(}q(hzh{h-]h,]h*]h+]h/]uh1K0h2hh]qh;Xgm = pyGPs.mean.Zero() k = pyGPs.cov.RBF() model.setPrior(mean=m,kernel=k) model.useInference("Laplace")qꅁq}q(hUh hubaubh?)q}q(hXXFor more information on how to use non-default settings see `demo_GPC`_ and `demo_GPR`_.qh hh!h$h&hCh(}q(h*]h+]h,]h-]h/]uh1K5h2hh]q(h;X<For more information on how to use non-default settings see qq}q(hX<For more information on how to use non-default settings see h hubcdocutils.nodes reference q)q}q(hX `demo_GPC`_hjKh hh&U referenceqh(}q(UnameXdemo_GPCUrefuriqXGPC.htmlqh-]h,]h*]h+]h/]uh]qh;Xdemo_GPCqq}q(hUh hubaubh;X and rr}r(hX and h hubh)r}r(hX `demo_GPR`_hjKh hh&hh(}r(UnameXdemo_GPRhXGPR.htmlrh-]h,]h*]h+]h/]uh]rh;Xdemo_GPRrr }r (hUh jubaubh;X.r }r (hX.h hubeubcdocutils.nodes target r )r}r(hX.. _demo_GPC: GPC.htmlU referencedrKh hh!h$h&Utargetrh(}r(hhh-]rhah,]h*]h+]h/]rh auh1K8h2hh]ubj )r}r(hX.. _demo_GPR: GPR.htmljKh hh!h$h&jh(}r(hjh-]rhah,]h*]h+]h/]rhauh1K9h2hh]ubh?)r}r(hXvBeside *optimizeAndPredict(xs)*, there is also an option to perform prediction without hyperparameter optimization: ::h hh!h$h&hCh(}r(h*]h+]h,]h-]h/]uh1K(hUh(}r?(h*]h+]h,]h-]h/]uh j3h]r@h;X1rA}rB(hUh j=ubah&UlabelrCubh?)rD}rE(hj5h j3h!h$h&hCh(}rF(h*]h+]h,]h-]h/]uh1KAh]rGh;X`A Database for Handwritten Text Recognition Research, J. J. Hull, IEEE PAMI 16(5) 550-554, 1994.rHrI}rJ(hj5h jDubaubeubeubeubahUU transformerrKNU footnote_refsrL}rMX1]rNhhasUrefnamesrO}rP(X1]rQhhaXdemo_gpc]rRhaXdemo_gpr]rSjauUsymbol_footnotesrT]rUUautofootnote_refsrV]rWUsymbol_footnote_refsrX]rYU citationsrZ]r[h2hU current_liner\NUtransform_messagesr]]r^Ureporterr_NUid_startr`KU autofootnotesra]rbU citation_refsrc}rdUindirect_targetsre]rfUsettingsrg(cdocutils.frontend Values rhori}rj(Ufootnote_backlinksrkKUrecord_dependenciesrlNU rfc_base_urlrmUhttp://tools.ietf.org/html/rnU tracebackroUpep_referencesrpNUstrip_commentsrqNU toc_backlinksrrUentryrsU language_codertUenruU datestamprvNU report_levelrwKU _destinationrxNU halt_levelryKU strip_classesrzNh8NUerror_encoding_error_handlerr{Ubackslashreplacer|Udebugr}NUembed_stylesheetr~Uoutput_encoding_error_handlerrUstrictrU sectnum_xformrKUdump_transformsrNU docinfo_xformrKUwarning_streamrNUpep_file_url_templaterUpep-%04drUexit_status_levelrKUconfigrNUstrict_visitorrNUcloak_email_addressesrUtrim_footnote_reference_spacerUenvrNUdump_pseudo_xmlrNUexpose_internalsrNUsectsubtitle_xformrU source_linkrNUrfc_referencesrNUoutput_encodingrUutf-8rU source_urlrNUinput_encodingrU utf-8-sigrU_disable_configrNU id_prefixrUU tab_widthrKUerror_encodingrUUTF-8rU_sourcerU?/Users/marion/workspace/workspace_GPs/pyGPs/doc/source/GPMC.rstrUgettext_compactrU generatorrNUdump_internalsrNU smart_quotesrU pep_base_urlrUhttp://www.python.org/dev/peps/rUsyntax_highlightrUlongrUinput_encoding_error_handlerrjUauto_id_prefixrUidrUdoctitle_xformrUstrip_elements_with_classesrNU _config_filesr]Ufile_insertion_enabledrU raw_enabledrKU dump_settingsrNubUsymbol_footnote_startrKUidsr}r(hhhhhjhj3hnhhhhQhjuUsubstitution_namesr}rh&h2h(}r(h*]h-]h,]Usourceh$h+]h/]uU footnotesr]rj3aUrefidsr}rub.pyGPs-1.2/doc/build/doctrees/GPR.doctree000066400000000000000000000707471236346020200201060ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(Xa bit more things you can doqNXa more complicated exampleqNXbasic regressionqNX load dataq NXgpmlq Ximport packagesq NXa five-line toy exampleq NXkernels & meansq X optimizersquUsubstitution_defsq}qUparse_messagesq]qUcurrent_sourceqNU decorationqNUautofootnote_startqKUnameidsq}q(hUa-bit-more-things-you-can-doqhUa-more-complicated-exampleqhUbasic-regressionqh U load-dataqh Ugpmlqh Uimport-packagesqh Ua-five-line-toy-exampleqh U kernels-meansqhU optimizersq uUchildrenq!]q"cdocutils.nodes section q#)q$}q%(U rawsourceq&UUparentq'hUsourceq(cdocutils.nodes reprunicode q)X>/Users/marion/workspace/workspace_GPs/pyGPs/doc/source/GPR.rstq*q+}q,bUtagnameq-Usectionq.U attributesq/}q0(Udupnamesq1]Uclassesq2]Ubackrefsq3]Uidsq4]q5haUnamesq6]q7hauUlineq8KUdocumentq9hh!]q:(cdocutils.nodes title q;)q<}q=(h&XBasic Regressionq>h'h$h(h+h-Utitleq?h/}q@(h1]h2]h3]h4]h6]uh8Kh9hh!]qAcdocutils.nodes Text qBXBasic RegressionqCqD}qE(h&h>h'hhBX:You can specify non-default mean and covariance functions:r?r@}rA(h&X:You can specify non-default mean and covariance functions:h'j:ubaubhw)rB}rC(h&Xim = pyGPs.mean.Linear( D=x.shape[1] ) + mean.Const() k = pyGPs.cov.RBF() model.setPrior(mean=m, kernel=k)h'j$h(h+h-hzh/}rD(h|h}h4]h3]h1]h2]h6]uh8K7h9hh!]rEhBXim = pyGPs.mean.Linear( D=x.shape[1] ) + mean.Const() k = pyGPs.cov.RBF() model.setPrior(mean=m, kernel=k)rFrG}rH(h&Uh'jBubaubhF)rI}rJ(h&XiHere, we use a composite mean as the sum of a linear and a constant function, and an rbf kernel. The initial hyperparameters are left to their default values. See `Kernels & Means`_ for a complete documentation of kernel/mean specification and custom kernel/mean construction. Once kernel and mean are specified, they are passed to the prior using *setPrior()*.rKh'j$h(h+h-hJh/}rL(h1]h2]h3]h4]h6]uh8K;h9hh!]rM(hBXHere, we use a composite mean as the sum of a linear and a constant function, and an rbf kernel. The initial hyperparameters are left to their default values. See rNrO}rP(h&XHere, we use a composite mean as the sum of a linear and a constant function, and an rbf kernel. The initial hyperparameters are left to their default values. See h'jIubcdocutils.nodes reference rQ)rR}rS(h&X`Kernels & Means`_UresolvedrTKh'jIh-U referencerUh/}rV(UnameXKernels & MeansUrefurirWX Kernels.htmlrXh4]h3]h1]h2]h6]uh!]rYhBXKernels & MeansrZr[}r\(h&Uh'jRubaubhBX for a complete documentation of kernel/mean specification and custom kernel/mean construction. Once kernel and mean are specified, they are passed to the prior using r]r^}r_(h&X for a complete documentation of kernel/mean specification and custom kernel/mean construction. Once kernel and mean are specified, they are passed to the prior using h'jIubhP)r`}ra(h&X *setPrior()*h/}rb(h1]h2]h3]h4]h6]uh'jIh!]rchBX setPrior()rdre}rf(h&Uh'j`ubah-hXubhBX.rg}rh(h&X.h'jIubeubcdocutils.nodes target ri)rj}rk(h&X!.. _Kernels & Means: Kernels.htmlU referencedrlKh'j$h(h+h-Utargetrmh/}rn(jWjXh4]rohah3]h1]h2]h6]rph auh8K=h9hh!]ubhF)rq}rr(h&X/You can add the traning data to the model explicitly by using *setData()*. So, you avoid passing them into *fit()* or *optimize()* each time used. More importantly, the deafult mean will be adapted to the average value of the trainging labels :math:`y` (if you do not specify mean function by your own).rsh'j$h(h+h-hJh/}rt(h1]h2]h3]h4]h6]uh8K?h9hh!]ru(hBX>You can add the traning data to the model explicitly by using rvrw}rx(h&X>You can add the traning data to the model explicitly by using h'jqubhP)ry}rz(h&X *setData()*h/}r{(h1]h2]h3]h4]h6]uh'jqh!]r|hBX setData()r}r~}r(h&Uh'jyubah-hXubhBX". So, you avoid passing them into rr}r(h&X". So, you avoid passing them into h'jqubhP)r}r(h&X*fit()*h/}r(h1]h2]h3]h4]h6]uh'jqh!]rhBXfit()rr}r(h&Uh'jubah-hXubhBX or rr}r(h&X or h'jqubhP)r}r(h&X *optimize()*h/}r(h1]h2]h3]h4]h6]uh'jqh!]rhBX optimize()rr}r(h&Uh'jubah-hXubhBXq each time used. More importantly, the deafult mean will be adapted to the average value of the trainging labels rr}r(h&Xq each time used. More importantly, the deafult mean will be adapted to the average value of the trainging labels h'jqubh)r}r(h&Uh/}r(UlatexXyh4]h3]h1]h2]h6]uh'jqh!]h-hubhBX3 (if you do not specify mean function by your own).rr}r(h&X3 (if you do not specify mean function by your own).h'jqubeubhF)r}r(h&X2Further, you can plot the data in the 1-d case: ::rh'j$h(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8KAh9hh!]rhBX/Further, you can plot the data in the 1-d case:rr}r(h&X/Further, you can plot the data in the 1-d case:h'jubaubhw)r}r(h&Xmodel.plotData_1d()h'j$h(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8KCh9hh!]rhBXmodel.plotData_1d()rr}r(h&Uh'jubaubj)r}r(h&Uh'j$h(h+h-jh/}r(jXcenterh4]h3]h1]h2]h6]uh8Nh9hh!]rj)r}r(h&Xd.. figure:: _images/d1_2.png :height: 600 px :width: 800 px :align: center :scale: 70 % h/}r(UscaleKFh1]UuriX_images/d1_2.pngrh4]h3]UwidthX800pxh2]j!}rU*jsh6]UheightX600pxuh'jh!]h-j#ubaubhF)r}r(h&XYou can specify a optimization method different from the default, which is a single run of Rasmussen's minimize. For example, you can choose to rerun the optimization method several times with different random initializations: ::h'j$h(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8KKh9hh!]rhBXYou can specify a optimization method different from the default, which is a single run of Rasmussen's minimize. For example, you can choose to rerun the optimization method several times with different random initializations:rr}r(h&XYou can specify a optimization method different from the default, which is a single run of Rasmussen's minimize. For example, you can choose to rerun the optimization method several times with different random initializations:h'jubaubhw)r}r(h&X/model.setOptimizer("Minimize", num_restarts=30)h'j$h(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8KNh9hh!]rhBX/model.setOptimizer("Minimize", num_restarts=30)rr}r(h&Uh'jubaubhF)r}r(h&XThe optimized hyperparameters returned by *optimize()* are then set to be the ones obtained from the run with the best result. The whole functionality for optimization is introduced in detail in the documentation `Optimizers`_.h'j$h(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8KPh9hh!]r(hBX*The optimized hyperparameters returned by rr}r(h&X*The optimized hyperparameters returned by h'jubhP)r}r(h&X *optimize()*h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX optimize()rr}r(h&Uh'jubah-hXubhBX are then set to be the ones obtained from the run with the best result. The whole functionality for optimization is introduced in detail in the documentation rr}r(h&X are then set to be the ones obtained from the run with the best result. The whole functionality for optimization is introduced in detail in the documentation h'jubjQ)r}r(h&X `Optimizers`_jTKh'jh-jUh/}r(UnameX OptimizersjWX Opts.htmlrh4]h3]h1]h2]h6]uh!]rhBX Optimizersrr}r(h&Uh'jubaubhBX.r}r(h&X.h'jubeubji)r}r(h&X.. _Optimizers: Opts.htmljlKh'j$h(h+h-jmh/}r(jWjh4]rh ah3]h1]h2]h6]rhauh8KSh9hh!]ubhF)r}r(h&XInstead of *fit()*, which only fits data using given hyperparameters, *optimize()* will optimize hyperparamters based on marginal likelihood: ::rh'j$h(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8KUh9hh!]r(hBX Instead of rr}r(h&X Instead of h'jubhP)r}r(h&X*fit()*h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXfit()rr}r(h&Uh'jubah-hXubhBX4, which only fits data using given hyperparameters, rr}r(h&X4, which only fits data using given hyperparameters, h'jubhP)r}r(h&X *optimize()*h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX optimize()rr}r(h&Uh'jubah-hXubhBX; will optimize hyperparamters based on marginal likelihood:rr}r(h&X; will optimize hyperparamters based on marginal likelihood:h'jubeubhw)r}r(h&Xmodel.optimize()h'j$h(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8KWh9hh!]rhBXmodel.optimize()rr}r(h&Uh'jubaubhF)r}r(h&X;There are several properties you can get from the model: ::rh'j$h(h+h-hJh/}r (h1]h2]h3]h4]h6]uh8KZh9hh!]r hBX8There are several properties you can get from the model:r r }r (h&X8There are several properties you can get from the model:h'jubaubhw)r}r(h&X-model.nlZ # negative log marginal likelihood model.dnlZ.cov # direvatives of negative log marginal likelihood model.dnlZ.lik model.dnlZ.mean model.posterior.sW # posterior structure model.posterior.alpha model.posterior.L model.covfunc.hyp model.meanfunc.hyp model.likfunc.hyp model.fm # latent mean model.fs2 # latent variance model.ym # predictive mean model.ys2 # predictive variance model.lp # log predictive probabilityh'j$h(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8K\h9hh!]rhBX-model.nlZ # negative log marginal likelihood model.dnlZ.cov # direvatives of negative log marginal likelihood model.dnlZ.lik model.dnlZ.mean model.posterior.sW # posterior structure model.posterior.alpha model.posterior.L model.covfunc.hyp model.meanfunc.hyp model.likfunc.hyp model.fm # latent mean model.fs2 # latent variance model.ym # predictive mean model.ys2 # predictive variance model.lp # log predictive probabilityrr}r(h&Uh'jubaubhF)r}r(h&X7For example, to get the log marginal likelihood use: ::rh'j$h(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Klh9hh!]rhBX4For example, to get the log marginal likelihood use:rr}r(h&X4For example, to get the log marginal likelihood use:h'jubaubhw)r}r(h&XGprint 'Optimized negative log marginal likelihood:', round(model.nlZ,3)h'j$h(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8Knh9hh!]r hBXGprint 'Optimized negative log marginal likelihood:', round(model.nlZ,3)r!r"}r#(h&Uh'jubaubhF)r$}r%(h&XPrediction on the test data will return five values, which are output mean (ymu) resp. variance (ys2), latent mean (fmu) resp. variance (fs2), and log predictive probabilities (lp) ::h'j$h(h+h-hJh/}r&(h1]h2]h3]h4]h6]uh8Kqh9hh!]r'hBXPrediction on the test data will return five values, which are output mean (ymu) resp. variance (ys2), latent mean (fmu) resp. variance (fs2), and log predictive probabilities (lp)r(r)}r*(h&XPrediction on the test data will return five values, which are output mean (ymu) resp. variance (ys2), latent mean (fmu) resp. variance (fs2), and log predictive probabilities (lp)h'j$ubaubhw)r+}r,(h&X'ym, ys2, fm, fs2, lp = model.predict(z)h'j$h(h+h-hzh/}r-(h|h}h4]h3]h1]h2]h6]uh8Kth9hh!]r.hBX'ym, ys2, fm, fs2, lp = model.predict(z)r/r0}r1(h&Uh'j+ubaubhF)r2}r3(h&XPlot data. Note that *GPR.plot()* is a toy method only for visualising 1-d data. Here we got a different posterior by using a different prior other than in the default example. ::r4h'j$h(h+h-hJh/}r5(h1]h2]h3]h4]h6]uh8Kwh9hh!]r6(hBXPlot data. Note that r7r8}r9(h&XPlot data. Note that h'j2ubhP)r:}r;(h&X *GPR.plot()*h/}r<(h1]h2]h3]h4]h6]uh'j2h!]r=hBX GPR.plot()r>r?}r@(h&Uh'j:ubah-hXubhBX is a toy method only for visualising 1-d data. Here we got a different posterior by using a different prior other than in the default example.rArB}rC(h&X is a toy method only for visualising 1-d data. Here we got a different posterior by using a different prior other than in the default example.h'j2ubeubhw)rD}rE(h&X model.plot()h'j$h(h+h-hzh/}rF(h|h}h4]h3]h1]h2]h6]uh8Kyh9hh!]rGhBX model.plot()rHrI}rJ(h&Uh'jDubaubj)rK}rL(h&Uh'j$h(h+h-jh/}rM(jXcenterh4]h3]h1]h2]h6]uh8Nh9hh!]rNj)rO}rP(h&Xf.. figure:: _images/d1_3.png :height: 600 px :width: 800 px :align: center :scale: 70 % h/}rQ(UscaleKFh1]UuriX_images/d1_3.pngrRh4]h3]UwidthX800pxh2]j!}rSU*jRsh6]UheightX600pxuh'jKh!]h-j#ubaubeubh#)rT}rU(h&Uh'h$h(h+h-h.h/}rV(h1]h2]h3]h4]rWhah6]rXhauh8Kh9hh!]rY(h;)rZ}r[(h&XA bit more things you can dor\h'jTh(h+h-h?h/}r](h1]h2]h3]h4]h6]uh8Kh9hh!]r^hBXA bit more things you can dor_r`}ra(h&j\h'jZubaubhF)rb}rc(h&Xg**[For all Models]** Speed up computation time for prediction if you know posterior in advance. Posterior is passed as an object with three fields (attributes) post.alpha, post.sW and post.L. How to use these vectors to represent the posterior can be best seen from Algorithm 2.1 (page 19) in Chapeter 2 of the `GPML`_ book by Rasmussen and Williams, 2006. ::rdh'jTh(h+h-hJh/}re(h1]h2]h3]h4]h6]uh8Kh9hh!]rf(cdocutils.nodes strong rg)rh}ri(h&X**[For all Models]**h/}rj(h1]h2]h3]h4]h6]uh'jbh!]rkhBX[For all Models]rlrm}rn(h&Uh'jhubah-UstrongroubhBX# Speed up computation time for prediction if you know posterior in advance. Posterior is passed as an object with three fields (attributes) post.alpha, post.sW and post.L. How to use these vectors to represent the posterior can be best seen from Algorithm 2.1 (page 19) in Chapeter 2 of the rprq}rr(h&X# Speed up computation time for prediction if you know posterior in advance. Posterior is passed as an object with three fields (attributes) post.alpha, post.sW and post.L. How to use these vectors to represent the posterior can be best seen from Algorithm 2.1 (page 19) in Chapeter 2 of the h'jbubjQ)rs}rt(h&X`GPML`_jTKh'jbh-jUh/}ru(UnameXGPMLjWX4http://www.gaussianprocess.org/gpml/chapters/RW2.pdfrvh4]h3]h1]h2]h6]uh!]rwhBXGPMLrxry}rz(h&Uh'jsubaubhBX& book by Rasmussen and Williams, 2006.r{r|}r}(h&X& book by Rasmussen and Williams, 2006.h'jbubeubhw)r~}r(h&Xlpost = myPosterior() # known in advance ym, ys2, fm, fs2, lp = model.predict_with_posterior( post,z )h'jTh(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8Kh9hh!]rhBXlpost = myPosterior() # known in advance ym, ys2, fm, fs2, lp = model.predict_with_posterior( post,z )rr}r(h&Uh'j~ubaubji)r}r(h&X>.. _GPML: http://www.gaussianprocess.org/gpml/chapters/RW2.pdfjlKh'jTh(h+h-jmh/}r(jWjvh4]rhah3]h1]h2]h6]rh auh8Kh9hh!]ubhF)r}r(h&XX**[Only for Regression]** Specify noise of data (with :math:`\sigma=0.1` by default): ::rh'jTh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Kh9hh!]r(jg)r}r(h&X**[Only for Regression]**h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX[Only for Regression]rr}r(h&Uh'jubah-joubhBX Specify noise of data (with rr}r(h&X Specify noise of data (with h'jubh)r}r(h&Uh/}r(UlatexX \sigma=0.1h4]h3]h1]h2]h6]uh'jh!]h-hubhBX by default):rr}r(h&X by default):h'jubeubhw)r}r(h&X)model.setNoise( log_sigma = np.log(0.1) )h'jTh(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8Kh9hh!]rhBX)model.setNoise( log_sigma = np.log(0.1) )rr}r(h&Uh'jubaubhF)r}r(h&XeYou do not need to specify the noise parameter if you are optimizing the hyperparamters later anyhow.rh'jTh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Kh9hh!]rhBXeYou do not need to specify the noise parameter if you are optimizing the hyperparamters later anyhow.rr}r(h&jh'jubaubhF)r}r(h&XfAll plotting methods have keyword axisvals. You can adjust plotting range if you want. For example: ::rh'jTh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Kh9hh!]rhBXcAll plotting methods have keyword axisvals. You can adjust plotting range if you want. For example:rr}r(h&XcAll plotting methods have keyword axisvals. You can adjust plotting range if you want. For example:h'jubaubhw)r}r(h&X-model.plot(axisvals = [-1.9, 1.9, -0.9, 3.9])h'jTh(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8Kh9hh!]rhBX-model.plot(axisvals = [-1.9, 1.9, -0.9, 3.9])rr}r(h&Uh'jubaubhF)r}r(h&X6Switch to other Inference and Likelihood functions. ::rh'jTh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Kh9hh!]rhBX3Switch to other Inference and Likelihood functions.rr}r(h&X3Switch to other Inference and Likelihood functions.h'jubaubhw)r}r(h&X7model.useInference("EP") model.useLikelihood("Laplace")h'jTh(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8Kh9hh!]rhBX7model.useInference("EP") model.useLikelihood("Laplace")rr}r(h&Uh'jubaubeubeubah&UU transformerrNU footnote_refsr}rUrefnamesr}r(Xgpml]rjsaXkernels & means]rjRaX optimizers]rjauUsymbol_footnotesr]rUautofootnote_refsr]rUsymbol_footnote_refsr]rU citationsr]rh9hU current_linerNUtransform_messagesr]rUreporterrNUid_startrKU autofootnotesr]rU citation_refsr}rUindirect_targetsr]rUsettingsr(cdocutils.frontend Values ror}r(Ufootnote_backlinksrKUrecord_dependenciesrNU 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pep_base_urlr"Uhttp://www.python.org/dev/peps/r#Usyntax_highlightr$Ulongr%Uinput_encoding_error_handlerr&jUauto_id_prefixr'Uidr(Udoctitle_xformr)Ustrip_elements_with_classesr*NU _config_filesr+]r,Ufile_insertion_enabledr-U raw_enabledr.KU dump_settingsr/NubUsymbol_footnote_startr0KUidsr1}r2(hhhjjhhahjhhhjThj$hh$h juUsubstitution_namesr3}r4h-h9h/}r5(h1]h4]h3]Usourceh+h2]h6]uU footnotesr6]r7Urefidsr8}r9ub.pyGPs-1.2/doc/build/doctrees/GPR_FITC.doctree000066400000000000000000000426211236346020200207010ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(X%first example default inducing pointsqNXa bit more things you can doqNX+second example user-defined inducing pointsqNXsparse regressionq NXfitc classificationq uUsubstitution_defsq }q Uparse_messagesq ]qUcurrent_sourceqNU decorationqNUautofootnote_startqKUnameidsq}q(hU%first-example-default-inducing-pointsqhUa-bit-more-things-you-can-doqhU+second-example-user-defined-inducing-pointsqh Usparse-regressionqh Ufitc-classificationquUchildrenq]qcdocutils.nodes section q)q}q(U rawsourceqUUparentqhUsourceq cdocutils.nodes reprunicode q!XC/Users/marion/workspace/workspace_GPs/pyGPs/doc/source/GPR_FITC.rstq"q#}q$bUtagnameq%Usectionq&U attributesq'}q((Udupnamesq)]Uclassesq*]Ubackrefsq+]Uidsq,]q-haUnamesq.]q/h auUlineq0KUdocumentq1hh]q2(cdocutils.nodes title q3)q4}q5(hXSparse Regressionq6hhh h#h%Utitleq7h'}q8(h)]h*]h+]h,]h.]uh0Kh1hh]q9cdocutils.nodes Text q:XSparse Regressionq;q<}q=(hh6hh4ubaubcdocutils.nodes paragraph q>)q?}q@(hXThe code shown in this tutorial can be obtained by running *pyGPs/Demo/demo_GPR_FITC.py* This demo is more or less similar to the demo of `FITC classification`_.hhh h#h%U paragraphqAh'}qB(h)]h*]h+]h,]h.]uh0Kh1hh]qC(h:X;The code shown in this tutorial can be obtained by running qDqE}qF(hX;The code shown in this tutorial can be obtained by running hh?ubcdocutils.nodes emphasis qG)qH}qI(hX*pyGPs/Demo/demo_GPR_FITC.py*h'}qJ(h)]h*]h+]h,]h.]uhh?h]qKh:XpyGPs/Demo/demo_GPR_FITC.pyqLqM}qN(hUhhHubah%UemphasisqOubh:X2 This demo is more or less similar to the demo of qPqQ}qR(hX2 This demo is more or less similar to the demo of hh?ubcdocutils.nodes reference qS)qT}qU(hX`FITC classification`_UresolvedqVKhh?h%U referenceqWh'}qX(UnameXFITC classificationUrefuriqYX GPC_FITC.htmlqZh,]h+]h)]h*]h.]uh]q[h:XFITC classificationq\q]}q^(hUhhTubaubh:X.q_}q`(hX.hh?ubeubcdocutils.nodes target qa)qb}qc(hX&.. _FITC classification: GPC_FITC.htmlU referencedqdKhhh h#h%Utargetqeh'}qf(hYhZh,]qghah+]h)]h*]h.]qhh auh0Kh1hh]ubh)qi}qj(hUhhh h#h%h&h'}qk(h)]h*]h+]h,]qlhah.]qmhauh0K h1hh]qn(h3)qo}qp(hX9First example :math:`\rightarrow` default inducing pointsqqhhih h#h%h7h'}qr(h)]h*]h+]h,]h.]uh0K h1hh]qs(h:XFirst example qtqu}qv(hXFirst example qwhhoubcsphinx.ext.mathbase math qx)qy}qz(hUh'}q{(UlatexX \rightarrowq|h,]h+]h)]h*]h.]uhhoh]h%Umathq}ubh:X default inducing pointsq~q}q(hX default inducing pointsqhhoubeubh>)q}q(hX,First load the same data as in the GPR demo.qhhih h#h%hAh'}q(h)]h*]h+]h,]h.]uh0K h1hh]qh:X,First load the same data as in the GPR demo.qq}q(hhhhubaubh>)q}q(hX**[Theory]** In case the number of training inputs :math:`x` exceeds a few hundred, approximate inference using Laplace approximation or expectation propagation takes too long. We offer the FITC approximation based on a low-rank plus diagonal approximation to the exact covariance to deal with these cases. The general idea is to use inducing points :math:`u` and to base the computations on cross-covariances between training, test and inducing points only.hhih h#h%hAh'}q(h)]h*]h+]h,]h.]uh0K h1hh]q(cdocutils.nodes strong q)q}q(hX **[Theory]**h'}q(h)]h*]h+]h,]h.]uhhh]qh:X[Theory]qq}q(hUhhubah%Ustrongqubh:X' In case the number of training inputs qq}q(hX' In case the number of training inputs hhubhx)q}q(hUh'}q(UlatexXxh,]h+]h)]h*]h.]uhhh]h%h}ubh:X" exceeds a few hundred, approximate inference using Laplace approximation or expectation propagation takes too long. We offer the FITC approximation based on a low-rank plus diagonal approximation to the exact covariance to deal with these cases. The general idea is to use inducing points qq}q(hX" exceeds a few hundred, approximate inference using Laplace approximation or expectation propagation takes too long. We offer the FITC approximation based on a low-rank plus diagonal approximation to the exact covariance to deal with these cases. The general idea is to use inducing points hhubhx)q}q(hUh'}q(UlatexXuh,]h+]h)]h*]h.]uhhh]h%h}ubh:Xc and to base the computations on cross-covariances between training, test and inducing points only.qq}q(hXc and to base the computations on cross-covariances between training, test and inducing points only.hhubeubh>)q}q(hX(Okay, now the model is FITC regression::qhhih h#h%hAh'}q(h)]h*]h+]h,]h.]uh0Kh1hh]qh:X'Okay, now the model is FITC regression:qq}q(hX'Okay, now the model is FITC regression:hhubaubcdocutils.nodes literal_block q)q}q(hXmodel = pyGPs.GPR_FITC()hhih h#h%U literal_blockqh'}q(U xml:spaceqUpreserveqh,]h+]h)]h*]h.]uh0Kh1hh]qh:Xmodel = pyGPs.GPR_FITC()qq}q(hUhhubaubh>)q}q(hXThe difference between the usage of basic :math:`GP` regression is that we will have to specify inducing points. In the first example here, we will introduce you how to use the default settings.hhih h#h%hAh'}q(h)]h*]h+]h,]h.]uh0Kh1hh]q(h:X*The difference between the usage of basic qq}q(hX*The difference between the usage of basic hhubhx)q}q(hUh'}q(UlatexXGPh,]h+]h)]h*]h.]uhhh]h%h}ubh:X regression is that we will have to specify inducing points. In the first example here, we will introduce you how to use the default settings.qÅq}q(hX regression is that we will have to specify inducing points. In the first example here, we will introduce you how to use the default settings.hhubeubh>)q}q(hXGThe default inducing points are a grid (hypercube for higher dimensions), where each dimension has 5 values in equidistant steps in :math:`[min, max]`, where :math:`min` and :math:`max` are the minimum and maximum values of the input data by default. In order to specify the dimension of input data, we HAVE TO set data first::hhih h#h%hAh'}q(h)]h*]h+]h,]h.]uh0Kh1hh]q(h:XThe default inducing points are a grid (hypercube for higher dimensions), where each dimension has 5 values in equidistant steps in qʅq}q(hXThe default inducing points are a grid (hypercube for higher dimensions), where each dimension has 5 values in equidistant steps in hhubhx)q}q(hUh'}q(UlatexX [min, max]h,]h+]h)]h*]h.]uhhh]h%h}ubh:X, where qЅq}q(hX, where hhubhx)q}q(hUh'}q(UlatexXminh,]h+]h)]h*]h.]uhhh]h%h}ubh:X and qօq}q(hX and hhubhx)q}q(hUh'}q(UlatexXmaxh,]h+]h)]h*]h.]uhhh]h%h}ubh:X are the minimum and maximum values of the input data by default. In order to specify the dimension of input data, we HAVE TO set data first:q܅q}q(hX are the minimum and maximum values of the input data by default. In order to specify the dimension of input data, we HAVE TO set data first:hhubeubh)q}q(hXmodel.setData(x, y)hhih h#h%hh'}q(hhh,]h+]h)]h*]h.]uh0Kh1hh]qh:Xmodel.setData(x, y)qㅁq}q(hUhhubaubh>)q}q(hX@The number of inducing points per axis is :math:`5` per default.qhhih h#h%hAh'}q(h)]h*]h+]h,]h.]uh0Kh1hh]q(h:X*The number of inducing points per axis is q녁q}q(hX*The number of inducing points per axis is hhubhx)q}q(hUh'}q(UlatexX5h,]h+]h)]h*]h.]uhhh]h%h}ubh:X per default.qq}q(hX per default.hhubeubh>)q}q(hX<Now, the regular training and prediction routines follow: ::qhhih h#h%hAh'}q(h)]h*]h+]h,]h.]uh0K!h1hh]qh:X9Now, the regular training and prediction routines follow:qq}q(hX9Now, the regular training and prediction routines follow:hhubaubh)q}q(hX.model.optimize() model.predict(z) model.plot()hhih h#h%hh'}q(hhh,]h+]h)]h*]h.]uh0K#h1hh]qh:X.model.optimize() model.predict(z) model.plot()rr}r(hUhhubaubcdocutils.nodes figure r)r}r(hUhhih h#h%Ufigurerh'}r(UalignrXcenterh,]h+]h)]h*]h.]uh0Nh1hh]r cdocutils.nodes image r )r }r (hXd.. figure:: _images/d3_1.png :height: 600 px :width: 800 px :align: center :scale: 70 % h'}r (UscaleKFh)]UuriX_images/d3_1.pngrh,]h+]UwidthX800pxh*]U candidatesr}rU*jsh.]UheightX600pxuhjh]h%Uimagerubaubh>)r}r(hX\The equidistant default inducing points :math:`u` that are shown in the figure as black x's.rhhih h#h%hAh'}r(h)]h*]h+]h,]h.]uh0K-h1hh]r(h:X(The equidistant default inducing points rr}r(hX(The equidistant default inducing points hjubhx)r}r(hUh'}r(UlatexXuh,]h+]h)]h*]h.]uhjh]h%h}ubh:X+ that are shown in the figure as black x's.rr}r(hX+ that are shown in the figure as black x's.hjubeubh>)r }r!(hXZTo change the number of inducing points per axis just specify a different value per axis::r"hhih h#h%hAh'}r#(h)]h*]h+]h,]h.]uh0K/h1hh]r$h:XYTo change the number of inducing points per axis just specify a different value per axis:r%r&}r'(hXYTo change the number of inducing points per axis just specify a different value per axis:hj ubaubh)r(}r)(hX&model.setData(x, y, value_per_axis=10)hhih h#h%hh'}r*(hhh,]h+]h)]h*]h.]uh0K1h1hh]r+h:X&model.setData(x, y, value_per_axis=10)r,r-}r.(hUhj(ubaubeubh)r/}r0(hUhhh h#h%h&h'}r1(h)]h*]h+]h,]r2hah.]r3hauh0K5h1hh]r4(h3)r5}r6(hX?Second example :math:`\rightarrow` user-defined inducing pointsr7hj/h h#h%h7h'}r8(h)]h*]h+]h,]h.]uh0K5h1hh]r9(h:XSecond example r:r;}r<(hXSecond example r=hj5ubhx)r>}r?(hUh'}r@(UlatexX \rightarrowrAh,]h+]h)]h*]h.]uhj5h]h%h}ubh:X user-defined inducing pointsrBrC}rD(hX user-defined inducing pointsrEhj5ubeubh>)rF}rG(hXAlternatively, a random subset of the training points can be used as inducing points. Note, that there are plenty of methods to set these inducing points. So, in the second example let us use a user-defined set of inducing points.rHhj/h h#h%hAh'}rI(h)]h*]h+]h,]h.]uh0K7h1hh]rJh:XAlternatively, a random subset of the training points can be used as inducing points. Note, that there are plenty of methods to set these inducing points. So, in the second example let us use a user-defined set of inducing points.rKrL}rM(hjHhjFubaubh>)rN}rO(hX7You can pick a set of fixed inducing points by hand: ::rPhj/h h#h%hAh'}rQ(h)]h*]h+]h,]h.]uh0K:h1hh]rRh:X4You can pick a set of fixed inducing points by hand:rSrT}rU(hX4You can pick a set of fixed inducing points by hand:hjNubaubh)rV}rW(hX0u = np.array([[-1], [-0.8], [-0.5], [0.3],[1.]])hj/h h#h%hh'}rX(hhh,]h+]h)]h*]h.]uh0K)r]}r^(hXYou can also use equidistant inducing points :math:`u`, but without the values on the margin of the grid.(i.e. shrinking the range of values) ::r_hj/h h#h%hAh'}r`(h)]h*]h+]h,]h.]uh0K>h1hh]ra(h:X-You can also use equidistant inducing points rbrc}rd(hX-You can also use equidistant inducing points hj]ubhx)re}rf(hUh'}rg(UlatexXuh,]h+]h)]h*]h.]uhj]h]h%h}ubh:XW, but without the values on the margin of the grid.(i.e. shrinking the range of values)rhri}rj(hXW, but without the values on the margin of the grid.(i.e. shrinking the range of values)hj]ubeubh)rk}rl(hXZnum_u = np.fix(x.shape[0]/2) u = np.linspace(-1.3,1.3,num_u).T u = np.reshape(u,(num_u,1))hj/h h#h%hh'}rm(hhh,]h+]h)]h*]h.]uh0K@h1hh]rnh:XZnum_u = np.fix(x.shape[0]/2) u = np.linspace(-1.3,1.3,num_u).T u = np.reshape(u,(num_u,1))rorp}rq(hUhjkubaubh>)rr}rs(hX-Then pass :math:`u` when specifying prior. ::rthj/h h#h%hAh'}ru(h)]h*]h+]h,]h.]uh0KDh1hh]rv(h:X Then pass rwrx}ry(hX Then pass hjrubhx)rz}r{(hUh'}r|(UlatexXuh,]h+]h)]h*]h.]uhjrh]h%h}ubh:X when specifying prior.r}r~}r(hX when specifying prior.hjrubeubh)r}r(hXm = pyGPs.mean.Zero() k = pyGPs.cov.RBFard(log_ell_list=[0.05,0.17], log_sigma=1.) model.setPrior(mean=m, kernel=k, inducing_points=u)hj/h h#h%hh'}r(hhh,]h+]h)]h*]h.]uh0KFh1hh]rh:Xm = pyGPs.mean.Zero() k = pyGPs.cov.RBFard(log_ell_list=[0.05,0.17], log_sigma=1.) model.setPrior(mean=m, kernel=k, inducing_points=u)rr}r(hUhjubaubh>)r}r(hXThe left figure below shows the result of fixed inducing points, and the right figure shows the result for equidistant :math:`u`.rhj/h h#h%hAh'}r(h)]h*]h+]h,]h.]uh0KJh1hh]r(h:XwThe left figure below shows the result of fixed inducing points, and the right figure shows the result for equidistant rr}r(hXwThe left figure below shows the result of fixed inducing points, and the right figure shows the result for equidistant hjubhx)r}r(hUh'}r(UlatexXuh,]h+]h)]h*]h.]uhjh]h%h}ubh:X.r}r(hX.hjubeubj )r}r(hX+.. image:: _images/d3_2.png :width: 45% hj/h h#h%jh'}r(h)]UuriX_images/d3_2.pngrh,]h+]UwidthX45%h*]j}rU*jsh.]uh0Nh1hh]ubj )r}r(hX,.. image:: _images/d3_3.png :width: 45% hj/h h#h%jh'}r(h)]UuriX_images/d3_3.pngrh,]h+]UwidthX45%h*]j}rU*jsh.]uh0Nh1hh]ubh>)r}r(hX**[Theory]** Note that the predictive variance is overestimated outside the support of the inducing inputs. In a multivariate example where densely sampled inducing inputs are infeasible, one can also try to simply use a random subset of the training points.hj/h h#h%hAh'}r(h)]h*]h+]h,]h.]uh0KSh1hh]r(h)r}r(hX **[Theory]**h'}r(h)]h*]h+]h,]h.]uhjh]rh:X[Theory]rr}r(hUhjubah%hubh:X Note that the predictive variance is overestimated outside the support of the inducing inputs. In a multivariate example where densely sampled inducing inputs are infeasible, one can also try to simply use a random subset of the training points.rr}r(hX Note that the predictive variance is overestimated outside the support of the inducing inputs. In a multivariate example where densely sampled inducing inputs are infeasible, one can also try to simply use a random subset of the training points.hjubeubeubh)r}r(hUhhh h#h%h&h'}r(h)]h*]h+]h,]rhah.]rhauh0KZh1hh]r(h3)r}r(hXA bit more things you can dorhjh h#h%h7h'}r(h)]h*]h+]h,]h.]uh0KZh1hh]rh:XA bit more things you can dorr}r(hjhjubaubh>)r}r(hX6Switch to other Inference and Likelihood functions. ::rhjh h#h%hAh'}r(h)]h*]h+]h,]h.]uh0K[h1hh]rh:X3Switch to other Inference and Likelihood functions.rr}r(hX3Switch to other Inference and Likelihood functions.hjubaubh)r}r(hX7model.useInference("EP") model.useLikelihood("Laplace")hjh h#h%hh'}r(hhh,]h+]h)]h*]h.]uh0K]h1hh]rh:X7model.useInference("EP") model.useLikelihood("Laplace")rr}r(hUhjubaubeubeubahUU transformerrNU footnote_refsr}rUrefnamesr}rXfitc classification]rhTasUsymbol_footnotesr]rUautofootnote_refsr]rUsymbol_footnote_refsr]rU citationsr]rh1hU current_linerNUtransform_messagesr]rUreporterrNUid_startrKU autofootnotesr]rU 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q)q}q(h&Uh'hh(hh-Udesc_parameterlistqh/}q(h1]h2]h3]h4]h6]uh8Nh9hh!]q(csphinx.addnodes desc_parameter q)q}q(h&XAh/}q(h1]h2]h3]h4]h6]uh'hh!]qhBXAq}q(h&Uh'hubah-Udesc_parameterqubh)q}q(h&X alpha=0.5h/}q(h1]h2]h3]h4]h6]uh'hh!]qhBX alpha=0.5q܅q}q(h&Uh'hubah-hubeubcsphinx.addnodes only q)q}q(h&Uh'hh(Nh-Uonlyqh/}q(UexprUhtmlqh4]h3]h1]h2]h6]uh8Nh9hh!]qcsphinx.addnodes pending_xref q)q}q(h&Uh/}q(UreftypeUviewcodeqUrefdocXGraphqU refdomainUstdqh4]h3]U refexplicith1]h2]h6]U reftargetX*_modules/pyGPs/GraphExtensions/nodeKernelsUrefidhuh'hh!]qcdocutils.nodes inline q)q}q(h&Uh/}q(h1]h2]qU viewcode-linkqah3]h4]h6]uh'hh!]qhBX[source]qq}q(h&Uh'hubah-Uinlinequbah-U pending_xrefqubaubeubcsphinx.addnodes desc_content q)q}q(h&Uh'hh(hh-U desc_contentqh/}q(h1]h2]h3]h4]h6]uh8Nh9hh!]q(hF)r}r(h&XWVon Neumann Diffusion Kernel on graph (Zhou et al., 2004) (also label spreading kernel)rh'hh(X/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/GraphExtensions/nodeKernels.py:docstring of 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Parametersrkrl}rm(h&Uh'jgubah-j$ubj%)rn}ro(h&Uh/}rp(h1]h2]h3]h4]h6]uh'jch!]rqj*)rr}rs(h&Uh/}rt(h1]h2]h3]h4]h6]uh'jnh!]ru(j/)rv}rw(h&Uh/}rx(h1]h2]h3]h4]h6]uh'jrh!]ryhF)rz}r{(h&Uh/}r|(h1]h2]h3]h4]h6]uh'jvh!]r}(j8)r~}r(h&XAh/}r(h1]h2]h3]h4]h6]uh'jzh!]rhBXAr}r(h&Uh'j~ubah-j?ubhBX -- rr}r(h&Uh'jzubhBXadjacency matrixrr}r(h&Xadjacency matrixh'jzubeh-hJubah-jFubj/)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jrh!]rhF)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j8)r}r(h&Xbetah/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXbetarr}r(h&Uh'jubah-j?ubhBX -- rr}r(h&Uh'jubhBXhyperparameter betarr}r(h&Xhyperparameter betah'jubeh-hJubah-jFubeh-j\ubah-j]ubeh-j^ubj)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'j_h!]r(j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXReturnsrr}r(h&Uh'jubah-j$ubj%)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhF)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX kernel matrixrr}r(h&X kernel matrixh'jubah-hJubah-j]ubeh-j^ubeubeubeubh)r}r(h&Uh'hh(Nh-hh/}r(h4]h3]h1]h2]h6]Uentries]r(hX7normLap() (in module pyGPs.GraphExtensions.nodeKernels)h 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pyGPs.GraphExtensions.nodeKernels.normLaph-hJh/}r(h1]h2]h3]h4]h6]uh8Kh9hh!]rhBXNormalized Laplacianrr}r(h&jh'jubaubj)r}r(h&Uh'jh(Nh-jh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]r(j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX Parametersr r }r (h&Uh'jubah-j$ubj%)r }r (h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhF)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'j h!]r(j8)r}r(h&XAh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXAr}r(h&Uh'jubah-j?ubhBX -- rr}r(h&Uh'jubhBXadjacency matrixrr}r(h&Xadjacency matrixh'jubeh-hJubah-j]ubeh-j^ubj)r }r!(h&Uh/}r"(h1]h2]h3]h4]h6]uh'jh!]r#(j)r$}r%(h&Uh/}r&(h1]h2]h3]h4]h6]uh'j h!]r'hBXReturnsr(r)}r*(h&Uh'j$ubah-j$ubj%)r+}r,(h&Uh/}r-(h1]h2]h3]h4]h6]uh'j h!]r.hF)r/}r0(h&Uh/}r1(h1]h2]h3]h4]h6]uh'j+h!]r2hBX kernel matrixr3r4}r5(h&X kernel matrixh'j/ubah-hJubah-j]ubeh-j^ubeubeubeubh)r6}r7(h&Uh'hh(Nh-hh/}r8(h4]h3]h1]h2]h6]Uentries]r9(hX>psInvLapKernel() (in module pyGPs.GraphExtensions.nodeKernels)hUtr:auh8Nh9hh!]ubh)r;}r<(h&Uh'hh(Nh-hh/}r=(hhXpyh4]h3]h1]h2]h6]hXfunctionr>hj>uh8Nh9hh!]r?(h)r@}rA(h&XpsInvLapKernel(A)h'j;h(hh-hh/}rB(h4]rChahh)X!pyGPs.GraphExtensions.nodeKernelsrDrE}rFbh3]h1]h2]h6]rGhahXpsInvLapKernelrHhUhuh8Nh9hh!]rI(h)rJ}rK(h&X"pyGPs.GraphExtensions.nodeKernels.h'j@h(hh-hh/}rL(h1]h2]h3]h4]h6]uh8Nh9hh!]rMhBX"pyGPs.GraphExtensions.nodeKernels.rNrO}rP(h&Uh'jJubaubh)rQ}rR(h&jHh'j@h(hh-hh/}rS(h1]h2]h3]h4]h6]uh8Nh9hh!]rThBXpsInvLapKernelrUrV}rW(h&Uh'jQubaubh)rX}rY(h&Uh'j@h(hh-hh/}rZ(h1]h2]h3]h4]h6]uh8Nh9hh!]r[h)r\}r](h&XAh/}r^(h1]h2]h3]h4]h6]uh'jXh!]r_hBXAr`}ra(h&Uh'j\ubah-hubaubh)rb}rc(h&Uh'j@h(Nh-hh/}rd(Uexprhh4]h3]h1]h2]h6]uh8Nh9hh!]reh)rf}rg(h&Uh/}rh(UreftypehUrefdochU refdomainhh4]h3]U refexplicith1]h2]h6]U reftargetX*_modules/pyGPs/GraphExtensions/nodeKernelsUrefidjHuh'jbh!]rih)rj}rk(h&Uh/}rl(h1]h2]rmhah3]h4]h6]uh'jfh!]rnhBX[source]rorp}rq(h&Uh'jjubah-hubah-hubaubeubh)rr}rs(h&Uh'j;h(hh-hh/}rt(h1]h2]h3]h4]h6]uh8Nh9hh!]ru(hF)rv}rw(h&X+Pseudo inverse of the normalized Laplacian.rxh'jrh(X/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/GraphExtensions/nodeKernels.py:docstring of pyGPs.GraphExtensions.nodeKernels.psInvLapKernelh-hJh/}ry(h1]h2]h3]h4]h6]uh8Kh9hh!]rzhBX+Pseudo inverse of the normalized Laplacian.r{r|}r}(h&jxh'jvubaubj)r~}r(h&Uh'jrh(Nh-jh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]r(j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'j~h!]r(j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX Parametersrr}r(h&Uh'jubah-j$ubj%)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhF)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j8)r}r(h&XAh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXAr}r(h&Uh'jubah-j?ubhBX -- rr}r(h&Uh'jubhBXadjacency matrixrr}r(h&Xadjacency matrixh'jubeh-hJubah-j]ubeh-j^ubj)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'j~h!]r(j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXReturnsrr}r(h&Uh'jubah-j$ubj%)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhF)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX kernel matrixrr}r(h&X kernel matrixh'jubah-hJubah-j]ubeh-j^ubeubeubeubh)r}r(h&Uh'hh(Nh-hh/}r(h4]h3]h1]h2]h6]Uentries]r(hX<regLapKernel() (in module pyGPs.GraphExtensions.nodeKernels)hUtrauh8Nh9hh!]ubh)r}r(h&Uh'hh(Nh-hh/}r(hhXpyh4]h3]h1]h2]h6]hXfunctionrhjuh8Nh9hh!]r(h)r}r(h&XregLapKernel(A, sigma=1)h'jh(hh-hh/}r(h4]rhahh)X!pyGPs.GraphExtensions.nodeKernelsrr}rbh3]h1]h2]h6]rhahX regLapKernelrhUhuh8Nh9hh!]r(h)r}r(h&X"pyGPs.GraphExtensions.nodeKernels.h'jh(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]rhBX"pyGPs.GraphExtensions.nodeKernels.rr}r(h&Uh'jubaubh)r}r(h&jh'jh(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]rhBX regLapKernelrr}r(h&Uh'jubaubh)r}r(h&Uh'jh(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]r(h)r}r(h&XAh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXAr}r(h&Uh'jubah-hubh)r}r(h&Xsigma=1h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXsigma=1rr}r(h&Uh'jubah-hubeubh)r}r(h&Uh'jh(Nh-hh/}r(Uexprhh4]h3]h1]h2]h6]uh8Nh9hh!]rh)r}r(h&Uh/}r(UreftypehUrefdochU refdomainhh4]h3]U refexplicith1]h2]h6]U reftargetX*_modules/pyGPs/GraphExtensions/nodeKernelsUrefidjuh'jh!]rh)r}r(h&Uh/}r(h1]h2]rhah3]h4]h6]uh'jh!]rhBX[source]rr}r(h&Uh'jubah-hubah-hubaubeubh)r}r(h&Uh'jh(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]r(hF)r}r(h&XRegularized Laplacian Kernelrh'jh(X/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/GraphExtensions/nodeKernels.py:docstring of pyGPs.GraphExtensions.nodeKernels.regLapKernelh-hJh/}r(h1]h2]h3]h4]h6]uh8Kh9hh!]rhBXRegularized Laplacian Kernelrr}r(h&jh'jubaubj)r}r(h&Uh'jh(Nh-jh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]r (j)r }r (h&Uh/}r (h1]h2]h3]h4]h6]uh'jh!]r (j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'j h!]rhBX Parametersrr}r(h&Uh'jubah-j$ubj%)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'j h!]rj*)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j/)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r hF)r!}r"(h&Uh/}r#(h1]h2]h3]h4]h6]uh'jh!]r$(j8)r%}r&(h&XAh/}r'(h1]h2]h3]h4]h6]uh'j!h!]r(hBXAr)}r*(h&Uh'j%ubah-j?ubhBX -- r+r,}r-(h&Uh'j!ubhBXadjacency matrixr.r/}r0(h&Xadjacency matrixh'j!ubeh-hJubah-jFubj/)r1}r2(h&Uh/}r3(h1]h2]h3]h4]h6]uh'jh!]r4hF)r5}r6(h&Uh/}r7(h1]h2]h3]h4]h6]uh'j1h!]r8(j8)r9}r:(h&Xsigmah/}r;(h1]h2]h3]h4]h6]uh'j5h!]r<hBXsigmar=r>}r?(h&Uh'j9ubah-j?ubhBX -- r@rA}rB(h&Uh'j5ubhBXhyperparameter sigmarCrD}rE(h&Xhyperparameter sigmah'j5ubeh-hJubah-jFubeh-j\ubah-j]ubeh-j^ubj)rF}rG(h&Uh/}rH(h1]h2]h3]h4]h6]uh'jh!]rI(j)rJ}rK(h&Uh/}rL(h1]h2]h3]h4]h6]uh'jFh!]rMhBXReturnsrNrO}rP(h&Uh'jJubah-j$ubj%)rQ}rR(h&Uh/}rS(h1]h2]h3]h4]h6]uh'jFh!]rThF)rU}rV(h&Uh/}rW(h1]h2]h3]h4]h6]uh'jQh!]rXhBX kernel matrixrYrZ}r[(h&X kernel matrixh'jUubah-hJubah-j]ubeh-j^ubeubeubeubh)r\}r](h&Uh'hh(Nh-hh/}r^(h4]h3]h1]h2]h6]Uentries]r_(hX8rwKernel() (in module pyGPs.GraphExtensions.nodeKernels)hUtr`auh8Nh9hh!]ubh)ra}rb(h&Uh'hh(Nh-hh/}rc(hhXpyh4]h3]h1]h2]h6]hXfunctionrdhjduh8Nh9hh!]re(h)rf}rg(h&XrwKernel(A, p=1, a=2)h'jah(hh-hh/}rh(h4]rihahh)X!pyGPs.GraphExtensions.nodeKernelsrjrk}rlbh3]h1]h2]h6]rmhahXrwKernelrnhUhuh8Nh9hh!]ro(h)rp}rq(h&X"pyGPs.GraphExtensions.nodeKernels.h'jfh(hh-hh/}rr(h1]h2]h3]h4]h6]uh8Nh9hh!]rshBX"pyGPs.GraphExtensions.nodeKernels.rtru}rv(h&Uh'jpubaubh)rw}rx(h&jnh'jfh(hh-hh/}ry(h1]h2]h3]h4]h6]uh8Nh9hh!]rzhBXrwKernelr{r|}r}(h&Uh'jwubaubh)r~}r(h&Uh'jfh(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]r(h)r}r(h&XAh/}r(h1]h2]h3]h4]h6]uh'j~h!]rhBXAr}r(h&Uh'jubah-hubh)r}r(h&Xp=1h/}r(h1]h2]h3]h4]h6]uh'j~h!]rhBXp=1rr}r(h&Uh'jubah-hubh)r}r(h&Xa=2h/}r(h1]h2]h3]h4]h6]uh'j~h!]rhBXa=2rr}r(h&Uh'jubah-hubeubh)r}r(h&Uh'jfh(Nh-hh/}r(Uexprhh4]h3]h1]h2]h6]uh8Nh9hh!]rh)r}r(h&Uh/}r(UreftypehUrefdochU refdomainhh4]h3]U refexplicith1]h2]h6]U reftargetX*_modules/pyGPs/GraphExtensions/nodeKernelsUrefidjnuh'jh!]rh)r}r(h&Uh/}r(h1]h2]rhah3]h4]h6]uh'jh!]rhBX[source]rr}r(h&Uh'jubah-hubah-hubaubeubh)r}r(h&Uh'jah(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]r(hF)r}r(h&X"p-step Random Walk Kernel with a>1rh'jh(X/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/GraphExtensions/nodeKernels.py:docstring of pyGPs.GraphExtensions.nodeKernels.rwKernelrh-hJh/}r(h1]h2]h3]h4]h6]uh8Kh9hh!]rhBX"p-step Random Walk Kernel with a>1rr}r(h&jh'jubaubhF)r}r(h&X3K = (aI-L)^p, p>1 and L is the normalized Laplacianrh'jh(jh-hJh/}r(h1]h2]h3]h4]h6]uh8Kh9hh!]rhBX3K = (aI-L)^p, p>1 and L is the normalized Laplacianrr}r(h&jh'jubaubj)r}r(h&Uh'jh(Nh-jh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]r(j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX Parametersrr}r(h&Uh'jubah-j$ubj%)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rj*)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j/)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhF)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j8)r}r(h&XAh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXAr}r(h&Uh'jubah-j?ubhBX -- rr}r(h&Uh'jubhBXadjacency matrixrr}r(h&Xadjacency matrixrh'jubeh-hJubah-jFubj/)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhF)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j8)r}r(h&Xph/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXpr}r(h&Uh'jubah-j?ubhBX -- rr}r(h&Uh'jubhBXstep parameterrr}r(h&Xstep parameterrh'jubeh-hJubah-jFubj/)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhF)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j8)r}r(h&Xah/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXar}r (h&Uh'jubah-j?ubhBX -- r r }r (h&Uh'jubhBXhyperparameter ar r}r(h&Xhyperparameter arh'jubeh-hJubah-jFubeh-j\ubah-j]ubeh-j^ubj)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXReturnsrr}r(h&Uh'jubah-j$ubj%)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhF)r }r!(h&Uh/}r"(h1]h2]h3]h4]h6]uh'jh!]r#hBX kernel matrixr$r%}r&(h&X kernel matrixr'h'j ubah-hJubah-j]ubeh-j^ubeubeubeubeubh#)r(}r)(h&Uh'h$h(h+h-h.h/}r*(h1]h2]h3]h4]r+(X)module-pyGPs.GraphExtensions.graphKernelsr,h eh6]r-h auh8Kh9hh!]r.(h;)r/}r0(h&X Graph Kernelsr1h'j(h(h+h-h?h/}r2(h1]h2]h3]h4]h6]uh8Kh9hh!]r3hBX Graph Kernelsr4r5}r6(h&j1h'j/ubaubh)r7}r8(h&Uh'j(h(hh-hh/}r9(h4]h3]h1]h2]h6]Uentries]r:(hX+pyGPs.GraphExtensions.graphKernels (module)X)module-pyGPs.GraphExtensions.graphKernelsUtr;auh8Kh9hh!]ubh)r<}r=(h&Uh'j(h(Nh-hh/}r>(h4]h3]h1]h2]h6]Uentries]r?(hXBpropagationKernel() (in module pyGPs.GraphExtensions.graphKernels)hUtr@auh8Nh9hh!]ubh)rA}rB(h&Uh'j(h(Nh-hh/}rC(hhXpyh4]h3]h1]h2]h6]hXfunctionrDhjDuh8Nh9hh!]rE(h)rF}rG(h&X`propagationKernel(A, l, gr_id, h_max, w, p, ktype=None, SUM=True, VIS=False, 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In: P.A. Flach, T.D. Bie, N. Cristianini (eds.) ECML/PKDD, Notes in Computer Science, vol. 7523, pp. 378-393. Springer (2012).rh'jh(X/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/GraphExtensions/graphKernels.py:docstring of pyGPs.GraphExtensions.graphKernels.propagationKernelrh-hJh/}r(h1]h2]h3]h4]h6]uh8Kh9hh!]rhBXPropagation kernel for graphs as described in: Neumann, M., Patricia, N., Garnett, R., Kersting, K.: Efficient Graph Kernels by Randomization. In: P.A. Flach, T.D. Bie, N. Cristianini (eds.) ECML/PKDD, Notes in Computer Science, vol. 7523, pp. 378-393. 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kernels-meansq9hhhhhhhhhUsimple-kernel-meanq:hhhhhhhhhhhhhhhU$list-of-means-and-default-parametersq;hhhUprecomputed-kernel-matrixqh*U&list-of-kernels-and-default-parametersq?h+h+h,h,h-h-h.h.uUchildrenq@]qAcdocutils.nodes section qB)qC}qD(U rawsourceqEUUparentqFhUsourceqGcdocutils.nodes reprunicode qHXB/Users/marion/workspace/workspace_GPs/pyGPs/doc/source/Kernels.rstqIqJ}qKbUtagnameqLUsectionqMU attributesqN}qO(UdupnamesqP]UclassesqQ]UbackrefsqR]UidsqS]qTh9aUnamesqU]qVhauUlineqWKUdocumentqXhh@]qY(cdocutils.nodes title qZ)q[}q\(hEXKernels & Meansq]hFhChGhJhLUtitleq^hN}q_(hP]hQ]hR]hS]hU]uhWKhXhh@]q`cdocutils.nodes Text qaXKernels & Meansqbqc}qd(hEh]hFh[ubaubhB)qe}qf(hEUhFhChGhJhLhMhN}qg(hP]hQ]hR]hS]qhh:ahU]qihauhWKhXhh@]qj(hZ)qk}ql(hEXSimple Kernel & MeanqmhFhehGhJhLh^hN}qn(hP]hQ]hR]hS]hU]uhWKhXhh@]qohaXSimple Kernel & Meanqpqq}qr(hEhmhFhkubaubcdocutils.nodes paragraph qs)qt}qu(hEX]You may already seen, we can specify a kernel function like this(same for mean fucntions): ::qvhFhehGhJhLU paragraphqwhN}qx(hP]hQ]hR]hS]hU]uhWKhXhh@]qyhaXZYou may already seen, we can specify a kernel function like this(same for mean fucntions):qzq{}q|(hEXZYou may already seen, we can specify a kernel function like this(same for mean fucntions):hFhtubaubcdocutils.nodes literal_block q})q~}q(hEX.k = pyGPs.cov.RBF( log_ell=-1., log_sigma=0. )hFhehGhJhLU literal_blockqhN}q(U xml:spaceqUpreserveqhS]hR]hP]hQ]hU]uhWKhXhh@]qhaX.k = pyGPs.cov.RBF( log_ell=-1., log_sigma=0. )qq}q(hEUhFh~ubaubhs)q}q(hEX,There are several points need to be noticed:qhFhehGhJhLhwhN}q(hP]hQ]hR]hS]hU]uhWK hXhh@]qhaX,There are several points need to be noticed:qq}q(hEhhFhubaubcdocutils.nodes enumerated_list q)q}q(hEUhFhehGhJhLUenumerated_listqhN}q(UsuffixqU.hS]hR]hP]UprefixqUhQ]hU]UenumtypeqUarabicquhWK hXhh@]q(cdocutils.nodes list_item q)q}q(hEXMost parameters are initilized in their logorithms. This is because we need to make sure they are positive during optimization. e.g. Here length scale and signal variance should always be positive. hFhhGhJhLU list_itemqhN}q(hP]hQ]hR]hS]hU]uhWNhXhh@]qhs)q}q(hEXMost parameters are initilized in their logorithms. This is because we need to make sure they are positive during optimization. e.g. Here length scale and signal variance should always be positive.qhFhhGhJhLhwhN}q(hP]hQ]hR]hS]hU]uhWK h@]qhaXMost parameters are initilized in their logorithms. This is because we need to make sure they are positive during optimization. e.g. Here length scale and signal variance should always be positive.qq}q(hEhhFhubaubaubh)q}q(hEXWMost kernel functions have a scalar in front, namely signal variance(set by log_sigma) hFhhGhJhLhhN}q(hP]hQ]hR]hS]hU]uhWNhXhh@]qhs)q}q(hEXVMost kernel functions have a scalar in front, namely signal variance(set by log_sigma)qhFhhGhJhLhwhN}q(hP]hQ]hR]hS]hU]uhWKh@]qhaXVMost kernel functions have a scalar in front, namely signal variance(set by log_sigma)qq}q(hEhhFhubaubaubh)q}q(hEXWIf you will do optimization later anyway, you can just leave parameters to be default hFhhGhJhLhhN}q(hP]hQ]hR]hS]hU]uhWNhXhh@]qhs)q}q(hEXUIf you will do optimization later anyway, you can just leave parameters to be defaultqhFhhGhJhLhwhN}q(hP]hQ]hR]hS]hU]uhWKh@]qhaXUIf you will do optimization later anyway, you can just leave parameters to be defaultqq}q(hEhhFhubaubaubeubeubhB)q}q(hEUhFhChGhJhLhMhN}q(hP]hQ]hR]hS]qh=ahU]qh!auhWKhXhh@]q(hZ)q}q(hEXSome Special CasesqhFhhGhJhLh^hN}q(hP]hQ]hR]hS]hU]uhWKhXhh@]qhaXSome Special Casesq˅q}q(hEhhFhubaubh)q}q(hEUhFhhGhJhLhhN}q(hU.hS]hR]hP]hUhQ]hU]hhuhWKhXhh@]q(h)q}q(hEXFor some kernels/means, number of hyperparameters depends on the dimension of input data. You can either enter the dimension, which use default values: :: m = pyGPs.mean.Linear( D=x.shape[1] ) or you can initialze with the exact hyperparameters, you should enter as a list, one element for each dimension :: m = pyGPs.mean.Linear( alpha_list=[0.2, 0.4, 0.3] ) All these "hyp-dim-dependent" functions are: * *pyGPs.mean.Linear* * *pyGPs.cov.RBFard* * *pyGPs.cov.LINard* * *pyGPs.cov.RQard* hFhhGNhLhhN}q(hP]hQ]hR]hS]hU]uhWNhXhh@]q(hs)q}q(hEXFor some kernels/means, number of hyperparameters depends on the dimension of input data. You can either enter the dimension, which use default values: ::hFhhGhJhLhwhN}q(hP]hQ]hR]hS]hU]uhWKh@]qhaXFor some kernels/means, number of hyperparameters depends on the dimension of input data. You can either enter the dimension, which use default values:qڅq}q(hEXFor some kernels/means, number of hyperparameters depends on the dimension of input data. You can either enter the dimension, which use default values:hFhubaubh})q}q(hEX%m = pyGPs.mean.Linear( D=x.shape[1] )hFhhLhhN}q(hhhS]hR]hP]hQ]hU]uhWKh@]qhaX%m = pyGPs.mean.Linear( D=x.shape[1] )qᅁq}q(hEUhFhubaubhs)q}q(hEXror you can initialze with the exact hyperparameters, you should enter as a list, one element for each dimension ::hFhhGhJhLhwhN}q(hP]hQ]hR]hS]hU]uhWKh@]qhaXoor you can initialze with the exact hyperparameters, you should enter as a list, one element for each dimensionq腁q}q(hEXoor you can initialze with the exact hyperparameters, you should enter as a list, one element for each dimensionhFhubaubh})q}q(hEX3m = pyGPs.mean.Linear( alpha_list=[0.2, 0.4, 0.3] )hFhhLhhN}q(hhhS]hR]hP]hQ]hU]uhWKh@]qhaX3m = pyGPs.mean.Linear( alpha_list=[0.2, 0.4, 0.3] )qq}q(hEUhFhubaubcdocutils.nodes definition_list q)q}q(hEUhN}q(hP]hQ]hR]hS]hU]uhFhh@]qcdocutils.nodes definition_list_item q)q}q(hEXAll these "hyp-dim-dependent" functions are: * *pyGPs.mean.Linear* * *pyGPs.cov.RBFard* * *pyGPs.cov.LINard* * *pyGPs.cov.RQard* hFhhGhJhLUdefinition_list_itemqhN}q(hP]hQ]hR]hS]hU]uhWK$h@]q(cdocutils.nodes term q)q}q(hEX,All these "hyp-dim-dependent" functions are:rhFhhGhJhLUtermrhN}r(hP]hQ]hR]hS]hU]uhWK$h@]rhaX,All these "hyp-dim-dependent" functions are:rr}r(hEjhFhubaubcdocutils.nodes definition r)r}r (hEUhN}r (hP]hQ]hR]hS]hU]uhFhh@]r cdocutils.nodes bullet_list r )r }r(hEUhN}r(UbulletrX*hS]hR]hP]hQ]hU]uhFjh@]r(h)r}r(hEX*pyGPs.mean.Linear*rhN}r(hP]hQ]hR]hS]hU]uhFj h@]rhs)r}r(hEjhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWK h@]rcdocutils.nodes emphasis r)r}r(hEjhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXpyGPs.mean.Linearr r!}r"(hEUhFjubahLUemphasisr#ubaubahLhubh)r$}r%(hEX*pyGPs.cov.RBFard*r&hN}r'(hP]hQ]hR]hS]hU]uhFj h@]r(hs)r)}r*(hEj&hFj$hGhJhLhwhN}r+(hP]hQ]hR]hS]hU]uhWK!h@]r,j)r-}r.(hEj&hN}r/(hP]hQ]hR]hS]hU]uhFj)h@]r0haXpyGPs.cov.RBFardr1r2}r3(hEUhFj-ubahLj#ubaubahLhubh)r4}r5(hEX*pyGPs.cov.LINard*r6hN}r7(hP]hQ]hR]hS]hU]uhFj h@]r8hs)r9}r:(hEj6hFj4hGhJhLhwhN}r;(hP]hQ]hR]hS]hU]uhWK"h@]r<j)r=}r>(hEj6hN}r?(hP]hQ]hR]hS]hU]uhFj9h@]r@haXpyGPs.cov.LINardrArB}rC(hEUhFj=ubahLj#ubaubahLhubh)rD}rE(hEX*pyGPs.cov.RQard* hN}rF(hP]hQ]hR]hS]hU]uhFj h@]rGhs)rH}rI(hEX*pyGPs.cov.RQard*rJhFjDhGhJhLhwhN}rK(hP]hQ]hR]hS]hU]uhWK#h@]rLj)rM}rN(hEjJhN}rO(hP]hQ]hR]hS]hU]uhFjHh@]rPhaXpyGPs.cov.RQardrQrR}rS(hEUhFjMubahLj#ubaubahLhubehLU bullet_listrTubahLU definitionrUubeubahLUdefinition_listrVubeubh)rW}rX(hEXFor *pyGPs.cov.RBFunit()*, its signal variance is always 1 (because of unit magnitude). Therefore this function do not have a hyperparameter of "signal variance". hFhhGhJhLhhN}rY(hP]hQ]hR]hS]hU]uhWNhXhh@]rZhs)r[}r\(hEXFor *pyGPs.cov.RBFunit()*, its signal variance is always 1 (because of unit magnitude). Therefore this function do not have a hyperparameter of "signal variance".hFjWhGhJhLhwhN}r](hP]hQ]hR]hS]hU]uhWK&h@]r^(haXFor r_r`}ra(hEXFor hFj[ubj)rb}rc(hEX*pyGPs.cov.RBFunit()*hN}rd(hP]hQ]hR]hS]hU]uhFj[h@]rehaXpyGPs.cov.RBFunit()rfrg}rh(hEUhFjbubahLj#ubhaX, its signal variance is always 1 (because of unit magnitude). Therefore this function do not have a hyperparameter of "signal variance".rirj}rk(hEX, its signal variance is always 1 (because of unit magnitude). Therefore this function do not have a hyperparameter of "signal variance".hFj[ubeubaubh)rl}rm(hEX*pyGPs.cov.Poly()* has three parameters, where hyperparameters are: * c -> inhomogeneous offset * sigma -> signal deviation however, * d -> order of polynomial will be treated as normal parameter, i.e. will not be trained hFhhGNhLhhN}rn(hP]hQ]hR]hS]hU]uhWNhXhh@]roh)rp}rq(hEUhN}rr(hP]hQ]hR]hS]hU]uhFjlh@]rs(h)rt}ru(hEX*pyGPs.cov.Poly()* has three parameters, where hyperparameters are: * c -> inhomogeneous offset * sigma -> signal deviation hFjphGhJhLhhN}rv(hP]hQ]hR]hS]hU]uhWK+h@]rw(h)rx}ry(hEXC*pyGPs.cov.Poly()* has three parameters, where hyperparameters are:hFjthGhJhLjhN}rz(hP]hQ]hR]hS]hU]uhWK+h@]r{(j)r|}r}(hEX*pyGPs.cov.Poly()*hN}r~(hP]hQ]hR]hS]hU]uhFjxh@]rhaXpyGPs.cov.Poly()rr}r(hEUhFj|ubahLj#ubhaX1 has three parameters, where hyperparameters are:rr}r(hEX1 has three parameters, where hyperparameters are:hFjxubeubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjth@]rj )r}r(hEUhN}r(jX*hS]hR]hP]hQ]hU]uhFjh@]r(h)r}r(hEXc -> inhomogeneous offsetrhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEjhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWK*h@]rhaXc -> inhomogeneous offsetrr}r(hEjhFjubaubahLhubh)r}r(hEXsigma -> signal deviation hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEXsigma -> signal deviationrhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWK+h@]rhaXsigma -> signal deviationrr}r(hEjhFjubaubahLhubehLjTubahLjUubeubh)r}r(hEXihowever, * d -> order of polynomial will be treated as normal parameter, i.e. will not be trained hFjphGhJhLhhN}r(hP]hQ]hR]hS]hU]uhWK0h@]r(h)r}r(hEXhowever,rhFjhGhJhLjhN}r(hP]hQ]hR]hS]hU]uhWK0h@]rhaXhowever,rr}r(hEjhFjubaubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rj )r}r(hEUhN}r(jX*hS]hR]hP]hQ]hU]uhFjh@]rh)r}r(hEX\d -> order of polynomial will be treated as normal parameter, i.e. will not be trained hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEXZd -> order of polynomial will be treated as normal parameter, i.e. will not be trainedrhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWK.h@]rhaXZd -> order of polynomial will be treated as normal parameter, i.e. will not be trainedrr}r(hEjhFjubaubahLhubahLjTubahLjUubeubehLjVubaubh)r}r(hEXcExplicitly set *pyGPs.cov.Noise* is not necessary, because noise are already added in likelihood. hFhhGhJhLhhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhs)r}r(hEXaExplicitly set *pyGPs.cov.Noise* is not necessary, because noise are already added in likelihood.hFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWK2h@]r(haXExplicitly set rr}r(hEXExplicitly set hFjubj)r}r(hEX*pyGPs.cov.Noise*hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXpyGPs.cov.Noiserr}r(hEUhFjubahLj#ubhaXA is not necessary, because noise are already added in likelihood.rr}r(hEXA is not necessary, because noise are already added in likelihood.hFjubeubaubeubeubhB)r}r(hEUhFhChGhJhLhMhN}r(hP]hQ]hR]hS]rh8ahU]rh auhWK6hXhh@]r(hZ)r}r(hEXComposite Kernels & MeamsrhFjhGhJhLh^hN}r(hP]hQ]hR]hS]hU]uhWK6hXhh@]rhaXComposite Kernels & Meamsrr}r(hEjhFjubaubhs)r}r(hEX9Adding and muliplying Kernels(Means) is really simple: ::rhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWK7hXhh@]rhaX6Adding and muliplying Kernels(Means) is really simple:rr}r(hEX6Adding and muliplying Kernels(Means) is really simple:hFjubaubh})r}r(hEXYk = pyGPs.cov.Periodic() * pyGPs.cov.RBF() k = 0.5*pyGPs.cov.LIN() + pyGPs.cov.Periodic()hFjhGhJhLhhN}r(hhhS]hR]hP]hQ]hU]uhWK9hXhh@]rhaXYk = pyGPs.cov.Periodic() * pyGPs.cov.RBF() k = 0.5*pyGPs.cov.LIN() + pyGPs.cov.Periodic()rr}r(hEUhFjubaubhs)r}r(hEXScalar will also be treated as a hyperparameter. For example, k = s1 * k1 + s2 * k2, then the list of hyperparameters is hyp = [s1, k1.hyp, s2, k2.hyp]. Scalar is passed in logorithm domain such that it will always be positive during optimization.rhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWK(h)r?}r@(hEXM1 is a matrix with shape **number of training points plus 1** by **number of test points** - cross covariances matrix (train by test) - last row is self covariances (diagonal of test by test)hFj;hGhJhLhhN}rA(hP]hQ]hR]hS]hU]uhWKPh@]rB(h)rC}rD(hEX[M1 is a matrix with shape **number of training points plus 1** by **number of test points**rEhFj?hGhJhLjhN}rF(hP]hQ]hR]hS]hU]uhWKPh@]rG(haXM1 is a matrix with shape rHrI}rJ(hEXM1 is a matrix with shape hFjCubcdocutils.nodes strong rK)rL}rM(hEX$**number of training points plus 1**hN}rN(hP]hQ]hR]hS]hU]uhFjCh@]rOhaX number of training points plus 1rPrQ}rR(hEUhFjLubahLUstrongrSubhaX by rTrU}rV(hEX by hFjCubjK)rW}rX(hEX**number of test points**hN}rY(hP]hQ]hR]hS]hU]uhFjCh@]rZhaXnumber of test pointsr[r\}r](hEUhFjWubahLjSubeubj)r^}r_(hEUhN}r`(hP]hQ]hR]hS]hU]uhFj?h@]raj )rb}rc(hEUhN}rd(jX-hS]hR]hP]hQ]hU]uhFj^h@]re(h)rf}rg(hEX(cross covariances matrix (train by test)rhhN}ri(hP]hQ]hR]hS]hU]uhFjbh@]rjhs)rk}rl(hEjhhFjfhGhJhLhwhN}rm(hP]hQ]hR]hS]hU]uhWKPh@]rnhaX(cross covariances matrix (train by test)rorp}rq(hEjhhFjkubaubahLhubh)rr}rs(hEX7last row is self covariances (diagonal of test by test)rthN}ru(hP]hQ]hR]hS]hU]uhFjbh@]rvhs)rw}rx(hEjthFjrhGhJhLhwhN}ry(hP]hQ]hR]hS]hU]uhWKQh@]rzhaX7last row is self covariances (diagonal of test by test)r{r|}r}(hEjthFjwubaubahLhubehLjTubahLjUubeubh)r~}r(hEX~M2 is a square matrix with **number of training points** for each dimension - training set covariance matrix (train by train) hFj;hGhJhLhhN}r(hP]hQ]hR]hS]hU]uhWKShXhh@]r(h)r}r(hEXKM2 is a square matrix with **number of training points** for each dimensionrhFj~hGhJhLjhN}r(hP]hQ]hR]hS]hU]uhWKSh@]r(haXM2 is a square matrix with rr}r(hEXM2 is a square matrix with hFjubjK)r}r(hEX**number of training points**hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXnumber of training pointsrr}r(hEUhFjubahLjSubhaX for each dimensionrr}r(hEX for each dimensionhFjubeubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFj~h@]rj )r}r(hEUhN}r(jX-hS]hR]hP]hQ]hU]uhFjh@]rh)r}r(hEX0training set covariance matrix (train by train) hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEX/training set covariance matrix (train by train)rhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWKSh@]rhaX/training set covariance matrix (train by train)rr}r(hEjhFjubaubahLhubahLjTubahLjUubeubeubhs)r}r(hEXzA precomputed kernel can also be composited with other kernels. You need to explictly add scalar for *pyGPs.cov.Pre()*. ::rhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWKUhXhh@]r(haXeA precomputed kernel can also be composited with other kernels. You need to explictly add scalar for rr}r(hEXeA precomputed kernel can also be composited with other kernels. You need to explictly add scalar for hFjubj)r}r(hEX*pyGPs.cov.Pre()*hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXpyGPs.cov.Pre()rr}r(hEUhFjubahLj#ubhaX.r}r(hEX.hFjubeubh})r}r(hEX/k = 0.5*pyGPs.cov.Pre(M1, M2) + pyGPs.cov.RBF()hFjhGhJhLhhN}r(hhhS]hR]hP]hQ]hU]uhWKWhXhh@]rhaX/k = 0.5*pyGPs.cov.Pre(M1, M2) + pyGPs.cov.RBF()rr}r(hEUhFjubaubeubhB)r}r(hEUhFhChGhJhLhMhN}r(hP]hQ]hR]hS]rh>ahU]rh)auhWK[hXhh@]r(hZ)r}r(hEX&Developing New Kernel & Mean FunctionsrhFjhGhJhLh^hN}r(hP]hQ]hR]hS]hU]uhWK[hXhh@]rhaX&Developing New Kernel & Mean Functionsrr}r(hEjhFjubaubhs)r}r(hEXWe also support the development of new kernel/mean classes, your customized kernel class need to follow the template as below: ::rhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWK\hXhh@]rhaX~We also support the development of new kernel/mean classes, your customized kernel class need to follow the template as below:rr}r(hEX~We also support the development of new kernel/mean classes, your customized kernel class need to follow the template as below:hFjubaubh})r}r(hEXY# Your kernel class needs to inherit base class Kernel, # which is in the module of Core.cov class MyKernel(Kernel): def __init__(self, hyp): ''' Intialize hyperparameters for MyKernel. ''' self.hyp = hyp def getCovMatrix(self,x=None,z=None,mode=None): ''' Return the specific covariance matrix according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self covariance matrix of test data(test by 1). 'train' return training covariance matrix(train by train). 'cross' return cross covariance matrix between x and z(train by test) :return: the corresponding covariance matrix ''' pass def getDerMatrix(self,x=None,z=None,mode=None,der=None): ''' Compute derivatives wrt. hyperparameters according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self derivative matrix of test data(test by 1). 'train' return training derivative matrix(train by train). 'cross' return cross derivative matrix between x and z(train by test) :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' passhFjhGhJhLhhN}r(hhhS]hR]hP]hQ]hU]uhWK^hXhh@]rhaXY# Your kernel class needs to inherit base class Kernel, # which is in the module of Core.cov class MyKernel(Kernel): def __init__(self, hyp): ''' Intialize hyperparameters for MyKernel. ''' self.hyp = hyp def getCovMatrix(self,x=None,z=None,mode=None): ''' Return the specific covariance matrix according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self covariance matrix of test data(test by 1). 'train' return training covariance matrix(train by train). 'cross' return cross covariance matrix between x and z(train by test) :return: the corresponding covariance matrix ''' pass def getDerMatrix(self,x=None,z=None,mode=None,der=None): ''' Compute derivatives wrt. hyperparameters according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self derivative matrix of test data(test by 1). 'train' return training derivative matrix(train by train). 'cross' return cross derivative matrix between x and z(train by test) :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' passrr}r(hEUhFjubaubhs)r}r(hEX!and for customized mean class: ::rhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaXand for customized mean class:rr}r(hEXand for customized mean class:hFjubaubh})r}r(hEXS# Your mean class needs to inherit base class Mean, # which is in the module of Core.mean class MyMean(Mean): def __init__(self, hyp): ''' Intialize hyperparameters for MyMean. ''' self.hyp = hyp def getMean(self, x=None): ''' Get the mean vector. ''' pass def getDerMatrix(self, x=None, der=None): ''' Compute derivatives wrt. hyperparameters. :param x: training data :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' passhFjhGhJhLhhN}r(hhhS]hR]hP]hQ]hU]uhWKhXhh@]rhaXS# Your mean class needs to inherit base class Mean, # which is in the module of Core.mean class MyMean(Mean): def __init__(self, hyp): ''' Intialize hyperparameters for MyMean. ''' self.hyp = hyp def getMean(self, x=None): ''' Get the mean vector. ''' pass def getDerMatrix(self, x=None, der=None): ''' Compute derivatives wrt. hyperparameters. :param x: training data :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' passrr}r(hEUhFjubaubhs)r}r(hEXYou can test your customized mean/kernel function using our framework of unit test. Taking kernel test as an example, you can uncomment method *test_cov_new* in *pyGPs.Testing.unit_test_cov.py* to check the outputs of your kernel function. ::hFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]r(haXYou can test your customized mean/kernel function using our framework of unit test. Taking kernel test as an example, you can uncomment method rr}r(hEXYou can test your customized mean/kernel function using our framework of unit test. Taking kernel test as an example, you can uncomment method hFjubj)r}r(hEX*test_cov_new*hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX test_cov_newrr}r(hEUhFjubahLj#ubhaX in rr}r(hEX in hFjubj)r}r(hEX *pyGPs.Testing.unit_test_cov.py*hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXpyGPs.Testing.unit_test_cov.pyrr}r(hEUhFjubahLj#ubhaX. to check the outputs of your kernel function.rr}r(hEX. to check the outputs of your kernel function.hFjubeubh})r}r(hEX# Test your customized covariance function def test_cov_new(self): k = myKernel() # specify your covariance function self.checkCovariance(k)hFjhGhJhLhhN}r (hhhS]hR]hP]hQ]hU]uhWKhXhh@]r haX# Test your customized covariance function def test_cov_new(self): k = myKernel() # specify your covariance function self.checkCovariance(k)r r }r (hEUhFjubaubhs)r}r(hEXAand testing mean function in *pyGPs.Testing.unit_test_mean.py* ::rhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]r(haXand testing mean function in rr}r(hEXand testing mean function in hFjubj)r}r(hEX!*pyGPs.Testing.unit_test_mean.py*hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXpyGPs.Testing.unit_test_mean.pyrr}r(hEUhFjubahLj#ubeubh})r}r(hEX# Test your customized mean function def test_mean_new(self): m = myMean # specify your mean function self.checkMean(m)hFjhGhJhLhhN}r(hhhS]hR]hP]hQ]hU]uhWKhXhh@]r haX# Test your customized mean function def test_mean_new(self): m = myMean # specify your mean function self.checkMean(m)r!r"}r#(hEUhFjubaubeubhB)r$}r%(hEUhFhChGhJhLhMhN}r&(hP]hQ]hR]hS]r'(Xmodule-pyGPs.Core.covr(h?ehU]r)h*auhWKhXhh@]r*(hZ)r+}r,(hEX&List of Kernels and Default Parametersr-hFj$hGhJhLh^hN}r.(hP]hQ]hR]hS]hU]uhWKhXhh@]r/haX&List of Kernels and Default Parametersr0r1}r2(hEj-hFj+ubaubcsphinx.addnodes index r3)r4}r5(hEUhFj$hGU r6hLUindexr7hN}r8(hS]hR]hP]hQ]hU]Uentries]r9(Usingler:XpyGPs.Core.cov (module)Xmodule-pyGPs.Core.covUtr;auhWKhXhh@]ubj3)r<}r=(hEUhFj$hGNhLj7hN}r>(hS]hR]hP]hQ]hU]Uentries]r?(j:XConst (class in pyGPs.Core.cov)h-Utr@auhWNhXhh@]ubcsphinx.addnodes desc rA)rB}rC(hEUhFj$hGNhLUdescrDhN}rE(UnoindexrFUdomainrGXpyhS]hR]hP]hQ]hU]UobjtyperHXclassrIUdesctyperJjIuhWNhXhh@]rK(csphinx.addnodes desc_signature rL)rM}rN(hEXConst(log_sigma=0.0)hFjBhGU rOhLUdesc_signaturerPhN}rQ(hS]rRh-aUmodulerShHXpyGPs.Core.covrTrU}rVbhR]hP]hQ]hU]rWh-aUfullnamerXXConstrYUclassrZUUfirstr[uhWNhXhh@]r\(csphinx.addnodes desc_annotation r])r^}r_(hEXclass hFjMhGjOhLUdesc_annotationr`hN}ra(hP]hQ]hR]hS]hU]uhWNhXhh@]rbhaXclass rcrd}re(hEUhFj^ubaubcsphinx.addnodes desc_addname rf)rg}rh(hEXpyGPs.Core.cov.hFjMhGjOhLU desc_addnamerihN}rj(hP]hQ]hR]hS]hU]uhWNhXhh@]rkhaXpyGPs.Core.cov.rlrm}rn(hEUhFjgubaubcsphinx.addnodes desc_name ro)rp}rq(hEjYhFjMhGjOhLU desc_namerrhN}rs(hP]hQ]hR]hS]hU]uhWNhXhh@]rthaXConstrurv}rw(hEUhFjpubaubcsphinx.addnodes desc_parameterlist rx)ry}rz(hEUhFjMhGjOhLUdesc_parameterlistr{hN}r|(hP]hQ]hR]hS]hU]uhWNhXhh@]r}csphinx.addnodes desc_parameter r~)r}r(hEX log_sigma=0.0hN}r(hP]hQ]hR]hS]hU]uhFjyh@]rhaX log_sigma=0.0rr}r(hEUhFjubahLUdesc_parameterrubaubcsphinx.addnodes only r)r}r(hEUhFjMhGNhLUonlyrhN}r(UexprUhtmlrhS]hR]hP]hQ]hU]uhWNhXhh@]rcsphinx.addnodes pending_xref r)r}r(hEUhN}r(UreftypeUviewcoderUrefdocXKernelsrU refdomainUstdrhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjYuhFjh@]rcdocutils.nodes inline r)r}r(hEUhN}r(hP]hQ]rU viewcode-linkrahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLUinlinerubahLU pending_xrefrubaubeubcsphinx.addnodes desc_content r)r}r(hEUhFjBhGjOhLU desc_contentrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r}r(hEX$Constant kernel. hyp = [ log_sigma ]rhFjhGX_/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.ConsthLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaX$Constant kernel. hyp = [ log_sigma ]rr}r(hEjhFjubaubcdocutils.nodes field_list r)r}r(hEUhFjhGNhLU field_listrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rcdocutils.nodes field r)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(cdocutils.nodes field_name r)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX Parametersrr}r(hEUhFjubahLU field_namerubcdocutils.nodes field_body r)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r(hEX log_sigmahN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_sigmarr}r(hEUhFjubahLjSubhaX -- rr}r(hEUhFjubhaXsignal deviation.rr}r(hEXsignal deviation.hFjubehLhwubahLU field_bodyrubehLUfieldrubaubeubeubj3)r}r(hEUhFj$hGXf/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.FITCOfKernelrhLj7hN}r(hS]hR]hP]hQ]hU]Uentries]r(j:X&FITCOfKernel (class in pyGPs.Core.cov)hUtrauhWNhXhh@]ubjA)r}r(hEUhFj$hGjhLjDhN}r(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrjJjuhWNhXhh@]r(jL)r}r(hEX FITCOfKernel(cov, inducingInput)hFjhGjOhLjPhN}r(hS]rhajShHXpyGPs.Core.covrr}rbhR]hP]hQ]hU]rhajXX FITCOfKernelrjZUj[uhWNhXhh@]r(j])r}r(hEXclass hFjhGjOhLj`hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXclass rr}r(hEUhFjubaubjf)r}r(hEXpyGPs.Core.cov.hFjhGjOhLjihN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXpyGPs.Core.cov.rr}r(hEUhFjubaubjo)r}r(hEjhFjhGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaX FITCOfKernelrr}r(hEUhFjubaubjx)r}r(hEUhFjhGjOhLj{hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r (j~)r }r (hEXcovhN}r (hP]hQ]hR]hS]hU]uhFjh@]r haXcovrr}r(hEUhFj ubahLjubj~)r}r(hEX inducingInputhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX inducingInputrr}r(hEUhFjubahLjubeubj)r}r(hEUhFjhGNhLjhN}r(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rj)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjuhFjh@]rj)r }r!(hEUhN}r"(hP]hQ]r#jahR]hS]hU]uhFjh@]r$haX[source]r%r&}r'(hEUhFj ubahLjubahLjubaubeubj)r(}r)(hEUhFjhGjOhLjhN}r*(hP]hQ]hR]hS]hU]uhWNhXhh@]r+hs)r,}r-(hEXNCovariance function to be used together with the FITC approximation. The function allows for more than one output argument and does not respect the interface of a proper covariance function. Instead of outputing the full covariance, it returns cross-covariances between the inputs x, z and the inducing inputs xu as needed by infFITCr.hFj(hGjhLhwhN}r/(hP]hQ]hR]hS]hU]uhWKhXhh@]r0haXNCovariance function to be used together with the FITC approximation. The function allows for more than one output argument and does not respect the interface of a proper covariance function. 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Setting FITC gp model will implicitly call this method.rr}r(hEjhFjubaubj)r}r(hEUhFjhGNhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXReturnsrr}r(hEUhFjubahLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXan instance of FITCOfKernelrr}r(hEXan instance of FITCOfKernelhFjubahLhwubahLjubehLjubaubeubeubj3)r}r(hEUhFjmhGNhLj7hN}r(hS]hR]hP]hQ]hU]Uentries]r(j:X-getCovMatrix() (pyGPs.Core.cov.Kernel method)hUtrauhWNhXhh@]ubjA)r}r(hEUhFjmhGNhLjDhN}r(jFjGXpyrhS]hR]hP]hQ]hU]jHXmethodrjJjuhWNhXhh@]r(jL)r}r(hEX.Kernel.getCovMatrix(x=None, z=None, mode=None)hFjhGjOhLjPhN}r(hS]rhajShHXpyGPs.Core.covrr}rbhR]hP]hQ]hU]rhajXXKernel.getCovMatrixrjZjFj[uhWNhXhh@]r(jo)r}r(hEX getCovMatrixhFjhGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaX getCovMatrixrr}r(hEUhFjubaubjx)r}r(hEUhFjhGjOhLj{hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(j~)r}r(hEXx=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXx=Nonerr}r(hEUhFjubahLjubj~)r}r(hEXz=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXz=Nonerr}r(hEUhFjubahLjubj~)r}r(hEX mode=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX mode=Nonerr}r (hEUhFjubahLjubeubj)r }r (hEUhFjhGNhLjhN}r (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r j)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjuhFj h@]rj)r}r(hEUhN}r(hP]hQ]rjahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLjubahLjubaubeubj)r}r(hEUhFjhGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r}r(hEX=Return the specific covariance matrix according to input moder hFjhGXm/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.Kernel.getCovMatrixhLhwhN}r!(hP]hQ]hR]hS]hU]uhWKhXhh@]r"haX=Return the specific covariance matrix according to input moder#r$}r%(hEj hFjubaubj)r&}r'(hEUhFjhGNhLjhN}r((hP]hQ]hR]hS]hU]uhWNhXhh@]r)(j)r*}r+(hEUhN}r,(hP]hQ]hR]hS]hU]uhFj&h@]r-(j)r.}r/(hEUhN}r0(hP]hQ]hR]hS]hU]uhFj*h@]r1haX Parametersr2r3}r4(hEUhFj.ubahLjubj)r5}r6(hEUhN}r7(hP]hQ]hR]hS]hU]uhFj*h@]r8j )r9}r:(hEUhN}r;(hP]hQ]hR]hS]hU]uhFj5h@]r<(h)r=}r>(hEUhN}r?(hP]hQ]hR]hS]hU]uhFj9h@]r@hs)rA}rB(hEUhN}rC(hP]hQ]hR]hS]hU]uhFj=h@]rD(jK)rE}rF(hEXxhN}rG(hP]hQ]hR]hS]hU]uhFjAh@]rHhaXxrI}rJ(hEUhFjEubahLjSubhaX -- rKrL}rM(hEUhFjAubhaX training datarNrO}rP(hEX training datahFjAubehLhwubahLhubh)rQ}rR(hEUhN}rS(hP]hQ]hR]hS]hU]uhFj9h@]rThs)rU}rV(hEUhN}rW(hP]hQ]hR]hS]hU]uhFjQh@]rX(jK)rY}rZ(hEXzhN}r[(hP]hQ]hR]hS]hU]uhFjUh@]r\haXzr]}r^(hEUhFjYubahLjSubhaX -- r_r`}ra(hEUhFjUubhaX test datarbrc}rd(hEX test datahFjUubehLhwubahLhubh)re}rf(hEUhN}rg(hP]hQ]hR]hS]hU]uhFj9h@]rhhs)ri}rj(hEUhN}rk(hP]hQ]hR]hS]hU]uhFjeh@]rl(jK)rm}rn(hEXmodehN}ro(hP]hQ]hR]hS]hU]uhFjih@]rphaXmoderqrr}rs(hEUhFjmubahLjSubhaX (rtru}rv(hEUhFjiubj)rw}rx(hEUhN}ry(UreftypeUobjrzU reftargetXstrr{U refdomainjhS]hR]U refexplicithP]hQ]hU]uhFjih@]r|j)r}}r~(hEj{hN}r(hP]hQ]hR]hS]hU]uhFjwh@]rhaXstrrr}r(hEUhFj}ubahLj#ubahLjubhaX)r}r(hEUhFjiubhaX -- rr}r(hEUhFjiubhaX'self_test' return self covariance matrix of test data(test by 1). 'train' return training covariance matrix(train by train). 'cross' return cross covariance matrix between x and z(train by test)rr}r(hEX'self_test' return self covariance matrix of test data(test by 1). 'train' return training covariance matrix(train by train). 'cross' return cross covariance matrix between x and z(train by test)hFjiubehLhwubahLhubehLjTubahLjubehLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFj&h@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXReturnsrr}r(hEUhFjubahLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX#the corresponding covariance matrixrr}r(hEX#the corresponding covariance matrixhFjubahLhwubahLjubehLjubeubeubeubj3)r}r(hEUhFjmhGNhLj7hN}r(hS]hR]hP]hQ]hU]Uentries]r(j:X-getDerMatrix() (pyGPs.Core.cov.Kernel method)hUtrauhWNhXhh@]ubjA)r}r(hEUhFjmhGNhLjDhN}r(jFjGXpyrhS]hR]hP]hQ]hU]jHXmethodrjJjuhWNhXhh@]r(jL)r}r(hEX8Kernel.getDerMatrix(x=None, z=None, mode=None, der=None)hFjhGjOhLjPhN}r(hS]rhajShHXpyGPs.Core.covrr}rbhR]hP]hQ]hU]rhajXXKernel.getDerMatrixrjZjFj[uhWNhXhh@]r(jo)r}r(hEX getDerMatrixhFjhGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaX getDerMatrixrr}r(hEUhFjubaubjx)r}r(hEUhFjhGjOhLj{hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(j~)r}r(hEXx=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXx=Nonerr}r(hEUhFjubahLjubj~)r}r(hEXz=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXz=Nonerr}r(hEUhFjubahLjubj~)r}r(hEX mode=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX mode=Nonerr}r(hEUhFjubahLjubj~)r}r(hEXder=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXder=Nonerr}r(hEUhFjubahLjubeubj)r}r(hEUhFjhGNhLjhN}r(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rj)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjuhFjh@]rj)r}r(hEUhN}r(hP]hQ]rjahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLjubahLjubaubeubj)r}r(hEUhFjhGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r}r(hEX@Compute derivatives wrt. hyperparameters according to input moderhFjhGXm/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.Kernel.getDerMatrixrhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaX@Compute derivatives wrt. hyperparameters according to input moderr}r(hEjhFjubaubj)r}r(hEUhFjhGNhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX Parametersrr}r (hEUhFjubahLjubj)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFjh@]r j )r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFj h@]r(h)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r(hEXxhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXxr}r(hEUhFjubahLjSubhaX -- r r!}r"(hEUhFjubhaX training datar#r$}r%(hEX training datar&hFjubehLhwubahLhubh)r'}r((hEUhN}r)(hP]hQ]hR]hS]hU]uhFjh@]r*hs)r+}r,(hEUhN}r-(hP]hQ]hR]hS]hU]uhFj'h@]r.(jK)r/}r0(hEXzhN}r1(hP]hQ]hR]hS]hU]uhFj+h@]r2haXzr3}r4(hEUhFj/ubahLjSubhaX -- r5r6}r7(hEUhFj+ubhaX test datar8r9}r:(hEX test datar;hFj+ubehLhwubahLhubh)r<}r=(hEUhN}r>(hP]hQ]hR]hS]hU]uhFjh@]r?hs)r@}rA(hEUhN}rB(hP]hQ]hR]hS]hU]uhFj<h@]rC(jK)rD}rE(hEXmodehN}rF(hP]hQ]hR]hS]hU]uhFj@h@]rGhaXmoderHrI}rJ(hEUhFjDubahLjSubhaX (rKrL}rM(hEUhFj@ubj)rN}rO(hEUhN}rP(UreftypejzU reftargetXstrrQU refdomainjhS]hR]U refexplicithP]hQ]hU]uhFj@h@]rRj)rS}rT(hEjQhN}rU(hP]hQ]hR]hS]hU]uhFjNh@]rVhaXstrrWrX}rY(hEUhFjSubahLj#ubahLjubhaX)rZ}r[(hEUhFj@ubhaX -- r\r]}r^(hEUhFj@ubhaX'self_test' return self derivative matrix of test data(test by 1). 'train' return training derivative matrix(train by train). 'cross' return cross derivative matrix between x and z(train by test)r_r`}ra(hEX'self_test' return self derivative matrix of test data(test by 1). 'train' return training derivative matrix(train by train). 'cross' return cross derivative matrix between x and z(train by test)rbhFj@ubehLhwubahLhubh)rc}rd(hEUhN}re(hP]hQ]hR]hS]hU]uhFjh@]rfhs)rg}rh(hEUhN}ri(hP]hQ]hR]hS]hU]uhFjch@]rj(jK)rk}rl(hEXderhN}rm(hP]hQ]hR]hS]hU]uhFjgh@]rnhaXderrorp}rq(hEUhFjkubahLjSubhaX (rrrs}rt(hEUhFjgubj)ru}rv(hEUhN}rw(UreftypejzU reftargetXintrxU refdomainjhS]hR]U refexplicithP]hQ]hU]uhFjgh@]ryj)rz}r{(hEjxhN}r|(hP]hQ]hR]hS]hU]uhFjuh@]r}haXintr~r}r(hEUhFjzubahLj#ubahLjubhaX)r}r(hEUhFjgubhaX -- rr}r(hEUhFjgubhaX7index of hyperparameter whose derivative to be computedrr}r(hEX7index of hyperparameter whose derivative to be computedrhFjgubehLhwubahLhubehLjTubahLjubehLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXReturnsrr}r(hEUhFjubahLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX#the corresponding derivative matrixrr}r(hEX#the corresponding derivative matrixrhFjubahLhwubahLjubehLjubeubeubeubeubeubj3)r}r(hEUhFj$hGNhLj7hN}r(hS]hR]hP]hQ]hU]Uentries]r(j:X LINard (class in pyGPs.Core.cov)h UtrauhWNhXhh@]ubjA)r}r(hEUhFj$hGNhLjDhN}r(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrjJjuhWNhXhh@]r(jL)r}r(hEX!LINard(D=None, log_ell_list=None)hFjhGjOhLjPhN}r(hS]rh ajShHXpyGPs.Core.covrr}rbhR]hP]hQ]hU]rh ajXXLINardrjZUj[uhWNhXhh@]r(j])r}r(hEXclass hFjhGjOhLj`hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXclass rr}r(hEUhFjubaubjf)r}r(hEXpyGPs.Core.cov.hFjhGjOhLjihN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXpyGPs.Core.cov.rr}r(hEUhFjubaubjo)r}r(hEjhFjhGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXLINardrr}r(hEUhFjubaubjx)r}r(hEUhFjhGjOhLj{hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(j~)r}r(hEXD=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXD=Nonerr}r(hEUhFjubahLjubj~)r}r(hEXlog_ell_list=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXlog_ell_list=Nonerr}r(hEUhFjubahLjubeubj)r}r(hEUhFjhGNhLjhN}r(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rj)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjuhFjh@]rj)r}r(hEUhN}r(hP]hQ]rjahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLjubahLjubaubeubj)r}r(hEUhFjhGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r}r(hEXTLinear covariance function with Automatic Relevance Detemination. hyp = log_ell_listrhFjhGX`/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.LINardhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaXTLinear covariance function with Automatic Relevance Detemination. hyp = log_ell_listrr}r(hEjhFjubaubj)r}r(hEUhFjhGNhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX Parametersrr}r(hEUhFjubahLjubj)r}r(hEUhN}r (hP]hQ]hR]hS]hU]uhFjh@]r j )r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFjh@]r(h)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFj h@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r(hEXDhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXDr}r(hEUhFjubahLjSubhaX -- rr}r(hEUhFjubhaXWdimension of training data. Set if you want default ell, which is 1 for each dimension.r r!}r"(hEXWdimension of training data. Set if you want default ell, which is 1 for each dimension.hFjubehLhwubahLhubh)r#}r$(hEUhN}r%(hP]hQ]hR]hS]hU]uhFj h@]r&hs)r'}r((hEUhN}r)(hP]hQ]hR]hS]hU]uhFj#h@]r*(jK)r+}r,(hEX log_ell_listhN}r-(hP]hQ]hR]hS]hU]uhFj'h@]r.haX log_ell_listr/r0}r1(hEUhFj+ubahLjSubhaX -- r2r3}r4(hEUhFj'ubhaX/characteristic length scale for each dimension.r5r6}r7(hEX/characteristic length scale for each dimension.hFj'ubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)r8}r9(hEUhFj$hGNhLj7hN}r:(hS]hR]hP]hQ]hU]Uentries]r;(j:X Linear (class in pyGPs.Core.cov)hUtr<auhWNhXhh@]ubjA)r=}r>(hEUhFj$hGNhLjDhN}r?(jFjGXpyhS]hR]hP]hQ]hU]jHXclassr@jJj@uhWNhXhh@]rA(jL)rB}rC(hEXLinear(log_sigma=0.0)hFj=hGjOhLjPhN}rD(hS]rEhajShHXpyGPs.Core.covrFrG}rHbhR]hP]hQ]hU]rIhajXXLinearrJjZUj[uhWNhXhh@]rK(j])rL}rM(hEXclass hFjBhGjOhLj`hN}rN(hP]hQ]hR]hS]hU]uhWNhXhh@]rOhaXclass rPrQ}rR(hEUhFjLubaubjf)rS}rT(hEXpyGPs.Core.cov.hFjBhGjOhLjihN}rU(hP]hQ]hR]hS]hU]uhWNhXhh@]rVhaXpyGPs.Core.cov.rWrX}rY(hEUhFjSubaubjo)rZ}r[(hEjJhFjBhGjOhLjrhN}r\(hP]hQ]hR]hS]hU]uhWNhXhh@]r]haXLinearr^r_}r`(hEUhFjZubaubjx)ra}rb(hEUhFjBhGjOhLj{hN}rc(hP]hQ]hR]hS]hU]uhWNhXhh@]rdj~)re}rf(hEX log_sigma=0.0hN}rg(hP]hQ]hR]hS]hU]uhFjah@]rhhaX log_sigma=0.0rirj}rk(hEUhFjeubahLjubaubj)rl}rm(hEUhFjBhGNhLjhN}rn(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]roj)rp}rq(hEUhN}rr(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjJuhFjlh@]rsj)rt}ru(hEUhN}rv(hP]hQ]rwjahR]hS]hU]uhFjph@]rxhaX[source]ryrz}r{(hEUhFjtubahLjubahLjubaubeubj)r|}r}(hEUhFj=hGjOhLjhN}r~(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r}r(hEX#Linear kernel. hyp = [ log_sigma ].rhFj|hGX`/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.LinearhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaX#Linear kernel. hyp = [ log_sigma ].rr}r(hEjhFjubaubj)r}r(hEUhFj|hGNhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX Parametersrr}r(hEUhFjubahLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r(hEX log_sigmahN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_sigmarr}r(hEUhFjubahLjSubhaX -- rr}r(hEUhFjubhaXsignal deviation.rr}r(hEXsignal deviation.hFjubehLhwubahLjubehLjubaubeubeubj3)r}r(hEUhFj$hGNhLj7hN}r(hS]hR]hP]hQ]hU]Uentries]r(j:X Matern (class in pyGPs.Core.cov)hUtrauhWNhXhh@]ubjA)r}r(hEUhFj$hGNhLjDhN}r(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrjJjuhWNhXhh@]r(jL)r}r(hEX'Matern(log_ell=0.0, d=3, log_sigma=0.0)hFjhGjOhLjPhN}r(hS]rhajShHXpyGPs.Core.covrr}rbhR]hP]hQ]hU]rhajXXMaternrjZUj[uhWNhXhh@]r(j])r}r(hEXclass hFjhGjOhLj`hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXclass rr}r(hEUhFjubaubjf)r}r(hEXpyGPs.Core.cov.hFjhGjOhLjihN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXpyGPs.Core.cov.rr}r(hEUhFjubaubjo)r}r(hEjhFjhGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXMaternrr}r(hEUhFjubaubjx)r}r(hEUhFjhGjOhLj{hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(j~)r}r(hEX log_ell=0.0hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_ell=0.0rr}r(hEUhFjubahLjubj~)r}r(hEXd=3hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXd=3rr}r(hEUhFjubahLjubj~)r}r(hEX log_sigma=0.0hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_sigma=0.0rr}r(hEUhFjubahLjubeubj)r}r(hEUhFjhGNhLjhN}r(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rj)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjuhFjh@]rj)r}r(hEUhN}r(hP]hQ]rjahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLjubahLjubaubeubj)r}r(hEUhFjhGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r}r(hEXMatern covariance function with nu = d/2 and isotropic distance measure. For d=1 the function is also known as the exponential covariance function or the Ornstein-Uhlenbeck covariance in 1d. d will be rounded to 1, 3, or 5. hyp = [ log_ell, log_sigma]rhFjhGX`/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.MaternhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaXMatern covariance function with nu = d/2 and isotropic distance measure. For d=1 the function is also known as the exponential covariance function or the Ornstein-Uhlenbeck covariance in 1d. d will be rounded to 1, 3, or 5. hyp = [ log_ell, log_sigma]rr}r (hEjhFjubaubj)r }r (hEUhFjhGNhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFj h@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX Parametersrr}r(hEUhFjubahLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rj )r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r (h)r!}r"(hEUhN}r#(hP]hQ]hR]hS]hU]uhFjh@]r$hs)r%}r&(hEUhN}r'(hP]hQ]hR]hS]hU]uhFj!h@]r((jK)r)}r*(hEXdhN}r+(hP]hQ]hR]hS]hU]uhFj%h@]r,haXdr-}r.(hEUhFj)ubahLjSubhaX -- r/r0}r1(hEUhFj%ubhaX&d is 2 times nu. Can only be 1,3 or 5.r2r3}r4(hEX&d is 2 times nu. Can only be 1,3 or 5.hFj%ubehLhwubahLhubh)r5}r6(hEUhN}r7(hP]hQ]hR]hS]hU]uhFjh@]r8hs)r9}r:(hEUhN}r;(hP]hQ]hR]hS]hU]uhFj5h@]r<(jK)r=}r>(hEXlog_ellhN}r?(hP]hQ]hR]hS]hU]uhFj9h@]r@haXlog_ellrArB}rC(hEUhFj=ubahLjSubhaX -- rDrE}rF(hEUhFj9ubhaXcharacteristic length scale.rGrH}rI(hEXcharacteristic length scale.hFj9ubehLhwubahLhubh)rJ}rK(hEUhN}rL(hP]hQ]hR]hS]hU]uhFjh@]rMhs)rN}rO(hEUhN}rP(hP]hQ]hR]hS]hU]uhFjJh@]rQ(jK)rR}rS(hEX log_sigmahN}rT(hP]hQ]hR]hS]hU]uhFjNh@]rUhaX log_sigmarVrW}rX(hEUhFjRubahLjSubhaX -- rYrZ}r[(hEUhFjNubhaXsignal deviation.r\r]}r^(hEXsignal deviation.hFjNubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)r_}r`(hEUhFj$hGNhLj7hN}ra(hS]hR]hP]hQ]hU]Uentries]rb(j:XNoise (class in pyGPs.Core.cov)hUtrcauhWNhXhh@]ubjA)rd}re(hEUhFj$hGNhLjDhN}rf(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrgjJjguhWNhXhh@]rh(jL)ri}rj(hEXNoise(log_sigma=0.0)hFjdhGjOhLjPhN}rk(hS]rlhajShHXpyGPs.Core.covrmrn}robhR]hP]hQ]hU]rphajXXNoiserqjZUj[uhWNhXhh@]rr(j])rs}rt(hEXclass hFjihGjOhLj`hN}ru(hP]hQ]hR]hS]hU]uhWNhXhh@]rvhaXclass rwrx}ry(hEUhFjsubaubjf)rz}r{(hEXpyGPs.Core.cov.hFjihGjOhLjihN}r|(hP]hQ]hR]hS]hU]uhWNhXhh@]r}haXpyGPs.Core.cov.r~r}r(hEUhFjzubaubjo)r}r(hEjqhFjihGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXNoiserr}r(hEUhFjubaubjx)r}r(hEUhFjihGjOhLj{hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rj~)r}r(hEX log_sigma=0.0hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_sigma=0.0rr}r(hEUhFjubahLjubaubj)r}r(hEUhFjihGNhLjhN}r(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rj)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjquhFjh@]rj)r}r(hEUhN}r(hP]hQ]rjahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLjubahLjubaubeubj)r}r(hEUhFjdhGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r}r(hEXIndependent covariance function, i.e "white noise", with specified variance. Normally NOT used anymore since noise is now added in liklihood. hyp = [ log_sigma ]rhFjhGX_/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.NoisehLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaXIndependent covariance function, i.e "white noise", with specified variance. Normally NOT used anymore since noise is now added in liklihood. hyp = [ log_sigma ]rr}r(hEjhFjubaubj)r}r(hEUhFjhGNhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX Parametersrr}r(hEUhFjubahLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r(hEX log_sigmahN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_sigmarr}r(hEUhFjubahLjSubhaX -- rr}r(hEUhFjubhaXsignal deviation.rr}r(hEXsignal deviation.hFjubehLhwubahLjubehLjubaubeubeubj3)r}r(hEUhFj$hGNhLj7hN}r(hS]hR]hP]hQ]hU]Uentries]r(j:X"Periodic (class in pyGPs.Core.cov)h UtrauhWNhXhh@]ubjA)r}r(hEUhFj$hGNhLjDhN}r(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrjJjuhWNhXhh@]r(jL)r}r(hEX/Periodic(log_ell=0.0, log_p=0.0, log_sigma=0.0)hFjhGjOhLjPhN}r(hS]rh ajShHXpyGPs.Core.covrr}rbhR]hP]hQ]hU]rh ajXXPeriodicrjZUj[uhWNhXhh@]r(j])r}r(hEXclass hFjhGjOhLj`hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXclass rr}r(hEUhFjubaubjf)r}r(hEXpyGPs.Core.cov.hFjhGjOhLjihN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXpyGPs.Core.cov.rr}r(hEUhFjubaubjo)r}r(hEjhFjhGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXPeriodicrr}r(hEUhFjubaubjx)r}r(hEUhFjhGjOhLj{hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(j~)r }r (hEX log_ell=0.0hN}r (hP]hQ]hR]hS]hU]uhFjh@]r haX log_ell=0.0r r }r (hEUhFj ubahLjubj~)r }r (hEX log_p=0.0hN}r (hP]hQ]hR]hS]hU]uhFjh@]r haX log_p=0.0r r }r (hEUhFj ubahLjubj~)r }r (hEX log_sigma=0.0hN}r (hP]hQ]hR]hS]hU]uhFjh@]r haX log_sigma=0.0r r }r (hEUhFj ubahLjubeubj)r }r (hEUhFjhGNhLjhN}r (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r j)r }r (hEUhN}r (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjuhFj h@]r j)r }r (hEUhN}r (hP]hQ]r jahR]hS]hU]uhFj h@]r! haX[source]r" r# }r$ (hEUhFj ubahLjubahLjubaubeubj)r% }r& (hEUhFjhGjOhLjhN}r' (hP]hQ]hR]hS]hU]uhWNhXhh@]r( (hs)r) }r* (hEXUStationary kernel for a smooth periodic function. hyp = [ log_ell, log_p, log_sigma]r+ hFj% hGXb/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.PeriodichLhwhN}r, (hP]hQ]hR]hS]hU]uhWKhXhh@]r- haXUStationary kernel for a smooth periodic function. hyp = [ log_ell, log_p, log_sigma]r. r/ }r0 (hEj+ hFj) ubaubj)r1 }r2 (hEUhFj% hGNhLjhN}r3 (hP]hQ]hR]hS]hU]uhWNhXhh@]r4 j)r5 }r6 (hEUhN}r7 (hP]hQ]hR]hS]hU]uhFj1 h@]r8 (j)r9 }r: (hEUhN}r; (hP]hQ]hR]hS]hU]uhFj5 h@]r< haX Parametersr= r> }r? (hEUhFj9 ubahLjubj)r@ }rA (hEUhN}rB (hP]hQ]hR]hS]hU]uhFj5 h@]rC j )rD }rE (hEUhN}rF (hP]hQ]hR]hS]hU]uhFj@ h@]rG (h)rH }rI (hEUhN}rJ (hP]hQ]hR]hS]hU]uhFjD h@]rK hs)rL }rM (hEUhN}rN (hP]hQ]hR]hS]hU]uhFjH h@]rO (jK)rP }rQ (hEXlog_phN}rR (hP]hQ]hR]hS]hU]uhFjL h@]rS haXlog_prT rU }rV (hEUhFjP ubahLjSubhaX -- rW rX }rY (hEUhFjL ubhaXperiod.rZ r[ }r\ (hEXperiod.hFjL ubehLhwubahLhubh)r] }r^ (hEUhN}r_ (hP]hQ]hR]hS]hU]uhFjD h@]r` hs)ra }rb (hEUhN}rc (hP]hQ]hR]hS]hU]uhFj] h@]rd (jK)re }rf (hEXlog_ellhN}rg (hP]hQ]hR]hS]hU]uhFja h@]rh haXlog_ellri rj }rk (hEUhFje ubahLjSubhaX -- rl rm }rn (hEUhFja ubhaXcharacteristic length scale.ro rp }rq (hEXcharacteristic length scale.hFja ubehLhwubahLhubh)rr }rs (hEUhN}rt (hP]hQ]hR]hS]hU]uhFjD h@]ru hs)rv }rw (hEUhN}rx (hP]hQ]hR]hS]hU]uhFjr h@]ry (jK)rz }r{ (hEX log_sigmahN}r| (hP]hQ]hR]hS]hU]uhFjv h@]r} haX log_sigmar~ r }r (hEUhFjz ubahLjSubhaX -- r r }r (hEUhFjv ubhaXsignal deviation.r r }r (hEXsignal deviation.hFjv ubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)r }r (hEUhFj$hGNhLj7hN}r (hS]hR]hP]hQ]hU]Uentries]r (j:X#PiecePoly (class in pyGPs.Core.cov)hUtr auhWNhXhh@]ubjA)r }r (hEUhFj$hGNhLjDhN}r (jFjGXpyhS]hR]hP]hQ]hU]jHXclassr jJj uhWNhXhh@]r (jL)r }r (hEX*PiecePoly(log_ell=0.0, v=2, log_sigma=0.0)hFj hGjOhLjPhN}r (hS]r hajShHXpyGPs.Core.covr r }r bhR]hP]hQ]hU]r hajXX PiecePolyr jZUj[uhWNhXhh@]r (j])r }r (hEXclass hFj hGjOhLj`hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXclass r r }r (hEUhFj ubaubjf)r }r (hEXpyGPs.Core.cov.hFj hGjOhLjihN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXpyGPs.Core.cov.r r }r (hEUhFj ubaubjo)r }r (hEj hFj hGjOhLjrhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haX PiecePolyr r }r (hEUhFj ubaubjx)r }r (hEUhFj hGjOhLj{hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (j~)r }r (hEX log_ell=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_ell=0.0r r }r (hEUhFj ubahLjubj~)r }r (hEXv=2hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXv=2r r }r (hEUhFj ubahLjubj~)r }r (hEX log_sigma=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_sigma=0.0r r }r (hEUhFj ubahLjubeubj)r }r (hEUhFj hGNhLjhN}r (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r j)r }r (hEUhN}r (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidj uhFj h@]r j)r }r (hEUhN}r (hP]hQ]r jahR]hS]hU]uhFj h@]r haX[source]r r }r (hEUhFj ubahLjubahLjubaubeubj)r }r (hEUhFj hGjOhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (hs)r }r (hEXMPiecewise polynomial kernel with compact support. hyp = [log_ell, log_sigma]r hFj hGXc/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.PiecePolyhLhwhN}r (hP]hQ]hR]hS]hU]uhWKhXhh@]r haXMPiecewise polynomial kernel with compact support. hyp = [log_ell, log_sigma]r r }r (hEj hFj ubaubj)r }r (hEUhFj hGNhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r j)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (j)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX Parametersr r }r (hEUhFj ubahLjubj)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r j )r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (h)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (jK)r }r (hEXlog_ellhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXlog_ellr r }r (hEUhFj ubahLjSubhaX -- r r }r (hEUhFj ubhaXcharacteristic length scale.r r }r (hEXcharacteristic length scale.hFj ubehLhwubahLhubh)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (jK)r }r (hEX log_sigmahN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_sigmar r }r (hEUhFj ubahLjSubhaX -- r r! }r" (hEUhFj ubhaXsignal deviation.r# r$ }r% (hEXsignal deviation.hFj ubehLhwubahLhubh)r& }r' (hEUhN}r( (hP]hQ]hR]hS]hU]uhFj h@]r) hs)r* }r+ (hEUhN}r, (hP]hQ]hR]hS]hU]uhFj& h@]r- (jK)r. }r/ (hEXvhN}r0 (hP]hQ]hR]hS]hU]uhFj* h@]r1 haXvr2 }r3 (hEUhFj. ubahLjSubhaX -- r4 r5 }r6 (hEUhFj* ubhaXbdegree v will be rounded to 0,1,2,or 3. (not treated as hyperparameter, i.e. will not be trained).r7 r8 }r9 (hEXbdegree v will be rounded to 0,1,2,or 3. (not treated as hyperparameter, i.e. will not be trained).hFj* ubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)r: }r; (hEUhFj$hGNhLj7hN}r< (hS]hR]hP]hQ]hU]Uentries]r= (j:XPoly (class in pyGPs.Core.cov)hUtr> auhWNhXhh@]ubjA)r? }r@ (hEUhFj$hGNhLjDhN}rA (jFjGXpyhS]hR]hP]hQ]hU]jHXclassrB jJjB uhWNhXhh@]rC (jL)rD }rE (hEX#Poly(log_c=0.0, d=2, log_sigma=0.0)hFj? hGjOhLjPhN}rF (hS]rG hajShHXpyGPs.Core.covrH rI }rJ bhR]hP]hQ]hU]rK hajXXPolyrL jZUj[uhWNhXhh@]rM (j])rN }rO (hEXclass hFjD hGjOhLj`hN}rP (hP]hQ]hR]hS]hU]uhWNhXhh@]rQ haXclass rR rS }rT (hEUhFjN ubaubjf)rU }rV (hEXpyGPs.Core.cov.hFjD hGjOhLjihN}rW (hP]hQ]hR]hS]hU]uhWNhXhh@]rX haXpyGPs.Core.cov.rY rZ }r[ (hEUhFjU ubaubjo)r\ }r] (hEjL hFjD hGjOhLjrhN}r^ (hP]hQ]hR]hS]hU]uhWNhXhh@]r_ haXPolyr` ra }rb (hEUhFj\ ubaubjx)rc }rd (hEUhFjD hGjOhLj{hN}re (hP]hQ]hR]hS]hU]uhWNhXhh@]rf (j~)rg }rh (hEX log_c=0.0hN}ri (hP]hQ]hR]hS]hU]uhFjc h@]rj haX log_c=0.0rk rl }rm (hEUhFjg ubahLjubj~)rn }ro (hEXd=2hN}rp (hP]hQ]hR]hS]hU]uhFjc h@]rq haXd=2rr rs }rt (hEUhFjn ubahLjubj~)ru }rv (hEX log_sigma=0.0hN}rw (hP]hQ]hR]hS]hU]uhFjc h@]rx haX log_sigma=0.0ry rz }r{ (hEUhFju ubahLjubeubj)r| }r} (hEUhFjD hGNhLjhN}r~ (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r j)r }r (hEUhN}r (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjL uhFj| h@]r j)r }r (hEUhN}r (hP]hQ]r jahR]hS]hU]uhFj h@]r haX[source]r r }r (hEUhFj ubahLjubahLjubaubeubj)r }r (hEUhFj? hGjOhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (hs)r }r (hEX:Polynomial covariance function. hyp = [ log_c, log_sigma ]r hFj hGX^/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.PolyhLhwhN}r (hP]hQ]hR]hS]hU]uhWKhXhh@]r haX:Polynomial covariance function. hyp = [ log_c, log_sigma ]r r }r (hEj hFj ubaubj)r }r (hEUhFj hGNhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r j)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (j)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX Parametersr r }r (hEUhFj ubahLjubj)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r j )r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (h)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (jK)r }r (hEXlog_chN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXlog_cr r }r (hEUhFj ubahLjSubhaX -- r r }r (hEUhFj ubhaXinhomogeneous offset.r r }r (hEXinhomogeneous offset.hFj ubehLhwubahLhubh)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (jK)r }r (hEX log_sigmahN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_sigmar r }r (hEUhFj ubahLjSubhaX -- r r }r (hEUhFj ubhaXsignal deviation.r r }r (hEXsignal deviation.hFj ubehLhwubahLhubh)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (jK)r }r (hEXdhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXdr }r (hEUhFj ubahLjSubhaX -- r r }r (hEUhFj ubhaXOdegree of polynomial (not treated as hyperparameter, i.e. will not be trained).r r }r (hEXOdegree of polynomial (not treated as hyperparameter, i.e. will not be trained).hFj ubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)r }r (hEUhFj$hGNhLj7hN}r (hS]hR]hP]hQ]hU]Uentries]r (j:XPre (class in pyGPs.Core.cov)h(Utr auhWNhXhh@]ubjA)r }r (hEUhFj$hGNhLjDhN}r (jFjGXpyhS]hR]hP]hQ]hU]jHXclassr jJj uhWNhXhh@]r (jL)r }r (hEX Pre(M1, M2)hFj hGjOhLjPhN}r (hS]r h(ajShHXpyGPs.Core.covr r }r bhR]hP]hQ]hU]r h(ajXXPrer jZUj[uhWNhXhh@]r (j])r }r (hEXclass hFj hGjOhLj`hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXclass r r }r (hEUhFj ubaubjf)r }r (hEXpyGPs.Core.cov.hFj hGjOhLjihN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXpyGPs.Core.cov.r r }r (hEUhFj ubaubjo)r }r (hEj hFj hGjOhLjrhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXPrer r }r (hEUhFj ubaubjx)r }r (hEUhFj hGjOhLj{hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (j~)r }r (hEXM1hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXM1r r }r (hEUhFj ubahLjubj~)r! }r" (hEXM2hN}r# (hP]hQ]hR]hS]hU]uhFj h@]r$ haXM2r% r& }r' (hEUhFj! ubahLjubeubj)r( }r) (hEUhFj hGNhLjhN}r* (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r+ j)r, }r- (hEUhN}r. (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidj uhFj( h@]r/ j)r0 }r1 (hEUhN}r2 (hP]hQ]r3 jahR]hS]hU]uhFj, h@]r4 haX[source]r5 r6 }r7 (hEUhFj0 ubahLjubahLjubaubeubj)r8 }r9 (hEUhFj hGjOhLjhN}r: (hP]hQ]hR]hS]hU]uhWNhXhh@]r; (hs)r< }r= (hEXQPrecomputed kernel matrix. No hyperparameters and thus nothing will be optimised.r> hFj8 hGX]/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.PrehLhwhN}r? (hP]hQ]hR]hS]hU]uhWKhXhh@]r@ haXQPrecomputed kernel matrix. No hyperparameters and thus nothing will be optimised.rA rB }rC (hEj> hFj< ubaubj)rD }rE (hEUhFj8 hGNhLjhN}rF (hP]hQ]hR]hS]hU]uhWNhXhh@]rG j)rH }rI (hEUhN}rJ (hP]hQ]hR]hS]hU]uhFjD h@]rK (j)rL }rM (hEUhN}rN (hP]hQ]hR]hS]hU]uhFjH h@]rO haX ParametersrP rQ }rR (hEUhFjL ubahLjubj)rS }rT (hEUhN}rU (hP]hQ]hR]hS]hU]uhFjH h@]rV j )rW }rX (hEUhN}rY (hP]hQ]hR]hS]hU]uhFjS h@]rZ (h)r[ }r\ (hEUhN}r] (hP]hQ]hR]hS]hU]uhFjW h@]r^ hs)r_ }r` (hEUhN}ra (hP]hQ]hR]hS]hU]uhFj[ h@]rb (jK)rc }rd (hEXM1hN}re (hP]hQ]hR]hS]hU]uhFj_ h@]rf haXM1rg rh }ri (hEUhFjc ubahLjSubhaX -- rj rk }rl (hEUhFj_ ubhaXbcross covariances matrix(train+1 by test). last row is self covariances (diagonal of test by test)rm rn }ro (hEXbcross covariances matrix(train+1 by test). last row is self covariances (diagonal of test by test)hFj_ ubehLhwubahLhubh)rp }rq (hEUhN}rr (hP]hQ]hR]hS]hU]uhFjW h@]rs hs)rt }ru (hEUhN}rv (hP]hQ]hR]hS]hU]uhFjp h@]rw (jK)rx }ry (hEXM2hN}rz (hP]hQ]hR]hS]hU]uhFjt h@]r{ haXM2r| r} }r~ (hEUhFjx ubahLjSubhaX -- r r }r (hEUhFjt ubhaX/training set covariance matrix (train by train)r r }r (hEX/training set covariance matrix (train by train)hFjt ubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)r }r (hEUhFj$hGXi/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.ProductOfKernelr hLj7hN}r (hS]hR]hP]hQ]hU]Uentries]r (j:X)ProductOfKernel (class in pyGPs.Core.cov)hUtr auhWNhXhh@]ubjA)r }r (hEUhFj$hGj hLjDhN}r (jFjGXpyhS]hR]hP]hQ]hU]jHXclassr jJj uhWNhXhh@]r (jL)r }r (hEXProductOfKernel(cov1, cov2)hFj hGjOhLjPhN}r (hS]r hajShHXpyGPs.Core.covr r }r bhR]hP]hQ]hU]r hajXXProductOfKernelr jZUj[uhWNhXhh@]r (j])r }r (hEXclass hFj hGjOhLj`hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXclass r r }r (hEUhFj ubaubjf)r }r (hEXpyGPs.Core.cov.hFj hGjOhLjihN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXpyGPs.Core.cov.r r }r (hEUhFj ubaubjo)r }r (hEj hFj hGjOhLjrhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXProductOfKernelr r }r (hEUhFj ubaubjx)r }r (hEUhFj hGjOhLj{hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (j~)r }r (hEXcov1hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXcov1r r }r (hEUhFj ubahLjubj~)r }r (hEXcov2hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXcov2r r }r (hEUhFj ubahLjubeubj)r }r (hEUhFj hGNhLjhN}r (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r j)r }r (hEUhN}r (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidj uhFj h@]r j)r }r (hEUhN}r (hP]hQ]r jahR]hS]hU]uhFj h@]r haX[source]r r }r (hEUhFj ubahLjubahLjubaubeubj)r }r (hEUhFj hGjOhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r hs)r }r (hEXProduct of two kernel function.r hFj hGj hLhwhN}r (hP]hQ]hR]hS]hU]uhWKhXhh@]r haXProduct of two kernel function.r r }r (hEj hFj ubaubaubeubj3)r }r (hEUhFj$hGNhLj7hN}r (hS]hR]hP]hQ]hU]Uentries]r (j:XRBF (class in pyGPs.Core.cov)h#Utr auhWNhXhh@]ubjA)r }r (hEUhFj$hGNhLjDhN}r (jFjGXpyhS]hR]hP]hQ]hU]jHXclassr jJj uhWNhXhh@]r (jL)r }r (hEXRBF(log_ell=0.0, log_sigma=0.0)hFj hGjOhLjPhN}r (hS]r h#ajShHXpyGPs.Core.covr r }r bhR]hP]hQ]hU]r h#ajXXRBFr jZUj[uhWNhXhh@]r (j])r }r (hEXclass hFj hGjOhLj`hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXclass r r }r (hEUhFj ubaubjf)r }r (hEXpyGPs.Core.cov.hFj hGjOhLjihN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXpyGPs.Core.cov.r r }r (hEUhFj ubaubjo)r }r (hEj hFj hGjOhLjrhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXRBFr r }r (hEUhFj ubaubjx)r }r (hEUhFj hGjOhLj{hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (j~)r }r (hEX log_ell=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_ell=0.0r r }r (hEUhFj ubahLjubj~)r }r (hEX log_sigma=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_sigma=0.0r r }r (hEUhFj ubahLjubeubj)r }r (hEUhFj hGNhLjhN}r (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r j)r }r (hEUhN}r (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidj uhFj h@]r j)r }r! (hEUhN}r" (hP]hQ]r# jahR]hS]hU]uhFj h@]r$ haX[source]r% r& }r' (hEUhFj ubahLjubahLjubaubeubj)r( }r) (hEUhFj hGjOhLjhN}r* (hP]hQ]hR]hS]hU]uhWNhXhh@]r+ (hs)r, }r- (hEXVSquared Exponential kernel with isotropic distance measure. hyp = [log_ell, log_sigma]r. hFj( hGX]/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.RBFhLhwhN}r/ (hP]hQ]hR]hS]hU]uhWKhXhh@]r0 haXVSquared Exponential kernel with isotropic distance measure. hyp = [log_ell, log_sigma]r1 r2 }r3 (hEj. hFj, ubaubj)r4 }r5 (hEUhFj( hGNhLjhN}r6 (hP]hQ]hR]hS]hU]uhWNhXhh@]r7 j)r8 }r9 (hEUhN}r: (hP]hQ]hR]hS]hU]uhFj4 h@]r; (j)r< }r= (hEUhN}r> (hP]hQ]hR]hS]hU]uhFj8 h@]r? haX Parametersr@ rA }rB (hEUhFj< ubahLjubj)rC }rD (hEUhN}rE (hP]hQ]hR]hS]hU]uhFj8 h@]rF j )rG }rH (hEUhN}rI (hP]hQ]hR]hS]hU]uhFjC h@]rJ (h)rK }rL (hEUhN}rM (hP]hQ]hR]hS]hU]uhFjG h@]rN hs)rO }rP (hEUhN}rQ (hP]hQ]hR]hS]hU]uhFjK h@]rR (jK)rS }rT (hEXlog_ellhN}rU (hP]hQ]hR]hS]hU]uhFjO h@]rV haXlog_ellrW rX }rY (hEUhFjS ubahLjSubhaX -- rZ r[ }r\ (hEUhFjO ubhaXcharacteristic length scale.r] r^ }r_ (hEXcharacteristic length scale.hFjO ubehLhwubahLhubh)r` }ra (hEUhN}rb (hP]hQ]hR]hS]hU]uhFjG h@]rc hs)rd }re (hEUhN}rf (hP]hQ]hR]hS]hU]uhFj` h@]rg (jK)rh }ri (hEX log_sigmahN}rj (hP]hQ]hR]hS]hU]uhFjd h@]rk haX log_sigmarl rm }rn (hEUhFjh ubahLjSubhaX -- ro rp }rq (hEUhFjd ubhaXsignal deviation.rr rs }rt (hEXsignal deviation.hFjd ubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)ru }rv (hEUhFj$hGNhLj7hN}rw (hS]hR]hP]hQ]hU]Uentries]rx (j:X RBFard (class in pyGPs.Core.cov)hUtry auhWNhXhh@]ubjA)rz }r{ (hEUhFj$hGNhLjDhN}r| (jFjGXpyhS]hR]hP]hQ]hU]jHXclassr} jJj} uhWNhXhh@]r~ (jL)r }r (hEX0RBFard(D=None, log_ell_list=None, log_sigma=0.0)hFjz hGjOhLjPhN}r (hS]r hajShHXpyGPs.Core.covr r }r bhR]hP]hQ]hU]r hajXXRBFardr jZUj[uhWNhXhh@]r (j])r }r (hEXclass hFj hGjOhLj`hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXclass r r }r (hEUhFj ubaubjf)r }r (hEXpyGPs.Core.cov.hFj hGjOhLjihN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXpyGPs.Core.cov.r r }r (hEUhFj ubaubjo)r }r (hEj hFj hGjOhLjrhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXRBFardr r }r (hEUhFj ubaubjx)r }r (hEUhFj hGjOhLj{hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (j~)r }r (hEXD=NonehN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXD=Noner r }r (hEUhFj ubahLjubj~)r }r (hEXlog_ell_list=NonehN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXlog_ell_list=Noner r }r (hEUhFj ubahLjubj~)r }r (hEX log_sigma=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_sigma=0.0r r }r (hEUhFj ubahLjubeubj)r }r (hEUhFj hGNhLjhN}r (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r j)r }r (hEUhN}r (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidj uhFj h@]r j)r }r (hEUhN}r (hP]hQ]r jahR]hS]hU]uhFj h@]r haX[source]r r }r (hEUhFj ubahLjubahLjubaubeubj)r }r (hEUhFjz hGjOhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (hs)r }r (hEXcSquared Exponential kernel with Automatic Relevance Determination. hyp = log_ell_list + [log_sigma]r hFj hGX`/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.RBFardhLhwhN}r (hP]hQ]hR]hS]hU]uhWKhXhh@]r haXcSquared Exponential kernel with Automatic Relevance Determination. hyp = log_ell_list + [log_sigma]r r }r (hEj hFj ubaubj)r }r (hEUhFj hGNhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r j)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (j)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX Parametersr r }r (hEUhFj ubahLjubj)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r j )r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (h)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (jK)r }r (hEXDhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXDr }r (hEUhFj ubahLjSubhaX -- r r }r (hEUhFj ubhaXQdimension of pattern. set if you want default ell, which is 1 for each dimension.r r }r (hEXQdimension of pattern. set if you want default ell, which is 1 for each dimension.hFj ubehLhwubahLhubh)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (jK)r }r (hEX log_ell_listhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_ell_listr r }r (hEUhFj ubahLjSubhaX -- r r }r (hEUhFj ubhaX/characteristic length scale for each dimension.r r }r (hEX/characteristic length scale for each dimension.hFj ubehLhwubahLhubh)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (jK)r }r (hEX log_sigmahN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_sigmar r }r! (hEUhFj ubahLjSubhaX -- r" r# }r$ (hEUhFj ubhaXsignal deviation.r% r& }r' (hEXsignal deviation.hFj ubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)r( }r) (hEUhFj$hGNhLj7hN}r* (hS]hR]hP]hQ]hU]Uentries]r+ (j:X!RBFunit (class in pyGPs.Core.cov)hUtr, auhWNhXhh@]ubjA)r- }r. (hEUhFj$hGNhLjDhN}r/ (jFjGXpyhS]hR]hP]hQ]hU]jHXclassr0 jJj0 uhWNhXhh@]r1 (jL)r2 }r3 (hEXRBFunit(log_ell=0.0)hFj- hGjOhLjPhN}r4 (hS]r5 hajShHXpyGPs.Core.covr6 r7 }r8 bhR]hP]hQ]hU]r9 hajXXRBFunitr: jZUj[uhWNhXhh@]r; (j])r< }r= (hEXclass hFj2 hGjOhLj`hN}r> (hP]hQ]hR]hS]hU]uhWNhXhh@]r? haXclass r@ rA }rB (hEUhFj< ubaubjf)rC }rD (hEXpyGPs.Core.cov.hFj2 hGjOhLjihN}rE (hP]hQ]hR]hS]hU]uhWNhXhh@]rF haXpyGPs.Core.cov.rG rH }rI (hEUhFjC ubaubjo)rJ }rK (hEj: hFj2 hGjOhLjrhN}rL (hP]hQ]hR]hS]hU]uhWNhXhh@]rM haXRBFunitrN rO }rP (hEUhFjJ ubaubjx)rQ }rR (hEUhFj2 hGjOhLj{hN}rS (hP]hQ]hR]hS]hU]uhWNhXhh@]rT j~)rU }rV (hEX log_ell=0.0hN}rW (hP]hQ]hR]hS]hU]uhFjQ h@]rX haX log_ell=0.0rY rZ }r[ (hEUhFjU ubahLjubaubj)r\ }r] (hEUhFj2 hGNhLjhN}r^ (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r_ j)r` }ra (hEUhN}rb (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidj: uhFj\ h@]rc j)rd }re (hEUhN}rf (hP]hQ]rg jahR]hS]hU]uhFj` h@]rh haX[source]ri rj }rk (hEUhFjd ubahLjubahLjubaubeubj)rl }rm (hEUhFj- hGjOhLjhN}rn (hP]hQ]hR]hS]hU]uhWNhXhh@]ro (hs)rp }rq (hEXSquared Exponential kernel with isotropic distance measure with unit magnitude. i.e signal variance is always 1. hyp = [ log_ell ]rr hFjl hGXa/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.RBFunithLhwhN}rs (hP]hQ]hR]hS]hU]uhWKhXhh@]rt haXSquared Exponential kernel with isotropic distance measure with unit magnitude. i.e signal variance is always 1. hyp = [ log_ell ]ru rv }rw (hEjr hFjp ubaubj)rx }ry (hEUhFjl hGNhLjhN}rz (hP]hQ]hR]hS]hU]uhWNhXhh@]r{ j)r| }r} (hEUhN}r~ (hP]hQ]hR]hS]hU]uhFjx h@]r (j)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj| h@]r haX Parametersr r }r (hEUhFj ubahLjubj)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj| h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (jK)r }r (hEXlog_ellhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXlog_ellr r }r (hEUhFj ubahLjSubhaX -- r r }r (hEUhFj ubhaXcharacteristic length scale.r r }r (hEXcharacteristic length scale.hFj ubehLhwubahLjubehLjubaubeubeubj3)r }r (hEUhFj$hGNhLj7hN}r (hS]hR]hP]hQ]hU]Uentries]r (j:XRQ (class in pyGPs.Core.cov)hUtr auhWNhXhh@]ubjA)r }r (hEUhFj$hGNhLjDhN}r (jFjGXpyhS]hR]hP]hQ]hU]jHXclassr jJj uhWNhXhh@]r (jL)r }r (hEX-RQ(log_ell=0.0, log_sigma=0.0, log_alpha=0.0)hFj hGjOhLjPhN}r (hS]r hajShHXpyGPs.Core.covr r }r bhR]hP]hQ]hU]r hajXXRQr jZUj[uhWNhXhh@]r (j])r }r (hEXclass hFj hGjOhLj`hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXclass r r }r (hEUhFj ubaubjf)r }r (hEXpyGPs.Core.cov.hFj hGjOhLjihN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXpyGPs.Core.cov.r r }r (hEUhFj ubaubjo)r }r (hEj hFj hGjOhLjrhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXRQr r }r (hEUhFj ubaubjx)r }r (hEUhFj hGjOhLj{hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (j~)r }r (hEX log_ell=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_ell=0.0r r }r (hEUhFj ubahLjubj~)r }r (hEX log_sigma=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_sigma=0.0r r }r (hEUhFj ubahLjubj~)r }r (hEX log_alpha=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_alpha=0.0r r }r (hEUhFj ubahLjubeubj)r }r (hEUhFj hGNhLjhN}r (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r j)r }r (hEUhN}r (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidj uhFj h@]r j)r }r (hEUhN}r (hP]hQ]r jahR]hS]hU]uhFj h@]r haX[source]r r }r (hEUhFj ubahLjubahLjubaubeubj)r }r (hEUhFj hGjOhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (hs)r }r (hEXoRational Quadratic covariance function with isotropic distance measure. hyp = [ log_ell, log_sigma, log_alpha ]r hFj hGX\/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.RQhLhwhN}r (hP]hQ]hR]hS]hU]uhWKhXhh@]r haXoRational Quadratic covariance function with isotropic distance measure. hyp = [ log_ell, log_sigma, log_alpha ]r r }r (hEj hFj ubaubj)r }r (hEUhFj hGNhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r j)r }r (hEUhN}r(hP]hQ]hR]hS]hU]uhFj h@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFj h@]rhaX Parametersrr}r(hEUhFjubahLjubj)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r j )r }r(hEUhN}r(hP]hQ]hR]hS]hU]uhFj h@]r(h)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFj h@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r(hEXlog_ellhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXlog_ellrr}r(hEUhFjubahLjSubhaX -- r r!}r"(hEUhFjubhaXcharacteristic length scale.r#r$}r%(hEXcharacteristic length scale.hFjubehLhwubahLhubh)r&}r'(hEUhN}r((hP]hQ]hR]hS]hU]uhFj h@]r)hs)r*}r+(hEUhN}r,(hP]hQ]hR]hS]hU]uhFj&h@]r-(jK)r.}r/(hEX log_sigmahN}r0(hP]hQ]hR]hS]hU]uhFj*h@]r1haX log_sigmar2r3}r4(hEUhFj.ubahLjSubhaX -- r5r6}r7(hEUhFj*ubhaXsignal deviation.r8r9}r:(hEXsignal deviation.hFj*ubehLhwubahLhubh)r;}r<(hEUhN}r=(hP]hQ]hR]hS]hU]uhFj h@]r>hs)r?}r@(hEUhN}rA(hP]hQ]hR]hS]hU]uhFj;h@]rB(jK)rC}rD(hEX log_alphahN}rE(hP]hQ]hR]hS]hU]uhFj?h@]rFhaX log_alpharGrH}rI(hEUhFjCubahLjSubhaX -- rJrK}rL(hEUhFj?ubhaX&shape parameter for the RQ covariance.rMrN}rO(hEX&shape parameter for the RQ covariance.hFj?ubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)rP}rQ(hEUhFj$hGNhLj7hN}rR(hS]hR]hP]hQ]hU]Uentries]rS(j:XRQard (class in pyGPs.Core.cov)hUtrTauhWNhXhh@]ubjA)rU}rV(hEUhFj$hGNhLjDhN}rW(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrXjJjXuhWNhXhh@]rY(jL)rZ}r[(hEX>RQard(D=None, log_ell_list=None, log_sigma=0.0, log_alpha=0.0)hFjUhGjOhLjPhN}r\(hS]r]hajShHXpyGPs.Core.covr^r_}r`bhR]hP]hQ]hU]rahajXXRQardrbjZUj[uhWNhXhh@]rc(j])rd}re(hEXclass hFjZhGjOhLj`hN}rf(hP]hQ]hR]hS]hU]uhWNhXhh@]rghaXclass rhri}rj(hEUhFjdubaubjf)rk}rl(hEXpyGPs.Core.cov.hFjZhGjOhLjihN}rm(hP]hQ]hR]hS]hU]uhWNhXhh@]rnhaXpyGPs.Core.cov.rorp}rq(hEUhFjkubaubjo)rr}rs(hEjbhFjZhGjOhLjrhN}rt(hP]hQ]hR]hS]hU]uhWNhXhh@]ruhaXRQardrvrw}rx(hEUhFjrubaubjx)ry}rz(hEUhFjZhGjOhLj{hN}r{(hP]hQ]hR]hS]hU]uhWNhXhh@]r|(j~)r}}r~(hEXD=NonehN}r(hP]hQ]hR]hS]hU]uhFjyh@]rhaXD=Nonerr}r(hEUhFj}ubahLjubj~)r}r(hEXlog_ell_list=NonehN}r(hP]hQ]hR]hS]hU]uhFjyh@]rhaXlog_ell_list=Nonerr}r(hEUhFjubahLjubj~)r}r(hEX log_sigma=0.0hN}r(hP]hQ]hR]hS]hU]uhFjyh@]rhaX log_sigma=0.0rr}r(hEUhFjubahLjubj~)r}r(hEX log_alpha=0.0hN}r(hP]hQ]hR]hS]hU]uhFjyh@]rhaX log_alpha=0.0rr}r(hEUhFjubahLjubeubj)r}r(hEUhFjZhGNhLjhN}r(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rj)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjbuhFjh@]rj)r}r(hEUhN}r(hP]hQ]rjahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLjubahLjubaubeubj)r}r(hEUhFjUhGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r}r(hEXRational Quadratic covariance function with Automatic Relevance Detemination (ARD) distance measure. hyp = log_ell_list + [ log_sigma, log_alpha ]rhFjhGX_/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.RQardrhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaXRational Quadratic covariance function with Automatic Relevance Detemination (ARD) distance measure. hyp = log_ell_list + [ log_sigma, log_alpha ]rr}r(hEjhFjubaubj)r}r(hEUhFjhGNhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX Parametersrr}r(hEUhFjubahLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rj )r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(h)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r(hEXDhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXDr}r(hEUhFjubahLjSubhaX -- rr}r(hEUhFjubhaXSdimension of pattern. set if you want default ell, which is 0.5 for each dimension.rr}r(hEXSdimension of pattern. set if you want default ell, which is 0.5 for each dimension.rhFjubehLhwubahLhubh)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r(hEX log_ell_listhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_ell_listrr}r(hEUhFjubahLjSubhaX -- rr}r(hEUhFjubhaX/characteristic length scale for each dimension.rr}r(hEX/characteristic length scale for each dimension.rhFjubehLhwubahLhubh)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r(hEX log_sigmahN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_sigmarr}r(hEUhFjubahLjSubhaX -- rr}r (hEUhFjubhaXsignal deviation.r r }r (hEXsignal deviation.r 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definitionqubeubaubh|)q}q(h'XAYou can change to other likelihood or inference methods using: ::qh(hGh)h,h.hh0}q(h2]h3]h4]h5]h7]uh9K h:hh"]qhCX>You can change to other likelihood or inference methods using:qq}q(h'X>You can change to other likelihood or inference methods using:h(hubaubcdocutils.nodes literal_block q)q}q(h'X6model.useLikelihood(newLik) model.useInference(newInf)h(hGh)h,h.U literal_blockqh0}q(U xml:spaceqUpreserveqh5]h4]h2]h3]h7]uh9Kh:hh"]qhCX6model.useLikelihood(newLik) model.useInference(newInf)qq}q(h'Uh(hubaubhU)q}q(h'Uh(hGh)Nh.hXh0}q(h2]h3]h4]h5]h7]uh9Nh:hh"]qh[)q}q(h'X*newLik* and *newInf* are **Strings**. Currently the options are: 1. Regression model * newLik: **"Laplace"**. Note this will force inference method to be EP. * newInf: **"EP"**, **"Laplace"**. 2. Classification model (including GPMC) * newInf: **"Laplace"** h(hh)h,h.h^h0}q(h2]h3]h4]h5]h7]uh9Kh"]q(ha)q}q(h'XA*newLik* and *newInf* are **Strings**. Currently the options are:qh(hh)h,h.heh0}q(h2]h3]h4]h5]h7]uh9Kh"]q(cdocutils.nodes emphasis q)q}q(h'X*newLik*h0}q(h2]h3]h4]h5]h7]uh(hh"]qhCXnewLikqхq}q(h'Uh(hubah.UemphasisqubhCX and qՅq}q(h'X and h(hubh)q}q(h'X*newInf*h0}q(h2]h3]h4]h5]h7]uh(hh"]qhCXnewInfq܅q}q(h'Uh(hubah.hubhCX are q߅q}q(h'X are h(hubcdocutils.nodes strong q)q}q(h'X **Strings**h0}q(h2]h3]h4]h5]h7]uh(hh"]qhCXStringsq煁q}q(h'Uh(hubah.UstrongqubhCX. Currently the options are:q녁q}q(h'X. Currently the options are:h(hubeubhk)q}q(h'Uh0}q(h2]h3]h4]h5]h7]uh(hh"]qcdocutils.nodes enumerated_list q)q}q(h'Uh0}q(UsuffixqU.h5]h4]h2]UprefixqUh3]h7]UenumtypeqUarabicquh(hh"]q(hv)q}q(h'X~Regression model * newLik: **"Laplace"**. Note this will force inference method to be EP. * newInf: **"EP"**, **"Laplace"**. h0}q(h2]h3]h4]h5]h7]uh(hh"]q(h|)q}r(h'XRegression modelrh(hh)h,h.hh0}r(h2]h3]h4]h5]h7]uh9Kh"]rhCXRegression modelrr}r(h'jh(hubaubhp)r}r(h'Uh0}r (htX*h5]h4]h2]h3]h7]uh(hh"]r (hv)r }r (h'XFnewLik: **"Laplace"**. Note this will force inference method to be EP.r h0}r(h2]h3]h4]h5]h7]uh(jh"]rh|)r}r(h'j h(j h)h,h.hh0}r(h2]h3]h4]h5]h7]uh9Kh"]r(hCXnewLik: rr}r(h'XnewLik: h(jubh)r}r(h'X **"Laplace"**h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCX "Laplace"rr}r(h'Uh(jubah.hubhCX1. Note this will force inference method to be EP.rr}r (h'X1. Note this will force inference method to be EP.h(jubeubah.hubhv)r!}r"(h'X!newInf: **"EP"**, **"Laplace"**. h0}r#(h2]h3]h4]h5]h7]uh(jh"]r$h|)r%}r&(h'X newInf: **"EP"**, **"Laplace"**.h(j!h)h,h.hh0}r'(h2]h3]h4]h5]h7]uh9Kh"]r((hCXnewInf: r)r*}r+(h'XnewInf: h(j%ubh)r,}r-(h'X**"EP"**h0}r.(h2]h3]h4]h5]h7]uh(j%h"]r/hCX"EP"r0r1}r2(h'Uh(j,ubah.hubhCX, r3r4}r5(h'X, h(j%ubh)r6}r7(h'X **"Laplace"**h0}r8(h2]h3]h4]h5]h7]uh(j%h"]r9hCX "Laplace"r:r;}r<(h'Uh(j6ubah.hubhCX.r=}r>(h'X.h(j%ubeubah.hubeh.hubeh.hubhv)r?}r@(h'XAClassification model (including GPMC) * newInf: **"Laplace"** h0}rA(h2]h3]h4]h5]h7]uh(hh"]rB(h|)rC}rD(h'X%Classification model (including GPMC)rEh(j?h)h,h.hh0}rF(h2]h3]h4]h5]h7]uh9Kh"]rGhCX%Classification model (including GPMC)rHrI}rJ(h'jEh(jCubaubhp)rK}rL(h'Uh0}rM(htX*h5]h4]h2]h3]h7]uh(j?h"]rNhv)rO}rP(h'XnewInf: **"Laplace"** h0}rQ(h2]h3]h4]h5]h7]uh(jKh"]rRh|)rS}rT(h'XnewInf: **"Laplace"**rUh(jOh)h,h.hh0}rV(h2]h3]h4]h5]h7]uh9Kh"]rW(hCXnewInf: rXrY}rZ(h'XnewInf: h(jSubh)r[}r\(h'X **"Laplace"**h0}r](h2]h3]h4]h5]h7]uh(jSh"]r^hCX "Laplace"r_r`}ra(h'Uh(j[ubah.hubeubah.hubah.hubeh.hubeh.Uenumerated_listrbubah.hubeubaubeubh$)rc}rd(h'Uh(h%h)h,h.h/h0}re(h2]h3]h4]h5]rfhah7]rghauh9Kh:hh"]rh(h<)ri}rj(h'X/Developing New Likelihood & Inference Functionsrkh(jch)h,h.h@h0}rl(h2]h3]h4]h5]h7]uh9Kh:hh"]rmhCX/Developing New Likelihood & Inference Functionsrnro}rp(h'jkh(jiubaubh|)rq}rr(h'XWe also support the development of new likelihood/inference classes, your customized inference class need to follow the template as below: ::rsh(jch)h,h.hh0}rt(h2]h3]h4]h5]h7]uh9Kh:hh"]ruhCXWe also support the development of new likelihood/inference classes, your customized inference class need to follow the template as below:rvrw}rx(h'XWe also support the development of new likelihood/inference classes, your customized inference class need to follow the template as below:h(jqubaubh)ry}rz(h'X# Your inference class needs to inherit base class Inference, # which is in the module of Core.inf class MyKernel(Kernel): def __init__(self): pass def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): ''' Inference computation based on inputs. post, nlZ, dnlZ = inf.proceed(mean, cov, lik, x, y) INPUT: cov: name of the covariance function (see covFunctions.m) lik: name of the likelihood function (see likFunctions.m) x: n by D matrix of training inputs y: 1d array (of size n) of targets OUTPUT: post(instance of postStruct): struct representation of the (approximate) posterior containing: nlZ: returned value of the negative log marginal likelihood dnlZ(instance of dnlZStruct): struct representation for derivatives of the negative log marginal likelihood w.r.t. each hyperparameter. Usually, the approximate posterior to be returned admits the form: N(m=K*alpha, V=inv(inv(K)+W)), where alpha is a vector and W is diagonal; if not, then L contains instead -inv(K+inv(W)), and sW is unused. For more information on the individual approximation methods and their implementations, see the respective inference function below. See also gp.py :param meanfunc: mean function :param covfunc: covariance function :param likfunc: likelihood function :param x: training data :param y: training labels :param nargout: specify the number of output(1,2 or 3) :return: posterior, negative-log-marginal-likelihood, derivative for negative-log-marginal-likelihood-likelihood ''' passh(jch)h,h.hh0}r{(hhh5]h4]h2]h3]h7]uh9K!h:hh"]r|hCX# Your inference class needs to inherit base class Inference, # which is in the module of Core.inf class MyKernel(Kernel): def __init__(self): pass def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): ''' Inference computation based on inputs. post, nlZ, dnlZ = inf.proceed(mean, cov, lik, x, y) INPUT: cov: name of the covariance function (see covFunctions.m) lik: name of the likelihood function (see likFunctions.m) x: n by D matrix of training inputs y: 1d array (of size n) of targets OUTPUT: post(instance of postStruct): struct representation of the (approximate) posterior containing: nlZ: returned value of the negative log marginal likelihood dnlZ(instance of dnlZStruct): struct representation for derivatives of the negative log marginal likelihood w.r.t. each hyperparameter. Usually, the approximate posterior to be returned admits the form: N(m=K*alpha, V=inv(inv(K)+W)), where alpha is a vector and W is diagonal; if not, then L contains instead -inv(K+inv(W)), and sW is unused. For more information on the individual approximation methods and their implementations, see the respective inference function below. See also gp.py :param meanfunc: mean function :param covfunc: covariance function :param likfunc: likelihood function :param x: training data :param y: training labels :param nargout: specify the number of output(1,2 or 3) :return: posterior, negative-log-marginal-likelihood, derivative for negative-log-marginal-likelihood-likelihood ''' passr}r~}r(h'Uh(jyubaubh|)r}r(h'XIwhere **postStruct** and **dnlZStruct** is also defined in *Core.inf*. ::rh(jch)h,h.hh0}r(h2]h3]h4]h5]h7]uh9KJh:hh"]r(hCXwhere rr}r(h'Xwhere h(jubh)r}r(h'X**postStruct**h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCX postStructrr}r(h'Uh(jubah.hubhCX and rr}r(h'X and h(jubh)r}r(h'X**dnlZStruct**h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCX dnlZStructrr}r(h'Uh(jubah.hubhCX is also defined in rr}r(h'X is also defined in h(jubh)r}r(h'X *Core.inf*h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCXCore.infrr}r(h'Uh(jubah.hubhCX.r}r(h'X.h(jubeubh)r}r(h'Xclass postStruct(object): ''' Data structure for posterior post.alpha -> 1d array containing inv(K)*m, where K is the prior covariance matrix and m the approx posterior mean post.sW: -> 1d array containing diagonal of sqrt(W) the approximate posterior covariance matrix is inv(inv(K)+W) post.L -> 2d array, L = chol(sW*K*sW+identity(n)) ''' class dnlZStruct(object): ''' Data structure for the derivatives of mean, cov and lik functions. dnlZ.mean -> list of derivatives for each hyperparameters in mean function dnlZ.cov -> list of derivatives for each hyperparameters in covariance function dnlZ.lik -> list of derivatives for each hyperparameters in likelihood function '''h(jch)h,h.hh0}r(hhh5]h4]h2]h3]h7]uh9KLh:hh"]rhCXclass postStruct(object): ''' Data structure for posterior post.alpha -> 1d array containing inv(K)*m, where K is the prior covariance matrix and m the approx posterior mean post.sW: -> 1d array containing diagonal of sqrt(W) the approximate posterior covariance matrix is inv(inv(K)+W) post.L -> 2d array, L = chol(sW*K*sW+identity(n)) ''' class dnlZStruct(object): ''' Data structure for the derivatives of mean, cov and lik functions. dnlZ.mean -> list of derivatives for each hyperparameters in mean function dnlZ.cov -> list of derivatives for each hyperparameters in covariance function dnlZ.lik -> list of derivatives for each hyperparameters in likelihood function '''rr}r(h'Uh(jubaubh|)r}r(h'XCustomizing likelihood function is more complicated. We will omit it here to keep this this page not too long. However, you can find detailed explaination either in the **source code** *Core.lik* or in coresponding section of **manual**.h(jch)h,h.hh0}r(h2]h3]h4]h5]h7]uh9Kah:hh"]r(hCXCustomizing likelihood function is more complicated. We will omit it here to keep this this page not too long. However, you can find detailed explaination either in the rr}r(h'XCustomizing likelihood function is more complicated. We will omit it here to keep this this page not too long. However, you can find detailed explaination either in the h(jubh)r}r(h'X**source code**h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCX source coderr}r(h'Uh(jubah.hubhCX r}r(h'X h(jubh)r}r(h'X *Core.lik*h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCXCore.likrr}r(h'Uh(jubah.hubhCX or in coresponding section of rr}r(h'X or in coresponding section of h(jubh)r}r(h'X **manual**h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCXmanualrr}r(h'Uh(jubah.hubhCX.r}r(h'X.h(jubeubh|)r}r(h'XJust like testing kernel/mean fucntions, you can also find unit test module for likelihood and inference functions. To test your customized inference function, uncomment the following method in *pyGPs.Testing.unit_test_inf.py*. ::h(jch)h,h.hh0}r(h2]h3]h4]h5]h7]uh9Kdh:hh"]r(hCXJust like testing kernel/mean fucntions, you can also find unit test module for likelihood and inference functions. To test your customized inference function, uncomment the following method in rr}r(h'XJust like testing kernel/mean fucntions, you can also find unit test module for likelihood and inference functions. To test your customized inference function, uncomment the following method in h(jubh)r}r(h'X *pyGPs.Testing.unit_test_inf.py*h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCXpyGPs.Testing.unit_test_inf.pyrr}r(h'Uh(jubah.hubhCX.r}r(h'X.h(jubeubh)r}r(h'X # Test your customized inference function def test_inf_new(self): # specify your inf function # set mean/cov/lik functions post, nlZ, dnlZ = inffunc.proceed(meanfunc, covfunc, likfunc, self.x, self.y, nargout=3) self.checkFITCOutput(post, nlZ, dnlZ)h(jch)h,h.hh0}r(hhh5]h4]h2]h3]h7]uh9Kgh:hh"]rhCX # Test your customized inference function def test_inf_new(self): # specify your inf function # set mean/cov/lik functions post, nlZ, dnlZ = inffunc.proceed(meanfunc, covfunc, likfunc, self.x, self.y, nargout=3) self.checkFITCOutput(post, nlZ, dnlZ)rr}r(h'Uh(jubaubh|)r}r(h'XNand test customized likelihood function in *pyGPs.Testing.unit_test_lik.py* ::rh(jch)h,h.hh0}r(h2]h3]h4]h5]h7]uh9Knh:hh"]r(hCX+and test customized likelihood function in rr}r(h'X+and test customized likelihood function in h(jubh)r}r(h'X *pyGPs.Testing.unit_test_lik.py*h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCXpyGPs.Testing.unit_test_lik.pyrr}r(h'Uh(jubah.hubeubh)r}r(h'X# Test your customized likelihood function def test_cov_new(self): likelihood = myLikelihood() # specify your likelihood function self.checkLikelihood(likelihood)h(jch)h,h.hh0}r(hhh5]h4]h2]h3]h7]uh9Kph:hh"]rhCX# Test your customized likelihood function def test_cov_new(self): likelihood = myLikelihood() # specify your likelihood function self.checkLikelihood(likelihood)rr}r(h'Uh(jubaubeubh$)r}r(h'Uh(h%h)h,h.h/h0}r(h2]h3]h4]h5]rh!ah7]rhauh9Kxh:hh"]r(h<)r}r(h'XList of Likelihoodsrh(jh)h,h.h@h0}r(h2]h3]h4]h5]h7]uh9Kxh:hh"]rhCXList of Likelihoodsrr}r (h'jh(jubaubcsphinx.addnodes index r )r }r (h'Uh(jh)X]/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/lik.py:docstring of pyGPs.Core.lik.Erfr h.Uindexrh0}r(h5]h4]h2]h3]h7]Uentries]r(UsinglerXErf (class in pyGPs.Core.lik)h Utrauh9Nh:hh"]ubcsphinx.addnodes desc r)r}r(h'Uh(jh)j h.Udescrh0}r(UnoindexrUdomainrXpyh5]h4]h2]h3]h7]UobjtyperXclassrUdesctyperjuh9Nh:hh"]r(csphinx.addnodes desc_signature r)r}r (h'XErf()h(jh)U r!h.Udesc_signaturer"h0}r#(h5]r$h aUmoduler%h*XpyGPs.Core.likr&r'}r(bh4]h2]h3]h7]r)h aUfullnamer*XErfr+Uclassr,UUfirstr-uh9Nh:hh"]r.(csphinx.addnodes desc_annotation r/)r0}r1(h'Xclass h(jh)j!h.Udesc_annotationr2h0}r3(h2]h3]h4]h5]h7]uh9Nh:hh"]r4hCXclass r5r6}r7(h'Uh(j0ubaubcsphinx.addnodes desc_addname r8)r9}r:(h'XpyGPs.Core.lik.h(jh)j!h.U desc_addnamer;h0}r<(h2]h3]h4]h5]h7]uh9Nh:hh"]r=hCXpyGPs.Core.lik.r>r?}r@(h'Uh(j9ubaubcsphinx.addnodes desc_name rA)rB}rC(h'j+h(jh)j!h.U desc_namerDh0}rE(h2]h3]h4]h5]h7]uh9Nh:hh"]rFhCXErfrGrH}rI(h'Uh(jBubaubcsphinx.addnodes only rJ)rK}rL(h'Uh(jh)Nh.UonlyrMh0}rN(UexprUhtmlrOh5]h4]h2]h3]h7]uh9Nh:hh"]rPcsphinx.addnodes pending_xref rQ)rR}rS(h'Uh0}rT(UreftypeUviewcoderUUrefdocX LikelihoodsrVU refdomainUstdrWh5]h4]U refexplicith2]h3]h7]U reftargetX_modules/pyGPs/Core/likUrefidj+uh(jKh"]rXcdocutils.nodes inline rY)rZ}r[(h'Uh0}r\(h2]h3]r]U viewcode-linkr^ah4]h5]h7]uh(jRh"]r_hCX[source]r`ra}rb(h'Uh(jZubah.Uinlinercubah.U pending_xrefrdubaubeubcsphinx.addnodes desc_content re)rf}rg(h'Uh(jh)j!h.U desc_contentrhh0}ri(h2]h3]h4]h5]h7]uh9Nh:hh"]rj(h|)rk}rl(h'XiError function or cumulative Gaussian likelihood function for binary classification or probit regression.rmh(jfh)j h.hh0}rn(h2]h3]h4]h5]h7]uh9Kh:hh"]rohCXiError function or cumulative Gaussian likelihood function for binary classification or probit regression.rprq}rr(h'jmh(jkubaubh|)rs}rt(h'X@:math:`Erf(t)=\frac{1}{2}(1+erf(\frac{t}{\sqrt{2}}))=normcdf(t)`h(jfh)j h.hh0}ru(h2]h3]h4]h5]h7]uh9Kh:hh"]rvcsphinx.ext.mathbase math rw)rx}ry(h'Uh0}rz(UlatexX8Erf(t)=\frac{1}{2}(1+erf(\frac{t}{\sqrt{2}}))=normcdf(t)h5]h4]h2]h3]h7]uh(jsh"]h.Umathr{ubaubeubeubj )r|}r}(h'Uh(jh)X_/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/lik.py:docstring of pyGPs.Core.lik.Gaussr~h.jh0}r(h5]h4]h2]h3]h7]Uentries]r(jXGauss (class in pyGPs.Core.lik)hUtrauh9Nh:hh"]ubj)r}r(h'Uh(jh)j~h.jh0}r(jjXpyh5]h4]h2]h3]h7]jXclassrjjuh9Nh:hh"]r(j)r}r(h'X$Gauss(log_sigma=-2.3025850929940455)h(jh)j!h.j"h0}r(h5]rhaj%h*XpyGPs.Core.likrr}rbh4]h2]h3]h7]rhaj*XGaussrj,Uj-uh9Nh:hh"]r(j/)r}r(h'Xclass h(jh)j!h.j2h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXclass rr}r(h'Uh(jubaubj8)r}r(h'XpyGPs.Core.lik.h(jh)j!h.j;h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXpyGPs.Core.lik.rr}r(h'Uh(jubaubjA)r}r(h'jh(jh)j!h.jDh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXGaussrr}r(h'Uh(jubaubcsphinx.addnodes desc_parameterlist r)r}r(h'Uh(jh)j!h.Udesc_parameterlistrh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rcsphinx.addnodes desc_parameter r)r}r(h'Xlog_sigma=-2.3025850929940455h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCXlog_sigma=-2.3025850929940455rr}r(h'Uh(jubah.Udesc_parameterrubaubjJ)r}r(h'Uh(jh)Nh.jMh0}r(UexprjOh5]h4]h2]h3]h7]uh9Nh:hh"]rjQ)r}r(h'Uh0}r(UreftypejUUrefdocjVU refdomainjWh5]h4]U refexplicith2]h3]h7]U reftargetX_modules/pyGPs/Core/likUrefidjuh(jh"]rjY)r}r(h'Uh0}r(h2]h3]rj^ah4]h5]h7]uh(jh"]rhCX[source]rr}r(h'Uh(jubah.jcubah.jdubaubeubje)r}r(h'Uh(jh)j!h.jhh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]r(h|)r}r(h'X,Gaussian likelihood function for regression.rh(jh)j~h.hh0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]rhCX,Gaussian likelihood function for regression.rr}r(h'jh(jubaubh|)r}r(h'X:math:`Gauss(t)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(t-y)^2}{2\sigma^2}}`, where :math:`y` is the mean and :math:`\sigma` is the standard deviation.h(jh)j~h.hh0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]r(jw)r}r(h'Uh0}r(UlatexXDGauss(t)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(t-y)^2}{2\sigma^2}}h5]h4]h2]h3]h7]uh(jh"]h.j{ubhCX, where rr}r(h'X, where h(jubjw)r}r(h'Uh0}r(UlatexXyh5]h4]h2]h3]h7]uh(jh"]h.j{ubhCX is the mean and rr}r(h'X is the mean and h(jubjw)r}r(h'Uh0}r(UlatexX\sigmah5]h4]h2]h3]h7]uh(jh"]h.j{ubhCX is the standard deviation.rr}r(h'X is the standard deviation.h(jubeubh|)r}r(h'Xhyp = [ log_sigma ]rh(jh)j~h.hh0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]rhCXhyp = [ log_sigma ]rr}r(h'jh(jubaubeubeubj )r}r(h'Uh(jh)Xa/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/lik.py:docstring of pyGPs.Core.lik.Laplacerh.jh0}r(h5]h4]h2]h3]h7]Uentries]r(jX!Laplace (class in pyGPs.Core.lik)hUtrauh9Nh:hh"]ubj)r}r(h'Uh(jh)jh.jh0}r(jjXpyh5]h4]h2]h3]h7]jXclassrjjuh9Nh:hh"]r(j)r}r(h'X&Laplace(log_sigma=-2.3025850929940455)h(jh)j!h.j"h0}r(h5]rhaj%h*XpyGPs.Core.likrr}rbh4]h2]h3]h7]rhaj*XLaplacerj,Uj-uh9Nh:hh"]r(j/)r}r(h'Xclass h(jh)j!h.j2h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXclass rr }r (h'Uh(jubaubj8)r }r (h'XpyGPs.Core.lik.h(jh)j!h.j;h0}r (h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXpyGPs.Core.lik.rr}r(h'Uh(j ubaubjA)r}r(h'jh(jh)j!h.jDh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXLaplacerr}r(h'Uh(jubaubj)r}r(h'Uh(jh)j!h.jh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rj)r}r(h'Xlog_sigma=-2.3025850929940455h0}r(h2]h3]h4]h5]h7]uh(jh"]r hCXlog_sigma=-2.3025850929940455r!r"}r#(h'Uh(jubah.jubaubjJ)r$}r%(h'Uh(jh)Nh.jMh0}r&(UexprjOh5]h4]h2]h3]h7]uh9Nh:hh"]r'jQ)r(}r)(h'Uh0}r*(UreftypejUUrefdocjVU refdomainjWh5]h4]U refexplicith2]h3]h7]U reftargetX_modules/pyGPs/Core/likUrefidjuh(j$h"]r+jY)r,}r-(h'Uh0}r.(h2]h3]r/j^ah4]h5]h7]uh(j(h"]r0hCX[source]r1r2}r3(h'Uh(j,ubah.jcubah.jdubaubeubje)r4}r5(h'Uh(jh)j!h.jhh0}r6(h2]h3]h4]h5]h7]uh9Nh:hh"]r7(h|)r8}r9(h'XKLaplacian likelihood function for regression. ONLY works with EP inference!r:h(j4h)jh.hh0}r;(h2]h3]h4]h5]h7]uh9Kh:hh"]r<hCXKLaplacian likelihood function for regression. ONLY works with EP inference!r=r>}r?(h'j:h(j8ubaubh|)r@}rA(h'X:math:`Laplace(t) = \frac{1}{2b}e^{-\frac{|t-y|}{b}}` where :math:`b=\frac{\sigma}{\sqrt{2}}`, :math:`y` is the mean and :math:`\sigma` is the standard deviation.h(j4h)jh.hh0}rB(h2]h3]h4]h5]h7]uh9Kh:hh"]rC(jw)rD}rE(h'Uh0}rF(UlatexX-Laplace(t) = \frac{1}{2b}e^{-\frac{|t-y|}{b}}h5]h4]h2]h3]h7]uh(j@h"]h.j{ubhCX where rGrH}rI(h'X where h(j@ubjw)rJ}rK(h'Uh0}rL(UlatexXb=\frac{\sigma}{\sqrt{2}}h5]h4]h2]h3]h7]uh(j@h"]h.j{ubhCX, rMrN}rO(h'X, h(j@ubjw)rP}rQ(h'Uh0}rR(UlatexXyh5]h4]h2]h3]h7]uh(j@h"]h.j{ubhCX is the mean and rSrT}rU(h'X is the mean and h(j@ubjw)rV}rW(h'Uh0}rX(UlatexX\sigmah5]h4]h2]h3]h7]uh(j@h"]h.j{ubhCX is the standard deviation.rYrZ}r[(h'X is the standard deviation.h(j@ubeubh|)r\}r](h'Xhyp = [ log_sigma ]r^h(j4h)jh.hh0}r_(h2]h3]h4]h5]h7]uh9Kh:hh"]r`hCXhyp = [ log_sigma ]rarb}rc(h'j^h(j\ubaubeubeubeubh$)rd}re(h'Uh(h%h)h,h.h/h0}rf(h2]h3]h4]h5]rgh ah7]rhhauh9Kh:hh"]ri(h<)rj}rk(h'XList of Inferencerlh(jdh)h,h.h@h0}rm(h2]h3]h4]h5]h7]uh9Kh:hh"]rnhCXList of Inferencerorp}rq(h'jlh(jjubaubj )rr}rs(h'Uh(jdh)X_/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/inf.py:docstring of pyGPs.Core.inf.Exactrth.jh0}ru(h5]h4]h2]h3]h7]Uentries]rv(jXExact (class in pyGPs.Core.inf)hUtrwauh9Nh:hh"]ubj)rx}ry(h'Uh(jdh)jth.jh0}rz(jjXpyh5]h4]h2]h3]h7]jXclassr{jj{uh9Nh:hh"]r|(j)r}}r~(h'XExact()h(jxh)j!h.j"h0}r(h5]rhaj%h*XpyGPs.Core.infrr}rbh4]h2]h3]h7]rhaj*XExactrj,Uj-uh9Nh:hh"]r(j/)r}r(h'Xclass h(j}h)j!h.j2h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXclass rr}r(h'Uh(jubaubj8)r}r(h'XpyGPs.Core.inf.h(j}h)j!h.j;h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXpyGPs.Core.inf.rr}r(h'Uh(jubaubjA)r}r(h'jh(j}h)j!h.jDh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXExactrr}r(h'Uh(jubaubjJ)r}r(h'Uh(j}h)Nh.jMh0}r(UexprjOh5]h4]h2]h3]h7]uh9Nh:hh"]rjQ)r}r(h'Uh0}r(UreftypejUUrefdocjVU refdomainjWh5]h4]U refexplicith2]h3]h7]U reftargetX_modules/pyGPs/Core/infUrefidjuh(jh"]rjY)r}r(h'Uh0}r(h2]h3]rj^ah4]h5]h7]uh(jh"]rhCX[source]rr}r(h'Uh(jubah.jcubah.jdubaubeubje)r}r(h'Uh(jxh)j!h.jhh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rh|)r}r(h'XExact inference for a GP with Gaussian likelihood. Compute a parametrization of the posterior, the negative log marginal likelihood and its derivatives w.r.t. the hyperparameters.rh(jh)jth.hh0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]rhCXExact inference for a GP with Gaussian likelihood. Compute a parametrization of the posterior, the negative log marginal likelihood and its derivatives w.r.t. the hyperparameters.rr}r(h'jh(jubaubaubeubj )r}r(h'Uh(jdh)X\/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/inf.py:docstring of pyGPs.Core.inf.EPrh.jh0}r(h5]h4]h2]h3]h7]Uentries]r(jXEP (class in pyGPs.Core.inf)h Utrauh9Nh:hh"]ubj)r}r(h'Uh(jdh)jh.jh0}r(jjXpyh5]h4]h2]h3]h7]jXclassrjjuh9Nh:hh"]r(j)r}r(h'XEP()h(jh)j!h.j"h0}r(h5]rh aj%h*XpyGPs.Core.infrr}rbh4]h2]h3]h7]rh aj*XEPrj,Uj-uh9Nh:hh"]r(j/)r}r(h'Xclass h(jh)j!h.j2h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXclass rr}r(h'Uh(jubaubj8)r}r(h'XpyGPs.Core.inf.h(jh)j!h.j;h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXpyGPs.Core.inf.rr}r(h'Uh(jubaubjA)r}r(h'jh(jh)j!h.jDh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXEPrr}r(h'Uh(jubaubjJ)r}r(h'Uh(jh)Nh.jMh0}r(UexprjOh5]h4]h2]h3]h7]uh9Nh:hh"]rjQ)r}r(h'Uh0}r(UreftypejUUrefdocjVU refdomainjWh5]h4]U refexplicith2]h3]h7]U reftargetX_modules/pyGPs/Core/infUrefidjuh(jh"]rjY)r}r(h'Uh0}r(h2]h3]rj^ah4]h5]h7]uh(jh"]rhCX[source]rr}r(h'Uh(jubah.jcubah.jdubaubeubje)r}r(h'Uh(jh)j!h.jhh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rh|)r}r(h'XHExpectation Propagation approximation to the posterior Gaussian Process.rh(jh)jh.hh0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]rhCXHExpectation Propagation approximation to the posterior Gaussian Process.rr}r(h'jh(jubaubaubeubj )r}r(h'Uh(jdh)Xa/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/inf.py:docstring of pyGPs.Core.inf.Laplacerh.jh0}r(h5]h4]h2]h3]h7]Uentries]r(jX!Laplace (class in pyGPs.Core.inf)hUtrauh9Nh:hh"]ubj)r}r(h'Uh(jdh)jh.jh0}r(jjXpyh5]h4]h2]h3]h7]jXclassrjjuh9Nh:hh"]r(j)r }r (h'X Laplace()h(jh)j!h.j"h0}r (h5]r haj%h*XpyGPs.Core.infr r}rbh4]h2]h3]h7]rhaj*XLaplacerj,Uj-uh9Nh:hh"]r(j/)r}r(h'Xclass h(j h)j!h.j2h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXclass rr}r(h'Uh(jubaubj8)r}r(h'XpyGPs.Core.inf.h(j h)j!h.j;h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXpyGPs.Core.inf.rr}r (h'Uh(jubaubjA)r!}r"(h'jh(j h)j!h.jDh0}r#(h2]h3]h4]h5]h7]uh9Nh:hh"]r$hCXLaplacer%r&}r'(h'Uh(j!ubaubjJ)r(}r)(h'Uh(j h)Nh.jMh0}r*(UexprjOh5]h4]h2]h3]h7]uh9Nh:hh"]r+jQ)r,}r-(h'Uh0}r.(UreftypejUUrefdocjVU refdomainjWh5]h4]U refexplicith2]h3]h7]U reftargetX_modules/pyGPs/Core/infUrefidjuh(j(h"]r/jY)r0}r1(h'Uh0}r2(h2]h3]r3j^ah4]h5]h7]uh(j,h"]r4hCX[source]r5r6}r7(h'Uh(j0ubah.jcubah.jdubaubeubje)r8}r9(h'Uh(jh)j!h.jhh0}r:(h2]h3]h4]h5]h7]uh9Nh:hh"]r;h|)r<}r=(h'X:Laplace's Approximation to the posterior Gaussian process.r>h(j8h)jh.hh0}r?(h2]h3]h4]h5]h7]uh9Kh:hh"]r@hCX:Laplace's Approximation to the posterior Gaussian process.rArB}rC(h'j>h(j<ubaubaubeubj )rD}rE(h'Uh(jdh)Xd/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/inf.py:docstring of pyGPs.Core.inf.FITC_ExactrFh.jh0}rG(h5]h4]h2]h3]h7]Uentries]rH(jX$FITC_Exact (class in pyGPs.Core.inf)h UtrIauh9Nh:hh"]ubj)rJ}rK(h'Uh(jdh)jFh.jh0}rL(jjXpyh5]h4]h2]h3]h7]jXclassrMjjMuh9Nh:hh"]rN(j)rO}rP(h'X FITC_Exact()h(jJh)j!h.j"h0}rQ(h5]rRh aj%h*XpyGPs.Core.infrSrT}rUbh4]h2]h3]h7]rVh aj*X FITC_ExactrWj,Uj-uh9Nh:hh"]rX(j/)rY}rZ(h'Xclass h(jOh)j!h.j2h0}r[(h2]h3]h4]h5]h7]uh9Nh:hh"]r\hCXclass r]r^}r_(h'Uh(jYubaubj8)r`}ra(h'XpyGPs.Core.inf.h(jOh)j!h.j;h0}rb(h2]h3]h4]h5]h7]uh9Nh:hh"]rchCXpyGPs.Core.inf.rdre}rf(h'Uh(j`ubaubjA)rg}rh(h'jWh(jOh)j!h.jDh0}ri(h2]h3]h4]h5]h7]uh9Nh:hh"]rjhCX FITC_Exactrkrl}rm(h'Uh(jgubaubjJ)rn}ro(h'Uh(jOh)Nh.jMh0}rp(UexprjOh5]h4]h2]h3]h7]uh9Nh:hh"]rqjQ)rr}rs(h'Uh0}rt(UreftypejUUrefdocjVU refdomainjWh5]h4]U refexplicith2]h3]h7]U reftargetX_modules/pyGPs/Core/infUrefidjWuh(jnh"]rujY)rv}rw(h'Uh0}rx(h2]h3]ryj^ah4]h5]h7]uh(jrh"]rzhCX[source]r{r|}r}(h'Uh(jvubah.jcubah.jdubaubeubje)r~}r(h'Uh(jJh)j!h.jhh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rh|)r}r(h'XMFITC approximation to the posterior Gaussian process. The function is equivalent to infExact with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Quu) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu, snu2 = sn2/1e6 is the noise of the inducing inputs and Quu = Kuu + snu2*eye(nu).rh(j~h)jFh.hh0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]rhCXMFITC approximation to the posterior Gaussian process. The function is equivalent to infExact with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Quu) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu, snu2 = sn2/1e6 is the noise of the inducing inputs and Quu = Kuu + snu2*eye(nu).rr}r(h'jh(jubaubaubeubj )r}r(h'Uh(jdh)Xa/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/inf.py:docstring of pyGPs.Core.inf.FITC_EPrh.jh0}r(h5]h4]h2]h3]h7]Uentries]r(jX!FITC_EP (class in pyGPs.Core.inf)h Utrauh9Nh:hh"]ubj)r}r(h'Uh(jdh)jh.jh0}r(jjXpyh5]h4]h2]h3]h7]jXclassrjjuh9Nh:hh"]r(j)r}r(h'X FITC_EP()h(jh)j!h.j"h0}r(h5]rh aj%h*XpyGPs.Core.infrr}rbh4]h2]h3]h7]rh aj*XFITC_EPrj,Uj-uh9Nh:hh"]r(j/)r}r(h'Xclass h(jh)j!h.j2h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXclass rr}r(h'Uh(jubaubj8)r}r(h'XpyGPs.Core.inf.h(jh)j!h.j;h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXpyGPs.Core.inf.rr}r(h'Uh(jubaubjA)r}r(h'jh(jh)j!h.jDh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXFITC_EPrr}r(h'Uh(jubaubjJ)r}r(h'Uh(jh)Nh.jMh0}r(UexprjOh5]h4]h2]h3]h7]uh9Nh:hh"]rjQ)r}r(h'Uh0}r(UreftypejUUrefdocjVU refdomainjWh5]h4]U refexplicith2]h3]h7]U reftargetX_modules/pyGPs/Core/infUrefidjuh(jh"]rjY)r}r(h'Uh0}r(h2]h3]rj^ah4]h5]h7]uh(jh"]rhCX[source]rr}r(h'Uh(jubah.jcubah.jdubaubeubje)r}r(h'Uh(jh)j!h.jhh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rh|)r}r(h'X{FITC-EP approximation to the posterior Gaussian process. The function is equivalent to infEP with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Kuu + snu2 * eye(nu)) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu and snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard deviation of the inducing inputs snu to be a one per mil of the measurement noise's standard deviation sn. In case of a likelihood without noise parameter sn2, we simply use snu2 = 1e-6. For details, see The Generalized FITC Approximation, Andrew Naish-Guzman and Sean Holden, NIPS, 2007.rh(jh)jh.hh0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]rhCX{FITC-EP approximation to the posterior Gaussian process. The function is equivalent to infEP with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Kuu + snu2 * eye(nu)) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu and snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard deviation of the inducing inputs snu to be a one per mil of the measurement noise's standard deviation sn. In case of a likelihood without noise parameter sn2, we simply use snu2 = 1e-6. For details, see The Generalized FITC Approximation, Andrew Naish-Guzman and Sean Holden, NIPS, 2007.rr}r(h'jh(jubaubaubeubj )r}r(h'Uh(jdh)Xf/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/inf.py:docstring of pyGPs.Core.inf.FITC_Laplacerh.jh0}r(h5]h4]h2]h3]h7]Uentries]r(jX&FITC_Laplace (class in pyGPs.Core.inf)hUtrauh9Nh:hh"]ubj)r}r(h'Uh(jdh)jh.jh0}r(jjXpyh5]h4]h2]h3]h7]jXclassrjjuh9Nh:hh"]r(j)r}r(h'XFITC_Laplace()rh(jh)j!h.j"h0}r(h5]rhaj%h*XpyGPs.Core.infrr}rbh4]h2]h3]h7]rhaj*X FITC_Laplacerj,Uj-uh9Nh:hh"]r(j/)r}r(h'Xclass h(jh)j!h.j2h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXclass rr}r(h'Uh(jubaubj8)r}r(h'XpyGPs.Core.inf.h(jh)j!h.j;h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXpyGPs.Core.inf.rr}r(h'Uh(jubaubjA)r}r(h'jh(jh)j!h.jDh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCX FITC_Laplacerr}r(h'Uh(jubaubjJ)r}r(h'Uh(jh)Nh.jMh0}r(UexprjOh5]h4]h2]h3]h7]uh9Nh:hh"]rjQ)r}r(h'Uh0}r(UreftypejUUrefdocjVU refdomainjWh5]h4]U refexplicith2]h3]h7]U reftargetX_modules/pyGPs/Core/infUrefidjuh(jh"]rjY)r}r(h'Uh0}r(h2]h3]rj^ah4]h5]h7]uh(jh"]rhCX[source]rr }r (h'Uh(jubah.jcubah.jdubaubeubje)r }r (h'Uh(jh)j!h.jhh0}r (h2]h3]h4]h5]h7]uh9Nh:hh"]rh|)r}r(h'XFITC-Laplace approximation to the posterior Gaussian process. The function is equivalent to infLaplace with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Kuu + snu2 * eye(nu)) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu and snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard deviation of the inducing inputs snu to be a one per mil of the measurement noise's standard deviation sn. In case of a likelihood without noise parameter sn2, we simply use snu2 = 1e-6.rh(j h)jh.hh0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]rhCXFITC-Laplace approximation to the posterior Gaussian process. The function is equivalent to infLaplace with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Kuu + snu2 * eye(nu)) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu and snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard deviation of the inducing inputs snu to be a one per mil of the measurement noise's standard deviation sn. 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likRange=Noneqq}q(hUhhubah"hubeubcsphinx.addnodes only q)q}q(hUhhghNh"Uonlyqh$}q(UexprUhtmlqh)]h(]h&]h']h+]uh-Nh.hh]qcsphinx.addnodes pending_xref q)q}q(hUh$}q(UreftypeUviewcodeqUrefdocXOptsqU refdomainUstdqh)]h(]U refexplicith&]h']h+]U reftargetX_modules/pyGPs/Core/gpUrefidhsuhhh]qcdocutils.nodes inline q)q}q(hUh$}q(h&]h']qU viewcode-linkqah(]h)]h+]uhhh]qh7X[source]qхq}q(hUhhubah"Uinlinequbah"U pending_xrefqubaubeubcsphinx.addnodes desc_content q)q}q(hUhh\hhih"U desc_contentqh$}q(h&]h']h(]h)]h+]uh-Nh.hh]q(hI)q}q(hXhThis method is used to sepecify optimization configuration. By default, gp uses a single run "minimize".qhhhXg/Users/marion/workspace/workspace_GPs/pyGPs/pyGPs/Core/gp.py:docstring of pyGPs.Core.gp.GP.setOptimizerqh"hMh$}q(h&]h']h(]h)]h+]uh-Kh.hh]qh7XhThis method is used to sepecify optimization configuration. By default, gp uses a single run "minimize".q⅁q}q(hhhhubaubcdocutils.nodes field_list q)q}q(hUhhhNh"U field_listqh$}q(h&]h']h(]h)]h+]uh-Nh.hh]qcdocutils.nodes field q)q}q(hUh$}q(h&]h']h(]h)]h+]uhhh]q(cdocutils.nodes field_name q)q}q(hUh$}q(h&]h']h(]h)]h+]uhhh]qh7X Parametersqq}q(hUhhubah"U field_namequbcdocutils.nodes field_body q)q}q(hUh$}q(h&]h']h(]h)]h+]uhhh]qcdocutils.nodes bullet_list q)q}r(hUh$}r(h&]h']h(]h)]h+]uhhh]r(cdocutils.nodes list_item r)r}r(hUh$}r(h&]h']h(]h)]h+]uhhh]rhI)r}r (hUh$}r (h&]h']h(]h)]h+]uhjh]r (cdocutils.nodes strong r )r }r(hXmethodh$}r(h&]h']h(]h)]h+]uhjh]rh7Xmethodrr}r(hUhj ubah"Ustrongrubh7X -- rr}r(hUhjubhI)r}r(hX*Optimization methods. Possible values are:rhjhhh"hMh$}r(h&]h']h(]h)]h+]uh-Kh]rh7X*Optimization methods. Possible values are:rr}r(hjhjubaubhI)r }r!(hXX"Minimize" -> minimize by Carl Rasmussen (python implementation of "minimize" in GPML)r"hjhhh"hMh$}r#(h&]h']h(]h)]h+]uh-Kh]r$h7XX"Minimize" -> minimize by Carl Rasmussen (python implementation of "minimize" in GPML)r%r&}r'(hj"hj ubaubhI)r(}r)(hX""CG" -> conjugent gradientr*hjhhh"hMh$}r+(h&]h']h(]h)]h+]uh-Kh]r,h7X""CG" -> conjugent gradientr-r.}r/(hj*hj(ubaubhI)r0}r1(hXU"BFGS" -> quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS)r2hjhhh"hMh$}r3(h&]h']h(]h)]h+]uh-K h]r4h7XU"BFGS" -> quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS)r5r6}r7(hj2hj0ubaubhI)r8}r9(hX:"SCG" -> scaled conjugent gradient (faster than CG)r:hjhhh"hMh$}r;(h&]h']h(]h)]h+]uh-K h]r<h7X:"SCG" -> scaled conjugent gradient (faster than CG)r=r>}r?(hj:hj8ubaubeh"hMubah"U list_itemr@ubj)rA}rB(hUh$}rC(h&]h']h(]h)]h+]uhhh]rDhI)rE}rF(hUh$}rG(h&]h']h(]h)]h+]uhjAh]rH(j )rI}rJ(hX num_restartsh$}rK(h&]h']h(]h)]h+]uhjEh]rLh7X num_restartsrMrN}rO(hUhjIubah"jubh7X -- rPrQ}rR(hUhjEubh7XSet if you want to run mulitiple times of optimization with different initial guess. It specifys the maximum number of runs/restarts/trials.rSrT}rU(hXSet if you want to run mulitiple times of optimization with different initial guess. It specifys the maximum number of runs/restarts/trials.rVhjEubeh"hMubah"j@ubj)rW}rX(hUh$}rY(h&]h']h(]h)]h+]uhhh]rZhI)r[}r\(hUh$}r](h&]h']h(]h)]h+]uhjWh]r^(j )r_}r`(hX min_thresholdh$}ra(h&]h']h(]h)]h+]uhj[h]rbh7X min_thresholdrcrd}re(hUhj_ubah"jubh7X -- rfrg}rh(hUhj[ubh7XSet if you want to run mulitiple times of optimization with different initial guess. It specifys the threshold of objective function value. Stop optimization when this value is reached.rirj}rk(hXSet if you want to run mulitiple times of optimization with different initial guess. It specifys the threshold of objective function value. Stop optimization when this value is reached.rlhj[ubeh"hMubah"j@ubj)rm}rn(hUh$}ro(h&]h']h(]h)]h+]uhhh]rphI)rq}rr(hUh$}rs(h&]h']h(]h)]h+]uhjmh]rt(j )ru}rv(hX meanRangeh$}rw(h&]h']h(]h)]h+]uhjqh]rxh7X meanRangeryrz}r{(hUhjuubah"jubh7X -- r|r}}r~(hUhjqubhI)r}r(hXThe range of initial guess for mean hyperparameters. e.g. meanRange = [(-2,2), (-5,5), (0,1)]. Each tuple specifys the range (low, high) of this hyperparameter, This is only the range of initial guess, during optimization process, optimal hyperparameters may go out of this range.rhjqhhh"hMh$}r(h&]h']h(]h)]h+]uh-Kh]rh7XThe range of initial guess for mean hyperparameters. e.g. meanRange = [(-2,2), (-5,5), (0,1)]. Each tuple specifys the range (low, high) of this hyperparameter, This is only the range of initial guess, during optimization process, optimal hyperparameters may go out of this range.rr}r(hjhjubaubhI)r}r(hX*(-5,5) for each hyperparameter by default.rhjqhhh"hMh$}r(h&]h']h(]h)]h+]uh-Kh]rh7X*(-5,5) for each hyperparameter by default.rr}r(hjhjubaubeh"hMubah"j@ubj)r}r(hUh$}r(h&]h']h(]h)]h+]uhhh]rhI)r}r(hUh$}r(h&]h']h(]h)]h+]uhjh]r(j )r}r(hXcovRangeh$}r(h&]h']h(]h)]h+]uhjh]rh7XcovRangerr}r(hUhjubah"jubh7X -- rr}r(hUhjubh7XJThe range of initial guess for kernel hyperparameters. Usage see meanRangerr}r(hXJThe range of initial guess for kernel hyperparameters. Usage see meanRangerhjubeh"hMubah"j@ubj)r}r(hUh$}r(h&]h']h(]h)]h+]uhhh]rhI)r}r(hUh$}r(h&]h']h(]h)]h+]uhjh]r(j )r}r(hXlikRangeh$}r(h&]h']h(]h)]h+]uhjh]rh7XlikRangerr}r(hUhjubah"jubh7X -- rr}r(hUhjubh7XNThe range of initial guess for likelihood hyperparameters. Usage see meanRangerr}r(hXNThe range of initial guess for likelihood hyperparameters. Usage see meanRangerhjubeh"hMubah"j@ubeh"U bullet_listrubah"U field_bodyrubeh"Ufieldrubaubeubeubeubh)r}r(hUhhhh h"h#h$}r(h&]h']h(]h)]rhah+]rh auh-K h.hh]r(h0)r}r(hXDeveloping Optimization Methodsrhjhh h"h4h$}r(h&]h']h(]h)]h+]uh-K h.hh]rh7XDeveloping Optimization Methodsrr}r(hjhjubaubhI)r}r(hX2We also support the development of new optimizers.rhjhh h"hMh$}r(h&]h']h(]h)]h+]uh-K h.hh]rh7X2We also support the development of new optimizers.rr}r(hjhjubaubhI)r}r(hXpYour own optimizer should inherent base class Optimizer in *pyGPs.Core.opt* and follow the template as below: ::hjhh h"hMh$}r(h&]h']h(]h)]h+]uh-Kh.hh]r(h7X;Your own optimizer should inherent base class Optimizer in rr}r(hX;Your own optimizer should inherent base class Optimizer in hjubcdocutils.nodes emphasis r)r}r(hX*pyGPs.Core.opt*h$}r(h&]h']h(]h)]h+]uhjh]rh7XpyGPs.Core.optrr}r(hUhjubah"Uemphasisrubh7X" and follow the template as below:rr}r(hX" and follow the template as below:hjubeubcdocutils.nodes literal_block r)r}r(hXTclass MyOptimizer(Optimizer): def __init__(self, model=None, searchConfig = None): self.model = model def findMin(self, x, y): ''' Find minimal value based on negative-log-marginal-likelihood. optimalHyp, funcValue = findMin(x, y) where funcValue is the minimal negative-log-marginal-likelihood during optimization, and optimalHyp is a flattened numpy array (in sequence of meanfunc.hyp, covfunc.hyp, likfunc.hyp) of the hyparameters to achieve such value. You can achieve advanced search strategy by initializing Optimizer with searchConfig, which is an instance of pyGPs.Optimization.conf. See more in pyGPs.Optimization.conf and pyGPs.Core.gp.GP.setOptimizer, as well as in online documentation of section Optimizers. ''' passhjhh h"U literal_blockrh$}r(U xml:spacerUpreserverh)]h(]h&]h']h+]uh-Kh.hh]rh7XTclass MyOptimizer(Optimizer): def __init__(self, model=None, searchConfig = None): self.model = model def findMin(self, x, y): ''' Find minimal value based on negative-log-marginal-likelihood. optimalHyp, funcValue = findMin(x, y) where funcValue is the minimal negative-log-marginal-likelihood during optimization, and optimalHyp is a flattened numpy array (in sequence of meanfunc.hyp, covfunc.hyp, likfunc.hyp) of the hyparameters to achieve such value. You can achieve advanced search strategy by initializing Optimizer with searchConfig, which is an instance of pyGPs.Optimization.conf. See more in pyGPs.Optimization.conf and pyGPs.Core.gp.GP.setOptimizer, as well as in online documentation of section Optimizers. ''' passrr}r(hUhjubaubeubeubahUU transformerrNU footnote_refsr}rUrefnamesr}rUsymbol_footnotesr]rUautofootnote_refsr]rUsymbol_footnote_refsr]rU citationsr]rh.hU current_linerNUtransform_messagesr]rUreporterrNUid_startrKU autofootnotesr]rU citation_refsr}rUindirect_targetsr]r Usettingsr (cdocutils.frontend Values r or }r (Ufootnote_backlinksrKUrecord_dependenciesrNU rfc_base_urlrUhttp://tools.ietf.org/html/rU tracebackrUpep_referencesrNUstrip_commentsrNU toc_backlinksrUentryrU language_coderUenrU datestamprNU report_levelrKU _destinationrNU halt_levelrKU strip_classesrNh4NUerror_encoding_error_handlerrUbackslashreplacerUdebugr NUembed_stylesheetr!Uoutput_encoding_error_handlerr"Ustrictr#U sectnum_xformr$KUdump_transformsr%NU docinfo_xformr&KUwarning_streamr'NUpep_file_url_templater(Upep-%04dr)Uexit_status_levelr*KUconfigr+NUstrict_visitorr,NUcloak_email_addressesr-Utrim_footnote_reference_spacer.Uenvr/NUdump_pseudo_xmlr0NUexpose_internalsr1NUsectsubtitle_xformr2U source_linkr3NUrfc_referencesr4NUoutput_encodingr5Uutf-8r6U source_urlr7NUinput_encodingr8U utf-8-sigr9U_disable_configr:NU id_prefixr;UU tab_widthr<KUerror_encodingr=UUTF-8r>U_sourcer?U?/Users/marion/workspace/workspace_GPs/pyGPs/doc/source/Opts.rstr@Ugettext_compactrAU generatorrBNUdump_internalsrCNU smart_quotesrDU pep_base_urlrEUhttp://www.python.org/dev/peps/rFUsyntax_highlightrGUlongrHUinput_encoding_error_handlerrIj#Uauto_id_prefixrJUidrKUdoctitle_xformrLUstrip_elements_with_classesrMNU _config_filesrN]Ufile_insertion_enabledrOU raw_enabledrPKU dump_settingsrQNubUsymbol_footnote_startrRKUidsrS}rT(hhghjhh;hhuUsubstitution_namesrU}rVh"h.h$}rW(h&]h)]h(]Usourceh h']h+]uU footnotesrX]rYUrefidsrZ}r[ub.pyGPs-1.2/doc/build/doctrees/SemiSupervised.doctree000066400000000000000000000641511236346020200224150ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(X$semi-supervised learning with graphsqNXstructured kernelsqX load dataqNX1Xform a nearest neighbour graphq NX2Xresultq NXgp classificationq NXimportq NXkernel on graphq NuUsubstitution_defsq}qUparse_messagesq]qUcurrent_sourceqNU decorationqNUautofootnote_startqKUnameidsq}q(hU$semi-supervised-learning-with-graphsqhUstructured-kernelsqhU load-dataqX1Uid3qh Uform-a-nearest-neighbour-graphqX2Uid4qh Uresultqh Ugp-classificationqh Uimportqh Ukernel-on-graphq uUchildrenq!]q"cdocutils.nodes section q#)q$}q%(U rawsourceq&UUparentq'hUsourceq(cdocutils.nodes reprunicode q)XI/Users/marion/workspace/workspace_GPs/pyGPs/doc/source/SemiSupervised.rstq*q+}q,bUtagnameq-Usectionq.U attributesq/}q0(Udupnamesq1]Uclassesq2]Ubackrefsq3]Uidsq4]q5haUnamesq6]q7hauUlineq8KUdocumentq9hh!]q:(cdocutils.nodes title q;)q<}q=(h&X$Semi-supervised Learning with Graphsq>h'h$h(h+h-Utitleq?h/}q@(h1]h2]h3]h4]h6]uh8Kh9hh!]qAcdocutils.nodes Text qBX$Semi-supervised Learning with GraphsqCqD}qE(h&h>h'hr?}r@(h&X graph.h'j3ubeubhF)rA}rB(h&XBelow shows an example of such symmetrized Euclidean :math:`2-NN` graph on some 1s and 2s taking from Xiaojin Zhu's doctoral thesis [2]_.rCh'hh(h+h-hJh/}rD(h1]h2]h3]h4]h6]uh8K-h9hh!]rE(hBX5Below shows an example of such symmetrized Euclidean rFrG}rH(h&X5Below shows an example of such symmetrized Euclidean h'jAubh)rI}rJ(h&Uh/}rK(UlatexX2-NNh4]h3]h1]h2]h6]uh'jAh!]h-hubhBXC graph on some 1s and 2s taking from Xiaojin Zhu's doctoral thesis rLrM}rN(h&XC graph on some 1s and 2s taking from Xiaojin Zhu's doctoral thesis h'jAubh)rO}rP(h&X[2]_hKh'jAh-hh/}rQ(h4]rRUid2rSah3]h1]h2]h6]hhuh!]rThBX2rU}rV(h&Uh'jOubaubhBX.rW}rX(h&X.h'jAubeubcdocutils.nodes figure rY)rZ}r[(h&Uh'hh(h+h-Ufigurer\h/}r](Ualignr^Xcenterh4]h3]h1]h2]h6]uh8Nh9hh!]r_h)r`}ra(h&X4.. figure:: _images/2nnGraph.png :align: center h/}rb(UuriX_images/2nnGraph.pngrch4]h3]h1]h2]h}rdU*jcsh6]uh'jZh!]h-hubaubeubh#)re}rf(h&Uh'h$h(h+h-h.h/}rg(h1]h2]h3]h4]rhh ah6]rih auh8K4h9hh!]rj(h;)rk}rl(h&XKernel on graphrmh'jeh(h+h-h?h/}rn(h1]h2]h3]h4]h6]uh8K4h9hh!]rohBXKernel on graphrprq}rr(h&jmh'jkubaubhF)rs}rt(h&XSeveral classical kernels on graph described in `Structured Kernels`_ can be built from adjacency matrix :math:`A`. We use diffusion kernel for this example to get the precomputed kernel matrix. ::ruh'jeh(h+h-hJh/}rv(h1]h2]h3]h4]h6]uh8K5h9hh!]rw(hBX0Several classical kernels on graph described in rxry}rz(h&X0Several classical kernels on graph described in h'jsubcdocutils.nodes reference r{)r|}r}(h&X`Structured Kernels`_hKh'jsh-U referencer~h/}r(UnameXStructured KernelsUrefurirX Graph.htmlrh4]h3]h1]h2]h6]uh!]rhBXStructured Kernelsrr}r(h&Uh'j|ubaubhBX$ can be built from adjacency matrix rr}r(h&X$ can be built from adjacency matrix h'jsubh)r}r(h&Uh/}r(UlatexXAh4]h3]h1]h2]h6]uh'jsh!]h-hubhBXP. We use diffusion kernel for this example to get the precomputed kernel matrix.rr}r(h&XP. We use diffusion kernel for this example to get the precomputed kernel matrix.h'jsubeubho)r}r(h&X"Matrix = nodeKernels.diffKernel(A)h'jeh(h+h-hrh/}r(hthuh4]h3]h1]h2]h6]uh8K7h9hh!]rhBX"Matrix = nodeKernels.diffKernel(A)rr}r(h&Uh'jubaubcdocutils.nodes target r)r}r(h&X".. _Structured Kernels: Graph.htmlU referencedrKh'jeh(h+h-Utargetrh/}r(jjh4]rhah3]h1]h2]h6]rhauh8K9h9hh!]ubhF)r}r(h&XThis a big square matrix with all rows and columns of the number of data points. By specifying the indice of training data and test data, we will form two matrix M1 and M2 with the exact format which *pyGPs.Core.cov.Pre* needed. ::h'jeh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8K;h9hh!]r(hBXThis a big square matrix with all rows and columns of the number of data points. By specifying the indice of training data and test data, we will form two matrix M1 and M2 with the exact format which rr}r(h&XThis a big square matrix with all rows and columns of the number of data points. By specifying the indice of training data and test data, we will form two matrix M1 and M2 with the exact format which h'jubhP)r}r(h&X*pyGPs.Core.cov.Pre*h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXpyGPs.Core.cov.Prerr}r(h&Uh'jubah-hXubhBX needed.rr}r(h&X needed.h'jubeubho)r}r(h&XEM1,M2 = graphUtil.formKernelMatrix(Matrix, indice_train, indice_test)h'jeh(h+h-hrh/}r(hthuh4]h3]h1]h2]h6]uh8K>h9hh!]rhBXEM1,M2 = graphUtil.formKernelMatrix(Matrix, indice_train, indice_test)rr}r(h&Uh'jubaubcdocutils.nodes definition_list r)r}r(h&Uh'jeh(Nh-Udefinition_listrh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]r(cdocutils.nodes definition_list_item r)r}r(h&XM1 is a matrix with shape **number of training points plus 1** by **number of test points** - cross covariances matrix (train by test) - last row is self covariances (diagonal of test by test)h'jh(h+h-Udefinition_list_itemrh/}r(h1]h2]h3]h4]h6]uh8KAh!]r(cdocutils.nodes term r)r}r(h&X[M1 is a matrix with shape **number of training points plus 1** by **number of test points**rh'jh(h+h-Utermrh/}r(h1]h2]h3]h4]h6]uh8KAh!]r(hBXM1 is a matrix with shape rr}r(h&XM1 is a matrix with shape h'jubcdocutils.nodes strong r)r}r(h&X$**number of training points plus 1**h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX number of training points plus 1rr}r(h&Uh'jubah-UstrongrubhBX by rr}r(h&X by h'jubj)r}r(h&X**number of test points**h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXnumber of test pointsrr}r(h&Uh'jubah-jubeubcdocutils.nodes definition r)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rcdocutils.nodes bullet_list r)r}r(h&Uh/}r(UbulletrX-h4]h3]h1]h2]h6]uh'jh!]r(cdocutils.nodes list_item r)r}r(h&X(cross covariances matrix (train by test)rh/}r(h1]h2]h3]h4]h6]uh'jh!]rhF)r}r(h&jh'jh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8KAh!]rhBX(cross covariances matrix (train by test)rr}r(h&jh'jubaubah-U list_itemrubj)r}r(h&X7last row is self covariances (diagonal of test by test)rh/}r(h1]h2]h3]h4]h6]uh'jh!]rhF)r}r(h&jh'jh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8KBh!]rhBX7last row is self covariances (diagonal of test by test)rr}r(h&jh'jubaubah-jubeh-U bullet_listrubah-U definitionrubeubj)r}r(h&XM2 is a square matrix with **number of training points** for each dimension - training set covariance matrix (train by train) h'jh(h+h-jh/}r(h1]h2]h3]h4]h6]uh8KEh9hh!]r (j)r }r (h&XKM2 is a square matrix with **number of training points** for each dimensionr h'jh(h+h-jh/}r (h1]h2]h3]h4]h6]uh8KEh!]r(hBXM2 is a square matrix with rr}r(h&XM2 is a square matrix with h'j ubj)r}r(h&X**number of training points**h/}r(h1]h2]h3]h4]h6]uh'j h!]rhBXnumber of training pointsrr}r(h&Uh'jubah-jubhBX for each dimensionrr}r(h&X for each dimensionh'j ubeubj)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rj)r }r!(h&Uh/}r"(jX-h4]h3]h1]h2]h6]uh'jh!]r#j)r$}r%(h&X1training set covariance matrix (train by train) h/}r&(h1]h2]h3]h4]h6]uh'j h!]r'hF)r(}r)(h&X/training set covariance matrix (train by train)r*h'j$h(h+h-hJh/}r+(h1]h2]h3]h4]h6]uh8KDh!]r,hBX/training set covariance matrix (train by train)r-r.}r/(h&j*h'j(ubaubah-jubah-jubah-jubeubeubeubh#)r0}r1(h&Uh'h$h(h+h-h.h/}r2(h1]h2]h3]h4]r3hah6]r4h auh8KHh9hh!]r5(h;)r6}r7(h&XGP classificationr8h'j0h(h+h-h?h/}r9(h1]h2]h3]h4]h6]uh8KHh9hh!]r:hBXGP classificationr;r<}r=(h&j8h'j6ubaubhF)r>}r?(h&XEvery ingredients for a basic semi-supervised learning is prepared now. Lets see how to proceed for :math:`GP` classification. First, the normal way with rbf kernel we have seen several times ::r@h'j0h(h+h-hJh/}rA(h1]h2]h3]h4]h6]uh8KIh9hh!]rB(hBXeEvery ingredients for a basic semi-supervised learning is prepared now. Lets see how to proceed for rCrD}rE(h&XeEvery ingredients for a basic semi-supervised learning is prepared now. Lets see how to proceed for h'j>ubh)rF}rG(h&Uh/}rH(UlatexXGPh4]h3]h1]h2]h6]uh'j>h!]h-hubhBXQ classification. First, the normal way with rbf kernel we have seen several timesrIrJ}rK(h&XQ classification. First, the normal way with rbf kernel we have seen several timesh'j>ubeubho)rL}rM(h&X@model = pyGPs.GPC() k = pyGPs.cov.RBF() model.setPrior(kernel=k)h'j0h(h+h-hrh/}rN(hthuh4]h3]h1]h2]h6]uh8KKh9hh!]rOhBX@model = pyGPs.GPC() k = pyGPs.cov.RBF() model.setPrior(kernel=k)rPrQ}rR(h&Uh'jLubaubhF)rS}rT(h&XThen lets use our kernel precomputed matrix. If you only use precomputed kernel matrix, there is no training data. However you still need to specify :math:`x` just to fit in the usage of pyGPs for generality reason. You can create any :math:`x` as long as the dimension is correct. ::h'j0h(h+h-hJh/}rU(h1]h2]h3]h4]h6]uh8KOh9hh!]rV(hBXThen lets use our kernel precomputed matrix. If you only use precomputed kernel matrix, there is no training data. However you still need to specify rWrX}rY(h&XThen lets use our kernel precomputed matrix. If you only use precomputed kernel matrix, there is no training data. However you still need to specify h'jSubh)rZ}r[(h&Uh/}r\(UlatexXxh4]h3]h1]h2]h6]uh'jSh!]h-hubhBXM just to fit in the usage of pyGPs for generality reason. You can create any r]r^}r_(h&XM just to fit in the usage of pyGPs for generality reason. You can create any h'jSubh)r`}ra(h&Uh/}rb(UlatexXxh4]h3]h1]h2]h6]uh'jSh!]h-hubhBX% as long as the dimension is correct.rcrd}re(h&X% as long as the dimension is correct.h'jSubeubho)rf}rg(h&XWx = np.zeros((n,1)) k = pyGPs.cov.Pre(M1,M2) + pyGPs.cov.RBF() model.setPrior(kernel=k)h'j0h(h+h-hrh/}rh(hthuh4]h3]h1]h2]h6]uh8KSh9hh!]rihBXWx = np.zeros((n,1)) k = pyGPs.cov.Pre(M1,M2) + pyGPs.cov.RBF() model.setPrior(kernel=k)rjrk}rl(h&Uh'jfubaubhF)rm}rn(h&XmMoreover, you can composite a kernel for both precomputed matrix and regular kernel function if necessary. ::roh'j0h(h+h-hJh/}rp(h1]h2]h3]h4]h6]uh8KWh9hh!]rqhBXjMoreover, you can composite a kernel for both precomputed matrix and regular kernel function if necessary.rrrs}rt(h&XjMoreover, you can composite a kernel for both precomputed matrix and regular kernel function if necessary.h'jmubaubho)ru}rv(h&XGk = pyGPs.cov.Pre(M1,M2) + pyGPs.cov.RBFunit() model.setPrior(kernel=k)h'j0h(h+h-hrh/}rw(hthuh4]h3]h1]h2]h6]uh8KYh9hh!]rxhBXGk = pyGPs.cov.Pre(M1,M2) + pyGPs.cov.RBFunit() model.setPrior(kernel=k)ryrz}r{(h&Uh'juubaubhF)r|}r}(h&XQThe rest way of using pyGPs is exactly the same as the demo of GP classification.r~h'j0h(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8K\h9hh!]rhBXQThe rest way of using pyGPs is exactly the same as the demo of GP classification.rr}r(h&j~h'j|ubaubeubh#)r}r(h&Uh'h$h(h+h-h.h/}r(h1]h2]h3]h4]rhah6]rh auh8K`h9hh!]r(h;)r}r(h&XResultrh'jh(h+h-h?h/}r(h1]h2]h3]h4]h6]uh8K`h9hh!]rhBXResultrr}r(h&jh'jubaubhF)r}r(h&XFor our manually created graph data, an rbf kernel works better than a diffusion kernel on the graph (higher accuracy). The performance in general should depend on the application as well as features of data.rh'jh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Kah9hh!]rhBXFor our manually created graph data, an rbf kernel works better than a diffusion kernel on the graph (higher accuracy). The performance in general should depend on the application as well as features of data.rr}r(h&jh'jubaubhF)r}r(h&XCThe left image shows the digit that using diffusion kernel will predict the wrong result (should be :math:`2`), but rbf kernel does the job fine. The right image shows the digit that rbf kernel predicts the wrong class, diffusion kernel on the other hand, predicts correctly due to graph information! (should be :math:`1`).h'jh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Kch9hh!]r(hBXdThe left image shows the digit that using diffusion kernel will predict the wrong result (should be rr}r(h&XdThe left image shows the digit that using diffusion kernel will predict the wrong result (should be h'jubh)r}r(h&Uh/}r(UlatexX2h4]h3]h1]h2]h6]uh'jh!]h-hubhBX), but rbf kernel does the job fine. The right image shows the digit that rbf kernel predicts the wrong class, diffusion kernel on the other hand, predicts correctly due to graph information! (should be rr}r(h&X), but rbf kernel does the job fine. The right image shows the digit that rbf kernel predicts the wrong class, diffusion kernel on the other hand, predicts correctly due to graph information! (should be h'jubh)r}r(h&Uh/}r(UlatexX1h4]h3]h1]h2]h6]uh'jh!]h-hubhBX).rr}r(h&X).h'jubeubhF)r}r(h&XInterestingly, using a composite kernel with diffusion kernel on graph and an rbf kernel together. All test cases including the following are predicted correctly.rh'jh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Kfh9hh!]rhBXInterestingly, using a composite kernel with diffusion kernel on graph and an rbf kernel together. All test cases including the following are predicted correctly.rr}r(h&jh'jubaubh)r}r(h&X5.. image:: _images/digitDiffwrong.png :width: 50% h'jh(h+h-hh/}r(h1]UuriX_images/digitDiffwrong.pngrh4]h3]UwidthX50%h2]h}rU*jsh6]uh8Nh9hh!]ubh)r}r(h&X5.. image:: _images/digitRBFwrong.png :width: 50% h'jh(h+h-hh/}r(h1]UuriX_images/digitRBFwrong.pngrh4]h3]UwidthX50%h2]h}rU*jsh6]uh8Nh9hh!]ubcdocutils.nodes footnote r)r}r(h&X`A Database for Handwritten Text Recognition Research, J. J. Hull, IEEE PAMI 16(5) 550-554, 1994.rhKh'jh(h+h-Ufootnoterh/}r(h1]h2]h3]rhah4]rhah6]rX1auh8Koh9hh!]r(cdocutils.nodes label r)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX1r}r(h&Uh'jubah-UlabelrubhF)r}r(h&jh'jh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Koh!]rhBX`A Database for Handwritten Text Recognition Research, J. J. Hull, IEEE PAMI 16(5) 550-554, 1994.rr}r(h&jh'jubaubeubj)r}r(h&XGSemi-Supervised Learning with Graphs, Xiaojin Zhu, CMU-LTI-05-192, 2005rhKh'jh(h+h-jh/}r(h1]h2]h3]rjSah4]rhah6]rX2auh8Kph9hh!]r(j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX2r}r(h&Uh'jubah-jubhF)r}r(h&jh'jh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Kph!]rhBXGSemi-Supervised Learning with Graphs, Xiaojin Zhu, CMU-LTI-05-192, 2005rr}r(h&jh'jubaubeubeubeubah&UU transformerrNU footnote_refsr}r(X1]rhaX2]rjOauUrefnamesr}r(X1]rhaX2]rjOaXstructured kernels]rj|auUsymbol_footnotesr]rUautofootnote_refsr]rUsymbol_footnote_refsr]rU citationsr]rh9hU current_linerNUtransform_messagesr]rUreporterrNUid_startrKU autofootnotesr]rU citation_refsr}rUindirect_targetsr]r Usettingsr (cdocutils.frontend Values r or }r (Ufootnote_backlinksrKUrecord_dependenciesrNU rfc_base_urlrUhttp://tools.ietf.org/html/rU tracebackrUpep_referencesrNUstrip_commentsrNU toc_backlinksrUentryrU language_coderUenrU datestamprNU report_levelrKU _destinationrNU halt_levelrKU strip_classesrNh?NUerror_encoding_error_handlerrUbackslashreplacerUdebugr NUembed_stylesheetr!Uoutput_encoding_error_handlerr"Ustrictr#U sectnum_xformr$KUdump_transformsr%NU docinfo_xformr&KUwarning_streamr'NUpep_file_url_templater(Upep-%04dr)Uexit_status_levelr*KUconfigr+NUstrict_visitorr,NUcloak_email_addressesr-Utrim_footnote_reference_spacer.Uenvr/NUdump_pseudo_xmlr0NUexpose_internalsr1NUsectsubtitle_xformr2U source_linkr3NUrfc_referencesr4NUoutput_encodingr5Uutf-8r6U source_urlr7NUinput_encodingr8U utf-8-sigr9U_disable_configr:NU id_prefixr;UU tab_widthr<KUerror_encodingr=UUTF-8r>U_sourcer?UI/Users/marion/workspace/workspace_GPs/pyGPs/doc/source/SemiSupervised.rstr@Ugettext_compactrAU generatorrBNUdump_internalsrCNU smart_quotesrDU pep_base_urlrEUhttp://www.python.org/dev/peps/rFUsyntax_highlightrGUlongrHUinput_encoding_error_handlerrIj#Uauto_id_prefixrJUidrKUdoctitle_xformrLUstrip_elements_with_classesrMNU _config_filesrN]rOUfile_insertion_enabledrPU raw_enabledrQKU dump_settingsrRNubUsymbol_footnote_startrSKUidsrT}rU(hjhh$hjjSjOhjhhhhzhjh jehhYhhhj0uUsubstitution_namesrV}rWh-h9h/}rX(h1]h4]h3]Usourceh+h2]h6]uU footnotesrY]rZ(jjeUrefidsr[}r\ub.pyGPs-1.2/doc/build/doctrees/Theory.doctree000066400000000000000000000525061236346020200207210ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(Xgps & functionalityqNXgpmlqXlist of functionalityqNuUsubstitution_defsq }q Uparse_messagesq ]q Ucurrent_sourceq NU decorationqNUautofootnote_startqKUnameidsq}q(hUgps-functionalityqhUgpmlqhUlist-of-functionalityquUchildrenq]qcdocutils.nodes section q)q}q(U rawsourceqUUparentqhUsourceqcdocutils.nodes reprunicode qXA/Users/marion/workspace/workspace_GPs/pyGPs/doc/source/Theory.rstqq}q bUtagnameq!Usectionq"U attributesq#}q$(Udupnamesq%]Uclassesq&]Ubackrefsq']Uidsq(]q)haUnamesq*]q+hauUlineq,KUdocumentq-hh]q.(cdocutils.nodes title q/)q0}q1(hXGPs & Functionalityq2hhhhh!Utitleq3h#}q4(h%]h&]h']h(]h*]uh,Kh-hh]q5cdocutils.nodes Text q6XGPs & Functionalityq7q8}q9(hh2hh0ubaubcdocutils.nodes paragraph q:)q;}q<(hXGaussian Processes (GPs) can conveniently be used for Bayesian supervised learning, such as regression and classification. In its simplest form, GP inference can be implemented in a few lines of code. However, in practice, things typically get a little more complicated: you might want to use expressive covariance and mean functions, learn good values for hyperparameters, use non-Gaussian likelihood functions (rendering exact inference intractable), use approximate inference algorithms, or combinations of many or all of the above.q=hhhhh!U paragraphq>h#}q?(h%]h&]h']h(]h*]uh,Kh-hh]q@h6XGaussian Processes (GPs) can conveniently be used for Bayesian supervised learning, such as regression and classification. In its simplest form, GP inference can be implemented in a few lines of code. However, in practice, things typically get a little more complicated: you might want to use expressive covariance and mean functions, learn good values for hyperparameters, use non-Gaussian likelihood functions (rendering exact inference intractable), use approximate inference algorithms, or combinations of many or all of the above.qAqB}qC(hh=hh;ubaubh:)qD}qE(hXA comprehensive introduction to Gaussian Processes for Machine Learning is provided in the `GPML`_ book by Rasmussen and Williams, 2006.qFhhhhh!h>h#}qG(h%]h&]h']h(]h*]uh,K h-hh]qH(h6X[A comprehensive introduction to Gaussian Processes for Machine Learning is provided in the qIqJ}qK(hX[A comprehensive introduction to Gaussian Processes for Machine Learning is provided in the hhDubcdocutils.nodes reference qL)qM}qN(hX`GPML`_UresolvedqOKhhDh!U referenceqPh#}qQ(UnameXGPMLUrefuriqRX#http://www.gaussianprocess.org/gpmlqSh(]h']h%]h&]h*]uh]qTh6XGPMLqUqV}qW(hUhhMubaubh6X& book by Rasmussen and Williams, 2006.qXqY}qZ(hX& book by Rasmussen 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pyGPs.qhhIhhh!h]h#}q(h%]h&]h']h(]h*]uh,K h]qh6XThis file is part of pyGPs.qq}q(hhhhubaubeubeubh)q}q(hUhhhhh!h"h#}q(h%]h&]h']h(]qhah*]qhauh,K h-hh]q(h/)q}q(hXLicenseqhhhhh!h3h#}q(h%]h&]h']h(]h*]uh,K h-hh]qh6XLicenseqq}q(hhhhubaubhH)q}q(hUhhhhh!hKh#}q(h%]h&]h']h(]h*]uh,Nh-hh]q(hZ)q}q(hX>The code is released under the BSD 2-Clause (FreeBSD) License.qhhhhh!h]h#}q(h%]h&]h']h(]h*]uh,Kh]qh6X>The code is released under the BSD 2-Clause (FreeBSD) License.qq}q(hhhhubaubhZ)q}q(hXhCopyright (c) 2014 Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting All rights reserved.qhhhhh!h]h#}q(h%]h&]h']h(]h*]uh,Kh]qh6XhCopyright (c) 2014 Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting All rights reserved.qq}q(hhhhubaubhZ)q}q(hXRedistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:qhhhhh!h]h#}q(h%]h&]h']h(]h*]uh,Kh]qh6XRedistribution and use in source and binary forms, with or without 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Suttorp and C. Igel, Approximation of Gaussian process regression models after training. In M. Verleysen (Hrsg.), Proceedings of the 16th European Symposium on Artificial Neural Networks (ESANN 2008) , pp. 427–432 (2008).q Uresolvedq!Khcdocutils.nodes section q")q#}q$(hUhhUsourceq%hUtagnameq&Usectionq'h}q((h]h]h]h]q)Uregression-on-uci-housing-dataq*ah]q+hauUlineq,KUdocumentq-hUchildrenq.]q/(cdocutils.nodes title q0)q1}q2(hXRegression on UCI Housing dataq3hh#h%hh&Utitleq4h}q5(h]h]h]h]h]uh,Kh-hh.]q6cdocutils.nodes Text q7XRegression on UCI Housing dataq8q9}q:(hh3hh1ubaubcdocutils.nodes paragraph q;)q<}q=(hX5Boston Housing is a fairly standard dataset used for testing regression problems. It contains 506 data points with 12 numeric attributes, and one binary categorical attribute. The goal is to determine median home values, based on various census attributes. This dataset is available at the `UCI Repository`_.hh#h%hh&U paragraphq>h}q?(h]h]h]h]h]uh,Kh-hh.]q@(h7X#Boston Housing is a fairly standard dataset used for testing regression problems. It contains 506 data points with 12 numeric attributes, and one binary categorical attribute. The goal is to determine median home values, based on various census attributes. This dataset is available at the qAqB}qC(hX#Boston Housing is a fairly standard dataset used for testing regression problems. It contains 506 data points with 12 numeric attributes, and one binary categorical attribute. The goal is to determine median home values, based on various census attributes. This dataset is available at the hhh}qS(h]h]h]h]h]uh,Kh-hh.]qT(h7XThe demo follows that in qUqV}qW(hXThe demo follows that in hhQubcdocutils.nodes footnote_reference qX)qY}qZ(hX[1]_h!KhhQh&Ufootnote_referenceq[h}q\(h]q]Uid1q^ah]h]h]h]Urefidq_Uid2q`uh.]qah7X1qb}qc(hUhhYubaubh7X. The data set was preprocessed as follows: each continuous feature was transformed to zero mean and unit variance (The categorical variable was dropped). The data was partitioned into qdqe}qf(hX. The data set was preprocessed as follows: each continuous feature was transformed to zero mean and unit variance (The categorical variable was dropped). The data was partitioned into hhQubcsphinx.ext.mathbase math qg)qh}qi(hUh}qj(UlatexX481h]h]h]h]h]uhhQh.]h&Umathqkubh7X points for training and qlqm}qn(hX points for training and hhQubhg)qo}qp(hUh}qq(UlatexX25h]h]h]h]h]uhhQh.]h&hkubh7X points for testing.qrqs}qt(hX points for testing.hhQubeubh;)qu}qv(hXyThe mean function used was :func:`src.Core.means.meanZero` and the covariance (using the :func:`src.Core.kernels.covSum` function) was a composite of :func:`src.Core.kernels.covSEiso` and :func:`src.Core.kernels.covNoise`. The initial values of the hyperparameters were selected randomly from a zero-mean, unit-variance normal distribtion. The actual values were: :math:`[ -0.75, \; 0.59, \; -0.45 ]`. The initial likelihood hyperparameter was :math:`-2.30`. The regression started with initial negative log marginal likelihood of :math:` 752.46`. Note the initial zero mean and the variance that is uniform over the test set. ::hh#h%hh&h>h}qw(h]h]h]h]h]uh,K h-hh.]qx(h7XThe mean function used was qyqz}q{(hXThe mean function used was hhuubcsphinx.addnodes pending_xref q|)q}}q~(hX:func:`src.Core.means.meanZero`qhhuh%hh&U pending_xrefqh}q(UreftypeXfuncUrefwarnqU reftargetqXsrc.Core.means.meanZeroU refdomainXpyqh]h]U refexplicith]h]h]UrefdocqX demoHousingqUpy:classqNU py:moduleqNuh,K h.]qcdocutils.nodes literal q)q}q(hhh}q(h]h]q(UxrefqhXpy-funcqeh]h]h]uhh}h.]qh7Xsrc.Core.means.meanZero()qq}q(hUhhubah&Uliteralqubaubh7X and the covariance (using the qq}q(hX and the covariance (using the hhuubh|)q}q(hX:func:`src.Core.kernels.covSum`qhhuh%hh&hh}q(UreftypeXfunchhXsrc.Core.kernels.covSumU refdomainXpyqh]h]U refexplicith]h]h]hhhNhNuh,K h.]qh)q}q(hhh}q(h]h]q(hhXpy-funcqeh]h]h]uhhh.]qh7Xsrc.Core.kernels.covSum()qq}q(hUhhubah&hubaubh7X function) was a composite of qq}q(hX function) was a composite of hhuubh|)q}q(hX!:func:`src.Core.kernels.covSEiso`qhhuh%hh&hh}q(UreftypeXfunchhXsrc.Core.kernels.covSEisoU refdomainXpyqh]h]U refexplicith]h]h]hhhNhNuh,K h.]qh)q}q(hhh}q(h]h]q(hhXpy-funcqeh]h]h]uhhh.]qh7Xsrc.Core.kernels.covSEiso()qq}q(hUhhubah&hubaubh7X and qq}q(hX and hhuubh|)q}q(hX!:func:`src.Core.kernels.covNoise`qhhuh%hh&hh}q(UreftypeXfunchhXsrc.Core.kernels.covNoiseU refdomainXpyqh]h]U refexplicith]h]h]hhhNhNuh,K h.]qh)q}q(hhh}q(h]h]q(hhXpy-funcqeh]h]h]uhhh.]qh7Xsrc.Core.kernels.covNoise()qɅq}q(hUhhubah&hubaubh7X. The initial values of the hyperparameters were selected randomly from a zero-mean, unit-variance normal distribtion. The actual values were: q̅q}q(hX. The initial values of the hyperparameters were selected randomly from a zero-mean, unit-variance normal distribtion. The actual values were: hhuubhg)q}q(hUh}q(UlatexX[ -0.75, \; 0.59, \; -0.45 ]h]h]h]h]h]uhhuh.]h&hkubh7X,. The initial likelihood hyperparameter was q҅q}q(hX,. The initial likelihood hyperparameter was hhuubhg)q}q(hUh}q(UlatexX-2.30h]h]h]h]h]uhhuh.]h&hkubh7X. The regression started with initial negative log marginal likelihood of :math:` 752.46`. Note the initial zero mean and the variance that is uniform over the test set.q؅q}q(hX. The regression started with initial negative log marginal likelihood of :math:` 752.46`. Note the initial zero mean and the variance that is uniform over the test set.hhuubeubcdocutils.nodes literal_block q)q}q(hXmodel = pyGPs.GPR() model.optimize(x,y) ym, ys2, fm, fs2, lp = model.predict(xs) xa = np.concatenate((data[:,:4],data[:,5:-1]),axis=1) xa = (xa - np.mean(xa,axis=0))/(np.std(xa,axis=0)+1.e-16) ya, ys2a, fma, fs2a, lpa = model.predict(xa)hh#h%hh&U literal_blockqh}q(U xml:spaceqUpreserveqh]h]h]h]h]uh,Kh-hh.]qh7Xmodel = pyGPs.GPR() model.optimize(x,y) ym, ys2, fm, fs2, lp = model.predict(xs) xa = np.concatenate((data[:,:4],data[:,5:-1]),axis=1) xa = (xa - np.mean(xa,axis=0))/(np.std(xa,axis=0)+1.e-16) ya, ys2a, fma, fs2a, lpa = model.predict(xa)qㅁq}q(hUhhubaubcdocutils.nodes figure q)q}q(hUhh#h%hh&Ufigureqh}q(UalignqXcenterh]h]h]h]h]uh,Nh-hh.]qcdocutils.nodes image q)q}q(hXT.. figure:: _images/demoH1.png :align: center :width: 600pt :height: 300pt h}q(h]UuriX_images/demoH1.pngqh]h]UwidthX600pth]U candidatesq}qU*hsh]UheightX300ptuhhh.]h&Uimagequbaubh;)q}q(hXAfter hyperparameter optimization, the covariance hyperparameters were :math:`[ 1.17, \; 0.45, \; -1.41 ]` and the likelihood hyperparameter was :math:`-2.27`. The final negative log marginal likelihood (optimized) was :math:`214.46`.hh#h%hh&h>h}q(h]h]h]h]h]uh,Kh-hh.]q(h7XGAfter hyperparameter optimization, the covariance hyperparameters were qq}q(hXGAfter hyperparameter optimization, the covariance hyperparameters were hhubhg)q}q(hUh}q(UlatexX[ 1.17, \; 0.45, \; -1.41 ]h]h]h]h]h]uhhh.]h&hkubh7X' and the likelihood hyperparameter was qr}r(hX' and the likelihood hyperparameter was hhubhg)r}r(hUh}r(UlatexX-2.27h]h]h]h]h]uhhh.]h&hkubh7X?. The final negative log marginal likelihood (optimized) was rr}r(hX?. The final negative log marginal likelihood (optimized) was hhubhg)r}r (hUh}r (UlatexX214.46h]h]h]h]h]uhhh.]h&hkubh7X.r }r (hX.hhubeubh)r }r(hUhh#h%hh&hh}r(hXcenterh]h]h]h]h]uh,Nh-hh.]rh)r}r(hXT.. figure:: _images/demoH2.png :align: center :width: 600pt :height: 300pt h}r(h]UuriX_images/demoH2.pngrh]h]UwidthX600pth]h}rU*jsh]UheightX300ptuhj h.]h&hubaubcdocutils.nodes target r)r}r(hXB.. _UCI Repository: http://archive.ics.uci.edu/ml/datasets/HousingU referencedrKhh#h%hh&Utargetrh}r(hIhJh]rUuci-repositoryrah]h]h]h]rhauh,K%h-hh.]ubheubh%hh&Ufootnoterh}r (h]h]h]r!h^ah]r"h`ah]r#X1auh,K'h-hh.]r$(cdocutils.nodes label r%)r&}r'(hUh}r((h]h]h]h]h]uhhh.]r)h7X1r*}r+(hUhj&ubah&Ulabelr,ubcdocutils.nodes enumerated_list r-)r.}r/(hUh}r0(Usuffixr1U.Ustartr2Kh]h]h]Uprefixr3Uh]h]Uenumtyper4U upperalphar5uhhh.]r6cdocutils.nodes list_item r7)r8}r9(hXSuttorp and C. Igel, Approximation of Gaussian process regression models after training. In M. Verleysen (Hrsg.), Proceedings of the 16th European Symposium on Artificial Neural Networks (ESANN 2008) , pp. 427–432 (2008).r:h}r;(h]h]h]h]h]uhj.h.]r<h;)r=}r>(hj:hj8h%hh&h>h}r?(h]h]h]h]h]uh,K'h.]r@h7XSuttorp and C. Igel, Approximation of Gaussian process regression models after training. In M. Verleysen (Hrsg.), Proceedings of the 16th European Symposium on Artificial Neural Networks (ESANN 2008) , pp. 427–432 (2008).rArB}rC(hj:hj=ubaubah&U list_itemrDubah&Uenumerated_listrEubeubh.]rFh;)rG}rH(hUh}rI(h]h]h]h]h]uhh h.]rJh7X;Enumerated list start value not ordinal-1: "T" (ordinal 20)rKrL}rM(hUhjGubah&h>ubah&Usystem_messagerNubaUcurrent_sourcerONU decorationrPNUautofootnote_startrQKUnameidsrR}rS(X1h`hjhh*uh.]rTh#ahUU transformerrUNU footnote_refsrV}rWX1]rXhYasUrefnamesrY}rZ(X1]r[hYaXuci repository]r\hEauUsymbol_footnotesr]]r^Uautofootnote_refsr_]r`Usymbol_footnote_refsra]rbU citationsrc]rdh-hU current_linereNUtransform_messagesrf]rgUreporterrhNUid_startriKU autofootnotesrj]rkU citation_refsrl}rmUindirect_targetsrn]roUsettingsrp(cdocutils.frontend Values rqorr}rs(Ufootnote_backlinksrtKUrecord_dependenciesruNU rfc_base_urlrvUhttp://tools.ietf.org/html/rwU tracebackrxUpep_referencesryNUstrip_commentsrzNU toc_backlinksr{Uentryr|U language_coder}Uenr~U datestamprNU report_levelrKU _destinationrNU halt_levelrKU strip_classesrNh4NUerror_encoding_error_handlerrUbackslashreplacerUdebugrNUembed_stylesheetrUoutput_encoding_error_handlerrUstrictrU sectnum_xformrKUdump_transformsrNU docinfo_xformrKUwarning_streamrNUpep_file_url_templaterUpep-%04drUexit_status_levelrKUconfigrNUstrict_visitorrNUcloak_email_addressesrUtrim_footnote_reference_spacerUenvrNUdump_pseudo_xmlrNUexpose_internalsrNUsectsubtitle_xformrU source_linkrNUrfc_referencesrNUoutput_encodingrUutf-8rU source_urlrNUinput_encodingrU utf-8-sigrU_disable_configrNU id_prefixrUU tab_widthrKUerror_encodingrUUTF-8rU_sourcerUF/Users/marion/workspace/workspace_GPs/pyGPs/doc/source/demoHousing.rstrUgettext_compactrU generatorrNUdump_internalsrNU smart_quotesrU pep_base_urlrUhttp://www.python.org/dev/peps/rUsyntax_highlightrUlongrUinput_encoding_error_handlerrjUauto_id_prefixrUidrUdoctitle_xformrUstrip_elements_with_classesrNU _config_filesr]Ufile_insertion_enabledrU raw_enabledrKU dump_settingsrNubUsymbol_footnote_startrKUidsr}r(h`hjjh*h#h^hYuUsubstitution_namesr}rh&h-h}r(h]h]h]Usourcehh]h]uU footnotesr]rhaUrefidsr}rub.pyGPs-1.2/doc/build/doctrees/demoMaunaLoa.doctree000066400000000000000000000643171236346020200220140ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(X1XgpmlqXregression on mauna loa dataqNX[2]quUsubstitution_defsq }q Uparse_messagesq ]q Ucurrent_sourceq NU decorationqNUautofootnote_startqKUnameidsq}q(X1Uid2qhUgpmlqhUregression-on-mauna-loa-dataqhUid3quUchildrenq]qcdocutils.nodes section q)q}q(U rawsourceqUUparentqhUsourceqcdocutils.nodes reprunicode qXG/Users/marion/workspace/workspace_GPs/pyGPs/doc/source/demoMaunaLoa.rstqq }q!bUtagnameq"Usectionq#U attributesq$}q%(Udupnamesq&]Uclassesq']Ubackrefsq(]Uidsq)]q*haUnamesq+]q,hauUlineq-KUdocumentq.hh]q/(cdocutils.nodes title q0)q1}q2(hXRegression on Mauna Loa dataq3hhhh h"Utitleq4h$}q5(h&]h']h(]h)]h+]uh-Kh.hh]q6cdocutils.nodes Text q7XRegression on Mauna Loa dataq8q9}q:(hh3hh1ubaubcdocutils.nodes paragraph q;)q<}q=(hXThis example does regression on the Hawaiian Mauna Loa data (example taken from chapter :math:`5` of the `GPML`_ book by Rasmussen and Williams, 2006)q>hhhh h"U paragraphq?h$}q@(h&]h']h(]h)]h+]uh-Kh.hh]qA(h7XXThis example does regression on the Hawaiian Mauna Loa data (example taken from chapter qBqC}qD(hXXThis example does regression on the Hawaiian Mauna Loa data (example taken from chapter hhThe data is shown in the above plot. Our goal is to model the qq}q(hX>The data is shown in the above plot. Our goal is to model the hhubhE)q}q(hUh$}q(UlatexXCO_2h)]h(]h&]h']h+]uhhh]h"hIubh7X% concentration as a function of time qƅq}q(hX% concentration as a function of time hhubhE)q}q(hUh$}q(UlatexXth)]h(]h&]h']h+]uhhh]h"hIubh7X. Several features are immediately apparent: a long term rising trend, a pronounced seasonal variation and some smaller irregularities. In the following, contributions to a combined covariance function which takes care of these individual properties are suggessted. This is meant primarily to illustrate the power and flexibility of the Gaussian process framework—it is possible that other choices would be more appropriate for this data set.q̅q}q(hX. Several features are immediately apparent: a long term rising trend, a pronounced seasonal variation and some smaller irregularities. In the following, contributions to a combined covariance function which takes care of these individual properties are suggessted. This is meant primarily to illustrate the power and flexibility of the Gaussian process framework—it is possible that other choices would be more appropriate for this data set.hhubeubh;)q}q(hXTo model the long term smooth rising trend, a squared exponential (SE) covariance term with two hyperparameters controlling the amplitude :math:`\theta_1` and characteristic length-scale :math:`\theta_2` is used:hhhh h"h?h$}q(h&]h']h(]h)]h+]uh-K#h.hh]q(h7XTo model the long term smooth rising trend, a squared exponential (SE) covariance term with two hyperparameters controlling the amplitude qӅq}q(hXTo model the long term smooth rising trend, a squared exponential (SE) covariance term with two hyperparameters controlling the amplitude hhubhE)q}q(hUh$}q(UlatexX\theta_1h)]h(]h&]h']h+]uhhh]h"hIubh7X! and characteristic length-scale qمq}q(hX! and characteristic length-scale hhubhE)q}q(hUh$}q(UlatexX\theta_2h)]h(]h&]h']h+]uhhh]h"hIubh7X is used:q߅q}q(hX is used:hhubeubcsphinx.ext.mathbase displaymath q)q}q(hUhhhh h"U displaymathqh$}q(UlatexqXIk_1(x,x') = \theta_1^2 \exp \left(-\frac{(x-x')^2}{2\theta_2^2}\right). Unowrapqh)]h(]h&]h']h+]UdocnameqX demoMaunaLoaqUlabelqNuh-K'h.hh]ubh;)q}q(hXNote that we just use a smooth trend; actually enforcing the trend *a priori* to be increasing is probably not so simple and (hopefully) not desirable. We can use the periodic covariance function with a period of one year to model the seasonal variation. However, it is not clear that the seasonal trend is exactly periodic, so we modify it by taking the product with a squared exponential component to allow a decay away from exact periodicity:hhhh h"h?h$}q(h&]h']h(]h)]h+]uh-K)h.hh]q(h7XCNote that we just use a smooth trend; actually enforcing the trend qq}q(hXCNote that we just use a smooth trend; actually enforcing the trend hhubh)q}q(hX *a priori*h$}q(h&]h']h(]h)]h+]uhhh]qh7Xa prioriqq}q(hUhhubah"hubh7Xp to be increasing is probably not so simple and (hopefully) not desirable. We can use the periodic covariance function with a period of one year to model the seasonal variation. However, it is not clear that the seasonal trend is exactly periodic, so we modify it by taking the product with a squared exponential component to allow a decay away from exact periodicity:qq}q(hXp to be increasing is probably not so simple and (hopefully) not desirable. We can use the periodic covariance function with a period of one year to model the seasonal variation. However, it is not clear that the seasonal trend is exactly periodic, so we modify it by taking the product with a squared exponential component to allow a decay away from exact periodicity:hhubeubh)q}q(hUhhhh h"hh$}q(hXmk_2(x,x') = \theta_3^2 \exp\left(-\frac{(x-x')^2}{2\theta_4^2} \frac{2\sin^2(\pi(x-x'))}{\theta_5^2}\right).hh)]h(]h&]h']h+]hhhNuh-K/h.hh]ubh;)r}r(hXwhere :math:`\theta_3` gives the magnitude, :math:`\theta_4` the decay-time for the periodic component, and :math:`\theta_5` the smoothness of the periodic component; the period has been fixed to one (year). The seasonal component in the data is caused primarily by different rates of :math:`CO_2` uptake for plants depending on the season, and it is probably reasonable to assume that this pattern may itself change slowly over time, partially due to the elevation of the :math:`CO_2` level itself; if this effect turns out not to be relevant, then it can be effectively removed at the fitting stage by allowing :math:`\theta_4` to become very large.hhhh h"h?h$}r(h&]h']h(]h)]h+]uh-K3h.hh]r(h7Xwhere rr}r(hXwhere hjubhE)r}r(hUh$}r (UlatexX\theta_3h)]h(]h&]h']h+]uhjh]h"hIubh7X gives the magnitude, r r }r (hX gives the magnitude, hjubhE)r }r(hUh$}r(UlatexX\theta_4h)]h(]h&]h']h+]uhjh]h"hIubh7X0 the decay-time for the periodic component, and rr}r(hX0 the decay-time for the periodic component, and hjubhE)r}r(hUh$}r(UlatexX\theta_5h)]h(]h&]h']h+]uhjh]h"hIubh7X the smoothness of the periodic component; the period has been fixed to one (year). The seasonal component in the data is caused primarily by different rates of rr}r(hX the smoothness of the periodic component; the period has been fixed to one (year). The seasonal component in the data is caused primarily by different rates of hjubhE)r}r(hUh$}r(UlatexXCO_2h)]h(]h&]h']h+]uhjh]h"hIubh7X uptake for plants depending on the season, and it is probably reasonable to assume that this pattern may itself change slowly over time, partially due to the elevation of the rr}r(hX uptake for plants depending on the season, and it is probably reasonable to assume that this pattern may itself change slowly over time, partially due to the elevation of the hjubhE)r}r (hUh$}r!(UlatexXCO_2h)]h(]h&]h']h+]uhjh]h"hIubh7X level itself; if this effect turns out not to be relevant, then it can be effectively removed at the fitting stage by allowing r"r#}r$(hX level itself; if this effect turns out not to be relevant, then it can be effectively removed at the fitting stage by allowing hjubhE)r%}r&(hUh$}r'(UlatexX\theta_4h)]h(]h&]h']h+]uhjh]h"hIubh7X to become very large.r(r)}r*(hX to become very large.hjubeubh;)r+}r,(hXSTo model the (small) medium term irregularities, a rational quadratic term is used:r-hhhh h"h?h$}r.(h&]h']h(]h)]h+]uh-Kh.hh]ubh;)r6}r7(hXwhere :math:`\theta_6` is the magnitude, :math:`\theta_7` is the typical length-scale and :math:`\theta_8` is the shape parameter determining diffuseness of the length-scales.hhhh h"h?h$}r8(h&]h']h(]h)]h+]uh-KBh.hh]r9(h7Xwhere r:r;}r<(hXwhere hj6ubhE)r=}r>(hUh$}r?(UlatexX\theta_6h)]h(]h&]h']h+]uhj6h]h"hIubh7X is the magnitude, r@rA}rB(hX is the magnitude, hj6ubhE)rC}rD(hUh$}rE(UlatexX\theta_7h)]h(]h&]h']h+]uhj6h]h"hIubh7X! is the typical length-scale and rFrG}rH(hX! is the typical length-scale and hj6ubhE)rI}rJ(hUh$}rK(UlatexX\theta_8h)]h(]h&]h']h+]uhj6h]h"hIubh7XE is the shape parameter determining diffuseness of the length-scales.rLrM}rN(hXE is the shape parameter determining diffuseness of the length-scales.hj6ubeubh;)rO}rP(hXOne could also have used a squared exponential form for this component, but it turns out that the rational quadratic works better (gives higher marginal likelihood), probably because it can accommodate several length-scales simultaneously.rQhhhh h"h?h$}rR(h&]h']h(]h)]h+]uh-KEh.hh]rSh7XOne could also have used a squared exponential form for this component, but it turns out that the rational quadratic works better (gives higher marginal likelihood), probably because it can accommodate several length-scales simultaneously.rTrU}rV(hjQhjOubaubh;)rW}rX(hXoFinally we specify a noise model as the sum of a squared exponential contrubition and an independent component:rYhhhh h"h?h$}rZ(h&]h']h(]h)]h+]uh-KIh.hh]r[h7XoFinally we specify a noise model as the sum of a squared exponential contrubition and an independent component:r\r]}r^(hjYhjWubaubh)r_}r`(hUhhhh h"hh$}ra(hXkk_4(x_p,x_q) = \theta_9^2\exp\left(-\frac{(x_p - x_q)^2}{2\theta_{10}^2}\right) + \theta_{11}^2\delta_{pq}.hh)]h(]h&]h']h+]hhhNuh-KKh.hh]ubh;)rb}rc(hXwhere :math:`\theta_9` is the magnitude of the correlated noise component, :math:`\theta_{10}` is its length scale and :math:`\theta_{11}` is the magnitude of the independent noise component. Noise in the series could be caused by measurement inaccuracies, and by local short-term weather phenomena, so it is probably reasonable to assume at least a modest amount of correlation in time. Notice that the correlated noise component, the first term has an identical expression to the long term component in the trend covariance. When optimizing the hyperparameters, we will see that one of these components becomes large with a long length-scale (the long term trend), while the other remains small with a short length-scale (noise). The fact that we have chosen to call one of these components ‘signal’ and the other one ‘noise’ is only a question of interpretation. Presumably, we are less interested in very short-term effect, and thus call it noise; if on the other hand we were interested in this effect, we would call it signal.hhhh h"h?h$}rd(h&]h']h(]h)]h+]uh-KOh.hh]re(h7Xwhere rfrg}rh(hXwhere hjbubhE)ri}rj(hUh$}rk(UlatexX\theta_9h)]h(]h&]h']h+]uhjbh]h"hIubh7X5 is the magnitude of the correlated noise component, rlrm}rn(hX5 is the magnitude of the correlated noise component, hjbubhE)ro}rp(hUh$}rq(UlatexX \theta_{10}h)]h(]h&]h']h+]uhjbh]h"hIubh7X is its length scale and rrrs}rt(hX is its length scale and hjbubhE)ru}rv(hUh$}rw(UlatexX \theta_{11}h)]h(]h&]h']h+]uhjbh]h"hIubh7X is the magnitude of the independent noise component. Noise in the series could be caused by measurement inaccuracies, and by local short-term weather phenomena, so it is probably reasonable to assume at least a modest amount of correlation in time. Notice that the correlated noise component, the first term has an identical expression to the long term component in the trend covariance. When optimizing the hyperparameters, we will see that one of these components becomes large with a long length-scale (the long term trend), while the other remains small with a short length-scale (noise). The fact that we have chosen to call one of these components ‘signal’ and the other one ‘noise’ is only a question of interpretation. Presumably, we are less interested in very short-term effect, and thus call it noise; if on the other hand we were interested in this effect, we would call it signal.rxry}rz(hX is the magnitude of the independent noise component. Noise in the series could be caused by measurement inaccuracies, and by local short-term weather phenomena, so it is probably reasonable to assume at least a modest amount of correlation in time. Notice that the correlated noise component, the first term has an identical expression to the long term component in the trend covariance. When optimizing the hyperparameters, we will see that one of these components becomes large with a long length-scale (the long term trend), while the other remains small with a short length-scale (noise). The fact that we have chosen to call one of these components ‘signal’ and the other one ‘noise’ is only a question of interpretation. Presumably, we are less interested in very short-term effect, and thus call it noise; if on the other hand we were interested in this effect, we would call it signal.hjbubeubh;)r{}r|(hX!The final covariance function is:r}hhhh h"h?h$}r~(h&]h']h(]h)]h+]uh-K]h.hh]rh7X!The final covariance function is:rr}r(hj}hj{ubaubh)r}r(hUhhhh h"hh$}r(hX7k(x,x') = k_1(x,x') + k_2(x,x') + k_3(x,x') + k_4(x,x')hh)]h(]h&]h']h+]hhhNuh-K_h.hh]ubh;)r}r(hXFwith hyperparameters :math:`\theta = (\theta_1,\ldots,\theta_{11})` ::rhhhh h"h?h$}r(h&]h']h(]h)]h+]uh-Kch.hh]r(h7Xwith hyperparameters rr}r(hXwith hyperparameters hjubhE)r}r(hUh$}r(UlatexX&\theta = (\theta_1,\ldots,\theta_{11})h)]h(]h&]h']h+]uhjh]h"hIubeubcdocutils.nodes literal_block r)r}r(hXX# DEFINE parameterized covariance function k1 = pyGPs.cov.RBF(np.log(67.), np.log(66.)) k2 = pyGPs.cov.Periodic(np.log(1.3), np.log(1.0), np.log(2.4)) * cov.RBF(np.log(90.), np.log(2.4)) k3 = pyGPs.cov.RQ(np.log(1.2), np.log(0.66), np.log(0.78)) k4 = pyGPs.cov.RBF(np.log(1.6/12.), np.log(0.18)) + cov.Noise(np.log(0.19)) k = k1 + k2 + k3 + k4hhhh h"U literal_blockrh$}r(U xml:spacerUpreserverh)]h(]h&]h']h+]uh-Keh.hh]rh7XX# DEFINE parameterized covariance function k1 = pyGPs.cov.RBF(np.log(67.), np.log(66.)) k2 = pyGPs.cov.Periodic(np.log(1.3), np.log(1.0), np.log(2.4)) * cov.RBF(np.log(90.), np.log(2.4)) k3 = pyGPs.cov.RQ(np.log(1.2), np.log(0.66), np.log(0.78)) k4 = pyGPs.cov.RBF(np.log(1.6/12.), np.log(0.18)) + cov.Noise(np.log(0.19)) k = k1 + k2 + k3 + k4rr}r(hUhjubaubh;)r}r(hX"After running the minimization, ::rhhhh h"h?h$}r(h&]h']h(]h)]h+]uh-Kmh.hh]rh7XAfter running the minimization,rr}r(hXAfter running the minimization,hjubaubj)r}r(hX?t0 = clock() model.optimize(x,y) t1 = clock() model.predict(xs)hhhh h"jh$}r(jjh)]h(]h&]h']h+]uh-Koh.hh]rh7X?t0 = clock() model.optimize(x,y) t1 = clock() model.predict(xs)rr}r(hUhjubaubh;)r}r(hX!The extrapolated data looks like:rhhhh h"h?h$}r(h&]h']h(]h)]h+]uh-Kth.hh]rh7X!The extrapolated data looks like:rr}r(hjhjubaubh)r}r(hUhhhh h"hh$}r(hXcenterh)]h(]h&]h']h+]uh-Nh.hh]rh)r}r(hXg.. figure:: _images/demoML2.png :height: 600 px :width: 800 px :align: center :scale: 70 % h$}r(UscaleKFh&]UuriX_images/demoML2.pngrh)]h(]UwidthX800pxh']h}rU*jsh+]UheightX600pxuhjh]h"hubaubh;)r}r(hXzand the optimized values of the hyperparameters allow for a principled analysis of different components driving the model.rhhhh h"h?h$}r(h&]h']h(]h)]h+]uh-K|h.hh]rh7Xzand the optimized values of the hyperparameters allow for a principled analysis of different components driving the model.rr}r(hjhjubaubcdocutils.nodes footnote r)r}r(hX Keeling, C. D. and Whorf, T. P. (2004). Atmospheric :math:`CO_2` Records from Sites in the SIO Air Sampling Network. In Trends: A Compendium of Data on Global Change. Carbon Dioxide Information Analysis Center, Oak Ridge National Laboratory, Oak Ridge, Tenn., U.S.A. hPKhhhh h"Ufootnoterh$}r(h&]h']h(]rh|ah)]rhah+]rX1auh-K~h.hh]r(cdocutils.nodes label r)r}r(hUh$}r(h&]h']h(]h)]h+]uhjh]rh7X1r}r(hUhjubah"hubh;)r}r(hX Keeling, C. D. and Whorf, T. P. (2004). Atmospheric :math:`CO_2` Records from Sites in the SIO Air Sampling Network. In Trends: A Compendium of Data on Global Change. Carbon Dioxide Information Analysis Center, Oak Ridge National Laboratory, Oak Ridge, Tenn., U.S.A.rhjhh h"h?h$}r(h&]h']h(]h)]h+]uh-K~h]r(h7X4Keeling, C. D. and Whorf, T. P. (2004). Atmospheric rr}r(hX4Keeling, C. D. and Whorf, T. P. (2004). Atmospheric hjubhE)r}r(hUh$}r(UlatexXCO_2h)]h(]h&]h']h+]uhjh]h"hIubh7X Records from Sites in the SIO Air Sampling Network. In Trends: A Compendium of Data on Global Change. Carbon Dioxide Information Analysis Center, Oak Ridge National Laboratory, Oak Ridge, Tenn., U.S.A.rr}r(hX Records from Sites in the SIO Air Sampling Network. In Trends: A Compendium of Data on Global Change. 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}h}h}h}h}h}h}h}uUversionchangesrL}U_viewcode_modulesrL}rL(cdocutils.nodes reprunicode rLXpyGPs.Core.covrLrL}rLbXò#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ # covariance functions to be use by Gaussian process functions. There are two # different kinds of covariance functions: simple and composite: # # simple cov functions: # # Poly - Polynomial covariance function # PiecePoly - Piecewise polynomial kernel with compact support. # RBF - Squared Exponential kernel. # RBFunit - Squared Exponential kernel with unit magnitude. # RBFard - Squared Exponential kernel with Automatic Relevance Determination. # Const - Constant kernel. # Linear - Linear kernel. # LINard - Linear covariance function with Automatic Relevance Detemination. # Matern - Matern covariance function. # Periodic - Stationary kernel for a smooth periodic function. # Noise - Independent covariance function, i.e "white noise". # RQ - Rational Quadratic covariance function with isotropic distance measure. # RQard - Rational Quadratic covariance function with ARD distance measure. # Pre - Precomputed kernel matrix. # # composite covariance functions: # # ScaleOfKernel - scaled version of a covariance function # ProductOfKernel - products of covariance functions # SumOfKernel - sums of covariance functions # FITCOfKernel - Covariance function to be used together with the FITC approximation # # # This is a object-oriented python implementation of gpml functionality # (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18). # based on the functional-version of python implementation # (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013) # # Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013 import numpy as np import math import scipy.spatial.distance as spdist class Kernel(object): """ This is a base class of Kernel functions there is no computation in this class, it just defines rules about a kernel class should have each covariance function will inherit it and implement its own behaviour """ def __init__(self): self.hyp = [] self.para = [] def getCovMatrix(self,x=None,z=None,mode=None): ''' Return the specific covariance matrix according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self covariance matrix of test data(test by 1). 'train' return training covariance matrix(train by train). 'cross' return cross covariance matrix between x and z(train by test) :return: the corresponding covariance matrix ''' pass def getDerMatrix(self,x=None,z=None,mode=None,der=None): ''' Compute derivatives wrt. hyperparameters according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self derivative matrix of test data(test by 1). 'train' return training derivative matrix(train by train). 'cross' return cross derivative matrix between x and z(train by test) :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' pass # overloading def __add__(self,cov): ''' Overloading + operator. :param cov: covariance function :return: an instance of SumOfKernel ''' return SumOfKernel(self,cov) # overloading def __mul__(self,other): ''' Overloading * operator. Using * for both multiplication with scalar and product of kernels depending on the type of the two objects. :param other: covariance function as product or int/float as scalar :return: an instance of ScaleOfKernel or ProductOfKernel ''' if isinstance(other, int) or isinstance(other, float): return ScaleOfKernel(self,other) elif isinstance(other, Kernel): return ProductOfKernel(self,other) else: print "only numbers and Kernels are supported operand types for *" # overloading __rmul__ = __mul__ def fitc(self,inducingInput): ''' Covariance function to be used together with the FITC approximation. Setting FITC gp model will implicitly call this method. :return: an instance of FITCOfKernel ''' return FITCOfKernel(self,inducingInput) # can be replaced by spdist from scipy def _sq_dist(self, a, b=None): '''Compute a matrix of all pairwise squared distances between two sets of vectors, stored in the row of the two matrices: a (of size n by D) and b (of size m by D).''' n = a.shape[0] D = a.shape[1] m = n if b == None: b = a.transpose() else: m = b.shape[0] b = b.transpose() C = np.zeros((n,m)) for d in range(0,D): tt = a[:,d] tt = tt.reshape(n,1) tem = np.kron(np.ones((1,m)), tt) tem = tem - np.kron(np.ones((n,1)), b[d,:]) C = C + tem * tem return C class ProductOfKernel(Kernel): '''Product of two kernel function.''' def __init__(self,cov1,cov2): self.cov1 = cov1 self.cov2 = cov2 self._hyp = cov1.hyp + cov2.hyp def _setHyp(self,hyp): assert len(hyp) == len(self._hyp) len1 = len(self.cov1.hyp) self._hyp = hyp self.cov1.hyp = self._hyp[:len1] self.cov2.hyp = self._hyp[len1:] def _getHyp(self): return self._hyp hyp = property(_getHyp,_setHyp) def getCovMatrix(self,x=None,z=None,mode=None): A = self.cov1.getCovMatrix(x,z,mode) * self.cov2.getCovMatrix(x,z,mode) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): if der < len(self.cov1.hyp): A = self.cov1.getDerMatrix(x,z,mode,der) * self.cov2.getCovMatrix(x,z,mode) elif der < len(self.hyp): der2 = der - len(self.cov1.hyp) A = self.cov2.getDerMatrix(x,z,mode,der2) * self.cov1.getCovMatrix(x,z,mode) else: raise Exception("Error: der out of range for covProduct") return A class SumOfKernel(Kernel): '''Sum of two kernel function.''' def __init__(self,cov1,cov2): self.cov1 = cov1 self.cov2 = cov2 self._hyp = cov1.hyp + cov2.hyp def _setHyp(self,hyp): assert len(hyp) == len(self._hyp) len1 = len(self.cov1.hyp) self._hyp = hyp self.cov1.hyp = self._hyp[:len1] self.cov2.hyp = self._hyp[len1:] def _getHyp(self): return self._hyp hyp = property(_getHyp,_setHyp) def getCovMatrix(self,x=None,z=None,mode=None): A = self.cov1.getCovMatrix(x,z,mode) + self.cov2.getCovMatrix(x,z,mode) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): if der < len(self.cov1.hyp): A = self.cov1.getDerMatrix(x,z,mode,der) elif der < len(self.hyp): der2 = der - len(self.cov1.hyp) A = self.cov2.getDerMatrix(x,z,mode,der2) else: raise Exception("Error: der out of range for covSum") return A class ScaleOfKernel(Kernel): '''Scale of a kernel function.''' def __init__(self,cov,scalar): self.cov = cov if cov.hyp: self._hyp = [scalar] + cov.hyp else: self._hyp = [scalar] def _setHyp(self,hyp): assert len(hyp) == len(self._hyp) self._hyp = hyp self.cov.hyp = self._hyp[1:] def _getHyp(self): return self._hyp hyp = property(_getHyp,_setHyp) def getCovMatrix(self,x=None,z=None,mode=None): sf2 = np.exp(self.hyp[0]) # scale parameter A = sf2 * self.cov.getCovMatrix(x,z,mode) # accumulate cov return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): sf2 = np.exp(self.hyp[0]) # scale parameter if der == 0: # compute derivative w.r.t. sf2 A = 2. * sf2 * self.cov.getCovMatrix(x,z,mode) else: A = sf2 * self.cov.getDerMatrix(x,z,mode,der-1) return A class FITCOfKernel(Kernel): ''' Covariance function to be used together with the FITC approximation. The function allows for more than one output argument and does not respect the interface of a proper covariance function. Instead of outputing the full covariance, it returns cross-covariances between the inputs x, z and the inducing inputs xu as needed by infFITC ''' def __init__(self,cov,inducingInput): self.inducingInput = inducingInput self.covfunc = cov self._hyp = cov.hyp def _getHyp(self): return self._hyp def _setHyp(self, hyp): self._hyp = hyp self.covfunc.hyp = hyp hyp = property(_getHyp,_setHyp) def getCovMatrix(self,x=None,z=None,mode=None): xu = self.inducingInput if x != None: try: assert(xu.shape[1] == x.shape[1]) except AssertionError: raise Exception('Dimensionality of inducing inputs must match training inputs') if mode == 'self_test': # self covariances for the test cases K = self.covfunc.getCovMatrix(z=z,mode='self_test') return K elif mode == 'train': # compute covariance matix for training set K = self.covfunc.getCovMatrix(z=x,mode='self_test') Kuu = self.covfunc.getCovMatrix(x=xu,mode='train') Ku = self.covfunc.getCovMatrix(x=xu,z=x,mode='cross') return K, Kuu, Ku elif mode == 'cross': # compute covariance between data sets x and z K = self.covfunc.getCovMatrix(x=xu,z=z,mode='cross') return K def getDerMatrix(self,x=None,z=None,mode=None,der=None): xu = self.inducingInput if x!= None: try: assert(xu.shape[1] == x.shape[1]) except AssertionError: raise Exception('Dimensionality of inducing inputs must match training inputs') if mode == 'self_test': # self covariances for the test cases K = self.covfunc.getDerMatrix(z=z,mode='self_test',der=der) return K elif mode == 'train': # compute covariance matix for training set K = self.covfunc.getDerMatrix(z=x,mode='self_test',der=der) Kuu = self.covfunc.getDerMatrix(x=xu,mode='train',der=der) Ku = self.covfunc.getDerMatrix(x=xu,z=x,mode='cross',der=der) return K, Kuu, Ku elif mode == 'cross': # compute covariance between data sets x and z K = self.covfunc.getDerMatrix(x=xu,z=z,mode='cross',der=der) return K class Poly(Kernel): ''' Polynomial covariance function. hyp = [ log_c, log_sigma ] :param log_c: inhomogeneous offset. :param log_sigma: signal deviation. :param d: degree of polynomial (not treated as hyperparameter, i.e. will not be trained). ''' def __init__(self, log_c=0., d=2, log_sigma=0. ): self.hyp = [log_c, log_sigma] self.para = [d] def getCovMatrix(self,x=None,z=None,mode=None): c = np.exp(self.hyp[0]) # inhomogeneous offset sf2 = np.exp(2.*self.hyp[1]) # signal variance ord = self.para[0] # order of polynomial if np.abs(ord-np.round(ord)) < 1e-8: # remove numerical error from format of parameter ord = int(round(ord)) assert(ord >= 1.) # only nonzero integers for ord ord = int(ord) if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.reshape(np.sum(z*z,1), (nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.dot(x,x.T) elif mode == 'cross': # compute covariance between data sets x and z A = np.dot(x,z.T) A = sf2 * (c + A)**ord return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): c = np.exp(self.hyp[0]) # inhomogeneous offset sf2 = np.exp(2.*self.hyp[1]) # signal variance ord = self.para[0] # order of polynomial if np.abs(ord-np.round(ord)) < 1e-8: # remove numerical error from format of parameter ord = int(round(ord)) assert(ord >= 1.) # only nonzero integers for ord ord = int(ord) if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.reshape(np.sum(z*z,1), (nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.dot(x,x.T) elif mode == 'cross': # compute covariance between data sets x and z A = np.dot(x,z.T) if der == 0: # compute derivative matrix wrt 1st parameter A = c * ord * sf2 * (c+A)**(ord-1) elif der == 1: # compute derivative matrix wrt 2nd parameter A = 2. * sf2 * (c + A)**ord elif der == 2: # no derivative wrt 3rd parameter A = np.zeros_like(A) # do nothing (d is not learned) else: raise Exception("Wrong derivative entry in Poly") return A class PiecePoly(Kernel): ''' Piecewise polynomial kernel with compact support. hyp = [log_ell, log_sigma] :param log_ell: characteristic length scale. :param log_sigma: signal deviation. :param v: degree v will be rounded to 0,1,2,or 3. (not treated as hyperparameter, i.e. will not be trained). ''' def __init__(self, log_ell=0., v=2, log_sigma=0. ): self.hyp = [log_ell, log_sigma] self.para = [v] def ppmax(self,A,B): return np.maximum(A,B*np.ones_like(A)) def func(self,v,r,j): if v == 0: return 1 elif v == 1: return ( 1. + (j+1) * r ) elif v == 2: return ( 1. + (j+2)*r + (j*j + 4.*j+ 3)/3.*r*r ) elif v == 3: return ( 1. + (j+3)*r + (6.*j*j+36.*j+45.)/15.*r*r + (j*j*j+9.*j*j+23.*j+15.)/15.*r*r*r ) else: raise Exception (["Wrong degree in PiecePoly. Should be 0,1,2 or 3, is " + str(v)]) def dfunc(self,v,r,j): if v == 0: return 0 elif v == 1: return ( j+1 ) elif v == 2: return ( (j+2) + 2.*(j*j+ 4.*j+ 3.)/3.*r ) elif v == 3: return ( (j+3) + 2.*(6.*j*j+36.*j+45.)/15.*r + (j*j*j+9.*j*j+23.*j+15.)/5.*r*r ) else: raise Exception (["Wrong degree in PiecePoly. Should be 0,1,2 or 3, is " + str(v)]) def pp(self,r,j,v,func): return func(v,r,j)*(self.ppmax(1-r,0)**(j+v)) def dpp(self,r,j,v,func,dfunc): return self.ppmax(1-r,0)**(j+v-1) * r * ( (j+v)*self.func(v,r,j) - self.ppmax(1-r,0) * self.dfunc(v,r,j) ) def getCovMatrix(self,x=None,z=None,mode=None): if x != None: n, D = x.shape if z != None: nn, D = z.shape ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.*self.hyp[1]) # signal variance v = self.para[0] # degree (v = 0,1,2 or 3 only) if np.abs(v-np.round(v)) < 1e-8: # remove numerical error from format of parameter v = int(round(v)) assert(int(v) in range(4)) # Only allowed degrees: 0,1,2 or 3 v = int(v) j = np.floor(0.5*D) + v + 1 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.sqrt( spdist.cdist(x/ell, x/ell, 'sqeuclidean') ) elif mode == 'cross': # compute covariance between data sets x and z A = np.sqrt( spdist.cdist(x/ell, z/ell, 'sqeuclidean') ) A = sf2 * self.pp(A,j,v,self.func) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): if x != None: n, D = x.shape if z != None: nn, D = z.shape ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.*self.hyp[1]) # signal variance v = self.para[0] # degree (v = 0,1,2 or 3 only) if np.abs(v-np.round(v)) < 1e-8: # remove numerical error from format of parameter v = int(round(v)) assert(int(v) in range(4)) # Only allowed degrees: 0,1,2 or 3 v = int(v) j = np.floor(0.5*D) + v + 1 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.sqrt( spdist.cdist(x/ell, x/ell, 'sqeuclidean') ) elif mode == 'cross': # compute covariance between data sets x and z A = np.sqrt( spdist.cdist(x/ell, z/ell, 'sqeuclidean') ) if der == 0: # compute derivative matrix wrt 1st parameter A = sf2 * self.dpp(A,j,v,self.func,self.dfunc) elif der == 1: # compute derivative matrix wrt 2nd parameter A = 2. * sf2 * self.pp(A,j,v,self.func) elif der == 2: # wants to compute derivative wrt order A = np.zeros_like(A) else: raise Exception("Wrong derivative entry in PiecePoly") return A class RBF(Kernel): ''' Squared Exponential kernel with isotropic distance measure. hyp = [log_ell, log_sigma] :param log_ell: characteristic length scale. :param log_sigma: signal deviation. ''' def __init__(self, log_ell=0., log_sigma=0.): self.hyp = [log_ell, log_sigma] def getCovMatrix(self,x=None,z=None,mode=None): ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.*self.hyp[1]) # signal variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for training set A = spdist.cdist(x/ell,x/ell,'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z A = spdist.cdist(x/ell,z/ell,'sqeuclidean') A = sf2 * np.exp(-0.5*A) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.*self.hyp[1]) # signal variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = spdist.cdist(x/ell,x/ell,'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z A = spdist.cdist(x/ell,z/ell,'sqeuclidean') if der == 0: # compute derivative matrix wrt 1st parameter A = sf2 * np.exp(-0.5*A) * A elif der == 1: # compute derivative matrix wrt 2nd parameter A = 2. * sf2 * np.exp(-0.5*A) else: raise Exception("Calling for a derivative in RBF that does not exist") return A class RBFunit(Kernel): ''' Squared Exponential kernel with isotropic distance measure with unit magnitude. i.e signal variance is always 1. hyp = [ log_ell ] :param log_ell: characteristic length scale. ''' def __init__(self, log_ell=0.): self.hyp = [log_ell] def getCovMatrix(self,x=None,z=None,mode=None): ell = np.exp(self.hyp[0]) # characteristic length scale if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = spdist.cdist(x/ell,x/ell,'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z A = spdist.cdist(x/ell,z/ell,'sqeuclidean') A = np.exp(-0.5*A) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): ell = np.exp(self.hyp[0]) # characteristic length scale if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = spdist.cdist(x/ell,x/ell,'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z A = spdist.cdist(x/ell,z/ell,'sqeuclidean') if der == 0: # compute derivative matrix wrt 1st parameter A = np.exp(-0.5*A) * A else: raise Exception("Wrong derivative index in RDFunit") return A class RBFard(Kernel): ''' Squared Exponential kernel with Automatic Relevance Determination. hyp = log_ell_list + [log_sigma] :param D: dimension of pattern. set if you want default ell, which is 1 for each dimension. :param log_ell_list: characteristic length scale for each dimension. :param log_sigma: signal deviation. ''' def __init__(self, D=None, log_ell_list=None, log_sigma=0.): if log_ell_list == None: self.hyp = [0. for i in xrange(D)] + [log_sigma] else: self.hyp = log_ell_list + [log_sigma] def getCovMatrix(self,x=None,z=None,mode=None): if x != None: n, D = x.shape if z != None: nn, D = z.shape ell = 1./np.exp(self.hyp[0:D]) # characteristic length scale sf2 = np.exp(2.*self.hyp[D]) # signal variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x tem = np.dot(np.diag(ell),x.T).T A = spdist.cdist(tem,tem,'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z A = spdist.cdist(np.dot(np.diag(ell),x.T).T,np.dot(np.diag(ell),z.T).T,'sqeuclidean') A = sf2*np.exp(-0.5*A) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): if x != None: n, D = x.shape if z != None: nn, D = z.shape ell = 1./np.exp(self.hyp[0:D]) # characteristic length scale sf2 = np.exp(2.*self.hyp[D]) # signal variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x tem = np.dot(np.diag(ell),x.T).T A = spdist.cdist(tem,tem,'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z A = spdist.cdist(np.dot(np.diag(ell),x.T).T,np.dot(np.diag(ell),z.T).T,'sqeuclidean') A = sf2*np.exp(-0.5*A) if der < D: # compute derivative matrix wrt length scale parameters if mode == 'self_test': A = A*0 elif mode == 'train': tem = np.atleast_2d(x[:,der])/ell[der] A *= spdist.cdist(tem,tem,'sqeuclidean') elif mode == 'cross': A *= spdist.cdist(np.atleast_2d(x[:,der]).T/ell[der],np.atleast_2d(z[:,der]).T/ell[der],'sqeuclidean') elif der == D: # compute derivative matrix wrt magnitude parameter A = 2.*A else: raise Exception("Wrong derivative index in RDFard") return A class Const(Kernel): ''' Constant kernel. hyp = [ log_sigma ] :param log_sigma: signal deviation. ''' def __init__(self, log_sigma=0.): self.hyp = [log_sigma] def getCovMatrix(self,x=None,z=None,mode=None): sf2 = np.exp(self.hyp[0]) # s2 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = sf2 * np.ones((nn,1)) elif mode == 'train': # compute covariance matix for dataset x n,D = x.shape A = sf2 * np.ones((n,n)) elif mode == 'cross': # compute covariance between data sets x and z n,D = x.shape nn,D = z.shape A = sf2 * np.ones((n,nn)) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): sf2 = np.exp(self.hyp[0]) # s2 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = sf2 * np.ones((nn,1)) elif mode == 'train': # compute covariance matix for dataset x n,D = x.shape A = sf2 * np.ones((n,n)) elif mode == 'cross': # compute covariance between data sets x and z n,D = x.shape nn,D = z.shape A = sf2 * np.ones((n,nn)) if der == 0: # compute derivative matrix wrt sf2 A = 2. * A else: raise Exception("Wrong derivative entry in covConst") return A class Linear(Kernel): ''' Linear kernel. hyp = [ log_sigma ]. :param log_sigma: signal deviation. ''' def __init__(self, log_sigma=0.): self.hyp = [ log_sigma ] def getCovMatrix(self,x=None,z=None,mode=None): sf2 = np.exp(self.hyp[0]) # s2 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.reshape(np.sum(z*z,1), (nn,1)) elif mode == 'train': # compute covariance matix for dataset x n,D = x.shape A = np.dot(x,x.T) + np.eye(n)*1e-16 # required for numerical accuracy elif mode == 'cross': # compute covariance between data sets x and z A = np.dot(x,z.T) A = sf2 * A return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): sf2 = np.exp(self.hyp[0]) # s2 if der == 0: if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.reshape(np.sum(z*z,1), (nn,1)) elif mode == 'train': # compute covariance matix for dataset x n,D = x.shape A = np.dot(x,x.T) + np.eye(n)*1e-16 # required for numerical accuracy elif mode == 'cross': # compute covariance between data sets x and z A = np.dot(x,z.T) A = 2 * sf2 * A else: raise Exception("Wrong derivative index in covLinear") return A class LINard(Kernel): ''' Linear covariance function with Automatic Relevance Detemination. hyp = log_ell_list :param D: dimension of training data. Set if you want default ell, which is 1 for each dimension. :param log_ell_list: characteristic length scale for each dimension. ''' def __init__(self, D=None, log_ell_list=None): if log_ell_list == None: self.hyp = [0. for i in xrange(D)] else: self.hyp = log_ell_list def getCovMatrix(self,x=None,z=None,mode=None): ell = np.exp(self.hyp) # ARD parameters if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.reshape(np.sum(z*z,1), (nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.dot(x,x.T) elif mode == 'cross': # compute covariance between data sets x and z z = np.dot(z,np.diag(1./ell)) A = np.dot(x,z.T) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): ell = np.exp(self.hyp) # ARD parameters if x != None: n, D = x.shape if z != None: nn, D = z.shape if der < D: if mode == 'self_test': tem = np.atleast_2d(z[:,der]).T A = -2.* tem * tem elif mode == 'train': A = -2.*np.dot(np.atleast_2d(x[:,der]).T,np.atleast_2d(x[:,der])) elif mode == 'cross': z = np.dot(z,np.diag(1./ell)) A = -2.*np.dot(np.atleast_2d(x[:,der]).T, np.atleast_2d(z[:,der])) # cross covariances else: raise Exception("Wrong derivative index in covLINard") return A class Matern(Kernel): ''' Matern covariance function with nu = d/2 and isotropic distance measure. For d=1 the function is also known as the exponential covariance function or the Ornstein-Uhlenbeck covariance in 1d. d will be rounded to 1, 3, or 5. hyp = [ log_ell, log_sigma] :param d: d is 2 times nu. Can only be 1,3 or 5. :param log_ell: characteristic length scale. :param log_sigma: signal deviation. ''' def __init__(self, log_ell=0., d=3, log_sigma=0. ): self.hyp = [ log_ell, log_sigma ] self.para = [d] def func(self,d,t): if d == 1: return 1 elif d == 3: return 1 + t elif d == 5: return 1 + t*(1+t/3.) else: raise Exception("Wrong value for d in Matern") def dfunc(self,d,t): if d == 1: return 1 elif d == 3: return t elif d == 5: return t*(1+t/3.) else: raise Exception("Wrong value for d in Matern") def mfunc(self,d,t): return self.func(d,t)*np.exp(-1.*t) def dmfunc(self,d,t): return self.dfunc(d,t)*t*np.exp(-1.*t) def getCovMatrix(self,x=None,z=None,mode=None): ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.* self.hyp[1]) # signal variance d = self.para[0] # 2 times nu if np.abs(d-np.round(d)) < 1e-8: # remove numerical error from format of parameter d = int(round(d)) d = int(d) try: assert(d in [1,3,5]) # check for valid values of d except AssertionError: print "Warning: You specified d to be neither 1,3 nor 5. We set to d=3. " d = 3 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x x = np.sqrt(d)*x/ell A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean')) elif mode == 'cross': # compute covariance between data sets x and z x = np.sqrt(d)*x/ell z = np.sqrt(d)*z/ell A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean')) A = sf2 * self.mfunc(d,A) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.* self.hyp[1]) # signal variance d = self.para[0] # 2 times nu if np.abs(d-np.round(d)) < 1e-8: # remove numerical error from format of parameter d = int(round(d)) d = int(d) try: assert(d in [1,3,5]) # check for valid values of d except AssertionError: print "Warning: You specified d to be neither 1,3 nor 5. We set to d=3. " d = 3 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x x = np.sqrt(d)*x/ell A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean')) elif mode == 'cross': # compute covariance between data sets x and z x = np.sqrt(d)*x/ell z = np.sqrt(d)*z/ell A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean')) A = sf2 * self.mfunc(d,A) if der == 0: # compute derivative matrix wrt 1st parameter A = sf2 * self.dmfunc(d,A) elif der == 1: # compute derivative matrix wrt 2nd parameter A = 2 * sf2 * self.mfunc(d,A) elif der == 2: # no derivative wrt 3rd parameter A = np.zeros_like(A) # do nothing (d is not learned) else: raise Exception("Wrong derivative value in Matern") return A class Periodic(Kernel): ''' Stationary kernel for a smooth periodic function. hyp = [ log_ell, log_p, log_sigma] :param log_p: period. :param log_ell: characteristic length scale. :param log_sigma: signal deviation. ''' def __init__(self, log_ell=0., log_p=0., log_sigma=0. ): self.hyp = [ log_ell, log_p, log_sigma] def getCovMatrix(self,x=None,z=None,mode=None): ell = np.exp(self.hyp[0]) # characteristic length scale p = np.exp(self.hyp[1]) # period sf2 = np.exp(2.*self.hyp[2]) # signal variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean')) elif mode == 'cross': # compute covariance between data sets x and z A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean')) A = np.pi*A/p A = np.sin(A)/ell A = A * A A = sf2 *np.exp(-2.*A) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): ell = np.exp(self.hyp[0]) # characteristic length scale p = np.exp(self.hyp[1]) # period sf2 = np.exp(2.*self.hyp[2]) # signal variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean')) elif mode == 'cross': # compute covariance between data sets x and z A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean')) A = np.pi*A/p if der == 0: # compute derivative matrix wrt 1st parameter A = np.sin(A)/ell A = A * A A = 4. *sf2 *np.exp(-2.*A) * A elif der == 1: # compute derivative matrix wrt 2nd parameter R = np.sin(A)/ell A = 4 * sf2/ell * np.exp(-2.*R*R)*R*np.cos(A)*A elif der == 2: # compute derivative matrix wrt 3rd parameter A = np.sin(A)/ell A = A * A A = 2. * sf2 * np.exp(-2.*A) else: raise Exception("Wrong derivative index in covPeriodic") return A class Noise(Kernel): ''' Independent covariance function, i.e "white noise", with specified variance. Normally NOT used anymore since noise is now added in liklihood. hyp = [ log_sigma ] :param log_sigma: signal deviation. ''' def __init__(self, log_sigma=0.): self.hyp = [log_sigma] def getCovMatrix(self,x=None,z=None,mode=None): tol = 1.e-9 # Tolerance for declaring two vectors "equal" s2 = np.exp(2.*self.hyp[0]) # noise variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x n,D = x.shape A = np.eye(n) elif mode == 'cross': # compute covariance between data sets x and z M = spdist.cdist(x, z, 'sqeuclidean') A = np.zeros_like(M,dtype=np.float) A[M < tol] = 1. A = s2*A return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): tol = 1.e-9 # Tolerance for declaring two vectors "equal" s2 = np.exp(2.*self.hyp[0]) # noise variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x n,D = x.shape A = np.eye(n) elif mode == 'cross': # compute covariance between data sets x and z M = spdist.cdist(x, z, 'sqeuclidean') A = np.zeros_like(M,dtype=np.float) A[M < tol] = 1. if der == 0: A = 2.*s2*A else: raise Exception("Wrong derivative index in covNoise") return A class RQ(Kernel): ''' Rational Quadratic covariance function with isotropic distance measure. hyp = [ log_ell, log_sigma, log_alpha ] :param log_ell: characteristic length scale. :param log_sigma: signal deviation. :param log_alpha: shape parameter for the RQ covariance. ''' def __init__(self, log_ell=0., log_sigma=0., log_alpha=0.): self.hyp = [ log_ell, log_sigma, log_alpha ] def getCovMatrix(self,x=None,z=None,mode=None): ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.*self.hyp[1]) # signal variance alpha = np.exp(self.hyp[2]) if mode == 'self_test': # self covariances for the test cases nn,D = z.shape D2 = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x D2 = spdist.cdist(x/ell, x/ell, 'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z D2 = spdist.cdist(x/ell, z/ell, 'sqeuclidean') A = sf2 * ( ( 1.0 + 0.5*D2/alpha )**(-alpha) ) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.*self.hyp[1]) # signal variance alpha = np.exp(self.hyp[2]) if mode == 'self_test': # self covariances for the test cases nn,D = z.shape D2 = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x D2 = spdist.cdist(x/ell, x/ell, 'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z D2 = spdist.cdist(x/ell, z/ell, 'sqeuclidean') if der == 0: # compute derivative matrix wrt 1st parameter A = sf2 * ( 1.0 + 0.5*D2/alpha )**(-alpha-1) * D2 elif der == 1: # compute derivative matrix wrt 2nd parameter A = 2.* sf2 * ( ( 1.0 + 0.5*D2/alpha )**(-alpha) ) elif der == 2: # compute derivative matrix wrt 3rd parameter K = ( 1.0 + 0.5*D2/alpha ) A = sf2 * K**(-alpha) * (0.5*D2/K - alpha*np.log(K) ) else: raise Exception("Wrong derivative index in covRQ") return A class RQard(Kernel): ''' Rational Quadratic covariance function with Automatic Relevance Detemination (ARD) distance measure. hyp = log_ell_list + [ log_sigma, log_alpha ] :param D: dimension of pattern. set if you want default ell, which is 0.5 for each dimension. :param log_ell_list: characteristic length scale for each dimension. :param log_sigma: signal deviation. :param log_alpha: shape parameter for the RQ covariance. ''' def __init__(self, D=None, log_ell_list=None, log_sigma=0., log_alpha=0.): if log_ell_list == None: self.hyp = [0. for i in xrange(D)] + [ log_sigma, log_alpha ] else: self.hyp = log_ell_list + [ log_sigma, log_alpha ] def getCovMatrix(self,x=None,z=None,mode=None): if x != None: n, D = x.shape if z != None: nn, D = z.shape ell = 1./np.exp(self.hyp[0:D]) # characteristic length scale sf2 = np.exp(2.*self.hyp[D]) # signal variance alpha = np.exp(self.hyp[D+1]) if mode == 'self_test': # self covariances for the test cases nn,D = z.shape D2 = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x tmp = np.dot(np.diag(ell),x.T).T D2 = spdist.cdist(tmp, tmp, 'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z D2 = spdist.cdist(np.dot(np.diag(ell),x.T).T, np.dot(np.diag(ell),z.T).T, 'sqeuclidean') A = sf2 * ( ( 1.0 + 0.5*D2/alpha )**(-alpha) ) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): if x != None: n, D = x.shape if z != None: nn, D = z.shape ell = 1./np.exp(self.hyp[0:D]) # characteristic length scale sf2 = np.exp(2.*self.hyp[D]) # signal variance alpha = np.exp(self.hyp[D+1]) if mode == 'self_test': # self covariances for the test cases nn,D = z.shape D2 = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x tmp = np.dot(np.diag(ell),x.T).T D2 = spdist.cdist(tmp, tmp, 'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z D2 = spdist.cdist(np.dot(np.diag(ell),x.T).T, np.dot(np.diag(ell),z.T).T, 'sqeuclidean') if der < D: if mode == 'self_test': A = D2*0 elif mode == 'train': tmp = np.atleast_2d(x[:,der])/ell[der] A = sf2 * ( 1.0 + 0.5*D2/alpha )**(-alpha-1) * spdist.cdist(tmp, tmp, 'sqeuclidean') elif mode == 'cross': A = sf2 * ( 1.0 + 0.5*D2/alpha )**(-alpha-1) * spdist.cdist(np.atleast_2d(x[:,der]).T/ell[der], np.atleast_2d(z[:,der]).T/ell[der], 'sqeuclidean') elif der==D: # compute derivative matrix wrt magnitude parameter A = 2. * sf2 * ( ( 1.0 + 0.5*D2/alpha )**(-alpha) ) elif der==(D+1): # compute derivative matrix wrt magnitude parameter K = ( 1.0 + 0.5*D2/alpha ) A = sf2 * K**(-alpha) * ( 0.5*D2/K - alpha*np.log(K) ) else: raise Exception("Wrong derivative index in covRQard") return A class Pre(Kernel): ''' Precomputed kernel matrix. No hyperparameters and thus nothing will be optimised. :param M1: cross covariances matrix(train+1 by test). last row is self covariances (diagonal of test by test) :param M2: training set covariance matrix (train by train) ''' def __init__(self,M1,M2): self.M1 = M1 self.M2 = M2 self.hyp = [] def getCovMatrix(self,x=None,z=None,mode=None): if mode == 'self_test': # diagonal covariance between test_test A = self.M1[-1,:] # self covariances for the test cases (last row) A = np.reshape(A, (A.shape[0],1)) elif mode == 'train': # covariance between train_train A = self.M2 elif mode == 'cross': # covariance between train_test A = self.M1[:-1,:] return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): if der != None: raise Exception("Error: NO optimization in precomputed kernel matrix") return 0 if __name__ == '__main__': pass rL}rL(XConstrLXclassrLMjMXPre.getDerMatrixXdefrLMLMSXPeriodic.getDerMatrixXdefrLMoMX PiecePolyrLXclassrLMqMXLinear.__init__XdefrLMMX Kernel.fitcXdefrLK{KXLinearrLXclassrLMMX RQ.__init__XdefrLMMXSumOfKernel.getCovMatrixXdefrLKKчXRQard.getCovMatrixXdefrLMMX Pre.__init__XdefrLM=MBXProductOfKernel._getHypXdefrLKKXKernel.getCovMatrixXdefrLK@KNX PiecePoly.ppXdefrLMMXLinear.getCovMatrixXdefrMMMXKernel.getDerMatrixXdefrMKNK^X ScaleOfKernelrMXclassrMKKXLinear.getDerMatrixXdefrMMMXProductOfKernel.getDerMatrixXdefrMKKXKernel.__init__XdefrMKMMpMsXLINardr?MXclassr@MMMXNoise.getDerMatrixXdefrAMMMXProductOfKernelrBMXclassrCMKKXPiecePoly.ppmaxXdefrDMM~MXConst.getCovMatrixXdefrEMMsMXPoly.getCovMatrixXdefrFMMDMVXSumOfKernel._setHypXdefrGMKKɇX Matern.funcXdefrHMMMX PiecePoly.dppXdefrIMMMXRQrJMXclassrKMMMXPeriodic.getCovMatrixXdefrLMM^MoXRBFard.getDerMatrixXdefrMMMJMjXMatern.__init__XdefrNMMMXRQardrOMXclassrPMMM5XRQard.__init__XdefrQMMMXLINard.getCovMatrixXdefrRMMMXMatern.getCovMatrixXdefrSMMM/X SumOfKernelrTMXclassrUMKK݇XPoly.getDerMatrixXdefrVMMVMqXRBFrWMXclassrXMMMXScaleOfKernel.__init__XdefrYMKKXFITCOfKernel._setHypXdefrZMM MXConst.getDerMatrixXdefr[MMMXRQard.getDerMatrixXdefr\MMM5XFITCOfKernel.getCovMatrixXdefr]MMM#XLINard.__init__Xdefr^MMMXScaleOfKernel.getDerMatrixXdefr_MKKXSumOfKernel.getDerMatrixXdefr`MKK݇XPre.getCovMatrixXdefraMMBMLXPiecePoly.funcXdefrbMMMu}rcM(XConstrdMhXXProductOfKernelreMhXX PiecePolyrfMhXX Kernel.fitcrgMhXXNoiserhMhXXLinearriMhXXKernel.getCovMatrixrjMhXXKernel.getDerMatrixrkMhXX ScaleOfKernelrlMhXX FITCOfKernelrmMhXXRQardrnMhXXMaternroMhXXPeriodicrpMhXX SumOfKernelrqMhXXRBFrrMhXXRBFardrsMhXXRQrtMhXXPreruMhXXKernelrvMhXXPolyrwMhXXRBFunitrxMhXXLINardryMhXujLXpyGPs.Core.infrzMr{M}r|MbX #================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ # Inference methods: Compute the (approximate) posterior for a Gaussian process. # Methods currently implemented include: # # infExact Exact inference (only possible with Gaussian likelihood) # infLaplace Laplace's Approximation # infEP Expectation Propagation # infVB [NOT IMPLEMENTED!] Variational Bayes Approximation # # infFITC Large scale regression with approximate covariance matrix # infFITC_Laplace Large scale inference with approximate covariance matrix # infFITC_EP Large scale inference with approximate covariance matrix # # infMCMC [NOT IMPLEMENTED!] # Markov Chain Monte Carlo and Annealed Importance Sampling # We offer two samplers. # - hmc: Hybrid Monte Carlo # - ess: Elliptical Slice Sampling # No derivatives w.r.t. to hyperparameters are provided. # # infLOO [NOT IMPLEMENTED!] # Leave-One-Out predictive probability and Least-Squares Approxim. # # # This is a object-oriented python implementation of gpml functionality # (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18). # based on the functional-version of python implementation # (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013) # # Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013 import numpy as np import lik, cov from copy import copy,deepcopy from tools import solve_chol,brentmin,cholupdate np.seterr(all='ignore') class postStruct(object): ''' Data structure for posterior | post.alpha: 1d array containing inv(K)*m, | where K is the prior covariance matrix and m the approx posterior mean | post.sW: 1d array containing diagonal of sqrt(W) | the approximate posterior covariance matrix is inv(inv(K)+W) | post.L : 2d array, L = chol(sW*K*sW+identity(n)) ''' def __init__(self): self.alpha = np.array([]) self.L = np.array([]) self.sW = np.array([]) class dnlZStruct(object): ''' Data structure for the derivatives of mean, cov and lik functions. |dnlZ.mean: list of derivatives for each hyperparameters in mean function |dnlZ.cov: list of derivatives for each hyperparameters in covariance function |dnlZ.lik: list of derivatives for each hyperparameters in likelihood function ''' def __init__(self, m, c, l): self.mean = [] self.cov = [] self.lik = [] if m.hyp: self.mean = [0 for i in xrange(len(m.hyp))] if c.hyp: self.cov = [0 for i in xrange(len(c.hyp))] if l.hyp: self.lik = [0 for i in xrange(len(l.hyp))] class Inference(object): ''' Base class for inference. Defined several tool methods in it. ''' def __init__(self): pass def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): ''' Inference computation based on inputs. post, nlZ, dnlZ = inf.proceed(mean, cov, lik, x, y) | INPUT: | cov: name of the covariance function (see covFunctions.m) | lik: name of the likelihood function (see likFunctions.m) | x: n by D matrix of training inputs | y: 1d array (of size n) of targets | OUTPUT: | post(postStruct): struct representation of the (approximate) posterior containing: | nlZ: returned value of the negative log marginal likelihood | dnlZ(dnlZStruct): struct representation for derivatives of the negative log marginal likelihood | w.r.t. each hyperparameter. Usually, the approximate posterior to be returned admits the form: N(m=K*alpha, V=inv(inv(K)+W)), where alpha is a vector and W is diagonal; if not, then L contains instead -inv(K+inv(W)), and sW is unused. For more information on the individual approximation methods and their implementations, see the respective inference function below. See also gp.py :param meanfunc: mean function :param covfunc: covariance function :param likfunc: likelihood function :param x: training data :param y: training labels :param nargout: specify the number of output(1,2 or 3) :return: posterior, negative-log-marginal-likelihood, derivative for negative-log-marginal-likelihood-likelihood ''' pass def _epComputeParams(self, K, y, ttau, tnu, likfunc, m, inffunc): n = len(y) # number of training cases ssi = np.sqrt(ttau) # compute Sigma and mu L = np.linalg.cholesky(np.eye(n)+np.dot(ssi,ssi.T)*K).T # L'*L=B=eye(n)+sW*K*sW V = np.linalg.solve(L.T,np.tile(ssi,(1,n))*K) Sigma = K - np.dot(V.T,V) mu = np.dot(Sigma,tnu) Dsigma = np.reshape(np.diag(Sigma),(np.diag(Sigma).shape[0],1)) tau_n = 1/Dsigma - ttau # compute the log marginal likelihood nu_n = mu/Dsigma-tnu + m*tau_n # vectors of cavity parameters lZ = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc) nlZ = np.log(np.diag(L)).sum() - lZ.sum() - np.dot(tnu.T,np.dot(Sigma,tnu))/2 \ - np.dot((nu_n-m*tau_n).T,((ttau/tau_n*(nu_n-m*tau_n)-2*tnu) / (ttau+tau_n)))/2 \ + (tnu**2/(tau_n+ttau)).sum()/2.- np.log(1.+ttau/tau_n).sum()/2. return Sigma, mu, nlZ[0], L def _logdetA(self,K,w,nargout): ''' Compute the log determinant ldA and the inverse iA of a square nxn matrix A = eye(n) + K*diag(w) from its LU decomposition; for negative definite A, we return ldA = Inf. We also return mwiA = -diag(w)*inv(A). [ldA,iA,mwiA] = _logdetA(K,w)''' n = K.shape[0] assert(K.shape[0] == K.shape[1]) A = np.eye(n) + K*np.tile(w.T,(n,1)) [L,U,P] = np.linalg.lu(A) # compute LU decomposition, A = P'*L*U u = np.diag(U) signU = np.prod(np.sign(u)) # sign of U detP = 1 # compute sign (and det) of the permutation matrix P p = np.dot(P,np.array(range(n)).T) for ii in range(n): if ii != p[ii]: detP = -detP j= [jj for jj,val in enumerate(p) if val == ii] p[ii,j[0]] = p[j[0],ii] if signU != detP: # log becomes complex for negative values, encoded by infinity ldA = np.inf else: # det(L) = 1 and U triangular => det(A) = det(P)*prod(diag(U)) ldA = np.log(np.abs(u)).sum() if nargout>1: iA = inv(U)*inv(L)*P; if nargout>2: mwiA = -np.tile(w,(1,n))*iA return ldA,iA,mwiA else: return ldA,iA else: return ldA def _Psi_line(self,s,dalpha,alpha,K,m,likfunc,y,inffunc): '''Criterion Psi at alpha + s*dalpha for line search [Psi,alpha,f,dlp,W] = _Psi_line(s,dalpha,alpha,hyp,K,m,lik,y,inf)''' alpha = alpha + s*dalpha f = np.dot(K,alpha) + m [lp,dlp,d2lp] = likfunc.proceed(y,f,None,inffunc,None,3) W = -d2lp Psi = np.dot(alpha.T,(f-m))/2. - lp.sum() return Psi[0],alpha,f,dlp,W def _epfitcZ(self,d,P,R,nn,gg,ttau,tnu,d0,R0,P0,y,likfunc,m,inffunc): '''Compute the marginal likelihood approximation effort is O(n*nu^2) provided that nu0: R = cholupdate(R,np.sqrt(r)*np.dot(R.T,v),'-') else: R = cholupdate(R,np.sqrt(-r)*np.dot(R.T,v),'+') ttemp = np.dot(R0.T,np.dot(R.T,np.dot(R,np.dot(R0,P_i)))) gg = gg + ((dbi-dwi*(hi-m[ii]))/t) * np.reshape(ttemp,(ttemp.shape[0],1)) # O(nu^2) w[ii] = w_i; b[ii] = b_i; # update site parameters O(1) P_i = P_i/t # O(nu) return d,P_i,R,nn,gg,w,b def _mvmZ(self,x,RVdd,t): '''Matrix vector multiplication with Z=inv(K+inv(W))''' Zx = t*x - np.dot(RVdd.T,np.dot(RVdd,x)) return Zx def _mvmK(self,al,V,d0): '''Matrix vector multiplication with approximate covariance matrix''' Kal = np.dot(V.T,np.dot(V,al)) + d0*al return Kal def _Psi_lineFITC(self,s,dalpha,alpha,V,d0,m,likfunc,y,inffunc): '''Criterion Psi at alpha + s*dalpha for line search''' alpha = alpha + s*dalpha f = self._mvmK(alpha,V,d0) + m vargout = likfunc.proceed(y,f,None,inffunc,None,3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W = -d2lp Psi = np.dot(alpha.T,(f-m))/2. - lp.sum() return Psi[0],alpha,f,dlp,W def _fitcRefresh(self,d0,P0,R0,R0P0, w): '''Refresh the representation of the posterior from initial and site parameters to prevent possible loss of numerical precision after many epfitcUpdates effort is O(n*nu^2) provided that nu1: # do we want the marginal likelihood? nlZ = np.dot((y-m).T,alpha)/2. + np.log(np.diag(L)).sum() + n*np.log(2*np.pi*sn2)/2. # -log marg lik if nargout>2: # do we want derivatives? dnlZ = dnlZStruct(meanfunc, covfunc, likfunc) # allocate space for derivatives Q = solve_chol(L,np.eye(n))/sn2 - np.dot(alpha,alpha.T) # precompute for convenience dnlZ.lik = [sn2*np.trace(Q)] if covfunc.hyp: for ii in range(len(covfunc.hyp)): dnlZ.cov[ii] = (Q*covfunc.getDerMatrix(x=x, mode='train', der=ii)).sum()/2. if meanfunc.hyp: for ii in range(len(meanfunc.hyp)): dnlZ.mean[ii] = np.dot(-meanfunc.getDerMatrix(x, ii).T,alpha) dnlZ.mean[ii] = dnlZ.mean[ii][0,0] return post, nlZ[0,0], dnlZ return post, nlZ[0,0] return post class FITC_Exact(Inference): ''' FITC approximation to the posterior Gaussian process. The function is equivalent to infExact with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Quu) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu, snu2 = sn2/1e6 is the noise of the inducing inputs and Quu = Kuu + snu2*eye(nu). ''' def __init__(self): self.name = 'FICT exact inference' def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): if not isinstance(likfunc, lik.Gauss): # NOTE: no explicit call to likGauss raise Exception ('Exact inference only possible with Gaussian likelihood') if not isinstance(covfunc, cov.FITCOfKernel): raise Exception('Only covFITC supported.') # check cov diagK,Kuu,Ku = covfunc.getCovMatrix(x=x, mode='train') # evaluate covariance matrix m = meanfunc.getMean(x) # evaluate mean vector n, D = x.shape nu = Kuu.shape[0] sn2 = np.exp(2*likfunc.hyp[0]) # noise variance of likGauss snu2 = 1.e-6*sn2 # hard coded inducing inputs noise Luu = np.linalg.cholesky(Kuu+snu2*np.eye(nu)).T # Kuu + snu2*I = Luu'*Luu V = np.linalg.solve(Luu.T,Ku) # V = inv(Luu')*Ku => V'*V = Q g_sn2 = diagK + sn2 - np.array([(V*V).sum(axis=0)]).T # g + sn2 = diag(K) + sn2 - diag(Q) Lu = np.linalg.cholesky(np.eye(nu) + np.dot(V/np.tile(g_sn2.T,(nu,1)),V.T)).T # Lu'*Lu=I+V*diag(1/g_sn2)*V' r = (y-m)/np.sqrt(g_sn2) be = np.linalg.solve(Lu.T,np.dot(V,r/np.sqrt(g_sn2))) iKuu = solve_chol(Luu,np.eye(nu)) # inv(Kuu + snu2*I) = iKuu post = postStruct() post.alpha = np.linalg.solve(Luu,np.linalg.solve(Lu,be))# return the posterior parameters post.L = solve_chol(np.dot(Lu,Luu),np.eye(nu)) - iKuu # Sigma-inv(Kuu) post.sW = np.ones((n,1))/np.sqrt(sn2) # unused for FITC prediction with gp.m if nargout>1: # do we want the marginal likelihood nlZ = np.log(np.diag(Lu)).sum() + (np.log(g_sn2).sum() + n*np.log(2*np.pi) + np.dot(r.T,r) - np.dot(be.T,be))/2. if nargout>2: # do we want derivatives? dnlZ = dnlZStruct(meanfunc, covfunc, likfunc) # allocate space for derivatives al = r/np.sqrt(g_sn2) - np.dot(V.T,np.linalg.solve(Lu,be))/g_sn2 # al = (Kt+sn2*eye(n))\y B = np.dot(iKuu,Ku) w = np.dot(B,al) W = np.linalg.solve(Lu.T,V/np.tile(g_sn2.T,(nu,1))) for ii in range(len(covfunc.hyp)): [ddiagKi,dKuui,dKui] = covfunc.getDerMatrix(x=x, mode='train', der=ii) # eval cov deriv R = 2.*dKui-np.dot(dKuui,B) v = ddiagKi - np.array([(R*B).sum(axis=0)]).T # diag part of cov deriv dnlZ.cov[ii] = ( np.dot(ddiagKi.T,1./g_sn2) + np.dot(w.T,(np.dot(dKuui,w)-2.*np.dot(dKui,al))) \ - np.dot(al.T,(v*al)) - np.dot(np.array([(W*W).sum(axis=0)]),v) - (np.dot(R,W.T)*np.dot(B,W.T)).sum() )/2. dnlZ.cov[ii] = dnlZ.cov[ii][0,0] dnlZ.lik = sn2*((1./g_sn2).sum() - (np.array([(W*W).sum(axis=0)])).sum() - np.dot(al.T,al)) dKuui = 2*snu2 R = -dKuui*B v = -np.array([(R*B).sum(axis=0)]).T # diag part of cov deriv dnlZ.lik += (np.dot(w.T,np.dot(dKuui,w)) -np.dot(al.T,(v*al)) \ - np.dot(np.array([(W*W).sum(axis=0)]),v) - (np.dot(R,W.T)*np.dot(B,W.T)).sum() )/2. dnlZ.lik = list(dnlZ.lik[0]) for ii in range(len(meanfunc.hyp)): dnlZ.mean[ii] = np.dot(-meanfunc.getDerMatrix(x, ii).T, al) dnlZ.mean[ii] = dnlZ.mean[ii][0,0] return post, nlZ[0,0], dnlZ return post, nlZ[0,0] return post class Laplace(Inference): ''' Laplace's Approximation to the posterior Gaussian process. ''' def __init__(self): self.last_alpha = None def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): tol = 1e-6 # tolerance for when to stop the Newton iterations smax = 2; Nline = 20; thr = 1e-4 # line search parameters maxit = 20 # max number of Newton steps in f inffunc = self K = covfunc.getCovMatrix(x=x, mode='train') # evaluate the covariance matrix m = meanfunc.getMean(x) # evaluate the mean vector n, D = x.shape Psi_old = np.inf # make sure while loop starts by the largest old objective val if self.last_alpha == None: # find a good starting point for alpha and f alpha = np.zeros((n,1)) f = np.dot(K,alpha) + m # start at mean if sizes not match vargout = likfunc.proceed(y, f, None, inffunc, None, 3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W= -d2lp Psi_new = -lp.sum() else: alpha = self.last_alpha f = np.dot(K,alpha) + m # try last one vargout = likfunc.proceed(y, f, None, inffunc, None, 3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W= -d2lp Psi_new = np.dot(alpha.T,(f-m))/2. - lp.sum() # objective for last alpha vargout = - likfunc.proceed(y, m, None, inffunc, None, 1) Psi_def = vargout[0] # objective for default init f==m if Psi_def < Psi_new: # if default is better, we use it alpha = np.zeros((n,1)) f = np.dot(K,alpha) + m vargout = likfunc.proceed(y, f, None, inffunc, None, 3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W=-d2lp; Psi_new = -lp.sum() isWneg = np.any(W<0) # flag indicating whether we found negative values of W it = 0 # this happens for the Student's t likelihood while (Psi_old - Psi_new > tol) and it2: # do we want derivatives? dnlZ = dnlZStruct(meanfunc, covfunc, likfunc) # allocate space for derivatives if isWneg: # switch between Cholesky and LU decomposition mode Z = -post.L # inv(K+inv(W)) g = np.atleast_2d((iA*K).sum(axis=1)).T /2 # deriv. of ln|B| wrt W; g = diag(inv(inv(K)+diag(W)))/2 else: Z = np.tile(sW,(1,n))*solve_chol(post.L,np.diag(np.reshape(sW,(sW.shape[0],)))) #sW*inv(B)*sW=inv(K+inv(W)) C = np.linalg.solve(post.L.T,np.tile(sW,(1,n))*K) # deriv. of ln|B| wrt W g = np.atleast_2d((np.diag(K)-(C**2).sum(axis=0).T)).T /2. # g = diag(inv(inv(K)+W))/2 dfhat = g* d3lp # deriv. of nlZ wrt. fhat: dfhat=diag(inv(inv(K)+W)).*d3lp/2 for ii in range(len(covfunc.hyp)): # covariance hypers dK = covfunc.getDerMatrix(x=x, mode='train', der=ii) dnlZ.cov[ii] = (Z*dK).sum()/2. - np.dot(alpha.T,np.dot(dK,alpha))/2. # explicit part b = np.dot(dK,dlp) # b-K*(Z*b) = inv(eye(n)+K*diag(W))*b dnlZ.cov[ii] -= np.dot(dfhat.T,b-np.dot(K,np.dot(Z,b))) # implicit part dnlZ.cov[ii] = dnlZ.cov[ii][0,0] for ii in range(len(likfunc.hyp)): # likelihood hypers [lp_dhyp,dlp_dhyp,d2lp_dhyp] = likfunc.proceed(y,f,None,inffunc,ii,3) dnlZ.lik[ii] = -np.dot(g.T,d2lp_dhyp) - lp_dhyp.sum() # explicit part b = np.dot(K,dlp_dhyp) # b-K*(Z*b) = inv(eye(n)+K*diag(W))*b dnlZ.lik[ii] -= np.dot(dfhat.T,b-np.dot(K,np.dot(Z,b))) # implicit part dnlZ.lik[ii] = dnlZ.lik[ii][0,0] for ii in range(len(meanfunc.hyp)): # mean hypers dm = meanfunc.getDerMatrix(x, ii) dnlZ.mean[ii] = -np.dot(alpha.T,dm) # explicit part dnlZ.mean[ii] -= np.dot(dfhat.T,dm-np.dot(K,np.dot(Z,dm))) # implicit part dnlZ.mean[ii] = dnlZ.mean[ii][0,0] return post,nlZ[0],dnlZ else: return post, nlZ[0] class FITC_Laplace(Inference): ''' FITC-Laplace approximation to the posterior Gaussian process. The function is equivalent to infLaplace with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Kuu + snu2 * eye(nu)) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu and snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard deviation of the inducing inputs snu to be a one per mil of the measurement noise's standard deviation sn. In case of a likelihood without noise parameter sn2, we simply use snu2 = 1e-6. ''' def __init__(self): self.last_alpha = None def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): if not isinstance(covfunc, cov.FITCOfKernel): raise Exception('Only covFITC supported.') tol = 1e-6 # tolerance for when to stop the Newton iterations smax = 2; Nline = 100; thr = 1e-4 # line search parameters maxit = 20 # max number of Newton steps in f inffunc = Laplace() diagK,Kuu,Ku = covfunc.getCovMatrix(x=x, mode='train') # evaluate the covariance matrix m = meanfunc.getMean(x) # evaluate the mean vector if likfunc.hyp: # hard coded inducing inputs noise sn2 = np.exp(2.*likfunc.hyp[-1]) snu2 = 1.e-6*sn2 # similar to infFITC else: snu2 = 1.e-6 n, D = x.shape nu = Kuu.shape[0] rot180 = lambda A: np.rot90(np.rot90(A)) # little helper functions chol_inv = lambda A: np.linalg.solve( rot180( np.linalg.cholesky(rot180(A)) ),np.eye(nu)) # chol(inv(A)) R0 = chol_inv(Kuu+snu2*np.eye(nu)) # initial R, used for refresh O(nu^3) V = np.dot(R0,Ku); d0 = diagK - np.array([(V*V).sum(axis=0)]).T # initial d, needed Psi_old = np.inf # make sure while loop starts by the largest old objective val if self.last_alpha == None: # find a good starting point for alpha and f alpha = np.zeros((n,1)) f = self._mvmK(alpha,V,d0) + m # start at mean if sizes not match vargout = likfunc.proceed(y, f, None, inffunc, None, 3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W=-d2lp; Psi_new = -lp.sum() else: alpha = self.last_alpha f = self._mvmK(alpha,V,d0) + m # try last one vargout = likfunc.proceed(y, f, None, inffunc, None, 3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W=-d2lp Psi_new = np.dot(alpha.T,(f-m))/2. - lp.sum() # objective for last alpha vargout = - likfunc.proceed(y, m, None, inffunc, None, 1) Psi_def = vargout[0] # objective for default init f==m if Psi_def < Psi_new: # if default is better, we use it alpha = np.zeros((n,1)) f = self._mvmK(alpha,V,d0) + m vargout = likfunc.proceed(y, f, None, inffunc, None, 3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W=-d2lp; Psi_new = -lp.sum() isWneg = np.any(W<0) # flag indicating whether we found negative values of W it = 0 # this happens for the Student's t likelihood while (Psi_old - Psi_new > tol) and it2: # do we want derivatives? dnlZ = dnlZStruct(meanfunc, covfunc, likfunc) # allocate space for derivatives [d,P,R] = self._fitcRefresh(d0,Ku,R0,V,W) # g = diag(inv(inv(K)+W))/2 g = d/2 + 0.5*np.atleast_2d((np.dot(np.dot(R,R0),P)**2).sum(axis=0)).T t = W/(1+W*d0) dfhat = g*d3lp # deriv. of nlZ wrt. fhat: dfhat=diag(inv(inv(K)+W)).*d3lp/2 for ii in range(len(covfunc.hyp)): # covariance hypers ddiagK,dKuu,dKu = covfunc.getDerMatrix(x=x, mode='train', der=ii) # eval cov derivatives dA = 2.*dKu.T-np.dot(R0tV.T,dKuu) # dQ = dA*R0tV w = np.atleast_2d((dA*R0tV.T).sum(axis=1)).T # w = diag(dQ) v = ddiagK-w # v = diag(dK)-diag(dQ); dnlZ.cov[ii] = np.dot(ddiagK.T,t) - np.dot((RVdd*RVdd).sum(axis=0),v) # explicit part dnlZ.cov[ii] -= (np.dot(RVdd,dA)*np.dot(RVdd,R0tV.T)).sum() # explicit part dnlZ.cov[ii] = 0.5*dnlZ.cov[ii] - np.dot(alpha.T,np.dot(dA,np.dot(R0tV,alpha))+v*alpha)/2. # explicit b = np.dot(dA,np.dot(R0tV,dlp)) + v*dlp # b-K*(Z*b) = inv(eye(n)+K*diag(W))*b KZb = self._mvmK(self._mvmZ(b,RVdd,t),V,d0) dnlZ.cov[ii] -= np.dot(dfhat.T,(b-KZb)) # implicit part dnlZ.cov[ii] = dnlZ.cov[ii][0,0] for ii in range(len(likfunc.hyp)): # likelihood hypers vargout = likfunc.proceed(y,f,None,inffunc,ii,3) lp_dhyp = vargout[0]; dlp_dhyp = vargout[1]; d2lp_dhyp = vargout[2] dnlZ.lik[ii] = -np.dot(g.T,d2lp_dhyp) - lp_dhyp.sum() # explicit part b = self._mvmK(dlp_dhyp,V,d0) # implicit part dnlZ.lik[ii] -= np.dot(dfhat.T,b-self._mvmK(self._mvmZ(b,RVdd,t),V,d0)) if ii == len(likfunc.hyp)-1: # since snu2 is a fixed fraction of sn2, there is a covariance-like term # in the derivative as well snu = np.sqrt(snu2); T = chol_inv(Kuu + snu2*np.eye(nu)); T = np.dot(T.T,np.dot(T,snu*Ku)); t = np.array([(T*T).sum(axis=0)]).T z = np.dot(alpha.T,np.dot(T.T,np.dot(T,alpha))-t*alpha) - np.dot(np.array([(RVdd*RVdd).sum(axis=0)]),t) z += (np.dot(RVdd,T.T)**2).sum() b = (t*dlp-np.dot(T.T,np.dot(T,dlp)))/2. KZb = self._mvmK(self._mvmZ(b,RVdd,t),V,d0) z -= np.dot(dfhat.T,b-KZb) dnlZ.lik[ii] += z dnlZ.lik[ii] = dnlZ.lik[ii][0,0] for ii in range(len(meanfunc.hyp)): # mean hypers dm = meanfunc.getDerMatrix(x, ii) dnlZ.mean[ii] = -np.dot(alpha.T,dm) # explicit part Zdm = self._mvmZ(dm,RVdd,t) dnlZ.mean[ii] -= np.dot(dfhat.T,(dm-self._mvmK(Zdm,V,d0))) # implicit part dnlZ.mean[ii] = dnlZ.mean[ii][0,0] return post,nlZ[0,0],dnlZ else: return post, nlZ[0,0] class EP(Inference): ''' Expectation Propagation approximation to the posterior Gaussian Process. ''' def __init__(self): self.name = 'Expectation Propagation' self.last_ttau = None self.last_tnu = None def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): tol = 1e-4; max_sweep = 10; min_sweep = 2 # tolerance to stop EP iterations n = x.shape[0] inffunc = self K = covfunc.getCovMatrix(x=x, mode='train') # evaluate the covariance matrix m = meanfunc.getMean(x) # evaluate the mean vector nlZ0 = -likfunc.proceed(y, m, np.reshape(np.diag(K),(np.diag(K).shape[0],1)), inffunc).sum() if self.last_ttau == None: # find starting point for tilde parameters ttau = np.zeros((n,1)) # initialize to zero if we have no better guess tnu = np.zeros((n,1)) Sigma = K # initialize Sigma and mu, the parameters of .. mu = np.zeros((n,1)) # .. the Gaussian posterior approximation nlZ = nlZ0 else: ttau = self.last_ttau # try the tilde values from previous call tnu = self.last_tnu Sigma, mu, nlZ, L = self._epComputeParams(K, y, ttau, tnu, likfunc, m, inffunc) if nlZ > nlZ0: # if zero is better .. ttau = np.zeros((n,1)) # .. then initialize with zero instead tnu = np.zeros((n,1)) Sigma = K # initialize Sigma and mu, the parameters of .. mu = np.zeros((n,1)) # .. the Gaussian posterior approximation nlZ = nlZ0 nlZ_old = np.inf; sweep = 0 # converged, max. sweeps or min. sweeps? while (np.abs(nlZ-nlZ_old) > tol and sweep < max_sweep) or (sweep < min_sweep): nlZ_old = nlZ; sweep += 1 rperm = xrange(n) # randperm(n) for ii in rperm: # iterate EP updates (in random order) over examples tau_ni = 1/Sigma[ii,ii] - ttau[ii]# first find the cavity distribution .. nu_ni = mu[ii]/Sigma[ii,ii] + m[ii]*tau_ni - tnu[ii] # .. params tau_ni and nu_ni # compute the desired derivatives of the indivdual log partition function lZ,dlZ,d2lZ = likfunc.proceed(y[ii], nu_ni/tau_ni, 1/tau_ni, inffunc, None, 3) ttau_old = copy(ttau[ii]) # then find the new tilde parameters, keep copy of old ttau[ii] = -d2lZ /(1.+d2lZ/tau_ni) ttau[ii] = max(ttau[ii],0) # enforce positivity i.e. lower bound ttau by zero tnu[ii] = ( dlZ + (m[ii]-nu_ni/tau_ni)*d2lZ )/(1.+d2lZ/tau_ni) ds2 = ttau[ii] - ttau_old # finally rank-1 update Sigma .. si = np.reshape(Sigma[:,ii],(Sigma.shape[0],1)) Sigma = Sigma - ds2/(1.+ds2*si[ii])*np.dot(si,si.T) # takes 70# of total time mu = np.dot(Sigma,tnu) # .. and recompute mu # recompute since repeated rank-one updates can destroy numerical precision Sigma, mu, nlZ, L = self._epComputeParams(K, y, ttau, tnu, likfunc, m, inffunc) if sweep == max_sweep: pass # print '[warning] maximum number of sweeps reached in function infEP' self.last_ttau = ttau; self.last_tnu = tnu # remember for next call sW = np.sqrt(ttau); alpha = tnu-sW*solve_chol(L,sW*np.dot(K,tnu)) post = postStruct() post.alpha = alpha # return the posterior params post.sW = sW post.L = L if nargout>2: # do we want derivatives? dnlZ = dnlZStruct(meanfunc, covfunc, likfunc) # allocate space for derivatives ssi = np.sqrt(ttau) V = np.linalg.solve(L.T,np.tile(ssi,(1,n))*K) Sigma = K - np.dot(V.T,V) mu = np.dot(Sigma,tnu) Dsigma = np.reshape(np.diag(Sigma),(np.diag(Sigma).shape[0],1)) tau_n = 1/Dsigma-ttau # compute the log marginal likelihood nu_n = mu/Dsigma-tnu # vectors of cavity parameters F = np.dot(alpha,alpha.T) - np.tile(sW,(1,n))* \ solve_chol(L,np.diag(np.reshape(sW,(sW.shape[0],)))) # covariance hypers for jj in range(len(covfunc.hyp)): dK = covfunc.getDerMatrix(x=x, mode='train', der=jj) dnlZ.cov[jj] = -(F*dK).sum()/2. for ii in range(len(likfunc.hyp)): dlik = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc, ii) dnlZ.lik[ii] = -dlik.sum() junk,dlZ = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc, None, 2) # mean hyps for ii in range(len(meanfunc.hyp)): dm = meanfunc.getDerMatrix(x, ii) dnlZ.mean[ii] = -np.dot(dlZ.T,dm) dnlZ.mean[ii] = dnlZ.mean[ii][0,0] return post, nlZ[0], dnlZ else: return post, nlZ[0] class FITC_EP(Inference): ''' FITC-EP approximation to the posterior Gaussian process. The function is equivalent to infEP with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Kuu + snu2 * eye(nu)) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu and snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard deviation of the inducing inputs snu to be a one per mil of the measurement noise's standard deviation sn. In case of a likelihood without noise parameter sn2, we simply use snu2 = 1e-6. For details, see The Generalized FITC Approximation, Andrew Naish-Guzman and Sean Holden, NIPS, 2007. ''' def __init__(self): self.name = 'FITC Expectation Propagation' self.last_ttau = None self.last_tnu = None def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): if not isinstance(covfunc, cov.FITCOfKernel): raise Exception('Only covFITC supported.') # check cov tol = 1e-4; max_sweep = 10; min_sweep = 2 # tolerance to stop EP iterations inffunc = EP() diagK,Kuu,Ku = covfunc.getCovMatrix(x=x, mode='train') # evaluate the covariance matrix m = meanfunc.getMean(x) # evaluate the mean vector if likfunc.hyp: # hard coded inducing inputs noise sn2 = np.exp(2.*likfunc.hyp[-1]) snu2 = 1.e-6*sn2 # similar to infFITC else: snu2 = 1.e-6 n, D = x.shape; nu = Kuu.shape[0] rot180 = lambda A: np.rot90(np.rot90(A)) # little helper functions chol_inv = lambda A: np.linalg.solve( rot180( np.linalg.cholesky(rot180(A)) ),np.eye(nu)) # chol(inv(A)) R0 = chol_inv(Kuu+snu2*np.eye(nu)) # initial R, used for refresh O(nu^3) V = np.dot(R0,Ku); d0 = diagK - np.array([(V*V).sum(axis=0)]).T # initial d, needed for refresh O(n*nu^2) # A note on naming: variables are given short but descriptive names in # accordance with Rasmussen & Williams "GPs for Machine Learning" (2006): mu # and s2 are mean and variance, nu and tau are natural parameters. A leading t # means tilde, a subscript _ni means "not i" (for cavity parameters), or _n # for a vector of cavity parameters. # marginal likelihood for ttau = tnu = zeros(n,1); equals n*log(2) for likCum* nlZ0 = -1.* likfunc.proceed(y, m, np.reshape(diagK,(diagK.shape[0],1)), inffunc).sum() if self.last_ttau == None: # find starting point for tilde parameters ttau = np.zeros((n,1)) # initialize to zero if we have no better guess tnu = np.zeros((n,1)) [d,P,R,nn,gg] = self._epfitcRefresh(d0,Ku,R0,V,ttau,tnu) # compute initial repres. nlZ = nlZ0 else: ttau = self.last_ttau # try the tilde values from previous call tnu = self.last_tnu [d,P,R,nn,gg] = self._epfitcRefresh(d0,Ku,R0,V,ttau,tnu) # compute initial repres. nlZ = self._epfitcZ(d,P,R,nn,gg,ttau,tnu,d0,R0,Ku,y,likfunc,m,inffunc)[0] if nlZ > nlZ0: # if zero is better .. ttau = np.zeros((n,1)) # .. then initialize with zero instead tnu = np.zeros((n,1)) [d,P,R,nn,gg] = self._epfitcRefresh(d0,Ku,R0,V,ttau,tnu) # initial repres. nlZ = nlZ0 nlZ_old = np.inf; sweep = 0 # converged, max. sweeps or min. sweeps? while (np.abs(nlZ-nlZ_old) > tol and sweep < max_sweep) or (sweep < min_sweep): nlZ_old = nlZ sweep += 1 rperm = range(n) # randperm(n) for ii in rperm: # iterate EP updates (in random order) over examples p_i = np.reshape(P[:,ii],(P.shape[0],1)) t = np.dot(R,np.dot(R0,p_i)) # temporary variables sigma_i = d[ii] + np.dot(t.T,t); mu_i = nn[ii] + np.dot(p_i.T,gg) # post moments O(nu^2) tau_ni = 1/sigma_i - ttau[ii] # first find the cavity distribution .. nu_ni = mu_i/sigma_i + m[ii]*tau_ni - tnu[ii] # .. params tau_ni and nu_ni # compute the desired derivatives of the indivdual log partition function vargout = likfunc.proceed(y[ii], nu_ni/tau_ni, 1/tau_ni, inffunc, None, 3) lZ = vargout[0]; dlZ = vargout[1]; d2lZ = vargout[2] ttau_i = -d2lZ /(1.+d2lZ/tau_ni) ttau_i = max(ttau_i,0) # enforce positivity i.e. lower bound ttau by zero tnu_i = ( dlZ + (m[ii]-nu_ni/tau_ni)*d2lZ )/(1.+d2lZ/tau_ni) [d,P[:,ii],R,nn,gg,ttau,tnu] = self._epfitcUpdate(d,P[:,ii],R,nn,gg,ttau,tnu,ii,ttau_i,tnu_i,m,d0,Ku,R0)# update representation # recompute since repeated rank-one updates can destroy numerical precision [d,P,R,nn,gg] = self._epfitcRefresh(d0,Ku,R0,V,ttau,tnu) [nlZ,nu_n,tau_n] = self._epfitcZ(d,P,R,nn,gg,ttau,tnu,d0,R0,Ku,y,likfunc,m,inffunc) if sweep == max_sweep: pass # print '[warning] maximum number of sweeps reached in function infEP' self.last_ttau = ttau self.last_tnu = tnu # remember for next call post = postStruct() post.sW = np.sqrt(ttau) # unused for FITC_EP prediction with gp.m dd = 1/(d0+1/ttau) alpha = tnu/ttau*dd RV = np.dot(R,V) R0tV = np.dot(R0.T,V) alpha = alpha - np.dot(RV.T,np.dot(RV,alpha))*dd # long alpha vector for ordinary infEP post.alpha = np.dot(R0tV,alpha) # alpha = R0'*V*inv(Kt+diag(1./ttau))*(tnu./ttau) B = R0tV*np.tile(dd.T,(nu,1)); L = np.dot(B,R0tV.T); B = np.dot(B,RV.T) post.L = np.dot(B,B.T) - L # L = -R0'*V*inv(Kt+diag(1./ttau))*V'*R0 if nargout>2: # do we want derivatives? dnlZ = dnlZStruct(meanfunc, covfunc, likfunc) # allocate space for derivatives RVdd = RV*np.tile(dd.T,(nu,1)) for ii in range(len(covfunc.hyp)): ddiagK,dKuu,dKu = covfunc.getDerMatrix(x=x, mode='train', der=ii) dA = 2*dKu.T - np.dot(R0tV.T,dKuu) # dQ = dA*R0tV w = np.atleast_2d((dA*R0tV.T).sum(axis=1)).T # w = diag(dQ) v = ddiagK - w # v = diag(dK)-diag(dQ) z = np.dot(dd.T,(v+w)) - np.dot(np.atleast_2d((RVdd*RVdd).sum(axis=0)), v) \ - (np.dot(RVdd,dA).T * np.dot(R0tV,RVdd.T)).sum() dnlZ.cov[ii] = (z - np.dot(alpha.T,(alpha*v)) - np.dot(np.dot(alpha.T,dA),np.dot(R0tV,alpha)))/2. dnlZ.cov[ii] = dnlZ.cov[ii][0,0] for ii in range(len(likfunc.hyp)): # likelihood hypers dlik = likfunc.proceed(y, nu_n/tau_n+m, 1/tau_n, inffunc, ii, 1) dnlZ.lik[ii] = -dlik.sum() if ii == len(likfunc.hyp)-1: # since snu2 is a fixed fraction of sn2, there is a covariance-like term # in the derivative as well v = np.atleast_2d((R0tV*R0tV).sum(axis=0)).T z = (np.dot(RVdd,R0tV.T)**2).sum() - np.dot(np.atleast_2d((RVdd*RVdd).sum(axis=0)),v) z = z + np.dot(post.alpha.T,post.alpha) - np.dot(alpha.T,(v*alpha)) dnlZ.lik[ii] += snu2*z dnlZ.lik[ii] = dnlZ.lik[ii][0,0] [junk,dlZ] = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc, None, 2) # mean hyps for ii in range(len(meanfunc.hyp)): dm = meanfunc.getDerMatrix(x, ii) dnlZ.mean[ii] = -np.dot(dlZ.T,dm) dnlZ.mean[ii] = dnlZ.mean[ii][0,0] return post, nlZ[0,0], dnlZ else: return post, nlZ[0,0] if __name__ == '__main__': pass r}M}r~M(XInference._epfitcZXdefrMKK͇XEPrMXclassrMMMXInference._mvmZXdefrMKKXFITC_Laplace.proceedXdefrMMMXInference._logdetAXdefrMKKXExact.__init__XdefrMMM XInference._epComputeParamsXdefrMKKX postStructrMXclassrMK4KCXInference._mvmKXdefrMKKXdnlZStruct.__init__XdefrMKKKVX Exact.proceedXdefrMM MAXFITC_Exact.proceedXdefrMMLMXInference._Psi_lineXdefrMKKXInference.__init__XdefrMKZK]XLaplace.__init__XdefrMMMXInference._epfitcUpdateXdefrMKKXFITC_Laplace.__init__XdefrMMMX FITC_ExactrMXclassrMMAMXFITC_EP.proceedXdefrMMMjXExactrMXclassrMMMAXInference._Psi_lineFITCXdefrMKMX FITC_LaplacerMXclassrMMMXpostStruct.__init__XdefrMK>KCXFITC_EPrMXclassrMMMjXLaplace.proceedXdefrMMMXInference._fitcRefreshXdefrMMMXInference._epfitcRefreshXdefrMKKއXLaplacerMXclassrMMMX dnlZStructrMXclassrMKCKVXInference.proceedXdefrMK]KXFITC_Exact.__init__XdefrMMIMLX InferencerMXclassrMKVMX EP.proceedXdefrMMMXFITC_EP.__init__XdefrMMMX EP.__init__XdefrMMMu}rM(X FITC_LaplacehXFITC_EPhXLaplacehX FITC_ExacthXExacthXEPhujLXpyGPs.Core.meanrMrM}rMbXx&#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ # mean functions to be use by Gaussian process functions. There are two # different kinds of mean functions: simple and composite: # # simple mean functions: # # Zero - zero mean function # One - one mean function # Const - constant mean function # Linear - linear mean function # # composite mean functions: # # ScaleOfMean - scaled version of a mean function # PowerOfMean - power of a mean function # ProductOfMean - products of mean functions # SumOfMean - sums of mean functions # # # This is a object-oriented python implementation of gpml functionality # (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18). # based on the functional-version of python implementation # (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013) # # # Copyright (c) by Marion Neumann and Shan Huang, Sep.2013 import numpy as np import math class Mean(object): """ The base function for mean function """ def __init__(self): super(Mean, self).__init__() self.hyp = [] self.para = [] # overloading def __add__(self,mean): ''' Overloading + operator. :param mean: mean function :return: an instance of SumOfMean ''' return SumOfMean(self,mean) # overloading def __mul__(self,other): ''' Overloading * operator. Using * for both multiplication with scalar and product of means depending on the type of the two objects. :param other: mean function as product or int/float as scalar :return: an instance of ScaleOfMean or ProductOfMean ''' if isinstance(other, int) or isinstance(other, float): return ScaleOfMean(self,other) elif isinstance(other, Mean): return ProductOfMean(self,other) else: print "only numbers and Means are allowed for *" # overloading __rmul__ = __mul__ # overloading def __pow__(self,number): ''' Overloading ** operator. :param int number: power of the mean function :return: an instance of PowerOfMean ''' if isinstance(number, int) and number > 0: return PowerOfMean(self,number) else: print "only non-zero integers are supported for **" def getMean(self, x=None): ''' Get the mean vector. ''' pass def getDerMatrix(self, x=None, der=None): ''' Compute derivatives wrt. hyperparameters. :param x: training data :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' pass class ProductOfMean(Mean): '''Product of two mean fucntions.''' def __init__(self,mean1,mean2): self.mean1 = mean1 self.mean2 = mean2 if mean1.hyp and mean2.hyp: self._hyp = mean1.hyp + mean2.hyp elif not mean1.hyp: self._hyp = mean2.hyp elif not mean2.hyp: self._hyp = mean1.hyp def _setHyp(self,hyp): assert len(hyp) == len(self._hyp) len1 = len(self.mean1.hyp) self._hyp = hyp self.mean1.hyp = self._hyp[:len1] self.mean2.hyp = self._hyp[len1:] def _getHyp(self): return self._hyp hyp = property(_getHyp,_setHyp) def getMean(self, x=None): A = self.mean1.getMean(x) * self.mean2.getMean(x) return A def getDerMatrix(self, x=None, der=None): if der < len(self.mean1.hyp): A = self.mean1.getDerMatrix(x, der) * self.mean2.getMean(x) elif der < len(self.hyp): der2 = der - len(self.mean1.hyp) A = self.mean2.getDerMatrix(x, der2) * self.mean1.getMean(x) else: raise Exception("Error: der out of range for meanProduct") return A class SumOfMean(Mean): '''Sum of two mean functions.''' def __init__(self,mean1,mean2): self.mean1 = mean1 self.mean2 = mean2 if mean1.hyp and mean2.hyp: self._hyp = mean1.hyp + mean2.hyp elif not mean1.hyp: self._hyp = mean2.hyp elif not mean2.hyp: self._hyp = mean1.hyp def _setHyp(self,hyp): assert len(hyp) == len(self._hyp) len1 = len(self.mean1.hyp) self._hyp = hyp self.mean1.hyp = self._hyp[:len1] self.mean2.hyp = self._hyp[len1:] def _getHyp(self): return self._hyp hyp = property(_getHyp,_setHyp) def getMean(self, x=None): A = self.mean1.getMean(x) + self.mean2.getMean(x) return A def getDerMatrix(self, x=None, der=None): if der < len(self.mean1.hyp): A = self.mean1.getDerMatrix(x, der) elif der < len(self.hyp): der2 = der - len(self.mean1.hyp) A = self.mean2.getDerMatrix(x, der2) else: raise Exception("Error: der out of range for meanSum") return A class ScaleOfMean(Mean): '''Scale of a mean function.''' def __init__(self,mean,scalar): self.mean = mean if mean.hyp: self._hyp = [scalar] + mean.hyp else: self._hyp = [scalar] def _setHyp(self,hyp): assert len(hyp) == len(self._hyp) self._hyp = hyp self.mean.hyp = self._hyp[1:] def _getHyp(self): return self._hyp hyp = property(_getHyp,_setHyp) def getMean(self, x=None): c = self.hyp[0] # scale parameter A = c * self.mean.getMean(x) # accumulate means return A def getDerMatrix(self, x=None, der=None): c = self.hyp[0] # scale parameter if der == 0: # compute derivative w.r.t. c A = self.mean.getMean(x) else: A = c * self.mean.getDerMatrix(x,der-1) return A class PowerOfMean(Mean): '''Power of a mean fucntion.''' def __init__(self, mean, d): self.mean = mean if mean.hyp: self._hyp = [d] + mean.hyp else: self._hyp = [d] def _setHyp(self,hyp): assert len(hyp) == len(self._hyp) self._hyp = hyp self.mean.hyp = self._hyp[1:] def _getHyp(self): return self._hyp hyp = property(_getHyp,_setHyp) def getMean(self, x=None): d = np.abs(np.floor(self.hyp[0])) d = max(d,1) A = self.mean.getMean(x) **d # accumulate means return A def getDerMatrix(self, x=None, der=None): d = np.abs(np.floor(self.hyp[0])) d = max(d,1) if der == 0: # compute derivative w.r.t. c a = self.mean.getMean(x) A = a**d * np.log(a) else: A = d * self.mean.getMean(x) ** (d-1) * self.mean.getDerMatrix(x, der-1) return A class Zero(Mean): '''Zero mean.''' def __init__(self): self.hyp = [] self.name = '0' def getMean(self, x=None): n, D = x.shape A = np.zeros((n,1)) return A def getDerMatrix(self, x=None, der=None): n, D = x.shape A = np.zeros((n,1)) return A class One(Mean): ''' One mean.''' def __init__(self): self.hyp = [] self.name = '1' def getMean(self, x=None): n, D = x.shape A = np.ones((n,1)) return A def getDerMatrix(self, x=None, der=None): n, D = x.shape A = np.zeros((n,1)) return A class Const(Mean): ''' Constant mean function. hyp = [c] :param c: constant value for mean ''' def __init__(self, c=5.): self.hyp = [c] def getMean(self, x=None): n,D = x.shape A = self.hyp[0] * np.ones((n,1)) return A def getDerMatrix(self, x=None, der=None): n,D = x.shape if der == 0: # compute derivative vector wrt c A = np.ones((n,1)) else: A = np.zeros((n,1)) return A class Linear(Mean): ''' Linear mean function. self.hyp = alpha_list :param D: dimension of training data. Set if you want default alpha, which is 0.5 for each dimension. :alpha_list: scalar alpha for each dimension ''' def __init__(self, D=None, alpha_list=None): if alpha_list == None: self.hyp = [0.5 for i in xrange(D)] else: self.hyp = alpha_list def getMean(self, x=None): n, D = x.shape c = np.array(self.hyp) c = np.reshape(c,(len(c),1)) A = np.dot(x,c) return A def getDerMatrix(self, x=None, der=None): n, D = x.shape c = np.array(self.hyp) c = np.reshape(c,(len(c),1)) if isinstance(der, int) and der < D: # compute derivative vector wrt meanparameters A = np.reshape(x[:,der], (len(x[:,der]),1) ) else: A = np.zeros((n,1)) return A if __name__ == '__main__': pass rM}rM(XScaleOfMean.__init__XdefrMKKćXScaleOfMean.getMeanXdefrMKKчX One.getMeanXdefrMMMX Zero.getMeanXdefrMMMX One.__init__XdefrMMMXSumOfMean.getMeanXdefrMKKXProductOfMean._setHypXdefrMK}KXMean.getDerMatrixXdefrMKeKrXScaleOfMean.getDerMatrixXdefrMKKۇX Mean.__add__XdefrMK5K?XProductOfMean.getDerMatrixXdefrMKKXOne.getDerMatrixXdefrMMM!XSumOfMean._getHypXdefrMKKXConstrMXclassrMM!M9XOnerMXclassrMMM!XProductOfMean.getMeanXdefrMKKXLinearrMXclassrMM9MZXSumOfMean.getDerMatrixXdefrMKKX Mean.getMeanXdefrMK_KeXZero.getDerMatrixXdefrMMMX ScaleOfMeanrMXclassrMKKۇXProductOfMean._getHypXdefrMKKX ProductOfMeanrMXclassrMKrKXPowerOfMean.getMeanXdefrMKKX Mean.__pow__XdefrMKSK_XPowerOfMean._getHypXdefrMKKXLinear.getDerMatrixXdefrMMMMZX SumOfMeanrMXclassrMKKXPowerOfMean.getDerMatrixXdefrMKKX Mean.__init__XdefrMK/K5X Zero.__init__XdefrMKMXLinear.getMeanXdefrMMFMMXScaleOfMean._setHypXdefrMKKȇX Const.getMeanXdefrMM*M/XProductOfMean.__init__XdefrMKtK}XMeanrMXclassrMK+KrXSumOfMean.__init__XdefrMKKX PowerOfMeanrMXclassrMKKXConst.getDerMatrixXdefrMM/M9XPowerOfMean._setHypXdefrMKKX Mean.__mul__XdefrMK?KPXZerorMXclassrMKMXConst.__init__XdefrMM'M*XLinear.__init__XdefrMM@MFXScaleOfMean._getHypXdefrMKKʇXPowerOfMean.__init__XdefrMKKXSumOfMean._setHypXdefrMKKu}rM(XConstrMhXXLinearrMhXX SumOfMeanrMhXX PowerOfMeanrMhXX Mean.getMeanrMhXXMean.getDerMatrixrMhXXOnerMhXX ScaleOfMeanrMhXXZerorMhXX ProductOfMeanrMhXXMeanrMhXujLX"pyGPs.GraphExtensions.graphKernelsrMrM}rMbXE#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ # Created on April, 2014 # @author: marion, shan import scipy.sparse as spsp import numpy as np import matplotlib.pyplot as plt def propagationKernel(A, l, gr_id, h_max, w, p, ktype=None, SUM=True, VIS=False, showEachStep=False): ''' Propagation kernel for graphs as described in: Neumann, M., Patricia, N., Garnett, R., Kersting, K.: Efficient Graph Kernels by Randomization. In: P.A. Flach, T.D. Bie, N. Cristianini (eds.) ECML/PKDD, Notes in Computer Science, vol. 7523, pp. 378-393. Springer (2012). :param A: adjacency matrix (num_nodes x num_nodes) :param l: label array (num_nodes x 1); values [1,...,k] or -1 for unlabeled nodes OR label array (num_nodes x num_labels); values [0,1], unlabeled nodes have only 0 entries :param gr_id: graph indicator array (num_nodes x 1); values [0,..,n] :param h_max: number of iterations :param w: bin widths parameter :param p: distance ('tv', 'hellinger', 'L1', 'L2') :param ktype: type of propagation kernel ['diffusion', 'label_propagation', 'label_spreading', 'belief_propagation'] :return: kernel matrix ''' # ToDO: SPLIT Varible label array and label probability # CAUTION: number of labels (k) > 1 !!! #=========================================================================== # ## Propagation Kernel COMPUTATION #=========================================================================== num_graphs = gr_id.max().astype(int) num_nodes = A.shape[0] if l.shape[1]==1: num_labels = l.max().astype(int) else: num_labels = l.shape[1] #=========================================================================== # ## INITIALIZATION #=========================================================================== ## CHECK and ADJUST sparseness if not spsp.issparse(A): A = spsp.csr_matrix(A) else: if not spsp.isspmatrix_csr(A): A = A.tocsr() # MAKE sure A is float! A = A.astype(float) ## INITIALIZE kernel matrix K = np.ndarray(shape=(num_graphs,num_graphs,h_max+1), dtype=float) ## INITIALIZE label probabilities of labeled nodes (num_nodes x num_labels) if l.shape[1]==1: lab_prob = np.zeros((num_nodes,num_labels),dtype=np.float64) for i in xrange(num_nodes): if l[i,0] > 0: lab_prob[i,l[i,0]-1] = 1 else: lab_prob = l if spsp.issparse(lab_prob): lab_prob = lab_prob.todense() lab_orig = lab_prob.copy() # lab_orig is the orignal label information ## GET indices of labeled and unlabeled nodes idx_unlab = np.where(np.max(lab_prob, axis=1)!=1)[0] idx_all = np.array(np.arange(0,A.shape[0]), ndmin=2) idx = np.array(np.setdiff1d(idx_all,idx_unlab), ndmin=2) ## INITIALIZE unlabeled/uninitialized nodes UNIFORMLY idx_unif = np.where(np.sum(lab_prob, axis=1)==0)[0] if idx_unif.shape[0] != 0: lab_prob[idx_unif,:] = 1./lab_prob.shape[1] ## row normalize A -> transition matrix T if (ktype=='label_propagation') or (ktype=='label_diffusion'): row_sums = np.array(A.sum(axis=1))[:,0] T = A.copy() T.data /= row_sums[A.nonzero()[0]] # so A and T are sparse matrix #=========================================================================== # ## PROPAGATION KERNEL ITERATIONS #=========================================================================== for h in xrange(h_max+1): print 'ITERATION: ', h if h > 0: ## LABEL UPDATE if showEachStep: print '...computing LABEL UPDATE' if ktype == 'label_propagation': lab_prob[idx,:] = lab_orig[idx,:] # PUSH BACK original LABELS lab_prob = T*lab_prob elif ktype == 'label_diffusion': lab_prob = T*lab_prob elif ktype == 'label_spreading': # S = D^(-1/2)*W*D^(-1/2) # y(t+1) = alpha*S*y(t)+(1-alpha)*y(0) alpha = 0.8 # compute S diag = np.array(A.sum(axis=1)).T**(-1/2) D = spsp.lil_matrix((A.shape[0], A.shape[1]), dtype=float) # D.setdiag(diag[0,:]) D = D.tocsr() S = D*A*D lab_prob = np.dot((alpha*S.todense()),lab_prob) + (1-alpha)*lab_orig ## COMPUTE hashvalues if showEachStep: print '...computing hashvalues' # determine path to take depending on chosen distance use_cauchy = (p =='L1') or (p =='tv') take_sqrt = (p =='hellinger') if take_sqrt: lab_prob = np.sqrt(lab_prob) # generate random projection vector v = np.random.normal(size=(lab_prob.shape[1], 1)) if use_cauchy: # if X, Y are N(0, 1), then X / Y has a standard Cauchy distribution v /= np.random.normal(size=(lab_prob.shape[1], 1)) # random offset b = w * np.random.rand() # compute hashes # hashLabels is a Vector with length: number of nodes (hashvalue for respective node) hashLabels = (np.dot(lab_prob,v) + b) / w hashLabels = np.floor(hashLabels) # take floor uniqueHash, hashLabels = np.unique(hashLabels, return_inverse=True) # map to consecutive integer from 0 ## COMPUTE kernel contribution if showEachStep: print '...computing KERNEL contribution' # aggregate counts on graphs # counts is a matrix: number of graphs x number of hashlabels num_bins = len(uniqueHash) counts = np.zeros((num_graphs, num_bins)) # init counts matrix # accumulate counts of hash labels for i in xrange(num_nodes): counts[ (gr_id[i,0]-1), hashLabels[i] ] +=1 # compute base kernel (here: LINEAR kernel) K_h = np.dot(counts,counts.T) ## SUM kernel contributions if SUM: if h == 0: K[:,:,h] = K_h else: K[:,:,h] = K[:,:,h-1] + K_h else: K[:,:,h] = K_h if showEachStep: print K[:,:,h] ## VISUALIZE KERNELS if VIS: for h in range(h_max+1): K_h = K[:,:,h] plt.figure() plt.title('height:'+str(h)) imgplot = plt.imshow(K_h) imgplot.set_interpolation('nearest') plt.colorbar() plt.show() return K rM}rMXpropagationKernelrMXdefrMKKs}rMXpropagationKernelhasjLX pyGPs.Core.gprMrM}rMbX #================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ # MEANING OF NOTATION: # # inffunc function specifying the inference method # covfunc prior covariance function (see below) # meanfunc prior mean function # likfunc likelihood function # x n by D matrix of training inputs # y column vector of length n of training targets # xs n by D matrix of test inputs # ys column vector of length nn of true test targets (optional) # nlZ returned value of the negative log marginal likelihood # dnlZ column vector of partial derivatives of the negative # log marginal likelihood w.r.t. each hyperparameter # ym column vector (of length ns) of predictive output means # ys2 column vector (of length ns) of predictive output variances # fm column vector (of length ns) of predictive latent means # fs2 column vector (of length ns) of predictive latent variances # lp column vector (of length ns) of log predictive probabilities # post struct representation of the (approximate) posterior # post consists of post.alpha, post.L, post.sW # # This is a object-oriented python implementation of gpml functionality # (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18). # based on the functional-version of python implementation # (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013) # # Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013 import itertools import numpy as np import matplotlib.pyplot as plt import inf, mean, lik, cov, opt from tools import unique from copy import deepcopy import pyGPs SHADEDCOLOR = [0.7539, 0.89453125, 0.62890625, 1.0] MEANCOLOR = [ 0.2109375, 0.63385, 0.1796875, 1.0] DATACOLOR = [0.12109375, 0.46875, 1., 1.0] class GP(object): '''Base class for GP model''' def __init__(self): super(GP, self).__init__() self.usingDefaultMean = True # was using default mean function now? self.meanfunc = None self.covfunc = None self.likfunc = None self.inffunc = None self.optimizer = None self.nlZ = None # negative log marginal likelihood self.dnlZ = None # column vector of partial derivatives of the negative # log marginal likelihood w.r.t. each hyperparameter self.posterior = None # struct representation of the (approximate) posterior self.x = None # n by D matrix of training inputs self.y = None # column vector of length n of training targets self.xs = None # n by D matrix of test inputs self.ys = None # column vector of length nn of true test targets (optional) self.ym = None # column vector (of length ns) of predictive output means self.ys2 = None # column vector (of length ns) of predictive output variances self.fm = None # column vector (of length ns) of predictive latent means self.fs2 = None # column vector (of length ns) of predictive latent variances self.lp = None # column vector (of length ns) of log predictive probabilities def setData(self, x, y): '''Pass training data and traning labels to model.''' if x.ndim == 1: x = np.reshape(x, (x.shape[0],1)) if y.ndim == 1: y = np.reshape(y, (y.shape[0],1)) self.x = x self.y = y if self.usingDefaultMean: c = np.mean(y) self.meanfunc = mean.Const(c) # adapt default prior mean wrt. training labels def plotData_1d(self, axisvals=None): '''Plot 1d (toy) data.''' plt.figure() plt.plot(self.x, self.y, ls='None', marker='+', color=DATACOLOR, ms=12, mew=2) if axisvals: plt.axis(axisvals) plt.grid() plt.xlabel('input x') plt.ylabel('target y') plt.show() def plotData_2d(self,x1,x2,t1,t2,p1,p2,axisvals=None): '''Plot 2d (toy) data.''' fig = plt.figure() plt.plot(x1[:,0], x1[:,1], 'b+', markersize = 12) plt.plot(x2[:,0], x2[:,1], 'r+', markersize = 12) pc = plt.contour(t1, t2, np.reshape(p2/(p1+p2), (t1.shape[0],t1.shape[1]) )) fig.colorbar(pc) plt.grid() if axisvals: plt.axis(axisvals) plt.show() def setPrior(self, mean=None, kernel=None): '''Set prior mean and cov''' if mean != None: self.meanfunc = mean self.usingDefaultMean = False if kernel != None: self.covfunc = kernel if type(kernel) is cov.Pre: self.usingDefaultMean = False def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None): '''This method is used to sepecify optimization configuration. By default, gp uses a single run "minimize". :param method: Optimization methods. Possible values are: "Minimize" -> minimize by Carl Rasmussen (python implementation of "minimize" in GPML) "CG" -> conjugent gradient "BFGS" -> quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS) "SCG" -> scaled conjugent gradient (faster than CG) :param num_restarts: Set if you want to run mulitiple times of optimization with different initial guess. It specifys the maximum number of runs/restarts/trials. :param min_threshold: Set if you want to run mulitiple times of optimization with different initial guess. It specifys the threshold of objective function value. Stop optimization when this value is reached. :param meanRange: The range of initial guess for mean hyperparameters. e.g. meanRange = [(-2,2), (-5,5), (0,1)]. Each tuple specifys the range (low, high) of this hyperparameter, This is only the range of initial guess, during optimization process, optimal hyperparameters may go out of this range. (-5,5) for each hyperparameter by default. :param covRange: The range of initial guess for kernel hyperparameters. Usage see meanRange :param likRange: The range of initial guess for likelihood hyperparameters. Usage see meanRange ''' pass def optimize(self, x=None, y=None): ''' Train optimal hyperparameters based on training data, adjust new hyperparameters to all mean/cov/lik functions ''' if x != None: if x.ndim == 1: x = np.reshape(x, (x.shape[0],1)) self.x = x if y != None: if y.ndim == 1: y = np.reshape(y, (y.shape[0],1)) self.y = y if self.usingDefaultMean and self.meanfunc == None: c = np.mean(y) self.meanfunc = mean.Const(c) # adapt default prior mean wrt. training labels # optimize optimalHyp, optimalNlZ = self.optimizer.findMin(self.x, self.y) self.nlZ= optimalNlZ # apply optimal hyp to all mean/cov/lik functions here self.optimizer._apply_in_objects(optimalHyp) self.fit() def fit(self, x=None, y=None, der=True): ''' Fit the training data. Update negative log marginal likelihood(nlZ), partial derivatives of nlZ w.r.t. each hyperparameter(dnlZ), and struct representation of the (approximate) posterior(post), which consists of post.alpha, post.L, post.sW. ''' if x != None: if x.ndim == 1: x = np.reshape(x, (x.shape[0],1)) self.x = x if y != None: if y.ndim == 1: y = np.reshape(y, (y.shape[0],1)) self.y = y if self.usingDefaultMean and self.meanfunc == None: c = np.mean(y) self.meanfunc = mean.Const(c) # adapt default prior mean wrt. training labels # call inference method if isinstance(self.likfunc, lik.Erf): #or is instance(self.likfunc, lik.Logistic): uy = unique(self.y) ind = ( uy != 1 ) if any( uy[ind] != -1): raise Exception('You attempt classification using labels different from {+1,-1}') if not der: post, nlZ = self.inffunc.proceed(self.meanfunc, self.covfunc, self.likfunc, self.x, self.y, 2) self.nlZ = nlZ self.posterior = deepcopy(post) return nlZ, post else: post, nlZ, dnlZ = self.inffunc.proceed(self.meanfunc, self.covfunc, self.likfunc, self.x, self.y, 3) self.nlZ = nlZ self.dnlZ = deepcopy(dnlZ) self.posterior = deepcopy(post) return nlZ, dnlZ, post def predict(self, xs, ys=None): ''' Prediction according to given inputs. Get predictive output means(ym), predictive output variances(ys2), predictive latent means(fm), predictive latent variances(fs2), log predictive probabilities(lp). :param xs: test input :param ys: test target(optional) :return: ym, ys2, fm, fs2, lp ''' meanfunc = self.meanfunc covfunc = self.covfunc likfunc = self.likfunc inffunc = self.inffunc x = self.x y = self.y if xs.ndim == 1: xs = np.reshape(xs, (xs.shape[0],1)) self.xs = xs if ys != None: if ys.ndim == 1: ys = np.reshape(ys, (ys.shape[0],1)) self.ys = ys if self.posterior == None: self.fit() alpha = self.posterior.alpha L = self.posterior.L sW = self.posterior.sW nz = range(len(alpha[:,0])) # non-sparse representation if L == []: # in case L is not provided, we compute it K = covfunc.getCovMatrix(x=x[nz,:], mode='train') L = np.linalg.cholesky( (np.eye(nz) + np.dot(sW,sW.T)*K).T ) Ltril = np.all( np.tril(L,-1) == 0 ) # is L an upper triangular matrix? ns = xs.shape[0] # number of data points nperbatch = 1000 # number of data points per mini batch nact = 0 # number of already processed test data points ymu = np.zeros((ns,1)) ys2 = np.zeros((ns,1)) fmu = np.zeros((ns,1)) fs2 = np.zeros((ns,1)) lp = np.zeros((ns,1)) while nact<=ns-1: # process minibatches of test cases to save memory id = range(nact,min(nact+nperbatch,ns)) # data points to process kss = covfunc.getCovMatrix(z=xs[id,:], mode='self_test') # self-variances Ks = covfunc.getCovMatrix(x=x[nz,:], z=xs[id,:], mode='cross') # cross-covariances ms = meanfunc.getMean(xs[id,:]) N = (alpha.shape)[1] # number of alphas (usually 1; more in case of sampling) Fmu = np.tile(ms,(1,N)) + np.dot(Ks.T,alpha[nz]) # conditional mean fs|f fmu[id] = np.reshape(Fmu.sum(axis=1)/N,(len(id),1)) # predictive means if Ltril: # L is triangular => use Cholesky parameters (alpha,sW,L) V = np.linalg.solve(L.T,np.tile(sW,(1,len(id)))*Ks) fs2[id] = kss - np.array([(V*V).sum(axis=0)]).T # predictive variances else: # L is not triangular => use alternative parametrization fs2[id] = kss + np.array([(Ks*np.dot(L,Ks)).sum(axis=0)]).T # predictive variances fs2[id] = np.maximum(fs2[id],0) # remove numerical noise i.e. negative variances Fs2 = np.tile(fs2[id],(1,N)) # we have multiple values in case of sampling if ys == None: [Lp, Ymu, Ys2] = likfunc.proceed(None,Fmu[:],Fs2[:],None,None,3) else: [Lp, Ymu, Ys2] = likfunc.proceed(np.tile(ys[id],(1,N)), Fmu[:], Fs2[:],None,None,3) lp[id] = np.reshape( np.reshape(Lp,(np.prod(Lp.shape),N)).sum(axis=1)/N , (len(id),1) ) # log probability; sample averaging ymu[id] = np.reshape( np.reshape(Ymu,(np.prod(Ymu.shape),N)).sum(axis=1)/N ,(len(id),1) ) # predictive mean ys|y and ... ys2[id] = np.reshape( np.reshape(Ys2,(np.prod(Ys2.shape),N)).sum(axis=1)/N , (len(id),1) ) # .. variance nact = id[-1]+1 # set counter to index of next data point self.ym = ymu self.ys2 = ys2 self.lp = lp self.fm = fmu self.fs2 = fs2 if ys == None: return ymu, ys2, fmu, fs2, None else: return ymu, ys2, fmu, fs2, lp def predict_with_posterior(self, post, xs, ys=None): ''' Prediction with provided posterior (i.e. you already have the posterior and thus don't need fitting/training phases) Get predictive output means(ym), predictive output variances(ys2), predictive latent means(fm), predictive latent variances(fs2), log predictive probabilities(lp). :param post: struct representation of posterior :param xs: test input :param ys: test target(optional) :return: ym, ys2, fm, fs2, lp ''' meanfunc = self.meanfunc covfunc = self.covfunc likfunc = self.likfunc inffunc = self.inffunc x = self.x y = self.y if xs.ndim == 1: xs = np.reshape(xs, (xs.shape[0],1)) self.xs = xs if ys != None: if ys.ndim == 1: ys = np.reshape(ys, (ys.shape[0],1)) self.ys = ys self.posterior = deepcopy(post) alpha = post.alpha L = post.L sW = post.sW nz = range(len(alpha[:,0])) # non-sparse representation if L == []: # in case L is not provided, we compute it K = covfunc.getCovMatrix(x=x[nz,:], mode='train') L = np.linalg.cholesky( (np.eye(nz) + np.dot(sW,sW.T)*K).T ) Ltril = np.all( np.tril(L,-1) == 0 ) # is L an upper triangular matrix? ns = xs.shape[0] # number of data points nperbatch = 1000 # number of data points per mini batch nact = 0 # number of already processed test data points ymu = np.zeros((ns,1)) ys2 = np.zeros((ns,1)) fmu = np.zeros((ns,1)) fs2 = np.zeros((ns,1)) lp = np.zeros((ns,1)) while nact<=ns-1: # process minibatches of test cases to save memory id = range(nact,min(nact+nperbatch,ns)) # data points to process kss = covfunc.getCovMatrix(z=xs[id,:], mode='self_test') # self-variances Ks = covfunc.getCovMatrix(x=x[nz,:], z=xs[id,:], mode='cross') # cross-covariances ms = meanfunc.getMean(xs[id,:]) N = (alpha.shape)[1] # number of alphas (usually 1; more in case of sampling) Fmu = np.tile(ms,(1,N)) + np.dot(Ks.T,alpha[nz]) # conditional mean fs|f fmu[id] = np.reshape(Fmu.sum(axis=1)/N,(len(id),1)) # predictive means if Ltril: # L is triangular => use Cholesky parameters (alpha,sW,L) V = np.linalg.solve(L.T,np.tile(sW,(1,len(id)))*Ks) fs2[id] = kss - np.array([(V*V).sum(axis=0)]).T # predictive variances else: # L is not triangular => use alternative parametrization fs2[id] = kss + np.array([(Ks*np.dot(L,Ks)).sum(axis=0)]).T # predictive variances fs2[id] = np.maximum(fs2[id],0) # remove numerical noise i.e. negative variances Fs2 = np.tile(fs2[id],(1,N)) # we have multiple values in case of sampling if ys == None: [Lp, Ymu, Ys2] = likfunc.proceed(None,Fmu[:],Fs2[:],None,None,3) else: [Lp, Ymu, Ys2] = likfunc.proceed(np.tile(ys[id],(1,N)), Fmu[:], Fs2[:],None,None,3) lp[id] = np.reshape( np.reshape(Lp,(np.prod(Lp.shape),N)).sum(axis=1)/N , (len(id),1) ) # log probability; sample averaging ymu[id] = np.reshape( np.reshape(Ymu,(np.prod(Ymu.shape),N)).sum(axis=1)/N ,(len(id),1) ) # predictive mean ys|y and ... ys2[id] = np.reshape( np.reshape(Ys2,(np.prod(Ys2.shape),N)).sum(axis=1)/N , (len(id),1) ) # .. variance nact = id[-1]+1 # set counter to index of next data point self.ym = ymu self.ys2 = ys2 self.lp = lp self.fm = fmu self.fs2 = fs2 if ys == None: return ymu, ys2, fmu, fs2, None else: return ymu, ys2, fmu, fs2, lp class GPR(GP): '''Gaussian Process Regression''' def __init__(self): super(GPR, self).__init__() self.meanfunc = mean.Zero() # default prior mean self.covfunc = cov.RBF() # default prior covariance self.likfunc = lik.Gauss() # likihood with default noise variance 0.1 self.inffunc = inf.Exact() # inference method self.optimizer = opt.Minimize(self) # default optimizer def setNoise(self,log_sigma): '''Set noise variance other than default''' self.likfunc = lik.Gauss(log_sigma) def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None): '''Set Optimizer. See base class.''' conf = None if (num_restarts!=None) or (min_threshold!=None): conf = pyGPs.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc) conf.num_restarts = num_restarts conf.min_threshold = min_threshold if meanRange != None: conf.meanRange = meanRange if covRange != None: conf.covRange = covRange if likRange != None: conf.likRange = likRange if method == "Minimize": self.optimizer = opt.Minimize(self,conf) elif method == "SCG": self.optimizer = opt.SCG(self,conf) elif method == "CG": self.optimizer = opt.CG(self,conf) elif method == "BFGS": self.optimizer = opt.BFGS(self,conf) def plot(self,axisvals=None): '''Plot 1d GP regression.''' xs = self.xs x = self.x y = self.y ym = self.ym ys2 = self.ys2 plt.figure() xss = np.reshape(xs,(xs.shape[0],)) ymm = np.reshape(ym,(ym.shape[0],)) ys22 = np.reshape(ys2,(ys2.shape[0],)) plt.plot(x, y, color=DATACOLOR, ls='None', marker='+',ms=12, mew=2) plt.plot(xs, ym, color=MEANCOLOR, ls='-', lw=3.) plt.fill_between(xss,ymm + 2.*np.sqrt(ys22), ymm - 2.*np.sqrt(ys22), facecolor=SHADEDCOLOR,linewidths=0.0) plt.grid() if axisvals != None: plt.axis(axisvals) plt.xlabel('input x') plt.ylabel('target y') plt.show() def useInference(self, newInf): ''' Use another inference techinique other than default exact inference. :param str newInf: 'Laplace' or 'EP' ''' if newInf == "Laplace": self.inffunc = inf.Laplace() elif newInf == "EP": self.inffunc = inf.EP() else: raise Exception('Possible inf values are "Laplace", "EP".') def useLikelihood(self,newLik): if newLik == "Laplace": self.likfunc = lik.Laplace() self.inffunc = inf.EP() else: raise Exception('Possible lik values are "Laplace".') class GPC(GP): '''Gaussian Process Classification''' def __init__(self): super(GPC, self).__init__() self.meanfunc = mean.Zero() # default prior mean self.covfunc = cov.RBF() # default prior covariance self.likfunc = lik.Erf() # erf likihood self.inffunc = inf.EP() # default inference method self.optimizer = opt.Minimize(self) # default optimizer def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None): '''Set Optimizer. See base class.''' conf = None if (num_restarts!=None) or (min_threshold!=None): conf = pyGPs.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc) conf.num_restarts = num_restarts conf.min_threshold = min_threshold if meanRange != None: conf.meanRange = meanRange if covRange != None: conf.covRange = covRange if likRange != None: conf.likRange = likRange if method == "Minimize": self.optimizer = opt.Minimize(self,conf) elif method == "SCG": self.optimizer = opt.SCG(self,conf) elif method == "CG": self.optimizer = opt.CG(self,conf) elif method == "BFGS": self.optimizer = opt.BFGS(self,conf) def plot(self,x1,x2,t1,t2,axisvals=None): '''Plot 2d gp classification.''' fig = plt.figure() plt.plot(x1[:,0], x1[:,1], 'b+', markersize = 12) plt.plot(x2[:,0], x2[:,1], 'r+', markersize = 12) pc = plt.contour(t1, t2, np.reshape(np.exp(self.lp), (t1.shape[0],t1.shape[1]) )) fig.colorbar(pc) plt.grid() if axisvals != None: plt.axis(axisvals) plt.show() def useInference(self, newInf): ''' Use another inference techinique other than default EP inference. :param str newInf: 'Laplace' ''' if newInf == "Laplace": self.inffunc = inf.Laplace() else: raise Exception('Possible inf values are "Laplace".') def useLikelihood(self,newLik): ''' Use another likelihood function other than default Erf inference. (Not used in this version) :param str newLik: 'Logistic' ''' if newLik == "Logistic": pass #self.likfunc = lik.Logistic() else: raise Exception('Possible lik values are "Logistic".') class GPMC(object): '''This is a one vs. one classification wrapper for GPC''' def __init__(self, n_class): self.meanfunc = mean.Zero() # default prior mean self.covfunc = cov.RBF() # default prior covariance self.n_class = n_class # number of different classes self.x_all = None self.y_all = None self.newInf = None # new inference? -> call useInference self.newLik = None # new likelihood? -> call useLikelihood self.newPrior = False def setPrior(self, mean=None, kernel=None): '''set prior mean function and covariance function''' if mean != None: self.meanfunc = mean if kernel != None: self.covfunc = kernel if type(kernel) is cov.Pre: self.usingDefaultMean = False self.newPrior = True def useInference(self, newInf): '''Use another inference techinique other than default EP inference.''' self.newInf = newInf def useLikelihood(self,newLik): '''Use another likelihood function other than default Erf inference.''' self.newLik = newLik def setData(self,x,y): '''Set data for multi-class model''' if x.ndim == 1: x = np.reshape(x, (x.shape[0],1)) if y.ndim == 1: y = np.reshape(y, (y.shape[0],1)) self.x_all = x self.y_all = y def fitAndPredict(self, xs): ''' Fitting model and prediction. predictive_vote is a matrix where row i is each test point i column j is the probability for being class j :return: predictive_vote ''' if xs.ndim == 1: xs = np.reshape(xs, (xs.shape[0],1)) predictive_vote = np.zeros((xs.shape[0],self.n_class)) for i in xrange(self.n_class): # classifier for class i... for j in xrange(i+1,self.n_class): # ...and class j x,y = self.createBinaryClass(i,j) model = GPC() if self.newPrior: model.setPrior(mean=self.meanfunc, kernel=self.covfunc) if self.newInf: model.useInference(self.newInf) if self.newLik: model.useLikelihood(self.newLik) model.fit(x,y) # fitting ym = model.predict(xs)[0] ym += 1 # now scale into 0 to 2, ym=0 is class j, ym=2 is class i vote_i = np.zeros((xs.shape[0],self.n_class)) vote_j = np.zeros((xs.shape[0],self.n_class)) vote_i[:,i:i+1] = ym vote_j[:,j:j+1] = 2-ym predictive_vote += vote_i predictive_vote += vote_j predictive_vote /= predictive_vote.sum(axis=1)[:,np.newaxis] return predictive_vote def optimizeAndPredict(self, xs): ''' Optimizing model and prediction. predictive_vote is a matrix where row i is each test point i column j is the probability for being class j :return: predictive_vote ''' if xs.ndim == 1: xs = np.reshape(xs, (xs.shape[0],1)) predictive_vote = np.zeros((xs.shape[0],self.n_class)) for i in xrange(self.n_class): # classifier for class i... for j in xrange(i+1,self.n_class): # ...and class j x,y = self.createBinaryClass(i,j) model = GPC() if self.newPrior: model.setPrior(mean=self.meanfunc, kernel=self.covfunc) if self.newInf: model.useInference(self.newInf) if self.newLik: model.useLikelihood(self.newLik) model.optimize(x,y) # training ym = model.predict(xs)[0] ym += 1 # now scale into 0 to 2, ym=0 is class j, ym=2 is class i vote_i = np.zeros((xs.shape[0],self.n_class)) vote_j = np.zeros((xs.shape[0],self.n_class)) vote_i[:,i:i+1] = ym vote_j[:,j:j+1] = 2-ym predictive_vote += vote_i predictive_vote += vote_j predictive_vote /= predictive_vote.sum(axis=1)[:,np.newaxis] return predictive_vote def createBinaryClass(self, i,j): ''' Create dataset x(data) and y(label) which only contains class i and j. Relabel class i to +1 and class j to -1''' class_i = [] class_j = [] for index in xrange(len(self.y_all)): # check all classes target = self.y_all[index] if target == i: class_i.append(index) elif target == j: class_j.append(index) n1 = len(class_i) n2 = len(class_j) class_i.extend(class_j) x = self.x_all[class_i,:] y = np.concatenate((np.ones((1,n1)),-np.ones((1,n2))),axis=1).T return x,y class GP_FITC(GP): '''FITC GP base class''' def __init__(self): super(GP_FITC, self).__init__() self.u = None # overriding def setData(self, x, y, value_per_axis=5): '''Set training data and derive deault inducing_points. :param x: training data :param y: training target :param int value_per_axis: number of value in each dimension when using a uni-distant default inducing points''' if x.ndim == 1: x = np.reshape(x, (x.shape[0],1)) if y.ndim == 1: y = np.reshape(y, (y.shape[0],1)) self.x = x self.y = y if self.usingDefaultMean: c = np.mean(y) self.meanfunc = mean.Const(c) # adapt default prior mean wrt. training labels # get range of x in each dimension # 5 uniformally selected value for each dimension gridAxis=[] for d in xrange(x.shape[1]): column = x[:,d] mini = np.min(column) maxi = np.max(column) axis = np.linspace(mini,maxi,value_per_axis) gridAxis.append(axis) # default inducing points-> a grid if self.u == None: self.u = np.array(list(itertools.product(*gridAxis))) self.covfunc = self.covfunc.fitc(self.u) # overriding def setPrior(self, mean=None, kernel=None, inducing_points=None): '''Set GP prior and inducing points''' if kernel != None: if inducing_points != None: self.covfunc = kernel.fitc(inducing_points) self.u = inducing_points else: if self.u != None: self.covfunc = kernel.fitc(self.u) else: raise error("To use default inducing points, please call setData() first!") if type(kernel) is cov.Pre: self.usingDefaultMean = False if mean != None: self.meanfunc = mean self.usingDefaultMean = False class GPR_FITC(GP_FITC): '''Gaussian Process Regression FITC''' def __init__(self): super(GPR_FITC, self).__init__() self.meanfunc = mean.Zero() # default prior mean self.covfunc = cov.RBF() # default prior covariance self.likfunc = lik.Gauss() # likihood with default noise variance 0.1 self.inffunc = inf.FITC_Exact() # inference method self.optimizer = opt.Minimize(self) # default optimizer self.u = None # no default inducing points def setNoise(self,log_sigma): '''Set noise variance other than default value''' self.likfunc = lik.Gauss(log_sigma) def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None): '''Set Optimizer. See base class.''' conf = None if (num_restarts!=None) or (min_threshold!=None): conf = pyGPs.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc) conf.num_restarts = num_restarts conf.min_threshold = min_threshold if meanRange != None: conf.meanRange = meanRange if covRange != None: conf.covRange = covRange if likRange != None: conf.likRange = likRange if method == "Minimize": self.optimizer = opt.Minimize(self,conf) elif method == "SCG": self.optimizer = opt.SCG(self,conf) elif method == "CG": self.optimizer = opt.CG(self,conf) elif method == "BFGS": self.optimizer = opt.BFGS(self,conf) def plot(self,axisvals=None): '''Plot 1d GP FITC regression.''' plt.figure() xss = np.reshape(self.xs,(self.xs.shape[0],)) ymm = np.reshape(self.ym,(self.ym.shape[0],)) ys22 = np.reshape(self.ys2,(self.ys2.shape[0],)) plt.plot(self.x, self.y, color=DATACOLOR, ls='None', marker='+',ms=12, mew=2) plt.plot(self.xs, self.ym, color=MEANCOLOR, ls='-', lw=3.) plt.fill_between(xss,ymm + 2.*np.sqrt(ys22), ymm - 2.*np.sqrt(ys22), facecolor=SHADEDCOLOR,linewidths=0.0) plt.grid() if axisvals != None: plt.axis(axisvals) plt.xlabel('input x') plt.ylabel('output y') plt.plot(self.u,np.ones_like(self.u), ls='None', color='k',marker='x',markersize=12,mew=2) plt.show() def useInference(self, newInf): ''' Use another inference techinique other than default exact inference. :param str newInf: 'Laplace' or 'EP' ''' if newInf == "Laplace": self.inffunc = inf.FITC_Laplace() elif newInf == "EP": self.inffunc = inf.FITC_EP() else: raise Exception('Possible inf values are "Laplace", "EP".') def useLikelihood(self,newLik): ''' Use another inference techinique other than default Gaussian likelihood. :param str newLik: 'Laplace' ''' if newLik == "Laplace": self.likfunc = lik.Laplace() self.inffunc = inf.FITC_EP() else: raise Exception('Possible lik values are "Laplace".') class GPC_FITC(GP_FITC): '''Gaussian Process Classification FITC''' def __init__(self): super(GPC_FITC, self).__init__() self.meanfunc = mean.Zero() # default prior mean self.covfunc = cov.RBF() # default prior covariance self.likfunc = lik.Erf() # erf liklihood self.inffunc = inf.FITC_EP() # default inference method self.optimizer = opt.Minimize(self) # default optimizer self.u = None # no default inducing points def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None): '''Set optimizer. See base class.''' conf = None if (num_restarts!=None) or (min_threshold!=None): conf = pyGPs.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc) conf.num_restarts = num_restarts conf.min_threshold = min_threshold if meanRange != None: conf.meanRange = meanRange if covRange != None: conf.covRange = covRange if likRange != None: conf.likRange = likRange if method == "Minimize": self.optimizer = opt.Minimize(self,conf) elif method == "SCG": self.optimizer = opt.SCG(self,conf) elif method == "CG": self.optimizer = opt.CG(self,conf) elif method == "BFGS": self.optimizer = opt.BFGS(self,conf) def plot(self,x1,x2,t1,t2,axisvals=None): '''Plot 2d GP FITC classification.''' fig = plt.figure() plt.plot(x1[:,0], x1[:,1], 'b+', markersize = 12) plt.plot(x2[:,0], x2[:,1], 'r+', markersize = 12) plt.plot(self.u[:,0],self.u[:,1],'ko', markersize=12) pc = plt.contour(t1, t2, np.reshape(np.exp(self.lp), (t1.shape[0],t1.shape[1]) )) fig.colorbar(pc) plt.grid() if axisvals != None: plt.axis(axisvals) plt.show() def useInference(self, newInf): ''' Use another inference techinique other than default exact inference. :param str newInf: 'Laplace' or 'EP' ''' if newInf == "Laplace": self.inffunc = inf.FITC_Laplace() else: raise Exception('Possible inf values are "Laplace".') def useLikelihood(self,newLik): ''' Use another inference techinique other than default Erf likelihood. (Not used in this version) :param str newLik: 'Logistic' ''' if newLik == "Logistic": pass #self.likfunc = lik.Logistic() else: raise Exception('Possible lik values are "Logistic".') rN}rN(XGPMCrNXclassrNMMXGP_FITC.__init__XdefrNMMXGPC.setOptimizerXdefrNMMXGPR_FITC.setOptimizerXdefrNMMX GPR.setNoiseXdefrNMMXGPMC.createBinaryClassXdefrNMMXGPR.setOptimizerXdefr NMMX GPMC.__init__Xdefr NMM$XGPR_FITC.useLikelihoodXdefr NMM'XGPC_FITC.__init__Xdefr NM)M2XGPMC.optimizeAndPredictXdefr NMbMXGPC_FITC.useLikelihoodXdefrNM`MpXGPR.useInferenceXdefrNMMXGPC_FITC.useInferenceXdefrNMUM`X GP.setPriorXdefrNKrK|XGPC.plotXdefrNMMXGPC.useLikelihoodXdefrNMMX GP.setDataXdefrNKOK[XGPR.plotXdefrNMMXGP.setOptimizerXdefrNK|KXGP.plotData_1dXdefrNK[KfXGPC_FITCrNXclassrNM'MpXGP.fitXdefrNKKڇX GPC_FITC.plotXdefrNMHMUXGPrNXclassrNK6MXGPR_FITC.setNoiseXdefrNMMXGPR_FITCrNXclassr NMM'X GP.predictXdefr!NKM+XGPCr"NXclassr#NMMXGPC.useInferenceXdefr$NMMXGPMC.useLikelihoodXdefr%NM2M6X GP.__init__Xdefr&NK8KOX GPR_FITC.plotXdefr'NMM XGPR_FITC.useInferenceXdefr(NM MXGPRr)NXclassr*NMMXGPC_FITC.setOptimizerXdefr+NM2MHX GP.optimizeXdefr,NKKX GPC.__init__Xdefr-NMMX GPR.__init__Xdefr.NMMX GPMC.setPriorXdefr/NM$M.XGP_FITC.setDataXdefr0NMMXGP_FITC.setPriorXdefr1NMMXGP.plotData_2dXdefr2NKfKrXGPMC.fitAndPredictXdefr3NM?MbX GPMC.setDataXdefr4NM6M?XGPR.useLikelihoodXdefr5NMMXGPMC.useInferenceXdefr6NM.M2XGP_FITCr7NXclassr8NMMXGP.predict_with_posteriorXdefr9NM+MXGPR_FITC.__init__Xdefr:NMMu}r;NXGP.setOptimizerh=sjLXpyGPs.Core.likrNbXqm# Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ # likelihood functions are provided to be used by the gp.py function: # # Erf (Error function, classification, probit regression) # Logistic [NOT IMPLEMENTED!] (Logistic, classification, logit regression) # Uni [NOT IMPLEMENTED!] (Uniform likelihood, classification) # # Gauss (Gaussian, regression) # Laplace (Laplacian or double exponential, regression) # Sech2 [NOT IMPLEMENTED!] (Sech-square, regression) # T [NOT IMPLEMENTED!] (Student's t, regression) # # Poisson [NOT IMPLEMENTED!] (Poisson regression, count data) # # Mix [NOT IMPLEMENTED!] (Mixture of individual covariance functions) # # See the documentation for the individual likelihood for the computations specific # to each likelihood function. # # # This is a object-oriented python implementation of gpml functionality # (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18). # based on the functional-version of python implementation # (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013) # # Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013 import numpy as np from scipy.special import erf import inf class Likelihood(object): """Base function for Likelihood function""" def __init__(self): self.hyp = [] def proceed(self, y=None, mu=None, s2=None, inffunc=None, der=None, nargout=1): ''' The likelihood functions have two possible modes, the mode being selected as follows (where "lik" stands for "proceed" method for any likelihood function): 1) With two or three input arguments: [PREDICTION MODE] lp = lik(y, mu) OR lp, ymu, ys2 = lik(y, mu, s2) This allows to evaluate the predictive distribution. Let p(y_*|f_*) be the likelihood of a test point and N(f_*|mu,s2) an approximation to the posterior marginal p(f_*|x_*,x,y) as returned by an inference method. The predictive distribution p(y_*|x_*,x,y) is approximated by: q(y_*) = \int N(f_*|mu,s2) p(y_*|f_*) df_* lp = log( q(y) ) for a particular value of y, if s2 is [] or 0, this corresponds to log( p(y|mu) ). ymu and ys2 are the mean and variance of the predictive marginal q(y) note that these two numbers do not depend on a particular value of y. All vectors have the same size. 2) With four or five input arguments, the fouth being an object of class "Inference" [INFERENCE MODE] lik(y, mu, s2, inf) OR lik(y, mu, s2, inf, i) There are two cases for inf, namely a) infLaplace, b) infEP The last input i, refers to derivatives w.r.t. the ith hyperparameter. | a1) | lp,dlp,d2lp,d3lp = lik(y, f, [], 'infLaplace'). | lp, dlp, d2lp and d3lp correspond to derivatives of the log likelihood. | log(p(y|f)) w.r.t. to the latent location f. | lp = log( p(y|f) ) | dlp = d log( p(y|f) ) / df | d2lp = d^2 log( p(y|f) ) / df^2 | d3lp = d^3 log( p(y|f) ) / df^3 | a2) | lp_dhyp,dlp_dhyp,d2lp_dhyp = lik(y, f, [], 'infLaplace', i) | returns derivatives w.r.t. to the ith hyperparameter | lp_dhyp = d log( p(y|f) ) / (dhyp_i) | dlp_dhyp = d^2 log( p(y|f) ) / (df dhyp_i) | d2lp_dhyp = d^3 log( p(y|f) ) / (df^2 dhyp_i) | b1) | lZ,dlZ,d2lZ = lik(y, mu, s2, 'infEP') | let Z = \int p(y|f) N(f|mu,s2) df then | lZ = log(Z) | dlZ = d log(Z) / dmu | d2lZ = d^2 log(Z) / dmu^2 | b2) | dlZhyp = lik(y, mu, s2, 'infEP', i) | returns derivatives w.r.t. to the ith hyperparameter | dlZhyp = d log(Z) / dhyp_i Cumulative likelihoods are designed for binary classification. Therefore, they only look at the sign of the targets y; zero values are treated as +1. Some examples for valid likelihood functions: | lik = likGauss([0.1]) | lik = likErf() ''' pass class Gauss(Likelihood): ''' Gaussian likelihood function for regression. :math:`Gauss(t)=\\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-\\frac{(t-y)^2}{2\\sigma^2}}`, where :math:`y` is the mean and :math:`\\sigma` is the standard deviation. hyp = [ log_sigma ] ''' def __init__(self, log_sigma=np.log(0.1) ): self.hyp = [log_sigma] def proceed(self, y=None, mu=None, s2=None, inffunc=None, der=None, nargout=1): sn2 = np.exp(2. * self.hyp[0]) if inffunc == None: # prediction mode if y == None: y = np.zeros_like(mu) s2zero = True if (not s2==None) and np.linalg.norm(s2) > 0: s2zero = False if s2zero: # log probability lp = -(y-mu)**2 /sn2/2 - np.log(2.*np.pi*sn2)/2. s2 = np.zeros_like(s2) else: inf_func = inf.EP() # prediction lp = self.proceed(y, mu, s2, inf_func) if nargout>1: ymu = mu # first y moment if nargout>2: ys2 = s2 + sn2 # second y moment return lp,ymu,ys2 else: return lp,ymu else: return lp else: if isinstance(inffunc, inf.EP): if der == None: # no derivative mode lZ = -(y-mu)**2/(sn2+s2)/2. - np.log(2*np.pi*(sn2+s2))/2. # log part function if nargout>1: dlZ = (y-mu)/(sn2+s2) # 1st derivative w.r.t. mean if nargout>2: d2lZ = -1/(sn2+s2) # 2nd derivative w.r.t. mean return lZ,dlZ,d2lZ else: return lZ,dlZ else: return lZ else: # derivative mode dlZhyp = ((y-mu)**2/(sn2+s2)-1) / (1+s2/sn2) # deriv. w.r.t. hyp.lik return dlZhyp elif isinstance(inffunc, inf.Laplace): if der == None: # no derivative mode if y == None: y=0 ymmu = y-mu lp = -ymmu**2/(2*sn2) - np.log(2*np.pi*sn2)/2. if nargout>1: dlp = ymmu/sn2 # dlp, derivative of log likelihood if nargout>2: # d2lp, 2nd derivative of log likelihood d2lp = -np.ones_like(ymmu)/sn2 if nargout>3: # d3lp, 3rd derivative of log likelihood d3lp = np.zeros_like(ymmu) return lp,dlp,d2lp,d3lp else: return lp,dlp,d2lp else: return lp,dlp else: return lp else: # derivative mode lp_dhyp = (y-mu)**2/sn2 - 1 # derivative of log likelihood w.r.t. hypers dlp_dhyp = 2*(mu-y)/sn2 # first derivative, d2lp_dhyp = 2*np.ones_like(mu)/sn2 # and also of the second mu derivative return lp_dhyp,dlp_dhyp,d2lp_dhyp ''' elif isinstance(inffunc, infVB): if der == None: # variational lower site bound # t(s) = exp(-(y-s)^2/2sn2)/sqrt(2*pi*sn2) # the bound has the form: b*s - s.^2/(2*ga) - h(ga)/2 with b=y/ga ga = s2 n = len(ga) b = y/ga y = y*np.ones((n,1)) db = -y/ga**2 d2b = 2*y/ga**3 h = np.zeros((n,1)) dh = h d2h = h # allocate memory for return args id = (ga <= sn2 + 1e-8) # OK below noise variance h[id] = y[id]**2/ga[id] + np.log(2*np.pi*sn2) h[np.logical_not(id)] = np.inf dh[id] = -y[id]**2/ga[id]**2 d2h[id] = 2*y[id]**2/ga[id]**3 id = ga < 0 h[id] = np.inf dh[id] = 0 d2h[id] = 0 # neg. var. treatment varargout = [h,b,dh,db,d2h,d2b] else: ga = s2 n = len(ga) dhhyp = np.zeros((n,1)) dhhyp[ga<=sn2] = 2 dhhyp[ga<0] = 0 # negative variances get a special treatment varargout = dhhyp # deriv. w.r.t. hyp.lik else: raise Exception('Incorrect inference in lik.Gauss\n') ''' class Erf(Likelihood): ''' Error function or cumulative Gaussian likelihood function for binary classification or probit regression. :math:`Erf(t)=\\frac{1}{2}(1+erf(\\frac{t}{\\sqrt{2}}))=normcdf(t)` ''' def __init__(self): self.hyp = [] def proceed(self, y=None, mu=None, s2=None, inffunc=None, der=None, nargout=1): if not y == None: y = np.sign(y) y[y==0] = 1 else: y = 1 # allow only +/- 1 values if inffunc == None: # prediction mode if inf is not present y = y*np.ones_like(mu) # make y a vector s2zero = True; if not s2 == None: if np.linalg.norm(s2)>0: s2zero = False # s2==0? if s2zero: # log probability evaluation p,lp = self.cumGauss(y,mu,2) else: # prediction lp = self.proceed(y, mu, s2, inf.EP()) p = np.exp(lp) if nargout>1: ymu = 2*p-1 # first y moment if nargout>2: ys2 = 4*p*(1-p) # second y moment return lp,ymu,ys2 else: return lp,ymu else: return lp else: # inference mode if isinstance(inffunc, inf.Laplace): if der == None: # no derivative mode f = mu; yf = y*f # product latents and labels p,lp = self.cumGauss(y,f,2) if nargout>1: # derivative of log likelihood n_p = self.gauOverCumGauss(yf,p) dlp = y*n_p # derivative of log likelihood if nargout>2: # 2nd derivative of log likelihood d2lp = -n_p**2 - yf*n_p if nargout>3: # 3rd derivative of log likelihood d3lp = 2*y*n_p**3 + 3*f*n_p**2 + y*(f**2-1)*n_p return lp,dlp,d2lp,d3lp else: return lp,dlp,d2lp else: return lp,dlp else: return lp else: # derivative mode return [] # derivative w.r.t. hypers elif isinstance(inffunc, inf.EP): if der == None: # no derivative mode z = mu/np.sqrt(1+s2) junk,lZ = self.cumGauss(y,z,2) # log part function if not y == None: z = z*y if nargout>1: if y == None: y = 1 n_p = self.gauOverCumGauss(z,np.exp(lZ)) dlZ = y*n_p/np.sqrt(1.+s2) # 1st derivative wrt mean if nargout>2: d2lZ = -n_p*(z+n_p)/(1.+s2) # 2nd derivative wrt mean return lZ,dlZ,d2lZ else: return lZ,dlZ else: return lZ else: # derivative mode return [] # deriv. wrt hyp.lik ''' if inffunc == 'inf.infVB': if der == None: # no derivative mode # naive variational lower bound based on asymptotical properties of lik # normcdf(t) -> -(t*A_hat^2-2dt+c)/2 for t->-np.inf (tight lower bound) d = 0.158482605320942; c = -1.785873318175113; ga = s2; n = len(ga); b = d*y*np.ones((n,1)); db = np.zeros((n,1)); d2b = db h = -2.*c*np.ones((n,1)); h[ga>1] = np.inf; dh = np.zeros((n,1)); d2h = dh varargout = [h,b,dh,db,d2h,d2b] else: # derivative mode varargout = [] # deriv. wrt hyp.lik ''' def cumGauss(self, y=None, f=None, nargout=1): # return [p,lp] = cumGauss(y,f) if not y == None: yf = y*f else: yf = f p = (1. + erf(yf/np.sqrt(2.)))/2. # likelihood if nargout>1: lp = self.logphi(yf,p) return p,lp else: return p def gauOverCumGauss(self,f,p): # return n_p = gauOverCumGauss(f,p) n_p = np.zeros_like(f) # safely compute Gaussian over cumulative Gaussian ok = f>-5 # naive evaluation for large values of f n_p[ok] = (np.exp(-f[ok]**2/2)/np.sqrt(2*np.pi)) / p[ok] bd = f<-6 # tight upper bound evaluation n_p[bd] = np.sqrt(f[bd]**2/4+1)-f[bd]/2 interp = np.logical_and(np.logical_not(ok),np.logical_not(bd)) # linearly interpolate between both of them tmp = f[interp] lam = -5. - f[interp] n_p[interp] = (1-lam)*(np.exp(-tmp**2/2)/np.sqrt(2*np.pi))/p[interp] + lam *(np.sqrt(tmp**2/4+1)-tmp/2); return n_p def logphi(self,z,p): # return lp = logphi(z,p) lp = np.zeros_like(z) # allocate memory zmin = -6.2; zmax = -5.5; ok = z>zmax # safe evaluation for large values bd = z0: s2zero = False # s2==0? if s2zero: # log probability evaluation lp = -np.abs(y-mu)/b -np.log(2*b); s2 = 0 else: # prediction lp = self.proceed(y, mu, s2, inf.EP()) if nargout>1: ymu = mu # first y moment if nargout>2: ys2 = s2 + sn**2 # second y moment return lp,ymu,ys2 else: return lp,ymu else: return lp else: # inference mode if isinstance(inffunc, inf.Laplace): if der == None: # no derivative mode if y == None: y = np.zeros_like(mu) ymmu = y-mu lp = np.abs(ymmu)/b - np.log(2*b) if nargout>1: # derivative of log likelihood dlp = np.sign(ymmu)/b if nargout>2: # 2nd derivative of log likelihood d2lp = np.zeros_like(ymmu) if nargout>3: # 3rd derivative of log likelihood d3lp = np.zeros_like(ymmu) return lp,dlp,d2lp,d3lp else: return lp,dlp,d2lp else: return lp,dlp else: return lp else: # derivative w.r.t. hypers lp_dhyp = np.abs(y-mu)/b - 1 # derivative of log likelihood w.r.t. hypers dlp_dhyp = np.sign(mu-y)/b # first derivative, d2lp_dhyp = np.zeros(mu.shape) # and also of the second mu derivative return lp_dhyp, dlp_dhyp, d2lp_dhyp elif isinstance(inffunc, inf.EP): n = np.max([len(y.flatten()),len(mu.flatten()),len(s2.flatten()),len(sn.flatten())]) on = np.ones((n,1)) y = y*on; mu = mu*on; s2 = s2*on; sn = sn*on; fac = 1e3; # factor between the widths of the two distributions ... # ... from when one considered a delta peak, we use 3 orders of magnitude #idlik = np.reshape( (fac*sn) < np.sqrt(s2) , (sn.shape[0],) ) # Likelihood is a delta peak #idgau = np.reshape( (fac*np.sqrt(s2)) < sn , (sn.shape[0],) ) # Gaussian is a delta peak idlik = (fac*sn) < np.sqrt(s2) idgau = (fac*np.sqrt(s2)) < sn id = np.logical_and(np.logical_not(idgau),np.logical_not(idlik)) # interesting case in between if der == None: # no derivative mode lZ = np.zeros((n,1)) dlZ = np.zeros((n,1)) d2lZ = np.zeros((n,1)) if np.any(idlik): l = Gauss(log_sigma=np.log(s2[idlik])/2) a = l.proceed(mu[idlik], y[idlik]) lZ[idlik] = a[0]; dlZ[idlik] = a[1]; d2lZ[idlik] = a[2] if np.any(idgau): l = Laplace(log_hyp=np.log(sn[idgau])) a = l.proceed(mu=mu[idgau], y=y[idgau]) lZ[idgau] = a[0]; dlZ[idgau] = a[1]; d2lZ[idgau] = a[2] if np.any(id): # substitution to obtain unit variance, zero mean Laplacian tvar = s2[id]/(sn[id]**2+1e-16) tmu = (mu[id]-y[id])/(sn[id]+1e-16) # an implementation based on logphi(t) = log(normcdf(t)) zp = (tmu+np.sqrt(2)*tvar)/np.sqrt(tvar) zm = (tmu-np.sqrt(2)*tvar)/np.sqrt(tvar) ap = self._logphi(-zp)+np.sqrt(2)*tmu am = self._logphi( zm)-np.sqrt(2)*tmu apam = np.vstack((ap,am)).T lZ[id] = self._logsum2exp(apam) + tvar - np.log(sn[id]*np.sqrt(2.)) if nargout>1: lqp = -0.5*zp**2 - 0.5*np.log(2*np.pi) - self._logphi(-zp); # log( N(z)/Phi(z) ) lqm = -0.5*zm**2 - 0.5*np.log(2*np.pi) - self._logphi( zm); dap = -np.exp(lqp-0.5*np.log(s2[id])) + np.sqrt(2)/sn[id] dam = np.exp(lqm-0.5*np.log(s2[id])) - np.sqrt(2)/sn[id] _z1 = np.vstack((ap,am)).T _z2 = np.vstack((dap,dam)).T _x = np.array([[1],[1]]) dlZ[id] = self._expABz_expAx(_z1, _x, _z2, _x) if nargout>2: a = np.sqrt(8.)/sn[id]/np.sqrt(s2[id]); bp = 2./sn[id]**2 - (a - zp/s2[id])*np.exp(lqp) bm = 2./sn[id]**2 - (a + zm/s2[id])*np.exp(lqm) _x = np.reshape(np.array([1,1]),(2,1)) _z1 = np.reshape(np.array([ap,am]),(1,2)) _z2 = np.reshape(np.array([bp,bm]),(1,2)) d2lZ[id] = self._expABz_expAx(_z1, _x, _z2, _x) - dlZ[id]**2 return lZ,dlZ,d2lZ else: return lZ,dlZ else: return lZ else: # derivative mode dlZhyp = np.zeros((n,1)) if np.any(idlik): dlZhyp[idlik] = 0 if np.any(idgau): l = Laplace(log_hyp=np.log(sn[idgau])) a = l.proceed(mu=mu[idgau], y=y[idgau], inffunc='inf.Laplace', nargout=1) dlZhyp[idgau] = a[0] if np.any(id): # substitution to obtain unit variance, zero mean Laplacian tmu = (mu[id]-y[id])/(sn[id]+1e-16); tvar = s2[id]/(sn[id]**2+1e-16) zp = (tvar+tmu/np.sqrt(2))/np.sqrt(tvar); vp = tvar+np.sqrt(2)*tmu zm = (tvar-tmu/np.sqrt(2))/np.sqrt(tvar); vm = tvar-np.sqrt(2)*tmu dzp = (-s2[id]/sn[id]+tmu*sn[id]/np.sqrt(2)) / np.sqrt(s2[id]) dvp = -2*tvar - np.sqrt(2)*tmu dzm = (-s2[id]/sn[id]-tmu*sn[id]/np.sqrt(2)) / np.sqrt(s2[id]) dvm = -2*tvar + np.sqrt(2)*tmu lezp = self._lerfc(zp); # ap = exp(vp).*ezp lezm = self._lerfc(zm); # am = exp(vm).*ezm vmax = np.max(np.array([vp+lezp,vm+lezm]),axis=0); # subtract max to avoid numerical pb ep = np.exp(vp+lezp-vmax) em = np.exp(vm+lezm-vmax) dap = ep*(dvp - 2/np.sqrt(np.pi)*np.exp(-zp**2-lezp)*dzp) dam = em*(dvm - 2/np.sqrt(np.pi)*np.exp(-zm**2-lezm)*dzm) dlZhyp[id] = (dap+dam)/(ep+em) - 1; return dlZhyp # deriv. wrt hyp.lik elif isinstance(inffunc, inf.VB): n = len(s2.flatten()); b = np.zeros((n,1)); y = y*np.ones((n,1)); z = y return b,z def _lerfc(self,t): ''' numerically safe implementation of f(t) = log(1-erf(t)) = log(erfc(t))''' from scipy.special import erfc f = np.zeros_like(t) tmin = 20; tmax = 25 ok = ttmax # evaluate tight bound nok = np.logical_not(ok) interp = np.logical_and(nok,np.logical_not(bd)) # interpolate between both of them f[nok] = np.log(2/np.sqrt(np.pi)) -t[nok]**2 -np.log(t[nok]+np.sqrt( t[nok]**2+4/np.pi )) lam = 1/(1+np.exp( 12*(0.5-(t[interp]-tmin)/(tmax-tmin)) )) # interp. weights f[interp] = lam*f[interp] + (1-lam)*np.log(erfc( t[interp] )) f[ok] += np.log(erfc( t[ok] )) # safe eval return f def _expABz_expAx(self,A,x,B,z): ''' Computes y = ( (exp(A).*B)*z ) ./ ( exp(A)*x ) in a numerically safe way The function is not general in the sense that it yields correct values for all types of inputs. 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GA_hGAthGAhGAAhGA1MMhGAxhGAsLbhGA|}q?(hX(pyGPs - A Package for Gaussian Processesq@hh6hh!h#UtitleqAh%}qB(h+]h,]h*]h)]h-]uh.Kh/hh]qCh1X(pyGPs - A Package for Gaussian ProcessesqDqE}qF(hh@hh>ubaubh5)qG}qH(hUhh6hh!h#h8h%}qI(h+]h,]h*]h)]qJhah-]qKhauh.Kh/hh]qL(h=)qM}qN(hXAbout the packageqOhhGhh!h#hAh%}qP(h+]h,]h*]h)]h-]uh.Kh/hh]qQh1XAbout the packageqRqS}qT(hhOhhMubaubcdocutils.nodes paragraph qU)qV}qW(hXpyGPs is a library hosting Python implementations of Gaussian processes (GPs) for machine learning. pyGPs bridges the gap between systems designed primarily for users, who mainly want to apply GPs and need basic machine learning routines for model training, evaluation, and visualiztion, and expressive systems for developers, who focus on extending the core functionalities as covariance and likelihood functions, as well as inference techniques.qXhhGhh!h#U paragraphqYh%}qZ(h+]h,]h*]h)]h-]uh.Kh/hh]q[h1XpyGPs is a library hosting Python implementations of Gaussian processes (GPs) for machine learning. pyGPs bridges the gap between systems designed primarily for users, who mainly want to apply GPs and need basic machine learning routines for model training, evaluation, and visualiztion, and expressive systems for developers, who focus on extending the core functionalities as covariance and likelihood functions, as well as inference techniques.q\q]}q^(hhXhhVubaubhU)q_}q`(hXThe software package is released under the BSD 2-Clause (FreeBSD) License. :doc:`Copyright <_static/copyright>` (c) by Marion Neumann, Shan Huang, Daniel Marthaler, & Kristian Kersting, Feb.2014hhGhh!h#hYh%}qa(h+]h,]h*]h)]h-]uh.Kh/hh]qb(h1XKThe software package is released under the BSD 2-Clause (FreeBSD) License. qcqd}qe(hXKThe software package is released under the BSD 2-Clause (FreeBSD) License. hh_ubcsphinx.addnodes pending_xref qf)qg}qh(hX$:doc:`Copyright <_static/copyright>`qihh_hh!h#U pending_xrefqjh%}qk(UreftypeXdocqlUrefwarnqmU reftargetqnX_static/copyrightU refdomainUh)]h*]U 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(1999-2006))qhhhh!h#hYh%}q(h+]h,]h*]h)]h-]uh.Kh]qh1Xminimize.py implemented in python by Roland Memisevic 2008, following minimize.m (Copyright (c) Carl Edward Rasmussen (1999-2006))qq}q(hhhhubaubaubh)q}q(hX0scg.py (Copyright (c) Ian T Nabney (1996-2001)) hhhh!h#hh%}q(h+]h,]h*]h)]h-]uh.Nh/hh]qhU)q}q(hX/scg.py (Copyright (c) Ian T Nabney (1996-2001))qhhhh!h#hYh%}q(h+]h,]h*]h)]h-]uh.Kh]qh1X/scg.py (Copyright (c) Ian T Nabney (1996-2001))qq}q(hhhhubaubaubh)q}q(hX7brentmin.py (Copyright (c) Hannes Nickisch 2010-01-10) hhhh!h#hh%}q(h+]h,]h*]h)]h-]uh.Nh/hh]qhU)q}q(hX6brentmin.py (Copyright (c) Hannes Nickisch 2010-01-10)qhhhh!h#hYh%}q(h+]h,]h*]h)]h-]uh.Kh]qh1X6brentmin.py (Copyright (c) Hannes Nickisch 2010-01-10)qq}q(hhhhubaubaubh)q}q(hXFITC functionality (following matlab implementations under Copyright (c) by Ed Snelson, Carl Edward Rasmussen and Hannes Nickisch, 2011-11-02) hhhh!h#hh%}q(h+]h,]h*]h)]h-]uh.Nh/hh]qhU)q}q(hXFITC functionality (following matlab implementations under Copyright (c) by Ed Snelson, Carl Edward Rasmussen and Hannes Nickisch, 2011-11-02)qhhhh!h#hYh%}q(h+]h,]h*]h)]h-]uh.Kh]qh1XFITC functionality (following matlab implementations under Copyright (c) by Ed Snelson, Carl Edward Rasmussen and Hannes Nickisch, 2011-11-02)qq}q(hhhhubaubaubeubhU)q}q(hXThe most recent stable release is pyGPs v1.2. See :doc:`changelog <_static/changelog>` for a list of changes to the previous release. If you observe problems or bugs, please contact us. You can download the developer's guide and manual containing an API here: :download:`API and manual <_static/pyGPs_API.pdf>`. You can also download a `procedual implementation`_ of GP functionality from Github. However, the procedual version will not be supported in future.hhGhh!h#hYh%}q(h+]h,]h*]h)]h-]uh.Kh/hh]q(h1X2The most recent stable release is pyGPs v1.2. See qŅq}q(hX2The most recent stable release is pyGPs v1.2. See hhubhf)q}q(hX$:doc:`changelog <_static/changelog>`qhhhh!h#hjh%}q(UreftypeXdocqhmhnX_static/changelogU refdomainUh)]h*]U refexplicith+]h,]h-]hohpuh.Kh]qhr)q}q(hhh%}q(h+]h,]q(hwheh*]h)]h-]uhhh]qh1X changelogqӅq}q(hUhhubah#h|ubaubh1X for a list of changes to the previous release. If you observe problems or bugs, please contact us. You can download the developer's guide and manual containing an API here: qօq}q(hX for a list of changes to the previous release. If you observe problems or bugs, please contact us. You can download the developer's guide and manual containing an API here: hhubcsphinx.addnodes download_reference q)q}q(hX2:download:`API and manual <_static/pyGPs_API.pdf>`qhhhh!h#Udownload_referenceqh%}q(UreftypeXdownloadqhmhnX_static/pyGPs_API.pdfqU refdomainUh)]h*]U refexplicith+]h,]h-]hohpUfilenameqX pyGPs_API.pdfquh.Kh]qhr)q}q(hhh%}q(h+]h,]q(hwheh*]h)]h-]uhhh]qh1XAPI and manualq酁q}q(hUhhubah#h|ubaubh1X. You can also download a q셁q}q(hX. You can also download a hhubcdocutils.nodes reference q)q}q(hX`procedual implementation`_UresolvedqKhhh#U referenceqh%}q(UnameXprocedual implementationUrefuriqX&https://github.com/marionmari/pyGP_PR/qh)]h*]h+]h,]h-]uh]qh1Xprocedual implementationqq}q(hUhhubaubh1Xa of GP functionality from Github. However, the procedual version will not be supported in future.qq}q(hXa of GP functionality from Github. However, the procedual version will not be supported in future.hhubeubcdocutils.nodes target q)q}r(hXD.. _procedual implementation: https://github.com/marionmari/pyGP_PR/U referencedrKhhGhh!h#Utargetrh%}r(hhh)]rhah*]h+]h,]h-]rhauh.K#h/hh]ubcdocutils.nodes definition_list r)r}r(hUhhGhNh#Udefinition_listr h%}r (h+]h,]h*]h)]h-]uh.Nh/hh]r cdocutils.nodes definition_list_item r )r }r(hXAuthors: - Marion Neumann [marion dot neumann at uni-bonn dot de] - Shan Huang [shan dot huang at iais dot fraunhofer dot de] - Daniel Marthaler [dan dot marthaler at gmail dot com] - Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] hjhh!h#Udefinition_list_itemrh%}r(h+]h,]h*]h)]h-]uh.K+h]r(cdocutils.nodes term r)r}r(hXAuthors:rhj hh!h#Utermrh%}r(h+]h,]h*]h)]h-]uh.K+h]rh1XAuthors:rr}r(hjhjubaubcdocutils.nodes definition r)r}r(hUh%}r(h+]h,]h*]h)]h-]uhj h]r h)r!}r"(hUh%}r#(hX-h)]h*]h+]h,]h-]uhjh]r$(h)r%}r&(hX6Marion Neumann [marion dot neumann at uni-bonn dot de]r'h%}r((h+]h,]h*]h)]h-]uhj!h]r)hU)r*}r+(hj'hj%hh!h#hYh%}r,(h+]h,]h*]h)]h-]uh.K(h]r-h1X6Marion Neumann [marion dot neumann at uni-bonn dot de]r.r/}r0(hj'hj*ubaubah#hubh)r1}r2(hX9Shan Huang [shan dot huang at iais dot fraunhofer dot de]r3h%}r4(h+]h,]h*]h)]h-]uhj!h]r5hU)r6}r7(hj3hj1hh!h#hYh%}r8(h+]h,]h*]h)]h-]uh.K)h]r9h1X9Shan Huang [shan dot huang at iais dot fraunhofer dot de]r:r;}r<(hj3hj6ubaubah#hubh)r=}r>(hX5Daniel Marthaler [dan dot marthaler at gmail dot com]r?h%}r@(h+]h,]h*]h)]h-]uhj!h]rAhU)rB}rC(hj?hj=hh!h#hYh%}rD(h+]h,]h*]h)]h-]uh.K*h]rEh1X5Daniel Marthaler [dan dot marthaler at gmail dot com]rFrG}rH(hj?hjBubaubah#hubh)rI}rJ(hXGKristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] h%}rK(h+]h,]h*]h)]h-]uhj!h]rLhU)rM}rN(hXFKristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]rOhjIhh!h#hYh%}rP(h+]h,]h*]h)]h-]uh.K+h]rQh1XFKristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]rRrS}rT(hjOhjMubaubah#hubeh#hubah#U definitionrUubeubaubhU)rV}rW(hXThe following persons helped to improve this software: Roman Garnett, Maciej Kurek, Hannes Nickisch, Zhao Xu, and Alejandro Molina.rXhhGhh!h#hYh%}rY(h+]h,]h*]h)]h-]uh.K-h/hh]rZh1XThe following persons helped to improve this software: Roman Garnett, Maciej Kurek, Hannes Nickisch, Zhao Xu, and Alejandro Molina.r[r\}r](hjXhjVubaubhU)r^}r_(hXJThis work is partly supported by the Fraunhofer ATTRACT fellowship STREAM.r`hhGhh!h#hYh%}ra(h+]h,]h*]h)]h-]uh.K/h/hh]rbh1XJThis work is partly supported by the Fraunhofer ATTRACT fellowship STREAM.rcrd}re(hj`hj^ubaubeubh5)rf}rg(hUhh6hh!h#h8h%}rh(h+]h,]h*]h)]rihah-]rjhauh.K3h/hh]rk(h=)rl}rm(hXGetting startedrnhjfhh!h#hAh%}ro(h+]h,]h*]h)]h-]uh.K3h/hh]rph1XGetting startedrqrr}rs(hjnhjlubaubcdocutils.nodes compound rt)ru}rv(hUhjfhh!h#Ucompoundrwh%}rx(h+]h,]ryUtoctree-wrapperrzah*]h)]h-]uh.Nh/hh]r{csphinx.addnodes toctree r|)r}}r~(hUhjuhh!h#Utoctreerh%}r(UnumberedrKU includehiddenrhhpU titlesonlyrUglobrh)]h*]h+]h,]h-]Uentriesr]r(NXInstallrrNXTheoryrrNXExamplesrrNXKernelsrrNX LikelihoodsrrNXOptsrrNXGraphrreUhiddenrU includefilesr]r(jjjjjjjeUmaxdepthrKuh.K5h]ubaubeubeubehUU transformerrNU footnote_refsr}rUrefnamesr}rXprocedual implementation]rhasUsymbol_footnotesr]rUautofootnote_refsr]rUsymbol_footnote_refsr]rU citationsr]rh/hU current_linerNUtransform_messagesr]rUreporterrNUid_startrKU autofootnotesr]rU citation_refsr}rUindirect_targetsr]rUsettingsr(cdocutils.frontend Values ror}r(Ufootnote_backlinksrKUrecord_dependenciesrNU rfc_base_urlrUhttp://tools.ietf.org/html/rU tracebackrUpep_referencesrNUstrip_commentsrNU toc_backlinksrUentryrU language_coderUenrU datestamprNU report_levelrKU _destinationrNU halt_levelrKU strip_classesrNhANUerror_encoding_error_handlerrUbackslashreplacerUdebugrNUembed_stylesheetrUoutput_encoding_error_handlerrUstrictrU sectnum_xformrKUdump_transformsrNU docinfo_xformrKUwarning_streamrNUpep_file_url_templaterUpep-%04drUexit_status_levelrKUconfigrNUstrict_visitorrNUcloak_email_addressesrUtrim_footnote_reference_spacerUenvrNUdump_pseudo_xmlrNUexpose_internalsrNUsectsubtitle_xformrU source_linkrNUrfc_referencesrNUoutput_encodingrUutf-8rU source_urlrNUinput_encodingrU utf-8-sigrU_disable_configrNU id_prefixrUU tab_widthrKUerror_encodingrUUTF-8rU_sourcerU@/Users/marion/workspace/workspace_GPs/pyGPs/doc/source/index.rstrUgettext_compactrU generatorrNUdump_internalsrNU smart_quotesrU pep_base_urlrUhttp://www.python.org/dev/peps/rUsyntax_highlightrUlongrUinput_encoding_error_handlerrjUauto_id_prefixrUidrUdoctitle_xformrUstrip_elements_with_classesrNU _config_filesr]Ufile_insertion_enabledrU raw_enabledrKU dump_settingsrNubUsymbol_footnote_startrKUidsr}r(hhhjfhh6hhGuUsubstitution_namesr}rh#h/h%}r(h+]h)]h*]Usourceh!h,]h-]uU footnotesr]rUrefidsr}rub.pyGPs-1.2/doc/build/html/000077500000000000000000000000001236346020200152245ustar00rootroot00000000000000pyGPs-1.2/doc/build/html/.buildinfo000066400000000000000000000003461236346020200172030ustar00rootroot00000000000000# Sphinx build info version 1 # 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When it is not found, a full rebuild will be done. config: 671334558ad756f088d7c8b8718316be tags: 645f666f9bcd5a90fca523b33c5a78b7 pyGPs-1.2/doc/build/html/.doctrees/000077500000000000000000000000001236346020200171125ustar00rootroot00000000000000pyGPs-1.2/doc/build/html/.doctrees/CV.doctree000066400000000000000000000241301236346020200207710ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(Xk-fold cross-validationqNXcross validationqNXevaluation measuresqNX1X load dataq NuUsubstitution_defsq }q Uparse_messagesq ]q Ucurrent_sourceqNU decorationqNUautofootnote_startqKUnameidsq}q(hUk-fold-cross-validationqhUcross-validationqhUevaluation-measuresqX1Uid2qh U load-dataquUchildrenq]qcdocutils.nodes section q)q}q(U rawsourceqUUparentqhUsourceqcdocutils.nodes reprunicode q X-/Users/hs/Program/git/pyGPs/doc/source/CV.rstq!q"}q#bUtagnameq$Usectionq%U attributesq&}q'(Udupnamesq(]Uclassesq)]Ubackrefsq*]Uidsq+]q,haUnamesq-]q.hauUlineq/KUdocumentq0hh]q1(cdocutils.nodes title q2)q3}q4(hXK-fold Cross-Validationq5hhhh"h$Utitleq6h&}q7(h(]h)]h*]h+]h-]uh/Kh0hh]q8cdocutils.nodes Text q9XK-fold Cross-Validationq:q;}q<(hh5hh3ubaubcdocutils.nodes paragraph q=)q>}q?(hXIn this demo, we'll show you the typical process of using GP for machine learning from loading data, preprocessing, training, predicting to validation and evaluation.q@hhhh"h$U paragraphqAh&}qB(h(]h)]h*]h+]h-]uh/Kh0hh]qCh9XIn this demo, we'll show you the typical process of using GP for machine learning from loading data, preprocessing, training, predicting to validation and evaluation.qDqE}qF(hh@hh>ubaubh)qG}qH(hUhhhh"h$h%h&}qI(h(]h)]h*]h+]qJhah-]qKh auh/Kh0hh]qL(h2)qM}qN(hX Load dataqOhhGhh"h$h6h&}qP(h(]h)]h*]h+]h-]uh/Kh0hh]qQh9X Load dataqRqS}qT(hhOhhMubaubh=)qU}qV(hXDWe use the ionosphere dataset [1]_ from Johns Hopkins University Ionosphere database. It is available in UCI machine learning repository. Then we need to do some data cleaning. Here we deal with label in ionosphere data, change "b" to"-1", and "g" to "+1". These preprocessing implementation are availabe in the source code.hhGhh"h$hAh&}qW(h(]h)]h*]h+]h-]uh/Kh0hh]qX(h9XWe use the ionosphere dataset qYqZ}q[(hXWe use the ionosphere dataset hhUubcdocutils.nodes footnote_reference q\)q]}q^(hX[1]_Uresolvedq_KhhUh$Ufootnote_referenceq`h&}qa(h+]qbUid1qcah*]h(]h)]h-]Urefidqdhuh]qeh9X1qf}qg(hUhh]ubaubh9X" from Johns Hopkins University Ionosphere database. It is available in UCI machine learning repository. Then we need to do some data cleaning. Here we deal with label in ionosphere data, change "b" to"-1", and "g" to "+1". These preprocessing implementation are availabe in the source code.qhqi}qj(hX" from Johns Hopkins University Ionosphere database. It is available in UCI machine learning repository. Then we need to do some data cleaning. Here we deal with label in ionosphere data, change "b" to"-1", and "g" to "+1". These preprocessing implementation are availabe in the source code.hhUubeubeubh)qk}ql(hUhhhh"h$h%h&}qm(h(]h)]h*]h+]qnhah-]qohauh/Kh0hh]qp(h2)qq}qr(hXCross Validationqshhkhh"h$h6h&}qt(h(]h)]h*]h+]h-]uh/Kh0hh]quh9XCross Validationqvqw}qx(hhshhqubaubh=)qy}qz(hX2Now, lets focus on the use of cross-validation. ::q{hhkhh"h$hAh&}q|(h(]h)]h*]h+]h-]uh/Kh0hh]q}h9X/Now, lets focus on the use of cross-validation.q~q}q(hX/Now, lets focus on the use of cross-validation.hhyubaubcdocutils.nodes literal_block q)q}q(hXK = 10 # number of folds for x_train, x_test, y_train, y_test in valid.k_fold_validation(x, y, K): # This is a binary classification problem model = pyGPs.GPC() # Since no prior knowldege, leave everything default model.optimize(x_train, y_train) # Predition ymu, ys2, fmu, fs2, lp = model.predict(x_test, ys=y_test) # ymu for classification is a continuous value over -1 to +1 # If you want predicting result to either one of the classes, take a sign of ymu. ymu_class = np.sign(ymu) # Evluation acc = valid.ACC(ymu_class, y_test) # accuracy rmse = valid.RMSE(ymu_class, y_test) # root-mean-square errorhhkhh"h$U literal_blockqh&}q(U xml:spaceqUpreserveqh+]h*]h(]h)]h-]uh/Kh0hh]qh9XK = 10 # number of folds for x_train, x_test, y_train, y_test in valid.k_fold_validation(x, y, K): # This is a binary classification problem model = pyGPs.GPC() # Since no prior knowldege, leave everything default model.optimize(x_train, y_train) # Predition ymu, ys2, fmu, fs2, lp = model.predict(x_test, ys=y_test) # ymu for classification is a continuous value over -1 to +1 # If you want predicting result to either one of the classes, take a sign of ymu. ymu_class = np.sign(ymu) # Evluation acc = valid.ACC(ymu_class, y_test) # accuracy rmse = valid.RMSE(ymu_class, y_test) # root-mean-square errorqq}q(hUhhubaubeubh)q}q(hUhhhh"h$h%h&}q(h(]h)]h*]h+]qhah-]qhauh/K"h0hh]q(h2)q}q(hXEvaluation measuresqhhhh"h$h6h&}q(h(]h)]h*]h+]h-]uh/K"h0hh]qh9XEvaluation measuresqq}q(hhhhubaubcdocutils.nodes definition_list q)q}q(hUhhhNh$Udefinition_listqh&}q(h(]h)]h*]h+]h-]uh/Nh0hh]qcdocutils.nodes definition_list_item q)q}q(hX)We implemented some classical evaluation measures. - RMSE - root mean squared error - ACC - classification/regression accuracy - Prec - classification precision for class +1 - Recall - classification recall for class +1 - NLPD - negative log predictive density in transformed observation space hhhh"h$Udefinition_list_itemqh&}q(h(]h)]h*]h+]h-]uh/K+h]q(cdocutils.nodes term q)q}q(hX2We implemented some classical evaluation measures.qhhhh"h$Utermqh&}q(h(]h)]h*]h+]h-]uh/K+h]qh9X2We implemented some classical evaluation measures.qq}q(hhhhubaubcdocutils.nodes definition q)q}q(hUh&}q(h(]h)]h*]h+]h-]uhhh]qcdocutils.nodes bullet_list q)q}q(hUh&}q(UbulletqX-h+]h*]h(]h)]h-]uhhh]q(cdocutils.nodes list_item q)q}q(hXRMSE - root mean squared errorqh&}q(h(]h)]h*]h+]h-]uhhh]qh=)q}q(hhhhhh"h$hAh&}q(h(]h)]h*]h+]h-]uh/K$h]qh9XRMSE - root mean squared errorqŅq}q(hhhhubaubah$U list_itemqubh)q}q(hX(ACC - classification/regression accuracyqh&}q(h(]h)]h*]h+]h-]uhhh]qh=)q}q(hhhhhh"h$hAh&}q(h(]h)]h*]h+]h-]uh/K%h]qh9X(ACC - classification/regression accuracyq҅q}q(hhhhubaubah$hubh)q}q(hX,Prec - classification precision for class +1qh&}q(h(]h)]h*]h+]h-]uhhh]qh=)q}q(hhhhhh"h$hAh&}q(h(]h)]h*]h+]h-]uh/K&h]qh9X,Prec - classification precision for class +1qޅq}q(hhhhubaubah$hubh)q}q(hX+Recall - classification recall for class +1qh&}q(h(]h)]h*]h+]h-]uhhh]qh=)q}q(hhhhhh"h$hAh&}q(h(]h)]h*]h+]h-]uh/K'h]qh9X+Recall - classification recall for class +1qꅁq}q(hhhhubaubah$hubh)q}q(hXKNLPD - negative log predictive density in transformed observation space h&}q(h(]h)]h*]h+]h-]uhhh]qh=)q}q(hXGNLPD - negative log predictive density in transformed observation spaceqhhhh"h$hAh&}q(h(]h)]h*]h+]h-]uh/K(h]qh9XGNLPD - negative log predictive density in transformed observation spaceqq}q(hhhhubaubah$hubeh$U bullet_listqubah$U definitionqubeubaubcdocutils.nodes footnote q)q}q(hXSigillito, V. G., Wing, S. P., Hutton, L. V., \& Baker, K. B. (1989). Classification of radar returns from the ionosphere using neural networks. Johns Hopkins APL Technical Digest, 10, 262-266.qh_Khhhh"h$Ufootnoteqh&}r(h(]h)]h*]rhcah+]rhah-]rX1auh/K-h0hh]r(cdocutils.nodes label r)r}r(hUh&}r(h(]h)]h*]h+]h-]uhhh]r h9X1r }r (hUhjubah$Ulabelr ubh=)r }r(hhhhhh"h$hAh&}r(h(]h)]h*]h+]h-]uh/K-h]rh9XSigillito, V. G., Wing, S. P., Hutton, L. V., & Baker, K. B. (1989). Classification of radar returns from the ionosphere using neural networks. Johns Hopkins APL Technical Digest, 10, 262-266.rr}r(hXSigillito, V. G., Wing, S. P., Hutton, L. V., \& Baker, K. B. (1989). Classification of radar returns from the ionosphere using neural networks. Johns Hopkins APL Technical Digest, 10, 262-266.hj ubaubeubeubeubahUU transformerrNU footnote_refsr}rX1]rh]asUrefnamesr}rX1]rh]asUsymbol_footnotesr]rUautofootnote_refsr]rUsymbol_footnote_refsr]r U citationsr!]r"h0hU current_liner#NUtransform_messagesr$]r%Ureporterr&NUid_startr'KU autofootnotesr(]r)U citation_refsr*}r+Uindirect_targetsr,]r-Usettingsr.(cdocutils.frontend Values r/or0}r1(Ufootnote_backlinksr2KUrecord_dependenciesr3NU rfc_base_urlr4Uhttp://tools.ietf.org/html/r5U tracebackr6Upep_referencesr7NUstrip_commentsr8NU toc_backlinksr9Uentryr:U language_coder;Uenr<U datestampr=NU report_levelr>KU _destinationr?NU halt_levelr@KU strip_classesrANh6NUerror_encoding_error_handlerrBUbackslashreplacerCUdebugrDNUembed_stylesheetrEUoutput_encoding_error_handlerrFUstrictrGU sectnum_xformrHKUdump_transformsrINU docinfo_xformrJKUwarning_streamrKNUpep_file_url_templaterLUpep-%04drMUexit_status_levelrNKUconfigrONUstrict_visitorrPNUcloak_email_addressesrQUtrim_footnote_reference_spacerRUenvrSNUdump_pseudo_xmlrTNUexpose_internalsrUNUsectsubtitle_xformrVU source_linkrWNUrfc_referencesrXNUoutput_encodingrYUutf-8rZU source_urlr[NUinput_encodingr\U utf-8-sigr]U_disable_configr^NU id_prefixr_UU tab_widthr`KUerror_encodingraUUTF-8rbU_sourcercU-/Users/hs/Program/git/pyGPs/doc/source/CV.rstrdUgettext_compactreU generatorrfNUdump_internalsrgNU smart_quotesrhU pep_base_urlriUhttp://www.python.org/dev/peps/rjUsyntax_highlightrkUlongrlUinput_encoding_error_handlerrmjGUauto_id_prefixrnUidroUdoctitle_xformrpUstrip_elements_with_classesrqNU _config_filesrr]Ufile_insertion_enabledrsU raw_enabledrtKU dump_settingsruNubUsymbol_footnote_startrvKUidsrw}rx(hhkhhhhhch]hhGhhuUsubstitution_namesry}rzh$h0h&}r{(h(]h+]h*]Usourceh"h)]h-]uU footnotesr|]r}haUrefidsr~}rub.pyGPs-1.2/doc/build/html/.doctrees/Examples.doctree000066400000000000000000000177671236346020200222610ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(XsourceqXdemosqNXkernels & meansqX optimizersq X pygps/demoq uUsubstitution_defsq }q Uparse_messagesq ]qUcurrent_sourceqNU decorationqNUautofootnote_startqKUnameidsq}q(hUsourceqhUdemosqhU kernels-meansqh U optimizersqh U pygps-demoquUchildrenq]qcdocutils.nodes section q)q}q(U rawsourceqUUparentqhUsourceq cdocutils.nodes reprunicode q!X3/Users/hs/Program/git/pyGPs/doc/source/Examples.rstq"q#}q$bUtagnameq%Usectionq&U attributesq'}q((Udupnamesq)]Uclassesq*]Ubackrefsq+]Uidsq,]q-haUnamesq.]q/hauUlineq0KUdocumentq1hh]q2(cdocutils.nodes title q3)q4}q5(hXDemosq6hhh h#h%Utitleq7h'}q8(h)]h*]h+]h,]h.]uh0Kh1hh]q9cdocutils.nodes Text q:XDemosq;q<}q=(hh6hh4ubaubcdocutils.nodes paragraph q>)q?}q@(hXThere are several demos exemplifying the use of pyGPs for various Gaussian process (:math:`GP`) tasks. We recommend to first go through *Basic GP Regression* which introduces the :math:`GP` regression model. Basic regression is the most intuitive and simplest learning task feasable with :math:`GPs`. The other demos will then provide a general insight into more advanced functionalities of the package. You will also find the implementation of the demos in the source_ folder under `pyGPs/Demo`_.hhh h#h%U paragraphqAh'}qB(h)]h*]h+]h,]h.]uh0Kh1hh]qC(h:XTThere are several demos exemplifying the use of pyGPs for various Gaussian process (qDqE}qF(hXTThere are several demos exemplifying the use of pyGPs for various Gaussian process (hh?ubcsphinx.ext.mathbase math qG)qH}qI(hUh'}qJ(UlatexXGPh,]h+]h)]h*]h.]uhh?h]h%UmathqKubh:X*) tasks. We recommend to first go through qLqM}qN(hX*) tasks. We recommend to first go through hh?ubcdocutils.nodes emphasis qO)qP}qQ(hX*Basic GP Regression*h'}qR(h)]h*]h+]h,]h.]uhh?h]qSh:XBasic GP RegressionqTqU}qV(hUhhPubah%UemphasisqWubh:X which introduces the qXqY}qZ(hX which introduces the hh?ubhG)q[}q\(hUh'}q](UlatexXGPh,]h+]h)]h*]h.]uhh?h]h%hKubh:Xc regression model. Basic regression is the most intuitive and simplest learning task feasable with q^q_}q`(hXc regression model. Basic regression is the most intuitive and simplest learning task feasable with hh?ubhG)qa}qb(hUh'}qc(UlatexXGPsh,]h+]h)]h*]h.]uhh?h]h%hKubh:X. The other demos will then provide a general insight into more advanced functionalities of the package. You will also find the implementation of the demos in the qdqe}qf(hX. The other demos will then provide a general insight into more advanced functionalities of the package. You will also find the implementation of the demos in the hh?ubcdocutils.nodes reference qg)qh}qi(hXsource_UresolvedqjKhh?h%U referenceqkh'}ql(UnameXsourceqmUrefuriqnX#https://github.com/marionmari/pyGPsqoh,]h+]h)]h*]h.]uh]qph:Xsourceqqqr}qs(hUhhhubaubh:X folder under qtqu}qv(hX folder under hh?ubhg)qw}qx(hX `pyGPs/Demo`_hjKhh?h%hkh'}qy(UnameX pyGPs/DemohnX:https://github.com/marionmari/pyGPs/tree/master/pyGPs/Demoqzh,]h+]h)]h*]h.]uh]q{h:X pyGPs/Demoq|q}}q~(hUhhwubaubh:X.q}q(hX.hh?ubeubcdocutils.nodes target q)q}q(hXJ.. _pyGPs/Demo: https://github.com/marionmari/pyGPs/tree/master/pyGPs/DemoU referencedqKhhh h#h%Utargetqh'}q(hnhzh,]qhah+]h)]h*]h.]qh auh0K h1hh]ubh>)q}q(hXThe Demos give some theoretical explanations. Further, it is useful to have a look at our documentation on `Kernels & Means`_ and `Optimizers`_.qhhh h#h%hAh'}q(h)]h*]h+]h,]h.]uh0K h1hh]q(h:XkThe Demos give some theoretical explanations. Further, it is useful to have a look at our documentation on qq}q(hXkThe Demos give some theoretical explanations. Further, it is useful to have a look at our documentation on hhubhg)q}q(hX`Kernels & Means`_hjKhhh%hkh'}q(UnameXKernels & MeanshnX Kernels.htmlqh,]h+]h)]h*]h.]uh]qh:XKernels & Meansqq}q(hUhhubaubh:X and qq}q(hX and hhubhg)q}q(hX `Optimizers`_hjKhhh%hkh'}q(UnameX OptimizershnX Opts.htmlqh,]h+]h)]h*]h.]uh]qh:X Optimizersqq}q(hUhhubaubh:X.q}q(hX.hhubeubh)q}q(hX!.. _Kernels & Means: Kernels.htmlhKhhh h#h%hh'}q(hnhh,]qhah+]h)]h*]h.]qhauh0K h1hh]ubh)q}q(hX.. _Optimizers: Opts.htmlhKhhh h#h%hh'}q(hnhh,]qhah+]h)]h*]h.]qh auh0Kh1hh]ubh)q}q(hX/.. _source: https://github.com/marionmari/pyGPshKhhh h#h%hh'}q(hnhoh,]qhah+]h)]h*]h.]qhauh0Kh1hh]ubh>)q}q(hX Regressionqhhh h#h%hAh'}q(h)]h*]h+]h,]h.]uh0Kh1hh]qh:X Regressionqq}q(hhhhubaubcdocutils.nodes compound q)q}q(hUhhh h#h%Ucompoundqh'}q(h)]h*]qUtoctree-wrapperqah+]h,]h.]uh0Nh1hh]qcsphinx.addnodes toctree q)q}q(hUhhh h#h%Utoctreeqh'}q(UnumberedqKU includehiddenqˉhXExamplesqU titlesonlyq͉UglobqΉh,]h+]h)]h*]h.]Uentriesq]q(NXGPRqцqNXGPR_FITCqӆqeUhiddenqՉU includefilesq]q(hheUmaxdepthqKuh0Kh]ubaubh>)q}q(hXClassificationqhhh h#h%hAh'}q(h)]h*]h+]h,]h.]uh0Kh1hh]qh:XClassificationqޅq}q(hhhhubaubh)q}q(hUhhh h#h%hh'}q(h)]h*]qhah+]h,]h.]uh0Nh1hh]qh)q}q(hUhhh h#h%hh'}q(hKhˉhhh͉hΉh,]h+]h)]h*]h.]h]q(NXGPCqqNXGPC_FITCqqNXGPMCqqehՉh]q(hhhehKuh0Kh]ubaubh>)q}q(hX!Some examples for real-world dataqhhh h#h%hAh'}q(h)]h*]h+]h,]h.]uh0K#h1hh]qh:X!Some examples for real-world dataqq}q(hhhhubaubh)q}q(hUhhh h#h%hh'}q(h)]h*]qhah+]h,]h.]uh0Nh1hh]qh)q}q(hUhhh h#h%hh'}r(hKhˉhhh͉hΉh,]h+]h)]h*]h.]h]r(NXCVrrNX demoMaunaLoarrNX demoHousingrrNXSemiSupervisedrr NX GraphKernelr r ehՉh]r (jjjjj ehKuh0K%h]ubaubeubahUU transformerr NU footnote_refsr}rUrefnamesr}r(hm]rhhaXkernels & means]rhaX optimizers]rhaX pygps/demo]rhwauUsymbol_footnotesr]rUautofootnote_refsr]rUsymbol_footnote_refsr]rU citationsr]rh1hU current_linerNUtransform_messagesr]r Ureporterr!NUid_startr"KU autofootnotesr#]r$U citation_refsr%}r&Uindirect_targetsr']r(Usettingsr)(cdocutils.frontend Values r*or+}r,(Ufootnote_backlinksr-KUrecord_dependenciesr.NU rfc_base_urlr/Uhttp://tools.ietf.org/html/r0U tracebackr1Upep_referencesr2NUstrip_commentsr3NU toc_backlinksr4Uentryr5U language_coder6Uenr7U datestampr8NU report_levelr9KU _destinationr:NU halt_levelr;KU strip_classesr<Nh7NUerror_encoding_error_handlerr=Ubackslashreplacer>Udebugr?NUembed_stylesheetr@Uoutput_encoding_error_handlerrAUstrictrBU sectnum_xformrCKUdump_transformsrDNU docinfo_xformrEKUwarning_streamrFNUpep_file_url_templaterGUpep-%04drHUexit_status_levelrIKUconfigrJNUstrict_visitorrKNUcloak_email_addressesrLUtrim_footnote_reference_spacerMUenvrNNUdump_pseudo_xmlrONUexpose_internalsrPNUsectsubtitle_xformrQU source_linkrRNUrfc_referencesrSNUoutput_encodingrTUutf-8rUU source_urlrVNUinput_encodingrWU utf-8-sigrXU_disable_configrYNU id_prefixrZUU tab_widthr[KUerror_encodingr\UUTF-8r]U_sourcer^U3/Users/hs/Program/git/pyGPs/doc/source/Examples.rstr_Ugettext_compactr`U generatorraNUdump_internalsrbNU smart_quotesrcU pep_base_urlrdUhttp://www.python.org/dev/peps/reUsyntax_highlightrfUlongrgUinput_encoding_error_handlerrhjBUauto_id_prefixriUidrjUdoctitle_xformrkUstrip_elements_with_classesrlNU _config_filesrm]Ufile_insertion_enabledrnU raw_enabledroKU dump_settingsrpNubUsymbol_footnote_startrqKUidsrr}rs(hhhhhhhhhhuUsubstitution_namesrt}ruh%h1h'}rv(h)]h,]h+]Usourceh#h*]h.]uU footnotesrw]rxUrefidsry}rzub.pyGPs-1.2/doc/build/html/.doctrees/GPC.doctree000066400000000000000000000430201236346020200210710ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(Xa bit more things you can doqNX"first example state default valuesqNX load dataqNX more_on_gpcq X second example gp classificationq NXbasic classificationq NuUsubstitution_defsq }q Uparse_messagesq]qUcurrent_sourceqNU decorationqNUautofootnote_startqKUnameidsq}q(hUa-bit-more-things-you-can-doqhU"first-example-state-default-valuesqhU load-dataqh U more-on-gpcqh U second-example-gp-classificationqh Ubasic-classificationquUchildrenq]qcdocutils.nodes section q)q}q(U rawsourceq UUparentq!hUsourceq"cdocutils.nodes reprunicode q#X./Users/hs/Program/git/pyGPs/doc/source/GPC.rstq$q%}q&bUtagnameq'Usectionq(U attributesq)}q*(Udupnamesq+]Uclassesq,]Ubackrefsq-]Uidsq.]q/haUnamesq0]q1h auUlineq2KUdocumentq3hh]q4(cdocutils.nodes title q5)q6}q7(h XBasic Classificationq8h!hh"h%h'Utitleq9h)}q:(h+]h,]h-]h.]h0]uh2Kh3hh]q;cdocutils.nodes Text q}q?(h h8h!h6ubaubcdocutils.nodes paragraph q@)qA}qB(h XTThe demo shown in this tutorial can be obtained by running *pyGPs/Demo/demo_GPC.py*.qCh!hh"h%h'U paragraphqDh)}qE(h+]h,]h-]h.]h0]uh2Kh3hh]qF(hAgain, lets see the simplest use of gp classification at firstqЅq}q(h X>Again, lets see the simplest use of gp classification at firsth!hubaubhk)q}q(h XBmodel = pyGPs.gp.GPC() # binary classification (default inference method: EP) model.fit(x, y) # fit default model (mean zero & rbf kernel) with data model.optimize(x, y) # optimize hyperparamters (default optimizer: single run minimize) model.predict(z) # predict test casesh!hh"h%h'hnh)}q(hphqh.]h-]h+]h,]h0]uh2K"h3hh]qhr?}r@(h Uh!j:ubah'UstrongrAubh}q?(h h8h!h6ubaubcdocutils.nodes paragraph q@)qA}qB(h XThe demo in this tutorial can be obtained by running *pyGPs/Demo/demo_GPC_FITC.py*. This demo is more or less a repetition of the demo of `FITC regression`_.h!hh"h%h'U paragraphqCh)}qD(h+]h,]h-]h.]h0]uh2Kh3hh]qE(h}r?(h X( are shown as black circles in the plot.h!j2ubeubeubh)r@}rA(h Uh!hh"h%h'h(h)}rB(h+]h,]h-]h.]rChah0]rDhauh2K1h3hh]rE(h5)rF}rG(h X?Second example :math:`\rightarrow` user-defined inducing pointsrHh!j@h"h%h'h9h)}rI(h+]h,]h-]h.]h0]uh2K1h3hh]rJ(h(hh7h h5ubaubcdocutils.nodes paragraph q?)q@}qA(hXsGPMC is NOT based on multi-class Laplace approximation. It works as a one vs. one classification wrapper. In other words, GPMC trains a GPC model for each pair of two classes, and uses a majority voting scheme over all results to determine the final class. The method only returns the predictive class with highest rating; no other values (such as variance) are returned.qBh hh!h$h&U paragraphqCh(}qD(h*]h+]h,]h-]h/]uh1Kh2hh]qEh;XsGPMC is NOT based on multi-class Laplace approximation. It works as a one vs. one classification wrapper. In other words, GPMC trains a GPC model for each pair of two classes, and uses a majority voting scheme over all results to determine the final class. The method only returns the predictive class with highest rating; no other values (such as variance) are returned.qFqG}qH(hhBh h@ubaubh?)qI}qJ(hXlLets see a practical example to classify the 10 (0,1,2,...9) hand-written digits in the USPS digits dataset.qKh hh!h$h&hCh(}qL(h*]h+]h,]h-]h/]uh1K h2hh]qMh;XlLets see a practical example to classify the 10 (0,1,2,...9) hand-written digits in the USPS digits dataset.qNqO}qP(hhKh hIubaubh)qQ}qR(hUh hh!h$h&h'h(}qS(h*]h+]h,]h-]qThah/]qUhauh1Kh2hh]qV(h4)qW}qX(hX Load dataqYh hQh!h$h&h8h(}qZ(h*]h+]h,]h-]h/]uh1Kh2hh]q[h;X Load dataq\q]}q^(hhYh hWubaubh?)q_}q`(hXThe USPS digits data were gathered at the Center of Excellence in Document Analysis and Recognition (CEDAR) at SUNY Buffalo, as part of a project sponsored by the US Postal Service. The dataset is described in [1]_. ::qah hQh!h$h&hCh(}qb(h*]h+]h,]h-]h/]uh1Kh2hh]qc(h;XThe USPS digits data were gathered at the Center of Excellence in Document Analysis and Recognition (CEDAR) at SUNY Buffalo, as part of a project sponsored by the US Postal Service. The dataset is described in qdqe}qf(hXThe USPS digits data were gathered at the Center of Excellence in Document Analysis and Recognition (CEDAR) at SUNY Buffalo, as part of a project sponsored by the US Postal Service. The dataset is described in h h_ubcdocutils.nodes footnote_reference qg)qh}qi(hX[1]_UresolvedqjKh h_h&Ufootnote_referenceqkh(}ql(h-]qmUid1qnah,]h*]h+]h/]Urefidqohuh]qph;X1qq}qr(hUh hhubaubh;X.qs}qt(hX.h h_ubeubcdocutils.nodes literal_block qu)qv}qw(hXdata = loadmat('data_for_demo/usps_resampled.mat') x = data['train_patterns'].T # train patterns y = data['train_labels'].T # train labels xs = data['test_patterns'].T # test patterns ys = data['test_labels'].T # test labelsh hQh!h$h&U literal_blockqxh(}qy(U xml:spaceqzUpreserveq{h-]h,]h*]h+]h/]uh1Kh2hh]q|h;Xdata = loadmat('data_for_demo/usps_resampled.mat') x = data['train_patterns'].T # train patterns y = data['train_labels'].T # train labels xs = data['test_patterns'].T # test patterns ys = data['test_labels'].T # test labelsq}q~}q(hUh hvubaubh?)q}q(hXLTo be used in GPMC, labels should start from 0 to k (k = number of classes).qh hQh!h$h&hCh(}q(h*]h+]h,]h-]h/]uh1Kh2hh]qh;XLTo be used in GPMC, labels should start from 0 to k (k = number of classes).qq}q(hhh hubaubeubh)q}q(hUh hh!h$h&h'h(}q(h*]h+]h,]h-]qhah/]qh auh1Kh2hh]q(h4)q}q(hX GPMC exampleqh hh!h$h&h8h(}q(h*]h+]h,]h-]h/]uh1Kh2hh]qh;X GPMC exampleqq}q(hhh hubaubh?)q}q(hX4State model with 10-class classification problem: ::qh hh!h$h&hCh(}q(h*]h+]h,]h-]h/]uh1Kh2hh]qh;X1State model with 10-class classification problem:qq}q(hX1State model with 10-class classification problem:h hubaubhu)q}q(hXmodel = pyGPs.GPMC(10)h hh!h$h&hxh(}q(hzh{h-]h,]h*]h+]h/]uh1K h2hh]qh;Xmodel = pyGPs.GPMC(10)qq}q(hUh hubaubh?)q}q(hXPass data to model: ::qh hh!h$h&hCh(}q(h*]h+]h,]h-]h/]uh1K"h2hh]qh;XPass data to model:qq}q(hXPass data to model:h hubaubhu)q}q(hXmodel.setData(x,y)h hh!h$h&hxh(}q(hzh{h-]h,]h*]h+]h/]uh1K$h2hh]qh;Xmodel.setData(x,y)qq}q(hUh hubaubh?)q}q(hXTrain default GPC model for each binary classification problem, and decide label for test patterns of hand-writen digits. The return value *predictive_vote[i,j]* is the probability of being class *j* for test pattern *i*. ::h hh!h$h&hCh(}q(h*]h+]h,]h-]h/]uh1K&h2hh]q(h;XTrain default GPC model for each binary classification problem, and decide label for test patterns of hand-writen digits. The return value qq}q(hXTrain default GPC model for each binary classification problem, and decide label for test patterns of hand-writen digits. The return value h hubcdocutils.nodes emphasis q)q}q(hX*predictive_vote[i,j]*h(}q(h*]h+]h,]h-]h/]uh hh]qh;Xpredictive_vote[i,j]qq}q(hUh hubah&Uemphasisqubh;X# is the probability of being class qąq}q(hX# is the probability of being class h hubh)q}q(hX*j*h(}q(h*]h+]h,]h-]h/]uh hh]qh;Xjq}q(hUh hubah&hubh;X for test pattern qͅq}q(hX for test pattern h hubh)q}q(hX*i*h(}q(h*]h+]h,]h-]h/]uh hh]qh;Xiq}q(hUh hubah&hubh;X.q}q(hX.h hubeubhu)q}q(hXdpredictive_vote = model.optimizeAndPredict(xs) predictive_class = np.argmax(predictive_vote, axis=1)h hh!h$h&hxh(}q(hzh{h-]h,]h*]h+]h/]uh1K*h2hh]qh;Xdpredictive_vote = model.optimizeAndPredict(xs) predictive_class = np.argmax(predictive_vote, axis=1)q܅q}q(hUh hubaubh?)q}q(hXJust like we did for GP classification, you can use specific settings (other than default) for all binary classificiation problem for example by: ::h hh!h$h&hCh(}q(h*]h+]h,]h-]h/]uh1K-h2hh]qh;XJust like we did for GP classification, you can use specific settings (other than default) for all binary classificiation problem for example by:qㅁq}q(hXJust like we did for GP classification, you can use specific settings (other than default) for all binary classificiation problem for example by:h hubaubhu)q}q(hXgm = pyGPs.mean.Zero() k = pyGPs.cov.RBF() model.setPrior(mean=m,kernel=k) model.useInference("Laplace")h hh!h$h&hxh(}q(hzh{h-]h,]h*]h+]h/]uh1K0h2hh]qh;Xgm = pyGPs.mean.Zero() k = pyGPs.cov.RBF() model.setPrior(mean=m,kernel=k) model.useInference("Laplace")qꅁq}q(hUh hubaubh?)q}q(hXXFor more information on how to use non-default settings see `demo_GPC`_ and `demo_GPR`_.qh hh!h$h&hCh(}q(h*]h+]h,]h-]h/]uh1K5h2hh]q(h;X<For more information on how to use non-default settings see qq}q(hX<For more information on how to use non-default settings see h hubcdocutils.nodes reference q)q}q(hX `demo_GPC`_hjKh hh&U referenceqh(}q(UnameXdemo_GPCUrefuriqXGPC.htmlqh-]h,]h*]h+]h/]uh]qh;Xdemo_GPCqq}q(hUh hubaubh;X and rr}r(hX and h hubh)r}r(hX `demo_GPR`_hjKh hh&hh(}r(UnameXdemo_GPRhXGPR.htmlrh-]h,]h*]h+]h/]uh]rh;Xdemo_GPRrr }r (hUh jubaubh;X.r }r (hX.h hubeubcdocutils.nodes target r )r}r(hX.. _demo_GPC: GPC.htmlU referencedrKh hh!h$h&Utargetrh(}r(hhh-]rhah,]h*]h+]h/]rh auh1K8h2hh]ubj )r}r(hX.. _demo_GPR: GPR.htmljKh hh!h$h&jh(}r(hjh-]rhah,]h*]h+]h/]rhauh1K9h2hh]ubh?)r}r(hXvBeside *optimizeAndPredict(xs)*, there is also an option to perform prediction without hyperparameter optimization: ::h hh!h$h&hCh(}r(h*]h+]h,]h-]h/]uh1K(hUh(}r?(h*]h+]h,]h-]h/]uh j3h]r@h;X1rA}rB(hUh j=ubah&UlabelrCubh?)rD}rE(hj5h j3h!h$h&hCh(}rF(h*]h+]h,]h-]h/]uh1KAh]rGh;X`A Database for Handwritten Text Recognition Research, J. 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Hull, IEEE PAMI 16(5) 550-554, 1994.rHrI}rJ(hj5h jDubaubeubeubeubahUU transformerrKNU footnote_refsrL}rMX1]rNhhasUrefnamesrO}rP(X1]rQhhaXdemo_gpc]rRhaXdemo_gpr]rSjauUsymbol_footnotesrT]rUUautofootnote_refsrV]rWUsymbol_footnote_refsrX]rYU citationsrZ]r[h2hU current_liner\NUtransform_messagesr]]r^Ureporterr_NUid_startr`KU autofootnotesra]rbU citation_refsrc}rdUindirect_targetsre]rfUsettingsrg(cdocutils.frontend Values rhori}rj(Ufootnote_backlinksrkKUrecord_dependenciesrlNU rfc_base_urlrmUhttp://tools.ietf.org/html/rnU tracebackroUpep_referencesrpNUstrip_commentsrqNU toc_backlinksrrUentryrsU language_codertUenruU datestamprvNU report_levelrwKU _destinationrxNU halt_levelryKU strip_classesrzNh8NUerror_encoding_error_handlerr{Ubackslashreplacer|Udebugr}NUembed_stylesheetr~Uoutput_encoding_error_handlerrUstrictrU sectnum_xformrKUdump_transformsrNU docinfo_xformrKUwarning_streamrNUpep_file_url_templaterUpep-%04drUexit_status_levelrKUconfigrNUstrict_visitorrNUcloak_email_addressesrUtrim_footnote_reference_spacerUenvrNUdump_pseudo_xmlrNUexpose_internalsrNUsectsubtitle_xformrU source_linkrNUrfc_referencesrNUoutput_encodingrUutf-8rU source_urlrNUinput_encodingrU utf-8-sigrU_disable_configrNU id_prefixrUU tab_widthrKUerror_encodingrUUTF-8rU_sourcerU//Users/hs/Program/git/pyGPs/doc/source/GPMC.rstrUgettext_compactrU generatorrNUdump_internalsrNU smart_quotesrU pep_base_urlrUhttp://www.python.org/dev/peps/rUsyntax_highlightrUlongrUinput_encoding_error_handlerrjUauto_id_prefixrUidrUdoctitle_xformrUstrip_elements_with_classesrNU _config_filesr]Ufile_insertion_enabledrU raw_enabledrKU dump_settingsrNubUsymbol_footnote_startrKUidsr}r(hhhhhjhj3hnhhhhQhjuUsubstitution_namesr}rh&h2h(}r(h*]h-]h,]Usourceh$h+]h/]uU footnotesr]rj3aUrefidsr}rub.pyGPs-1.2/doc/build/html/.doctrees/GPR.doctree000066400000000000000000000707021236346020200211170ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(Xa bit more things you can doqNXa more complicated exampleqNXbasic regressionqNX load dataq NXgpmlq Ximport packagesq NXa five-line toy exampleq NXkernels & meansq X optimizersquUsubstitution_defsq}qUparse_messagesq]qUcurrent_sourceqNU decorationqNUautofootnote_startqKUnameidsq}q(hUa-bit-more-things-you-can-doqhUa-more-complicated-exampleqhUbasic-regressionqh U load-dataqh Ugpmlqh Uimport-packagesqh Ua-five-line-toy-exampleqh U kernels-meansqhU optimizersq uUchildrenq!]q"cdocutils.nodes section q#)q$}q%(U rawsourceq&UUparentq'hUsourceq(cdocutils.nodes reprunicode q)X./Users/hs/Program/git/pyGPs/doc/source/GPR.rstq*q+}q,bUtagnameq-Usectionq.U attributesq/}q0(Udupnamesq1]Uclassesq2]Ubackrefsq3]Uidsq4]q5haUnamesq6]q7hauUlineq8KUdocumentq9hh!]q:(cdocutils.nodes title q;)q<}q=(h&XBasic Regressionq>h'h$h(h+h-Utitleq?h/}q@(h1]h2]h3]h4]h6]uh8Kh9hh!]qAcdocutils.nodes Text qBXBasic RegressionqCqD}qE(h&h>h'hhBX:You can specify non-default mean and covariance functions:r?r@}rA(h&X:You can specify non-default mean and covariance functions:h'j:ubaubhw)rB}rC(h&Xim = pyGPs.mean.Linear( D=x.shape[1] ) + mean.Const() k = pyGPs.cov.RBF() model.setPrior(mean=m, kernel=k)h'j$h(h+h-hzh/}rD(h|h}h4]h3]h1]h2]h6]uh8K7h9hh!]rEhBXim = pyGPs.mean.Linear( D=x.shape[1] ) + mean.Const() k = pyGPs.cov.RBF() model.setPrior(mean=m, kernel=k)rFrG}rH(h&Uh'jBubaubhF)rI}rJ(h&XiHere, we use a composite mean as the sum of a linear and a constant function, and an rbf kernel. The initial hyperparameters are left to their default values. See `Kernels & Means`_ for a complete documentation of kernel/mean specification and custom kernel/mean construction. Once kernel and mean are specified, they are passed to the prior using *setPrior()*.rKh'j$h(h+h-hJh/}rL(h1]h2]h3]h4]h6]uh8K;h9hh!]rM(hBXHere, we use a composite mean as the sum of a linear and a constant function, and an rbf kernel. The initial hyperparameters are left to their default values. See rNrO}rP(h&XHere, we use a composite mean as the sum of a linear and a constant function, and an rbf kernel. The initial hyperparameters are left to their default values. See h'jIubcdocutils.nodes reference rQ)rR}rS(h&X`Kernels & Means`_UresolvedrTKh'jIh-U referencerUh/}rV(UnameXKernels & MeansUrefurirWX Kernels.htmlrXh4]h3]h1]h2]h6]uh!]rYhBXKernels & MeansrZr[}r\(h&Uh'jRubaubhBX for a complete documentation of kernel/mean specification and custom kernel/mean construction. Once kernel and mean are specified, they are passed to the prior using r]r^}r_(h&X for a complete documentation of kernel/mean specification and custom kernel/mean construction. Once kernel and mean are specified, they are passed to the prior using h'jIubhP)r`}ra(h&X *setPrior()*h/}rb(h1]h2]h3]h4]h6]uh'jIh!]rchBX setPrior()rdre}rf(h&Uh'j`ubah-hXubhBX.rg}rh(h&X.h'jIubeubcdocutils.nodes target ri)rj}rk(h&X!.. _Kernels & Means: Kernels.htmlU referencedrlKh'j$h(h+h-Utargetrmh/}rn(jWjXh4]rohah3]h1]h2]h6]rph auh8K=h9hh!]ubhF)rq}rr(h&X/You can add the traning data to the model explicitly by using *setData()*. So, you avoid passing them into *fit()* or *optimize()* each time used. More importantly, the deafult mean will be adapted to the average value of the trainging labels :math:`y` (if you do not specify mean function by your own).rsh'j$h(h+h-hJh/}rt(h1]h2]h3]h4]h6]uh8K?h9hh!]ru(hBX>You can add the traning data to the model explicitly by using rvrw}rx(h&X>You can add the traning data to the model explicitly by using h'jqubhP)ry}rz(h&X *setData()*h/}r{(h1]h2]h3]h4]h6]uh'jqh!]r|hBX setData()r}r~}r(h&Uh'jyubah-hXubhBX". So, you avoid passing them into rr}r(h&X". So, you avoid passing them into h'jqubhP)r}r(h&X*fit()*h/}r(h1]h2]h3]h4]h6]uh'jqh!]rhBXfit()rr}r(h&Uh'jubah-hXubhBX or rr}r(h&X or h'jqubhP)r}r(h&X *optimize()*h/}r(h1]h2]h3]h4]h6]uh'jqh!]rhBX optimize()rr}r(h&Uh'jubah-hXubhBXq each time used. More importantly, the deafult mean will be adapted to the average value of the trainging labels rr}r(h&Xq each time used. More importantly, the deafult mean will be adapted to the average value of the trainging labels h'jqubh)r}r(h&Uh/}r(UlatexXyh4]h3]h1]h2]h6]uh'jqh!]h-hubhBX3 (if you do not specify mean function by your own).rr}r(h&X3 (if you do not specify mean function by your own).h'jqubeubhF)r}r(h&X2Further, you can plot the data in the 1-d case: ::rh'j$h(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8KAh9hh!]rhBX/Further, you can plot the data in the 1-d case:rr}r(h&X/Further, you can plot the data in the 1-d case:h'jubaubhw)r}r(h&Xmodel.plotData_1d()h'j$h(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8KCh9hh!]rhBXmodel.plotData_1d()rr}r(h&Uh'jubaubj)r}r(h&Uh'j$h(h+h-jh/}r(jXcenterh4]h3]h1]h2]h6]uh8Nh9hh!]rj)r}r(h&Xd.. figure:: _images/d1_2.png :height: 600 px :width: 800 px :align: center :scale: 70 % h/}r(UscaleKFh1]UuriX_images/d1_2.pngrh4]h3]UwidthX800pxh2]j!}rU*jsh6]UheightX600pxuh'jh!]h-j#ubaubhF)r}r(h&XYou can specify a optimization method different from the default, which is a single run of Rasmussen's minimize. For example, you can choose to rerun the optimization method several times with different random initializations: ::h'j$h(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8KKh9hh!]rhBXYou can specify a optimization method different from the default, which is a single run of Rasmussen's minimize. For example, you can choose to rerun the optimization method several times with different random initializations:rr}r(h&XYou can specify a optimization method different from the default, which is a single run of Rasmussen's minimize. For example, you can choose to rerun the optimization method several times with different random initializations:h'jubaubhw)r}r(h&X/model.setOptimizer("Minimize", num_restarts=30)h'j$h(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8KNh9hh!]rhBX/model.setOptimizer("Minimize", num_restarts=30)rr}r(h&Uh'jubaubhF)r}r(h&XThe optimized hyperparameters returned by *optimize()* are then set to be the ones obtained from the run with the best result. The whole functionality for optimization is introduced in detail in the documentation `Optimizers`_.h'j$h(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8KPh9hh!]r(hBX*The optimized hyperparameters returned by rr}r(h&X*The optimized hyperparameters returned by h'jubhP)r}r(h&X *optimize()*h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX optimize()rr}r(h&Uh'jubah-hXubhBX are then set to be the ones obtained from the run with the best result. The whole functionality for optimization is introduced in detail in the documentation rr}r(h&X are then set to be the ones obtained from the run with the best result. The whole functionality for optimization is introduced in detail in the documentation h'jubjQ)r}r(h&X `Optimizers`_jTKh'jh-jUh/}r(UnameX OptimizersjWX Opts.htmlrh4]h3]h1]h2]h6]uh!]rhBX Optimizersrr}r(h&Uh'jubaubhBX.r}r(h&X.h'jubeubji)r}r(h&X.. _Optimizers: Opts.htmljlKh'j$h(h+h-jmh/}r(jWjh4]rh ah3]h1]h2]h6]rhauh8KSh9hh!]ubhF)r}r(h&XInstead of *fit()*, which only fits data using given hyperparameters, *optimize()* will optimize hyperparamters based on marginal likelihood: ::rh'j$h(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8KUh9hh!]r(hBX Instead of rr}r(h&X Instead of h'jubhP)r}r(h&X*fit()*h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXfit()rr}r(h&Uh'jubah-hXubhBX4, which only fits data using given hyperparameters, rr}r(h&X4, which only fits data using given hyperparameters, h'jubhP)r}r(h&X *optimize()*h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX optimize()rr}r(h&Uh'jubah-hXubhBX; will optimize hyperparamters based on marginal likelihood:rr}r(h&X; will optimize hyperparamters based on marginal likelihood:h'jubeubhw)r}r(h&Xmodel.optimize()h'j$h(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8KWh9hh!]rhBXmodel.optimize()rr}r(h&Uh'jubaubhF)r}r(h&X;There are several properties you can get from the model: ::rh'j$h(h+h-hJh/}r (h1]h2]h3]h4]h6]uh8KZh9hh!]r hBX8There are several properties you can get from the model:r r }r (h&X8There are several properties you can get from the model:h'jubaubhw)r}r(h&X-model.nlZ # negative log marginal likelihood model.dnlZ.cov # direvatives of negative log marginal likelihood model.dnlZ.lik model.dnlZ.mean model.posterior.sW # posterior structure model.posterior.alpha model.posterior.L model.covfunc.hyp model.meanfunc.hyp model.likfunc.hyp model.fm # latent mean model.fs2 # latent variance model.ym # predictive mean model.ys2 # predictive variance model.lp # log predictive probabilityh'j$h(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8K\h9hh!]rhBX-model.nlZ # negative log marginal likelihood model.dnlZ.cov # direvatives of negative log marginal likelihood model.dnlZ.lik model.dnlZ.mean model.posterior.sW # posterior structure model.posterior.alpha model.posterior.L model.covfunc.hyp model.meanfunc.hyp model.likfunc.hyp model.fm # latent mean model.fs2 # latent variance model.ym # predictive mean model.ys2 # predictive variance model.lp # log predictive probabilityrr}r(h&Uh'jubaubhF)r}r(h&X7For example, to get the log marginal likelihood use: ::rh'j$h(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Klh9hh!]rhBX4For example, to get the log marginal likelihood use:rr}r(h&X4For example, to get the log marginal likelihood use:h'jubaubhw)r}r(h&XGprint 'Optimized negative log marginal likelihood:', round(model.nlZ,3)h'j$h(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8Knh9hh!]r hBXGprint 'Optimized negative log marginal likelihood:', round(model.nlZ,3)r!r"}r#(h&Uh'jubaubhF)r$}r%(h&XPrediction on the test data will return five values, which are output mean (ymu) resp. variance (ys2), latent mean (fmu) resp. variance (fs2), and log predictive probabilities (lp) ::h'j$h(h+h-hJh/}r&(h1]h2]h3]h4]h6]uh8Kqh9hh!]r'hBXPrediction on the test data will return five values, which are output mean (ymu) resp. variance (ys2), latent mean (fmu) resp. variance (fs2), and log predictive probabilities (lp)r(r)}r*(h&XPrediction on the test data will return five values, which are output mean (ymu) resp. variance (ys2), latent mean (fmu) resp. variance (fs2), and log predictive probabilities (lp)h'j$ubaubhw)r+}r,(h&X'ym, ys2, fm, fs2, lp = model.predict(z)h'j$h(h+h-hzh/}r-(h|h}h4]h3]h1]h2]h6]uh8Kth9hh!]r.hBX'ym, ys2, fm, fs2, lp = model.predict(z)r/r0}r1(h&Uh'j+ubaubhF)r2}r3(h&XPlot data. Note that *GPR.plot()* is a toy method only for visualising 1-d data. Here we got a different posterior by using a different prior other than in the default example. ::r4h'j$h(h+h-hJh/}r5(h1]h2]h3]h4]h6]uh8Kwh9hh!]r6(hBXPlot data. Note that r7r8}r9(h&XPlot data. Note that h'j2ubhP)r:}r;(h&X *GPR.plot()*h/}r<(h1]h2]h3]h4]h6]uh'j2h!]r=hBX GPR.plot()r>r?}r@(h&Uh'j:ubah-hXubhBX is a toy method only for visualising 1-d data. Here we got a different posterior by using a different prior other than in the default example.rArB}rC(h&X is a toy method only for visualising 1-d data. Here we got a different posterior by using a different prior other than in the default example.h'j2ubeubhw)rD}rE(h&X model.plot()h'j$h(h+h-hzh/}rF(h|h}h4]h3]h1]h2]h6]uh8Kyh9hh!]rGhBX model.plot()rHrI}rJ(h&Uh'jDubaubj)rK}rL(h&Uh'j$h(h+h-jh/}rM(jXcenterh4]h3]h1]h2]h6]uh8Nh9hh!]rNj)rO}rP(h&Xf.. figure:: _images/d1_3.png :height: 600 px :width: 800 px :align: center :scale: 70 % h/}rQ(UscaleKFh1]UuriX_images/d1_3.pngrRh4]h3]UwidthX800pxh2]j!}rSU*jRsh6]UheightX600pxuh'jKh!]h-j#ubaubeubh#)rT}rU(h&Uh'h$h(h+h-h.h/}rV(h1]h2]h3]h4]rWhah6]rXhauh8Kh9hh!]rY(h;)rZ}r[(h&XA bit more things you can dor\h'jTh(h+h-h?h/}r](h1]h2]h3]h4]h6]uh8Kh9hh!]r^hBXA bit more things you can dor_r`}ra(h&j\h'jZubaubhF)rb}rc(h&Xg**[For all Models]** Speed up computation time for prediction if you know posterior in advance. Posterior is passed as an object with three fields (attributes) post.alpha, post.sW and post.L. How to use these vectors to represent the posterior can be best seen from Algorithm 2.1 (page 19) in Chapeter 2 of the `GPML`_ book by Rasmussen and Williams, 2006. ::rdh'jTh(h+h-hJh/}re(h1]h2]h3]h4]h6]uh8Kh9hh!]rf(cdocutils.nodes strong rg)rh}ri(h&X**[For all Models]**h/}rj(h1]h2]h3]h4]h6]uh'jbh!]rkhBX[For all Models]rlrm}rn(h&Uh'jhubah-UstrongroubhBX# Speed up computation time for prediction if you know posterior in advance. Posterior is passed as an object with three fields (attributes) post.alpha, post.sW and post.L. How to use these vectors to represent the posterior can be best seen from Algorithm 2.1 (page 19) in Chapeter 2 of the rprq}rr(h&X# Speed up computation time for prediction if you know posterior in advance. Posterior is passed as an object with three fields (attributes) post.alpha, post.sW and post.L. How to use these vectors to represent the posterior can be best seen from Algorithm 2.1 (page 19) in Chapeter 2 of the h'jbubjQ)rs}rt(h&X`GPML`_jTKh'jbh-jUh/}ru(UnameXGPMLjWX4http://www.gaussianprocess.org/gpml/chapters/RW2.pdfrvh4]h3]h1]h2]h6]uh!]rwhBXGPMLrxry}rz(h&Uh'jsubaubhBX& book by Rasmussen and Williams, 2006.r{r|}r}(h&X& book by Rasmussen and Williams, 2006.h'jbubeubhw)r~}r(h&Xlpost = myPosterior() # known in advance ym, ys2, fm, fs2, lp = model.predict_with_posterior( post,z )h'jTh(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8Kh9hh!]rhBXlpost = myPosterior() # known in advance ym, ys2, fm, fs2, lp = model.predict_with_posterior( post,z )rr}r(h&Uh'j~ubaubji)r}r(h&X>.. _GPML: http://www.gaussianprocess.org/gpml/chapters/RW2.pdfjlKh'jTh(h+h-jmh/}r(jWjvh4]rhah3]h1]h2]h6]rh auh8Kh9hh!]ubhF)r}r(h&XX**[Only for Regression]** Specify noise of data (with :math:`\sigma=0.1` by default): ::rh'jTh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Kh9hh!]r(jg)r}r(h&X**[Only for Regression]**h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX[Only for Regression]rr}r(h&Uh'jubah-joubhBX Specify noise of data (with rr}r(h&X Specify noise of data (with h'jubh)r}r(h&Uh/}r(UlatexX \sigma=0.1h4]h3]h1]h2]h6]uh'jh!]h-hubhBX by default):rr}r(h&X by default):h'jubeubhw)r}r(h&X)model.setNoise( log_sigma = np.log(0.1) )h'jTh(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8Kh9hh!]rhBX)model.setNoise( log_sigma = np.log(0.1) )rr}r(h&Uh'jubaubhF)r}r(h&XeYou do not need to specify the noise parameter if you are optimizing the hyperparamters later anyhow.rh'jTh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Kh9hh!]rhBXeYou do not need to specify the noise parameter if you are optimizing the hyperparamters later anyhow.rr}r(h&jh'jubaubhF)r}r(h&XfAll plotting methods have keyword axisvals. You can adjust plotting range if you want. For example: ::rh'jTh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Kh9hh!]rhBXcAll plotting methods have keyword axisvals. You can adjust plotting range if you want. For example:rr}r(h&XcAll plotting methods have keyword axisvals. You can adjust plotting range if you want. For example:h'jubaubhw)r}r(h&X-model.plot(axisvals = [-1.9, 1.9, -0.9, 3.9])h'jTh(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8Kh9hh!]rhBX-model.plot(axisvals = [-1.9, 1.9, -0.9, 3.9])rr}r(h&Uh'jubaubhF)r}r(h&X6Switch to other Inference and Likelihood functions. ::rh'jTh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Kh9hh!]rhBX3Switch to other Inference and Likelihood functions.rr}r(h&X3Switch to other Inference and Likelihood functions.h'jubaubhw)r}r(h&X7model.useInference("EP") model.useLikelihood("Laplace")h'jTh(h+h-hzh/}r(h|h}h4]h3]h1]h2]h6]uh8Kh9hh!]rhBX7model.useInference("EP") model.useLikelihood("Laplace")rr}r(h&Uh'jubaubeubeubah&UU transformerrNU footnote_refsr}rUrefnamesr}r(Xgpml]rjsaXkernels & means]rjRaX optimizers]rjauUsymbol_footnotesr]rUautofootnote_refsr]rUsymbol_footnote_refsr]rU citationsr]rh9hU current_linerNUtransform_messagesr]rUreporterrNUid_startrKU autofootnotesr]rU citation_refsr}rUindirect_targetsr]rUsettingsr(cdocutils.frontend Values ror}r(Ufootnote_backlinksrKUrecord_dependenciesrNU 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footnotesq]qUrefidsq}qub.pyGPs-1.2/doc/build/html/.doctrees/GPR_FITC.doctree000066400000000000000000000425611236346020200217260ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(X%first example default inducing pointsqNXa bit more things you can doqNX+second example user-defined inducing pointsqNXsparse regressionq NXfitc classificationq uUsubstitution_defsq }q Uparse_messagesq ]qUcurrent_sourceqNU decorationqNUautofootnote_startqKUnameidsq}q(hU%first-example-default-inducing-pointsqhUa-bit-more-things-you-can-doqhU+second-example-user-defined-inducing-pointsqh Usparse-regressionqh Ufitc-classificationquUchildrenq]qcdocutils.nodes section q)q}q(U rawsourceqUUparentqhUsourceq cdocutils.nodes reprunicode q!X3/Users/hs/Program/git/pyGPs/doc/source/GPR_FITC.rstq"q#}q$bUtagnameq%Usectionq&U attributesq'}q((Udupnamesq)]Uclassesq*]Ubackrefsq+]Uidsq,]q-haUnamesq.]q/h auUlineq0KUdocumentq1hh]q2(cdocutils.nodes title q3)q4}q5(hXSparse Regressionq6hhh h#h%Utitleq7h'}q8(h)]h*]h+]h,]h.]uh0Kh1hh]q9cdocutils.nodes Text q:XSparse Regressionq;q<}q=(hh6hh4ubaubcdocutils.nodes paragraph q>)q?}q@(hXThe code shown in this tutorial can be obtained by running *pyGPs/Demo/demo_GPR_FITC.py* This demo is more or less similar to the demo of `FITC classification`_.hhh h#h%U paragraphqAh'}qB(h)]h*]h+]h,]h.]uh0Kh1hh]qC(h:X;The code shown in this tutorial can be obtained by running qDqE}qF(hX;The code shown in this tutorial can be obtained by running hh?ubcdocutils.nodes emphasis qG)qH}qI(hX*pyGPs/Demo/demo_GPR_FITC.py*h'}qJ(h)]h*]h+]h,]h.]uhh?h]qKh:XpyGPs/Demo/demo_GPR_FITC.pyqLqM}qN(hUhhHubah%UemphasisqOubh:X2 This demo is more or less similar to the demo of qPqQ}qR(hX2 This demo is more or less similar to the demo of hh?ubcdocutils.nodes reference qS)qT}qU(hX`FITC classification`_UresolvedqVKhh?h%U referenceqWh'}qX(UnameXFITC classificationUrefuriqYX GPC_FITC.htmlqZh,]h+]h)]h*]h.]uh]q[h:XFITC classificationq\q]}q^(hUhhTubaubh:X.q_}q`(hX.hh?ubeubcdocutils.nodes target qa)qb}qc(hX&.. _FITC classification: GPC_FITC.htmlU referencedqdKhhh h#h%Utargetqeh'}qf(hYhZh,]qghah+]h)]h*]h.]qhh auh0Kh1hh]ubh)qi}qj(hUhhh h#h%h&h'}qk(h)]h*]h+]h,]qlhah.]qmhauh0K h1hh]qn(h3)qo}qp(hX9First example :math:`\rightarrow` default inducing pointsqqhhih h#h%h7h'}qr(h)]h*]h+]h,]h.]uh0K h1hh]qs(h:XFirst example qtqu}qv(hXFirst example qwhhoubcsphinx.ext.mathbase math qx)qy}qz(hUh'}q{(UlatexX \rightarrowq|h,]h+]h)]h*]h.]uhhoh]h%Umathq}ubh:X default inducing pointsq~q}q(hX default inducing pointsqhhoubeubh>)q}q(hX,First load the same data as in the GPR demo.qhhih h#h%hAh'}q(h)]h*]h+]h,]h.]uh0K h1hh]qh:X,First load the same data as in the GPR demo.qq}q(hhhhubaubh>)q}q(hX**[Theory]** In case the number of training inputs :math:`x` exceeds a few hundred, approximate inference using Laplace approximation or expectation propagation takes too long. We offer the FITC approximation based on a low-rank plus diagonal approximation to the exact covariance to deal with these cases. The general idea is to use inducing points :math:`u` and to base the computations on cross-covariances between training, test and inducing points only.hhih h#h%hAh'}q(h)]h*]h+]h,]h.]uh0K h1hh]q(cdocutils.nodes strong q)q}q(hX **[Theory]**h'}q(h)]h*]h+]h,]h.]uhhh]qh:X[Theory]qq}q(hUhhubah%Ustrongqubh:X' In case the number of training inputs qq}q(hX' In case the number of training inputs hhubhx)q}q(hUh'}q(UlatexXxh,]h+]h)]h*]h.]uhhh]h%h}ubh:X" exceeds a few hundred, approximate inference using Laplace approximation or expectation propagation takes too long. We offer the FITC approximation based on a low-rank plus diagonal approximation to the exact covariance to deal with these cases. The general idea is to use inducing points qq}q(hX" exceeds a few hundred, approximate inference using Laplace approximation or expectation propagation takes too long. We offer the FITC approximation based on a low-rank plus diagonal approximation to the exact covariance to deal with these cases. The general idea is to use inducing points hhubhx)q}q(hUh'}q(UlatexXuh,]h+]h)]h*]h.]uhhh]h%h}ubh:Xc and to base the computations on cross-covariances between training, test and inducing points only.qq}q(hXc and to base the computations on cross-covariances between training, test and inducing points only.hhubeubh>)q}q(hX(Okay, now the model is FITC regression::qhhih h#h%hAh'}q(h)]h*]h+]h,]h.]uh0Kh1hh]qh:X'Okay, now the model is FITC regression:qq}q(hX'Okay, now the model is FITC regression:hhubaubcdocutils.nodes literal_block q)q}q(hXmodel = pyGPs.GPR_FITC()hhih h#h%U literal_blockqh'}q(U xml:spaceqUpreserveqh,]h+]h)]h*]h.]uh0Kh1hh]qh:Xmodel = pyGPs.GPR_FITC()qq}q(hUhhubaubh>)q}q(hXThe difference between the usage of basic :math:`GP` regression is that we will have to specify inducing points. In the first example here, we will introduce you how to use the default settings.hhih h#h%hAh'}q(h)]h*]h+]h,]h.]uh0Kh1hh]q(h:X*The difference between the usage of basic qq}q(hX*The difference between the usage of basic hhubhx)q}q(hUh'}q(UlatexXGPh,]h+]h)]h*]h.]uhhh]h%h}ubh:X regression is that we will have to specify inducing points. In the first example here, we will introduce you how to use the default settings.qÅq}q(hX regression is that we will have to specify inducing points. In the first example here, we will introduce you how to use the default settings.hhubeubh>)q}q(hXGThe default inducing points are a grid (hypercube for higher dimensions), where each dimension has 5 values in equidistant steps in :math:`[min, max]`, where :math:`min` and :math:`max` are the minimum and maximum values of the input data by default. In order to specify the dimension of input data, we HAVE TO set data first::hhih h#h%hAh'}q(h)]h*]h+]h,]h.]uh0Kh1hh]q(h:XThe default inducing points are a grid (hypercube for higher dimensions), where each dimension has 5 values in equidistant steps in qʅq}q(hXThe default inducing points are a grid (hypercube for higher dimensions), where each dimension has 5 values in equidistant steps in hhubhx)q}q(hUh'}q(UlatexX [min, max]h,]h+]h)]h*]h.]uhhh]h%h}ubh:X, where qЅq}q(hX, where hhubhx)q}q(hUh'}q(UlatexXminh,]h+]h)]h*]h.]uhhh]h%h}ubh:X and qօq}q(hX and hhubhx)q}q(hUh'}q(UlatexXmaxh,]h+]h)]h*]h.]uhhh]h%h}ubh:X are the minimum and maximum values of the input data by default. In order to specify the dimension of input data, we HAVE TO set data first:q܅q}q(hX are the minimum and maximum values of the input data by default. In order to specify the dimension of input data, we HAVE TO set data first:hhubeubh)q}q(hXmodel.setData(x, y)hhih h#h%hh'}q(hhh,]h+]h)]h*]h.]uh0Kh1hh]qh:Xmodel.setData(x, y)qㅁq}q(hUhhubaubh>)q}q(hX@The number of inducing points per axis is :math:`5` per default.qhhih h#h%hAh'}q(h)]h*]h+]h,]h.]uh0Kh1hh]q(h:X*The number of inducing points per axis is q녁q}q(hX*The number of inducing points per axis is hhubhx)q}q(hUh'}q(UlatexX5h,]h+]h)]h*]h.]uhhh]h%h}ubh:X per default.qq}q(hX per default.hhubeubh>)q}q(hX<Now, the regular training and prediction routines follow: ::qhhih h#h%hAh'}q(h)]h*]h+]h,]h.]uh0K!h1hh]qh:X9Now, the regular training and prediction routines follow:qq}q(hX9Now, the regular training and prediction routines follow:hhubaubh)q}q(hX.model.optimize() model.predict(z) model.plot()hhih h#h%hh'}q(hhh,]h+]h)]h*]h.]uh0K#h1hh]qh:X.model.optimize() model.predict(z) model.plot()rr}r(hUhhubaubcdocutils.nodes figure r)r}r(hUhhih h#h%Ufigurerh'}r(UalignrXcenterh,]h+]h)]h*]h.]uh0Nh1hh]r cdocutils.nodes image r )r }r (hXd.. figure:: _images/d3_1.png :height: 600 px :width: 800 px :align: center :scale: 70 % h'}r (UscaleKFh)]UuriX_images/d3_1.pngrh,]h+]UwidthX800pxh*]U candidatesr}rU*jsh.]UheightX600pxuhjh]h%Uimagerubaubh>)r}r(hX\The equidistant default inducing points :math:`u` that are shown in the figure as black x's.rhhih h#h%hAh'}r(h)]h*]h+]h,]h.]uh0K-h1hh]r(h:X(The equidistant default inducing points rr}r(hX(The equidistant default inducing points hjubhx)r}r(hUh'}r(UlatexXuh,]h+]h)]h*]h.]uhjh]h%h}ubh:X+ that are shown in the figure as black x's.rr}r(hX+ that are shown in the figure as black x's.hjubeubh>)r }r!(hXZTo change the number of inducing points per axis just specify a different value per axis::r"hhih h#h%hAh'}r#(h)]h*]h+]h,]h.]uh0K/h1hh]r$h:XYTo change the number of inducing points per axis just specify a different value per axis:r%r&}r'(hXYTo change the number of inducing points per axis just specify a different value per axis:hj ubaubh)r(}r)(hX&model.setData(x, y, value_per_axis=10)hhih h#h%hh'}r*(hhh,]h+]h)]h*]h.]uh0K1h1hh]r+h:X&model.setData(x, y, value_per_axis=10)r,r-}r.(hUhj(ubaubeubh)r/}r0(hUhhh h#h%h&h'}r1(h)]h*]h+]h,]r2hah.]r3hauh0K5h1hh]r4(h3)r5}r6(hX?Second example :math:`\rightarrow` user-defined inducing pointsr7hj/h h#h%h7h'}r8(h)]h*]h+]h,]h.]uh0K5h1hh]r9(h:XSecond example r:r;}r<(hXSecond example r=hj5ubhx)r>}r?(hUh'}r@(UlatexX \rightarrowrAh,]h+]h)]h*]h.]uhj5h]h%h}ubh:X user-defined inducing pointsrBrC}rD(hX user-defined inducing pointsrEhj5ubeubh>)rF}rG(hXAlternatively, a random subset of the training points can be used as inducing points. Note, that there are plenty of methods to set these inducing points. So, in the second example let us use a user-defined set of inducing points.rHhj/h h#h%hAh'}rI(h)]h*]h+]h,]h.]uh0K7h1hh]rJh:XAlternatively, a random subset of the training points can be used as inducing points. Note, that there are plenty of methods to set these inducing points. So, in the second example let us use a user-defined set of inducing points.rKrL}rM(hjHhjFubaubh>)rN}rO(hX7You can pick a set of fixed inducing points by hand: ::rPhj/h h#h%hAh'}rQ(h)]h*]h+]h,]h.]uh0K:h1hh]rRh:X4You can pick a set of fixed inducing points by hand:rSrT}rU(hX4You can pick a set of fixed inducing points by hand:hjNubaubh)rV}rW(hX0u = np.array([[-1], [-0.8], [-0.5], [0.3],[1.]])hj/h h#h%hh'}rX(hhh,]h+]h)]h*]h.]uh0K)r]}r^(hXYou can also use equidistant inducing points :math:`u`, but without the values on the margin of the grid.(i.e. shrinking the range of values) ::r_hj/h h#h%hAh'}r`(h)]h*]h+]h,]h.]uh0K>h1hh]ra(h:X-You can also use equidistant inducing points rbrc}rd(hX-You can also use equidistant inducing points hj]ubhx)re}rf(hUh'}rg(UlatexXuh,]h+]h)]h*]h.]uhj]h]h%h}ubh:XW, but without the values on the margin of the grid.(i.e. shrinking the range of values)rhri}rj(hXW, but without the values on the margin of the grid.(i.e. shrinking the range of values)hj]ubeubh)rk}rl(hXZnum_u = np.fix(x.shape[0]/2) u = np.linspace(-1.3,1.3,num_u).T u = np.reshape(u,(num_u,1))hj/h h#h%hh'}rm(hhh,]h+]h)]h*]h.]uh0K@h1hh]rnh:XZnum_u = np.fix(x.shape[0]/2) u = np.linspace(-1.3,1.3,num_u).T u = np.reshape(u,(num_u,1))rorp}rq(hUhjkubaubh>)rr}rs(hX-Then pass :math:`u` when specifying prior. ::rthj/h h#h%hAh'}ru(h)]h*]h+]h,]h.]uh0KDh1hh]rv(h:X Then pass rwrx}ry(hX Then pass hjrubhx)rz}r{(hUh'}r|(UlatexXuh,]h+]h)]h*]h.]uhjrh]h%h}ubh:X when specifying prior.r}r~}r(hX when specifying prior.hjrubeubh)r}r(hXm = pyGPs.mean.Zero() k = pyGPs.cov.RBFard(log_ell_list=[0.05,0.17], log_sigma=1.) model.setPrior(mean=m, kernel=k, inducing_points=u)hj/h h#h%hh'}r(hhh,]h+]h)]h*]h.]uh0KFh1hh]rh:Xm = pyGPs.mean.Zero() k = pyGPs.cov.RBFard(log_ell_list=[0.05,0.17], log_sigma=1.) model.setPrior(mean=m, kernel=k, inducing_points=u)rr}r(hUhjubaubh>)r}r(hXThe left figure below shows the result of fixed inducing points, and the right figure shows the result for equidistant :math:`u`.rhj/h h#h%hAh'}r(h)]h*]h+]h,]h.]uh0KJh1hh]r(h:XwThe left figure below shows the result of fixed inducing points, and the right figure shows the result for equidistant rr}r(hXwThe left figure below shows the result of fixed inducing points, and the right figure shows the result for equidistant hjubhx)r}r(hUh'}r(UlatexXuh,]h+]h)]h*]h.]uhjh]h%h}ubh:X.r}r(hX.hjubeubj )r}r(hX+.. image:: _images/d3_2.png :width: 45% hj/h h#h%jh'}r(h)]UuriX_images/d3_2.pngrh,]h+]UwidthX45%h*]j}rU*jsh.]uh0Nh1hh]ubj )r}r(hX,.. image:: _images/d3_3.png :width: 45% hj/h h#h%jh'}r(h)]UuriX_images/d3_3.pngrh,]h+]UwidthX45%h*]j}rU*jsh.]uh0Nh1hh]ubh>)r}r(hX**[Theory]** Note that the predictive variance is overestimated outside the support of the inducing inputs. In a multivariate example where densely sampled inducing inputs are infeasible, one can also try to simply use a random subset of the training points.hj/h h#h%hAh'}r(h)]h*]h+]h,]h.]uh0KSh1hh]r(h)r}r(hX **[Theory]**h'}r(h)]h*]h+]h,]h.]uhjh]rh:X[Theory]rr}r(hUhjubah%hubh:X Note that the predictive variance is overestimated outside the support of the inducing inputs. In a multivariate example where densely sampled inducing inputs are infeasible, one can also try to simply use a random subset of the training points.rr}r(hX Note that the predictive variance is overestimated outside the support of the inducing inputs. 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pyGPs.GraphExtensions.nodeKernels)hUtr>auh8Nh9hh!]ubh)r?}r@(h&Uh'hh(Nh-hh/}rA(hhXpyh4]h3]h1]h2]h6]hXfunctionrBhjBuh8Nh9hh!]rC(h)rD}rE(h&XpsInvLapKernel(A)h'j?h(hh-hh/}rF(h4]rGhahh)X!pyGPs.GraphExtensions.nodeKernelsrHrI}rJbh3]h1]h2]h6]rKhahXpsInvLapKernelrLhUhuh8Nh9hh!]rM(h)rN}rO(h&X"pyGPs.GraphExtensions.nodeKernels.h'jDh(hh-hh/}rP(h1]h2]h3]h4]h6]uh8Nh9hh!]rQhBX"pyGPs.GraphExtensions.nodeKernels.rRrS}rT(h&Uh'jNubaubh)rU}rV(h&jLh'jDh(hh-hh/}rW(h1]h2]h3]h4]h6]uh8Nh9hh!]rXhBXpsInvLapKernelrYrZ}r[(h&Uh'jUubaubh)r\}r](h&Uh'jDh(hh-hh/}r^(h1]h2]h3]h4]h6]uh8Nh9hh!]r_h)r`}ra(h&XAh/}rb(h1]h2]h3]h4]h6]uh'j\h!]rchBXArd}re(h&Uh'j`ubah-hubaubh)rf}rg(h&Uh'jDh(Nh-hh/}rh(Uexprhh4]h3]h1]h2]h6]uh8Nh9hh!]rih)rj}rk(h&Uh/}rl(UreftypehUrefdochU refdomainhh4]h3]U refexplicith1]h2]h6]U reftargetX*_modules/pyGPs/GraphExtensions/nodeKernelsUrefidjLuh'jfh!]rmh)rn}ro(h&Uh/}rp(h1]h2]rqhah3]h4]h6]uh'jjh!]rrhBX[source]rsrt}ru(h&Uh'jnubah-hubah-hubaubeubh)rv}rw(h&Uh'j?h(hh-hh/}rx(h1]h2]h3]h4]h6]uh8Nh9hh!]ry(hF)rz}r{(h&X+Pseudo inverse of the normalized Laplacian.r|h'jvh(X~/Users/hs/Program/git/pyGPs/pyGPs/GraphExtensions/nodeKernels.py:docstring of pyGPs.GraphExtensions.nodeKernels.psInvLapKernelr}h-hJh/}r~(h1]h2]h3]h4]h6]uh8Kh9hh!]rhBX+Pseudo inverse of the normalized Laplacian.rr}r(h&j|h'jzubaubj)r}r(h&Uh'jvh(Nh-jh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]r(j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX Parametersrr}r(h&Uh'jubah-j$ubj%)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhF)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j8)r}r(h&XAh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXAr}r(h&Uh'jubah-j?ubhBX -- rr}r(h&Uh'jubhBXadjacency matrixrr}r(h&Xadjacency matrixh'jubeh-hJubah-j_ubeh-j`ubj)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXReturnsrr}r(h&Uh'jubah-j$ubj%)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhF)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX kernel matrixrr}r(h&X kernel matrixh'jubah-hJubah-j_ubeh-j`ubeubeubeubh)r}r(h&Uh'hh(Nh-hh/}r(h4]h3]h1]h2]h6]Uentries]r(hX<regLapKernel() (in module pyGPs.GraphExtensions.nodeKernels)hUtrauh8Nh9hh!]ubh)r}r(h&Uh'hh(Nh-hh/}r(hhXpyh4]h3]h1]h2]h6]hXfunctionrhjuh8Nh9hh!]r(h)r}r(h&XregLapKernel(A, sigma=1)h'jh(hh-hh/}r(h4]rhahh)X!pyGPs.GraphExtensions.nodeKernelsrr}rbh3]h1]h2]h6]rhahX regLapKernelrhUhuh8Nh9hh!]r(h)r}r(h&X"pyGPs.GraphExtensions.nodeKernels.h'jh(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]rhBX"pyGPs.GraphExtensions.nodeKernels.rr}r(h&Uh'jubaubh)r}r(h&jh'jh(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]rhBX regLapKernelrr}r(h&Uh'jubaubh)r}r(h&Uh'jh(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]r(h)r}r(h&XAh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXAr}r(h&Uh'jubah-hubh)r}r(h&Xsigma=1h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXsigma=1rr}r(h&Uh'jubah-hubeubh)r}r(h&Uh'jh(Nh-hh/}r(Uexprhh4]h3]h1]h2]h6]uh8Nh9hh!]rh)r}r(h&Uh/}r(UreftypehUrefdochU refdomainhh4]h3]U refexplicith1]h2]h6]U reftargetX*_modules/pyGPs/GraphExtensions/nodeKernelsUrefidjuh'jh!]rh)r}r(h&Uh/}r(h1]h2]rhah3]h4]h6]uh'jh!]rhBX[source]rr}r(h&Uh'jubah-hubah-hubaubeubh)r}r(h&Uh'jh(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]r(hF)r}r(h&XRegularized Laplacian Kernelrh'jh(X|/Users/hs/Program/git/pyGPs/pyGPs/GraphExtensions/nodeKernels.py:docstring of pyGPs.GraphExtensions.nodeKernels.regLapKernelrh-hJh/}r(h1]h2]h3]h4]h6]uh8Kh9hh!]rhBXRegularized Laplacian Kernelr r }r (h&jh'jubaubj)r }r (h&Uh'jh(Nh-jh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]r(j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'j h!]r(j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX Parametersrr}r(h&Uh'jubah-j$ubj%)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rj*)r}r (h&Uh/}r!(h1]h2]h3]h4]h6]uh'jh!]r"(j/)r#}r$(h&Uh/}r%(h1]h2]h3]h4]h6]uh'jh!]r&hF)r'}r((h&Uh/}r)(h1]h2]h3]h4]h6]uh'j#h!]r*(j8)r+}r,(h&XAh/}r-(h1]h2]h3]h4]h6]uh'j'h!]r.hBXAr/}r0(h&Uh'j+ubah-j?ubhBX -- r1r2}r3(h&Uh'j'ubhBXadjacency matrixr4r5}r6(h&Xadjacency matrixh'j'ubeh-hJubah-jGubj/)r7}r8(h&Uh/}r9(h1]h2]h3]h4]h6]uh'jh!]r:hF)r;}r<(h&Uh/}r=(h1]h2]h3]h4]h6]uh'j7h!]r>(j8)r?}r@(h&Xsigmah/}rA(h1]h2]h3]h4]h6]uh'j;h!]rBhBXsigmarCrD}rE(h&Uh'j?ubah-j?ubhBX -- rFrG}rH(h&Uh'j;ubhBXhyperparameter sigmarIrJ}rK(h&Xhyperparameter sigmah'j;ubeh-hJubah-jGubeh-j^ubah-j_ubeh-j`ubj)rL}rM(h&Uh/}rN(h1]h2]h3]h4]h6]uh'j h!]rO(j)rP}rQ(h&Uh/}rR(h1]h2]h3]h4]h6]uh'jLh!]rShBXReturnsrTrU}rV(h&Uh'jPubah-j$ubj%)rW}rX(h&Uh/}rY(h1]h2]h3]h4]h6]uh'jLh!]rZhF)r[}r\(h&Uh/}r](h1]h2]h3]h4]h6]uh'jWh!]r^hBX kernel matrixr_r`}ra(h&X kernel matrixh'j[ubah-hJubah-j_ubeh-j`ubeubeubeubh)rb}rc(h&Uh'hh(Nh-hh/}rd(h4]h3]h1]h2]h6]Uentries]re(hX8rwKernel() (in module pyGPs.GraphExtensions.nodeKernels)hUtrfauh8Nh9hh!]ubh)rg}rh(h&Uh'hh(Nh-hh/}ri(hhXpyh4]h3]h1]h2]h6]hXfunctionrjhjjuh8Nh9hh!]rk(h)rl}rm(h&XrwKernel(A, p=1, a=2)h'jgh(hh-hh/}rn(h4]rohahh)X!pyGPs.GraphExtensions.nodeKernelsrprq}rrbh3]h1]h2]h6]rshahXrwKernelrthUhuh8Nh9hh!]ru(h)rv}rw(h&X"pyGPs.GraphExtensions.nodeKernels.h'jlh(hh-hh/}rx(h1]h2]h3]h4]h6]uh8Nh9hh!]ryhBX"pyGPs.GraphExtensions.nodeKernels.rzr{}r|(h&Uh'jvubaubh)r}}r~(h&jth'jlh(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]rhBXrwKernelrr}r(h&Uh'j}ubaubh)r}r(h&Uh'jlh(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]r(h)r}r(h&XAh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXAr}r(h&Uh'jubah-hubh)r}r(h&Xp=1h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXp=1rr}r(h&Uh'jubah-hubh)r}r(h&Xa=2h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXa=2rr}r(h&Uh'jubah-hubeubh)r}r(h&Uh'jlh(Nh-hh/}r(Uexprhh4]h3]h1]h2]h6]uh8Nh9hh!]rh)r}r(h&Uh/}r(UreftypehUrefdochU refdomainhh4]h3]U refexplicith1]h2]h6]U reftargetX*_modules/pyGPs/GraphExtensions/nodeKernelsUrefidjtuh'jh!]rh)r}r(h&Uh/}r(h1]h2]rhah3]h4]h6]uh'jh!]rhBX[source]rr}r(h&Uh'jubah-hubah-hubaubeubh)r}r(h&Uh'jgh(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]r(hF)r}r(h&X"p-step Random Walk Kernel with a>1rh'jh(Xx/Users/hs/Program/git/pyGPs/pyGPs/GraphExtensions/nodeKernels.py:docstring of pyGPs.GraphExtensions.nodeKernels.rwKernelrh-hJh/}r(h1]h2]h3]h4]h6]uh8Kh9hh!]rhBX"p-step Random Walk Kernel with a>1rr}r(h&jh'jubaubhF)r}r(h&X3K = (aI-L)^p, p>1 and L is the normalized Laplacianrh'jh(jh-hJh/}r(h1]h2]h3]h4]h6]uh8Kh9hh!]rhBX3K = (aI-L)^p, p>1 and L is the normalized Laplacianrr}r(h&jh'jubaubj)r}r(h&Uh'jh(Nh-jh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]r(j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX Parametersrr}r(h&Uh'jubah-j$ubj%)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rj*)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j/)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhF)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j8)r}r(h&XAh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXAr}r(h&Uh'jubah-j?ubhBX -- rr}r(h&Uh'jubhBXadjacency matrixrr}r(h&Xadjacency matrixrh'jubeh-hJubah-jGubj/)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhF)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j8)r}r(h&Xph/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXpr}r(h&Uh'jubah-j?ubhBX -- rr}r(h&Uh'jubhBXstep parameterrr}r(h&Xstep parameterrh'jubeh-hJubah-jGubj/)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhF)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r (j8)r }r (h&Xah/}r (h1]h2]h3]h4]h6]uh'jh!]r hBXar}r(h&Uh'j ubah-j?ubhBX -- rr}r(h&Uh'jubhBXhyperparameter arr}r(h&Xhyperparameter arh'jubeh-hJubah-jGubeh-j^ubah-j_ubeh-j`ubj)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXReturnsrr }r!(h&Uh'jubah-j$ubj%)r"}r#(h&Uh/}r$(h1]h2]h3]h4]h6]uh'jh!]r%hF)r&}r'(h&Uh/}r((h1]h2]h3]h4]h6]uh'j"h!]r)hBX kernel matrixr*r+}r,(h&X kernel matrixr-h'j&ubah-hJubah-j_ubeh-j`ubeubeubeubeubh#)r.}r/(h&Uh'h$h(h+h-h.h/}r0(h1]h2]h3]h4]r1(X)module-pyGPs.GraphExtensions.graphKernelsr2h eh6]r3h auh8Kh9hh!]r4(h;)r5}r6(h&X Graph Kernelsr7h'j.h(h+h-h?h/}r8(h1]h2]h3]h4]h6]uh8Kh9hh!]r9hBX Graph Kernelsr:r;}r<(h&j7h'j5ubaubh)r=}r>(h&Uh'j.h(hh-hh/}r?(h4]h3]h1]h2]h6]Uentries]r@(hX+pyGPs.GraphExtensions.graphKernels (module)X)module-pyGPs.GraphExtensions.graphKernelsUtrAauh8Kh9hh!]ubh)rB}rC(h&Uh'j.h(Nh-hh/}rD(h4]h3]h1]h2]h6]Uentries]rE(hXBpropagationKernel() (in module pyGPs.GraphExtensions.graphKernels)hUtrFauh8Nh9hh!]ubh)rG}rH(h&Uh'j.h(Nh-hh/}rI(hhXpyh4]h3]h1]h2]h6]hXfunctionrJhjJuh8Nh9hh!]rK(h)rL}rM(h&X`propagationKernel(A, l, gr_id, h_max, w, p, ktype=None, SUM=True, VIS=False, showEachStep=False)h'jGh(hh-hh/}rN(h4]rOhahh)X"pyGPs.GraphExtensions.graphKernelsrPrQ}rRbh3]h1]h2]h6]rShahXpropagationKernelrThUhuh8Nh9hh!]rU(h)rV}rW(h&X#pyGPs.GraphExtensions.graphKernels.h'jLh(hh-hh/}rX(h1]h2]h3]h4]h6]uh8Nh9hh!]rYhBX#pyGPs.GraphExtensions.graphKernels.rZr[}r\(h&Uh'jVubaubh)r]}r^(h&jTh'jLh(hh-hh/}r_(h1]h2]h3]h4]h6]uh8Nh9hh!]r`hBXpropagationKernelrarb}rc(h&Uh'j]ubaubh)rd}re(h&Uh'jLh(hh-hh/}rf(h1]h2]h3]h4]h6]uh8Nh9hh!]rg(h)rh}ri(h&XAh/}rj(h1]h2]h3]h4]h6]uh'jdh!]rkhBXArl}rm(h&Uh'jhubah-hubh)rn}ro(h&Xlh/}rp(h1]h2]h3]h4]h6]uh'jdh!]rqhBXlrr}rs(h&Uh'jnubah-hubh)rt}ru(h&Xgr_idh/}rv(h1]h2]h3]h4]h6]uh'jdh!]rwhBXgr_idrxry}rz(h&Uh'jtubah-hubh)r{}r|(h&Xh_maxh/}r}(h1]h2]h3]h4]h6]uh'jdh!]r~hBXh_maxrr}r(h&Uh'j{ubah-hubh)r}r(h&Xwh/}r(h1]h2]h3]h4]h6]uh'jdh!]rhBXwr}r(h&Uh'jubah-hubh)r}r(h&Xph/}r(h1]h2]h3]h4]h6]uh'jdh!]rhBXpr}r(h&Uh'jubah-hubh)r}r(h&X ktype=Noneh/}r(h1]h2]h3]h4]h6]uh'jdh!]rhBX ktype=Nonerr}r(h&Uh'jubah-hubh)r}r(h&XSUM=Trueh/}r(h1]h2]h3]h4]h6]uh'jdh!]rhBXSUM=Truerr}r(h&Uh'jubah-hubh)r}r(h&X VIS=Falseh/}r(h1]h2]h3]h4]h6]uh'jdh!]rhBX VIS=Falserr}r(h&Uh'jubah-hubh)r}r(h&XshowEachStep=Falseh/}r(h1]h2]h3]h4]h6]uh'jdh!]rhBXshowEachStep=Falserr}r(h&Uh'jubah-hubeubh)r}r(h&Uh'jLh(Nh-hh/}r(Uexprhh4]h3]h1]h2]h6]uh8Nh9hh!]rh)r}r(h&Uh/}r(UreftypehUrefdochU refdomainhh4]h3]U refexplicith1]h2]h6]U reftargetX+_modules/pyGPs/GraphExtensions/graphKernelsUrefidjTuh'jh!]rh)r}r(h&Uh/}r(h1]h2]rhah3]h4]h6]uh'jh!]rhBX[source]rr}r(h&Uh'jubah-hubah-hubaubeubh)r}r(h&Uh'jGh(hh-hh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]r(hF)r}r(h&XPropagation kernel for graphs as described in: Neumann, M., Patricia, N., Garnett, R., Kersting, K.: Efficient Graph Kernels by Randomization. In: P.A. Flach, T.D. Bie, N. Cristianini (eds.) ECML/PKDD, Notes in Computer Science, vol. 7523, pp. 378-393. Springer (2012).rh'jh(X/Users/hs/Program/git/pyGPs/pyGPs/GraphExtensions/graphKernels.py:docstring of pyGPs.GraphExtensions.graphKernels.propagationKernelrh-hJh/}r(h1]h2]h3]h4]h6]uh8Kh9hh!]rhBXPropagation kernel for graphs as described in: Neumann, M., Patricia, N., Garnett, R., Kersting, K.: Efficient Graph Kernels by Randomization. In: P.A. Flach, T.D. Bie, N. Cristianini (eds.) ECML/PKDD, Notes in Computer Science, vol. 7523, pp. 378-393. Springer (2012).rr}r(h&jh'jubaubj)r}r(h&Uh'jh(Nh-jh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]r(j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX Parametersrr}r(h&Uh'jubah-j$ubj%)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rj*)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j/)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhF)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j8)r}r(h&XAh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXAr}r(h&Uh'jubah-j?ubhBX -- rr}r(h&Uh'jubhBX(adjacency matrix (num_nodes x num_nodes)rr}r(h&X(adjacency matrix (num_nodes x num_nodes)rh'jubeh-hJubah-jGubj/)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhF)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j8)r}r(h&Xlh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXlr}r(h&Uh'jubah-j?ubhBX -- rr}r(h&Uh'jubhBXlabel array (num_nodes x 1); values [1,...,k] or -1 for unlabeled nodes OR label array (num_nodes x num_labels); values [0,1], unlabeled nodes have only 0 entriesrr}r(h&Xlabel array (num_nodes x 1); values [1,...,k] or -1 for unlabeled nodes OR label array (num_nodes x num_labels); values [0,1], unlabeled nodes have only 0 entriesrh'jubeh-hJubah-jGubj/)r}r (h&Uh/}r (h1]h2]h3]h4]h6]uh'jh!]r hF)r }r (h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]r(j8)r}r(h&Xgr_idh/}r(h1]h2]h3]h4]h6]uh'j h!]rhBXgr_idrr}r(h&Uh'jubah-j?ubhBX -- rr}r(h&Uh'j ubhBX6graph indicator array (num_nodes x 1); values [0,..,n]rr}r(h&X6graph indicator array (num_nodes x 1); values [0,..,n]rh'j ubeh-hJubah-jGubj/)r}r(h&Uh/}r (h1]h2]h3]h4]h6]uh'jh!]r!hF)r"}r#(h&Uh/}r$(h1]h2]h3]h4]h6]uh'jh!]r%(j8)r&}r'(h&Xh_maxh/}r((h1]h2]h3]h4]h6]uh'j"h!]r)hBXh_maxr*r+}r,(h&Uh'j&ubah-j?ubhBX -- r-r.}r/(h&Uh'j"ubhBXnumber of iterationsr0r1}r2(h&Xnumber of iterationsr3h'j"ubeh-hJubah-jGubj/)r4}r5(h&Uh/}r6(h1]h2]h3]h4]h6]uh'jh!]r7hF)r8}r9(h&Uh/}r:(h1]h2]h3]h4]h6]uh'j4h!]r;(j8)r<}r=(h&Xwh/}r>(h1]h2]h3]h4]h6]uh'j8h!]r?hBXwr@}rA(h&Uh'j<ubah-j?ubhBX -- rBrC}rD(h&Uh'j8ubhBXbin widths parameterrErF}rG(h&Xbin widths parameterrHh'j8ubeh-hJubah-jGubj/)rI}rJ(h&Uh/}rK(h1]h2]h3]h4]h6]uh'jh!]rLhF)rM}rN(h&Uh/}rO(h1]h2]h3]h4]h6]uh'jIh!]rP(j8)rQ}rR(h&Xph/}rS(h1]h2]h3]h4]h6]uh'jMh!]rThBXprU}rV(h&Uh'jQubah-j?ubhBX -- rWrX}rY(h&Uh'jMubhBX(distance ('tv', 'hellinger', 'L1', 'L2')rZr[}r\(h&X(distance ('tv', 'hellinger', 'L1', 'L2')r]h'jMubeh-hJubah-jGubj/)r^}r_(h&Uh/}r`(h1]h2]h3]h4]h6]uh'jh!]rahF)rb}rc(h&Uh/}rd(h1]h2]h3]h4]h6]uh'j^h!]re(j8)rf}rg(h&Xktypeh/}rh(h1]h2]h3]h4]h6]uh'jbh!]rihBXktyperjrk}rl(h&Uh'jfubah-j?ubhBX -- rmrn}ro(h&Uh'jbubhBXftype of propagation kernel ['diffusion', 'label_propagation', 'label_spreading', 'belief_propagation']rprq}rr(h&Xftype of propagation kernel ['diffusion', 'label_propagation', 'label_spreading', 'belief_propagation']rsh'jbubeh-hJubah-jGubeh-j^ubah-j_ubeh-j`ubj)rt}ru(h&Uh/}rv(h1]h2]h3]h4]h6]uh'jh!]rw(j)rx}ry(h&Uh/}rz(h1]h2]h3]h4]h6]uh'jth!]r{hBXReturnsr|r}}r~(h&Uh'jxubah-j$ubj%)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jth!]rhF)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX kernel 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j.hjLhh$hhuUsubstitution_namesr}rh-h9h/}r(h1]h4]h3]Usourceh+h2]h6]uU footnotesr]rUrefidsr}rub.pyGPs-1.2/doc/build/html/.doctrees/GraphKernel.doctree000066400000000000000000000400451236346020200226660ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(Xstandard gp classificationqNX load dataqNX graph kernelsqNX1X2X!compute propagation kernel matrixq NuUsubstitution_defsq }q Uparse_messagesq ]q Ucurrent_sourceqNU decorationqNUautofootnote_startqKUnameidsq}q(hUstandard-gp-classificationqhU load-dataqhU graph-kernelsqX1Uid3qX2Uid4qh U!compute-propagation-kernel-matrixquUchildrenq]qcdocutils.nodes section q)q}q(U rawsourceqUUparentqhUsourceq cdocutils.nodes reprunicode q!X6/Users/hs/Program/git/pyGPs/doc/source/GraphKernel.rstq"q#}q$bUtagnameq%Usectionq&U attributesq'}q((Udupnamesq)]Uclassesq*]Ubackrefsq+]Uidsq,]q-haUnamesq.]q/hauUlineq0KUdocumentq1hh]q2(cdocutils.nodes title q3)q4}q5(hX Graph Kernelsq6hhh h#h%Utitleq7h'}q8(h)]h*]h+]h,]h.]uh0Kh1hh]q9cdocutils.nodes Text q:X Graph Kernelsq;q<}q=(hh6hh4ubaubcdocutils.nodes paragraph q>)q?}q@(hX[The code shown in this tutorial can be executed by running *pyGPs/Demo/demo_GraphKernel.py*qAhhh h#h%U paragraphqBh'}qC(h)]h*]h+]h,]h.]uh0Kh1hh]qD(h:X;The code shown in this tutorial can be executed by running qEqF}qG(hX;The code shown in this tutorial can be executed by running hh?ubcdocutils.nodes emphasis qH)qI}qJ(hX *pyGPs/Demo/demo_GraphKernel.py*h'}qK(h)]h*]h+]h,]h.]uhh?h]qLh:XpyGPs/Demo/demo_GraphKernel.pyqMqN}qO(hUhhIubah%UemphasisqPubeubh)qQ}qR(hUhhh h#h%h&h'}qS(h)]h*]h+]h,]qThah.]qUhauh0Kh1hh]qV(h3)qW}qX(hX Load dataqYhhQh h#h%h7h'}qZ(h)]h*]h+]h,]h.]uh0Kh1hh]q[h:X Load dataq\q]}q^(hhYhhWubaubh>)q_}q`(hXMUTAG [1]_ is a data set of 188 mutagenic aromatic and heteroaromatic nitro compounds labeled according to whether or not they have a mutagenic effect on the Gram-negative bacterium Salmonella typhimurium. MUTAG is a simple but popular dataset to benchmark graph kernels. ::hhQh h#h%hBh'}qa(h)]h*]h+]h,]h.]uh0Kh1hh]qb(h:XMUTAG qcqd}qe(hXMUTAG hh_ubcdocutils.nodes footnote_reference qf)qg}qh(hX[1]_UresolvedqiKhh_h%Ufootnote_referenceqjh'}qk(h,]qlUid1qmah+]h)]h*]h.]Urefidqnhuh]qoh:X1qp}qq(hUhhgubaubh:X is a data set of 188 mutagenic aromatic and heteroaromatic nitro compounds labeled according to whether or not they have a mutagenic effect on the Gram-negative bacterium Salmonella typhimurium. MUTAG is a simple but popular dataset to benchmark graph kernels.qrqs}qt(hX is a data set of 188 mutagenic aromatic and heteroaromatic nitro compounds labeled according to whether or not they have a mutagenic effect on the Gram-negative bacterium Salmonella typhimurium. MUTAG is a simple but popular dataset to benchmark graph kernels.hh_ubeubcdocutils.nodes literal_block qu)qv}qw(hXdata = np.load('MUTAG.npz') A = csc_matrix( (data['adj_data'], data['adj_indice'], \ data['adj_indptr']), shape=data['adj_shape']) # n x n adjancy array (sparse matrix) gr_id = data['graph_ind'] # n x 1 graph id array node_label = data['responses'] # n x 1 node label array graph_label = data['labels'] # N x 1 graph label arrayhhQh h#h%U literal_blockqxh'}qy(U xml:spaceqzUpreserveq{h,]h+]h)]h*]h.]uh0K h1hh]q|h:Xdata = np.load('MUTAG.npz') A = csc_matrix( (data['adj_data'], data['adj_indice'], \ data['adj_indptr']), shape=data['adj_shape']) # n x n adjancy array (sparse matrix) gr_id = data['graph_ind'] # n x 1 graph id array node_label = data['responses'] # n x 1 node label array graph_label = data['labels'] # N x 1 graph label arrayq}q~}q(hUhhvubaubh>)q}q(hXNote that adjacency matrix for all graphs is usually too big to fit into memory when using :math:`GP`. Therefore, we generate :math:`A` using sparse matrix *csc_matrix* provided by *scipy*.hhQh h#h%hBh'}q(h)]h*]h+]h,]h.]uh0Kh1hh]q(h:X[Note that adjacency matrix for all graphs is usually too big to fit into memory when using qq}q(hX[Note that adjacency matrix for all graphs is usually too big to fit into memory when using hhubcsphinx.ext.mathbase math q)q}q(hUh'}q(UlatexXGPh,]h+]h)]h*]h.]uhhh]h%Umathqubh:X. Therefore, we generate qq}q(hX. Therefore, we generate hhubh)q}q(hUh'}q(UlatexXAh,]h+]h)]h*]h.]uhhh]h%hubh:X using sparse matrix qq}q(hX using sparse matrix hhubhH)q}q(hX *csc_matrix*h'}q(h)]h*]h+]h,]h.]uhhh]qh:X csc_matrixqq}q(hUhhubah%hPubh:X provided by qq}q(hX provided by hhubhH)q}q(hX*scipy*h'}q(h)]h*]h+]h,]h.]uhhh]qh:Xscipyqq}q(hUhhubah%hPubh:X.q}q(hX.hhubeubeubh)q}q(hUhhh h#h%h&h'}q(h)]h*]h+]h,]qhah.]qh auh0Kh1hh]q(h3)q}q(hX!Compute propagation kernel matrixqhhh h#h%h7h'}q(h)]h*]h+]h,]h.]uh0Kh1hh]qh:X!Compute propagation kernel matrixqq}q(hhhhubaubh>)q}q(hXPropagation kernels [2]_ are recently introduced fast and flexible graph kernels. Generate prapagation kernel based on given graph data. ::hhh h#h%hBh'}q(h)]h*]h+]h,]h.]uh0Kh1hh]q(h:XPropagation kernels qq}q(hXPropagation kernels hhubhf)q}q(hX[2]_hiKhhh%hjh'}q(h,]qUid2qah+]h)]h*]h.]hnhuh]qh:X2q}q(hUhhubaubh:Xp are recently introduced fast and flexible graph kernels. Generate prapagation kernel based on given graph data.qŅq}q(hXp are recently introduced fast and flexible graph kernels. Generate prapagation kernel based on given graph data.hhubeubhu)q}q(hXnum_Iteration = 10 w = 1e-4 dist = 'tv' # possible values: 'tv', 'hellinger' K = graphKernels.propagationKernel(A, node_label, gr_id, num_Iteration, w, dist, 'label_diffusion')hhh h#h%hxh'}q(hzh{h,]h+]h)]h*]h.]uh0Kh1hh]qh:Xnum_Iteration = 10 w = 1e-4 dist = 'tv' # possible values: 'tv', 'hellinger' K = graphKernels.propagationKernel(A, node_label, gr_id, num_Iteration, w, dist, 'label_diffusion')q̅q}q(hUhhubaubh>)q}q(hX]Adjacency matrix :math:`A` can either be in format of standard numpy matrix or sparse matrix.qhhh h#h%hBh'}q(h)]h*]h+]h,]h.]uh0K!h1hh]q(h:XAdjacency matrix qԅq}q(hXAdjacency matrix hhubh)q}q(hUh'}q(UlatexXAh,]h+]h)]h*]h.]uhhh]h%hubh:XC can either be in format of standard numpy matrix or sparse matrix.qڅq}q(hXC can either be in format of standard numpy matrix or sparse matrix.hhubeubeubh)q}q(hUhhh h#h%h&h'}q(h)]h*]h+]h,]qhah.]qhauh0K%h1hh]q(h3)q}q(hXStandard GP Classificationqhhh h#h%h7h'}q(h)]h*]h+]h,]h.]uh0K%h1hh]qh:XStandard GP Classificationq腁q}q(hhhhubaubh>)q}q(hX:math:`K` is a big square matrix with all rows and columns of the number of data points. By specifying the indice of training data and test data, we will form two matrix M1 and M2 with the exact format which *pyGPs.Core.cov.Pre* needed. ::hhh h#h%hBh'}q(h)]h*]h+]h,]h.]uh0K&h1hh]q(h)q}q(hUh'}q(UlatexXKh,]h+]h)]h*]h.]uhhh]h%hubh:X is a big square matrix with all rows and columns of the number of data points. By specifying the indice of training data and test data, we will form two matrix M1 and M2 with the exact format which qq}q(hX is a big square matrix with all rows and columns of the number of data points. By specifying the indice of training data and test data, we will form two matrix M1 and M2 with the exact format which hhubhH)q}q(hX*pyGPs.Core.cov.Pre*h'}q(h)]h*]h+]h,]h.]uhhh]qh:XpyGPs.Core.cov.Preqq}q(hUhhubah%hPubh:X needed.qq}q(hX needed.hhubeubhu)q}r(hX^M1,M2 = graphUtil.formKernelMatrix(Matrix, indice_train, indice_test) k = pyGPs.cov.Pre(M1,M2)hhh h#h%hxh'}r(hzh{h,]h+]h)]h*]h.]uh0K)h1hh]rh:X^M1,M2 = graphUtil.formKernelMatrix(Matrix, indice_train, indice_test) k = pyGPs.cov.Pre(M1,M2)rr}r(hUhhubaubcdocutils.nodes definition_list r)r}r(hUhhh Nh%Udefinition_listr h'}r (h)]h*]h+]h,]h.]uh0Nh1hh]r (cdocutils.nodes definition_list_item r )r }r(hXM1 is a matrix with shape **number of training points plus 1** by **number of test points** - cross covariances matrix (train by test) - last row is self covariances (diagonal of test by test)hjh h#h%Udefinition_list_itemrh'}r(h)]h*]h+]h,]h.]uh0K-h]r(cdocutils.nodes term r)r}r(hX[M1 is a matrix with shape **number of training points plus 1** by **number of test points**rhj h h#h%Utermrh'}r(h)]h*]h+]h,]h.]uh0K-h]r(h:XM1 is a matrix with shape rr}r(hXM1 is a matrix with shape hjubcdocutils.nodes strong r)r}r(hX$**number of training points plus 1**h'}r(h)]h*]h+]h,]h.]uhjh]r h:X number of training points plus 1r!r"}r#(hUhjubah%Ustrongr$ubh:X by r%r&}r'(hX by hjubj)r(}r)(hX**number of test points**h'}r*(h)]h*]h+]h,]h.]uhjh]r+h:Xnumber of test pointsr,r-}r.(hUhj(ubah%j$ubeubcdocutils.nodes definition r/)r0}r1(hUh'}r2(h)]h*]h+]h,]h.]uhj h]r3cdocutils.nodes bullet_list r4)r5}r6(hUh'}r7(Ubulletr8X-h,]h+]h)]h*]h.]uhj0h]r9(cdocutils.nodes list_item r:)r;}r<(hX(cross covariances matrix (train by test)r=h'}r>(h)]h*]h+]h,]h.]uhj5h]r?h>)r@}rA(hj=hj;h h#h%hBh'}rB(h)]h*]h+]h,]h.]uh0K-h]rCh:X(cross covariances matrix (train by test)rDrE}rF(hj=hj@ubaubah%U list_itemrGubj:)rH}rI(hX7last row is self covariances (diagonal of test by test)rJh'}rK(h)]h*]h+]h,]h.]uhj5h]rLh>)rM}rN(hjJhjHh h#h%hBh'}rO(h)]h*]h+]h,]h.]uh0K.h]rPh:X7last row is self covariances (diagonal of test by test)rQrR}rS(hjJhjMubaubah%jGubeh%U bullet_listrTubah%U definitionrUubeubj )rV}rW(hX~M2 is a square matrix with **number of training points** for each dimension - training set covariance matrix (train by train) hjh h#h%jh'}rX(h)]h*]h+]h,]h.]uh0K0h1hh]rY(j)rZ}r[(hXKM2 is a square matrix with **number of training points** for each dimensionr\hjVh h#h%jh'}r](h)]h*]h+]h,]h.]uh0K0h]r^(h:XM2 is a square matrix with r_r`}ra(hXM2 is a square matrix with hjZubj)rb}rc(hX**number of training points**h'}rd(h)]h*]h+]h,]h.]uhjZh]reh:Xnumber of training pointsrfrg}rh(hUhjbubah%j$ubh:X for each dimensionrirj}rk(hX for each dimensionhjZubeubj/)rl}rm(hUh'}rn(h)]h*]h+]h,]h.]uhjVh]roj4)rp}rq(hUh'}rr(j8X-h,]h+]h)]h*]h.]uhjlh]rsj:)rt}ru(hX0training set covariance matrix (train by train) h'}rv(h)]h*]h+]h,]h.]uhjph]rwh>)rx}ry(hX/training set covariance matrix (train by train)rzhjth h#h%hBh'}r{(h)]h*]h+]h,]h.]uh0K0h]r|h:X/training set covariance matrix (train by train)r}r~}r(hjzhjxubaubah%jGubah%jTubah%jUubeubeubh>)r}r(hX<The following is the standard way to do GP classification ::rhhh h#h%hBh'}r(h)]h*]h+]h,]h.]uh0K2h1hh]rh:X9The following is the standard way to do GP classificationrr}r(hX9The following is the standard way to do GP classificationhjubaubhu)r}r(hX^model = pyGPs.GPC() model.setPrior(kernel=k) model.fit(x_train, y_train) model.predict(x_test)hhh h#h%hxh'}r(hzh{h,]h+]h)]h*]h.]uh0K4h1hh]rh:X^model = pyGPs.GPC() model.setPrior(kernel=k) model.fit(x_train, y_train) model.predict(x_test)rr}r(hUhjubaubcdocutils.nodes footnote r)r}r(hXDebnath, A., Lopez de Compadre, R., Debnath, G., Shusterman, A., and Hansch, C.. Structure-activity relationship of mutagenic aromatic and heteroaromatic nitro compounds. Correlation with molecular orbital energies and hydrophobicity. J. Med. Chem., 34:786–797, 1991. hiKhhh h#h%Ufootnoterh'}r(h)]h*]h+]rhmah,]rhah.]rX1auh0KEh1hh]r(cdocutils.nodes label r)r}r(hUh'}r(h)]h*]h+]h,]h.]uhjh]rh:X1r}r(hUhjubah%Ulabelrubh>)r}r(hX Debnath, A., Lopez de Compadre, R., Debnath, G., Shusterman, A., and Hansch, C.. Structure-activity relationship of mutagenic aromatic and heteroaromatic nitro compounds. Correlation with molecular orbital energies and hydrophobicity. J. Med. Chem., 34:786–797, 1991.rhjh h#h%hBh'}r(h)]h*]h+]h,]h.]uh0KEh]rh:X Debnath, A., Lopez de Compadre, R., Debnath, G., Shusterman, A., and Hansch, C.. Structure-activity relationship of mutagenic aromatic and heteroaromatic nitro compounds. Correlation with molecular orbital energies and hydrophobicity. J. Med. Chem., 34:786–797, 1991.rr}r(hjhjubaubeubj)r}r(hXNeumann, M., Patricia, N., Garnett, R., Kersting, K.: Efficient Graph Kernels by Randomization. In: P.A. Flach, T.D. Bie, N. Cristianini (eds.) ECML/PKDD, Notes in Computer Science, vol. 7523, pp. 378-393. Springer (2012). hiKhhh h#h%jh'}r(h)]h*]h+]rhah,]rhah.]rX2auh0KGh1hh]r(j)r}r(hUh'}r(h)]h*]h+]h,]h.]uhjh]rh:X2r}r(hUhjubah%jubh>)r}r(hXNeumann, M., Patricia, N., Garnett, R., Kersting, K.: Efficient Graph Kernels by Randomization. In: P.A. Flach, T.D. Bie, N. Cristianini (eds.) ECML/PKDD, Notes in Computer Science, vol. 7523, pp. 378-393. Springer (2012).rhjh h#h%hBh'}r(h)]h*]h+]h,]h.]uh0KGh]rh:XNeumann, M., Patricia, N., Garnett, R., Kersting, K.: Efficient Graph Kernels by Randomization. In: P.A. Flach, T.D. Bie, N. Cristianini (eds.) ECML/PKDD, Notes in Computer Science, vol. 7523, pp. 378-393. 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}qUparse_messagesq]qcdocutils.nodes system_message q)q}q(U rawsourceqUUparentqcdocutils.nodes section q)q}q(hUhhUsourceqcdocutils.nodes reprunicode qX2/Users/hs/Program/git/pyGPs/doc/source/Install.rstqq}qbUtagnameqUsectionqU attributesq }q!(Udupnamesq"]Uclassesq#]Ubackrefsq$]Uidsq%]q&U installationq'aUnamesq(]q)h auUlineq*KUdocumentq+hUchildrenq,]q-(cdocutils.nodes title q.)q/}q0(hX Installationq1hhhhhUtitleq2h }q3(h"]h#]h$]h%]h(]uh*Kh+hh,]q4cdocutils.nodes Text q5X Installationq6q7}q8(hh1hh/ubaubcdocutils.nodes enumerated_list q9)q:}q;(hUhhhhhUenumerated_listqU.h%]h$]h"]Uprefixq?Uh#]h(]Uenumtypeq@UarabicqAuh*Kh+hh,]qBcdocutils.nodes list_item qC)qD}qE(hXPFirst, download_ the archive from github and extract it to any local directory. hh:hhhU list_itemqFh }qG(h"]h#]h$]h%]h(]uh*Nh+hh,]qHcdocutils.nodes paragraph qI)qJ}qK(hXOFirst, download_ the archive from github and extract it to any local directory.hhDhhhU paragraphqLh }qM(h"]h#]h$]h%]h(]uh*Kh,]qN(h5XFirst, qOqP}qQ(hXFirst, hhJubcdocutils.nodes reference qR)qS}qT(hX download_UresolvedqUKhhJhU referenceqVh }qW(UnameXdownloadqXUrefuriqYX5https://github.com/marionmari/pyGPs/releases/tag/v1.1qZh%]h$]h"]h#]h(]uh,]q[h5Xdownloadq\q]}q^(hUhhSubaubh5X? the archive from github and extract it to any local directory.q_q`}qa(hX? the archive from github and extract it to any local directory.hhJubeubaubaubcdocutils.nodes target qb)qc}qd(hXC.. _download: https://github.com/marionmari/pyGPs/releases/tag/v1.1U referencedqeKhhhhhUtargetqfh }qg(hYhZh%]qhUdownloadqiah$]h"]h#]h(]qjh auh*Kh+hh,]ubh9)qk}ql(hUhhhhhhU.UstartqnKh%]h$]h"]h?Uh#]h(]h@hAuh*Kh+hh,]qo(hC)qp}qq(hXYou can either add the local directory to your PYTHONPATH :: export PYTHONPATH=$PYTHONPATH:/path/to/local/directory/../parent_folder_of_pyGPs hhkhhhhFh }qr(h"]h#]h$]h%]h(]uh*Nh+hh,]qs(hI)qt}qu(hX<You can either add the local directory to your PYTHONPATH ::hhphhhhLh }qv(h"]h#]h$]h%]h(]uh*Kh,]qwh5X9You can either add the local directory to your PYTHONPATHqxqy}qz(hX9You can either add the local directory to your PYTHONPATHhhtubaubcdocutils.nodes literal_block q{)q|}q}(hXPexport PYTHONPATH=$PYTHONPATH:/path/to/local/directory/../parent_folder_of_pyGPshhphU literal_blockq~h }q(U xml:spaceqUpreserveqh%]h$]h"]h#]h(]uh*K h,]qh5XPexport PYTHONPATH=$PYTHONPATH:/path/to/local/directory/../parent_folder_of_pyGPsqq}q(hUhh|ubaubeubhC)q}q(hXKOr install the package using setup.py:: sudo python setup.py install hhkhhhhFh }q(h"]h#]h$]h%]h(]uh*Nh+hh,]q(hI)q}q(hX'Or install the package using setup.py::hhhhhhLh }q(h"]h#]h$]h%]h(]uh*K h,]qh5X&Or install the package using setup.py:qq}q(hX&Or install the package using setup.py:hhubaubh{)q}q(hXsudo python setup.py installhhhh~h }q(hhh%]h$]h"]h#]h(]uh*Kh,]qh5Xsudo python setup.py installqq}q(hUhhubaubeubeubh)q}q(hUhhhhhhh }q(h"]h#]h$]h%]qU requirementsqah(]qhauh*Kh+hh,]q(h.)q}q(hX Requirementsqhhhhhh2h }q(h"]h#]h$]h%]h(]uh*Kh+hh,]qh5X Requirementsqq}q(hhhhubaubcdocutils.nodes bullet_list q)q}q(hUhhhhhU bullet_listqh }q(UbulletqX*h%]h$]h"]h#]h(]uh*Kh+hh,]q(hC)q}q(hX`python 2.6 or 2.7`_qhhhhhhFh }q(h"]h#]h$]h%]h(]uh*Nh+hh,]qhI)q}q(hhhhhhhhLh }q(h"]h#]h$]h%]h(]uh*Kh,]qhR)q}q(hhhUKhhhhVh }q(UnameXpython 2.6 or 2.7hYXhttp://www.python.org/qh%]h$]h"]h#]h(]uh,]qh5Xpython 2.6 or 2.7qq}q(hUhhubaubaubaubhC)q}q(hX``scipy`_, `numpy`_, and `matplotlib`_: open-source packages for scientific computing in Python. hhhhhhFh }q(h"]h#]h$]h%]h(]uh*Nh+hh,]qhI)q}q(hX_`scipy`_, `numpy`_, and `matplotlib`_: open-source packages for scientific computing in Python.qhhhhhhLh }q(h"]h#]h$]h%]h(]uh*Kh,]q(hR)q}q(hX`scipy`_hUKhhhhVh }q(UnameXscipyqhYXhttp://www.scipy.org/qh%]h$]h"]h#]h(]uh,]qh5Xscipyq΅q}q(hUhhubaubh5X, qхq}q(hX, hhubhR)q}q(hX`numpy`_hUKhhhhVh }q(UnameXnumpyqhYXhttp://www.numpy.org/qh%]h$]h"]h#]h(]uh,]qh5Xnumpyqڅq}q(hUhhubaubh5X, and q݅q}q(hX, and hhubhR)q}q(hX `matplotlib`_hUKhhhhVh }q(UnameX matplotlibqhYXhttp://matplotlib.org/qh%]h$]h"]h#]h(]uh,]qh5X matplotlibq慁q}q(hUhhubaubh5X:: 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footnotesr]rUrefidsr}rub.pyGPs-1.2/doc/build/html/.doctrees/Kernels.doctree000066400000000000000000004024231236346020200220710ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(XpyGPs.Core.cov.RQqX"pyGPs.Core.cov.Kernel.getCovMatrixqXpyGPs.Core.mean.SumOfMeanqX!pyGPs.Core.mean.Mean.getDerMatrixq Xcomposite kernels & meamsq NXpyGPs.Core.mean.Zeroq XpyGPs.Core.mean.Meanq XpyGPs.Core.cov.Periodicq XpyGPs.Core.cov.PolyqXpyGPs.Core.cov.ProductOfKernelqXkernels & meansqNXpyGPs.Core.cov.MaternqXpyGPs.Core.cov.Kernel.fitcqXpyGPs.Core.cov.FITCOfKernelqX"pyGPs.Core.cov.Kernel.getDerMatrixqXsimple kernel & meanqNXpyGPs.Core.cov.ScaleOfKernelqXpyGPs.Core.mean.PowerOfMeanqXpyGPs.Core.cov.RQardqXpyGPs.Core.cov.NoiseqXpyGPs.Core.cov.RBFardqXpyGPs.Core.cov.LinearqXpyGPs.Core.cov.PiecePolyqX$list of means and default parametersqNXpyGPs.Core.cov.RBFunitqXprecomputed kernel matrixqNXpyGPs.Core.cov.LINardq Xsome special casesq!NXpyGPs.Core.mean.Constq"XpyGPs.Core.cov.RBFq#XpyGPs.Core.cov.Kernelq$XpyGPs.Core.mean.ProductOfMeanq%XpyGPs.Core.mean.ScaleOfMeanq&XpyGPs.Core.mean.Linearq'XpyGPs.Core.cov.Preq(X&developing new kernel & mean functionsq)NX&list of kernels and default parametersq*NXpyGPs.Core.mean.Oneq+XpyGPs.Core.mean.Mean.getMeanq,XpyGPs.Core.cov.Constq-XpyGPs.Core.cov.SumOfKernelq.uUsubstitution_defsq/}q0Uparse_messagesq1]q2Ucurrent_sourceq3NU decorationq4NUautofootnote_startq5KUnameidsq6}q7(hhhhhhh h h Ucomposite-kernels-meamsq8h h h h h h hhhhhU kernels-meansq9hhhhhhhhhUsimple-kernel-meanq:hhhhhhhhhhhhhhhU$list-of-means-and-default-parametersq;hhhUprecomputed-kernel-matrixqh*U&list-of-kernels-and-default-parametersq?h+h+h,h,h-h-h.h.uUchildrenq@]qAcdocutils.nodes section qB)qC}qD(U rawsourceqEUUparentqFhUsourceqGcdocutils.nodes reprunicode qHX2/Users/hs/Program/git/pyGPs/doc/source/Kernels.rstqIqJ}qKbUtagnameqLUsectionqMU attributesqN}qO(UdupnamesqP]UclassesqQ]UbackrefsqR]UidsqS]qTh9aUnamesqU]qVhauUlineqWKUdocumentqXhh@]qY(cdocutils.nodes title qZ)q[}q\(hEXKernels & Meansq]hFhChGhJhLUtitleq^hN}q_(hP]hQ]hR]hS]hU]uhWKhXhh@]q`cdocutils.nodes Text qaXKernels & Meansqbqc}qd(hEh]hFh[ubaubhB)qe}qf(hEUhFhChGhJhLhMhN}qg(hP]hQ]hR]hS]qhh:ahU]qihauhWKhXhh@]qj(hZ)qk}ql(hEXSimple Kernel & MeanqmhFhehGhJhLh^hN}qn(hP]hQ]hR]hS]hU]uhWKhXhh@]qohaXSimple Kernel & Meanqpqq}qr(hEhmhFhkubaubcdocutils.nodes paragraph qs)qt}qu(hEX]You may already seen, we can specify a kernel function like this(same for mean fucntions): ::qvhFhehGhJhLU paragraphqwhN}qx(hP]hQ]hR]hS]hU]uhWKhXhh@]qyhaXZYou may already seen, we can specify a kernel function like this(same for mean fucntions):qzq{}q|(hEXZYou may already seen, we can specify a kernel function like this(same for mean fucntions):hFhtubaubcdocutils.nodes literal_block q})q~}q(hEX.k = pyGPs.cov.RBF( log_ell=-1., log_sigma=0. )hFhehGhJhLU literal_blockqhN}q(U xml:spaceqUpreserveqhS]hR]hP]hQ]hU]uhWKhXhh@]qhaX.k = pyGPs.cov.RBF( log_ell=-1., log_sigma=0. )qq}q(hEUhFh~ubaubhs)q}q(hEX,There are several points need to be noticed:qhFhehGhJhLhwhN}q(hP]hQ]hR]hS]hU]uhWK hXhh@]qhaX,There are several points need to be noticed:qq}q(hEhhFhubaubcdocutils.nodes enumerated_list q)q}q(hEUhFhehGhJhLUenumerated_listqhN}q(UsuffixqU.hS]hR]hP]UprefixqUhQ]hU]UenumtypeqUarabicquhWK hXhh@]q(cdocutils.nodes list_item q)q}q(hEXMost parameters are initilized in their logorithms. This is because we need to make sure they are positive during optimization. e.g. Here length scale and signal variance should always be positive. hFhhGhJhLU list_itemqhN}q(hP]hQ]hR]hS]hU]uhWNhXhh@]qhs)q}q(hEXMost parameters are initilized in their logorithms. This is because we need to make sure they are positive during optimization. e.g. Here length scale and signal variance should always be positive.qhFhhGhJhLhwhN}q(hP]hQ]hR]hS]hU]uhWK h@]qhaXMost parameters are initilized in their logorithms. This is because we need to make sure they are positive during optimization. e.g. Here length scale and signal variance should always be positive.qq}q(hEhhFhubaubaubh)q}q(hEXWMost kernel functions have a scalar in front, namely signal variance(set by log_sigma) hFhhGhJhLhhN}q(hP]hQ]hR]hS]hU]uhWNhXhh@]qhs)q}q(hEXVMost kernel functions have a scalar in front, namely signal variance(set by log_sigma)qhFhhGhJhLhwhN}q(hP]hQ]hR]hS]hU]uhWKh@]qhaXVMost kernel functions have a scalar in front, namely signal variance(set by log_sigma)qq}q(hEhhFhubaubaubh)q}q(hEXWIf you will do optimization later anyway, you can just leave parameters to be default hFhhGhJhLhhN}q(hP]hQ]hR]hS]hU]uhWNhXhh@]qhs)q}q(hEXUIf you will do optimization later anyway, you can just leave parameters to be defaultqhFhhGhJhLhwhN}q(hP]hQ]hR]hS]hU]uhWKh@]qhaXUIf you will do optimization later anyway, you can just leave parameters to be defaultqq}q(hEhhFhubaubaubeubeubhB)q}q(hEUhFhChGhJhLhMhN}q(hP]hQ]hR]hS]qh=ahU]qh!auhWKhXhh@]q(hZ)q}q(hEXSome Special CasesqhFhhGhJhLh^hN}q(hP]hQ]hR]hS]hU]uhWKhXhh@]qhaXSome Special Casesq˅q}q(hEhhFhubaubh)q}q(hEUhFhhGhJhLhhN}q(hU.hS]hR]hP]hUhQ]hU]hhuhWKhXhh@]q(h)q}q(hEXFor some kernels/means, number of hyperparameters depends on the dimension of input data. You can either enter the dimension, which use default values: :: m = pyGPs.mean.Linear( D=x.shape[1] ) or you can initialze with the exact hyperparameters, you should enter as a list, one element for each dimension :: m = pyGPs.mean.Linear( alpha_list=[0.2, 0.4, 0.3] ) All these "hyp-dim-dependent" functions are: * *pyGPs.mean.Linear* * *pyGPs.cov.RBFard* * *pyGPs.cov.LINard* * *pyGPs.cov.RQard* hFhhGNhLhhN}q(hP]hQ]hR]hS]hU]uhWNhXhh@]q(hs)q}q(hEXFor some kernels/means, number of hyperparameters depends on the dimension of input data. You can either enter the dimension, which use default values: ::hFhhGhJhLhwhN}q(hP]hQ]hR]hS]hU]uhWKh@]qhaXFor some kernels/means, number of hyperparameters depends on the dimension of input data. You can either enter the dimension, which use default values:qڅq}q(hEXFor some kernels/means, number of hyperparameters depends on the dimension of input data. You can either enter the dimension, which use default values:hFhubaubh})q}q(hEX%m = pyGPs.mean.Linear( D=x.shape[1] )hFhhLhhN}q(hhhS]hR]hP]hQ]hU]uhWKh@]qhaX%m = pyGPs.mean.Linear( D=x.shape[1] )qᅁq}q(hEUhFhubaubhs)q}q(hEXror you can initialze with the exact hyperparameters, you should enter as a list, one element for each dimension ::hFhhGhJhLhwhN}q(hP]hQ]hR]hS]hU]uhWKh@]qhaXoor you can initialze with the exact hyperparameters, you should enter as a list, one element for each dimensionq腁q}q(hEXoor you can initialze with the exact hyperparameters, you should enter as a list, one element for each dimensionhFhubaubh})q}q(hEX3m = pyGPs.mean.Linear( alpha_list=[0.2, 0.4, 0.3] )hFhhLhhN}q(hhhS]hR]hP]hQ]hU]uhWKh@]qhaX3m = pyGPs.mean.Linear( alpha_list=[0.2, 0.4, 0.3] )qq}q(hEUhFhubaubcdocutils.nodes definition_list q)q}q(hEUhN}q(hP]hQ]hR]hS]hU]uhFhh@]qcdocutils.nodes definition_list_item q)q}q(hEXAll these "hyp-dim-dependent" functions are: * *pyGPs.mean.Linear* * *pyGPs.cov.RBFard* * *pyGPs.cov.LINard* * *pyGPs.cov.RQard* hFhhGhJhLUdefinition_list_itemqhN}q(hP]hQ]hR]hS]hU]uhWK$h@]q(cdocutils.nodes term q)q}q(hEX,All these "hyp-dim-dependent" functions are:rhFhhGhJhLUtermrhN}r(hP]hQ]hR]hS]hU]uhWK$h@]rhaX,All these "hyp-dim-dependent" functions are:rr}r(hEjhFhubaubcdocutils.nodes definition r)r}r (hEUhN}r (hP]hQ]hR]hS]hU]uhFhh@]r cdocutils.nodes bullet_list r )r }r(hEUhN}r(UbulletrX*hS]hR]hP]hQ]hU]uhFjh@]r(h)r}r(hEX*pyGPs.mean.Linear*rhN}r(hP]hQ]hR]hS]hU]uhFj h@]rhs)r}r(hEjhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWK h@]rcdocutils.nodes emphasis r)r}r(hEjhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXpyGPs.mean.Linearr r!}r"(hEUhFjubahLUemphasisr#ubaubahLhubh)r$}r%(hEX*pyGPs.cov.RBFard*r&hN}r'(hP]hQ]hR]hS]hU]uhFj h@]r(hs)r)}r*(hEj&hFj$hGhJhLhwhN}r+(hP]hQ]hR]hS]hU]uhWK!h@]r,j)r-}r.(hEj&hN}r/(hP]hQ]hR]hS]hU]uhFj)h@]r0haXpyGPs.cov.RBFardr1r2}r3(hEUhFj-ubahLj#ubaubahLhubh)r4}r5(hEX*pyGPs.cov.LINard*r6hN}r7(hP]hQ]hR]hS]hU]uhFj h@]r8hs)r9}r:(hEj6hFj4hGhJhLhwhN}r;(hP]hQ]hR]hS]hU]uhWK"h@]r<j)r=}r>(hEj6hN}r?(hP]hQ]hR]hS]hU]uhFj9h@]r@haXpyGPs.cov.LINardrArB}rC(hEUhFj=ubahLj#ubaubahLhubh)rD}rE(hEX*pyGPs.cov.RQard* hN}rF(hP]hQ]hR]hS]hU]uhFj h@]rGhs)rH}rI(hEX*pyGPs.cov.RQard*rJhFjDhGhJhLhwhN}rK(hP]hQ]hR]hS]hU]uhWK#h@]rLj)rM}rN(hEjJhN}rO(hP]hQ]hR]hS]hU]uhFjHh@]rPhaXpyGPs.cov.RQardrQrR}rS(hEUhFjMubahLj#ubaubahLhubehLU bullet_listrTubahLU definitionrUubeubahLUdefinition_listrVubeubh)rW}rX(hEXFor *pyGPs.cov.RBFunit()*, its signal variance is always 1 (because of unit magnitude). Therefore this function do not have a hyperparameter of "signal variance". hFhhGhJhLhhN}rY(hP]hQ]hR]hS]hU]uhWNhXhh@]rZhs)r[}r\(hEXFor *pyGPs.cov.RBFunit()*, its signal variance is always 1 (because of unit magnitude). Therefore this function do not have a hyperparameter of "signal variance".hFjWhGhJhLhwhN}r](hP]hQ]hR]hS]hU]uhWK&h@]r^(haXFor r_r`}ra(hEXFor hFj[ubj)rb}rc(hEX*pyGPs.cov.RBFunit()*hN}rd(hP]hQ]hR]hS]hU]uhFj[h@]rehaXpyGPs.cov.RBFunit()rfrg}rh(hEUhFjbubahLj#ubhaX, its signal variance is always 1 (because of unit magnitude). Therefore this function do not have a hyperparameter of "signal variance".rirj}rk(hEX, its signal variance is always 1 (because of unit magnitude). Therefore this function do not have a hyperparameter of "signal variance".hFj[ubeubaubh)rl}rm(hEX*pyGPs.cov.Poly()* has three parameters, where hyperparameters are: * c -> inhomogeneous offset * sigma -> signal deviation however, * d -> order of polynomial will be treated as normal parameter, i.e. will not be trained hFhhGNhLhhN}rn(hP]hQ]hR]hS]hU]uhWNhXhh@]roh)rp}rq(hEUhN}rr(hP]hQ]hR]hS]hU]uhFjlh@]rs(h)rt}ru(hEX*pyGPs.cov.Poly()* has three parameters, where hyperparameters are: * c -> inhomogeneous offset * sigma -> signal deviation hFjphGhJhLhhN}rv(hP]hQ]hR]hS]hU]uhWK+h@]rw(h)rx}ry(hEXC*pyGPs.cov.Poly()* has three parameters, where hyperparameters are:hFjthGhJhLjhN}rz(hP]hQ]hR]hS]hU]uhWK+h@]r{(j)r|}r}(hEX*pyGPs.cov.Poly()*hN}r~(hP]hQ]hR]hS]hU]uhFjxh@]rhaXpyGPs.cov.Poly()rr}r(hEUhFj|ubahLj#ubhaX1 has three parameters, where hyperparameters are:rr}r(hEX1 has three parameters, where hyperparameters are:hFjxubeubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjth@]rj )r}r(hEUhN}r(jX*hS]hR]hP]hQ]hU]uhFjh@]r(h)r}r(hEXc -> inhomogeneous offsetrhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEjhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWK*h@]rhaXc -> inhomogeneous offsetrr}r(hEjhFjubaubahLhubh)r}r(hEXsigma -> signal deviation hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEXsigma -> signal deviationrhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWK+h@]rhaXsigma -> signal deviationrr}r(hEjhFjubaubahLhubehLjTubahLjUubeubh)r}r(hEXihowever, * d -> order of polynomial will be treated as normal parameter, i.e. will not be trained hFjphGhJhLhhN}r(hP]hQ]hR]hS]hU]uhWK0h@]r(h)r}r(hEXhowever,rhFjhGhJhLjhN}r(hP]hQ]hR]hS]hU]uhWK0h@]rhaXhowever,rr}r(hEjhFjubaubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rj )r}r(hEUhN}r(jX*hS]hR]hP]hQ]hU]uhFjh@]rh)r}r(hEX\d -> order of polynomial will be treated as normal parameter, i.e. will not be trained hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEXZd -> order of polynomial will be treated as normal parameter, i.e. will not be trainedrhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWK.h@]rhaXZd -> order of polynomial will be treated as normal parameter, i.e. will not be trainedrr}r(hEjhFjubaubahLhubahLjTubahLjUubeubehLjVubaubh)r}r(hEXcExplicitly set *pyGPs.cov.Noise* is not necessary, because noise are already added in likelihood. hFhhGhJhLhhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhs)r}r(hEXaExplicitly set *pyGPs.cov.Noise* is not necessary, because noise are already added in likelihood.hFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWK2h@]r(haXExplicitly set rr}r(hEXExplicitly set hFjubj)r}r(hEX*pyGPs.cov.Noise*hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXpyGPs.cov.Noiserr}r(hEUhFjubahLj#ubhaXA is not necessary, because noise are already added in likelihood.rr}r(hEXA is not necessary, because noise are already added in likelihood.hFjubeubaubeubeubhB)r}r(hEUhFhChGhJhLhMhN}r(hP]hQ]hR]hS]rh8ahU]rh auhWK6hXhh@]r(hZ)r}r(hEXComposite Kernels & MeamsrhFjhGhJhLh^hN}r(hP]hQ]hR]hS]hU]uhWK6hXhh@]rhaXComposite Kernels & Meamsrr}r(hEjhFjubaubhs)r}r(hEX9Adding and muliplying Kernels(Means) is really simple: ::rhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWK7hXhh@]rhaX6Adding and muliplying Kernels(Means) is really simple:rr}r(hEX6Adding and muliplying Kernels(Means) is really simple:hFjubaubh})r}r(hEXYk = pyGPs.cov.Periodic() * pyGPs.cov.RBF() k = 0.5*pyGPs.cov.LIN() + pyGPs.cov.Periodic()hFjhGhJhLhhN}r(hhhS]hR]hP]hQ]hU]uhWK9hXhh@]rhaXYk = pyGPs.cov.Periodic() * pyGPs.cov.RBF() k = 0.5*pyGPs.cov.LIN() + pyGPs.cov.Periodic()rr}r(hEUhFjubaubhs)r}r(hEXScalar will also be treated as a hyperparameter. For example, k = s1 * k1 + s2 * k2, then the list of hyperparameters is hyp = [s1, k1.hyp, s2, k2.hyp]. Scalar is passed in logorithm domain such that it will always be positive during optimization.rhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWK(h)r?}r@(hEXM1 is a matrix with shape **number of training points plus 1** by **number of test points** - cross covariances matrix (train by test) - last row is self covariances (diagonal of test by test)hFj;hGhJhLhhN}rA(hP]hQ]hR]hS]hU]uhWKPh@]rB(h)rC}rD(hEX[M1 is a matrix with shape **number of training points plus 1** by **number of test points**rEhFj?hGhJhLjhN}rF(hP]hQ]hR]hS]hU]uhWKPh@]rG(haXM1 is a matrix with shape rHrI}rJ(hEXM1 is a matrix with shape hFjCubcdocutils.nodes strong rK)rL}rM(hEX$**number of training points plus 1**hN}rN(hP]hQ]hR]hS]hU]uhFjCh@]rOhaX number of training points plus 1rPrQ}rR(hEUhFjLubahLUstrongrSubhaX by rTrU}rV(hEX by hFjCubjK)rW}rX(hEX**number of test points**hN}rY(hP]hQ]hR]hS]hU]uhFjCh@]rZhaXnumber of test pointsr[r\}r](hEUhFjWubahLjSubeubj)r^}r_(hEUhN}r`(hP]hQ]hR]hS]hU]uhFj?h@]raj )rb}rc(hEUhN}rd(jX-hS]hR]hP]hQ]hU]uhFj^h@]re(h)rf}rg(hEX(cross covariances matrix (train by test)rhhN}ri(hP]hQ]hR]hS]hU]uhFjbh@]rjhs)rk}rl(hEjhhFjfhGhJhLhwhN}rm(hP]hQ]hR]hS]hU]uhWKPh@]rnhaX(cross covariances matrix (train by test)rorp}rq(hEjhhFjkubaubahLhubh)rr}rs(hEX7last row is self covariances (diagonal of test by test)rthN}ru(hP]hQ]hR]hS]hU]uhFjbh@]rvhs)rw}rx(hEjthFjrhGhJhLhwhN}ry(hP]hQ]hR]hS]hU]uhWKQh@]rzhaX7last row is self covariances (diagonal of test by test)r{r|}r}(hEjthFjwubaubahLhubehLjTubahLjUubeubh)r~}r(hEX~M2 is a square matrix with **number of training points** for each dimension - training set covariance matrix (train by train) hFj;hGhJhLhhN}r(hP]hQ]hR]hS]hU]uhWKShXhh@]r(h)r}r(hEXKM2 is a square matrix with **number of training points** for each dimensionrhFj~hGhJhLjhN}r(hP]hQ]hR]hS]hU]uhWKSh@]r(haXM2 is a square matrix with rr}r(hEXM2 is a square matrix with hFjubjK)r}r(hEX**number of training points**hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXnumber of training pointsrr}r(hEUhFjubahLjSubhaX for each dimensionrr}r(hEX for each dimensionhFjubeubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFj~h@]rj )r}r(hEUhN}r(jX-hS]hR]hP]hQ]hU]uhFjh@]rh)r}r(hEX0training set covariance matrix (train by train) hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEX/training set covariance matrix (train by train)rhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWKSh@]rhaX/training set covariance matrix (train by train)rr}r(hEjhFjubaubahLhubahLjTubahLjUubeubeubhs)r}r(hEXzA precomputed kernel can also be composited with other kernels. You need to explictly add scalar for *pyGPs.cov.Pre()*. ::rhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWKUhXhh@]r(haXeA precomputed kernel can also be composited with other kernels. You need to explictly add scalar for rr}r(hEXeA precomputed kernel can also be composited with other kernels. You need to explictly add scalar for hFjubj)r}r(hEX*pyGPs.cov.Pre()*hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXpyGPs.cov.Pre()rr}r(hEUhFjubahLj#ubhaX.r}r(hEX.hFjubeubh})r}r(hEX/k = 0.5*pyGPs.cov.Pre(M1, M2) + pyGPs.cov.RBF()hFjhGhJhLhhN}r(hhhS]hR]hP]hQ]hU]uhWKWhXhh@]rhaX/k = 0.5*pyGPs.cov.Pre(M1, M2) + pyGPs.cov.RBF()rr}r(hEUhFjubaubeubhB)r}r(hEUhFhChGhJhLhMhN}r(hP]hQ]hR]hS]rh>ahU]rh)auhWK[hXhh@]r(hZ)r}r(hEX&Developing New Kernel & Mean FunctionsrhFjhGhJhLh^hN}r(hP]hQ]hR]hS]hU]uhWK[hXhh@]rhaX&Developing New Kernel & Mean Functionsrr}r(hEjhFjubaubhs)r}r(hEXWe also support the development of new kernel/mean classes, your customized kernel class need to follow the template as below: ::rhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWK\hXhh@]rhaX~We also support the development of new kernel/mean classes, your customized kernel class need to follow the template as below:rr}r(hEX~We also support the development of new kernel/mean classes, your customized kernel class need to follow the template as below:hFjubaubh})r}r(hEXY# Your kernel class needs to inherit base class Kernel, # which is in the module of Core.cov class MyKernel(Kernel): def __init__(self, hyp): ''' Intialize hyperparameters for MyKernel. ''' self.hyp = hyp def getCovMatrix(self,x=None,z=None,mode=None): ''' Return the specific covariance matrix according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self covariance matrix of test data(test by 1). 'train' return training covariance matrix(train by train). 'cross' return cross covariance matrix between x and z(train by test) :return: the corresponding covariance matrix ''' pass def getDerMatrix(self,x=None,z=None,mode=None,der=None): ''' Compute derivatives wrt. hyperparameters according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self derivative matrix of test data(test by 1). 'train' return training derivative matrix(train by train). 'cross' return cross derivative matrix between x and z(train by test) :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' passhFjhGhJhLhhN}r(hhhS]hR]hP]hQ]hU]uhWK^hXhh@]rhaXY# Your kernel class needs to inherit base class Kernel, # which is in the module of Core.cov class MyKernel(Kernel): def __init__(self, hyp): ''' Intialize hyperparameters for MyKernel. ''' self.hyp = hyp def getCovMatrix(self,x=None,z=None,mode=None): ''' Return the specific covariance matrix according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self covariance matrix of test data(test by 1). 'train' return training covariance matrix(train by train). 'cross' return cross covariance matrix between x and z(train by test) :return: the corresponding covariance matrix ''' pass def getDerMatrix(self,x=None,z=None,mode=None,der=None): ''' Compute derivatives wrt. hyperparameters according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self derivative matrix of test data(test by 1). 'train' return training derivative matrix(train by train). 'cross' return cross derivative matrix between x and z(train by test) :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' passrr}r(hEUhFjubaubhs)r}r(hEX!and for customized mean class: ::rhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaXand for customized mean class:rr}r(hEXand for customized mean class:hFjubaubh})r}r(hEXS# Your mean class needs to inherit base class Mean, # which is in the module of Core.mean class MyMean(Mean): def __init__(self, hyp): ''' Intialize hyperparameters for MyMean. ''' self.hyp = hyp def getMean(self, x=None): ''' Get the mean vector. ''' pass def getDerMatrix(self, x=None, der=None): ''' Compute derivatives wrt. hyperparameters. :param x: training data :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' passhFjhGhJhLhhN}r(hhhS]hR]hP]hQ]hU]uhWKhXhh@]rhaXS# Your mean class needs to inherit base class Mean, # which is in the module of Core.mean class MyMean(Mean): def __init__(self, hyp): ''' Intialize hyperparameters for MyMean. ''' self.hyp = hyp def getMean(self, x=None): ''' Get the mean vector. ''' pass def getDerMatrix(self, x=None, der=None): ''' Compute derivatives wrt. hyperparameters. :param x: training data :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' passrr}r(hEUhFjubaubhs)r}r(hEXYou can test your customized mean/kernel function using our framework of unit test. Taking kernel test as an example, you can uncomment method *test_cov_new* in *pyGPs.Testing.unit_test_cov.py* to check the outputs of your kernel function. ::hFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]r(haXYou can test your customized mean/kernel function using our framework of unit test. Taking kernel test as an example, you can uncomment method rr}r(hEXYou can test your customized mean/kernel function using our framework of unit test. Taking kernel test as an example, you can uncomment method hFjubj)r}r(hEX*test_cov_new*hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX test_cov_newrr}r(hEUhFjubahLj#ubhaX in rr}r(hEX in hFjubj)r}r(hEX *pyGPs.Testing.unit_test_cov.py*hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXpyGPs.Testing.unit_test_cov.pyrr}r(hEUhFjubahLj#ubhaX. to check the outputs of your kernel function.rr}r(hEX. to check the outputs of your kernel function.hFjubeubh})r}r(hEX# Test your customized covariance function def test_cov_new(self): k = myKernel() # specify your covariance function self.checkCovariance(k)hFjhGhJhLhhN}r (hhhS]hR]hP]hQ]hU]uhWKhXhh@]r haX# Test your customized covariance function def test_cov_new(self): k = myKernel() # specify your covariance function self.checkCovariance(k)r r }r (hEUhFjubaubhs)r}r(hEXAand testing mean function in *pyGPs.Testing.unit_test_mean.py* ::rhFjhGhJhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]r(haXand testing mean function in rr}r(hEXand testing mean function in hFjubj)r}r(hEX!*pyGPs.Testing.unit_test_mean.py*hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXpyGPs.Testing.unit_test_mean.pyrr}r(hEUhFjubahLj#ubeubh})r}r(hEX# Test your customized mean function def test_mean_new(self): m = myMean # specify your mean function self.checkMean(m)hFjhGhJhLhhN}r(hhhS]hR]hP]hQ]hU]uhWKhXhh@]r haX# Test your customized mean function def test_mean_new(self): m = myMean # specify your mean function self.checkMean(m)r!r"}r#(hEUhFjubaubeubhB)r$}r%(hEUhFhChGhJhLhMhN}r&(hP]hQ]hR]hS]r'(Xmodule-pyGPs.Core.covr(h?ehU]r)h*auhWKhXhh@]r*(hZ)r+}r,(hEX&List of Kernels and Default Parametersr-hFj$hGhJhLh^hN}r.(hP]hQ]hR]hS]hU]uhWKhXhh@]r/haX&List of Kernels and Default Parametersr0r1}r2(hEj-hFj+ubaubcsphinx.addnodes index r3)r4}r5(hEUhFj$hGU r6hLUindexr7hN}r8(hS]hR]hP]hQ]hU]Uentries]r9(Usingler:XpyGPs.Core.cov (module)Xmodule-pyGPs.Core.covUtr;auhWKhXhh@]ubj3)r<}r=(hEUhFj$hGNhLj7hN}r>(hS]hR]hP]hQ]hU]Uentries]r?(j:XConst (class in pyGPs.Core.cov)h-Utr@auhWNhXhh@]ubcsphinx.addnodes desc rA)rB}rC(hEUhFj$hGNhLUdescrDhN}rE(UnoindexrFUdomainrGXpyhS]hR]hP]hQ]hU]UobjtyperHXclassrIUdesctyperJjIuhWNhXhh@]rK(csphinx.addnodes desc_signature rL)rM}rN(hEXConst(log_sigma=0.0)hFjBhGU rOhLUdesc_signaturerPhN}rQ(hS]rRh-aUmodulerShHXpyGPs.Core.covrTrU}rVbhR]hP]hQ]hU]rWh-aUfullnamerXXConstrYUclassrZUUfirstr[uhWNhXhh@]r\(csphinx.addnodes desc_annotation r])r^}r_(hEXclass hFjMhGjOhLUdesc_annotationr`hN}ra(hP]hQ]hR]hS]hU]uhWNhXhh@]rbhaXclass rcrd}re(hEUhFj^ubaubcsphinx.addnodes desc_addname rf)rg}rh(hEXpyGPs.Core.cov.hFjMhGjOhLU desc_addnamerihN}rj(hP]hQ]hR]hS]hU]uhWNhXhh@]rkhaXpyGPs.Core.cov.rlrm}rn(hEUhFjgubaubcsphinx.addnodes desc_name ro)rp}rq(hEjYhFjMhGjOhLU desc_namerrhN}rs(hP]hQ]hR]hS]hU]uhWNhXhh@]rthaXConstrurv}rw(hEUhFjpubaubcsphinx.addnodes desc_parameterlist rx)ry}rz(hEUhFjMhGjOhLUdesc_parameterlistr{hN}r|(hP]hQ]hR]hS]hU]uhWNhXhh@]r}csphinx.addnodes desc_parameter r~)r}r(hEX log_sigma=0.0hN}r(hP]hQ]hR]hS]hU]uhFjyh@]rhaX log_sigma=0.0rr}r(hEUhFjubahLUdesc_parameterrubaubcsphinx.addnodes only r)r}r(hEUhFjMhGNhLUonlyrhN}r(UexprUhtmlrhS]hR]hP]hQ]hU]uhWNhXhh@]rcsphinx.addnodes pending_xref r)r}r(hEUhN}r(UreftypeUviewcoderUrefdocXKernelsrU refdomainUstdrhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjYuhFjh@]rcdocutils.nodes inline r)r}r(hEUhN}r(hP]hQ]rU viewcode-linkrahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLUinlinerubahLU pending_xrefrubaubeubcsphinx.addnodes desc_content r)r}r(hEUhFjBhGjOhLU desc_contentrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r}r(hEX$Constant kernel. hyp = [ log_sigma ]rhFjhGXO/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.ConstrhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaX$Constant kernel. hyp = [ log_sigma ]rr}r(hEjhFjubaubcdocutils.nodes field_list r)r}r(hEUhFjhGNhLU field_listrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rcdocutils.nodes field r)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(cdocutils.nodes field_name r)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX Parametersrr}r(hEUhFjubahLU field_namerubcdocutils.nodes field_body r)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r(hEX log_sigmahN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_sigmarr}r(hEUhFjubahLjSubhaX -- rr}r(hEUhFjubhaXsignal deviation.rr}r(hEXsignal deviation.hFjubehLhwubahLU field_bodyrubehLUfieldrubaubeubeubj3)r}r(hEUhFj$hGXV/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.FITCOfKernelrhLj7hN}r(hS]hR]hP]hQ]hU]Uentries]r(j:X&FITCOfKernel (class in pyGPs.Core.cov)hUtrauhWNhXhh@]ubjA)r}r(hEUhFj$hGjhLjDhN}r(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrjJjuhWNhXhh@]r(jL)r}r(hEX FITCOfKernel(cov, inducingInput)hFjhGjOhLjPhN}r(hS]rhajShHXpyGPs.Core.covrr}rbhR]hP]hQ]hU]rhajXX FITCOfKernelrjZUj[uhWNhXhh@]r(j])r}r(hEXclass hFjhGjOhLj`hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXclass rr}r(hEUhFjubaubjf)r}r(hEXpyGPs.Core.cov.hFjhGjOhLjihN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXpyGPs.Core.cov.rr}r(hEUhFjubaubjo)r}r(hEjhFjhGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaX FITCOfKernelrr}r(hEUhFjubaubjx)r}r(hEUhFjhGjOhLj{hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (j~)r }r (hEXcovhN}r (hP]hQ]hR]hS]hU]uhFjh@]rhaXcovrr}r(hEUhFj ubahLjubj~)r}r(hEX inducingInputhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX inducingInputrr}r(hEUhFjubahLjubeubj)r}r(hEUhFjhGNhLjhN}r(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rj)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjuhFjh@]r j)r!}r"(hEUhN}r#(hP]hQ]r$jahR]hS]hU]uhFjh@]r%haX[source]r&r'}r((hEUhFj!ubahLjubahLjubaubeubj)r)}r*(hEUhFjhGjOhLjhN}r+(hP]hQ]hR]hS]hU]uhWNhXhh@]r,hs)r-}r.(hEXNCovariance function to be used together with the FITC approximation. The function allows for more than one output argument and does not respect the interface of a proper covariance function. Instead of outputing the full covariance, it returns cross-covariances between the inputs x, z and the inducing inputs xu as needed by infFITCr/hFj)hGjhLhwhN}r0(hP]hQ]hR]hS]hU]uhWKhXhh@]r1haXNCovariance function to be used together with the FITC approximation. The function allows for more than one output argument and does not respect the interface of a proper covariance function. Instead of outputing the full covariance, it returns cross-covariances between the inputs x, z and the inducing inputs xu as needed by infFITCr2r3}r4(hEj/hFj-ubaubaubeubj3)r5}r6(hEUhFj$hGNhLj7hN}r7(hS]hR]hP]hQ]hU]Uentries]r8(j:X Kernel (class in pyGPs.Core.cov)h$Utr9auhWNhXhh@]ubjA)r:}r;(hEUhFj$hGNhLjDhN}r<(jFjGXpyhS]hR]hP]hQ]hU]jHXclassr=jJj=uhWNhXhh@]r>(jL)r?}r@(hEXKernel()hFj:hGjOhLjPhN}rA(hS]rBh$ajShHXpyGPs.Core.covrCrD}rEbhR]hP]hQ]hU]rFh$ajXXKernelrGjZUj[uhWNhXhh@]rH(j])rI}rJ(hEXclass hFj?hGjOhLj`hN}rK(hP]hQ]hR]hS]hU]uhWNhXhh@]rLhaXclass rMrN}rO(hEUhFjIubaubjf)rP}rQ(hEXpyGPs.Core.cov.hFj?hGjOhLjihN}rR(hP]hQ]hR]hS]hU]uhWNhXhh@]rShaXpyGPs.Core.cov.rTrU}rV(hEUhFjPubaubjo)rW}rX(hEjGhFj?hGjOhLjrhN}rY(hP]hQ]hR]hS]hU]uhWNhXhh@]rZhaXKernelr[r\}r](hEUhFjWubaubj)r^}r_(hEUhFj?hGNhLjhN}r`(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]raj)rb}rc(hEUhN}rd(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjGuhFj^h@]rej)rf}rg(hEUhN}rh(hP]hQ]rijahR]hS]hU]uhFjbh@]rjhaX[source]rkrl}rm(hEUhFjfubahLjubahLjubaubeubj)rn}ro(hEUhFj:hGjOhLjhN}rp(hP]hQ]hR]hS]hU]uhWNhXhh@]rq(hs)rr}rs(hEXThis is a base class of Kernel functions there is no computation in this class, it just defines rules about a kernel class should have each covariance function will inherit it and implement its own behaviourrthFjnhGXP/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.KernelruhLhwhN}rv(hP]hQ]hR]hS]hU]uhWKhXhh@]rwhaXThis is a base class of Kernel functions there is no computation in this class, it just defines rules about a kernel class should have each covariance function will inherit it and implement its own behaviourrxry}rz(hEjthFjrubaubj3)r{}r|(hEUhFjnhGNhLj7hN}r}(hS]hR]hP]hQ]hU]Uentries]r~(j:X%fitc() (pyGPs.Core.cov.Kernel method)hUtrauhWNhXhh@]ubjA)r}r(hEUhFjnhGNhLjDhN}r(jFjGXpyhS]hR]hP]hQ]hU]jHXmethodrjJjuhWNhXhh@]r(jL)r}r(hEXKernel.fitc(inducingInput)hFjhGjOhLjPhN}r(hS]rhajShHXpyGPs.Core.covrr}rbhR]hP]hQ]hU]rhajXX Kernel.fitcrjZjGj[uhWNhXhh@]r(jo)r}r(hEXfitchFjhGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXfitcrr}r(hEUhFjubaubjx)r}r(hEUhFjhGjOhLj{hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rj~)r}r(hEX inducingInputhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX inducingInputrr}r(hEUhFjubahLjubaubj)r}r(hEUhFjhGNhLjhN}r(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rj)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjuhFjh@]rj)r}r(hEUhN}r(hP]hQ]rjahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLjubahLjubaubeubj)r}r(hEUhFjhGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r}r(hEX|Covariance function to be used together with the FITC approximation. Setting FITC gp model will implicitly call this method.rhFjhGXU/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.Kernel.fitcrhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaX|Covariance function to be used together with the FITC approximation. Setting FITC gp model will implicitly call this method.rr}r(hEjhFjubaubj)r}r(hEUhFjhGNhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXReturnsrr}r(hEUhFjubahLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXan instance of FITCOfKernelrr}r(hEXan instance of FITCOfKernelrhFjubahLhwubahLjubehLjubaubeubeubj3)r}r(hEUhFjnhGNhLj7hN}r(hS]hR]hP]hQ]hU]Uentries]r(j:X-getCovMatrix() (pyGPs.Core.cov.Kernel method)hUtrauhWNhXhh@]ubjA)r}r(hEUhFjnhGNhLjDhN}r(jFjGXpyrhS]hR]hP]hQ]hU]jHXmethodrjJjuhWNhXhh@]r(jL)r}r(hEX.Kernel.getCovMatrix(x=None, z=None, mode=None)hFjhGjOhLjPhN}r(hS]rhajShHXpyGPs.Core.covrr}rbhR]hP]hQ]hU]rhajXXKernel.getCovMatrixrjZjGj[uhWNhXhh@]r(jo)r}r(hEX getCovMatrixhFjhGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaX getCovMatrixrr}r(hEUhFjubaubjx)r}r(hEUhFjhGjOhLj{hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(j~)r}r(hEXx=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXx=Nonerr}r(hEUhFjubahLjubj~)r}r(hEXz=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXz=Nonerr}r(hEUhFjubahLjubj~)r}r(hEX mode=NonehN}r (hP]hQ]hR]hS]hU]uhFjh@]r haX mode=Noner r }r (hEUhFjubahLjubeubj)r}r(hEUhFjhGNhLjhN}r(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rj)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjuhFjh@]rj)r}r(hEUhN}r(hP]hQ]rjahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLjubahLjubaubeubj)r}r(hEUhFjhGjOhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r!(hs)r"}r#(hEX=Return the specific covariance matrix according to input moder$hFjhGX]/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.Kernel.getCovMatrixr%hLhwhN}r&(hP]hQ]hR]hS]hU]uhWKhXhh@]r'haX=Return the specific covariance matrix according to input moder(r)}r*(hEj$hFj"ubaubj)r+}r,(hEUhFjhGNhLjhN}r-(hP]hQ]hR]hS]hU]uhWNhXhh@]r.(j)r/}r0(hEUhN}r1(hP]hQ]hR]hS]hU]uhFj+h@]r2(j)r3}r4(hEUhN}r5(hP]hQ]hR]hS]hU]uhFj/h@]r6haX Parametersr7r8}r9(hEUhFj3ubahLjubj)r:}r;(hEUhN}r<(hP]hQ]hR]hS]hU]uhFj/h@]r=j )r>}r?(hEUhN}r@(hP]hQ]hR]hS]hU]uhFj:h@]rA(h)rB}rC(hEUhN}rD(hP]hQ]hR]hS]hU]uhFj>h@]rEhs)rF}rG(hEUhN}rH(hP]hQ]hR]hS]hU]uhFjBh@]rI(jK)rJ}rK(hEXxhN}rL(hP]hQ]hR]hS]hU]uhFjFh@]rMhaXxrN}rO(hEUhFjJubahLjSubhaX -- rPrQ}rR(hEUhFjFubhaX training datarSrT}rU(hEX training datahFjFubehLhwubahLhubh)rV}rW(hEUhN}rX(hP]hQ]hR]hS]hU]uhFj>h@]rYhs)rZ}r[(hEUhN}r\(hP]hQ]hR]hS]hU]uhFjVh@]r](jK)r^}r_(hEXzhN}r`(hP]hQ]hR]hS]hU]uhFjZh@]rahaXzrb}rc(hEUhFj^ubahLjSubhaX -- rdre}rf(hEUhFjZubhaX test datargrh}ri(hEX test datahFjZubehLhwubahLhubh)rj}rk(hEUhN}rl(hP]hQ]hR]hS]hU]uhFj>h@]rmhs)rn}ro(hEUhN}rp(hP]hQ]hR]hS]hU]uhFjjh@]rq(jK)rr}rs(hEXmodehN}rt(hP]hQ]hR]hS]hU]uhFjnh@]ruhaXmodervrw}rx(hEUhFjrubahLjSubhaX (ryrz}r{(hEUhFjnubj)r|}r}(hEUhN}r~(UreftypeUobjrU reftargetXstrrU refdomainjhS]hR]U refexplicithP]hQ]hU]uhFjnh@]rj)r}r(hEjhN}r(hP]hQ]hR]hS]hU]uhFj|h@]rhaXstrrr}r(hEUhFjubahLj#ubahLjubhaX)r}r(hEUhFjnubhaX -- rr}r(hEUhFjnubhaX'self_test' return self covariance matrix of test data(test by 1). 'train' return training covariance matrix(train by train). 'cross' return cross covariance matrix between x and z(train by test)rr}r(hEX'self_test' return self covariance matrix of test data(test by 1). 'train' return training covariance matrix(train by train). 'cross' return cross covariance matrix between x and z(train by test)hFjnubehLhwubahLhubehLjTubahLjubehLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFj+h@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXReturnsrr}r(hEUhFjubahLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX#the corresponding covariance matrixrr}r(hEX#the corresponding covariance matrixhFjubahLhwubahLjubehLjubeubeubeubj3)r}r(hEUhFjnhGNhLj7hN}r(hS]hR]hP]hQ]hU]Uentries]r(j:X-getDerMatrix() (pyGPs.Core.cov.Kernel method)hUtrauhWNhXhh@]ubjA)r}r(hEUhFjnhGNhLjDhN}r(jFjGXpyrhS]hR]hP]hQ]hU]jHXmethodrjJjuhWNhXhh@]r(jL)r}r(hEX8Kernel.getDerMatrix(x=None, z=None, mode=None, der=None)hFjhGjOhLjPhN}r(hS]rhajShHXpyGPs.Core.covrr}rbhR]hP]hQ]hU]rhajXXKernel.getDerMatrixrjZjGj[uhWNhXhh@]r(jo)r}r(hEX getDerMatrixhFjhGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaX getDerMatrixrr}r(hEUhFjubaubjx)r}r(hEUhFjhGjOhLj{hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(j~)r}r(hEXx=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXx=Nonerr}r(hEUhFjubahLjubj~)r}r(hEXz=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXz=Nonerr}r(hEUhFjubahLjubj~)r}r(hEX mode=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX mode=Nonerr}r(hEUhFjubahLjubj~)r}r(hEXder=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXder=Nonerr}r(hEUhFjubahLjubeubj)r}r(hEUhFjhGNhLjhN}r(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rj)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjuhFjh@]rj)r}r(hEUhN}r(hP]hQ]rjahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLjubahLjubaubeubj)r}r(hEUhFjhGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r}r(hEX@Compute derivatives wrt. hyperparameters according to input moderhFjhGX]/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.Kernel.getDerMatrixrhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaX@Compute derivatives wrt. hyperparameters according to input moderr}r(hEjhFjubaubj)r}r(hEUhFjhGNhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(j)r}r (hEUhN}r (hP]hQ]hR]hS]hU]uhFjh@]r haX Parametersr r }r(hEUhFjubahLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rj )r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(h)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r (hEXxhN}r!(hP]hQ]hR]hS]hU]uhFjh@]r"haXxr#}r$(hEUhFjubahLjSubhaX -- r%r&}r'(hEUhFjubhaX training datar(r)}r*(hEX training datar+hFjubehLhwubahLhubh)r,}r-(hEUhN}r.(hP]hQ]hR]hS]hU]uhFjh@]r/hs)r0}r1(hEUhN}r2(hP]hQ]hR]hS]hU]uhFj,h@]r3(jK)r4}r5(hEXzhN}r6(hP]hQ]hR]hS]hU]uhFj0h@]r7haXzr8}r9(hEUhFj4ubahLjSubhaX -- r:r;}r<(hEUhFj0ubhaX test datar=r>}r?(hEX test datar@hFj0ubehLhwubahLhubh)rA}rB(hEUhN}rC(hP]hQ]hR]hS]hU]uhFjh@]rDhs)rE}rF(hEUhN}rG(hP]hQ]hR]hS]hU]uhFjAh@]rH(jK)rI}rJ(hEXmodehN}rK(hP]hQ]hR]hS]hU]uhFjEh@]rLhaXmoderMrN}rO(hEUhFjIubahLjSubhaX (rPrQ}rR(hEUhFjEubj)rS}rT(hEUhN}rU(UreftypejU reftargetXstrrVU refdomainjhS]hR]U refexplicithP]hQ]hU]uhFjEh@]rWj)rX}rY(hEjVhN}rZ(hP]hQ]hR]hS]hU]uhFjSh@]r[haXstrr\r]}r^(hEUhFjXubahLj#ubahLjubhaX)r_}r`(hEUhFjEubhaX -- rarb}rc(hEUhFjEubhaX'self_test' return self derivative matrix of test data(test by 1). 'train' return training derivative matrix(train by train). 'cross' return cross derivative matrix between x and z(train by test)rdre}rf(hEX'self_test' return self derivative matrix of test data(test by 1). 'train' return training derivative matrix(train by train). 'cross' return cross derivative matrix between x and z(train by test)rghFjEubehLhwubahLhubh)rh}ri(hEUhN}rj(hP]hQ]hR]hS]hU]uhFjh@]rkhs)rl}rm(hEUhN}rn(hP]hQ]hR]hS]hU]uhFjhh@]ro(jK)rp}rq(hEXderhN}rr(hP]hQ]hR]hS]hU]uhFjlh@]rshaXderrtru}rv(hEUhFjpubahLjSubhaX (rwrx}ry(hEUhFjlubj)rz}r{(hEUhN}r|(UreftypejU reftargetXintr}U refdomainjhS]hR]U refexplicithP]hQ]hU]uhFjlh@]r~j)r}r(hEj}hN}r(hP]hQ]hR]hS]hU]uhFjzh@]rhaXintrr}r(hEUhFjubahLj#ubahLjubhaX)r}r(hEUhFjlubhaX -- rr}r(hEUhFjlubhaX7index of hyperparameter whose derivative to be computedrr}r(hEX7index of hyperparameter whose derivative to be computedrhFjlubehLhwubahLhubehLjTubahLjubehLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXReturnsrr}r(hEUhFjubahLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX#the corresponding derivative matrixrr}r(hEX#the corresponding derivative matrixrhFjubahLhwubahLjubehLjubeubeubeubeubeubj3)r}r(hEUhFj$hGNhLj7hN}r(hS]hR]hP]hQ]hU]Uentries]r(j:X LINard (class in pyGPs.Core.cov)h UtrauhWNhXhh@]ubjA)r}r(hEUhFj$hGNhLjDhN}r(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrjJjuhWNhXhh@]r(jL)r}r(hEX!LINard(D=None, log_ell_list=None)hFjhGjOhLjPhN}r(hS]rh ajShHXpyGPs.Core.covrr}rbhR]hP]hQ]hU]rh ajXXLINardrjZUj[uhWNhXhh@]r(j])r}r(hEXclass hFjhGjOhLj`hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXclass rr}r(hEUhFjubaubjf)r}r(hEXpyGPs.Core.cov.hFjhGjOhLjihN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXpyGPs.Core.cov.rr}r(hEUhFjubaubjo)r}r(hEjhFjhGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXLINardrr}r(hEUhFjubaubjx)r}r(hEUhFjhGjOhLj{hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(j~)r}r(hEXD=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXD=Nonerr}r(hEUhFjubahLjubj~)r}r(hEXlog_ell_list=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXlog_ell_list=Nonerr}r(hEUhFjubahLjubeubj)r}r(hEUhFjhGNhLjhN}r(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rj)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjuhFjh@]rj)r}r(hEUhN}r(hP]hQ]rjahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLjubahLjubaubeubj)r}r(hEUhFjhGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r}r(hEXTLinear covariance function with Automatic Relevance Detemination. hyp = log_ell_listrhFjhGXP/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.LINardrhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaXTLinear covariance function with Automatic Relevance Detemination. hyp = log_ell_listrr}r(hEjhFjubaubj)r}r(hEUhFjhGNhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r haX Parametersr r }r (hEUhFjubahLjubj)r }r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rj )r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFj h@]r(h)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r(hEXDhN}r(hP]hQ]hR]hS]hU]uhFjh@]r haXDr!}r"(hEUhFjubahLjSubhaX -- r#r$}r%(hEUhFjubhaXWdimension of training data. Set if you want default ell, which is 1 for each dimension.r&r'}r((hEXWdimension of training data. Set if you want default ell, which is 1 for each dimension.hFjubehLhwubahLhubh)r)}r*(hEUhN}r+(hP]hQ]hR]hS]hU]uhFjh@]r,hs)r-}r.(hEUhN}r/(hP]hQ]hR]hS]hU]uhFj)h@]r0(jK)r1}r2(hEX log_ell_listhN}r3(hP]hQ]hR]hS]hU]uhFj-h@]r4haX log_ell_listr5r6}r7(hEUhFj1ubahLjSubhaX -- r8r9}r:(hEUhFj-ubhaX/characteristic length scale for each dimension.r;r<}r=(hEX/characteristic length scale for each dimension.hFj-ubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)r>}r?(hEUhFj$hGNhLj7hN}r@(hS]hR]hP]hQ]hU]Uentries]rA(j:X Linear (class in pyGPs.Core.cov)hUtrBauhWNhXhh@]ubjA)rC}rD(hEUhFj$hGNhLjDhN}rE(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrFjJjFuhWNhXhh@]rG(jL)rH}rI(hEXLinear(log_sigma=0.0)hFjChGjOhLjPhN}rJ(hS]rKhajShHXpyGPs.Core.covrLrM}rNbhR]hP]hQ]hU]rOhajXXLinearrPjZUj[uhWNhXhh@]rQ(j])rR}rS(hEXclass hFjHhGjOhLj`hN}rT(hP]hQ]hR]hS]hU]uhWNhXhh@]rUhaXclass rVrW}rX(hEUhFjRubaubjf)rY}rZ(hEXpyGPs.Core.cov.hFjHhGjOhLjihN}r[(hP]hQ]hR]hS]hU]uhWNhXhh@]r\haXpyGPs.Core.cov.r]r^}r_(hEUhFjYubaubjo)r`}ra(hEjPhFjHhGjOhLjrhN}rb(hP]hQ]hR]hS]hU]uhWNhXhh@]rchaXLinearrdre}rf(hEUhFj`ubaubjx)rg}rh(hEUhFjHhGjOhLj{hN}ri(hP]hQ]hR]hS]hU]uhWNhXhh@]rjj~)rk}rl(hEX log_sigma=0.0hN}rm(hP]hQ]hR]hS]hU]uhFjgh@]rnhaX log_sigma=0.0rorp}rq(hEUhFjkubahLjubaubj)rr}rs(hEUhFjHhGNhLjhN}rt(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]ruj)rv}rw(hEUhN}rx(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjPuhFjrh@]ryj)rz}r{(hEUhN}r|(hP]hQ]r}jahR]hS]hU]uhFjvh@]r~haX[source]rr}r(hEUhFjzubahLjubahLjubaubeubj)r}r(hEUhFjChGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r}r(hEX#Linear kernel. hyp = [ log_sigma ].rhFjhGXP/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.LinearrhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaX#Linear kernel. hyp = [ log_sigma ].rr}r(hEjhFjubaubj)r}r(hEUhFjhGNhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX Parametersrr}r(hEUhFjubahLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r(hEX log_sigmahN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_sigmarr}r(hEUhFjubahLjSubhaX -- rr}r(hEUhFjubhaXsignal deviation.rr}r(hEXsignal deviation.hFjubehLhwubahLjubehLjubaubeubeubj3)r}r(hEUhFj$hGNhLj7hN}r(hS]hR]hP]hQ]hU]Uentries]r(j:X Matern (class in pyGPs.Core.cov)hUtrauhWNhXhh@]ubjA)r}r(hEUhFj$hGNhLjDhN}r(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrjJjuhWNhXhh@]r(jL)r}r(hEX'Matern(log_ell=0.0, d=3, log_sigma=0.0)hFjhGjOhLjPhN}r(hS]rhajShHXpyGPs.Core.covrr}rbhR]hP]hQ]hU]rhajXXMaternrjZUj[uhWNhXhh@]r(j])r}r(hEXclass hFjhGjOhLj`hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXclass rr}r(hEUhFjubaubjf)r}r(hEXpyGPs.Core.cov.hFjhGjOhLjihN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXpyGPs.Core.cov.rr}r(hEUhFjubaubjo)r}r(hEjhFjhGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXMaternrr}r(hEUhFjubaubjx)r}r(hEUhFjhGjOhLj{hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(j~)r}r(hEX log_ell=0.0hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_ell=0.0rr}r(hEUhFjubahLjubj~)r}r(hEXd=3hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXd=3rr}r(hEUhFjubahLjubj~)r}r(hEX log_sigma=0.0hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_sigma=0.0rr}r(hEUhFjubahLjubeubj)r}r(hEUhFjhGNhLjhN}r(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rj)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjuhFjh@]rj)r}r(hEUhN}r(hP]hQ]rjahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLjubahLjubaubeubj)r}r(hEUhFjhGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r }r (hEXMatern covariance function with nu = d/2 and isotropic distance measure. For d=1 the function is also known as the exponential covariance function or the Ornstein-Uhlenbeck covariance in 1d. d will be rounded to 1, 3, or 5. hyp = [ log_ell, log_sigma]r hFjhGXP/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.Maternr hLhwhN}r (hP]hQ]hR]hS]hU]uhWKhXhh@]rhaXMatern covariance function with nu = d/2 and isotropic distance measure. For d=1 the function is also known as the exponential covariance function or the Ornstein-Uhlenbeck covariance in 1d. d will be rounded to 1, 3, or 5. hyp = [ log_ell, log_sigma]rr}r(hEj hFj ubaubj)r}r(hEUhFjhGNhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX Parametersrr}r (hEUhFjubahLjubj)r!}r"(hEUhN}r#(hP]hQ]hR]hS]hU]uhFjh@]r$j )r%}r&(hEUhN}r'(hP]hQ]hR]hS]hU]uhFj!h@]r((h)r)}r*(hEUhN}r+(hP]hQ]hR]hS]hU]uhFj%h@]r,hs)r-}r.(hEUhN}r/(hP]hQ]hR]hS]hU]uhFj)h@]r0(jK)r1}r2(hEXdhN}r3(hP]hQ]hR]hS]hU]uhFj-h@]r4haXdr5}r6(hEUhFj1ubahLjSubhaX -- r7r8}r9(hEUhFj-ubhaX&d is 2 times nu. Can only be 1,3 or 5.r:r;}r<(hEX&d is 2 times nu. Can only be 1,3 or 5.hFj-ubehLhwubahLhubh)r=}r>(hEUhN}r?(hP]hQ]hR]hS]hU]uhFj%h@]r@hs)rA}rB(hEUhN}rC(hP]hQ]hR]hS]hU]uhFj=h@]rD(jK)rE}rF(hEXlog_ellhN}rG(hP]hQ]hR]hS]hU]uhFjAh@]rHhaXlog_ellrIrJ}rK(hEUhFjEubahLjSubhaX -- rLrM}rN(hEUhFjAubhaXcharacteristic length scale.rOrP}rQ(hEXcharacteristic length scale.hFjAubehLhwubahLhubh)rR}rS(hEUhN}rT(hP]hQ]hR]hS]hU]uhFj%h@]rUhs)rV}rW(hEUhN}rX(hP]hQ]hR]hS]hU]uhFjRh@]rY(jK)rZ}r[(hEX log_sigmahN}r\(hP]hQ]hR]hS]hU]uhFjVh@]r]haX log_sigmar^r_}r`(hEUhFjZubahLjSubhaX -- rarb}rc(hEUhFjVubhaXsignal deviation.rdre}rf(hEXsignal deviation.hFjVubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)rg}rh(hEUhFj$hGNhLj7hN}ri(hS]hR]hP]hQ]hU]Uentries]rj(j:XNoise (class in pyGPs.Core.cov)hUtrkauhWNhXhh@]ubjA)rl}rm(hEUhFj$hGNhLjDhN}rn(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrojJjouhWNhXhh@]rp(jL)rq}rr(hEXNoise(log_sigma=0.0)hFjlhGjOhLjPhN}rs(hS]rthajShHXpyGPs.Core.covrurv}rwbhR]hP]hQ]hU]rxhajXXNoiseryjZUj[uhWNhXhh@]rz(j])r{}r|(hEXclass hFjqhGjOhLj`hN}r}(hP]hQ]hR]hS]hU]uhWNhXhh@]r~haXclass rr}r(hEUhFj{ubaubjf)r}r(hEXpyGPs.Core.cov.hFjqhGjOhLjihN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXpyGPs.Core.cov.rr}r(hEUhFjubaubjo)r}r(hEjyhFjqhGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXNoiserr}r(hEUhFjubaubjx)r}r(hEUhFjqhGjOhLj{hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rj~)r}r(hEX log_sigma=0.0hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_sigma=0.0rr}r(hEUhFjubahLjubaubj)r}r(hEUhFjqhGNhLjhN}r(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rj)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjyuhFjh@]rj)r}r(hEUhN}r(hP]hQ]rjahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLjubahLjubaubeubj)r}r(hEUhFjlhGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r}r(hEXIndependent covariance function, i.e "white noise", with specified variance. Normally NOT used anymore since noise is now added in liklihood. hyp = [ log_sigma ]rhFjhGXO/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.NoiserhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaXIndependent covariance function, i.e "white noise", with specified variance. Normally NOT used anymore since noise is now added in liklihood. hyp = [ log_sigma ]rr}r(hEjhFjubaubj)r}r(hEUhFjhGNhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX Parametersrr}r(hEUhFjubahLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r(hEX log_sigmahN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_sigmarr}r(hEUhFjubahLjSubhaX -- rr}r(hEUhFjubhaXsignal deviation.rr}r(hEXsignal deviation.hFjubehLhwubahLjubehLjubaubeubeubj3)r}r(hEUhFj$hGNhLj7hN}r(hS]hR]hP]hQ]hU]Uentries]r(j:X"Periodic (class in pyGPs.Core.cov)h UtrauhWNhXhh@]ubjA)r}r(hEUhFj$hGNhLjDhN}r(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrjJjuhWNhXhh@]r(jL)r}r(hEX/Periodic(log_ell=0.0, log_p=0.0, log_sigma=0.0)hFjhGjOhLjPhN}r(hS]rh ajShHXpyGPs.Core.covrr}rbhR]hP]hQ]hU]rh ajXXPeriodicrjZUj[uhWNhXhh@]r(j])r}r(hEXclass hFjhGjOhLj`hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXclass rr}r(hEUhFjubaubjf)r}r(hEXpyGPs.Core.cov.hFjhGjOhLjihN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXpyGPs.Core.cov.rr}r(hEUhFjubaubjo)r}r(hEjhFjhGjOhLjrhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXPeriodicr r }r (hEUhFjubaubjx)r }r (hEUhFjhGjOhLj{hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (j~)r }r (hEX log_ell=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_ell=0.0r r }r (hEUhFj ubahLjubj~)r }r (hEX log_p=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_p=0.0r r }r (hEUhFj ubahLjubj~)r }r (hEX log_sigma=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_sigma=0.0r r }r (hEUhFj ubahLjubeubj)r }r (hEUhFjhGNhLjhN}r (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r! j)r" }r# (hEUhN}r$ (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjuhFj h@]r% j)r& }r' (hEUhN}r( (hP]hQ]r) jahR]hS]hU]uhFj" h@]r* haX[source]r+ r, }r- (hEUhFj& ubahLjubahLjubaubeubj)r. }r/ (hEUhFjhGjOhLjhN}r0 (hP]hQ]hR]hS]hU]uhWNhXhh@]r1 (hs)r2 }r3 (hEXUStationary kernel for a smooth periodic function. hyp = [ log_ell, log_p, log_sigma]r4 hFj. hGXR/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.Periodicr5 hLhwhN}r6 (hP]hQ]hR]hS]hU]uhWKhXhh@]r7 haXUStationary kernel for a smooth periodic function. hyp = [ log_ell, log_p, log_sigma]r8 r9 }r: (hEj4 hFj2 ubaubj)r; }r< (hEUhFj. hGNhLjhN}r= (hP]hQ]hR]hS]hU]uhWNhXhh@]r> j)r? }r@ (hEUhN}rA (hP]hQ]hR]hS]hU]uhFj; h@]rB (j)rC }rD (hEUhN}rE (hP]hQ]hR]hS]hU]uhFj? h@]rF haX ParametersrG rH }rI (hEUhFjC ubahLjubj)rJ }rK (hEUhN}rL (hP]hQ]hR]hS]hU]uhFj? h@]rM j )rN }rO (hEUhN}rP (hP]hQ]hR]hS]hU]uhFjJ h@]rQ (h)rR }rS (hEUhN}rT (hP]hQ]hR]hS]hU]uhFjN h@]rU hs)rV }rW (hEUhN}rX (hP]hQ]hR]hS]hU]uhFjR h@]rY (jK)rZ }r[ (hEXlog_phN}r\ (hP]hQ]hR]hS]hU]uhFjV h@]r] haXlog_pr^ r_ }r` (hEUhFjZ ubahLjSubhaX -- ra rb }rc (hEUhFjV ubhaXperiod.rd re }rf (hEXperiod.hFjV ubehLhwubahLhubh)rg }rh (hEUhN}ri (hP]hQ]hR]hS]hU]uhFjN h@]rj hs)rk }rl (hEUhN}rm (hP]hQ]hR]hS]hU]uhFjg h@]rn (jK)ro }rp (hEXlog_ellhN}rq (hP]hQ]hR]hS]hU]uhFjk h@]rr haXlog_ellrs rt }ru (hEUhFjo ubahLjSubhaX -- rv rw }rx (hEUhFjk ubhaXcharacteristic length scale.ry rz }r{ (hEXcharacteristic length scale.hFjk ubehLhwubahLhubh)r| }r} (hEUhN}r~ (hP]hQ]hR]hS]hU]uhFjN h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj| h@]r (jK)r }r (hEX log_sigmahN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_sigmar r }r (hEUhFj ubahLjSubhaX -- r r }r (hEUhFj ubhaXsignal deviation.r r }r (hEXsignal deviation.hFj ubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)r }r (hEUhFj$hGNhLj7hN}r (hS]hR]hP]hQ]hU]Uentries]r (j:X#PiecePoly (class in pyGPs.Core.cov)hUtr auhWNhXhh@]ubjA)r }r (hEUhFj$hGNhLjDhN}r (jFjGXpyhS]hR]hP]hQ]hU]jHXclassr jJj uhWNhXhh@]r (jL)r }r (hEX*PiecePoly(log_ell=0.0, v=2, log_sigma=0.0)hFj hGjOhLjPhN}r (hS]r hajShHXpyGPs.Core.covr r }r bhR]hP]hQ]hU]r hajXX PiecePolyr jZUj[uhWNhXhh@]r (j])r }r (hEXclass hFj hGjOhLj`hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXclass r r }r (hEUhFj ubaubjf)r }r (hEXpyGPs.Core.cov.hFj hGjOhLjihN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXpyGPs.Core.cov.r r }r (hEUhFj ubaubjo)r }r (hEj hFj hGjOhLjrhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haX PiecePolyr r }r (hEUhFj ubaubjx)r }r (hEUhFj hGjOhLj{hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (j~)r }r (hEX log_ell=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_ell=0.0r r }r (hEUhFj ubahLjubj~)r }r (hEXv=2hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXv=2r r }r (hEUhFj ubahLjubj~)r }r (hEX log_sigma=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_sigma=0.0r r }r (hEUhFj ubahLjubeubj)r }r (hEUhFj hGNhLjhN}r (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r j)r }r (hEUhN}r (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidj uhFj h@]r j)r }r (hEUhN}r (hP]hQ]r jahR]hS]hU]uhFj h@]r haX[source]r r }r (hEUhFj ubahLjubahLjubaubeubj)r }r (hEUhFj hGjOhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (hs)r }r (hEXMPiecewise polynomial kernel with compact support. hyp = [log_ell, log_sigma]r hFj hGXS/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.PiecePolyr hLhwhN}r (hP]hQ]hR]hS]hU]uhWKhXhh@]r haXMPiecewise polynomial kernel with compact support. hyp = [log_ell, log_sigma]r r }r (hEj hFj ubaubj)r }r (hEUhFj hGNhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r j)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (j)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX Parametersr r }r (hEUhFj ubahLjubj)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r j )r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (h)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (jK)r }r (hEXlog_ellhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXlog_ellr r }r (hEUhFj ubahLjSubhaX -- r r }r (hEUhFj ubhaXcharacteristic length scale.r r }r (hEXcharacteristic length scale.hFj ubehLhwubahLhubh)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r hs)r }r! (hEUhN}r" (hP]hQ]hR]hS]hU]uhFj h@]r# (jK)r$ }r% (hEX log_sigmahN}r& (hP]hQ]hR]hS]hU]uhFj h@]r' haX log_sigmar( r) }r* (hEUhFj$ ubahLjSubhaX -- r+ r, }r- (hEUhFj ubhaXsignal deviation.r. r/ }r0 (hEXsignal deviation.hFj ubehLhwubahLhubh)r1 }r2 (hEUhN}r3 (hP]hQ]hR]hS]hU]uhFj h@]r4 hs)r5 }r6 (hEUhN}r7 (hP]hQ]hR]hS]hU]uhFj1 h@]r8 (jK)r9 }r: (hEXvhN}r; (hP]hQ]hR]hS]hU]uhFj5 h@]r< haXvr= }r> (hEUhFj9 ubahLjSubhaX -- r? r@ }rA (hEUhFj5 ubhaXbdegree v will be rounded to 0,1,2,or 3. (not treated as hyperparameter, i.e. will not be trained).rB rC }rD (hEXbdegree v will be rounded to 0,1,2,or 3. (not treated as hyperparameter, i.e. will not be trained).hFj5 ubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)rE }rF (hEUhFj$hGNhLj7hN}rG (hS]hR]hP]hQ]hU]Uentries]rH (j:XPoly (class in pyGPs.Core.cov)hUtrI auhWNhXhh@]ubjA)rJ }rK (hEUhFj$hGNhLjDhN}rL (jFjGXpyhS]hR]hP]hQ]hU]jHXclassrM jJjM uhWNhXhh@]rN (jL)rO }rP (hEX#Poly(log_c=0.0, d=2, log_sigma=0.0)hFjJ hGjOhLjPhN}rQ (hS]rR hajShHXpyGPs.Core.covrS rT }rU bhR]hP]hQ]hU]rV hajXXPolyrW jZUj[uhWNhXhh@]rX (j])rY }rZ (hEXclass hFjO hGjOhLj`hN}r[ (hP]hQ]hR]hS]hU]uhWNhXhh@]r\ haXclass r] r^ }r_ (hEUhFjY ubaubjf)r` }ra (hEXpyGPs.Core.cov.hFjO hGjOhLjihN}rb (hP]hQ]hR]hS]hU]uhWNhXhh@]rc haXpyGPs.Core.cov.rd re }rf (hEUhFj` ubaubjo)rg }rh (hEjW hFjO hGjOhLjrhN}ri (hP]hQ]hR]hS]hU]uhWNhXhh@]rj haXPolyrk rl }rm (hEUhFjg ubaubjx)rn }ro (hEUhFjO hGjOhLj{hN}rp (hP]hQ]hR]hS]hU]uhWNhXhh@]rq (j~)rr }rs (hEX log_c=0.0hN}rt (hP]hQ]hR]hS]hU]uhFjn h@]ru haX log_c=0.0rv rw }rx (hEUhFjr ubahLjubj~)ry }rz (hEXd=2hN}r{ (hP]hQ]hR]hS]hU]uhFjn h@]r| haXd=2r} r~ }r (hEUhFjy ubahLjubj~)r }r (hEX log_sigma=0.0hN}r (hP]hQ]hR]hS]hU]uhFjn h@]r haX log_sigma=0.0r r }r (hEUhFj ubahLjubeubj)r }r (hEUhFjO hGNhLjhN}r (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r j)r }r (hEUhN}r (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjW uhFj h@]r j)r }r (hEUhN}r (hP]hQ]r jahR]hS]hU]uhFj h@]r haX[source]r r }r (hEUhFj ubahLjubahLjubaubeubj)r }r (hEUhFjJ hGjOhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (hs)r }r (hEX:Polynomial covariance function. hyp = [ log_c, log_sigma ]r hFj hGXN/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.Polyr hLhwhN}r (hP]hQ]hR]hS]hU]uhWKhXhh@]r haX:Polynomial covariance function. hyp = [ log_c, log_sigma ]r r }r (hEj hFj ubaubj)r }r (hEUhFj hGNhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r j)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (j)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX Parametersr r }r (hEUhFj ubahLjubj)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r j )r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (h)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (jK)r }r (hEXlog_chN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXlog_cr r }r (hEUhFj ubahLjSubhaX -- r r }r (hEUhFj ubhaXinhomogeneous offset.r r }r (hEXinhomogeneous offset.hFj ubehLhwubahLhubh)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (jK)r }r (hEX log_sigmahN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_sigmar r }r (hEUhFj ubahLjSubhaX -- r r }r (hEUhFj ubhaXsignal deviation.r r }r (hEXsignal deviation.hFj ubehLhwubahLhubh)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (jK)r }r (hEXdhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXdr }r (hEUhFj ubahLjSubhaX -- r r }r (hEUhFj ubhaXOdegree of polynomial (not treated as hyperparameter, i.e. will not be trained).r r }r (hEXOdegree of polynomial (not treated as hyperparameter, i.e. will not be trained).hFj ubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)r }r (hEUhFj$hGNhLj7hN}r (hS]hR]hP]hQ]hU]Uentries]r (j:XPre (class in pyGPs.Core.cov)h(Utr auhWNhXhh@]ubjA)r }r (hEUhFj$hGNhLjDhN}r (jFjGXpyhS]hR]hP]hQ]hU]jHXclassr jJj uhWNhXhh@]r (jL)r }r (hEX Pre(M1, M2)hFj hGjOhLjPhN}r (hS]r h(ajShHXpyGPs.Core.covr r }r bhR]hP]hQ]hU]r h(ajXXPrer jZUj[uhWNhXhh@]r (j])r }r (hEXclass hFj hGjOhLj`hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXclass r r }r (hEUhFj ubaubjf)r }r (hEXpyGPs.Core.cov.hFj hGjOhLjihN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXpyGPs.Core.cov.r r }r (hEUhFj ubaubjo)r }r (hEj hFj hGjOhLjrhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXPrer r }r! (hEUhFj ubaubjx)r" }r# (hEUhFj hGjOhLj{hN}r$ (hP]hQ]hR]hS]hU]uhWNhXhh@]r% (j~)r& }r' (hEXM1hN}r( (hP]hQ]hR]hS]hU]uhFj" h@]r) haXM1r* r+ }r, (hEUhFj& ubahLjubj~)r- }r. (hEXM2hN}r/ (hP]hQ]hR]hS]hU]uhFj" h@]r0 haXM2r1 r2 }r3 (hEUhFj- ubahLjubeubj)r4 }r5 (hEUhFj hGNhLjhN}r6 (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r7 j)r8 }r9 (hEUhN}r: (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidj uhFj4 h@]r; j)r< }r= (hEUhN}r> (hP]hQ]r? jahR]hS]hU]uhFj8 h@]r@ haX[source]rA rB }rC (hEUhFj< ubahLjubahLjubaubeubj)rD }rE (hEUhFj hGjOhLjhN}rF (hP]hQ]hR]hS]hU]uhWNhXhh@]rG (hs)rH }rI (hEXQPrecomputed kernel matrix. No hyperparameters and thus nothing will be optimised.rJ hFjD hGXM/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.PrerK hLhwhN}rL (hP]hQ]hR]hS]hU]uhWKhXhh@]rM haXQPrecomputed kernel matrix. No hyperparameters and thus nothing will be optimised.rN rO }rP (hEjJ hFjH ubaubj)rQ }rR (hEUhFjD hGNhLjhN}rS (hP]hQ]hR]hS]hU]uhWNhXhh@]rT j)rU }rV (hEUhN}rW (hP]hQ]hR]hS]hU]uhFjQ h@]rX (j)rY }rZ (hEUhN}r[ (hP]hQ]hR]hS]hU]uhFjU h@]r\ haX Parametersr] r^ }r_ (hEUhFjY ubahLjubj)r` }ra (hEUhN}rb (hP]hQ]hR]hS]hU]uhFjU h@]rc j )rd }re (hEUhN}rf (hP]hQ]hR]hS]hU]uhFj` h@]rg (h)rh }ri (hEUhN}rj (hP]hQ]hR]hS]hU]uhFjd h@]rk hs)rl }rm (hEUhN}rn (hP]hQ]hR]hS]hU]uhFjh h@]ro (jK)rp }rq (hEXM1hN}rr (hP]hQ]hR]hS]hU]uhFjl h@]rs haXM1rt ru }rv (hEUhFjp ubahLjSubhaX -- rw rx }ry (hEUhFjl ubhaXbcross covariances matrix(train+1 by test). last row is self covariances (diagonal of test by test)rz r{ }r| (hEXbcross covariances matrix(train+1 by test). last row is self covariances (diagonal of test by test)hFjl ubehLhwubahLhubh)r} }r~ (hEUhN}r (hP]hQ]hR]hS]hU]uhFjd h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj} h@]r (jK)r }r (hEXM2hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXM2r r }r (hEUhFj ubahLjSubhaX -- r r }r (hEUhFj ubhaX/training set covariance matrix (train by train)r r }r (hEX/training set covariance matrix (train by train)hFj ubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)r }r (hEUhFj$hGXY/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.ProductOfKernelr hLj7hN}r (hS]hR]hP]hQ]hU]Uentries]r (j:X)ProductOfKernel (class in pyGPs.Core.cov)hUtr auhWNhXhh@]ubjA)r }r (hEUhFj$hGj hLjDhN}r (jFjGXpyhS]hR]hP]hQ]hU]jHXclassr jJj uhWNhXhh@]r (jL)r }r (hEXProductOfKernel(cov1, cov2)hFj hGjOhLjPhN}r (hS]r hajShHXpyGPs.Core.covr r }r bhR]hP]hQ]hU]r hajXXProductOfKernelr jZUj[uhWNhXhh@]r (j])r }r (hEXclass hFj hGjOhLj`hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXclass r r }r (hEUhFj ubaubjf)r }r (hEXpyGPs.Core.cov.hFj hGjOhLjihN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXpyGPs.Core.cov.r r }r (hEUhFj ubaubjo)r }r (hEj hFj hGjOhLjrhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXProductOfKernelr r }r (hEUhFj ubaubjx)r }r (hEUhFj hGjOhLj{hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (j~)r }r (hEXcov1hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXcov1r r }r (hEUhFj ubahLjubj~)r }r (hEXcov2hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXcov2r r }r (hEUhFj ubahLjubeubj)r }r (hEUhFj hGNhLjhN}r (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r j)r }r (hEUhN}r (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidj uhFj h@]r j)r }r (hEUhN}r (hP]hQ]r jahR]hS]hU]uhFj h@]r haX[source]r r }r (hEUhFj ubahLjubahLjubaubeubj)r }r (hEUhFj hGjOhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r hs)r }r (hEXProduct of two kernel function.r hFj hGj hLhwhN}r (hP]hQ]hR]hS]hU]uhWKhXhh@]r haXProduct of two kernel function.r r }r (hEj hFj ubaubaubeubj3)r }r (hEUhFj$hGNhLj7hN}r (hS]hR]hP]hQ]hU]Uentries]r (j:XRBF (class in pyGPs.Core.cov)h#Utr auhWNhXhh@]ubjA)r }r (hEUhFj$hGNhLjDhN}r (jFjGXpyhS]hR]hP]hQ]hU]jHXclassr jJj uhWNhXhh@]r (jL)r }r (hEXRBF(log_ell=0.0, log_sigma=0.0)hFj hGjOhLjPhN}r (hS]r h#ajShHXpyGPs.Core.covr r }r bhR]hP]hQ]hU]r h#ajXXRBFr jZUj[uhWNhXhh@]r (j])r }r (hEXclass hFj hGjOhLj`hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXclass r r }r (hEUhFj ubaubjf)r }r (hEXpyGPs.Core.cov.hFj hGjOhLjihN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXpyGPs.Core.cov.r r }r (hEUhFj ubaubjo)r }r (hEj hFj hGjOhLjrhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXRBFr r }r (hEUhFj ubaubjx)r }r (hEUhFj hGjOhLj{hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (j~)r }r (hEX log_ell=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_ell=0.0r r }r (hEUhFj ubahLjubj~)r }r (hEX log_sigma=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r! haX log_sigma=0.0r" r# }r$ (hEUhFj ubahLjubeubj)r% }r& (hEUhFj hGNhLjhN}r' (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r( j)r) }r* (hEUhN}r+ (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidj uhFj% h@]r, j)r- }r. (hEUhN}r/ (hP]hQ]r0 jahR]hS]hU]uhFj) h@]r1 haX[source]r2 r3 }r4 (hEUhFj- ubahLjubahLjubaubeubj)r5 }r6 (hEUhFj hGjOhLjhN}r7 (hP]hQ]hR]hS]hU]uhWNhXhh@]r8 (hs)r9 }r: (hEXVSquared Exponential kernel with isotropic distance measure. hyp = [log_ell, log_sigma]r; hFj5 hGXM/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.RBFr< hLhwhN}r= (hP]hQ]hR]hS]hU]uhWKhXhh@]r> haXVSquared Exponential kernel with isotropic distance measure. hyp = [log_ell, log_sigma]r? r@ }rA (hEj; hFj9 ubaubj)rB }rC (hEUhFj5 hGNhLjhN}rD (hP]hQ]hR]hS]hU]uhWNhXhh@]rE j)rF }rG (hEUhN}rH (hP]hQ]hR]hS]hU]uhFjB h@]rI (j)rJ }rK (hEUhN}rL (hP]hQ]hR]hS]hU]uhFjF h@]rM haX ParametersrN rO }rP (hEUhFjJ ubahLjubj)rQ }rR (hEUhN}rS (hP]hQ]hR]hS]hU]uhFjF h@]rT j )rU }rV (hEUhN}rW (hP]hQ]hR]hS]hU]uhFjQ h@]rX (h)rY }rZ (hEUhN}r[ (hP]hQ]hR]hS]hU]uhFjU h@]r\ hs)r] }r^ (hEUhN}r_ (hP]hQ]hR]hS]hU]uhFjY h@]r` (jK)ra }rb (hEXlog_ellhN}rc (hP]hQ]hR]hS]hU]uhFj] h@]rd haXlog_ellre rf }rg (hEUhFja ubahLjSubhaX -- rh ri }rj (hEUhFj] ubhaXcharacteristic length scale.rk rl }rm (hEXcharacteristic length scale.rn hFj] ubehLhwubahLhubh)ro }rp (hEUhN}rq (hP]hQ]hR]hS]hU]uhFjU h@]rr hs)rs }rt (hEUhN}ru (hP]hQ]hR]hS]hU]uhFjo h@]rv (jK)rw }rx (hEX log_sigmahN}ry (hP]hQ]hR]hS]hU]uhFjs h@]rz haX log_sigmar{ r| }r} (hEUhFjw ubahLjSubhaX -- r~ r }r (hEUhFjs ubhaXsignal deviation.r r }r (hEXsignal deviation.r hFjs ubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)r }r (hEUhFj$hGNhLj7hN}r (hS]hR]hP]hQ]hU]Uentries]r (j:X RBFard (class in pyGPs.Core.cov)hUtr auhWNhXhh@]ubjA)r }r (hEUhFj$hGNhLjDhN}r (jFjGXpyhS]hR]hP]hQ]hU]jHXclassr jJj uhWNhXhh@]r (jL)r }r (hEX0RBFard(D=None, log_ell_list=None, log_sigma=0.0)hFj hGjOhLjPhN}r (hS]r hajShHXpyGPs.Core.covr r }r bhR]hP]hQ]hU]r hajXXRBFardr jZUj[uhWNhXhh@]r (j])r }r (hEXclass hFj hGjOhLj`hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXclass r r }r (hEUhFj ubaubjf)r }r (hEXpyGPs.Core.cov.hFj hGjOhLjihN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXpyGPs.Core.cov.r r }r (hEUhFj ubaubjo)r }r (hEj hFj hGjOhLjrhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXRBFardr r }r (hEUhFj ubaubjx)r }r (hEUhFj hGjOhLj{hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (j~)r }r (hEXD=NonehN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXD=Noner r }r (hEUhFj ubahLjubj~)r }r (hEXlog_ell_list=NonehN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXlog_ell_list=Noner r }r (hEUhFj ubahLjubj~)r }r (hEX log_sigma=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_sigma=0.0r r }r (hEUhFj ubahLjubeubj)r }r (hEUhFj hGNhLjhN}r (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r j)r }r (hEUhN}r (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidj uhFj h@]r j)r }r (hEUhN}r (hP]hQ]r jahR]hS]hU]uhFj h@]r haX[source]r r }r (hEUhFj ubahLjubahLjubaubeubj)r }r (hEUhFj hGjOhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (hs)r }r (hEXcSquared Exponential kernel with Automatic Relevance Determination. hyp = log_ell_list + [log_sigma]r hFj hGXP/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.RBFardr hLhwhN}r (hP]hQ]hR]hS]hU]uhWKhXhh@]r haXcSquared Exponential kernel with Automatic Relevance Determination. hyp = log_ell_list + [log_sigma]r r }r (hEj hFj ubaubj)r }r (hEUhFj hGNhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r j)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (j)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX Parametersr r }r (hEUhFj ubahLjubj)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r j )r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (h)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (jK)r }r (hEXDhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXDr }r (hEUhFj ubahLjSubhaX -- r r }r (hEUhFj ubhaXQdimension of pattern. set if you want default ell, which is 1 for each dimension.r r }r (hEXQdimension of pattern. set if you want default ell, which is 1 for each dimension.hFj ubehLhwubahLhubh)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (jK)r }r (hEX log_ell_listhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_ell_listr r }r (hEUhFj ubahLjSubhaX -- r r }r (hEUhFj ubhaX/characteristic length scale for each dimension.r! r" }r# (hEX/characteristic length scale for each dimension.hFj ubehLhwubahLhubh)r$ }r% (hEUhN}r& (hP]hQ]hR]hS]hU]uhFj h@]r' hs)r( }r) (hEUhN}r* (hP]hQ]hR]hS]hU]uhFj$ h@]r+ (jK)r, }r- (hEX log_sigmahN}r. (hP]hQ]hR]hS]hU]uhFj( h@]r/ haX log_sigmar0 r1 }r2 (hEUhFj, ubahLjSubhaX -- r3 r4 }r5 (hEUhFj( ubhaXsignal deviation.r6 r7 }r8 (hEXsignal deviation.hFj( ubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)r9 }r: (hEUhFj$hGNhLj7hN}r; (hS]hR]hP]hQ]hU]Uentries]r< (j:X!RBFunit (class in pyGPs.Core.cov)hUtr= auhWNhXhh@]ubjA)r> }r? (hEUhFj$hGNhLjDhN}r@ (jFjGXpyhS]hR]hP]hQ]hU]jHXclassrA jJjA uhWNhXhh@]rB (jL)rC }rD (hEXRBFunit(log_ell=0.0)hFj> hGjOhLjPhN}rE (hS]rF hajShHXpyGPs.Core.covrG rH }rI bhR]hP]hQ]hU]rJ hajXXRBFunitrK jZUj[uhWNhXhh@]rL (j])rM }rN (hEXclass hFjC hGjOhLj`hN}rO (hP]hQ]hR]hS]hU]uhWNhXhh@]rP haXclass rQ rR }rS (hEUhFjM ubaubjf)rT }rU (hEXpyGPs.Core.cov.hFjC hGjOhLjihN}rV (hP]hQ]hR]hS]hU]uhWNhXhh@]rW haXpyGPs.Core.cov.rX rY }rZ (hEUhFjT ubaubjo)r[ }r\ (hEjK hFjC hGjOhLjrhN}r] (hP]hQ]hR]hS]hU]uhWNhXhh@]r^ haXRBFunitr_ r` }ra (hEUhFj[ ubaubjx)rb }rc (hEUhFjC hGjOhLj{hN}rd (hP]hQ]hR]hS]hU]uhWNhXhh@]re j~)rf }rg (hEX log_ell=0.0hN}rh (hP]hQ]hR]hS]hU]uhFjb h@]ri haX log_ell=0.0rj rk }rl (hEUhFjf ubahLjubaubj)rm }rn (hEUhFjC hGNhLjhN}ro (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rp j)rq }rr (hEUhN}rs (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjK uhFjm h@]rt j)ru }rv (hEUhN}rw (hP]hQ]rx jahR]hS]hU]uhFjq h@]ry haX[source]rz r{ }r| (hEUhFju ubahLjubahLjubaubeubj)r} }r~ (hEUhFj> hGjOhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (hs)r }r (hEXSquared Exponential kernel with isotropic distance measure with unit magnitude. i.e signal variance is always 1. hyp = [ log_ell ]r hFj} hGXQ/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.RBFunitr hLhwhN}r (hP]hQ]hR]hS]hU]uhWKhXhh@]r haXSquared Exponential kernel with isotropic distance measure with unit magnitude. i.e signal variance is always 1. hyp = [ log_ell ]r r }r (hEj hFj ubaubj)r }r (hEUhFj} hGNhLjhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r j)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (j)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX Parametersr r }r (hEUhFj ubahLjubj)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r hs)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFj h@]r (jK)r }r (hEXlog_ellhN}r (hP]hQ]hR]hS]hU]uhFj h@]r haXlog_ellr r }r (hEUhFj ubahLjSubhaX -- r r }r (hEUhFj ubhaXcharacteristic length scale.r r }r (hEXcharacteristic length scale.hFj ubehLhwubahLjubehLjubaubeubeubj3)r }r (hEUhFj$hGNhLj7hN}r (hS]hR]hP]hQ]hU]Uentries]r (j:XRQ (class in pyGPs.Core.cov)hUtr auhWNhXhh@]ubjA)r }r (hEUhFj$hGNhLjDhN}r (jFjGXpyhS]hR]hP]hQ]hU]jHXclassr jJj uhWNhXhh@]r (jL)r }r (hEX-RQ(log_ell=0.0, log_sigma=0.0, log_alpha=0.0)hFj hGjOhLjPhN}r (hS]r hajShHXpyGPs.Core.covr r }r bhR]hP]hQ]hU]r hajXXRQr jZUj[uhWNhXhh@]r (j])r }r (hEXclass hFj hGjOhLj`hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXclass r r }r (hEUhFj ubaubjf)r }r (hEXpyGPs.Core.cov.hFj hGjOhLjihN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXpyGPs.Core.cov.r r }r (hEUhFj ubaubjo)r }r (hEj hFj hGjOhLjrhN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r haXRQr r }r (hEUhFj ubaubjx)r }r (hEUhFj hGjOhLj{hN}r (hP]hQ]hR]hS]hU]uhWNhXhh@]r (j~)r }r (hEX log_ell=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_ell=0.0r r }r (hEUhFj ubahLjubj~)r }r (hEX log_sigma=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_sigma=0.0r r }r (hEUhFj ubahLjubj~)r }r (hEX log_alpha=0.0hN}r (hP]hQ]hR]hS]hU]uhFj h@]r haX log_alpha=0.0r r }r (hEUhFj ubahLjubeubj)r }r (hEUhFj hGNhLjhN}r (UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r j)r }r (hEUhN}r (UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidj uhFj h@]r j)r }r (hEUhN}r (hP]hQ]r jahR]hS]hU]uhFj h@]r haX[source]r r }r (hEUhFj ubahLjubahLjubaubeubj)r}r(hEUhFj hGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r}r(hEXoRational Quadratic covariance function with isotropic distance measure. hyp = [ log_ell, log_sigma, log_alpha ]rhFjhGXL/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.RQrhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]r haXoRational Quadratic covariance function with isotropic distance measure. hyp = [ log_ell, log_sigma, log_alpha ]r r }r (hEjhFjubaubj)r }r(hEUhFjhGNhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFj h@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX Parametersrr}r(hEUhFjubahLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rj )r }r!(hEUhN}r"(hP]hQ]hR]hS]hU]uhFjh@]r#(h)r$}r%(hEUhN}r&(hP]hQ]hR]hS]hU]uhFj h@]r'hs)r(}r)(hEUhN}r*(hP]hQ]hR]hS]hU]uhFj$h@]r+(jK)r,}r-(hEXlog_ellhN}r.(hP]hQ]hR]hS]hU]uhFj(h@]r/haXlog_ellr0r1}r2(hEUhFj,ubahLjSubhaX -- r3r4}r5(hEUhFj(ubhaXcharacteristic length scale.r6r7}r8(hEXcharacteristic length scale.hFj(ubehLhwubahLhubh)r9}r:(hEUhN}r;(hP]hQ]hR]hS]hU]uhFj h@]r<hs)r=}r>(hEUhN}r?(hP]hQ]hR]hS]hU]uhFj9h@]r@(jK)rA}rB(hEX log_sigmahN}rC(hP]hQ]hR]hS]hU]uhFj=h@]rDhaX log_sigmarErF}rG(hEUhFjAubahLjSubhaX -- rHrI}rJ(hEUhFj=ubhaXsignal deviation.rKrL}rM(hEXsignal deviation.hFj=ubehLhwubahLhubh)rN}rO(hEUhN}rP(hP]hQ]hR]hS]hU]uhFj h@]rQhs)rR}rS(hEUhN}rT(hP]hQ]hR]hS]hU]uhFjNh@]rU(jK)rV}rW(hEX log_alphahN}rX(hP]hQ]hR]hS]hU]uhFjRh@]rYhaX log_alpharZr[}r\(hEUhFjVubahLjSubhaX -- r]r^}r_(hEUhFjRubhaX&shape parameter for the RQ covariance.r`ra}rb(hEX&shape parameter for the RQ covariance.hFjRubehLhwubahLhubehLjTubahLjubehLjubaubeubeubj3)rc}rd(hEUhFj$hGNhLj7hN}re(hS]hR]hP]hQ]hU]Uentries]rf(j:XRQard (class in pyGPs.Core.cov)hUtrgauhWNhXhh@]ubjA)rh}ri(hEUhFj$hGNhLjDhN}rj(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrkjJjkuhWNhXhh@]rl(jL)rm}rn(hEX>RQard(D=None, log_ell_list=None, log_sigma=0.0, log_alpha=0.0)hFjhhGjOhLjPhN}ro(hS]rphajShHXpyGPs.Core.covrqrr}rsbhR]hP]hQ]hU]rthajXXRQardrujZUj[uhWNhXhh@]rv(j])rw}rx(hEXclass hFjmhGjOhLj`hN}ry(hP]hQ]hR]hS]hU]uhWNhXhh@]rzhaXclass r{r|}r}(hEUhFjwubaubjf)r~}r(hEXpyGPs.Core.cov.hFjmhGjOhLjihN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXpyGPs.Core.cov.rr}r(hEUhFj~ubaubjo)r}r(hEjuhFjmhGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXRQardrr}r(hEUhFjubaubjx)r}r(hEUhFjmhGjOhLj{hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(j~)r}r(hEXD=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXD=Nonerr}r(hEUhFjubahLjubj~)r}r(hEXlog_ell_list=NonehN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXlog_ell_list=Nonerr}r(hEUhFjubahLjubj~)r}r(hEX log_sigma=0.0hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_sigma=0.0rr}r(hEUhFjubahLjubj~)r}r(hEX log_alpha=0.0hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_alpha=0.0rr}r(hEUhFjubahLjubeubj)r}r(hEUhFjmhGNhLjhN}r(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rj)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjuuhFjh@]rj)r}r(hEUhN}r(hP]hQ]rjahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLjubahLjubaubeubj)r}r(hEUhFjhhGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(hs)r}r(hEXRational Quadratic covariance function with Automatic Relevance Detemination (ARD) distance measure. hyp = log_ell_list + [ log_sigma, log_alpha ]rhFjhGXO/Users/hs/Program/git/pyGPs/pyGPs/Core/cov.py:docstring of pyGPs.Core.cov.RQardrhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaXRational Quadratic covariance function with Automatic Relevance Detemination (ARD) distance measure. hyp = log_ell_list + [ log_sigma, log_alpha ]rr}r(hEjhFjubaubj)r}r(hEUhFjhGNhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(j)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX Parametersrr}r(hEUhFjubahLjubj)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rj )r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(h)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r(hEXDhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXDr}r(hEUhFjubahLjSubhaX -- rr}r(hEUhFjubhaXSdimension of pattern. set if you want default ell, which is 0.5 for each dimension.rr}r(hEXSdimension of pattern. set if you want default ell, which is 0.5 for each dimension.rhFjubehLhwubahLhubh)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFjh@]r(jK)r}r(hEX log_ell_listhN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_ell_listrr}r(hEUhFjubahLjSubhaX -- rr}r(hEUhFjubhaX/characteristic length scale for each dimension.rr}r (hEX/characteristic length scale for each dimension.r hFjubehLhwubahLhubh)r }r (hEUhN}r (hP]hQ]hR]hS]hU]uhFjh@]rhs)r}r(hEUhN}r(hP]hQ]hR]hS]hU]uhFj h@]r(jK)r}r(hEX log_sigmahN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaX log_sigmarr}r(hEUhFjubahLjSubhaX -- rr}r(hEUhFjubhaXsignal deviation.rr}r(hEXsignal deviation.r 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pyGPs.Core.cov.SumOfKernelrhLj7hN}r(hS]hR]hP]hQ]hU]Uentries]r(j:X%SumOfKernel (class in pyGPs.Core.cov)h.UtrauhWNhXhh@]ubjA)r}r(hEUhFj$hGjhLjDhN}r(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrjJjuhWNhXhh@]r(jL)r}r(hEXSumOfKernel(cov1, cov2)hFjhGjOhLjPhN}r(hS]rh.ajShHXpyGPs.Core.covrr}rbhR]hP]hQ]hU]rh.ajXX SumOfKernelrjZUj[uhWNhXhh@]r(j])r}r(hEXclass hFjhGjOhLj`hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXclass rr}r(hEUhFjubaubjf)r}r(hEXpyGPs.Core.cov.hFjhGjOhLjihN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXpyGPs.Core.cov.rr}r(hEUhFjubaubjo)r}r(hEjhFjhGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaX SumOfKernelrr}r(hEUhFjubaubjx)r}r(hEUhFjhGjOhLj{hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r(j~)r}r(hEXcov1hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXcov1rr}r(hEUhFjubahLjubj~)r}r(hEXcov2hN}r(hP]hQ]hR]hS]hU]uhFjh@]rhaXcov2rr}r(hEUhFjubahLjubeubj)r}r(hEUhFjhGNhLjhN}r(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]rj)r}r(hEUhN}r(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/covUrefidjuhFjh@]rj)r}r(hEUhN}r(hP]hQ]rjahR]hS]hU]uhFjh@]rhaX[source]rr}r(hEUhFjubahLjubahLjubaubeubj)r}r(hEUhFjhGjOhLjhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhs)r}r(hEXSum of two kernel function.rhFjhGjhLhwhN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaXSum of two kernel function.rr}r(hEjhFjubaubaubeubeubhB)r}r(hEUhFhChGhJhLhMhN}r(hP]hQ]hR]hS]r(Xmodule-pyGPs.Core.meanrh;ehU]rhauhWKhXhh@]r(hZ)r}r(hEX$List of Means and Default ParametersrhFjhGhJhLh^hN}r(hP]hQ]hR]hS]hU]uhWKhXhh@]rhaX$List of Means and Default Parametersrr}r(hEjhFjubaubj3)r}r(hEUhFjhGj6hLj7hN}r(hS]hR]hP]hQ]hU]Uentries]r(j:XpyGPs.Core.mean (module)Xmodule-pyGPs.Core.meanUtrauhWKhXhh@]ubj3)r}r(hEUhFjhGNhLj7hN}r(hS]hR]hP]hQ]hU]Uentries]r(j:X Const (class in pyGPs.Core.mean)h"UtrauhWNhXhh@]ubjA)r}r(hEUhFjhGNhLjDhN}r(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrjJjuhWNhXhh@]r(jL)r}r(hEX Const(c=5.0)hFjhGjOhLjPhN}r(hS]rh"ajShHXpyGPs.Core.meanr r }r bhR]hP]hQ]hU]r h"ajXXConstr jZUj[uhWNhXhh@]r(j])r}r(hEXclass hFjhGjOhLj`hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXclass rr}r(hEUhFjubaubjf)r}r(hEXpyGPs.Core.mean.hFjhGjOhLjihN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXpyGPs.Core.mean.rr}r(hEUhFjubaubjo)r}r(hEj hFjhGjOhLjrhN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]r haXConstr!r"}r#(hEUhFjubaubjx)r$}r%(hEUhFjhGjOhLj{hN}r&(hP]hQ]hR]hS]hU]uhWNhXhh@]r'j~)r(}r)(hEXc=5.0hN}r*(hP]hQ]hR]hS]hU]uhFj$h@]r+haXc=5.0r,r-}r.(hEUhFj(ubahLjubaubj)r/}r0(hEUhFjhGNhLjhN}r1(UexprjhS]hR]hP]hQ]hU]uhWNhXhh@]r2j)r3}r4(hEUhN}r5(UreftypejUrefdocjU refdomainjhS]hR]U refexplicithP]hQ]hU]U reftargetX_modules/pyGPs/Core/meanUrefidj uhFj/h@]r6j)r7}r8(hEUhN}r9(hP]hQ]r:jahR]hS]hU]uhFj3h@]r;haX[source]r<r=}r>(hEUhFj7ubahLjubahLjubaubeubj)r?}r@(hEUhFjhGjOhLjhN}rA(hP]hQ]hR]hS]hU]uhWNhXhh@]rB(hs)rC}rD(hEX!Constant mean function. hyp = [c]rEhFj?hGXQ/Users/hs/Program/git/pyGPs/pyGPs/Core/mean.py:docstring of pyGPs.Core.mean.ConstrFhLhwhN}rG(hP]hQ]hR]hS]hU]uhWKhXhh@]rHhaX!Constant mean function. hyp = [c]rIrJ}rK(hEjEhFjCubaubj)rL}rM(hEUhFj?hGNhLjhN}rN(hP]hQ]hR]hS]hU]uhWNhXhh@]rOj)rP}rQ(hEUhN}rR(hP]hQ]hR]hS]hU]uhFjLh@]rS(j)rT}rU(hEUhN}rV(hP]hQ]hR]hS]hU]uhFjPh@]rWhaX ParametersrXrY}rZ(hEUhFjTubahLjubj)r[}r\(hEUhN}r](hP]hQ]hR]hS]hU]uhFjPh@]r^hs)r_}r`(hEUhN}ra(hP]hQ]hR]hS]hU]uhFj[h@]rb(jK)rc}rd(hEXchN}re(hP]hQ]hR]hS]hU]uhFj_h@]rfhaXcrg}rh(hEUhFjcubahLjSubhaX -- rirj}rk(hEUhFj_ubhaXconstant value for meanrlrm}rn(hEXconstant value for meanhFj_ubehLhwubahLjubehLjubaubeubeubj3)ro}rp(hEUhFjhGNhLj7hN}rq(hS]hR]hP]hQ]hU]Uentries]rr(j:X!Linear (class in pyGPs.Core.mean)h'UtrsauhWNhXhh@]ubjA)rt}ru(hEUhFjhGNhLjDhN}rv(jFjGXpyhS]hR]hP]hQ]hU]jHXclassrwjJjwuhWNhXhh@]rx(jL)ry}rz(hEXLinear(D=None, alpha_list=None)hFjthGjOhLjPhN}r{(hS]r|h'ajShHXpyGPs.Core.meanr}r~}rbhR]hP]hQ]hU]rh'ajXXLinearrjZUj[uhWNhXhh@]r(j])r}r(hEXclass hFjyhGjOhLj`hN}r(hP]hQ]hR]hS]hU]uhWNhXhh@]rhaXclass 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inference.qh0}q(h2]h3]h4]h5]h7]uh(hqh"]qh|)q}q(h'hh(hh)h,h.hh0}q(h2]h3]h4]h5]h7]uh9Kh"]qhCX3GPC uses Error functionlikelihood and EP inference.qq}q(h'hh(hubaubah.hubhv)q}q(h'X?FITC model uses same default with corresponding FITC inference.qh0}q(h2]h3]h4]h5]h7]uh(hqh"]qh|)q}q(h'hh(hh)h,h.hh0}q(h2]h3]h4]h5]h7]uh9K h"]qhCX?FITC model uses same default with corresponding FITC inference.qq}q(h'hh(hubaubah.hubhv)q}q(h'X8GPMC calls GPC and thus uses the default setting of GPC h0}q(h2]h3]h4]h5]h7]uh(hqh"]qh|)q}q(h'X7GPMC calls GPC and thus uses the default setting of GPCqh(hh)h,h.hh0}q(h2]h3]h4]h5]h7]uh9K h"]qhCX7GPMC calls GPC and thus uses the default setting of GPCqq}q(h'hh(hubaubah.hubeh.U bullet_listqubah.U definitionqubeubaubh|)q}q(h'XAYou can change to other likelihood or inference methods using: ::qh(hGh)h,h.hh0}q(h2]h3]h4]h5]h7]uh9K h:hh"]qhCX>You can change to other likelihood or inference methods using:qq}q(h'X>You can change to other likelihood or inference methods using:h(hubaubcdocutils.nodes literal_block q)q}q(h'X6model.useLikelihood(newLik) model.useInference(newInf)h(hGh)h,h.U literal_blockqh0}q(U xml:spaceqUpreserveqh5]h4]h2]h3]h7]uh9Kh:hh"]qhCX6model.useLikelihood(newLik) model.useInference(newInf)qq}q(h'Uh(hubaubhU)q}q(h'Uh(hGh)Nh.hXh0}q(h2]h3]h4]h5]h7]uh9Nh:hh"]qh[)q}q(h'X*newLik* and *newInf* are **Strings**. Currently the options are: 1. Regression model * newLik: **"Laplace"**. Note this will force inference method to be EP. * newInf: **"EP"**, **"Laplace"**. 2. Classification model (including GPMC) * newInf: **"Laplace"** h(hh)h,h.h^h0}q(h2]h3]h4]h5]h7]uh9Kh"]q(ha)q}q(h'XA*newLik* and *newInf* are **Strings**. Currently the options are:qh(hh)h,h.heh0}q(h2]h3]h4]h5]h7]uh9Kh"]q(cdocutils.nodes emphasis q)q}q(h'X*newLik*h0}q(h2]h3]h4]h5]h7]uh(hh"]qhCXnewLikqхq}q(h'Uh(hubah.UemphasisqubhCX and qՅq}q(h'X and h(hubh)q}q(h'X*newInf*h0}q(h2]h3]h4]h5]h7]uh(hh"]qhCXnewInfq܅q}q(h'Uh(hubah.hubhCX are q߅q}q(h'X are h(hubcdocutils.nodes strong q)q}q(h'X **Strings**h0}q(h2]h3]h4]h5]h7]uh(hh"]qhCXStringsq煁q}q(h'Uh(hubah.UstrongqubhCX. Currently the options are:q녁q}q(h'X. Currently the options are:h(hubeubhk)q}q(h'Uh0}q(h2]h3]h4]h5]h7]uh(hh"]qcdocutils.nodes enumerated_list q)q}q(h'Uh0}q(UsuffixqU.h5]h4]h2]UprefixqUh3]h7]UenumtypeqUarabicquh(hh"]q(hv)q}q(h'X~Regression model * newLik: **"Laplace"**. Note this will force inference method to be EP. * newInf: **"EP"**, **"Laplace"**. h0}q(h2]h3]h4]h5]h7]uh(hh"]q(h|)q}r(h'XRegression modelrh(hh)h,h.hh0}r(h2]h3]h4]h5]h7]uh9Kh"]rhCXRegression modelrr}r(h'jh(hubaubhp)r}r(h'Uh0}r (htX*h5]h4]h2]h3]h7]uh(hh"]r (hv)r }r (h'XFnewLik: **"Laplace"**. Note this will force inference method to be EP.r h0}r(h2]h3]h4]h5]h7]uh(jh"]rh|)r}r(h'j h(j h)h,h.hh0}r(h2]h3]h4]h5]h7]uh9Kh"]r(hCXnewLik: rr}r(h'XnewLik: h(jubh)r}r(h'X **"Laplace"**h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCX "Laplace"rr}r(h'Uh(jubah.hubhCX1. Note this will force inference method to be EP.rr}r (h'X1. Note this will force inference method to be EP.h(jubeubah.hubhv)r!}r"(h'X!newInf: **"EP"**, **"Laplace"**. h0}r#(h2]h3]h4]h5]h7]uh(jh"]r$h|)r%}r&(h'X newInf: **"EP"**, **"Laplace"**.h(j!h)h,h.hh0}r'(h2]h3]h4]h5]h7]uh9Kh"]r((hCXnewInf: r)r*}r+(h'XnewInf: h(j%ubh)r,}r-(h'X**"EP"**h0}r.(h2]h3]h4]h5]h7]uh(j%h"]r/hCX"EP"r0r1}r2(h'Uh(j,ubah.hubhCX, r3r4}r5(h'X, h(j%ubh)r6}r7(h'X **"Laplace"**h0}r8(h2]h3]h4]h5]h7]uh(j%h"]r9hCX "Laplace"r:r;}r<(h'Uh(j6ubah.hubhCX.r=}r>(h'X.h(j%ubeubah.hubeh.hubeh.hubhv)r?}r@(h'XAClassification model (including GPMC) * newInf: **"Laplace"** h0}rA(h2]h3]h4]h5]h7]uh(hh"]rB(h|)rC}rD(h'X%Classification model (including GPMC)rEh(j?h)h,h.hh0}rF(h2]h3]h4]h5]h7]uh9Kh"]rGhCX%Classification model (including GPMC)rHrI}rJ(h'jEh(jCubaubhp)rK}rL(h'Uh0}rM(htX*h5]h4]h2]h3]h7]uh(j?h"]rNhv)rO}rP(h'XnewInf: **"Laplace"** h0}rQ(h2]h3]h4]h5]h7]uh(jKh"]rRh|)rS}rT(h'XnewInf: **"Laplace"**rUh(jOh)h,h.hh0}rV(h2]h3]h4]h5]h7]uh9Kh"]rW(hCXnewInf: rXrY}rZ(h'XnewInf: h(jSubh)r[}r\(h'X **"Laplace"**h0}r](h2]h3]h4]h5]h7]uh(jSh"]r^hCX "Laplace"r_r`}ra(h'Uh(j[ubah.hubeubah.hubah.hubeh.hubeh.Uenumerated_listrbubah.hubeubaubeubh$)rc}rd(h'Uh(h%h)h,h.h/h0}re(h2]h3]h4]h5]rfhah7]rghauh9Kh:hh"]rh(h<)ri}rj(h'X/Developing New Likelihood & Inference Functionsrkh(jch)h,h.h@h0}rl(h2]h3]h4]h5]h7]uh9Kh:hh"]rmhCX/Developing New Likelihood & Inference Functionsrnro}rp(h'jkh(jiubaubh|)rq}rr(h'XWe also support the development of new likelihood/inference classes, your customized inference class need to follow the template as below: ::rsh(jch)h,h.hh0}rt(h2]h3]h4]h5]h7]uh9Kh:hh"]ruhCXWe also support the development of new likelihood/inference classes, your customized inference class need to follow the template as below:rvrw}rx(h'XWe also support the development of new likelihood/inference classes, your customized inference class need to follow the template as below:h(jqubaubh)ry}rz(h'X# Your inference class needs to inherit base class Inference, # which is in the module of Core.inf class MyKernel(Kernel): def __init__(self): pass def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): ''' Inference computation based on inputs. post, nlZ, dnlZ = inf.proceed(mean, cov, lik, x, y) INPUT: cov: name of the covariance function (see covFunctions.m) lik: name of the likelihood function (see likFunctions.m) x: n by D matrix of training inputs y: 1d array (of size n) of targets OUTPUT: post(instance of postStruct): struct representation of the (approximate) posterior containing: nlZ: returned value of the negative log marginal likelihood dnlZ(instance of dnlZStruct): struct representation for derivatives of the negative log marginal likelihood w.r.t. each hyperparameter. Usually, the approximate posterior to be returned admits the form: N(m=K*alpha, V=inv(inv(K)+W)), where alpha is a vector and W is diagonal; if not, then L contains instead -inv(K+inv(W)), and sW is unused. For more information on the individual approximation methods and their implementations, see the respective inference function below. See also gp.py :param meanfunc: mean function :param covfunc: covariance function :param likfunc: likelihood function :param x: training data :param y: training labels :param nargout: specify the number of output(1,2 or 3) :return: posterior, negative-log-marginal-likelihood, derivative for negative-log-marginal-likelihood-likelihood ''' passh(jch)h,h.hh0}r{(hhh5]h4]h2]h3]h7]uh9K!h:hh"]r|hCX# Your inference class needs to inherit base class Inference, # which is in the module of Core.inf class MyKernel(Kernel): def __init__(self): pass def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): ''' Inference computation based on inputs. post, nlZ, dnlZ = inf.proceed(mean, cov, lik, x, y) INPUT: cov: name of the covariance function (see covFunctions.m) lik: name of the likelihood function (see likFunctions.m) x: n by D matrix of training inputs y: 1d array (of size n) of targets OUTPUT: post(instance of postStruct): struct representation of the (approximate) posterior containing: nlZ: returned value of the negative log marginal likelihood dnlZ(instance of dnlZStruct): struct representation for derivatives of the negative log marginal likelihood w.r.t. each hyperparameter. Usually, the approximate posterior to be returned admits the form: N(m=K*alpha, V=inv(inv(K)+W)), where alpha is a vector and W is diagonal; if not, then L contains instead -inv(K+inv(W)), and sW is unused. For more information on the individual approximation methods and their implementations, see the respective inference function below. See also gp.py :param meanfunc: mean function :param covfunc: covariance function :param likfunc: likelihood function :param x: training data :param y: training labels :param nargout: specify the number of output(1,2 or 3) :return: posterior, negative-log-marginal-likelihood, derivative for negative-log-marginal-likelihood-likelihood ''' passr}r~}r(h'Uh(jyubaubh|)r}r(h'XIwhere **postStruct** and **dnlZStruct** is also defined in *Core.inf*. ::rh(jch)h,h.hh0}r(h2]h3]h4]h5]h7]uh9KJh:hh"]r(hCXwhere rr}r(h'Xwhere h(jubh)r}r(h'X**postStruct**h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCX postStructrr}r(h'Uh(jubah.hubhCX and rr}r(h'X and h(jubh)r}r(h'X**dnlZStruct**h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCX dnlZStructrr}r(h'Uh(jubah.hubhCX is also defined in rr}r(h'X is also defined in h(jubh)r}r(h'X *Core.inf*h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCXCore.infrr}r(h'Uh(jubah.hubhCX.r}r(h'X.h(jubeubh)r}r(h'Xclass postStruct(object): ''' Data structure for posterior post.alpha -> 1d array containing inv(K)*m, where K is the prior covariance matrix and m the approx posterior mean post.sW: -> 1d array containing diagonal of sqrt(W) the approximate posterior covariance matrix is inv(inv(K)+W) post.L -> 2d array, L = chol(sW*K*sW+identity(n)) ''' class dnlZStruct(object): ''' Data structure for the derivatives of mean, cov and lik functions. dnlZ.mean -> list of derivatives for each hyperparameters in mean function dnlZ.cov -> list of derivatives for each hyperparameters in covariance function dnlZ.lik -> list of derivatives for each hyperparameters in likelihood function '''h(jch)h,h.hh0}r(hhh5]h4]h2]h3]h7]uh9KLh:hh"]rhCXclass postStruct(object): ''' Data structure for posterior post.alpha -> 1d array containing inv(K)*m, where K is the prior covariance matrix and m the approx posterior mean post.sW: -> 1d array containing diagonal of sqrt(W) the approximate posterior covariance matrix is inv(inv(K)+W) post.L -> 2d array, L = chol(sW*K*sW+identity(n)) ''' class dnlZStruct(object): ''' Data structure for the derivatives of mean, cov and lik functions. dnlZ.mean -> list of derivatives for each hyperparameters in mean function dnlZ.cov -> list of derivatives for each hyperparameters in covariance function dnlZ.lik -> list of derivatives for each hyperparameters in likelihood function '''rr}r(h'Uh(jubaubh|)r}r(h'XCustomizing likelihood function is more complicated. We will omit it here to keep this this page not too long. However, you can find detailed explaination either in the **source code** *Core.lik* or in coresponding section of **manual**.h(jch)h,h.hh0}r(h2]h3]h4]h5]h7]uh9Kah:hh"]r(hCXCustomizing likelihood function is more complicated. We will omit it here to keep this this page not too long. However, you can find detailed explaination either in the rr}r(h'XCustomizing likelihood function is more complicated. We will omit it here to keep this this page not too long. However, you can find detailed explaination either in the h(jubh)r}r(h'X**source code**h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCX source coderr}r(h'Uh(jubah.hubhCX r}r(h'X h(jubh)r}r(h'X *Core.lik*h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCXCore.likrr}r(h'Uh(jubah.hubhCX or in coresponding section of rr}r(h'X or in coresponding section of h(jubh)r}r(h'X **manual**h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCXmanualrr}r(h'Uh(jubah.hubhCX.r}r(h'X.h(jubeubh|)r}r(h'XJust like testing kernel/mean fucntions, you can also find unit test module for likelihood and inference functions. To test your customized inference function, uncomment the following method in *pyGPs.Testing.unit_test_inf.py*. ::h(jch)h,h.hh0}r(h2]h3]h4]h5]h7]uh9Kdh:hh"]r(hCXJust like testing kernel/mean fucntions, you can also find unit test module for likelihood and inference functions. To test your customized inference function, uncomment the following method in rr}r(h'XJust like testing kernel/mean fucntions, you can also find unit test module for likelihood and inference functions. To test your customized inference function, uncomment the following method in h(jubh)r}r(h'X *pyGPs.Testing.unit_test_inf.py*h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCXpyGPs.Testing.unit_test_inf.pyrr}r(h'Uh(jubah.hubhCX.r}r(h'X.h(jubeubh)r}r(h'X # Test your customized inference function def test_inf_new(self): # specify your inf function # set mean/cov/lik functions post, nlZ, dnlZ = inffunc.proceed(meanfunc, covfunc, likfunc, self.x, self.y, nargout=3) self.checkFITCOutput(post, nlZ, dnlZ)h(jch)h,h.hh0}r(hhh5]h4]h2]h3]h7]uh9Kgh:hh"]rhCX # Test your customized inference function def test_inf_new(self): # specify your inf function # set mean/cov/lik functions post, nlZ, dnlZ = inffunc.proceed(meanfunc, covfunc, likfunc, self.x, self.y, nargout=3) self.checkFITCOutput(post, nlZ, dnlZ)rr}r(h'Uh(jubaubh|)r}r(h'XNand test customized likelihood function in *pyGPs.Testing.unit_test_lik.py* ::rh(jch)h,h.hh0}r(h2]h3]h4]h5]h7]uh9Knh:hh"]r(hCX+and test customized likelihood function in rr}r(h'X+and test customized likelihood function in h(jubh)r}r(h'X *pyGPs.Testing.unit_test_lik.py*h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCXpyGPs.Testing.unit_test_lik.pyrr}r(h'Uh(jubah.hubeubh)r}r(h'X# Test your customized likelihood function def test_cov_new(self): likelihood = myLikelihood() # specify your likelihood function self.checkLikelihood(likelihood)h(jch)h,h.hh0}r(hhh5]h4]h2]h3]h7]uh9Kph:hh"]rhCX# Test your customized likelihood function def test_cov_new(self): likelihood = myLikelihood() # specify your likelihood function self.checkLikelihood(likelihood)rr}r(h'Uh(jubaubeubh$)r}r(h'Uh(h%h)h,h.h/h0}r(h2]h3]h4]h5]rh!ah7]rhauh9Kxh:hh"]r(h<)r}r(h'XList of Likelihoodsrh(jh)h,h.h@h0}r(h2]h3]h4]h5]h7]uh9Kxh:hh"]rhCXList of Likelihoodsrr}r (h'jh(jubaubcsphinx.addnodes index r )r }r (h'Uh(jh)XM/Users/hs/Program/git/pyGPs/pyGPs/Core/lik.py:docstring of pyGPs.Core.lik.Erfr h.Uindexrh0}r(h5]h4]h2]h3]h7]Uentries]r(UsinglerXErf (class in pyGPs.Core.lik)h Utrauh9Nh:hh"]ubcsphinx.addnodes desc r)r}r(h'Uh(jh)j h.Udescrh0}r(UnoindexrUdomainrXpyh5]h4]h2]h3]h7]UobjtyperXclassrUdesctyperjuh9Nh:hh"]r(csphinx.addnodes desc_signature r)r}r (h'XErf()h(jh)U r!h.Udesc_signaturer"h0}r#(h5]r$h aUmoduler%h*XpyGPs.Core.likr&r'}r(bh4]h2]h3]h7]r)h aUfullnamer*XErfr+Uclassr,UUfirstr-uh9Nh:hh"]r.(csphinx.addnodes desc_annotation r/)r0}r1(h'Xclass h(jh)j!h.Udesc_annotationr2h0}r3(h2]h3]h4]h5]h7]uh9Nh:hh"]r4hCXclass r5r6}r7(h'Uh(j0ubaubcsphinx.addnodes desc_addname r8)r9}r:(h'XpyGPs.Core.lik.h(jh)j!h.U desc_addnamer;h0}r<(h2]h3]h4]h5]h7]uh9Nh:hh"]r=hCXpyGPs.Core.lik.r>r?}r@(h'Uh(j9ubaubcsphinx.addnodes desc_name rA)rB}rC(h'j+h(jh)j!h.U desc_namerDh0}rE(h2]h3]h4]h5]h7]uh9Nh:hh"]rFhCXErfrGrH}rI(h'Uh(jBubaubcsphinx.addnodes only rJ)rK}rL(h'Uh(jh)Nh.UonlyrMh0}rN(UexprUhtmlrOh5]h4]h2]h3]h7]uh9Nh:hh"]rPcsphinx.addnodes pending_xref rQ)rR}rS(h'Uh0}rT(UreftypeUviewcoderUUrefdocX LikelihoodsrVU refdomainUstdrWh5]h4]U refexplicith2]h3]h7]U reftargetX_modules/pyGPs/Core/likUrefidj+uh(jKh"]rXcdocutils.nodes inline rY)rZ}r[(h'Uh0}r\(h2]h3]r]U viewcode-linkr^ah4]h5]h7]uh(jRh"]r_hCX[source]r`ra}rb(h'Uh(jZubah.Uinlinercubah.U pending_xrefrdubaubeubcsphinx.addnodes desc_content re)rf}rg(h'Uh(jh)j!h.U desc_contentrhh0}ri(h2]h3]h4]h5]h7]uh9Nh:hh"]rj(h|)rk}rl(h'XiError function or cumulative Gaussian likelihood function for binary classification or probit regression.rmh(jfh)j h.hh0}rn(h2]h3]h4]h5]h7]uh9Kh:hh"]rohCXiError function or cumulative Gaussian likelihood function for binary classification or probit regression.rprq}rr(h'jmh(jkubaubh|)rs}rt(h'X@:math:`Erf(t)=\frac{1}{2}(1+erf(\frac{t}{\sqrt{2}}))=normcdf(t)`h(jfh)j h.hh0}ru(h2]h3]h4]h5]h7]uh9Kh:hh"]rvcsphinx.ext.mathbase math rw)rx}ry(h'Uh0}rz(UlatexX8Erf(t)=\frac{1}{2}(1+erf(\frac{t}{\sqrt{2}}))=normcdf(t)h5]h4]h2]h3]h7]uh(jsh"]h.Umathr{ubaubeubeubj )r|}r}(h'Uh(jh)XO/Users/hs/Program/git/pyGPs/pyGPs/Core/lik.py:docstring of pyGPs.Core.lik.Gaussr~h.jh0}r(h5]h4]h2]h3]h7]Uentries]r(jXGauss (class in pyGPs.Core.lik)hUtrauh9Nh:hh"]ubj)r}r(h'Uh(jh)j~h.jh0}r(jjXpyh5]h4]h2]h3]h7]jXclassrjjuh9Nh:hh"]r(j)r}r(h'X$Gauss(log_sigma=-2.3025850929940455)h(jh)j!h.j"h0}r(h5]rhaj%h*XpyGPs.Core.likrr}rbh4]h2]h3]h7]rhaj*XGaussrj,Uj-uh9Nh:hh"]r(j/)r}r(h'Xclass h(jh)j!h.j2h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXclass rr}r(h'Uh(jubaubj8)r}r(h'XpyGPs.Core.lik.h(jh)j!h.j;h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXpyGPs.Core.lik.rr}r(h'Uh(jubaubjA)r}r(h'jh(jh)j!h.jDh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXGaussrr}r(h'Uh(jubaubcsphinx.addnodes desc_parameterlist r)r}r(h'Uh(jh)j!h.Udesc_parameterlistrh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rcsphinx.addnodes desc_parameter r)r}r(h'Xlog_sigma=-2.3025850929940455h0}r(h2]h3]h4]h5]h7]uh(jh"]rhCXlog_sigma=-2.3025850929940455rr}r(h'Uh(jubah.Udesc_parameterrubaubjJ)r}r(h'Uh(jh)Nh.jMh0}r(UexprjOh5]h4]h2]h3]h7]uh9Nh:hh"]rjQ)r}r(h'Uh0}r(UreftypejUUrefdocjVU refdomainjWh5]h4]U refexplicith2]h3]h7]U reftargetX_modules/pyGPs/Core/likUrefidjuh(jh"]rjY)r}r(h'Uh0}r(h2]h3]rj^ah4]h5]h7]uh(jh"]rhCX[source]rr}r(h'Uh(jubah.jcubah.jdubaubeubje)r}r(h'Uh(jh)j!h.jhh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]r(h|)r}r(h'X,Gaussian likelihood function for regression.rh(jh)j~h.hh0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]rhCX,Gaussian likelihood function for regression.rr}r(h'jh(jubaubh|)r}r(h'X:math:`Gauss(t)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(t-y)^2}{2\sigma^2}}`, where :math:`y` is the mean and :math:`\sigma` is the standard deviation.h(jh)j~h.hh0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]r(jw)r}r(h'Uh0}r(UlatexXDGauss(t)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(t-y)^2}{2\sigma^2}}h5]h4]h2]h3]h7]uh(jh"]h.j{ubhCX, where rr}r(h'X, where h(jubjw)r}r(h'Uh0}r(UlatexXyh5]h4]h2]h3]h7]uh(jh"]h.j{ubhCX is the mean and rr}r(h'X is the mean and h(jubjw)r}r(h'Uh0}r(UlatexX\sigmah5]h4]h2]h3]h7]uh(jh"]h.j{ubhCX is the standard deviation.rr}r(h'X is the standard deviation.h(jubeubh|)r}r(h'Xhyp = [ log_sigma ]rh(jh)j~h.hh0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]rhCXhyp = [ log_sigma ]rr}r(h'jh(jubaubeubeubj )r}r(h'Uh(jh)XQ/Users/hs/Program/git/pyGPs/pyGPs/Core/lik.py:docstring of pyGPs.Core.lik.Laplacerh.jh0}r(h5]h4]h2]h3]h7]Uentries]r(jX!Laplace (class in pyGPs.Core.lik)hUtrauh9Nh:hh"]ubj)r}r(h'Uh(jh)jh.jh0}r(jjXpyh5]h4]h2]h3]h7]jXclassrjjuh9Nh:hh"]r(j)r}r(h'X&Laplace(log_sigma=-2.3025850929940455)h(jh)j!h.j"h0}r(h5]rhaj%h*XpyGPs.Core.likrr}rbh4]h2]h3]h7]rhaj*XLaplacerj,Uj-uh9Nh:hh"]r(j/)r}r(h'Xclass h(jh)j!h.j2h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXclass rr }r (h'Uh(jubaubj8)r }r (h'XpyGPs.Core.lik.h(jh)j!h.j;h0}r (h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXpyGPs.Core.lik.rr}r(h'Uh(j ubaubjA)r}r(h'jh(jh)j!h.jDh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXLaplacerr}r(h'Uh(jubaubj)r}r(h'Uh(jh)j!h.jh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rj)r}r(h'Xlog_sigma=-2.3025850929940455h0}r(h2]h3]h4]h5]h7]uh(jh"]r hCXlog_sigma=-2.3025850929940455r!r"}r#(h'Uh(jubah.jubaubjJ)r$}r%(h'Uh(jh)Nh.jMh0}r&(UexprjOh5]h4]h2]h3]h7]uh9Nh:hh"]r'jQ)r(}r)(h'Uh0}r*(UreftypejUUrefdocjVU refdomainjWh5]h4]U refexplicith2]h3]h7]U reftargetX_modules/pyGPs/Core/likUrefidjuh(j$h"]r+jY)r,}r-(h'Uh0}r.(h2]h3]r/j^ah4]h5]h7]uh(j(h"]r0hCX[source]r1r2}r3(h'Uh(j,ubah.jcubah.jdubaubeubje)r4}r5(h'Uh(jh)j!h.jhh0}r6(h2]h3]h4]h5]h7]uh9Nh:hh"]r7(h|)r8}r9(h'XKLaplacian likelihood function for regression. ONLY works with EP inference!r:h(j4h)jh.hh0}r;(h2]h3]h4]h5]h7]uh9Kh:hh"]r<hCXKLaplacian likelihood function for regression. ONLY works with EP inference!r=r>}r?(h'j:h(j8ubaubh|)r@}rA(h'X:math:`Laplace(t) = \frac{1}{2b}e^{-\frac{|t-y|}{b}}` where :math:`b=\frac{\sigma}{\sqrt{2}}`, :math:`y` is the mean and :math:`\sigma` is the standard deviation.h(j4h)jh.hh0}rB(h2]h3]h4]h5]h7]uh9Kh:hh"]rC(jw)rD}rE(h'Uh0}rF(UlatexX-Laplace(t) = \frac{1}{2b}e^{-\frac{|t-y|}{b}}h5]h4]h2]h3]h7]uh(j@h"]h.j{ubhCX where rGrH}rI(h'X where h(j@ubjw)rJ}rK(h'Uh0}rL(UlatexXb=\frac{\sigma}{\sqrt{2}}h5]h4]h2]h3]h7]uh(j@h"]h.j{ubhCX, rMrN}rO(h'X, h(j@ubjw)rP}rQ(h'Uh0}rR(UlatexXyh5]h4]h2]h3]h7]uh(j@h"]h.j{ubhCX is the mean and rSrT}rU(h'X is the mean and h(j@ubjw)rV}rW(h'Uh0}rX(UlatexX\sigmah5]h4]h2]h3]h7]uh(j@h"]h.j{ubhCX is the standard deviation.rYrZ}r[(h'X is the standard deviation.h(j@ubeubh|)r\}r](h'Xhyp = [ log_sigma ]r^h(j4h)jh.hh0}r_(h2]h3]h4]h5]h7]uh9Kh:hh"]r`hCXhyp = [ log_sigma ]rarb}rc(h'j^h(j\ubaubeubeubeubh$)rd}re(h'Uh(h%h)h,h.h/h0}rf(h2]h3]h4]h5]rgh ah7]rhhauh9Kh:hh"]ri(h<)rj}rk(h'XList of Inferencerlh(jdh)h,h.h@h0}rm(h2]h3]h4]h5]h7]uh9Kh:hh"]rnhCXList of Inferencerorp}rq(h'jlh(jjubaubj )rr}rs(h'Uh(jdh)XO/Users/hs/Program/git/pyGPs/pyGPs/Core/inf.py:docstring of pyGPs.Core.inf.Exactrth.jh0}ru(h5]h4]h2]h3]h7]Uentries]rv(jXExact (class in pyGPs.Core.inf)hUtrwauh9Nh:hh"]ubj)rx}ry(h'Uh(jdh)jth.jh0}rz(jjXpyh5]h4]h2]h3]h7]jXclassr{jj{uh9Nh:hh"]r|(j)r}}r~(h'XExact()h(jxh)j!h.j"h0}r(h5]rhaj%h*XpyGPs.Core.infrr}rbh4]h2]h3]h7]rhaj*XExactrj,Uj-uh9Nh:hh"]r(j/)r}r(h'Xclass h(j}h)j!h.j2h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXclass rr}r(h'Uh(jubaubj8)r}r(h'XpyGPs.Core.inf.h(j}h)j!h.j;h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXpyGPs.Core.inf.rr}r(h'Uh(jubaubjA)r}r(h'jh(j}h)j!h.jDh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXExactrr}r(h'Uh(jubaubjJ)r}r(h'Uh(j}h)Nh.jMh0}r(UexprjOh5]h4]h2]h3]h7]uh9Nh:hh"]rjQ)r}r(h'Uh0}r(UreftypejUUrefdocjVU refdomainjWh5]h4]U refexplicith2]h3]h7]U reftargetX_modules/pyGPs/Core/infUrefidjuh(jh"]rjY)r}r(h'Uh0}r(h2]h3]rj^ah4]h5]h7]uh(jh"]rhCX[source]rr}r(h'Uh(jubah.jcubah.jdubaubeubje)r}r(h'Uh(jxh)j!h.jhh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rh|)r}r(h'XExact inference for a GP with Gaussian likelihood. Compute a parametrization of the posterior, the negative log marginal likelihood and its derivatives w.r.t. the hyperparameters.rh(jh)jth.hh0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]rhCXExact inference for a GP with Gaussian likelihood. Compute a parametrization of the posterior, the negative log marginal likelihood and its derivatives w.r.t. the hyperparameters.rr}r(h'jh(jubaubaubeubj )r}r(h'Uh(jdh)XL/Users/hs/Program/git/pyGPs/pyGPs/Core/inf.py:docstring of pyGPs.Core.inf.EPrh.jh0}r(h5]h4]h2]h3]h7]Uentries]r(jXEP (class in pyGPs.Core.inf)h Utrauh9Nh:hh"]ubj)r}r(h'Uh(jdh)jh.jh0}r(jjXpyh5]h4]h2]h3]h7]jXclassrjjuh9Nh:hh"]r(j)r}r(h'XEP()h(jh)j!h.j"h0}r(h5]rh aj%h*XpyGPs.Core.infrr}rbh4]h2]h3]h7]rh aj*XEPrj,Uj-uh9Nh:hh"]r(j/)r}r(h'Xclass h(jh)j!h.j2h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXclass rr}r(h'Uh(jubaubj8)r}r(h'XpyGPs.Core.inf.h(jh)j!h.j;h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXpyGPs.Core.inf.rr}r(h'Uh(jubaubjA)r}r(h'jh(jh)j!h.jDh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXEPrr}r(h'Uh(jubaubjJ)r}r(h'Uh(jh)Nh.jMh0}r(UexprjOh5]h4]h2]h3]h7]uh9Nh:hh"]rjQ)r}r(h'Uh0}r(UreftypejUUrefdocjVU refdomainjWh5]h4]U refexplicith2]h3]h7]U reftargetX_modules/pyGPs/Core/infUrefidjuh(jh"]rjY)r}r(h'Uh0}r(h2]h3]rj^ah4]h5]h7]uh(jh"]rhCX[source]rr}r(h'Uh(jubah.jcubah.jdubaubeubje)r}r(h'Uh(jh)j!h.jhh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rh|)r}r(h'XHExpectation Propagation approximation to the posterior Gaussian Process.rh(jh)jh.hh0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]rhCXHExpectation Propagation approximation to the posterior Gaussian Process.rr}r(h'jh(jubaubaubeubj )r}r(h'Uh(jdh)XQ/Users/hs/Program/git/pyGPs/pyGPs/Core/inf.py:docstring of pyGPs.Core.inf.Laplacerh.jh0}r(h5]h4]h2]h3]h7]Uentries]r(jX!Laplace (class in pyGPs.Core.inf)hUtrauh9Nh:hh"]ubj)r}r(h'Uh(jdh)jh.jh0}r(jjXpyh5]h4]h2]h3]h7]jXclassrjjuh9Nh:hh"]r(j)r }r (h'X Laplace()h(jh)j!h.j"h0}r (h5]r haj%h*XpyGPs.Core.infr r}rbh4]h2]h3]h7]rhaj*XLaplacerj,Uj-uh9Nh:hh"]r(j/)r}r(h'Xclass h(j h)j!h.j2h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXclass rr}r(h'Uh(jubaubj8)r}r(h'XpyGPs.Core.inf.h(j h)j!h.j;h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXpyGPs.Core.inf.rr}r (h'Uh(jubaubjA)r!}r"(h'jh(j h)j!h.jDh0}r#(h2]h3]h4]h5]h7]uh9Nh:hh"]r$hCXLaplacer%r&}r'(h'Uh(j!ubaubjJ)r(}r)(h'Uh(j h)Nh.jMh0}r*(UexprjOh5]h4]h2]h3]h7]uh9Nh:hh"]r+jQ)r,}r-(h'Uh0}r.(UreftypejUUrefdocjVU refdomainjWh5]h4]U refexplicith2]h3]h7]U reftargetX_modules/pyGPs/Core/infUrefidjuh(j(h"]r/jY)r0}r1(h'Uh0}r2(h2]h3]r3j^ah4]h5]h7]uh(j,h"]r4hCX[source]r5r6}r7(h'Uh(j0ubah.jcubah.jdubaubeubje)r8}r9(h'Uh(jh)j!h.jhh0}r:(h2]h3]h4]h5]h7]uh9Nh:hh"]r;h|)r<}r=(h'X:Laplace's Approximation to the posterior Gaussian process.r>h(j8h)jh.hh0}r?(h2]h3]h4]h5]h7]uh9Kh:hh"]r@hCX:Laplace's Approximation to the posterior Gaussian process.rArB}rC(h'j>h(j<ubaubaubeubj )rD}rE(h'Uh(jdh)XT/Users/hs/Program/git/pyGPs/pyGPs/Core/inf.py:docstring of pyGPs.Core.inf.FITC_ExactrFh.jh0}rG(h5]h4]h2]h3]h7]Uentries]rH(jX$FITC_Exact (class in pyGPs.Core.inf)h UtrIauh9Nh:hh"]ubj)rJ}rK(h'Uh(jdh)jFh.jh0}rL(jjXpyh5]h4]h2]h3]h7]jXclassrMjjMuh9Nh:hh"]rN(j)rO}rP(h'X FITC_Exact()h(jJh)j!h.j"h0}rQ(h5]rRh aj%h*XpyGPs.Core.infrSrT}rUbh4]h2]h3]h7]rVh aj*X FITC_ExactrWj,Uj-uh9Nh:hh"]rX(j/)rY}rZ(h'Xclass h(jOh)j!h.j2h0}r[(h2]h3]h4]h5]h7]uh9Nh:hh"]r\hCXclass r]r^}r_(h'Uh(jYubaubj8)r`}ra(h'XpyGPs.Core.inf.h(jOh)j!h.j;h0}rb(h2]h3]h4]h5]h7]uh9Nh:hh"]rchCXpyGPs.Core.inf.rdre}rf(h'Uh(j`ubaubjA)rg}rh(h'jWh(jOh)j!h.jDh0}ri(h2]h3]h4]h5]h7]uh9Nh:hh"]rjhCX FITC_Exactrkrl}rm(h'Uh(jgubaubjJ)rn}ro(h'Uh(jOh)Nh.jMh0}rp(UexprjOh5]h4]h2]h3]h7]uh9Nh:hh"]rqjQ)rr}rs(h'Uh0}rt(UreftypejUUrefdocjVU refdomainjWh5]h4]U refexplicith2]h3]h7]U reftargetX_modules/pyGPs/Core/infUrefidjWuh(jnh"]rujY)rv}rw(h'Uh0}rx(h2]h3]ryj^ah4]h5]h7]uh(jrh"]rzhCX[source]r{r|}r}(h'Uh(jvubah.jcubah.jdubaubeubje)r~}r(h'Uh(jJh)j!h.jhh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rh|)r}r(h'XMFITC approximation to the posterior Gaussian process. The function is equivalent to infExact with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Quu) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu, snu2 = sn2/1e6 is the noise of the inducing inputs and Quu = Kuu + snu2*eye(nu).rh(j~h)jFh.hh0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]rhCXMFITC approximation to the posterior Gaussian process. The function is equivalent to infExact with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Quu) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu, snu2 = sn2/1e6 is the noise of the inducing inputs and Quu = Kuu + snu2*eye(nu).rr}r(h'jh(jubaubaubeubj )r}r(h'Uh(jdh)XQ/Users/hs/Program/git/pyGPs/pyGPs/Core/inf.py:docstring of pyGPs.Core.inf.FITC_EPrh.jh0}r(h5]h4]h2]h3]h7]Uentries]r(jX!FITC_EP (class in pyGPs.Core.inf)h Utrauh9Nh:hh"]ubj)r}r(h'Uh(jdh)jh.jh0}r(jjXpyh5]h4]h2]h3]h7]jXclassrjjuh9Nh:hh"]r(j)r}r(h'X FITC_EP()h(jh)j!h.j"h0}r(h5]rh aj%h*XpyGPs.Core.infrr}rbh4]h2]h3]h7]rh aj*XFITC_EPrj,Uj-uh9Nh:hh"]r(j/)r}r(h'Xclass h(jh)j!h.j2h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXclass rr}r(h'Uh(jubaubj8)r}r(h'XpyGPs.Core.inf.h(jh)j!h.j;h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXpyGPs.Core.inf.rr}r(h'Uh(jubaubjA)r}r(h'jh(jh)j!h.jDh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXFITC_EPrr}r(h'Uh(jubaubjJ)r}r(h'Uh(jh)Nh.jMh0}r(UexprjOh5]h4]h2]h3]h7]uh9Nh:hh"]rjQ)r}r(h'Uh0}r(UreftypejUUrefdocjVU refdomainjWh5]h4]U refexplicith2]h3]h7]U reftargetX_modules/pyGPs/Core/infUrefidjuh(jh"]rjY)r}r(h'Uh0}r(h2]h3]rj^ah4]h5]h7]uh(jh"]rhCX[source]rr}r(h'Uh(jubah.jcubah.jdubaubeubje)r}r(h'Uh(jh)j!h.jhh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rh|)r}r(h'X{FITC-EP approximation to the posterior Gaussian process. The function is equivalent to infEP with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Kuu + snu2 * eye(nu)) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu and snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard deviation of the inducing inputs snu to be a one per mil of the measurement noise's standard deviation sn. In case of a likelihood without noise parameter sn2, we simply use snu2 = 1e-6. For details, see The Generalized FITC Approximation, Andrew Naish-Guzman and Sean Holden, NIPS, 2007.rh(jh)jh.hh0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]rhCX{FITC-EP approximation to the posterior Gaussian process. The function is equivalent to infEP with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Kuu + snu2 * eye(nu)) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu and snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard deviation of the inducing inputs snu to be a one per mil of the measurement noise's standard deviation sn. In case of a likelihood without noise parameter sn2, we simply use snu2 = 1e-6. For details, see The Generalized FITC Approximation, Andrew Naish-Guzman and Sean Holden, NIPS, 2007.rr}r(h'jh(jubaubaubeubj )r}r(h'Uh(jdh)XV/Users/hs/Program/git/pyGPs/pyGPs/Core/inf.py:docstring of pyGPs.Core.inf.FITC_Laplacerh.jh0}r(h5]h4]h2]h3]h7]Uentries]r(jX&FITC_Laplace (class in pyGPs.Core.inf)hUtrauh9Nh:hh"]ubj)r}r(h'Uh(jdh)jh.jh0}r(jjXpyh5]h4]h2]h3]h7]jXclassrjjuh9Nh:hh"]r(j)r}r(h'XFITC_Laplace()rh(jh)j!h.j"h0}r(h5]rhaj%h*XpyGPs.Core.infrr}rbh4]h2]h3]h7]rhaj*X FITC_Laplacerj,Uj-uh9Nh:hh"]r(j/)r}r(h'Xclass h(jh)j!h.j2h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXclass rr}r(h'Uh(jubaubj8)r}r(h'XpyGPs.Core.inf.h(jh)j!h.j;h0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCXpyGPs.Core.inf.rr}r(h'Uh(jubaubjA)r}r(h'jh(jh)j!h.jDh0}r(h2]h3]h4]h5]h7]uh9Nh:hh"]rhCX FITC_Laplacerr}r(h'Uh(jubaubjJ)r}r(h'Uh(jh)Nh.jMh0}r(UexprjOh5]h4]h2]h3]h7]uh9Nh:hh"]rjQ)r}r(h'Uh0}r(UreftypejUUrefdocjVU refdomainjWh5]h4]U refexplicith2]h3]h7]U reftargetX_modules/pyGPs/Core/infUrefidjuh(jh"]rjY)r}r(h'Uh0}r(h2]h3]rj^ah4]h5]h7]uh(jh"]rhCX[source]rr }r (h'Uh(jubah.jcubah.jdubaubeubje)r }r (h'Uh(jh)j!h.jhh0}r (h2]h3]h4]h5]h7]uh9Nh:hh"]rh|)r}r(h'XFITC-Laplace approximation to the posterior Gaussian process. The function is equivalent to infLaplace with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Kuu + snu2 * eye(nu)) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu and snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard deviation of the inducing inputs snu to be a one per mil of the measurement noise's standard deviation sn. In case of a likelihood without noise parameter sn2, we simply use snu2 = 1e-6.rh(j h)jh.hh0}r(h2]h3]h4]h5]h7]uh9Kh:hh"]rhCXFITC-Laplace approximation to the posterior Gaussian process. The function is equivalent to infLaplace with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Kuu + snu2 * eye(nu)) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu and snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard deviation of the inducing inputs snu to be a one per mil of the measurement noise's standard deviation sn. 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q)q}q(hUh$}q(UreftypeUviewcodeqUrefdocXOptsqU refdomainUstdqh)]h(]U refexplicith&]h']h+]U reftargetX_modules/pyGPs/Core/gpUrefidhsuhhh]qcdocutils.nodes inline q)q}q(hUh$}q(h&]h']qU viewcode-linkqah(]h)]h+]uhhh]qh7X[source]qхq}q(hUhhubah"Uinlinequbah"U pending_xrefqubaubeubcsphinx.addnodes desc_content q)q}q(hUhh\hhih"U desc_contentqh$}q(h&]h']h(]h)]h+]uh-Nh.hh]q(hI)q}q(hXhThis method is used to sepecify optimization configuration. By default, gp uses a single run "minimize".qhhhXW/Users/hs/Program/git/pyGPs/pyGPs/Core/gp.py:docstring of pyGPs.Core.gp.GP.setOptimizerqh"hMh$}q(h&]h']h(]h)]h+]uh-Kh.hh]qh7XhThis method is used to sepecify optimization configuration. By default, gp uses a single run "minimize".q⅁q}q(hhhhubaubcdocutils.nodes field_list q)q}q(hUhhhNh"U field_listqh$}q(h&]h']h(]h)]h+]uh-Nh.hh]qcdocutils.nodes field q)q}q(hUh$}q(h&]h']h(]h)]h+]uhhh]q(cdocutils.nodes field_name q)q}q(hUh$}q(h&]h']h(]h)]h+]uhhh]qh7X Parametersqq}q(hUhhubah"U field_namequbcdocutils.nodes field_body q)q}q(hUh$}q(h&]h']h(]h)]h+]uhhh]qcdocutils.nodes bullet_list q)q}r(hUh$}r(h&]h']h(]h)]h+]uhhh]r(cdocutils.nodes list_item r)r}r(hUh$}r(h&]h']h(]h)]h+]uhhh]rhI)r}r (hUh$}r (h&]h']h(]h)]h+]uhjh]r (cdocutils.nodes strong r )r }r(hXmethodh$}r(h&]h']h(]h)]h+]uhjh]rh7Xmethodrr}r(hUhj ubah"Ustrongrubh7X -- rr}r(hUhjubhI)r}r(hX*Optimization methods. Possible values are:rhjhhh"hMh$}r(h&]h']h(]h)]h+]uh-Kh]rh7X*Optimization methods. Possible values are:rr}r(hjhjubaubhI)r }r!(hXX"Minimize" -> minimize by Carl Rasmussen (python implementation of "minimize" in GPML)r"hjhhh"hMh$}r#(h&]h']h(]h)]h+]uh-Kh]r$h7XX"Minimize" -> minimize by Carl Rasmussen (python implementation of "minimize" in GPML)r%r&}r'(hj"hj ubaubhI)r(}r)(hX""CG" -> conjugent gradientr*hjhhh"hMh$}r+(h&]h']h(]h)]h+]uh-Kh]r,h7X""CG" -> conjugent gradientr-r.}r/(hj*hj(ubaubhI)r0}r1(hXU"BFGS" -> quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS)r2hjhhh"hMh$}r3(h&]h']h(]h)]h+]uh-K h]r4h7XU"BFGS" -> quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS)r5r6}r7(hj2hj0ubaubhI)r8}r9(hX:"SCG" -> scaled conjugent gradient (faster than CG)r:hjhhh"hMh$}r;(h&]h']h(]h)]h+]uh-K h]r<h7X:"SCG" -> scaled conjugent gradient (faster than CG)r=r>}r?(hj:hj8ubaubeh"hMubah"U list_itemr@ubj)rA}rB(hUh$}rC(h&]h']h(]h)]h+]uhhh]rDhI)rE}rF(hUh$}rG(h&]h']h(]h)]h+]uhjAh]rH(j )rI}rJ(hX num_restartsh$}rK(h&]h']h(]h)]h+]uhjEh]rLh7X num_restartsrMrN}rO(hUhjIubah"jubh7X -- rPrQ}rR(hUhjEubh7XSet if you want to run mulitiple times of optimization with different initial guess. It specifys the maximum number of runs/restarts/trials.rSrT}rU(hXSet if you want to run mulitiple times of optimization with different initial guess. It specifys the maximum number of runs/restarts/trials.rVhjEubeh"hMubah"j@ubj)rW}rX(hUh$}rY(h&]h']h(]h)]h+]uhhh]rZhI)r[}r\(hUh$}r](h&]h']h(]h)]h+]uhjWh]r^(j )r_}r`(hX min_thresholdh$}ra(h&]h']h(]h)]h+]uhj[h]rbh7X min_thresholdrcrd}re(hUhj_ubah"jubh7X -- rfrg}rh(hUhj[ubh7XSet if you want to run mulitiple times of optimization with different initial guess. It specifys the threshold of objective function value. Stop optimization when this value is reached.rirj}rk(hXSet if you want to run mulitiple times of optimization with different initial guess. It specifys the threshold of objective function value. Stop optimization when this value is reached.rlhj[ubeh"hMubah"j@ubj)rm}rn(hUh$}ro(h&]h']h(]h)]h+]uhhh]rphI)rq}rr(hUh$}rs(h&]h']h(]h)]h+]uhjmh]rt(j )ru}rv(hX meanRangeh$}rw(h&]h']h(]h)]h+]uhjqh]rxh7X meanRangeryrz}r{(hUhjuubah"jubh7X -- r|r}}r~(hUhjqubhI)r}r(hXThe range of initial guess for mean hyperparameters. e.g. meanRange = [(-2,2), (-5,5), (0,1)]. Each tuple specifys the range (low, high) of this hyperparameter, This is only the range of initial guess, during optimization process, optimal hyperparameters may go out of this range.rhjqhhh"hMh$}r(h&]h']h(]h)]h+]uh-Kh]rh7XThe range of initial guess for mean hyperparameters. e.g. meanRange = [(-2,2), (-5,5), (0,1)]. Each tuple specifys the range (low, high) of this hyperparameter, This is only the range of initial guess, during optimization process, optimal hyperparameters may go out of this range.rr}r(hjhjubaubhI)r}r(hX*(-5,5) for each hyperparameter by default.rhjqhhh"hMh$}r(h&]h']h(]h)]h+]uh-Kh]rh7X*(-5,5) for each hyperparameter by default.rr}r(hjhjubaubeh"hMubah"j@ubj)r}r(hUh$}r(h&]h']h(]h)]h+]uhhh]rhI)r}r(hUh$}r(h&]h']h(]h)]h+]uhjh]r(j )r}r(hXcovRangeh$}r(h&]h']h(]h)]h+]uhjh]rh7XcovRangerr}r(hUhjubah"jubh7X -- rr}r(hUhjubh7XJThe range of initial guess for kernel hyperparameters. Usage see meanRangerr}r(hXJThe range of initial guess for kernel hyperparameters. Usage see meanRangerhjubeh"hMubah"j@ubj)r}r(hUh$}r(h&]h']h(]h)]h+]uhhh]rhI)r}r(hUh$}r(h&]h']h(]h)]h+]uhjh]r(j )r}r(hXlikRangeh$}r(h&]h']h(]h)]h+]uhjh]rh7XlikRangerr}r(hUhjubah"jubh7X -- rr}r(hUhjubh7XNThe range of initial guess for likelihood hyperparameters. Usage see meanRangerr}r(hXNThe range of initial guess for likelihood hyperparameters. Usage see meanRangerhjubeh"hMubah"j@ubeh"U bullet_listrubah"U field_bodyrubeh"Ufieldrubaubeubeubeubh)r}r(hUhhhh h"h#h$}r(h&]h']h(]h)]rhah+]rh auh-K h.hh]r(h0)r}r(hXDeveloping Optimization Methodsrhjhh h"h4h$}r(h&]h']h(]h)]h+]uh-K h.hh]rh7XDeveloping Optimization Methodsrr}r(hjhjubaubhI)r}r(hX2We also support the development of new optimizers.rhjhh h"hMh$}r(h&]h']h(]h)]h+]uh-K h.hh]rh7X2We also support the development of new optimizers.rr}r(hjhjubaubhI)r}r(hXpYour own optimizer should inherent base class Optimizer in *pyGPs.Core.opt* and follow the template as below: ::hjhh h"hMh$}r(h&]h']h(]h)]h+]uh-Kh.hh]r(h7X;Your own optimizer should inherent base class Optimizer in rr}r(hX;Your own optimizer should inherent base class Optimizer in hjubcdocutils.nodes emphasis r)r}r(hX*pyGPs.Core.opt*h$}r(h&]h']h(]h)]h+]uhjh]rh7XpyGPs.Core.optrr}r(hUhjubah"Uemphasisrubh7X" and follow the template as below:rr}r(hX" and follow the template as below:hjubeubcdocutils.nodes literal_block r)r}r(hXTclass MyOptimizer(Optimizer): def __init__(self, model=None, searchConfig = None): self.model = model def findMin(self, x, y): ''' Find minimal value based on negative-log-marginal-likelihood. optimalHyp, funcValue = findMin(x, y) where funcValue is the minimal negative-log-marginal-likelihood during optimization, and optimalHyp is a flattened numpy array (in sequence of meanfunc.hyp, covfunc.hyp, likfunc.hyp) of the hyparameters to achieve such value. You can achieve advanced search strategy by initializing Optimizer with searchConfig, which is an instance of pyGPs.Optimization.conf. See more in pyGPs.Optimization.conf and pyGPs.Core.gp.GP.setOptimizer, as well as in online documentation of section Optimizers. ''' passhjhh h"U literal_blockrh$}r(U xml:spacerUpreserverh)]h(]h&]h']h+]uh-Kh.hh]rh7XTclass MyOptimizer(Optimizer): def __init__(self, model=None, searchConfig = None): self.model = model def findMin(self, x, y): ''' Find minimal value based on negative-log-marginal-likelihood. optimalHyp, funcValue = findMin(x, y) where funcValue is the minimal negative-log-marginal-likelihood during optimization, and optimalHyp is a flattened numpy array (in sequence of meanfunc.hyp, covfunc.hyp, likfunc.hyp) of the hyparameters to achieve such value. You can achieve advanced search strategy by initializing Optimizer with searchConfig, which is an instance of pyGPs.Optimization.conf. See more in pyGPs.Optimization.conf and pyGPs.Core.gp.GP.setOptimizer, as well as in online documentation of section Optimizers. ''' passrr}r(hUhjubaubeubeubahUU transformerrNU footnote_refsr}rUrefnamesr}rUsymbol_footnotesr]rUautofootnote_refsr]rUsymbol_footnote_refsr]rU citationsr]rh.hU current_linerNUtransform_messagesr]rUreporterrNUid_startrKU autofootnotesr]rU citation_refsr}rUindirect_targetsr]r Usettingsr (cdocutils.frontend Values r or }r (Ufootnote_backlinksrKUrecord_dependenciesrNU rfc_base_urlrUhttp://tools.ietf.org/html/rU tracebackrUpep_referencesrNUstrip_commentsrNU toc_backlinksrUentryrU language_coderUenrU datestamprNU report_levelrKU _destinationrNU halt_levelrKU strip_classesrNh4NUerror_encoding_error_handlerrUbackslashreplacerUdebugr NUembed_stylesheetr!Uoutput_encoding_error_handlerr"Ustrictr#U sectnum_xformr$KUdump_transformsr%NU docinfo_xformr&KUwarning_streamr'NUpep_file_url_templater(Upep-%04dr)Uexit_status_levelr*KUconfigr+NUstrict_visitorr,NUcloak_email_addressesr-Utrim_footnote_reference_spacer.Uenvr/NUdump_pseudo_xmlr0NUexpose_internalsr1NUsectsubtitle_xformr2U source_linkr3NUrfc_referencesr4NUoutput_encodingr5Uutf-8r6U source_urlr7NUinput_encodingr8U utf-8-sigr9U_disable_configr:NU id_prefixr;UU tab_widthr<KUerror_encodingr=UUTF-8r>U_sourcer?U//Users/hs/Program/git/pyGPs/doc/source/Opts.rstr@Ugettext_compactrAU generatorrBNUdump_internalsrCNU smart_quotesrDU pep_base_urlrEUhttp://www.python.org/dev/peps/rFUsyntax_highlightrGUlongrHUinput_encoding_error_handlerrIj#Uauto_id_prefixrJUidrKUdoctitle_xformrLUstrip_elements_with_classesrMNU _config_filesrN]Ufile_insertion_enabledrOU raw_enabledrPKU dump_settingsrQNubUsymbol_footnote_startrRKUidsrS}rT(hhghjhh;hhuUsubstitution_namesrU}rVh"h.h$}rW(h&]h)]h(]Usourceh h']h+]uU footnotesrX]rYUrefidsrZ}r[ub.pyGPs-1.2/doc/build/html/.doctrees/SemiSupervised.doctree000066400000000000000000000641041236346020200234350ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(X$semi-supervised learning with graphsqNXstructured kernelsqX load dataqNX1Xform a nearest neighbour graphq NX2Xresultq NXgp classificationq NXimportq NXkernel on graphq NuUsubstitution_defsq}qUparse_messagesq]qUcurrent_sourceqNU decorationqNUautofootnote_startqKUnameidsq}q(hU$semi-supervised-learning-with-graphsqhUstructured-kernelsqhU load-dataqX1Uid3qh Uform-a-nearest-neighbour-graphqX2Uid4qh Uresultqh Ugp-classificationqh Uimportqh Ukernel-on-graphq uUchildrenq!]q"cdocutils.nodes section q#)q$}q%(U rawsourceq&UUparentq'hUsourceq(cdocutils.nodes reprunicode q)X9/Users/hs/Program/git/pyGPs/doc/source/SemiSupervised.rstq*q+}q,bUtagnameq-Usectionq.U attributesq/}q0(Udupnamesq1]Uclassesq2]Ubackrefsq3]Uidsq4]q5haUnamesq6]q7hauUlineq8KUdocumentq9hh!]q:(cdocutils.nodes title q;)q<}q=(h&X$Semi-supervised Learning with Graphsq>h'h$h(h+h-Utitleq?h/}q@(h1]h2]h3]h4]h6]uh8Kh9hh!]qAcdocutils.nodes Text qBX$Semi-supervised Learning with GraphsqCqD}qE(h&h>h'hr?}r@(h&X graph.h'j3ubeubhF)rA}rB(h&XBelow shows an example of such symmetrized Euclidean :math:`2-NN` graph on some 1s and 2s taking from Xiaojin Zhu's doctoral thesis [2]_.rCh'hh(h+h-hJh/}rD(h1]h2]h3]h4]h6]uh8K-h9hh!]rE(hBX5Below shows an example of such symmetrized Euclidean rFrG}rH(h&X5Below shows an example of such symmetrized Euclidean h'jAubh)rI}rJ(h&Uh/}rK(UlatexX2-NNh4]h3]h1]h2]h6]uh'jAh!]h-hubhBXC graph on some 1s and 2s taking from Xiaojin Zhu's doctoral thesis rLrM}rN(h&XC graph on some 1s and 2s taking from Xiaojin Zhu's doctoral thesis h'jAubh)rO}rP(h&X[2]_hKh'jAh-hh/}rQ(h4]rRUid2rSah3]h1]h2]h6]hhuh!]rThBX2rU}rV(h&Uh'jOubaubhBX.rW}rX(h&X.h'jAubeubcdocutils.nodes figure rY)rZ}r[(h&Uh'hh(h+h-Ufigurer\h/}r](Ualignr^Xcenterh4]h3]h1]h2]h6]uh8Nh9hh!]r_h)r`}ra(h&X4.. figure:: _images/2nnGraph.png :align: center h/}rb(UuriX_images/2nnGraph.pngrch4]h3]h1]h2]h}rdU*jcsh6]uh'jZh!]h-hubaubeubh#)re}rf(h&Uh'h$h(h+h-h.h/}rg(h1]h2]h3]h4]rhh ah6]rih auh8K4h9hh!]rj(h;)rk}rl(h&XKernel on graphrmh'jeh(h+h-h?h/}rn(h1]h2]h3]h4]h6]uh8K4h9hh!]rohBXKernel on graphrprq}rr(h&jmh'jkubaubhF)rs}rt(h&XSeveral classical kernels on graph described in `Structured Kernels`_ can be built from adjacency matrix :math:`A`. We use diffusion kernel for this example to get the precomputed kernel matrix. ::ruh'jeh(h+h-hJh/}rv(h1]h2]h3]h4]h6]uh8K5h9hh!]rw(hBX0Several classical kernels on graph described in rxry}rz(h&X0Several classical kernels on graph described in h'jsubcdocutils.nodes reference r{)r|}r}(h&X`Structured Kernels`_hKh'jsh-U referencer~h/}r(UnameXStructured KernelsUrefurirX Graph.htmlrh4]h3]h1]h2]h6]uh!]rhBXStructured Kernelsrr}r(h&Uh'j|ubaubhBX$ can be built from adjacency matrix rr}r(h&X$ can be built from adjacency matrix h'jsubh)r}r(h&Uh/}r(UlatexXAh4]h3]h1]h2]h6]uh'jsh!]h-hubhBXP. We use diffusion kernel for this example to get the precomputed kernel matrix.rr}r(h&XP. We use diffusion kernel for this example to get the precomputed kernel matrix.h'jsubeubho)r}r(h&X"Matrix = nodeKernels.diffKernel(A)h'jeh(h+h-hrh/}r(hthuh4]h3]h1]h2]h6]uh8K7h9hh!]rhBX"Matrix = nodeKernels.diffKernel(A)rr}r(h&Uh'jubaubcdocutils.nodes target r)r}r(h&X".. _Structured Kernels: Graph.htmlU referencedrKh'jeh(h+h-Utargetrh/}r(jjh4]rhah3]h1]h2]h6]rhauh8K9h9hh!]ubhF)r}r(h&XThis a big square matrix with all rows and columns of the number of data points. By specifying the indice of training data and test data, we will form two matrix M1 and M2 with the exact format which *pyGPs.Core.cov.Pre* needed. ::h'jeh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8K;h9hh!]r(hBXThis a big square matrix with all rows and columns of the number of data points. By specifying the indice of training data and test data, we will form two matrix M1 and M2 with the exact format which rr}r(h&XThis a big square matrix with all rows and columns of the number of data points. By specifying the indice of training data and test data, we will form two matrix M1 and M2 with the exact format which h'jubhP)r}r(h&X*pyGPs.Core.cov.Pre*h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXpyGPs.Core.cov.Prerr}r(h&Uh'jubah-hXubhBX needed.rr}r(h&X needed.h'jubeubho)r}r(h&XEM1,M2 = graphUtil.formKernelMatrix(Matrix, indice_train, indice_test)h'jeh(h+h-hrh/}r(hthuh4]h3]h1]h2]h6]uh8K>h9hh!]rhBXEM1,M2 = graphUtil.formKernelMatrix(Matrix, indice_train, indice_test)rr}r(h&Uh'jubaubcdocutils.nodes definition_list r)r}r(h&Uh'jeh(Nh-Udefinition_listrh/}r(h1]h2]h3]h4]h6]uh8Nh9hh!]r(cdocutils.nodes definition_list_item r)r}r(h&XM1 is a matrix with shape **number of training points plus 1** by **number of test points** - cross covariances matrix (train by test) - last row is self covariances (diagonal of test by test)h'jh(h+h-Udefinition_list_itemrh/}r(h1]h2]h3]h4]h6]uh8KAh!]r(cdocutils.nodes term r)r}r(h&X[M1 is a matrix with shape **number of training points plus 1** by **number of test points**rh'jh(h+h-Utermrh/}r(h1]h2]h3]h4]h6]uh8KAh!]r(hBXM1 is a matrix with shape rr}r(h&XM1 is a matrix with shape h'jubcdocutils.nodes strong r)r}r(h&X$**number of training points plus 1**h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX number of training points plus 1rr}r(h&Uh'jubah-UstrongrubhBX by rr}r(h&X by h'jubj)r}r(h&X**number of test points**h/}r(h1]h2]h3]h4]h6]uh'jh!]rhBXnumber of test pointsrr}r(h&Uh'jubah-jubeubcdocutils.nodes definition r)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rcdocutils.nodes bullet_list r)r}r(h&Uh/}r(UbulletrX-h4]h3]h1]h2]h6]uh'jh!]r(cdocutils.nodes list_item r)r}r(h&X(cross covariances matrix (train by test)rh/}r(h1]h2]h3]h4]h6]uh'jh!]rhF)r}r(h&jh'jh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8KAh!]rhBX(cross covariances matrix (train by test)rr}r(h&jh'jubaubah-U list_itemrubj)r}r(h&X7last row is self covariances (diagonal of test by test)rh/}r(h1]h2]h3]h4]h6]uh'jh!]rhF)r}r(h&jh'jh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8KBh!]rhBX7last row is self covariances (diagonal of test by test)rr}r(h&jh'jubaubah-jubeh-U bullet_listrubah-U definitionrubeubj)r}r(h&XM2 is a square matrix with **number of training points** for each dimension - training set covariance matrix (train by train) h'jh(h+h-jh/}r(h1]h2]h3]h4]h6]uh8KEh9hh!]r (j)r }r (h&XKM2 is a square matrix with **number of training points** for each dimensionr h'jh(h+h-jh/}r (h1]h2]h3]h4]h6]uh8KEh!]r(hBXM2 is a square matrix with rr}r(h&XM2 is a square matrix with h'j ubj)r}r(h&X**number of training points**h/}r(h1]h2]h3]h4]h6]uh'j h!]rhBXnumber of training pointsrr}r(h&Uh'jubah-jubhBX for each dimensionrr}r(h&X for each dimensionh'j ubeubj)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rj)r }r!(h&Uh/}r"(jX-h4]h3]h1]h2]h6]uh'jh!]r#j)r$}r%(h&X1training set covariance matrix (train by train) h/}r&(h1]h2]h3]h4]h6]uh'j h!]r'hF)r(}r)(h&X/training set covariance matrix (train by train)r*h'j$h(h+h-hJh/}r+(h1]h2]h3]h4]h6]uh8KDh!]r,hBX/training set covariance matrix (train by train)r-r.}r/(h&j*h'j(ubaubah-jubah-jubah-jubeubeubeubh#)r0}r1(h&Uh'h$h(h+h-h.h/}r2(h1]h2]h3]h4]r3hah6]r4h auh8KHh9hh!]r5(h;)r6}r7(h&XGP classificationr8h'j0h(h+h-h?h/}r9(h1]h2]h3]h4]h6]uh8KHh9hh!]r:hBXGP classificationr;r<}r=(h&j8h'j6ubaubhF)r>}r?(h&XEvery ingredients for a basic semi-supervised learning is prepared now. Lets see how to proceed for :math:`GP` classification. First, the normal way with rbf kernel we have seen several times ::r@h'j0h(h+h-hJh/}rA(h1]h2]h3]h4]h6]uh8KIh9hh!]rB(hBXeEvery ingredients for a basic semi-supervised learning is prepared now. Lets see how to proceed for rCrD}rE(h&XeEvery ingredients for a basic semi-supervised learning is prepared now. Lets see how to proceed for h'j>ubh)rF}rG(h&Uh/}rH(UlatexXGPh4]h3]h1]h2]h6]uh'j>h!]h-hubhBXQ classification. First, the normal way with rbf kernel we have seen several timesrIrJ}rK(h&XQ classification. First, the normal way with rbf kernel we have seen several timesh'j>ubeubho)rL}rM(h&X@model = pyGPs.GPC() k = pyGPs.cov.RBF() model.setPrior(kernel=k)h'j0h(h+h-hrh/}rN(hthuh4]h3]h1]h2]h6]uh8KKh9hh!]rOhBX@model = pyGPs.GPC() k = pyGPs.cov.RBF() model.setPrior(kernel=k)rPrQ}rR(h&Uh'jLubaubhF)rS}rT(h&XThen lets use our kernel precomputed matrix. If you only use precomputed kernel matrix, there is no training data. However you still need to specify :math:`x` just to fit in the usage of pyGPs for generality reason. You can create any :math:`x` as long as the dimension is correct. ::h'j0h(h+h-hJh/}rU(h1]h2]h3]h4]h6]uh8KOh9hh!]rV(hBXThen lets use our kernel precomputed matrix. If you only use precomputed kernel matrix, there is no training data. However you still need to specify rWrX}rY(h&XThen lets use our kernel precomputed matrix. If you only use precomputed kernel matrix, there is no training data. However you still need to specify h'jSubh)rZ}r[(h&Uh/}r\(UlatexXxh4]h3]h1]h2]h6]uh'jSh!]h-hubhBXM just to fit in the usage of pyGPs for generality reason. You can create any r]r^}r_(h&XM just to fit in the usage of pyGPs for generality reason. You can create any h'jSubh)r`}ra(h&Uh/}rb(UlatexXxh4]h3]h1]h2]h6]uh'jSh!]h-hubhBX% as long as the dimension is correct.rcrd}re(h&X% as long as the dimension is correct.h'jSubeubho)rf}rg(h&XWx = np.zeros((n,1)) k = pyGPs.cov.Pre(M1,M2) + pyGPs.cov.RBF() model.setPrior(kernel=k)h'j0h(h+h-hrh/}rh(hthuh4]h3]h1]h2]h6]uh8KSh9hh!]rihBXWx = np.zeros((n,1)) k = pyGPs.cov.Pre(M1,M2) + pyGPs.cov.RBF() model.setPrior(kernel=k)rjrk}rl(h&Uh'jfubaubhF)rm}rn(h&XmMoreover, you can composite a kernel for both precomputed matrix and regular kernel function if necessary. ::roh'j0h(h+h-hJh/}rp(h1]h2]h3]h4]h6]uh8KWh9hh!]rqhBXjMoreover, you can composite a kernel for both precomputed matrix and regular kernel function if necessary.rrrs}rt(h&XjMoreover, you can composite a kernel for both precomputed matrix and regular kernel function if necessary.h'jmubaubho)ru}rv(h&XGk = pyGPs.cov.Pre(M1,M2) + pyGPs.cov.RBFunit() model.setPrior(kernel=k)h'j0h(h+h-hrh/}rw(hthuh4]h3]h1]h2]h6]uh8KYh9hh!]rxhBXGk = pyGPs.cov.Pre(M1,M2) + pyGPs.cov.RBFunit() model.setPrior(kernel=k)ryrz}r{(h&Uh'juubaubhF)r|}r}(h&XQThe rest way of using pyGPs is exactly the same as the demo of GP classification.r~h'j0h(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8K\h9hh!]rhBXQThe rest way of using pyGPs is exactly the same as the demo of GP classification.rr}r(h&j~h'j|ubaubeubh#)r}r(h&Uh'h$h(h+h-h.h/}r(h1]h2]h3]h4]rhah6]rh auh8K`h9hh!]r(h;)r}r(h&XResultrh'jh(h+h-h?h/}r(h1]h2]h3]h4]h6]uh8K`h9hh!]rhBXResultrr}r(h&jh'jubaubhF)r}r(h&XFor our manually created graph data, an rbf kernel works better than a diffusion kernel on the graph (higher accuracy). The performance in general should depend on the application as well as features of data.rh'jh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Kah9hh!]rhBXFor our manually created graph data, an rbf kernel works better than a diffusion kernel on the graph (higher accuracy). The performance in general should depend on the application as well as features of data.rr}r(h&jh'jubaubhF)r}r(h&XCThe left image shows the digit that using diffusion kernel will predict the wrong result (should be :math:`2`), but rbf kernel does the job fine. The right image shows the digit that rbf kernel predicts the wrong class, diffusion kernel on the other hand, predicts correctly due to graph information! (should be :math:`1`).h'jh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Kch9hh!]r(hBXdThe left image shows the digit that using diffusion kernel will predict the wrong result (should be rr}r(h&XdThe left image shows the digit that using diffusion kernel will predict the wrong result (should be h'jubh)r}r(h&Uh/}r(UlatexX2h4]h3]h1]h2]h6]uh'jh!]h-hubhBX), but rbf kernel does the job fine. The right image shows the digit that rbf kernel predicts the wrong class, diffusion kernel on the other hand, predicts correctly due to graph information! (should be rr}r(h&X), but rbf kernel does the job fine. The right image shows the digit that rbf kernel predicts the wrong class, diffusion kernel on the other hand, predicts correctly due to graph information! (should be h'jubh)r}r(h&Uh/}r(UlatexX1h4]h3]h1]h2]h6]uh'jh!]h-hubhBX).rr}r(h&X).h'jubeubhF)r}r(h&XInterestingly, using a composite kernel with diffusion kernel on graph and an rbf kernel together. All test cases including the following are predicted correctly.rh'jh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Kfh9hh!]rhBXInterestingly, using a composite kernel with diffusion kernel on graph and an rbf kernel together. All test cases including the following are predicted correctly.rr}r(h&jh'jubaubh)r}r(h&X5.. image:: _images/digitDiffwrong.png :width: 50% h'jh(h+h-hh/}r(h1]UuriX_images/digitDiffwrong.pngrh4]h3]UwidthX50%h2]h}rU*jsh6]uh8Nh9hh!]ubh)r}r(h&X5.. image:: _images/digitRBFwrong.png :width: 50% h'jh(h+h-hh/}r(h1]UuriX_images/digitRBFwrong.pngrh4]h3]UwidthX50%h2]h}rU*jsh6]uh8Nh9hh!]ubcdocutils.nodes footnote r)r}r(h&X`A Database for Handwritten Text Recognition Research, J. J. Hull, IEEE PAMI 16(5) 550-554, 1994.rhKh'jh(h+h-Ufootnoterh/}r(h1]h2]h3]rhah4]rhah6]rX1auh8Koh9hh!]r(cdocutils.nodes label r)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX1r}r(h&Uh'jubah-UlabelrubhF)r}r(h&jh'jh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Koh!]rhBX`A Database for Handwritten Text Recognition Research, J. J. Hull, IEEE PAMI 16(5) 550-554, 1994.rr}r(h&jh'jubaubeubj)r}r(h&XGSemi-Supervised Learning with Graphs, Xiaojin Zhu, CMU-LTI-05-192, 2005rhKh'jh(h+h-jh/}r(h1]h2]h3]rjSah4]rhah6]rX2auh8Kph9hh!]r(j)r}r(h&Uh/}r(h1]h2]h3]h4]h6]uh'jh!]rhBX2r}r(h&Uh'jubah-jubhF)r}r(h&jh'jh(h+h-hJh/}r(h1]h2]h3]h4]h6]uh8Kph!]rhBXGSemi-Supervised Learning with Graphs, Xiaojin Zhu, CMU-LTI-05-192, 2005rr}r(h&jh'jubaubeubeubeubah&UU transformerrNU footnote_refsr}r(X1]rhaX2]rjOauUrefnamesr}r(X1]rhaX2]rjOaXstructured kernels]rj|auUsymbol_footnotesr]rUautofootnote_refsr]rUsymbol_footnote_refsr]rU citationsr]rh9hU current_linerNUtransform_messagesr]rUreporterrNUid_startrKU autofootnotesr]rU citation_refsr}rUindirect_targetsr]r Usettingsr (cdocutils.frontend Values r or }r (Ufootnote_backlinksrKUrecord_dependenciesrNU rfc_base_urlrUhttp://tools.ietf.org/html/rU tracebackrUpep_referencesrNUstrip_commentsrNU toc_backlinksrUentryrU language_coderUenrU datestamprNU report_levelrKU _destinationrNU halt_levelrKU strip_classesrNh?NUerror_encoding_error_handlerrUbackslashreplacerUdebugr NUembed_stylesheetr!Uoutput_encoding_error_handlerr"Ustrictr#U sectnum_xformr$KUdump_transformsr%NU docinfo_xformr&KUwarning_streamr'NUpep_file_url_templater(Upep-%04dr)Uexit_status_levelr*KUconfigr+NUstrict_visitorr,NUcloak_email_addressesr-Utrim_footnote_reference_spacer.Uenvr/NUdump_pseudo_xmlr0NUexpose_internalsr1NUsectsubtitle_xformr2U source_linkr3NUrfc_referencesr4NUoutput_encodingr5Uutf-8r6U source_urlr7NUinput_encodingr8U utf-8-sigr9U_disable_configr:NU id_prefixr;UU tab_widthr<KUerror_encodingr=UUTF-8r>U_sourcer?U9/Users/hs/Program/git/pyGPs/doc/source/SemiSupervised.rstr@Ugettext_compactrAU generatorrBNUdump_internalsrCNU smart_quotesrDU pep_base_urlrEUhttp://www.python.org/dev/peps/rFUsyntax_highlightrGUlongrHUinput_encoding_error_handlerrIj#Uauto_id_prefixrJUidrKUdoctitle_xformrLUstrip_elements_with_classesrMNU _config_filesrN]Ufile_insertion_enabledrOU raw_enabledrPKU dump_settingsrQNubUsymbol_footnote_startrRKUidsrS}rT(hjhh$hjjSjOhjhhhhzhjh jehhYhhhj0uUsubstitution_namesrU}rVh-h9h/}rW(h1]h4]h3]Usourceh+h2]h6]uU footnotesrX]rY(jjeUrefidsrZ}r[ub.pyGPs-1.2/doc/build/html/.doctrees/Theory.doctree000066400000000000000000000524461236346020200217460ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(Xgps & functionalityqNXgpmlqXlist of functionalityqNuUsubstitution_defsq }q Uparse_messagesq ]q Ucurrent_sourceq NU decorationqNUautofootnote_startqKUnameidsq}q(hUgps-functionalityqhUgpmlqhUlist-of-functionalityquUchildrenq]qcdocutils.nodes section q)q}q(U rawsourceqUUparentqhUsourceqcdocutils.nodes reprunicode qX1/Users/hs/Program/git/pyGPs/doc/source/Theory.rstqq}q bUtagnameq!Usectionq"U attributesq#}q$(Udupnamesq%]Uclassesq&]Ubackrefsq']Uidsq(]q)haUnamesq*]q+hauUlineq,KUdocumentq-hh]q.(cdocutils.nodes title q/)q0}q1(hXGPs & Functionalityq2hhhhh!Utitleq3h#}q4(h%]h&]h']h(]h*]uh,Kh-hh]q5cdocutils.nodes Text q6XGPs & Functionalityq7q8}q9(hh2hh0ubaubcdocutils.nodes paragraph q:)q;}q<(hXGaussian Processes (GPs) can conveniently be used for Bayesian supervised learning, such as regression and classification. In its simplest form, GP inference can be implemented in a few lines of code. However, in practice, things typically get a little more complicated: you might want to use expressive covariance and mean functions, learn good values for hyperparameters, use non-Gaussian likelihood functions (rendering exact inference intractable), use approximate inference algorithms, or combinations of many or all of the above.q=hhhhh!U paragraphq>h#}q?(h%]h&]h']h(]h*]uh,Kh-hh]q@h6XGaussian Processes (GPs) can conveniently be used for Bayesian supervised learning, such as regression and classification. In its simplest form, GP inference can be implemented in a few lines of code. However, in practice, things typically get a little more complicated: you might want to use expressive covariance and mean functions, learn good values for hyperparameters, use non-Gaussian likelihood functions (rendering exact inference intractable), use approximate inference algorithms, or combinations of many or all of the above.qAqB}qC(hh=hh;ubaubh:)qD}qE(hXA comprehensive introduction to Gaussian Processes for Machine Learning is provided in the `GPML`_ book by Rasmussen and Williams, 2006.qFhhhhh!h>h#}qG(h%]h&]h']h(]h*]uh,K h-hh]qH(h6X[A comprehensive introduction to Gaussian Processes for Machine Learning is provided in the qIqJ}qK(hX[A comprehensive introduction to Gaussian Processes for Machine Learning is provided in the hhDubcdocutils.nodes reference qL)qM}qN(hX`GPML`_UresolvedqOKhhDh!U referenceqPh#}qQ(UnameXGPMLUrefuriqRX#http://www.gaussianprocess.org/gpmlqSh(]h']h%]h&]h*]uh]qTh6XGPMLqUqV}qW(hUhhMubaubh6X& book by Rasmussen and Williams, 2006.qXqY}qZ(hX& book by Rasmussen 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qN)qO}qP(hUh#}qQ(UbulletqRX*h(]h']h%]h&]h*]uhhIh]qS(cdocutils.nodes list_item qT)qU}qV(hX6Marion Neumann [marion dot neumann at uni-bonn dot de]qWh#}qX(h%]h&]h']h(]h*]uhhOh]qYcdocutils.nodes paragraph qZ)q[}q\(hhWhhUhhh!U paragraphq]h#}q^(h%]h&]h']h(]h*]uh,Kh]q_h6X6Marion Neumann [marion dot neumann at uni-bonn dot de]q`qa}qb(hhWhh[ubaubah!U list_itemqcubhT)qd}qe(hX9Shan Huang [shan dot huang at iais dot fraunhofer dot de]qfh#}qg(h%]h&]h']h(]h*]uhhOh]qhhZ)qi}qj(hhfhhdhhh!h]h#}qk(h%]h&]h']h(]h*]uh,Kh]qlh6X9Shan Huang [shan dot huang at iais dot fraunhofer dot de]qmqn}qo(hhfhhiubaubah!hcubhT)qp}qq(hX*Daniel Marthaler [marthaler at ge dot com]qrh#}qs(h%]h&]h']h(]h*]uhhOh]qthZ)qu}qv(hhrhhphhh!h]h#}qw(h%]h&]h']h(]h*]uh,Kh]qxh6X*Daniel Marthaler [marthaler at ge dot com]qyqz}q{(hhrhhuubaubah!hcubhT)q|}q}(hXGKristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] h#}q~(h%]h&]h']h(]h*]uhhOh]qhZ)q}q(hXFKristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]qhh|hhh!h]h#}q(h%]h&]h']h(]h*]uh,Kh]qh6XFKristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]qq}q(hhhhubaubah!hcubeh!U bullet_listqubhZ)q}q(hXThis file is part of pyGPs.qhhIhhh!h]h#}q(h%]h&]h']h(]h*]uh,K h]qh6XThis file is part of pyGPs.qq}q(hhhhubaubeubeubh)q}q(hUhhhhh!h"h#}q(h%]h&]h']h(]qhah*]qhauh,K h-hh]q(h/)q}q(hXLicenseqhhhhh!h3h#}q(h%]h&]h']h(]h*]uh,K h-hh]qh6XLicenseqq}q(hhhhubaubhH)q}q(hUhhhhh!hKh#}q(h%]h&]h']h(]h*]uh,Nh-hh]q(hZ)q}q(hX>The code is released under the BSD 2-Clause (FreeBSD) License.qhhhhh!h]h#}q(h%]h&]h']h(]h*]uh,Kh]qh6X>The code is released under the BSD 2-Clause (FreeBSD) License.qq}q(hhhhubaubhZ)q}q(hXhCopyright (c) 2014 Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting All rights reserved.qhhhhh!h]h#}q(h%]h&]h']h(]h*]uh,Kh]qh6XhCopyright (c) 2014 Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting All rights reserved.qq}q(hhhhubaubhZ)q}q(hXRedistribution and use in source and binary forms, with or without 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Suttorp and C. Igel, Approximation of Gaussian process regression models after training. In M. Verleysen (Hrsg.), Proceedings of the 16th European Symposium on Artificial Neural Networks (ESANN 2008) , pp. 427–432 (2008).q Uresolvedq!Khcdocutils.nodes section q")q#}q$(hUhhUsourceq%hUtagnameq&Usectionq'h}q((h]h]h]h]q)Uregression-on-uci-housing-dataq*ah]q+hauUlineq,KUdocumentq-hUchildrenq.]q/(cdocutils.nodes title q0)q1}q2(hXRegression on UCI Housing dataq3hh#h%hh&Utitleq4h}q5(h]h]h]h]h]uh,Kh-hh.]q6cdocutils.nodes Text q7XRegression on UCI Housing dataq8q9}q:(hh3hh1ubaubcdocutils.nodes paragraph q;)q<}q=(hX5Boston Housing is a fairly standard dataset used for testing regression problems. It contains 506 data points with 12 numeric attributes, and one binary categorical attribute. The goal is to determine median home values, based on various census attributes. This dataset is available at the `UCI Repository`_.hh#h%hh&U paragraphq>h}q?(h]h]h]h]h]uh,Kh-hh.]q@(h7X#Boston Housing is a fairly standard dataset used for testing regression problems. It contains 506 data points with 12 numeric attributes, and one binary categorical attribute. The goal is to determine median home values, based on various census attributes. This dataset is available at the qAqB}qC(hX#Boston Housing is a fairly standard dataset used for testing regression problems. It contains 506 data points with 12 numeric attributes, and one binary categorical attribute. The goal is to determine median home values, based on various census attributes. This dataset is available at the hhh}qS(h]h]h]h]h]uh,Kh-hh.]qT(h7XThe demo follows that in qUqV}qW(hXThe demo follows that in hhQubcdocutils.nodes footnote_reference qX)qY}qZ(hX[1]_h!KhhQh&Ufootnote_referenceq[h}q\(h]q]Uid1q^ah]h]h]h]Urefidq_Uid2q`uh.]qah7X1qb}qc(hUhhYubaubh7X. The data set was preprocessed as follows: each continuous feature was transformed to zero mean and unit variance (The categorical variable was dropped). The data was partitioned into qdqe}qf(hX. The data set was preprocessed as follows: each continuous feature was transformed to zero mean and unit variance (The categorical variable was dropped). The data was partitioned into hhQubcsphinx.ext.mathbase math qg)qh}qi(hUh}qj(UlatexX481h]h]h]h]h]uhhQh.]h&Umathqkubh7X points for training and qlqm}qn(hX points for training and hhQubhg)qo}qp(hUh}qq(UlatexX25h]h]h]h]h]uhhQh.]h&hkubh7X points for testing.qrqs}qt(hX points for testing.hhQubeubh;)qu}qv(hXyThe mean function used was :func:`src.Core.means.meanZero` and the covariance (using the :func:`src.Core.kernels.covSum` function) was a composite of :func:`src.Core.kernels.covSEiso` and :func:`src.Core.kernels.covNoise`. The initial values of the hyperparameters were selected randomly from a zero-mean, unit-variance normal distribtion. The actual values were: :math:`[ -0.75, \; 0.59, \; -0.45 ]`. The initial likelihood hyperparameter was :math:`-2.30`. The regression started with initial negative log marginal likelihood of :math:` 752.46`. Note the initial zero mean and the variance that is uniform over the test set. ::hh#h%hh&h>h}qw(h]h]h]h]h]uh,K h-hh.]qx(h7XThe mean function used was qyqz}q{(hXThe mean function used was hhuubcsphinx.addnodes pending_xref q|)q}}q~(hX:func:`src.Core.means.meanZero`qhhuh%hh&U pending_xrefqh}q(UreftypeXfuncUrefwarnqU reftargetqXsrc.Core.means.meanZeroU refdomainXpyqh]h]U refexplicith]h]h]UrefdocqX demoHousingqUpy:classqNU py:moduleqNuh,K h.]qcdocutils.nodes literal q)q}q(hhh}q(h]h]q(UxrefqhXpy-funcqeh]h]h]uhh}h.]qh7Xsrc.Core.means.meanZero()qq}q(hUhhubah&Uliteralqubaubh7X and the covariance (using the qq}q(hX and the covariance (using the hhuubh|)q}q(hX:func:`src.Core.kernels.covSum`qhhuh%hh&hh}q(UreftypeXfunchhXsrc.Core.kernels.covSumU refdomainXpyqh]h]U refexplicith]h]h]hhhNhNuh,K h.]qh)q}q(hhh}q(h]h]q(hhXpy-funcqeh]h]h]uhhh.]qh7Xsrc.Core.kernels.covSum()qq}q(hUhhubah&hubaubh7X function) was a composite of qq}q(hX function) was a composite of hhuubh|)q}q(hX!:func:`src.Core.kernels.covSEiso`qhhuh%hh&hh}q(UreftypeXfunchhXsrc.Core.kernels.covSEisoU refdomainXpyqh]h]U refexplicith]h]h]hhhNhNuh,K h.]qh)q}q(hhh}q(h]h]q(hhXpy-funcqeh]h]h]uhhh.]qh7Xsrc.Core.kernels.covSEiso()qq}q(hUhhubah&hubaubh7X and qq}q(hX and hhuubh|)q}q(hX!:func:`src.Core.kernels.covNoise`qhhuh%hh&hh}q(UreftypeXfunchhXsrc.Core.kernels.covNoiseU refdomainXpyqh]h]U refexplicith]h]h]hhhNhNuh,K h.]qh)q}q(hhh}q(h]h]q(hhXpy-funcqeh]h]h]uhhh.]qh7Xsrc.Core.kernels.covNoise()qɅq}q(hUhhubah&hubaubh7X. The initial values of the hyperparameters were selected randomly from a zero-mean, unit-variance normal distribtion. The actual values were: q̅q}q(hX. The initial values of the hyperparameters were selected randomly from a zero-mean, unit-variance normal distribtion. The actual values were: hhuubhg)q}q(hUh}q(UlatexX[ -0.75, \; 0.59, \; -0.45 ]h]h]h]h]h]uhhuh.]h&hkubh7X,. The initial likelihood hyperparameter was q҅q}q(hX,. The initial likelihood hyperparameter was hhuubhg)q}q(hUh}q(UlatexX-2.30h]h]h]h]h]uhhuh.]h&hkubh7X. The regression started with initial negative log marginal likelihood of :math:` 752.46`. Note the initial zero mean and the variance that is uniform over the test set.q؅q}q(hX. The regression started with initial negative log marginal likelihood of :math:` 752.46`. Note the initial zero mean and the variance that is uniform over the test set.hhuubeubcdocutils.nodes literal_block q)q}q(hXmodel = pyGPs.GPR() model.optimize(x,y) ym, ys2, fm, fs2, lp = model.predict(xs) xa = np.concatenate((data[:,:4],data[:,5:-1]),axis=1) xa = (xa - np.mean(xa,axis=0))/(np.std(xa,axis=0)+1.e-16) ya, ys2a, fma, fs2a, lpa = model.predict(xa)hh#h%hh&U literal_blockqh}q(U xml:spaceqUpreserveqh]h]h]h]h]uh,Kh-hh.]qh7Xmodel = pyGPs.GPR() model.optimize(x,y) ym, ys2, fm, fs2, lp = model.predict(xs) xa = np.concatenate((data[:,:4],data[:,5:-1]),axis=1) xa = (xa - np.mean(xa,axis=0))/(np.std(xa,axis=0)+1.e-16) ya, ys2a, fma, fs2a, lpa = model.predict(xa)qㅁq}q(hUhhubaubcdocutils.nodes figure q)q}q(hUhh#h%hh&Ufigureqh}q(UalignqXcenterh]h]h]h]h]uh,Nh-hh.]qcdocutils.nodes image q)q}q(hXT.. figure:: _images/demoH1.png :align: center :width: 600pt :height: 300pt h}q(h]UuriX_images/demoH1.pngqh]h]UwidthX600pth]U candidatesq}qU*hsh]UheightX300ptuhhh.]h&Uimagequbaubh;)q}q(hXAfter hyperparameter optimization, the covariance hyperparameters were :math:`[ 1.17, \; 0.45, \; -1.41 ]` and the likelihood hyperparameter was :math:`-2.27`. The final negative log marginal likelihood (optimized) was :math:`214.46`.hh#h%hh&h>h}q(h]h]h]h]h]uh,Kh-hh.]q(h7XGAfter hyperparameter optimization, the covariance hyperparameters were qq}q(hXGAfter hyperparameter optimization, the covariance hyperparameters were hhubhg)q}q(hUh}q(UlatexX[ 1.17, \; 0.45, \; -1.41 ]h]h]h]h]h]uhhh.]h&hkubh7X' and the likelihood hyperparameter was qr}r(hX' and the likelihood hyperparameter was hhubhg)r}r(hUh}r(UlatexX-2.27h]h]h]h]h]uhhh.]h&hkubh7X?. The final negative log marginal likelihood (optimized) was rr}r(hX?. The final negative log marginal likelihood (optimized) was hhubhg)r}r (hUh}r (UlatexX214.46h]h]h]h]h]uhhh.]h&hkubh7X.r }r (hX.hhubeubh)r }r(hUhh#h%hh&hh}r(hXcenterh]h]h]h]h]uh,Nh-hh.]rh)r}r(hXT.. figure:: _images/demoH2.png :align: center :width: 600pt :height: 300pt h}r(h]UuriX_images/demoH2.pngrh]h]UwidthX600pth]h}rU*jsh]UheightX300ptuhj h.]h&hubaubcdocutils.nodes target r)r}r(hXB.. _UCI Repository: http://archive.ics.uci.edu/ml/datasets/HousingU referencedrKhh#h%hh&Utargetrh}r(hIhJh]rUuci-repositoryrah]h]h]h]rhauh,K%h-hh.]ubheubh%hh&Ufootnoterh}r (h]h]h]r!h^ah]r"h`ah]r#X1auh,K'h-hh.]r$(cdocutils.nodes label r%)r&}r'(hUh}r((h]h]h]h]h]uhhh.]r)h7X1r*}r+(hUhj&ubah&Ulabelr,ubcdocutils.nodes enumerated_list r-)r.}r/(hUh}r0(Usuffixr1U.Ustartr2Kh]h]h]Uprefixr3Uh]h]Uenumtyper4U upperalphar5uhhh.]r6cdocutils.nodes list_item r7)r8}r9(hXSuttorp and C. Igel, Approximation of Gaussian process regression models after training. In M. Verleysen (Hrsg.), Proceedings of the 16th European Symposium on Artificial Neural Networks (ESANN 2008) , pp. 427–432 (2008).r:h}r;(h]h]h]h]h]uhj.h.]r<h;)r=}r>(hj:hj8h%hh&h>h}r?(h]h]h]h]h]uh,K'h.]r@h7XSuttorp and C. Igel, Approximation of Gaussian process regression models after training. In M. Verleysen (Hrsg.), Proceedings of the 16th European Symposium on Artificial Neural Networks (ESANN 2008) , pp. 427–432 (2008).rArB}rC(hj:hj=ubaubah&U list_itemrDubah&Uenumerated_listrEubeubh.]rFh;)rG}rH(hUh}rI(h]h]h]h]h]uhh h.]rJh7X;Enumerated list start value not ordinal-1: "T" (ordinal 20)rKrL}rM(hUhjGubah&h>ubah&Usystem_messagerNubaUcurrent_sourcerONU decorationrPNUautofootnote_startrQKUnameidsrR}rS(X1h`hjhh*uh.]rTh#ahUU transformerrUNU footnote_refsrV}rWX1]rXhYasUrefnamesrY}rZ(X1]r[hYaXuci repository]r\hEauUsymbol_footnotesr]]r^Uautofootnote_refsr_]r`Usymbol_footnote_refsra]rbU citationsrc]rdh-hU current_linereNUtransform_messagesrf]rgUreporterrhNUid_startriKU autofootnotesrj]rkU citation_refsrl}rmUindirect_targetsrn]roUsettingsrp(cdocutils.frontend Values rqorr}rs(Ufootnote_backlinksrtKUrecord_dependenciesruNU rfc_base_urlrvUhttp://tools.ietf.org/html/rwU tracebackrxUpep_referencesryNUstrip_commentsrzNU toc_backlinksr{Uentryr|U language_coder}Uenr~U datestamprNU report_levelrKU _destinationrNU halt_levelrKU strip_classesrNh4NUerror_encoding_error_handlerrUbackslashreplacerUdebugrNUembed_stylesheetrUoutput_encoding_error_handlerrUstrictrU sectnum_xformrKUdump_transformsrNU docinfo_xformrKUwarning_streamrNUpep_file_url_templaterUpep-%04drUexit_status_levelrKUconfigrNUstrict_visitorrNUcloak_email_addressesrUtrim_footnote_reference_spacerUenvrNUdump_pseudo_xmlrNUexpose_internalsrNUsectsubtitle_xformrU source_linkrNUrfc_referencesrNUoutput_encodingrUutf-8rU source_urlrNUinput_encodingrU utf-8-sigrU_disable_configrNU id_prefixrUU tab_widthrKUerror_encodingrUUTF-8rU_sourcerU6/Users/hs/Program/git/pyGPs/doc/source/demoHousing.rstrUgettext_compactrU generatorrNUdump_internalsrNU smart_quotesrU pep_base_urlrUhttp://www.python.org/dev/peps/rUsyntax_highlightrUlongrUinput_encoding_error_handlerrjUauto_id_prefixrUidrUdoctitle_xformrUstrip_elements_with_classesrNU _config_filesr]Ufile_insertion_enabledrU raw_enabledrKU dump_settingsrNubUsymbol_footnote_startrKUidsr}r(h`hjjh*h#h^hYuUsubstitution_namesr}rh&h-h}r(h]h]h]Usourcehh]h]uU footnotesr]rhaUrefidsr}rub.pyGPs-1.2/doc/build/html/.doctrees/demoMaunaLoa.doctree000066400000000000000000000642571236346020200230410ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(X1XgpmlqXregression on mauna loa dataqNX[2]quUsubstitution_defsq }q Uparse_messagesq ]q Ucurrent_sourceq NU decorationqNUautofootnote_startqKUnameidsq}q(X1Uid2qhUgpmlqhUregression-on-mauna-loa-dataqhUid3quUchildrenq]qcdocutils.nodes section q)q}q(U rawsourceqUUparentqhUsourceqcdocutils.nodes reprunicode qX7/Users/hs/Program/git/pyGPs/doc/source/demoMaunaLoa.rstqq }q!bUtagnameq"Usectionq#U attributesq$}q%(Udupnamesq&]Uclassesq']Ubackrefsq(]Uidsq)]q*haUnamesq+]q,hauUlineq-KUdocumentq.hh]q/(cdocutils.nodes title q0)q1}q2(hXRegression on Mauna Loa dataq3hhhh h"Utitleq4h$}q5(h&]h']h(]h)]h+]uh-Kh.hh]q6cdocutils.nodes Text q7XRegression on Mauna Loa dataq8q9}q:(hh3hh1ubaubcdocutils.nodes paragraph q;)q<}q=(hXThis example does regression on the Hawaiian Mauna Loa data (example taken from chapter :math:`5` of the `GPML`_ book by Rasmussen and Williams, 2006)q>hhhh h"U paragraphq?h$}q@(h&]h']h(]h)]h+]uh-Kh.hh]qA(h7XXThis example does regression on the Hawaiian Mauna Loa data (example taken from chapter qBqC}qD(hXXThis example does regression on the Hawaiian Mauna Loa data (example taken from chapter hhThe data is shown in the above plot. Our goal is to model the qq}q(hX>The data is shown in the above plot. Our goal is to model the hhubhE)q}q(hUh$}q(UlatexXCO_2h)]h(]h&]h']h+]uhhh]h"hIubh7X% concentration as a function of time qƅq}q(hX% concentration as a function of time hhubhE)q}q(hUh$}q(UlatexXth)]h(]h&]h']h+]uhhh]h"hIubh7X. Several features are immediately apparent: a long term rising trend, a pronounced seasonal variation and some smaller irregularities. In the following, contributions to a combined covariance function which takes care of these individual properties are suggessted. This is meant primarily to illustrate the power and flexibility of the Gaussian process framework—it is possible that other choices would be more appropriate for this data set.q̅q}q(hX. Several features are immediately apparent: a long term rising trend, a pronounced seasonal variation and some smaller irregularities. In the following, contributions to a combined covariance function which takes care of these individual properties are suggessted. This is meant primarily to illustrate the power and flexibility of the Gaussian process framework—it is possible that other choices would be more appropriate for this data set.hhubeubh;)q}q(hXTo model the long term smooth rising trend, a squared exponential (SE) covariance term with two hyperparameters controlling the amplitude :math:`\theta_1` and characteristic length-scale :math:`\theta_2` is used:hhhh h"h?h$}q(h&]h']h(]h)]h+]uh-K#h.hh]q(h7XTo model the long term smooth rising trend, a squared exponential (SE) covariance term with two hyperparameters controlling the amplitude qӅq}q(hXTo model the long term smooth rising trend, a squared exponential (SE) covariance term with two hyperparameters controlling the amplitude hhubhE)q}q(hUh$}q(UlatexX\theta_1h)]h(]h&]h']h+]uhhh]h"hIubh7X! and characteristic length-scale qمq}q(hX! and characteristic length-scale hhubhE)q}q(hUh$}q(UlatexX\theta_2h)]h(]h&]h']h+]uhhh]h"hIubh7X is used:q߅q}q(hX is used:hhubeubcsphinx.ext.mathbase displaymath q)q}q(hUhhhh h"U displaymathqh$}q(UlatexqXIk_1(x,x') = \theta_1^2 \exp \left(-\frac{(x-x')^2}{2\theta_2^2}\right). Unowrapqh)]h(]h&]h']h+]UdocnameqX demoMaunaLoaqUlabelqNuh-K'h.hh]ubh;)q}q(hXNote that we just use a smooth trend; actually enforcing the trend *a priori* to be increasing is probably not so simple and (hopefully) not desirable. We can use the periodic covariance function with a period of one year to model the seasonal variation. However, it is not clear that the seasonal trend is exactly periodic, so we modify it by taking the product with a squared exponential component to allow a decay away from exact periodicity:hhhh h"h?h$}q(h&]h']h(]h)]h+]uh-K)h.hh]q(h7XCNote that we just use a smooth trend; actually enforcing the trend qq}q(hXCNote that we just use a smooth trend; actually enforcing the trend hhubh)q}q(hX *a priori*h$}q(h&]h']h(]h)]h+]uhhh]qh7Xa prioriqq}q(hUhhubah"hubh7Xp to be increasing is probably not so simple and (hopefully) not desirable. We can use the periodic covariance function with a period of one year to model the seasonal variation. However, it is not clear that the seasonal trend is exactly periodic, so we modify it by taking the product with a squared exponential component to allow a decay away from exact periodicity:qq}q(hXp to be increasing is probably not so simple and (hopefully) not desirable. We can use the periodic covariance function with a period of one year to model the seasonal variation. However, it is not clear that the seasonal trend is exactly periodic, so we modify it by taking the product with a squared exponential component to allow a decay away from exact periodicity:hhubeubh)q}q(hUhhhh h"hh$}q(hXmk_2(x,x') = \theta_3^2 \exp\left(-\frac{(x-x')^2}{2\theta_4^2} \frac{2\sin^2(\pi(x-x'))}{\theta_5^2}\right).hh)]h(]h&]h']h+]hhhNuh-K/h.hh]ubh;)r}r(hXwhere :math:`\theta_3` gives the magnitude, :math:`\theta_4` the decay-time for the periodic component, and :math:`\theta_5` the smoothness of the periodic component; the period has been fixed to one (year). The seasonal component in the data is caused primarily by different rates of :math:`CO_2` uptake for plants depending on the season, and it is probably reasonable to assume that this pattern may itself change slowly over time, partially due to the elevation of the :math:`CO_2` level itself; if this effect turns out not to be relevant, then it can be effectively removed at the fitting stage by allowing :math:`\theta_4` to become very large.hhhh h"h?h$}r(h&]h']h(]h)]h+]uh-K3h.hh]r(h7Xwhere rr}r(hXwhere hjubhE)r}r(hUh$}r (UlatexX\theta_3h)]h(]h&]h']h+]uhjh]h"hIubh7X gives the magnitude, r r }r (hX gives the magnitude, hjubhE)r }r(hUh$}r(UlatexX\theta_4h)]h(]h&]h']h+]uhjh]h"hIubh7X0 the decay-time for the periodic component, and rr}r(hX0 the decay-time for the periodic component, and hjubhE)r}r(hUh$}r(UlatexX\theta_5h)]h(]h&]h']h+]uhjh]h"hIubh7X the smoothness of the periodic component; the period has been fixed to one (year). The seasonal component in the data is caused primarily by different rates of rr}r(hX the smoothness of the periodic component; the period has been fixed to one (year). The seasonal component in the data is caused primarily by different rates of hjubhE)r}r(hUh$}r(UlatexXCO_2h)]h(]h&]h']h+]uhjh]h"hIubh7X uptake for plants depending on the season, and it is probably reasonable to assume that this pattern may itself change slowly over time, partially due to the elevation of the rr}r(hX uptake for plants depending on the season, and it is probably reasonable to assume that this pattern may itself change slowly over time, partially due to the elevation of the hjubhE)r}r (hUh$}r!(UlatexXCO_2h)]h(]h&]h']h+]uhjh]h"hIubh7X level itself; if this effect turns out not to be relevant, then it can be effectively removed at the fitting stage by allowing r"r#}r$(hX level itself; if this effect turns out not to be relevant, then it can be effectively removed at the fitting stage by allowing hjubhE)r%}r&(hUh$}r'(UlatexX\theta_4h)]h(]h&]h']h+]uhjh]h"hIubh7X to become very large.r(r)}r*(hX to become very large.hjubeubh;)r+}r,(hXSTo model the (small) medium term irregularities, a rational quadratic term is used:r-hhhh h"h?h$}r.(h&]h']h(]h)]h+]uh-Kh.hh]ubh;)r6}r7(hXwhere :math:`\theta_6` is the magnitude, :math:`\theta_7` is the typical length-scale and :math:`\theta_8` is the shape parameter determining diffuseness of the length-scales.hhhh h"h?h$}r8(h&]h']h(]h)]h+]uh-KBh.hh]r9(h7Xwhere r:r;}r<(hXwhere hj6ubhE)r=}r>(hUh$}r?(UlatexX\theta_6h)]h(]h&]h']h+]uhj6h]h"hIubh7X is the magnitude, r@rA}rB(hX is the magnitude, hj6ubhE)rC}rD(hUh$}rE(UlatexX\theta_7h)]h(]h&]h']h+]uhj6h]h"hIubh7X! is the typical length-scale and rFrG}rH(hX! is the typical length-scale and hj6ubhE)rI}rJ(hUh$}rK(UlatexX\theta_8h)]h(]h&]h']h+]uhj6h]h"hIubh7XE is the shape parameter determining diffuseness of the length-scales.rLrM}rN(hXE is the shape parameter determining diffuseness of the length-scales.hj6ubeubh;)rO}rP(hXOne could also have used a squared exponential form for this component, but it turns out that the rational quadratic works better (gives higher marginal likelihood), probably because it can accommodate several length-scales simultaneously.rQhhhh h"h?h$}rR(h&]h']h(]h)]h+]uh-KEh.hh]rSh7XOne could also have used a squared exponential form for this component, but it turns out that the rational quadratic works better (gives higher marginal likelihood), probably because it can accommodate several length-scales simultaneously.rTrU}rV(hjQhjOubaubh;)rW}rX(hXoFinally we specify a noise model as the sum of a squared exponential contrubition and an independent component:rYhhhh h"h?h$}rZ(h&]h']h(]h)]h+]uh-KIh.hh]r[h7XoFinally we specify a noise model as the sum of a squared exponential contrubition and an independent component:r\r]}r^(hjYhjWubaubh)r_}r`(hUhhhh h"hh$}ra(hXkk_4(x_p,x_q) = \theta_9^2\exp\left(-\frac{(x_p - x_q)^2}{2\theta_{10}^2}\right) + \theta_{11}^2\delta_{pq}.hh)]h(]h&]h']h+]hhhNuh-KKh.hh]ubh;)rb}rc(hXwhere :math:`\theta_9` is the magnitude of the correlated noise component, :math:`\theta_{10}` is its length scale and :math:`\theta_{11}` is the magnitude of the independent noise component. Noise in the series could be caused by measurement inaccuracies, and by local short-term weather phenomena, so it is probably reasonable to assume at least a modest amount of correlation in time. Notice that the correlated noise component, the first term has an identical expression to the long term component in the trend covariance. When optimizing the hyperparameters, we will see that one of these components becomes large with a long length-scale (the long term trend), while the other remains small with a short length-scale (noise). The fact that we have chosen to call one of these components ‘signal’ and the other one ‘noise’ is only a question of interpretation. Presumably, we are less interested in very short-term effect, and thus call it noise; if on the other hand we were interested in this effect, we would call it signal.hhhh h"h?h$}rd(h&]h']h(]h)]h+]uh-KOh.hh]re(h7Xwhere rfrg}rh(hXwhere hjbubhE)ri}rj(hUh$}rk(UlatexX\theta_9h)]h(]h&]h']h+]uhjbh]h"hIubh7X5 is the magnitude of the correlated noise component, rlrm}rn(hX5 is the magnitude of the correlated noise component, hjbubhE)ro}rp(hUh$}rq(UlatexX \theta_{10}h)]h(]h&]h']h+]uhjbh]h"hIubh7X is its length scale and rrrs}rt(hX is its length scale and hjbubhE)ru}rv(hUh$}rw(UlatexX \theta_{11}h)]h(]h&]h']h+]uhjbh]h"hIubh7X is the magnitude of the independent noise component. Noise in the series could be caused by measurement inaccuracies, and by local short-term weather phenomena, so it is probably reasonable to assume at least a modest amount of correlation in time. Notice that the correlated noise component, the first term has an identical expression to the long term component in the trend covariance. When optimizing the hyperparameters, we will see that one of these components becomes large with a long length-scale (the long term trend), while the other remains small with a short length-scale (noise). The fact that we have chosen to call one of these components ‘signal’ and the other one ‘noise’ is only a question of interpretation. Presumably, we are less interested in very short-term effect, and thus call it noise; if on the other hand we were interested in this effect, we would call it signal.rxry}rz(hX is the magnitude of the independent noise component. Noise in the series could be caused by measurement inaccuracies, and by local short-term weather phenomena, so it is probably reasonable to assume at least a modest amount of correlation in time. Notice that the correlated noise component, the first term has an identical expression to the long term component in the trend covariance. When optimizing the hyperparameters, we will see that one of these components becomes large with a long length-scale (the long term trend), while the other remains small with a short length-scale (noise). The fact that we have chosen to call one of these components ‘signal’ and the other one ‘noise’ is only a question of interpretation. Presumably, we are less interested in very short-term effect, and thus call it noise; if on the other hand we were interested in this effect, we would call it signal.hjbubeubh;)r{}r|(hX!The final covariance function is:r}hhhh h"h?h$}r~(h&]h']h(]h)]h+]uh-K]h.hh]rh7X!The final covariance function is:rr}r(hj}hj{ubaubh)r}r(hUhhhh h"hh$}r(hX7k(x,x') = k_1(x,x') + k_2(x,x') + k_3(x,x') + k_4(x,x')hh)]h(]h&]h']h+]hhhNuh-K_h.hh]ubh;)r}r(hXFwith hyperparameters :math:`\theta = (\theta_1,\ldots,\theta_{11})` ::rhhhh h"h?h$}r(h&]h']h(]h)]h+]uh-Kch.hh]r(h7Xwith hyperparameters rr}r(hXwith hyperparameters hjubhE)r}r(hUh$}r(UlatexX&\theta = (\theta_1,\ldots,\theta_{11})h)]h(]h&]h']h+]uhjh]h"hIubeubcdocutils.nodes literal_block r)r}r(hXX# DEFINE parameterized covariance function k1 = pyGPs.cov.RBF(np.log(67.), np.log(66.)) k2 = pyGPs.cov.Periodic(np.log(1.3), np.log(1.0), np.log(2.4)) * cov.RBF(np.log(90.), np.log(2.4)) k3 = pyGPs.cov.RQ(np.log(1.2), np.log(0.66), np.log(0.78)) k4 = pyGPs.cov.RBF(np.log(1.6/12.), np.log(0.18)) + cov.Noise(np.log(0.19)) k = k1 + k2 + k3 + k4hhhh h"U literal_blockrh$}r(U xml:spacerUpreserverh)]h(]h&]h']h+]uh-Keh.hh]rh7XX# DEFINE parameterized covariance function k1 = pyGPs.cov.RBF(np.log(67.), np.log(66.)) k2 = pyGPs.cov.Periodic(np.log(1.3), np.log(1.0), np.log(2.4)) * cov.RBF(np.log(90.), np.log(2.4)) k3 = pyGPs.cov.RQ(np.log(1.2), np.log(0.66), np.log(0.78)) k4 = pyGPs.cov.RBF(np.log(1.6/12.), np.log(0.18)) + cov.Noise(np.log(0.19)) k = k1 + k2 + k3 + k4rr}r(hUhjubaubh;)r}r(hX"After running the minimization, ::rhhhh h"h?h$}r(h&]h']h(]h)]h+]uh-Kmh.hh]rh7XAfter running the minimization,rr}r(hXAfter running the minimization,hjubaubj)r}r(hX?t0 = clock() model.optimize(x,y) t1 = clock() model.predict(xs)hhhh h"jh$}r(jjh)]h(]h&]h']h+]uh-Koh.hh]rh7X?t0 = clock() model.optimize(x,y) t1 = clock() model.predict(xs)rr}r(hUhjubaubh;)r}r(hX!The extrapolated data looks like:rhhhh h"h?h$}r(h&]h']h(]h)]h+]uh-Kth.hh]rh7X!The extrapolated data looks like:rr}r(hjhjubaubh)r}r(hUhhhh h"hh$}r(hXcenterh)]h(]h&]h']h+]uh-Nh.hh]rh)r}r(hXg.. figure:: _images/demoML2.png :height: 600 px :width: 800 px :align: center :scale: 70 % h$}r(UscaleKFh&]UuriX_images/demoML2.pngrh)]h(]UwidthX800pxh']h}rU*jsh+]UheightX600pxuhjh]h"hubaubh;)r}r(hXzand the optimized values of the hyperparameters allow for a principled analysis of different components driving the model.rhhhh h"h?h$}r(h&]h']h(]h)]h+]uh-K|h.hh]rh7Xzand the optimized values of the hyperparameters allow for a principled analysis of different components driving the model.rr}r(hjhjubaubcdocutils.nodes footnote r)r}r(hX Keeling, C. D. and Whorf, T. P. (2004). Atmospheric :math:`CO_2` Records from Sites in the SIO Air Sampling Network. In Trends: A Compendium of Data on Global Change. Carbon Dioxide Information Analysis Center, Oak Ridge National Laboratory, Oak Ridge, Tenn., U.S.A. hPKhhhh h"Ufootnoterh$}r(h&]h']h(]rh|ah)]rhah+]rX1auh-K~h.hh]r(cdocutils.nodes label r)r}r(hUh$}r(h&]h']h(]h)]h+]uhjh]rh7X1r}r(hUhjubah"hubh;)r}r(hX Keeling, C. D. and Whorf, T. P. (2004). Atmospheric :math:`CO_2` Records from Sites in the SIO Air Sampling Network. In Trends: A Compendium of Data on Global Change. Carbon Dioxide Information Analysis Center, Oak Ridge National Laboratory, Oak Ridge, Tenn., U.S.A.rhjhh h"h?h$}r(h&]h']h(]h)]h+]uh-K~h]r(h7X4Keeling, C. D. and Whorf, T. P. (2004). Atmospheric rr}r(hX4Keeling, C. D. and Whorf, T. P. (2004). Atmospheric hjubhE)r}r(hUh$}r(UlatexXCO_2h)]h(]h&]h']h+]uhjh]h"hIubh7X Records from Sites in the SIO Air Sampling Network. In Trends: A Compendium of Data on Global Change. Carbon Dioxide Information Analysis Center, Oak Ridge National Laboratory, Oak Ridge, Tenn., U.S.A.rr}r(hX Records from Sites in the SIO Air Sampling Network. In Trends: A Compendium of Data on Global Change. 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Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ # covariance functions to be use by Gaussian process functions. There are two # different kinds of covariance functions: simple and composite: # # simple cov functions: # # Poly - Polynomial covariance function # PiecePoly - Piecewise polynomial kernel with compact support. # RBF - Squared Exponential kernel. # RBFunit - Squared Exponential kernel with unit magnitude. # RBFard - Squared Exponential kernel with Automatic Relevance Determination. # Const - Constant kernel. # Linear - Linear kernel. # LINard - Linear covariance function with Automatic Relevance Detemination. # Matern - Matern covariance function. # Periodic - Stationary kernel for a smooth periodic function. # Noise - Independent covariance function, i.e "white noise". # RQ - Rational Quadratic covariance function with isotropic distance measure. # RQard - Rational Quadratic covariance function with ARD distance measure. # Pre - Precomputed kernel matrix. # # composite covariance functions: # # ScaleOfKernel - scaled version of a covariance function # ProductOfKernel - products of covariance functions # SumOfKernel - sums of covariance functions # FITCOfKernel - Covariance function to be used together with the FITC approximation # # # This is a object-oriented python implementation of gpml functionality # (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18). # based on the functional-version of python implementation # (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013) # # Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013 import numpy as np import math import scipy.spatial.distance as spdist class Kernel(object): """ This is a base class of Kernel functions there is no computation in this class, it just defines rules about a kernel class should have each covariance function will inherit it and implement its own behaviour """ def __init__(self): self.hyp = [] self.para = [] def getCovMatrix(self,x=None,z=None,mode=None): ''' Return the specific covariance matrix according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self covariance matrix of test data(test by 1). 'train' return training covariance matrix(train by train). 'cross' return cross covariance matrix between x and z(train by test) :return: the corresponding covariance matrix ''' pass def getDerMatrix(self,x=None,z=None,mode=None,der=None): ''' Compute derivatives wrt. hyperparameters according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self derivative matrix of test data(test by 1). 'train' return training derivative matrix(train by train). 'cross' return cross derivative matrix between x and z(train by test) :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' pass # overloading def __add__(self,cov): ''' Overloading + operator. :param cov: covariance function :return: an instance of SumOfKernel ''' return SumOfKernel(self,cov) # overloading def __mul__(self,other): ''' Overloading * operator. Using * for both multiplication with scalar and product of kernels depending on the type of the two objects. :param other: covariance function as product or int/float as scalar :return: an instance of ScaleOfKernel or ProductOfKernel ''' if isinstance(other, int) or isinstance(other, float): return ScaleOfKernel(self,other) elif isinstance(other, Kernel): return ProductOfKernel(self,other) else: print "only numbers and Kernels are supported operand types for *" # overloading __rmul__ = __mul__ def fitc(self,inducingInput): ''' Covariance function to be used together with the FITC approximation. Setting FITC gp model will implicitly call this method. :return: an instance of FITCOfKernel ''' return FITCOfKernel(self,inducingInput) # can be replaced by spdist from scipy def _sq_dist(self, a, b=None): '''Compute a matrix of all pairwise squared distances between two sets of vectors, stored in the row of the two matrices: a (of size n by D) and b (of size m by D).''' n = a.shape[0] D = a.shape[1] m = n if b == None: b = a.transpose() else: m = b.shape[0] b = b.transpose() C = np.zeros((n,m)) for d in range(0,D): tt = a[:,d] tt = tt.reshape(n,1) tem = np.kron(np.ones((1,m)), tt) tem = tem - np.kron(np.ones((n,1)), b[d,:]) C = C + tem * tem return C class ProductOfKernel(Kernel): '''Product of two kernel function.''' def __init__(self,cov1,cov2): self.cov1 = cov1 self.cov2 = cov2 self._hyp = cov1.hyp + cov2.hyp def _setHyp(self,hyp): assert len(hyp) == len(self._hyp) len1 = len(self.cov1.hyp) self._hyp = hyp self.cov1.hyp = self._hyp[:len1] self.cov2.hyp = self._hyp[len1:] def _getHyp(self): return self._hyp hyp = property(_getHyp,_setHyp) def getCovMatrix(self,x=None,z=None,mode=None): A = self.cov1.getCovMatrix(x,z,mode) * self.cov2.getCovMatrix(x,z,mode) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): if der < len(self.cov1.hyp): A = self.cov1.getDerMatrix(x,z,mode,der) * self.cov2.getCovMatrix(x,z,mode) elif der < len(self.hyp): der2 = der - len(self.cov1.hyp) A = self.cov2.getDerMatrix(x,z,mode,der2) * self.cov1.getCovMatrix(x,z,mode) else: raise Exception("Error: der out of range for covProduct") return A class SumOfKernel(Kernel): '''Sum of two kernel function.''' def __init__(self,cov1,cov2): self.cov1 = cov1 self.cov2 = cov2 self._hyp = cov1.hyp + cov2.hyp def _setHyp(self,hyp): assert len(hyp) == len(self._hyp) len1 = len(self.cov1.hyp) self._hyp = hyp self.cov1.hyp = self._hyp[:len1] self.cov2.hyp = self._hyp[len1:] def _getHyp(self): return self._hyp hyp = property(_getHyp,_setHyp) def getCovMatrix(self,x=None,z=None,mode=None): A = self.cov1.getCovMatrix(x,z,mode) + self.cov2.getCovMatrix(x,z,mode) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): if der < len(self.cov1.hyp): A = self.cov1.getDerMatrix(x,z,mode,der) elif der < len(self.hyp): der2 = der - len(self.cov1.hyp) A = self.cov2.getDerMatrix(x,z,mode,der2) else: raise Exception("Error: der out of range for covSum") return A class ScaleOfKernel(Kernel): '''Scale of a kernel function.''' def __init__(self,cov,scalar): self.cov = cov if cov.hyp: self._hyp = [scalar] + cov.hyp else: self._hyp = [scalar] def _setHyp(self,hyp): assert len(hyp) == len(self._hyp) self._hyp = hyp self.cov.hyp = self._hyp[1:] def _getHyp(self): return self._hyp hyp = property(_getHyp,_setHyp) def getCovMatrix(self,x=None,z=None,mode=None): sf2 = np.exp(self.hyp[0]) # scale parameter A = sf2 * self.cov.getCovMatrix(x,z,mode) # accumulate cov return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): sf2 = np.exp(self.hyp[0]) # scale parameter if der == 0: # compute derivative w.r.t. sf2 A = 2. * sf2 * self.cov.getCovMatrix(x,z,mode) else: A = sf2 * self.cov.getDerMatrix(x,z,mode,der-1) return A class FITCOfKernel(Kernel): ''' Covariance function to be used together with the FITC approximation. The function allows for more than one output argument and does not respect the interface of a proper covariance function. Instead of outputing the full covariance, it returns cross-covariances between the inputs x, z and the inducing inputs xu as needed by infFITC ''' def __init__(self,cov,inducingInput): self.inducingInput = inducingInput self.covfunc = cov self._hyp = cov.hyp def _getHyp(self): return self._hyp def _setHyp(self, hyp): self._hyp = hyp self.covfunc.hyp = hyp hyp = property(_getHyp,_setHyp) def getCovMatrix(self,x=None,z=None,mode=None): xu = self.inducingInput if x != None: try: assert(xu.shape[1] == x.shape[1]) except AssertionError: raise Exception('Dimensionality of inducing inputs must match training inputs') if mode == 'self_test': # self covariances for the test cases K = self.covfunc.getCovMatrix(z=z,mode='self_test') return K elif mode == 'train': # compute covariance matix for training set K = self.covfunc.getCovMatrix(z=x,mode='self_test') Kuu = self.covfunc.getCovMatrix(x=xu,mode='train') Ku = self.covfunc.getCovMatrix(x=xu,z=x,mode='cross') return K, Kuu, Ku elif mode == 'cross': # compute covariance between data sets x and z K = self.covfunc.getCovMatrix(x=xu,z=z,mode='cross') return K def getDerMatrix(self,x=None,z=None,mode=None,der=None): xu = self.inducingInput if x!= None: try: assert(xu.shape[1] == x.shape[1]) except AssertionError: raise Exception('Dimensionality of inducing inputs must match training inputs') if mode == 'self_test': # self covariances for the test cases K = self.covfunc.getDerMatrix(z=z,mode='self_test',der=der) return K elif mode == 'train': # compute covariance matix for training set K = self.covfunc.getDerMatrix(z=x,mode='self_test',der=der) Kuu = self.covfunc.getDerMatrix(x=xu,mode='train',der=der) Ku = self.covfunc.getDerMatrix(x=xu,z=x,mode='cross',der=der) return K, Kuu, Ku elif mode == 'cross': # compute covariance between data sets x and z K = self.covfunc.getDerMatrix(x=xu,z=z,mode='cross',der=der) return K class Poly(Kernel): ''' Polynomial covariance function. hyp = [ log_c, log_sigma ] :param log_c: inhomogeneous offset. :param log_sigma: signal deviation. :param d: degree of polynomial (not treated as hyperparameter, i.e. will not be trained). ''' def __init__(self, log_c=0., d=2, log_sigma=0. ): self.hyp = [log_c, log_sigma] self.para = [d] def getCovMatrix(self,x=None,z=None,mode=None): c = np.exp(self.hyp[0]) # inhomogeneous offset sf2 = np.exp(2.*self.hyp[1]) # signal variance ord = self.para[0] # order of polynomial if np.abs(ord-np.round(ord)) < 1e-8: # remove numerical error from format of parameter ord = int(round(ord)) assert(ord >= 1.) # only nonzero integers for ord ord = int(ord) if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.reshape(np.sum(z*z,1), (nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.dot(x,x.T) elif mode == 'cross': # compute covariance between data sets x and z A = np.dot(x,z.T) A = sf2 * (c + A)**ord return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): c = np.exp(self.hyp[0]) # inhomogeneous offset sf2 = np.exp(2.*self.hyp[1]) # signal variance ord = self.para[0] # order of polynomial if np.abs(ord-np.round(ord)) < 1e-8: # remove numerical error from format of parameter ord = int(round(ord)) assert(ord >= 1.) # only nonzero integers for ord ord = int(ord) if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.reshape(np.sum(z*z,1), (nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.dot(x,x.T) elif mode == 'cross': # compute covariance between data sets x and z A = np.dot(x,z.T) if der == 0: # compute derivative matrix wrt 1st parameter A = c * ord * sf2 * (c+A)**(ord-1) elif der == 1: # compute derivative matrix wrt 2nd parameter A = 2. * sf2 * (c + A)**ord elif der == 2: # no derivative wrt 3rd parameter A = np.zeros_like(A) # do nothing (d is not learned) else: raise Exception("Wrong derivative entry in Poly") return A class PiecePoly(Kernel): ''' Piecewise polynomial kernel with compact support. hyp = [log_ell, log_sigma] :param log_ell: characteristic length scale. :param log_sigma: signal deviation. :param v: degree v will be rounded to 0,1,2,or 3. (not treated as hyperparameter, i.e. will not be trained). ''' def __init__(self, log_ell=0., v=2, log_sigma=0. ): self.hyp = [log_ell, log_sigma] self.para = [v] def ppmax(self,A,B): return np.maximum(A,B*np.ones_like(A)) def func(self,v,r,j): if v == 0: return 1 elif v == 1: return ( 1. + (j+1) * r ) elif v == 2: return ( 1. + (j+2)*r + (j*j + 4.*j+ 3)/3.*r*r ) elif v == 3: return ( 1. + (j+3)*r + (6.*j*j+36.*j+45.)/15.*r*r + (j*j*j+9.*j*j+23.*j+15.)/15.*r*r*r ) else: raise Exception (["Wrong degree in PiecePoly. Should be 0,1,2 or 3, is " + str(v)]) def dfunc(self,v,r,j): if v == 0: return 0 elif v == 1: return ( j+1 ) elif v == 2: return ( (j+2) + 2.*(j*j+ 4.*j+ 3.)/3.*r ) elif v == 3: return ( (j+3) + 2.*(6.*j*j+36.*j+45.)/15.*r + (j*j*j+9.*j*j+23.*j+15.)/5.*r*r ) else: raise Exception (["Wrong degree in PiecePoly. Should be 0,1,2 or 3, is " + str(v)]) def pp(self,r,j,v,func): return func(v,r,j)*(self.ppmax(1-r,0)**(j+v)) def dpp(self,r,j,v,func,dfunc): return self.ppmax(1-r,0)**(j+v-1) * r * ( (j+v)*self.func(v,r,j) - self.ppmax(1-r,0) * self.dfunc(v,r,j) ) def getCovMatrix(self,x=None,z=None,mode=None): if x != None: n, D = x.shape if z != None: nn, D = z.shape ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.*self.hyp[1]) # signal variance v = self.para[0] # degree (v = 0,1,2 or 3 only) if np.abs(v-np.round(v)) < 1e-8: # remove numerical error from format of parameter v = int(round(v)) assert(int(v) in range(4)) # Only allowed degrees: 0,1,2 or 3 v = int(v) j = np.floor(0.5*D) + v + 1 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.sqrt( spdist.cdist(x/ell, x/ell, 'sqeuclidean') ) elif mode == 'cross': # compute covariance between data sets x and z A = np.sqrt( spdist.cdist(x/ell, z/ell, 'sqeuclidean') ) A = sf2 * self.pp(A,j,v,self.func) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): if x != None: n, D = x.shape if z != None: nn, D = z.shape ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.*self.hyp[1]) # signal variance v = self.para[0] # degree (v = 0,1,2 or 3 only) if np.abs(v-np.round(v)) < 1e-8: # remove numerical error from format of parameter v = int(round(v)) assert(int(v) in range(4)) # Only allowed degrees: 0,1,2 or 3 v = int(v) j = np.floor(0.5*D) + v + 1 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.sqrt( spdist.cdist(x/ell, x/ell, 'sqeuclidean') ) elif mode == 'cross': # compute covariance between data sets x and z A = np.sqrt( spdist.cdist(x/ell, z/ell, 'sqeuclidean') ) if der == 0: # compute derivative matrix wrt 1st parameter A = sf2 * self.dpp(A,j,v,self.func,self.dfunc) elif der == 1: # compute derivative matrix wrt 2nd parameter A = 2. * sf2 * self.pp(A,j,v,self.func) elif der == 2: # wants to compute derivative wrt order A = np.zeros_like(A) else: raise Exception("Wrong derivative entry in PiecePoly") return A class RBF(Kernel): ''' Squared Exponential kernel with isotropic distance measure. hyp = [log_ell, log_sigma] :param log_ell: characteristic length scale. :param log_sigma: signal deviation. ''' def __init__(self, log_ell=0., log_sigma=0.): self.hyp = [log_ell, log_sigma] def getCovMatrix(self,x=None,z=None,mode=None): ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.*self.hyp[1]) # signal variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for training set A = spdist.cdist(x/ell,x/ell,'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z A = spdist.cdist(x/ell,z/ell,'sqeuclidean') A = sf2 * np.exp(-0.5*A) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.*self.hyp[1]) # signal variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = spdist.cdist(x/ell,x/ell,'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z A = spdist.cdist(x/ell,z/ell,'sqeuclidean') if der == 0: # compute derivative matrix wrt 1st parameter A = sf2 * np.exp(-0.5*A) * A elif der == 1: # compute derivative matrix wrt 2nd parameter A = 2. * sf2 * np.exp(-0.5*A) else: raise Exception("Calling for a derivative in RBF that does not exist") return A class RBFunit(Kernel): ''' Squared Exponential kernel with isotropic distance measure with unit magnitude. i.e signal variance is always 1. hyp = [ log_ell ] :param log_ell: characteristic length scale. ''' def __init__(self, log_ell=0.): self.hyp = [log_ell] def getCovMatrix(self,x=None,z=None,mode=None): ell = np.exp(self.hyp[0]) # characteristic length scale if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = spdist.cdist(x/ell,x/ell,'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z A = spdist.cdist(x/ell,z/ell,'sqeuclidean') A = np.exp(-0.5*A) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): ell = np.exp(self.hyp[0]) # characteristic length scale if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = spdist.cdist(x/ell,x/ell,'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z A = spdist.cdist(x/ell,z/ell,'sqeuclidean') if der == 0: # compute derivative matrix wrt 1st parameter A = np.exp(-0.5*A) * A else: raise Exception("Wrong derivative index in RDFunit") return A class RBFard(Kernel): ''' Squared Exponential kernel with Automatic Relevance Determination. hyp = log_ell_list + [log_sigma] :param D: dimension of pattern. set if you want default ell, which is 1 for each dimension. :param log_ell_list: characteristic length scale for each dimension. :param log_sigma: signal deviation. ''' def __init__(self, D=None, log_ell_list=None, log_sigma=0.): if log_ell_list == None: self.hyp = [0. for i in xrange(D)] + [log_sigma] else: self.hyp = log_ell_list + [log_sigma] def getCovMatrix(self,x=None,z=None,mode=None): if x != None: n, D = x.shape if z != None: nn, D = z.shape ell = 1./np.exp(self.hyp[0:D]) # characteristic length scale sf2 = np.exp(2.*self.hyp[D]) # signal variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x tem = np.dot(np.diag(ell),x.T).T A = spdist.cdist(tem,tem,'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z A = spdist.cdist(np.dot(np.diag(ell),x.T).T,np.dot(np.diag(ell),z.T).T,'sqeuclidean') A = sf2*np.exp(-0.5*A) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): if x != None: n, D = x.shape if z != None: nn, D = z.shape ell = 1./np.exp(self.hyp[0:D]) # characteristic length scale sf2 = np.exp(2.*self.hyp[D]) # signal variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x tem = np.dot(np.diag(ell),x.T).T A = spdist.cdist(tem,tem,'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z A = spdist.cdist(np.dot(np.diag(ell),x.T).T,np.dot(np.diag(ell),z.T).T,'sqeuclidean') A = sf2*np.exp(-0.5*A) if der < D: # compute derivative matrix wrt length scale parameters if mode == 'self_test': A = A*0 elif mode == 'train': tem = np.atleast_2d(x[:,der])/ell[der] A *= spdist.cdist(tem,tem,'sqeuclidean') elif mode == 'cross': A *= spdist.cdist(np.atleast_2d(x[:,der]).T/ell[der],np.atleast_2d(z[:,der]).T/ell[der],'sqeuclidean') elif der == D: # compute derivative matrix wrt magnitude parameter A = 2.*A else: raise Exception("Wrong derivative index in RDFard") return A class Const(Kernel): ''' Constant kernel. hyp = [ log_sigma ] :param log_sigma: signal deviation. ''' def __init__(self, log_sigma=0.): self.hyp = [log_sigma] def getCovMatrix(self,x=None,z=None,mode=None): sf2 = np.exp(self.hyp[0]) # s2 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = sf2 * np.ones((nn,1)) elif mode == 'train': # compute covariance matix for dataset x n,D = x.shape A = sf2 * np.ones((n,n)) elif mode == 'cross': # compute covariance between data sets x and z n,D = x.shape nn,D = z.shape A = sf2 * np.ones((n,nn)) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): sf2 = np.exp(self.hyp[0]) # s2 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = sf2 * np.ones((nn,1)) elif mode == 'train': # compute covariance matix for dataset x n,D = x.shape A = sf2 * np.ones((n,n)) elif mode == 'cross': # compute covariance between data sets x and z n,D = x.shape nn,D = z.shape A = sf2 * np.ones((n,nn)) if der == 0: # compute derivative matrix wrt sf2 A = 2. * A else: raise Exception("Wrong derivative entry in covConst") return A class Linear(Kernel): ''' Linear kernel. hyp = [ log_sigma ]. ''' def __init__(self, log_sigma=0.): self.hyp = [ log_sigma ] def getCovMatrix(self,x=None,z=None,mode=None): sf2 = np.exp(self.hyp[0]) # s2 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.reshape(np.sum(z*z,1), (nn,1)) elif mode == 'train': # compute covariance matix for dataset x n,D = x.shape A = np.dot(x,x.T) + np.eye(n)*1e-16 # required for numerical accuracy elif mode == 'cross': # compute covariance between data sets x and z A = np.dot(x,z.T) A = sf2 * A return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): sf2 = np.exp(self.hyp[0]) # s2 if der == 0: if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.reshape(np.sum(z*z,1), (nn,1)) elif mode == 'train': # compute covariance matix for dataset x n,D = x.shape A = np.dot(x,x.T) + np.eye(n)*1e-16 # required for numerical accuracy elif mode == 'cross': # compute covariance between data sets x and z A = np.dot(x,z.T) A = 2 * sf2 * A else: raise Exception("Wrong derivative index in covLinear") return A class LINard(Kernel): ''' Linear covariance function with Automatic Relevance Detemination. hyp = log_ell_list :param D: dimension of training data. Set if you want default ell, which is 1 for each dimension. :param log_ell_list: characteristic length scale for each dimension. ''' def __init__(self, D=None, log_ell_list=None): if log_ell_list == None: self.hyp = [0. for i in xrange(D)] else: self.hyp = log_ell_list def getCovMatrix(self,x=None,z=None,mode=None): ell = np.exp(self.hyp) # ARD parameters if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.reshape(np.sum(z*z,1), (nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.dot(x,x.T) elif mode == 'cross': # compute covariance between data sets x and z z = np.dot(z,np.diag(1./ell)) A = np.dot(x,z.T) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): ell = np.exp(self.hyp) # ARD parameters if x != None: n, D = x.shape if z != None: nn, D = z.shape if der < D: if mode == 'self_test': tem = np.atleast_2d(z[:,der]).T A = -2.* tem * tem elif mode == 'train': A = -2.*np.dot(np.atleast_2d(x[:,der]).T,np.atleast_2d(x[:,der])) elif mode == 'cross': z = np.dot(z,np.diag(1./ell)) A = -2.*np.dot(np.atleast_2d(x[:,der]).T, np.atleast_2d(z[:,der])) # cross covariances else: raise Exception("Wrong derivative index in covLINard") return A class Matern(Kernel): ''' Matern covariance function with nu = d/2 and isotropic distance measure. For d=1 the function is also known as the exponential covariance function or the Ornstein-Uhlenbeck covariance in 1d. d will be rounded to 1, 3, or 5. hyp = [ log_ell, log_sigma] :param d: d is 2 times nu. Can only be 1,3 or 5. :param log_ell: characteristic length scale. :param log_sigma: signal deviation. ''' def __init__(self, log_ell=0., d=3, log_sigma=0. ): self.hyp = [ log_ell, log_sigma ] self.para = [d] def func(self,d,t): if d == 1: return 1 elif d == 3: return 1 + t elif d == 5: return 1 + t*(1+t/3.) else: raise Exception("Wrong value for d in Matern") def dfunc(self,d,t): if d == 1: return 1 elif d == 3: return t elif d == 5: return t*(1+t/3.) else: raise Exception("Wrong value for d in Matern") def mfunc(self,d,t): return self.func(d,t)*np.exp(-1.*t) def dmfunc(self,d,t): return self.dfunc(d,t)*t*np.exp(-1.*t) def getCovMatrix(self,x=None,z=None,mode=None): ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.* self.hyp[1]) # signal variance d = self.para[0] # 2 times nu if np.abs(d-np.round(d)) < 1e-8: # remove numerical error from format of parameter d = int(round(d)) d = int(d) try: assert(d in [1,3,5]) # check for valid values of d except AssertionError: print "Warning: You specified d to be neither 1,3 nor 5. We set to d=3. " d = 3 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x x = np.sqrt(d)*x/ell A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean')) elif mode == 'cross': # compute covariance between data sets x and z x = np.sqrt(d)*x/ell z = np.sqrt(d)*z/ell A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean')) A = sf2 * self.mfunc(d,A) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.* self.hyp[1]) # signal variance d = self.para[0] # 2 times nu if np.abs(d-np.round(d)) < 1e-8: # remove numerical error from format of parameter d = int(round(d)) d = int(d) try: assert(d in [1,3,5]) # check for valid values of d except AssertionError: print "Warning: You specified d to be neither 1,3 nor 5. We set to d=3. " d = 3 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x x = np.sqrt(d)*x/ell A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean')) elif mode == 'cross': # compute covariance between data sets x and z x = np.sqrt(d)*x/ell z = np.sqrt(d)*z/ell A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean')) A = sf2 * self.mfunc(d,A) if der == 0: # compute derivative matrix wrt 1st parameter A = sf2 * self.dmfunc(d,A) elif der == 1: # compute derivative matrix wrt 2nd parameter A = 2 * sf2 * self.mfunc(d,A) elif der == 2: # no derivative wrt 3rd parameter A = np.zeros_like(A) # do nothing (d is not learned) else: raise Exception("Wrong derivative value in Matern") return A class Periodic(Kernel): ''' Stationary kernel for a smooth periodic function. hyp = [ log_ell, log_p, log_sigma] :param log_p: period. :param log_ell: characteristic length scale. :param log_sigma: signal deviation. ''' def __init__(self, log_ell=0., log_p=0., log_sigma=0. ): self.hyp = [ log_ell, log_p, log_sigma] def getCovMatrix(self,x=None,z=None,mode=None): ell = np.exp(self.hyp[0]) # characteristic length scale p = np.exp(self.hyp[1]) # period sf2 = np.exp(2.*self.hyp[2]) # signal variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean')) elif mode == 'cross': # compute covariance between data sets x and z A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean')) A = np.pi*A/p A = np.sin(A)/ell A = A * A A = sf2 *np.exp(-2.*A) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): ell = np.exp(self.hyp[0]) # characteristic length scale p = np.exp(self.hyp[1]) # period sf2 = np.exp(2.*self.hyp[2]) # signal variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean')) elif mode == 'cross': # compute covariance between data sets x and z A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean')) A = np.pi*A/p if der == 0: # compute derivative matrix wrt 1st parameter A = np.sin(A)/ell A = A * A A = 4. *sf2 *np.exp(-2.*A) * A elif der == 1: # compute derivative matrix wrt 2nd parameter R = np.sin(A)/ell A = 4 * sf2/ell * np.exp(-2.*R*R)*R*np.cos(A)*A elif der == 2: # compute derivative matrix wrt 3rd parameter A = np.sin(A)/ell A = A * A A = 2. * sf2 * np.exp(-2.*A) else: raise Exception("Wrong derivative index in covPeriodic") return A class Noise(Kernel): ''' Independent covariance function, i.e "white noise", with specified variance. Normally NOT used anymore since noise is now added in liklihood. hyp = [ log_sigma ] :param log_sigma: signal deviation. ''' def __init__(self, log_sigma=0.): self.hyp = [log_sigma] def getCovMatrix(self,x=None,z=None,mode=None): tol = 1.e-9 # Tolerance for declaring two vectors "equal" s2 = np.exp(2.*self.hyp[0]) # noise variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x n,D = x.shape A = np.eye(n) elif mode == 'cross': # compute covariance between data sets x and z M = spdist.cdist(x, z, 'sqeuclidean') A = np.zeros_like(M,dtype=np.float) A[M < tol] = 1. A = s2*A return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): tol = 1.e-9 # Tolerance for declaring two vectors "equal" s2 = np.exp(2.*self.hyp[0]) # noise variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x n,D = x.shape A = np.eye(n) elif mode == 'cross': # compute covariance between data sets x and z M = spdist.cdist(x, z, 'sqeuclidean') A = np.zeros_like(M,dtype=np.float) A[M < tol] = 1. if der == 0: A = 2.*s2*A else: raise Exception("Wrong derivative index in covNoise") return A class RQ(Kernel): ''' Rational Quadratic covariance function with isotropic distance measure. hyp = [ log_ell, log_sigma, log_alpha ] :param log_ell: characteristic length scale. :param log_sigma: signal deviation. :param log_alpha: shape parameter for the RQ covariance. ''' def __init__(self, log_ell=0., log_sigma=0., log_alpha=0.): self.hyp = [ log_ell, log_sigma, log_alpha ] def getCovMatrix(self,x=None,z=None,mode=None): ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.*self.hyp[1]) # signal variance alpha = np.exp(self.hyp[2]) if mode == 'self_test': # self covariances for the test cases nn,D = z.shape D2 = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x D2 = spdist.cdist(x/ell, x/ell, 'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z D2 = spdist.cdist(x/ell, z/ell, 'sqeuclidean') A = sf2 * ( ( 1.0 + 0.5*D2/alpha )**(-alpha) ) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.*self.hyp[1]) # signal variance alpha = np.exp(self.hyp[2]) if mode == 'self_test': # self covariances for the test cases nn,D = z.shape D2 = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x D2 = spdist.cdist(x/ell, x/ell, 'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z D2 = spdist.cdist(x/ell, z/ell, 'sqeuclidean') if der == 0: # compute derivative matrix wrt 1st parameter A = sf2 * ( 1.0 + 0.5*D2/alpha )**(-alpha-1) * D2 elif der == 1: # compute derivative matrix wrt 2nd parameter A = 2.* sf2 * ( ( 1.0 + 0.5*D2/alpha )**(-alpha) ) elif der == 2: # compute derivative matrix wrt 3rd parameter K = ( 1.0 + 0.5*D2/alpha ) A = sf2 * K**(-alpha) * (0.5*D2/K - alpha*np.log(K) ) else: raise Exception("Wrong derivative index in covRQ") return A class RQard(Kernel): ''' Rational Quadratic covariance function with Automatic Relevance Detemination (ARD) distance measure. hyp = log_ell_list + [ log_sigma, log_alpha ] :param D: dimension of pattern. set if you want default ell, which is 0.5 for each dimension. :param log_ell_list: characteristic length scale for each dimension. :param log_sigma: signal deviation. :param log_alpha: shape parameter for the RQ covariance. ''' def __init__(self, D=None, log_ell_list=None, log_sigma=0., log_alpha=0.): if log_ell_list == None: self.hyp = [0. for i in xrange(D)] + [ log_sigma, log_alpha ] else: self.hyp = log_ell_list + [ log_sigma, log_alpha ] def getCovMatrix(self,x=None,z=None,mode=None): if x != None: n, D = x.shape if z != None: nn, D = z.shape ell = 1./np.exp(self.hyp[0:D]) # characteristic length scale sf2 = np.exp(2.*self.hyp[D]) # signal variance alpha = np.exp(self.hyp[D+1]) if mode == 'self_test': # self covariances for the test cases nn,D = z.shape D2 = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x tmp = np.dot(np.diag(ell),x.T).T D2 = spdist.cdist(tmp, tmp, 'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z D2 = spdist.cdist(np.dot(np.diag(ell),x.T).T, np.dot(np.diag(ell),z.T).T, 'sqeuclidean') A = sf2 * ( ( 1.0 + 0.5*D2/alpha )**(-alpha) ) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): if x != None: n, D = x.shape if z != None: nn, D = z.shape ell = 1./np.exp(self.hyp[0:D]) # characteristic length scale sf2 = np.exp(2.*self.hyp[D]) # signal variance alpha = np.exp(self.hyp[D+1]) if mode == 'self_test': # self covariances for the test cases nn,D = z.shape D2 = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x tmp = np.dot(np.diag(ell),x.T).T D2 = spdist.cdist(tmp, tmp, 'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z D2 = spdist.cdist(np.dot(np.diag(ell),x.T).T, np.dot(np.diag(ell),z.T).T, 'sqeuclidean') if der < D: if mode == 'self_test': A = D2*0 elif mode == 'train': tmp = np.atleast_2d(x[:,der])/ell[der] A = sf2 * ( 1.0 + 0.5*D2/alpha )**(-alpha-1) * spdist.cdist(tmp, tmp, 'sqeuclidean') elif mode == 'cross': A = sf2 * ( 1.0 + 0.5*D2/alpha )**(-alpha-1) * spdist.cdist(np.atleast_2d(x[:,der]).T/ell[der], np.atleast_2d(z[:,der]).T/ell[der], 'sqeuclidean') elif der==D: # compute derivative matrix wrt magnitude parameter A = 2. * sf2 * ( ( 1.0 + 0.5*D2/alpha )**(-alpha) ) elif der==(D+1): # compute derivative matrix wrt magnitude parameter K = ( 1.0 + 0.5*D2/alpha ) A = sf2 * K**(-alpha) * ( 0.5*D2/K - alpha*np.log(K) ) else: raise Exception("Wrong derivative index in covRQard") return A class Pre(Kernel): ''' Precomputed kernel matrix. No hyperparameters and thus nothing will be optimised. :param M1: cross covariances matrix(train+1 by test). last row is self covariances (diagonal of test by test) :param M2: training set covariance matrix (train by train) ''' def __init__(self,M1,M2): self.M1 = M1 self.M2 = M2 self.hyp = [] def getCovMatrix(self,x=None,z=None,mode=None): if mode == 'self_test': # diagonal covariance between test_test A = self.M1[-1,:] # self covariances for the test cases (last row) A = np.reshape(A, (A.shape[0],1)) elif mode == 'train': # covariance between train_train A = self.M2 elif mode == 'cross': # covariance between train_test A = self.M1[:-1,:] return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): if der != None: raise Exception("Error: NO optimization in precomputed kernel matrix") return 0 if __name__ == '__main__': pass }r;M(XConstXclassMjMXPre.getDerMatrixXdefMJMQXPeriodic.getDerMatrixXdefMmMX PiecePolyXclassMqMXLinear.__init__XdefMMX Kernel.fitcXdefK{KXLinearXclassMMX RQ.__init__XdefMMXSumOfKernel.getCovMatrixXdefKKчXRQard.getCovMatrixXdefMMX Pre.__init__XdefM;M@XProductOfKernel._getHypXdefKKXKernel.getCovMatrixXdefK@KNX PiecePoly.ppXdefMMXLinear.getCovMatrixXdefMMXKernel.getDerMatrixXdefKNK^X ScaleOfKernelXclassKKXLinear.getDerMatrixXdefMMXRQardXclassMM3XKernel.__init__XdefKM}r?MbXx&#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ # mean functions to be use by Gaussian process functions. There are two # different kinds of mean functions: simple and composite: # # simple mean functions: # # Zero - zero mean function # One - one mean function # Const - constant mean function # Linear - linear mean function # # composite mean functions: # # ScaleOfMean - scaled version of a mean function # PowerOfMean - power of a mean function # ProductOfMean - products of mean functions # SumOfMean - sums of mean functions # # # This is a object-oriented python implementation of gpml functionality # (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18). # based on the functional-version of python implementation # (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013) # # # Copyright (c) by Marion Neumann and Shan Huang, Sep.2013 import numpy as np import math class Mean(object): """ The base function for mean function """ def __init__(self): super(Mean, self).__init__() self.hyp = [] self.para = [] # overloading def __add__(self,mean): ''' Overloading + operator. :param mean: mean function :return: an instance of SumOfMean ''' return SumOfMean(self,mean) # overloading def __mul__(self,other): ''' Overloading * operator. Using * for both multiplication with scalar and product of means depending on the type of the two objects. :param other: mean function as product or int/float as scalar :return: an instance of ScaleOfMean or ProductOfMean ''' if isinstance(other, int) or isinstance(other, float): return ScaleOfMean(self,other) elif isinstance(other, Mean): return ProductOfMean(self,other) else: print "only numbers and Means are allowed for *" # overloading __rmul__ = __mul__ # overloading def __pow__(self,number): ''' Overloading ** operator. :param int number: power of the mean function :return: an instance of PowerOfMean ''' if isinstance(number, int) and number > 0: return PowerOfMean(self,number) else: print "only non-zero integers are supported for **" def getMean(self, x=None): ''' Get the mean vector. ''' pass def getDerMatrix(self, x=None, der=None): ''' Compute derivatives wrt. hyperparameters. :param x: training data :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' pass class ProductOfMean(Mean): '''Product of two mean fucntions.''' def __init__(self,mean1,mean2): self.mean1 = mean1 self.mean2 = mean2 if mean1.hyp and mean2.hyp: self._hyp = mean1.hyp + mean2.hyp elif not mean1.hyp: self._hyp = mean2.hyp elif not mean2.hyp: self._hyp = mean1.hyp def _setHyp(self,hyp): assert len(hyp) == len(self._hyp) len1 = len(self.mean1.hyp) self._hyp = hyp self.mean1.hyp = self._hyp[:len1] self.mean2.hyp = self._hyp[len1:] def _getHyp(self): return self._hyp hyp = property(_getHyp,_setHyp) def getMean(self, x=None): A = self.mean1.getMean(x) * self.mean2.getMean(x) return A def getDerMatrix(self, x=None, der=None): if der < len(self.mean1.hyp): A = self.mean1.getDerMatrix(x, der) * self.mean2.getMean(x) elif der < len(self.hyp): der2 = der - len(self.mean1.hyp) A = self.mean2.getDerMatrix(x, der2) * self.mean1.getMean(x) else: raise Exception("Error: der out of range for meanProduct") return A class SumOfMean(Mean): '''Sum of two mean functions.''' def __init__(self,mean1,mean2): self.mean1 = mean1 self.mean2 = mean2 if mean1.hyp and mean2.hyp: self._hyp = mean1.hyp + mean2.hyp elif not mean1.hyp: self._hyp = mean2.hyp elif not mean2.hyp: self._hyp = mean1.hyp def _setHyp(self,hyp): assert len(hyp) == len(self._hyp) len1 = len(self.mean1.hyp) self._hyp = hyp self.mean1.hyp = self._hyp[:len1] self.mean2.hyp = self._hyp[len1:] def _getHyp(self): return self._hyp hyp = property(_getHyp,_setHyp) def getMean(self, x=None): A = self.mean1.getMean(x) + self.mean2.getMean(x) return A def getDerMatrix(self, x=None, der=None): if der < len(self.mean1.hyp): A = self.mean1.getDerMatrix(x, der) elif der < len(self.hyp): der2 = der - len(self.mean1.hyp) A = self.mean2.getDerMatrix(x, der2) else: raise Exception("Error: der out of range for meanSum") return A class ScaleOfMean(Mean): '''Scale of a mean function.''' def __init__(self,mean,scalar): self.mean = mean if mean.hyp: self._hyp = [scalar] + mean.hyp else: self._hyp = [scalar] def _setHyp(self,hyp): assert len(hyp) == len(self._hyp) self._hyp = hyp self.mean.hyp = self._hyp[1:] def _getHyp(self): return self._hyp hyp = property(_getHyp,_setHyp) def getMean(self, x=None): c = self.hyp[0] # scale parameter A = c * self.mean.getMean(x) # accumulate means return A def getDerMatrix(self, x=None, der=None): c = self.hyp[0] # scale parameter if der == 0: # compute derivative w.r.t. c A = self.mean.getMean(x) else: A = c * self.mean.getDerMatrix(x,der-1) return A class PowerOfMean(Mean): '''Power of a mean fucntion.''' def __init__(self, mean, d): self.mean = mean if mean.hyp: self._hyp = [d] + mean.hyp else: self._hyp = [d] def _setHyp(self,hyp): assert len(hyp) == len(self._hyp) self._hyp = hyp self.mean.hyp = self._hyp[1:] def _getHyp(self): return self._hyp hyp = property(_getHyp,_setHyp) def getMean(self, x=None): d = np.abs(np.floor(self.hyp[0])) d = max(d,1) A = self.mean.getMean(x) **d # accumulate means return A def getDerMatrix(self, x=None, der=None): d = np.abs(np.floor(self.hyp[0])) d = max(d,1) if der == 0: # compute derivative w.r.t. c a = self.mean.getMean(x) A = a**d * np.log(a) else: A = d * self.mean.getMean(x) ** (d-1) * self.mean.getDerMatrix(x, der-1) return A class Zero(Mean): '''Zero mean.''' def __init__(self): self.hyp = [] self.name = '0' def getMean(self, x=None): n, D = x.shape A = np.zeros((n,1)) return A def getDerMatrix(self, x=None, der=None): n, D = x.shape A = np.zeros((n,1)) return A class One(Mean): ''' One mean.''' def __init__(self): self.hyp = [] self.name = '1' def getMean(self, x=None): n, D = x.shape A = np.ones((n,1)) return A def getDerMatrix(self, x=None, der=None): n, D = x.shape A = np.zeros((n,1)) return A class Const(Mean): ''' Constant mean function. hyp = [c] :param c: constant value for mean ''' def __init__(self, c=5.): self.hyp = [c] def getMean(self, x=None): n,D = x.shape A = self.hyp[0] * np.ones((n,1)) return A def getDerMatrix(self, x=None, der=None): n,D = x.shape if der == 0: # compute derivative vector wrt c A = np.ones((n,1)) else: A = np.zeros((n,1)) return A class Linear(Mean): ''' Linear mean function. self.hyp = alpha_list :param D: dimension of training data. Set if you want default alpha, which is 0.5 for each dimension. :alpha_list: scalar alpha for each dimension ''' def __init__(self, D=None, alpha_list=None): if alpha_list == None: self.hyp = [0.5 for i in xrange(D)] else: self.hyp = alpha_list def getMean(self, x=None): n, D = x.shape c = np.array(self.hyp) c = np.reshape(c,(len(c),1)) A = np.dot(x,c) return A def getDerMatrix(self, x=None, der=None): n, D = x.shape c = np.array(self.hyp) c = np.reshape(c,(len(c),1)) if isinstance(der, int) and der < D: # compute derivative vector wrt meanparameters A = np.reshape(x[:,der], (len(x[:,der]),1) ) else: A = np.zeros((n,1)) return A if __name__ == '__main__': pass }r@M(XScaleOfMean.__init__XdefKKćXScaleOfMean.getMeanXdefKKчX One.getMeanXdefMMX Zero.getMeanXdefMMX One.__init__XdefMMXSumOfMean.getMeanXdefKKXProductOfMean._setHypXdefK}KXMean.getDerMatrixXdefKeKrXScaleOfMean.getDerMatrixXdefKKۇX Mean.__add__XdefK5K?XProductOfMean.getDerMatrixXdefKKXSumOfMean._getHypXdefKKXConstXclassM!M9X Mean.__mul__XdefK?KPXProductOfMean.getMeanXdefKKXLinearXclassM9MZXSumOfMean.getDerMatrixXdefKKX Mean.getMeanXdefK_KeXZero.getDerMatrixXdefMMX ScaleOfMeanXclassKKۇXProductOfMean._getHypXdefKKXPowerOfMean.getDerMatrixXdefKKX ProductOfMeanXclassKrKXPowerOfMean.getMeanXdefKKX Mean.__pow__XdefKSK_XPowerOfMean._getHypXdefKKXLinear.getDerMatrixXdefMMMZX SumOfMeanXclassKKXOne.getDerMatrixXdefMM!XScaleOfMean._getHypXdefKKʇX Mean.__init__XdefK/K5X Zero.__init__XdefKMXLinear.getMeanXdefMFMMXScaleOfMean._setHypXdefKKȇX Const.getMeanXdefM*M/XProductOfMean.__init__XdefKtK}XMeanXclassK+KrX PowerOfMeanXclassKKXConst.getDerMatrixXdefM/M9XPowerOfMean._setHypXdefKKXOneXclassMM!XZeroXclassKMXConst.__init__XdefM'M*XLinear.__init__XdefM@MFXSumOfMean.__init__XdefKKXPowerOfMean.__init__XdefKKXSumOfMean._setHypXdefKKu}rAM(XConsthXLinearhX SumOfMeanhXMean.getDerMatrixhX Mean.getMeanhX PowerOfMeanhXOnehX ScaleOfMeanhXZerohX ProductOfMeanhXMeanhuj7MXpyGPs.Core.infrBMrCM}rDMbX #================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ # Inference methods: Compute the (approximate) posterior for a Gaussian process. # Methods currently implemented include: # # infExact Exact inference (only possible with Gaussian likelihood) # infLaplace Laplace's Approximation # infEP Expectation Propagation # infVB [NOT IMPLEMENTED!] Variational Bayes Approximation # # infFITC Large scale regression with approximate covariance matrix # infFITC_Laplace Large scale inference with approximate covariance matrix # infFITC_EP Large scale inference with approximate covariance matrix # # infMCMC [NOT IMPLEMENTED!] # Markov Chain Monte Carlo and Annealed Importance Sampling # We offer two samplers. # - hmc: Hybrid Monte Carlo # - ess: Elliptical Slice Sampling # No derivatives w.r.t. to hyperparameters are provided. # # infLOO [NOT IMPLEMENTED!] # Leave-One-Out predictive probability and Least-Squares Approxim. # # # This is a object-oriented python implementation of gpml functionality # (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18). # based on the functional-version of python implementation # (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013) # # Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013 import numpy as np import lik, cov from copy import copy,deepcopy from tools import solve_chol,brentmin,cholupdate np.seterr(all='ignore') class postStruct(object): ''' Data structure for posterior | post.alpha: 1d array containing inv(K)*m, | where K is the prior covariance matrix and m the approx posterior mean | post.sW: 1d array containing diagonal of sqrt(W) | the approximate posterior covariance matrix is inv(inv(K)+W) | post.L : 2d array, L = chol(sW*K*sW+identity(n)) ''' def __init__(self): self.alpha = np.array([]) self.L = np.array([]) self.sW = np.array([]) class dnlZStruct(object): ''' Data structure for the derivatives of mean, cov and lik functions. |dnlZ.mean: list of derivatives for each hyperparameters in mean function |dnlZ.cov: list of derivatives for each hyperparameters in covariance function |dnlZ.lik: list of derivatives for each hyperparameters in likelihood function ''' def __init__(self, m, c, l): self.mean = [] self.cov = [] self.lik = [] if m.hyp: self.mean = [0 for i in xrange(len(m.hyp))] if c.hyp: self.cov = [0 for i in xrange(len(c.hyp))] if l.hyp: self.lik = [0 for i in xrange(len(l.hyp))] class Inference(object): ''' Base class for inference. Defined several tool methods in it. ''' def __init__(self): pass def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): ''' Inference computation based on inputs. post, nlZ, dnlZ = inf.proceed(mean, cov, lik, x, y) | INPUT: | cov: name of the covariance function (see covFunctions.m) | lik: name of the likelihood function (see likFunctions.m) | x: n by D matrix of training inputs | y: 1d array (of size n) of targets | OUTPUT: | post(postStruct): struct representation of the (approximate) posterior containing: | nlZ: returned value of the negative log marginal likelihood | dnlZ(dnlZStruct): struct representation for derivatives of the negative log marginal likelihood | w.r.t. each hyperparameter. Usually, the approximate posterior to be returned admits the form: N(m=K*alpha, V=inv(inv(K)+W)), where alpha is a vector and W is diagonal; if not, then L contains instead -inv(K+inv(W)), and sW is unused. For more information on the individual approximation methods and their implementations, see the respective inference function below. See also gp.py :param meanfunc: mean function :param covfunc: covariance function :param likfunc: likelihood function :param x: training data :param y: training labels :param nargout: specify the number of output(1,2 or 3) :return: posterior, negative-log-marginal-likelihood, derivative for negative-log-marginal-likelihood-likelihood ''' pass def _epComputeParams(self, K, y, ttau, tnu, likfunc, m, inffunc): n = len(y) # number of training cases ssi = np.sqrt(ttau) # compute Sigma and mu L = np.linalg.cholesky(np.eye(n)+np.dot(ssi,ssi.T)*K).T # L'*L=B=eye(n)+sW*K*sW V = np.linalg.solve(L.T,np.tile(ssi,(1,n))*K) Sigma = K - np.dot(V.T,V) mu = np.dot(Sigma,tnu) Dsigma = np.reshape(np.diag(Sigma),(np.diag(Sigma).shape[0],1)) tau_n = 1/Dsigma - ttau # compute the log marginal likelihood nu_n = mu/Dsigma-tnu + m*tau_n # vectors of cavity parameters lZ = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc) nlZ = np.log(np.diag(L)).sum() - lZ.sum() - np.dot(tnu.T,np.dot(Sigma,tnu))/2 \ - np.dot((nu_n-m*tau_n).T,((ttau/tau_n*(nu_n-m*tau_n)-2*tnu) / (ttau+tau_n)))/2 \ + (tnu**2/(tau_n+ttau)).sum()/2.- np.log(1.+ttau/tau_n).sum()/2. return Sigma, mu, nlZ[0], L def _logdetA(self,K,w,nargout): ''' Compute the log determinant ldA and the inverse iA of a square nxn matrix A = eye(n) + K*diag(w) from its LU decomposition; for negative definite A, we return ldA = Inf. We also return mwiA = -diag(w)*inv(A). [ldA,iA,mwiA] = _logdetA(K,w)''' n = K.shape[0] assert(K.shape[0] == K.shape[1]) A = np.eye(n) + K*np.tile(w.T,(n,1)) [L,U,P] = np.linalg.lu(A) # compute LU decomposition, A = P'*L*U u = np.diag(U) signU = np.prod(np.sign(u)) # sign of U detP = 1 # compute sign (and det) of the permutation matrix P p = np.dot(P,np.array(range(n)).T) for ii in range(n): if ii != p[ii]: detP = -detP j= [jj for jj,val in enumerate(p) if val == ii] p[ii,j[0]] = p[j[0],ii] if signU != detP: # log becomes complex for negative values, encoded by infinity ldA = np.inf else: # det(L) = 1 and U triangular => det(A) = det(P)*prod(diag(U)) ldA = np.log(np.abs(u)).sum() if nargout>1: iA = inv(U)*inv(L)*P; if nargout>2: mwiA = -np.tile(w,(1,n))*iA return ldA,iA,mwiA else: return ldA,iA else: return ldA def _Psi_line(self,s,dalpha,alpha,K,m,likfunc,y,inffunc): '''Criterion Psi at alpha + s*dalpha for line search [Psi,alpha,f,dlp,W] = _Psi_line(s,dalpha,alpha,hyp,K,m,lik,y,inf)''' alpha = alpha + s*dalpha f = np.dot(K,alpha) + m [lp,dlp,d2lp] = likfunc.proceed(y,f,None,inffunc,None,3) W = -d2lp Psi = np.dot(alpha.T,(f-m))/2. - lp.sum() return Psi[0],alpha,f,dlp,W def _epfitcZ(self,d,P,R,nn,gg,ttau,tnu,d0,R0,P0,y,likfunc,m,inffunc): '''Compute the marginal likelihood approximation effort is O(n*nu^2) provided that nu0: R = cholupdate(R,np.sqrt(r)*np.dot(R.T,v),'-') else: R = cholupdate(R,np.sqrt(-r)*np.dot(R.T,v),'+') ttemp = np.dot(R0.T,np.dot(R.T,np.dot(R,np.dot(R0,P_i)))) gg = gg + ((dbi-dwi*(hi-m[ii]))/t) * np.reshape(ttemp,(ttemp.shape[0],1)) # O(nu^2) w[ii] = w_i; b[ii] = b_i; # update site parameters O(1) P_i = P_i/t # O(nu) return d,P_i,R,nn,gg,w,b def _mvmZ(self,x,RVdd,t): '''Matrix vector multiplication with Z=inv(K+inv(W))''' Zx = t*x - np.dot(RVdd.T,np.dot(RVdd,x)) return Zx def _mvmK(self,al,V,d0): '''Matrix vector multiplication with approximate covariance matrix''' Kal = np.dot(V.T,np.dot(V,al)) + d0*al return Kal def _Psi_lineFITC(self,s,dalpha,alpha,V,d0,m,likfunc,y,inffunc): '''Criterion Psi at alpha + s*dalpha for line search''' alpha = alpha + s*dalpha f = self._mvmK(alpha,V,d0) + m vargout = likfunc.proceed(y,f,None,inffunc,None,3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W = -d2lp Psi = np.dot(alpha.T,(f-m))/2. - lp.sum() return Psi[0],alpha,f,dlp,W def _fitcRefresh(self,d0,P0,R0,R0P0, w): '''Refresh the representation of the posterior from initial and site parameters to prevent possible loss of numerical precision after many epfitcUpdates effort is O(n*nu^2) provided that nu1: # do we want the marginal likelihood? nlZ = np.dot((y-m).T,alpha)/2. + np.log(np.diag(L)).sum() + n*np.log(2*np.pi*sn2)/2. # -log marg lik if nargout>2: # do we want derivatives? dnlZ = dnlZStruct(meanfunc, covfunc, likfunc) # allocate space for derivatives Q = solve_chol(L,np.eye(n))/sn2 - np.dot(alpha,alpha.T) # precompute for convenience dnlZ.lik = [sn2*np.trace(Q)] if covfunc.hyp: for ii in range(len(covfunc.hyp)): dnlZ.cov[ii] = (Q*covfunc.getDerMatrix(x=x, mode='train', der=ii)).sum()/2. if meanfunc.hyp: for ii in range(len(meanfunc.hyp)): dnlZ.mean[ii] = np.dot(-meanfunc.getDerMatrix(x, ii).T,alpha) dnlZ.mean[ii] = dnlZ.mean[ii][0,0] return post, nlZ[0,0], dnlZ return post, nlZ[0,0] return post class FITC_Exact(Inference): ''' FITC approximation to the posterior Gaussian process. The function is equivalent to infExact with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Quu) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu, snu2 = sn2/1e6 is the noise of the inducing inputs and Quu = Kuu + snu2*eye(nu). ''' def __init__(self): self.name = 'FICT exact inference' def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): if not isinstance(likfunc, lik.Gauss): # NOTE: no explicit call to likGauss raise Exception ('Exact inference only possible with Gaussian likelihood') if not isinstance(covfunc, cov.FITCOfKernel): raise Exception('Only covFITC supported.') # check cov diagK,Kuu,Ku = covfunc.getCovMatrix(x=x, mode='train') # evaluate covariance matrix m = meanfunc.getMean(x) # evaluate mean vector n, D = x.shape nu = Kuu.shape[0] sn2 = np.exp(2*likfunc.hyp[0]) # noise variance of likGauss snu2 = 1.e-6*sn2 # hard coded inducing inputs noise Luu = np.linalg.cholesky(Kuu+snu2*np.eye(nu)).T # Kuu + snu2*I = Luu'*Luu V = np.linalg.solve(Luu.T,Ku) # V = inv(Luu')*Ku => V'*V = Q g_sn2 = diagK + sn2 - np.array([(V*V).sum(axis=0)]).T # g + sn2 = diag(K) + sn2 - diag(Q) Lu = np.linalg.cholesky(np.eye(nu) + np.dot(V/np.tile(g_sn2.T,(nu,1)),V.T)).T # Lu'*Lu=I+V*diag(1/g_sn2)*V' r = (y-m)/np.sqrt(g_sn2) be = np.linalg.solve(Lu.T,np.dot(V,r/np.sqrt(g_sn2))) iKuu = solve_chol(Luu,np.eye(nu)) # inv(Kuu + snu2*I) = iKuu post = postStruct() post.alpha = np.linalg.solve(Luu,np.linalg.solve(Lu,be))# return the posterior parameters post.L = solve_chol(np.dot(Lu,Luu),np.eye(nu)) - iKuu # Sigma-inv(Kuu) post.sW = np.ones((n,1))/np.sqrt(sn2) # unused for FITC prediction with gp.m if nargout>1: # do we want the marginal likelihood nlZ = np.log(np.diag(Lu)).sum() + (np.log(g_sn2).sum() + n*np.log(2*np.pi) + np.dot(r.T,r) - np.dot(be.T,be))/2. if nargout>2: # do we want derivatives? dnlZ = dnlZStruct(meanfunc, covfunc, likfunc) # allocate space for derivatives al = r/np.sqrt(g_sn2) - np.dot(V.T,np.linalg.solve(Lu,be))/g_sn2 # al = (Kt+sn2*eye(n))\y B = np.dot(iKuu,Ku) w = np.dot(B,al) W = np.linalg.solve(Lu.T,V/np.tile(g_sn2.T,(nu,1))) for ii in range(len(covfunc.hyp)): [ddiagKi,dKuui,dKui] = covfunc.getDerMatrix(x=x, mode='train', der=ii) # eval cov deriv R = 2.*dKui-np.dot(dKuui,B) v = ddiagKi - np.array([(R*B).sum(axis=0)]).T # diag part of cov deriv dnlZ.cov[ii] = ( np.dot(ddiagKi.T,1./g_sn2) + np.dot(w.T,(np.dot(dKuui,w)-2.*np.dot(dKui,al))) \ - np.dot(al.T,(v*al)) - np.dot(np.array([(W*W).sum(axis=0)]),v) - (np.dot(R,W.T)*np.dot(B,W.T)).sum() )/2. dnlZ.cov[ii] = dnlZ.cov[ii][0,0] dnlZ.lik = sn2*((1./g_sn2).sum() - (np.array([(W*W).sum(axis=0)])).sum() - np.dot(al.T,al)) dKuui = 2*snu2 R = -dKuui*B v = -np.array([(R*B).sum(axis=0)]).T # diag part of cov deriv dnlZ.lik += (np.dot(w.T,np.dot(dKuui,w)) -np.dot(al.T,(v*al)) \ - np.dot(np.array([(W*W).sum(axis=0)]),v) - (np.dot(R,W.T)*np.dot(B,W.T)).sum() )/2. dnlZ.lik = list(dnlZ.lik[0]) for ii in range(len(meanfunc.hyp)): dnlZ.mean[ii] = np.dot(-meanfunc.getDerMatrix(x, ii).T, al) dnlZ.mean[ii] = dnlZ.mean[ii][0,0] return post, nlZ[0,0], dnlZ return post, nlZ[0,0] return post class Laplace(Inference): ''' Laplace's Approximation to the posterior Gaussian process. ''' def __init__(self): self.last_alpha = None def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): tol = 1e-6 # tolerance for when to stop the Newton iterations smax = 2; Nline = 20; thr = 1e-4 # line search parameters maxit = 20 # max number of Newton steps in f inffunc = self K = covfunc.getCovMatrix(x=x, mode='train') # evaluate the covariance matrix m = meanfunc.getMean(x) # evaluate the mean vector n, D = x.shape Psi_old = np.inf # make sure while loop starts by the largest old objective val if self.last_alpha == None: # find a good starting point for alpha and f alpha = np.zeros((n,1)) f = np.dot(K,alpha) + m # start at mean if sizes not match vargout = likfunc.proceed(y, f, None, inffunc, None, 3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W= -d2lp Psi_new = -lp.sum() else: alpha = self.last_alpha f = np.dot(K,alpha) + m # try last one vargout = likfunc.proceed(y, f, None, inffunc, None, 3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W= -d2lp Psi_new = np.dot(alpha.T,(f-m))/2. - lp.sum() # objective for last alpha vargout = - likfunc.proceed(y, m, None, inffunc, None, 1) Psi_def = vargout[0] # objective for default init f==m if Psi_def < Psi_new: # if default is better, we use it alpha = np.zeros((n,1)) f = np.dot(K,alpha) + m vargout = likfunc.proceed(y, f, None, inffunc, None, 3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W=-d2lp; Psi_new = -lp.sum() isWneg = np.any(W<0) # flag indicating whether we found negative values of W it = 0 # this happens for the Student's t likelihood while (Psi_old - Psi_new > tol) and it2: # do we want derivatives? dnlZ = dnlZStruct(meanfunc, covfunc, likfunc) # allocate space for derivatives if isWneg: # switch between Cholesky and LU decomposition mode Z = -post.L # inv(K+inv(W)) g = np.atleast_2d((iA*K).sum(axis=1)).T /2 # deriv. of ln|B| wrt W; g = diag(inv(inv(K)+diag(W)))/2 else: Z = np.tile(sW,(1,n))*solve_chol(post.L,np.diag(np.reshape(sW,(sW.shape[0],)))) #sW*inv(B)*sW=inv(K+inv(W)) C = np.linalg.solve(post.L.T,np.tile(sW,(1,n))*K) # deriv. of ln|B| wrt W g = np.atleast_2d((np.diag(K)-(C**2).sum(axis=0).T)).T /2. # g = diag(inv(inv(K)+W))/2 dfhat = g* d3lp # deriv. of nlZ wrt. fhat: dfhat=diag(inv(inv(K)+W)).*d3lp/2 for ii in range(len(covfunc.hyp)): # covariance hypers dK = covfunc.getDerMatrix(x=x, mode='train', der=ii) dnlZ.cov[ii] = (Z*dK).sum()/2. - np.dot(alpha.T,np.dot(dK,alpha))/2. # explicit part b = np.dot(dK,dlp) # b-K*(Z*b) = inv(eye(n)+K*diag(W))*b dnlZ.cov[ii] -= np.dot(dfhat.T,b-np.dot(K,np.dot(Z,b))) # implicit part dnlZ.cov[ii] = dnlZ.cov[ii][0,0] for ii in range(len(likfunc.hyp)): # likelihood hypers [lp_dhyp,dlp_dhyp,d2lp_dhyp] = likfunc.proceed(y,f,None,inffunc,ii,3) dnlZ.lik[ii] = -np.dot(g.T,d2lp_dhyp) - lp_dhyp.sum() # explicit part b = np.dot(K,dlp_dhyp) # b-K*(Z*b) = inv(eye(n)+K*diag(W))*b dnlZ.lik[ii] -= np.dot(dfhat.T,b-np.dot(K,np.dot(Z,b))) # implicit part dnlZ.lik[ii] = dnlZ.lik[ii][0,0] for ii in range(len(meanfunc.hyp)): # mean hypers dm = meanfunc.getDerMatrix(x, ii) dnlZ.mean[ii] = -np.dot(alpha.T,dm) # explicit part dnlZ.mean[ii] -= np.dot(dfhat.T,dm-np.dot(K,np.dot(Z,dm))) # implicit part dnlZ.mean[ii] = dnlZ.mean[ii][0,0] return post,nlZ[0],dnlZ else: return post, nlZ[0] class FITC_Laplace(Inference): ''' FITC-Laplace approximation to the posterior Gaussian process. The function is equivalent to infLaplace with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Kuu + snu2 * eye(nu)) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu and snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard deviation of the inducing inputs snu to be a one per mil of the measurement noise's standard deviation sn. In case of a likelihood without noise parameter sn2, we simply use snu2 = 1e-6. ''' def __init__(self): self.last_alpha = None def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): if not isinstance(covfunc, cov.FITCOfKernel): raise Exception('Only covFITC supported.') tol = 1e-6 # tolerance for when to stop the Newton iterations smax = 2; Nline = 100; thr = 1e-4 # line search parameters maxit = 20 # max number of Newton steps in f inffunc = Laplace() diagK,Kuu,Ku = covfunc.getCovMatrix(x=x, mode='train') # evaluate the covariance matrix m = meanfunc.getMean(x) # evaluate the mean vector if likfunc.hyp: # hard coded inducing inputs noise sn2 = np.exp(2.*likfunc.hyp[-1]) snu2 = 1.e-6*sn2 # similar to infFITC else: snu2 = 1.e-6 n, D = x.shape nu = Kuu.shape[0] rot180 = lambda A: np.rot90(np.rot90(A)) # little helper functions chol_inv = lambda A: np.linalg.solve( rot180( np.linalg.cholesky(rot180(A)) ),np.eye(nu)) # chol(inv(A)) R0 = chol_inv(Kuu+snu2*np.eye(nu)) # initial R, used for refresh O(nu^3) V = np.dot(R0,Ku); d0 = diagK - np.array([(V*V).sum(axis=0)]).T # initial d, needed Psi_old = np.inf # make sure while loop starts by the largest old objective val if self.last_alpha == None: # find a good starting point for alpha and f alpha = np.zeros((n,1)) f = self._mvmK(alpha,V,d0) + m # start at mean if sizes not match vargout = likfunc.proceed(y, f, None, inffunc, None, 3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W=-d2lp; Psi_new = -lp.sum() else: alpha = self.last_alpha f = self._mvmK(alpha,V,d0) + m # try last one vargout = likfunc.proceed(y, f, None, inffunc, None, 3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W=-d2lp Psi_new = np.dot(alpha.T,(f-m))/2. - lp.sum() # objective for last alpha vargout = - likfunc.proceed(y, m, None, inffunc, None, 1) Psi_def = vargout[0] # objective for default init f==m if Psi_def < Psi_new: # if default is better, we use it alpha = np.zeros((n,1)) f = self._mvmK(alpha,V,d0) + m vargout = likfunc.proceed(y, f, None, inffunc, None, 3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W=-d2lp; Psi_new = -lp.sum() isWneg = np.any(W<0) # flag indicating whether we found negative values of W it = 0 # this happens for the Student's t likelihood while (Psi_old - Psi_new > tol) and it2: # do we want derivatives? dnlZ = dnlZStruct(meanfunc, covfunc, likfunc) # allocate space for derivatives [d,P,R] = self._fitcRefresh(d0,Ku,R0,V,W) # g = diag(inv(inv(K)+W))/2 g = d/2 + 0.5*np.atleast_2d((np.dot(np.dot(R,R0),P)**2).sum(axis=0)).T t = W/(1+W*d0) dfhat = g*d3lp # deriv. of nlZ wrt. fhat: dfhat=diag(inv(inv(K)+W)).*d3lp/2 for ii in range(len(covfunc.hyp)): # covariance hypers ddiagK,dKuu,dKu = covfunc.getDerMatrix(x=x, mode='train', der=ii) # eval cov derivatives dA = 2.*dKu.T-np.dot(R0tV.T,dKuu) # dQ = dA*R0tV w = np.atleast_2d((dA*R0tV.T).sum(axis=1)).T # w = diag(dQ) v = ddiagK-w # v = diag(dK)-diag(dQ); dnlZ.cov[ii] = np.dot(ddiagK.T,t) - np.dot((RVdd*RVdd).sum(axis=0),v) # explicit part dnlZ.cov[ii] -= (np.dot(RVdd,dA)*np.dot(RVdd,R0tV.T)).sum() # explicit part dnlZ.cov[ii] = 0.5*dnlZ.cov[ii] - np.dot(alpha.T,np.dot(dA,np.dot(R0tV,alpha))+v*alpha)/2. # explicit b = np.dot(dA,np.dot(R0tV,dlp)) + v*dlp # b-K*(Z*b) = inv(eye(n)+K*diag(W))*b KZb = self._mvmK(self._mvmZ(b,RVdd,t),V,d0) dnlZ.cov[ii] -= np.dot(dfhat.T,(b-KZb)) # implicit part dnlZ.cov[ii] = dnlZ.cov[ii][0,0] for ii in range(len(likfunc.hyp)): # likelihood hypers vargout = likfunc.proceed(y,f,None,inffunc,ii,3) lp_dhyp = vargout[0]; dlp_dhyp = vargout[1]; d2lp_dhyp = vargout[2] dnlZ.lik[ii] = -np.dot(g.T,d2lp_dhyp) - lp_dhyp.sum() # explicit part b = self._mvmK(dlp_dhyp,V,d0) # implicit part dnlZ.lik[ii] -= np.dot(dfhat.T,b-self._mvmK(self._mvmZ(b,RVdd,t),V,d0)) if ii == len(likfunc.hyp)-1: # since snu2 is a fixed fraction of sn2, there is a covariance-like term # in the derivative as well snu = np.sqrt(snu2); T = chol_inv(Kuu + snu2*np.eye(nu)); T = np.dot(T.T,np.dot(T,snu*Ku)); t = np.array([(T*T).sum(axis=0)]).T z = np.dot(alpha.T,np.dot(T.T,np.dot(T,alpha))-t*alpha) - np.dot(np.array([(RVdd*RVdd).sum(axis=0)]),t) z += (np.dot(RVdd,T.T)**2).sum() b = (t*dlp-np.dot(T.T,np.dot(T,dlp)))/2. KZb = self._mvmK(self._mvmZ(b,RVdd,t),V,d0) z -= np.dot(dfhat.T,b-KZb) dnlZ.lik[ii] += z dnlZ.lik[ii] = dnlZ.lik[ii][0,0] for ii in range(len(meanfunc.hyp)): # mean hypers dm = meanfunc.getDerMatrix(x, ii) dnlZ.mean[ii] = -np.dot(alpha.T,dm) # explicit part Zdm = self._mvmZ(dm,RVdd,t) dnlZ.mean[ii] -= np.dot(dfhat.T,(dm-self._mvmK(Zdm,V,d0))) # implicit part dnlZ.mean[ii] = dnlZ.mean[ii][0,0] return post,nlZ[0,0],dnlZ else: return post, nlZ[0,0] class EP(Inference): ''' Expectation Propagation approximation to the posterior Gaussian Process. ''' def __init__(self): self.name = 'Expectation Propagation' self.last_ttau = None self.last_tnu = None def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): tol = 1e-4; max_sweep = 10; min_sweep = 2 # tolerance to stop EP iterations n = x.shape[0] inffunc = self K = covfunc.getCovMatrix(x=x, mode='train') # evaluate the covariance matrix m = meanfunc.getMean(x) # evaluate the mean vector nlZ0 = -likfunc.proceed(y, m, np.reshape(np.diag(K),(np.diag(K).shape[0],1)), inffunc).sum() if self.last_ttau == None: # find starting point for tilde parameters ttau = np.zeros((n,1)) # initialize to zero if we have no better guess tnu = np.zeros((n,1)) Sigma = K # initialize Sigma and mu, the parameters of .. mu = np.zeros((n,1)) # .. the Gaussian posterior approximation nlZ = nlZ0 else: ttau = self.last_ttau # try the tilde values from previous call tnu = self.last_tnu Sigma, mu, nlZ, L = self._epComputeParams(K, y, ttau, tnu, likfunc, m, inffunc) if nlZ > nlZ0: # if zero is better .. ttau = np.zeros((n,1)) # .. then initialize with zero instead tnu = np.zeros((n,1)) Sigma = K # initialize Sigma and mu, the parameters of .. mu = np.zeros((n,1)) # .. the Gaussian posterior approximation nlZ = nlZ0 nlZ_old = np.inf; sweep = 0 # converged, max. sweeps or min. sweeps? while (np.abs(nlZ-nlZ_old) > tol and sweep < max_sweep) or (sweep < min_sweep): nlZ_old = nlZ; sweep += 1 rperm = xrange(n) # randperm(n) for ii in rperm: # iterate EP updates (in random order) over examples tau_ni = 1/Sigma[ii,ii] - ttau[ii]# first find the cavity distribution .. nu_ni = mu[ii]/Sigma[ii,ii] + m[ii]*tau_ni - tnu[ii] # .. params tau_ni and nu_ni # compute the desired derivatives of the indivdual log partition function lZ,dlZ,d2lZ = likfunc.proceed(y[ii], nu_ni/tau_ni, 1/tau_ni, inffunc, None, 3) ttau_old = copy(ttau[ii]) # then find the new tilde parameters, keep copy of old ttau[ii] = -d2lZ /(1.+d2lZ/tau_ni) ttau[ii] = max(ttau[ii],0) # enforce positivity i.e. lower bound ttau by zero tnu[ii] = ( dlZ + (m[ii]-nu_ni/tau_ni)*d2lZ )/(1.+d2lZ/tau_ni) ds2 = ttau[ii] - ttau_old # finally rank-1 update Sigma .. si = np.reshape(Sigma[:,ii],(Sigma.shape[0],1)) Sigma = Sigma - ds2/(1.+ds2*si[ii])*np.dot(si,si.T) # takes 70# of total time mu = np.dot(Sigma,tnu) # .. and recompute mu # recompute since repeated rank-one updates can destroy numerical precision Sigma, mu, nlZ, L = self._epComputeParams(K, y, ttau, tnu, likfunc, m, inffunc) if sweep == max_sweep: pass # print '[warning] maximum number of sweeps reached in function infEP' self.last_ttau = ttau; self.last_tnu = tnu # remember for next call sW = np.sqrt(ttau); alpha = tnu-sW*solve_chol(L,sW*np.dot(K,tnu)) post = postStruct() post.alpha = alpha # return the posterior params post.sW = sW post.L = L if nargout>2: # do we want derivatives? dnlZ = dnlZStruct(meanfunc, covfunc, likfunc) # allocate space for derivatives ssi = np.sqrt(ttau) V = np.linalg.solve(L.T,np.tile(ssi,(1,n))*K) Sigma = K - np.dot(V.T,V) mu = np.dot(Sigma,tnu) Dsigma = np.reshape(np.diag(Sigma),(np.diag(Sigma).shape[0],1)) tau_n = 1/Dsigma-ttau # compute the log marginal likelihood nu_n = mu/Dsigma-tnu # vectors of cavity parameters F = np.dot(alpha,alpha.T) - np.tile(sW,(1,n))* \ solve_chol(L,np.diag(np.reshape(sW,(sW.shape[0],)))) # covariance hypers for jj in range(len(covfunc.hyp)): dK = covfunc.getDerMatrix(x=x, mode='train', der=jj) dnlZ.cov[jj] = -(F*dK).sum()/2. for ii in range(len(likfunc.hyp)): dlik = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc, ii) dnlZ.lik[ii] = -dlik.sum() junk,dlZ = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc, None, 2) # mean hyps for ii in range(len(meanfunc.hyp)): dm = meanfunc.getDerMatrix(x, ii) dnlZ.mean[ii] = -np.dot(dlZ.T,dm) dnlZ.mean[ii] = dnlZ.mean[ii][0,0] return post, nlZ[0], dnlZ else: return post, nlZ[0] class FITC_EP(Inference): ''' FITC-EP approximation to the posterior Gaussian process. The function is equivalent to infEP with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Kuu + snu2 * eye(nu)) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu and snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard deviation of the inducing inputs snu to be a one per mil of the measurement noise's standard deviation sn. In case of a likelihood without noise parameter sn2, we simply use snu2 = 1e-6. For details, see The Generalized FITC Approximation, Andrew Naish-Guzman and Sean Holden, NIPS, 2007. ''' def __init__(self): self.name = 'FITC Expectation Propagation' self.last_ttau = None self.last_tnu = None def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): if not isinstance(covfunc, cov.FITCOfKernel): raise Exception('Only covFITC supported.') # check cov tol = 1e-4; max_sweep = 10; min_sweep = 2 # tolerance to stop EP iterations inffunc = EP() diagK,Kuu,Ku = covfunc.getCovMatrix(x=x, mode='train') # evaluate the covariance matrix m = meanfunc.getMean(x) # evaluate the mean vector if likfunc.hyp: # hard coded inducing inputs noise sn2 = np.exp(2.*likfunc.hyp[-1]) snu2 = 1.e-6*sn2 # similar to infFITC else: snu2 = 1.e-6 n, D = x.shape; nu = Kuu.shape[0] rot180 = lambda A: np.rot90(np.rot90(A)) # little helper functions chol_inv = lambda A: np.linalg.solve( rot180( np.linalg.cholesky(rot180(A)) ),np.eye(nu)) # chol(inv(A)) R0 = chol_inv(Kuu+snu2*np.eye(nu)) # initial R, used for refresh O(nu^3) V = np.dot(R0,Ku); d0 = diagK - np.array([(V*V).sum(axis=0)]).T # initial d, needed for refresh O(n*nu^2) # A note on naming: variables are given short but descriptive names in # accordance with Rasmussen & Williams "GPs for Machine Learning" (2006): mu # and s2 are mean and variance, nu and tau are natural parameters. A leading t # means tilde, a subscript _ni means "not i" (for cavity parameters), or _n # for a vector of cavity parameters. # marginal likelihood for ttau = tnu = zeros(n,1); equals n*log(2) for likCum* nlZ0 = -1.* likfunc.proceed(y, m, np.reshape(diagK,(diagK.shape[0],1)), inffunc).sum() if self.last_ttau == None: # find starting point for tilde parameters ttau = np.zeros((n,1)) # initialize to zero if we have no better guess tnu = np.zeros((n,1)) [d,P,R,nn,gg] = self._epfitcRefresh(d0,Ku,R0,V,ttau,tnu) # compute initial repres. nlZ = nlZ0 else: ttau = self.last_ttau # try the tilde values from previous call tnu = self.last_tnu [d,P,R,nn,gg] = self._epfitcRefresh(d0,Ku,R0,V,ttau,tnu) # compute initial repres. nlZ = self._epfitcZ(d,P,R,nn,gg,ttau,tnu,d0,R0,Ku,y,likfunc,m,inffunc)[0] if nlZ > nlZ0: # if zero is better .. ttau = np.zeros((n,1)) # .. then initialize with zero instead tnu = np.zeros((n,1)) [d,P,R,nn,gg] = self._epfitcRefresh(d0,Ku,R0,V,ttau,tnu) # initial repres. nlZ = nlZ0 nlZ_old = np.inf; sweep = 0 # converged, max. sweeps or min. sweeps? while (np.abs(nlZ-nlZ_old) > tol and sweep < max_sweep) or (sweep < min_sweep): nlZ_old = nlZ sweep += 1 rperm = range(n) # randperm(n) for ii in rperm: # iterate EP updates (in random order) over examples p_i = np.reshape(P[:,ii],(P.shape[0],1)) t = np.dot(R,np.dot(R0,p_i)) # temporary variables sigma_i = d[ii] + np.dot(t.T,t); mu_i = nn[ii] + np.dot(p_i.T,gg) # post moments O(nu^2) tau_ni = 1/sigma_i - ttau[ii] # first find the cavity distribution .. nu_ni = mu_i/sigma_i + m[ii]*tau_ni - tnu[ii] # .. params tau_ni and nu_ni # compute the desired derivatives of the indivdual log partition function vargout = likfunc.proceed(y[ii], nu_ni/tau_ni, 1/tau_ni, inffunc, None, 3) lZ = vargout[0]; dlZ = vargout[1]; d2lZ = vargout[2] ttau_i = -d2lZ /(1.+d2lZ/tau_ni) ttau_i = max(ttau_i,0) # enforce positivity i.e. lower bound ttau by zero tnu_i = ( dlZ + (m[ii]-nu_ni/tau_ni)*d2lZ )/(1.+d2lZ/tau_ni) [d,P[:,ii],R,nn,gg,ttau,tnu] = self._epfitcUpdate(d,P[:,ii],R,nn,gg,ttau,tnu,ii,ttau_i,tnu_i,m,d0,Ku,R0)# update representation # recompute since repeated rank-one updates can destroy numerical precision [d,P,R,nn,gg] = self._epfitcRefresh(d0,Ku,R0,V,ttau,tnu) [nlZ,nu_n,tau_n] = self._epfitcZ(d,P,R,nn,gg,ttau,tnu,d0,R0,Ku,y,likfunc,m,inffunc) if sweep == max_sweep: pass # print '[warning] maximum number of sweeps reached in function infEP' self.last_ttau = ttau self.last_tnu = tnu # remember for next call post = postStruct() post.sW = np.sqrt(ttau) # unused for FITC_EP prediction with gp.m dd = 1/(d0+1/ttau) alpha = tnu/ttau*dd RV = np.dot(R,V) R0tV = np.dot(R0.T,V) alpha = alpha - np.dot(RV.T,np.dot(RV,alpha))*dd # long alpha vector for ordinary infEP post.alpha = np.dot(R0tV,alpha) # alpha = R0'*V*inv(Kt+diag(1./ttau))*(tnu./ttau) B = R0tV*np.tile(dd.T,(nu,1)); L = np.dot(B,R0tV.T); B = np.dot(B,RV.T) post.L = np.dot(B,B.T) - L # L = -R0'*V*inv(Kt+diag(1./ttau))*V'*R0 if nargout>2: # do we want derivatives? dnlZ = dnlZStruct(meanfunc, covfunc, likfunc) # allocate space for derivatives RVdd = RV*np.tile(dd.T,(nu,1)) for ii in range(len(covfunc.hyp)): ddiagK,dKuu,dKu = covfunc.getDerMatrix(x=x, mode='train', der=ii) dA = 2*dKu.T - np.dot(R0tV.T,dKuu) # dQ = dA*R0tV w = np.atleast_2d((dA*R0tV.T).sum(axis=1)).T # w = diag(dQ) v = ddiagK - w # v = diag(dK)-diag(dQ) z = np.dot(dd.T,(v+w)) - np.dot(np.atleast_2d((RVdd*RVdd).sum(axis=0)), v) \ - (np.dot(RVdd,dA).T * np.dot(R0tV,RVdd.T)).sum() dnlZ.cov[ii] = (z - np.dot(alpha.T,(alpha*v)) - np.dot(np.dot(alpha.T,dA),np.dot(R0tV,alpha)))/2. dnlZ.cov[ii] = dnlZ.cov[ii][0,0] for ii in range(len(likfunc.hyp)): # likelihood hypers dlik = likfunc.proceed(y, nu_n/tau_n+m, 1/tau_n, inffunc, ii, 1) dnlZ.lik[ii] = -dlik.sum() if ii == len(likfunc.hyp)-1: # since snu2 is a fixed fraction of sn2, there is a covariance-like term # in the derivative as well v = np.atleast_2d((R0tV*R0tV).sum(axis=0)).T z = (np.dot(RVdd,R0tV.T)**2).sum() - np.dot(np.atleast_2d((RVdd*RVdd).sum(axis=0)),v) z = z + np.dot(post.alpha.T,post.alpha) - np.dot(alpha.T,(v*alpha)) dnlZ.lik[ii] += snu2*z dnlZ.lik[ii] = dnlZ.lik[ii][0,0] [junk,dlZ] = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc, None, 2) # mean hyps for ii in range(len(meanfunc.hyp)): dm = meanfunc.getDerMatrix(x, ii) dnlZ.mean[ii] = -np.dot(dlZ.T,dm) dnlZ.mean[ii] = dnlZ.mean[ii][0,0] return post, nlZ[0,0], dnlZ else: return post, nlZ[0,0] if __name__ == '__main__': pass }rEM(XInference._epfitcZXdefKK͇XInference._mvmZXdefKKXFITC_Laplace.proceedXdefMMXInference._logdetAXdefKKXExact.__init__XdefMM XInference._epComputeParamsXdefKKX postStructXclassK4KCXInference._mvmKXdefKKXdnlZStruct.__init__XdefKKKVX Exact.proceedXdefM MAXInference._Psi_lineXdefKKXFITC_Exact.proceedXdefMLMXInference.__init__XdefKZK]XLaplace.__init__XdefMMXInference._epfitcUpdateXdefKKXFITC_Laplace.__init__XdefMMX FITC_ExactXclassMAMXFITC_EP.proceedXdefMMjXExactXclassMMAXInference._Psi_lineFITCXdefKMX FITC_LaplaceXclassMMXpostStruct.__init__XdefK>KCXFITC_EPXclassMMjXLaplace.proceedXdefMMXInference._fitcRefreshXdefMMXInference._epfitcRefreshXdefKKއXEPXclassMMX dnlZStructXclassKCKVXInference.proceedXdefK]KXFITC_Exact.__init__XdefMIMLX InferenceXclassKVMX EP.proceedXdefMMXFITC_EP.__init__XdefMMXLaplaceXclassMMX EP.__init__XdefMMu}rFM(X FITC_LaplacehXFITC_EPhXEPhX FITC_ExacthXExacthXLaplacehuj7MX pyGPs.Core.gprGMrHM}rIMbX #================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ # MEANING OF NOTATION: # # inffunc function specifying the inference method # covfunc prior covariance function (see below) # meanfunc prior mean function # likfunc likelihood function # x n by D matrix of training inputs # y column vector of length n of training targets # xs n by D matrix of test inputs # ys column vector of length nn of true test targets (optional) # nlZ returned value of the negative log marginal likelihood # dnlZ column vector of partial derivatives of the negative # log marginal likelihood w.r.t. each hyperparameter # ym column vector (of length ns) of predictive output means # ys2 column vector (of length ns) of predictive output variances # fm column vector (of length ns) of predictive latent means # fs2 column vector (of length ns) of predictive latent variances # lp column vector (of length ns) of log predictive probabilities # post struct representation of the (approximate) posterior # post consists of post.alpha, post.L, post.sW # # This is a object-oriented python implementation of gpml functionality # (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18). # based on the functional-version of python implementation # (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013) # # Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013 import itertools import numpy as np import matplotlib.pyplot as plt import inf, mean, lik, cov, opt from tools import unique from copy import deepcopy import pyGPs SHADEDCOLOR = [0.7539, 0.89453125, 0.62890625, 1.0] MEANCOLOR = [ 0.2109375, 0.63385, 0.1796875, 1.0] DATACOLOR = [0.12109375, 0.46875, 1., 1.0] class GP(object): '''Base class for GP model''' def __init__(self): super(GP, self).__init__() self.usingDefaultMean = True # was using default mean function now? self.meanfunc = None self.covfunc = None self.likfunc = None self.inffunc = None self.optimizer = None self.nlZ = None # negative log marginal likelihood self.dnlZ = None # column vector of partial derivatives of the negative # log marginal likelihood w.r.t. each hyperparameter self.posterior = None # struct representation of the (approximate) posterior self.x = None # n by D matrix of training inputs self.y = None # column vector of length n of training targets self.xs = None # n by D matrix of test inputs self.ys = None # column vector of length nn of true test targets (optional) self.ym = None # column vector (of length ns) of predictive output means self.ys2 = None # column vector (of length ns) of predictive output variances self.fm = None # column vector (of length ns) of predictive latent means self.fs2 = None # column vector (of length ns) of predictive latent variances self.lp = None # column vector (of length ns) of log predictive probabilities def setData(self, x, y): '''Pass training data and traning labels to model.''' if x.ndim == 1: x = np.reshape(x, (x.shape[0],1)) if y.ndim == 1: y = np.reshape(y, (y.shape[0],1)) self.x = x self.y = y if self.usingDefaultMean: c = np.mean(y) self.meanfunc = mean.Const(c) # adapt default prior mean wrt. training labels def plotData_1d(self, axisvals=None): '''Plot 1d (toy) data.''' plt.figure() plt.plot(self.x, self.y, ls='None', marker='+', color=DATACOLOR, ms=12, mew=2) if axisvals: plt.axis(axisvals) plt.grid() plt.xlabel('input x') plt.ylabel('target y') plt.show() def plotData_2d(self,x1,x2,t1,t2,p1,p2,axisvals=None): '''Plot 2d (toy) data.''' fig = plt.figure() plt.plot(x1[:,0], x1[:,1], 'b+', markersize = 12) plt.plot(x2[:,0], x2[:,1], 'r+', markersize = 12) pc = plt.contour(t1, t2, np.reshape(p2/(p1+p2), (t1.shape[0],t1.shape[1]) )) fig.colorbar(pc) plt.grid() if axisvals: plt.axis(axisvals) plt.show() def setPrior(self, mean=None, kernel=None): '''Set prior mean and cov''' if mean != None: self.meanfunc = mean self.usingDefaultMean = False if kernel != None: self.covfunc = kernel if type(kernel) is cov.Pre: self.usingDefaultMean = False def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None): '''This method is used to sepecify optimization configuration. By default, gp uses a single run "minimize". :param method: Optimization methods. Possible values are: "Minimize" -> minimize by Carl Rasmussen (python implementation of "minimize" in GPML) "CG" -> conjugent gradient "BFGS" -> quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS) "SCG" -> scaled conjugent gradient (faster than CG) :param num_restarts: Set if you want to run mulitiple times of optimization with different initial guess. It specifys the maximum number of runs/restarts/trials. :param min_threshold: Set if you want to run mulitiple times of optimization with different initial guess. It specifys the threshold of objective function value. Stop optimization when this value is reached. :param meanRange: The range of initial guess for mean hyperparameters. e.g. meanRange = [(-2,2), (-5,5), (0,1)]. Each tuple specifys the range (low, high) of this hyperparameter, This is only the range of initial guess, during optimization process, optimal hyperparameters may go out of this range. (-5,5) for each hyperparameter by default. :param covRange: The range of initial guess for kernel hyperparameters. Usage see meanRange :param likRange: The range of initial guess for likelihood hyperparameters. Usage see meanRange ''' pass def optimize(self, x=None, y=None): ''' Train optimal hyperparameters based on training data, adjust new hyperparameters to all mean/cov/lik functions ''' if x != None: if x.ndim == 1: x = np.reshape(x, (x.shape[0],1)) self.x = x if y != None: if y.ndim == 1: y = np.reshape(y, (y.shape[0],1)) self.y = y if self.usingDefaultMean and self.meanfunc == None: c = np.mean(y) self.meanfunc = mean.Const(c) # adapt default prior mean wrt. training labels # optimize optimalHyp, optimalNlZ = self.optimizer.findMin(self.x, self.y) self.nlZ= optimalNlZ # apply optimal hyp to all mean/cov/lik functions here self.optimizer._apply_in_objects(optimalHyp) self.fit() def fit(self, x=None, y=None, der=True): ''' Fit the training data. Update negative log marginal likelihood(nlZ), partial derivatives of nlZ w.r.t. each hyperparameter(dnlZ), and struct representation of the (approximate) posterior(post), which consists of post.alpha, post.L, post.sW. ''' if x != None: if x.ndim == 1: x = np.reshape(x, (x.shape[0],1)) self.x = x if y != None: if y.ndim == 1: y = np.reshape(y, (y.shape[0],1)) self.y = y if self.usingDefaultMean and self.meanfunc == None: c = np.mean(y) self.meanfunc = mean.Const(c) # adapt default prior mean wrt. training labels # call inference method if isinstance(self.likfunc, lik.Erf): #or is instance(self.likfunc, lik.Logistic): uy = unique(self.y) ind = ( uy != 1 ) if any( uy[ind] != -1): raise Exception('You attempt classification using labels different from {+1,-1}') if not der: post, nlZ = self.inffunc.proceed(self.meanfunc, self.covfunc, self.likfunc, self.x, self.y, 2) self.nlZ = nlZ self.posterior = deepcopy(post) return nlZ, post else: post, nlZ, dnlZ = self.inffunc.proceed(self.meanfunc, self.covfunc, self.likfunc, self.x, self.y, 3) self.nlZ = nlZ self.dnlZ = deepcopy(dnlZ) self.posterior = deepcopy(post) return nlZ, dnlZ, post def predict(self, xs, ys=None): ''' Prediction according to given inputs. Get predictive output means(ym), predictive output variances(ys2), predictive latent means(fm), predictive latent variances(fs2), log predictive probabilities(lp). :param xs: test input :param ys: test target(optional) :return: ym, ys2, fm, fs2, lp ''' meanfunc = self.meanfunc covfunc = self.covfunc likfunc = self.likfunc inffunc = self.inffunc x = self.x y = self.y if xs.ndim == 1: xs = np.reshape(xs, (xs.shape[0],1)) self.xs = xs if ys != None: if ys.ndim == 1: ys = np.reshape(ys, (ys.shape[0],1)) self.ys = ys if self.posterior == None: self.fit() alpha = self.posterior.alpha L = self.posterior.L sW = self.posterior.sW nz = range(len(alpha[:,0])) # non-sparse representation if L == []: # in case L is not provided, we compute it K = covfunc.getCovMatrix(x=x[nz,:], mode='train') L = np.linalg.cholesky( (np.eye(nz) + np.dot(sW,sW.T)*K).T ) Ltril = np.all( np.tril(L,-1) == 0 ) # is L an upper triangular matrix? ns = xs.shape[0] # number of data points nperbatch = 1000 # number of data points per mini batch nact = 0 # number of already processed test data points ymu = np.zeros((ns,1)) ys2 = np.zeros((ns,1)) fmu = np.zeros((ns,1)) fs2 = np.zeros((ns,1)) lp = np.zeros((ns,1)) while nact<=ns-1: # process minibatches of test cases to save memory id = range(nact,min(nact+nperbatch,ns)) # data points to process kss = covfunc.getCovMatrix(z=xs[id,:], mode='self_test') # self-variances Ks = covfunc.getCovMatrix(x=x[nz,:], z=xs[id,:], mode='cross') # cross-covariances ms = meanfunc.getMean(xs[id,:]) N = (alpha.shape)[1] # number of alphas (usually 1; more in case of sampling) Fmu = np.tile(ms,(1,N)) + np.dot(Ks.T,alpha[nz]) # conditional mean fs|f fmu[id] = np.reshape(Fmu.sum(axis=1)/N,(len(id),1)) # predictive means if Ltril: # L is triangular => use Cholesky parameters (alpha,sW,L) V = np.linalg.solve(L.T,np.tile(sW,(1,len(id)))*Ks) fs2[id] = kss - np.array([(V*V).sum(axis=0)]).T # predictive variances else: # L is not triangular => use alternative parametrization fs2[id] = kss + np.array([(Ks*np.dot(L,Ks)).sum(axis=0)]).T # predictive variances fs2[id] = np.maximum(fs2[id],0) # remove numerical noise i.e. negative variances Fs2 = np.tile(fs2[id],(1,N)) # we have multiple values in case of sampling if ys == None: [Lp, Ymu, Ys2] = likfunc.proceed(None,Fmu[:],Fs2[:],None,None,3) else: [Lp, Ymu, Ys2] = likfunc.proceed(np.tile(ys[id],(1,N)), Fmu[:], Fs2[:],None,None,3) lp[id] = np.reshape( np.reshape(Lp,(np.prod(Lp.shape),N)).sum(axis=1)/N , (len(id),1) ) # log probability; sample averaging ymu[id] = np.reshape( np.reshape(Ymu,(np.prod(Ymu.shape),N)).sum(axis=1)/N ,(len(id),1) ) # predictive mean ys|y and ... ys2[id] = np.reshape( np.reshape(Ys2,(np.prod(Ys2.shape),N)).sum(axis=1)/N , (len(id),1) ) # .. variance nact = id[-1]+1 # set counter to index of next data point self.ym = ymu self.ys2 = ys2 self.lp = lp self.fm = fmu self.fs2 = fs2 if ys == None: return ymu, ys2, fmu, fs2, None else: return ymu, ys2, fmu, fs2, lp def predict_with_posterior(self, post, xs, ys=None): ''' Prediction with provided posterior (i.e. you already have the posterior and thus don't need fitting/training phases) Get predictive output means(ym), predictive output variances(ys2), predictive latent means(fm), predictive latent variances(fs2), log predictive probabilities(lp). :param post: struct representation of posterior :param xs: test input :param ys: test target(optional) :return: ym, ys2, fm, fs2, lp ''' meanfunc = self.meanfunc covfunc = self.covfunc likfunc = self.likfunc inffunc = self.inffunc x = self.x y = self.y if xs.ndim == 1: xs = np.reshape(xs, (xs.shape[0],1)) self.xs = xs if ys != None: if ys.ndim == 1: ys = np.reshape(ys, (ys.shape[0],1)) self.ys = ys self.posterior = deepcopy(post) alpha = post.alpha L = post.L sW = post.sW nz = range(len(alpha[:,0])) # non-sparse representation if L == []: # in case L is not provided, we compute it K = covfunc.getCovMatrix(x=x[nz,:], mode='train') L = np.linalg.cholesky( (np.eye(nz) + np.dot(sW,sW.T)*K).T ) Ltril = np.all( np.tril(L,-1) == 0 ) # is L an upper triangular matrix? ns = xs.shape[0] # number of data points nperbatch = 1000 # number of data points per mini batch nact = 0 # number of already processed test data points ymu = np.zeros((ns,1)) ys2 = np.zeros((ns,1)) fmu = np.zeros((ns,1)) fs2 = np.zeros((ns,1)) lp = np.zeros((ns,1)) while nact<=ns-1: # process minibatches of test cases to save memory id = range(nact,min(nact+nperbatch,ns)) # data points to process kss = covfunc.getCovMatrix(z=xs[id,:], mode='self_test') # self-variances Ks = covfunc.getCovMatrix(x=x[nz,:], z=xs[id,:], mode='cross') # cross-covariances ms = meanfunc.getMean(xs[id,:]) N = (alpha.shape)[1] # number of alphas (usually 1; more in case of sampling) Fmu = np.tile(ms,(1,N)) + np.dot(Ks.T,alpha[nz]) # conditional mean fs|f fmu[id] = np.reshape(Fmu.sum(axis=1)/N,(len(id),1)) # predictive means if Ltril: # L is triangular => use Cholesky parameters (alpha,sW,L) V = np.linalg.solve(L.T,np.tile(sW,(1,len(id)))*Ks) fs2[id] = kss - np.array([(V*V).sum(axis=0)]).T # predictive variances else: # L is not triangular => use alternative parametrization fs2[id] = kss + np.array([(Ks*np.dot(L,Ks)).sum(axis=0)]).T # predictive variances fs2[id] = np.maximum(fs2[id],0) # remove numerical noise i.e. negative variances Fs2 = np.tile(fs2[id],(1,N)) # we have multiple values in case of sampling if ys == None: [Lp, Ymu, Ys2] = likfunc.proceed(None,Fmu[:],Fs2[:],None,None,3) else: [Lp, Ymu, Ys2] = likfunc.proceed(np.tile(ys[id],(1,N)), Fmu[:], Fs2[:],None,None,3) lp[id] = np.reshape( np.reshape(Lp,(np.prod(Lp.shape),N)).sum(axis=1)/N , (len(id),1) ) # log probability; sample averaging ymu[id] = np.reshape( np.reshape(Ymu,(np.prod(Ymu.shape),N)).sum(axis=1)/N ,(len(id),1) ) # predictive mean ys|y and ... ys2[id] = np.reshape( np.reshape(Ys2,(np.prod(Ys2.shape),N)).sum(axis=1)/N , (len(id),1) ) # .. variance nact = id[-1]+1 # set counter to index of next data point self.ym = ymu self.ys2 = ys2 self.lp = lp self.fm = fmu self.fs2 = fs2 if ys == None: return ymu, ys2, fmu, fs2, None else: return ymu, ys2, fmu, fs2, lp class GPR(GP): '''Gaussian Process Regression''' def __init__(self): super(GPR, self).__init__() self.meanfunc = mean.Zero() # default prior mean self.covfunc = cov.RBF() # default prior covariance self.likfunc = lik.Gauss() # likihood with default noise variance 0.1 self.inffunc = inf.Exact() # inference method self.optimizer = opt.Minimize(self) # default optimizer def setNoise(self,log_sigma): '''Set noise variance other than default''' self.likfunc = lik.Gauss(log_sigma) def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None): '''Set Optimizer. See base class.''' conf = None if (num_restarts!=None) or (min_threshold!=None): conf = pyGPs.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc) conf.num_restarts = num_restarts conf.min_threshold = min_threshold if meanRange != None: conf.meanRange = meanRange if covRange != None: conf.covRange = covRange if likRange != None: conf.likRange = likRange if method == "Minimize": self.optimizer = opt.Minimize(self,conf) elif method == "SCG": self.optimizer = opt.SCG(self,conf) elif method == "CG": self.optimizer = opt.CG(self,conf) elif method == "BFGS": self.optimizer = opt.BFGS(self,conf) def plot(self,axisvals=None): '''Plot 1d GP regression.''' xs = self.xs x = self.x y = self.y ym = self.ym ys2 = self.ys2 plt.figure() xss = np.reshape(xs,(xs.shape[0],)) ymm = np.reshape(ym,(ym.shape[0],)) ys22 = np.reshape(ys2,(ys2.shape[0],)) plt.plot(x, y, color=DATACOLOR, ls='None', marker='+',ms=12, mew=2) plt.plot(xs, ym, color=MEANCOLOR, ls='-', lw=3.) plt.fill_between(xss,ymm + 2.*np.sqrt(ys22), ymm - 2.*np.sqrt(ys22), facecolor=SHADEDCOLOR,linewidths=0.0) plt.grid() if axisvals != None: plt.axis(axisvals) plt.xlabel('input x') plt.ylabel('target y') plt.show() def useInference(self, newInf): ''' Use another inference techinique other than default exact inference. :param str newInf: 'Laplace' or 'EP' ''' if newInf == "Laplace": self.inffunc = inf.Laplace() elif newInf == "EP": self.inffunc = inf.EP() else: raise Exception('Possible inf values are "Laplace", "EP".') def useLikelihood(self,newLik): if newLik == "Laplace": self.likfunc = lik.Laplace() self.inffunc = inf.EP() else: raise Exception('Possible lik values are "Laplace".') class GPC(GP): '''Gaussian Process Classification''' def __init__(self): super(GPC, self).__init__() self.meanfunc = mean.Zero() # default prior mean self.covfunc = cov.RBF() # default prior covariance self.likfunc = lik.Erf() # erf likihood self.inffunc = inf.EP() # default inference method self.optimizer = opt.Minimize(self) # default optimizer def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None): '''Set Optimizer. See base class.''' conf = None if (num_restarts!=None) or (min_threshold!=None): conf = pyGPs.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc) conf.num_restarts = num_restarts conf.min_threshold = min_threshold if meanRange != None: conf.meanRange = meanRange if covRange != None: conf.covRange = covRange if likRange != None: conf.likRange = likRange if method == "Minimize": self.optimizer = opt.Minimize(self,conf) elif method == "SCG": self.optimizer = opt.SCG(self,conf) elif method == "CG": self.optimizer = opt.CG(self,conf) elif method == "BFGS": self.optimizer = opt.BFGS(self,conf) def plot(self,x1,x2,t1,t2,axisvals=None): '''Plot 2d gp classification.''' fig = plt.figure() plt.plot(x1[:,0], x1[:,1], 'b+', markersize = 12) plt.plot(x2[:,0], x2[:,1], 'r+', markersize = 12) pc = plt.contour(t1, t2, np.reshape(np.exp(self.lp), (t1.shape[0],t1.shape[1]) )) fig.colorbar(pc) plt.grid() if axisvals != None: plt.axis(axisvals) plt.show() def useInference(self, newInf): ''' Use another inference techinique other than default EP inference. :param str newInf: 'Laplace' ''' if newInf == "Laplace": self.inffunc = inf.Laplace() else: raise Exception('Possible inf values are "Laplace".') def useLikelihood(self,newLik): ''' Use another likelihood function other than default Erf inference. (Not used in this version) :param str newLik: 'Logistic' ''' if newLik == "Logistic": pass #self.likfunc = lik.Logistic() else: raise Exception('Possible lik values are "Logistic".') class GPMC(object): '''This is a one vs. one classification wrapper for GPC''' def __init__(self, n_class): self.meanfunc = mean.Zero() # default prior mean self.covfunc = cov.RBF() # default prior covariance self.n_class = n_class # number of different classes self.x_all = None self.y_all = None self.newInf = None # new inference? -> call useInference self.newLik = None # new likelihood? -> call useLikelihood self.newPrior = False def setPrior(self, mean=None, kernel=None): '''set prior mean function and covariance function''' if mean != None: self.meanfunc = mean if kernel != None: self.covfunc = kernel if type(kernel) is cov.Pre: self.usingDefaultMean = False self.newPrior = True def useInference(self, newInf): '''Use another inference techinique other than default EP inference.''' self.newInf = newInf def useLikelihood(self,newLik): '''Use another likelihood function other than default Erf inference.''' self.newLik = newLik def setData(self,x,y): '''Set data for multi-class model''' if x.ndim == 1: x = np.reshape(x, (x.shape[0],1)) if y.ndim == 1: y = np.reshape(y, (y.shape[0],1)) self.x_all = x self.y_all = y def fitAndPredict(self, xs): ''' Fitting model and prediction. predictive_vote is a matrix where row i is each test point i column j is the probability for being class j :return: predictive_vote ''' if xs.ndim == 1: xs = np.reshape(xs, (xs.shape[0],1)) predictive_vote = np.zeros((xs.shape[0],self.n_class)) for i in xrange(self.n_class): # classifier for class i... for j in xrange(i+1,self.n_class): # ...and class j x,y = self.createBinaryClass(i,j) model = GPC() if self.newPrior: model.setPrior(mean=self.meanfunc, kernel=self.covfunc) if self.newInf: model.useInference(self.newInf) if self.newLik: model.useLikelihood(self.newLik) model.fit(x,y) # fitting ym = model.predict(xs)[0] ym += 1 # now scale into 0 to 2, ym=0 is class j, ym=2 is class i vote_i = np.zeros((xs.shape[0],self.n_class)) vote_j = np.zeros((xs.shape[0],self.n_class)) vote_i[:,i:i+1] = ym vote_j[:,j:j+1] = 2-ym predictive_vote += vote_i predictive_vote += vote_j predictive_vote /= predictive_vote.sum(axis=1)[:,np.newaxis] return predictive_vote def optimizeAndPredict(self, xs): ''' Optimizing model and prediction. predictive_vote is a matrix where row i is each test point i column j is the probability for being class j :return: predictive_vote ''' if xs.ndim == 1: xs = np.reshape(xs, (xs.shape[0],1)) predictive_vote = np.zeros((xs.shape[0],self.n_class)) for i in xrange(self.n_class): # classifier for class i... for j in xrange(i+1,self.n_class): # ...and class j x,y = self.createBinaryClass(i,j) model = GPC() if self.newPrior: model.setPrior(mean=self.meanfunc, kernel=self.covfunc) if self.newInf: model.useInference(self.newInf) if self.newLik: model.useLikelihood(self.newLik) model.optimize(x,y) # training ym = model.predict(xs)[0] ym += 1 # now scale into 0 to 2, ym=0 is class j, ym=2 is class i vote_i = np.zeros((xs.shape[0],self.n_class)) vote_j = np.zeros((xs.shape[0],self.n_class)) vote_i[:,i:i+1] = ym vote_j[:,j:j+1] = 2-ym predictive_vote += vote_i predictive_vote += vote_j predictive_vote /= predictive_vote.sum(axis=1)[:,np.newaxis] return predictive_vote def createBinaryClass(self, i,j): ''' Create dataset x(data) and y(label) which only contains class i and j. Relabel class i to +1 and class j to -1''' class_i = [] class_j = [] for index in xrange(len(self.y_all)): # check all classes target = self.y_all[index] if target == i: class_i.append(index) elif target == j: class_j.append(index) n1 = len(class_i) n2 = len(class_j) class_i.extend(class_j) x = self.x_all[class_i,:] y = np.concatenate((np.ones((1,n1)),-np.ones((1,n2))),axis=1).T return x,y class GP_FITC(GP): '''FITC GP base class''' def __init__(self): super(GP_FITC, self).__init__() self.u = None # overriding def setData(self, x, y, value_per_axis=5): '''Set training data and derive deault inducing_points. :param x: training data :param y: training target :param int value_per_axis: number of value in each dimension when using a uni-distant default inducing points''' if x.ndim == 1: x = np.reshape(x, (x.shape[0],1)) if y.ndim == 1: y = np.reshape(y, (y.shape[0],1)) self.x = x self.y = y if self.usingDefaultMean: c = np.mean(y) self.meanfunc = mean.Const(c) # adapt default prior mean wrt. training labels # get range of x in each dimension # 5 uniformally selected value for each dimension gridAxis=[] for d in xrange(x.shape[1]): column = x[:,d] mini = np.min(column) maxi = np.max(column) axis = np.linspace(mini,maxi,value_per_axis) gridAxis.append(axis) # default inducing points-> a grid if self.u == None: self.u = np.array(list(itertools.product(*gridAxis))) self.covfunc = self.covfunc.fitc(self.u) # overriding def setPrior(self, mean=None, kernel=None, inducing_points=None): '''Set GP prior and inducing points''' if kernel != None: if inducing_points != None: self.covfunc = kernel.fitc(inducing_points) self.u = inducing_points else: if self.u != None: self.covfunc = kernel.fitc(self.u) else: raise error("To use default inducing points, please call setData() first!") if type(kernel) is cov.Pre: self.usingDefaultMean = False if mean != None: self.meanfunc = mean self.usingDefaultMean = False class GPR_FITC(GP_FITC): '''Gaussian Process Regression FITC''' def __init__(self): super(GPR_FITC, self).__init__() self.meanfunc = mean.Zero() # default prior mean self.covfunc = cov.RBF() # default prior covariance self.likfunc = lik.Gauss() # likihood with default noise variance 0.1 self.inffunc = inf.FITC_Exact() # inference method self.optimizer = opt.Minimize(self) # default optimizer self.u = None # no default inducing points def setNoise(self,log_sigma): '''Set noise variance other than default value''' self.likfunc = lik.Gauss(log_sigma) def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None): '''Set Optimizer. See base class.''' conf = None if (num_restarts!=None) or (min_threshold!=None): conf = pyGPs.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc) conf.num_restarts = num_restarts conf.min_threshold = min_threshold if meanRange != None: conf.meanRange = meanRange if covRange != None: conf.covRange = covRange if likRange != None: conf.likRange = likRange if method == "Minimize": self.optimizer = opt.Minimize(self,conf) elif method == "SCG": self.optimizer = opt.SCG(self,conf) elif method == "CG": self.optimizer = opt.CG(self,conf) elif method == "BFGS": self.optimizer = opt.BFGS(self,conf) def plot(self,axisvals=None): '''Plot 1d GP FITC regression.''' plt.figure() xss = np.reshape(self.xs,(self.xs.shape[0],)) ymm = np.reshape(self.ym,(self.ym.shape[0],)) ys22 = np.reshape(self.ys2,(self.ys2.shape[0],)) plt.plot(self.x, self.y, color=DATACOLOR, ls='None', marker='+',ms=12, mew=2) plt.plot(self.xs, self.ym, color=MEANCOLOR, ls='-', lw=3.) plt.fill_between(xss,ymm + 2.*np.sqrt(ys22), ymm - 2.*np.sqrt(ys22), facecolor=SHADEDCOLOR,linewidths=0.0) plt.grid() if axisvals != None: plt.axis(axisvals) plt.xlabel('input x') plt.ylabel('output y') plt.plot(self.u,np.ones_like(self.u), ls='None', color='k',marker='x',markersize=12,mew=2) plt.show() def useInference(self, newInf): ''' Use another inference techinique other than default exact inference. :param str newInf: 'Laplace' or 'EP' ''' if newInf == "Laplace": self.inffunc = inf.FITC_Laplace() elif newInf == "EP": self.inffunc = inf.FITC_EP() else: raise Exception('Possible inf values are "Laplace", "EP".') def useLikelihood(self,newLik): ''' Use another inference techinique other than default Gaussian likelihood. :param str newLik: 'Laplace' ''' if newLik == "Laplace": self.likfunc = lik.Laplace() self.inffunc = inf.FITC_EP() else: raise Exception('Possible lik values are "Laplace".') class GPC_FITC(GP_FITC): '''Gaussian Process Classification FITC''' def __init__(self): super(GPC_FITC, self).__init__() self.meanfunc = mean.Zero() # default prior mean self.covfunc = cov.RBF() # default prior covariance self.likfunc = lik.Erf() # erf liklihood self.inffunc = inf.FITC_EP() # default inference method self.optimizer = opt.Minimize(self) # default optimizer self.u = None # no default inducing points def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None): '''Set optimizer. See base class.''' conf = None if (num_restarts!=None) or (min_threshold!=None): conf = pyGPs.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc) conf.num_restarts = num_restarts conf.min_threshold = min_threshold if meanRange != None: conf.meanRange = meanRange if covRange != None: conf.covRange = covRange if likRange != None: conf.likRange = likRange if method == "Minimize": self.optimizer = opt.Minimize(self,conf) elif method == "SCG": self.optimizer = opt.SCG(self,conf) elif method == "CG": self.optimizer = opt.CG(self,conf) elif method == "BFGS": self.optimizer = opt.BFGS(self,conf) def plot(self,x1,x2,t1,t2,axisvals=None): '''Plot 2d GP FITC classification.''' fig = plt.figure() plt.plot(x1[:,0], x1[:,1], 'b+', markersize = 12) plt.plot(x2[:,0], x2[:,1], 'r+', markersize = 12) plt.plot(self.u[:,0],self.u[:,1],'ko', markersize=12) pc = plt.contour(t1, t2, np.reshape(np.exp(self.lp), (t1.shape[0],t1.shape[1]) )) fig.colorbar(pc) plt.grid() if axisvals != None: plt.axis(axisvals) plt.show() def useInference(self, newInf): ''' Use another inference techinique other than default exact inference. :param str newInf: 'Laplace' or 'EP' ''' if newInf == "Laplace": self.inffunc = inf.FITC_Laplace() else: raise Exception('Possible inf values are "Laplace".') def useLikelihood(self,newLik): ''' Use another inference techinique other than default Erf likelihood. (Not used in this version) :param str newLik: 'Logistic' ''' if newLik == "Logistic": pass #self.likfunc = lik.Logistic() else: raise Exception('Possible lik values are "Logistic".') }rJM(XGPMCXclassMMXGP_FITC.__init__XdefMMXGPC.setOptimizerXdefMMXGPR_FITC.setOptimizerXdefMMX GPR.setNoiseXdefMMXGPMC.createBinaryClassXdefMMXGPR.setOptimizerXdefMMX GPMC.__init__XdefMM$XGPR_FITC.useLikelihoodXdefMM'XGPC_FITC.__init__XdefM)M2XGPMC.optimizeAndPredictXdefMbMXGPC_FITC.useLikelihoodXdefM`MpXGP_FITCXclassMMXGPR.useInferenceXdefMMXGPC_FITC.useInferenceXdefMUM`X GP.setPriorXdefKrK|XGPC.plotXdefMMXGPC.useLikelihoodXdefMMX GP.setDataXdefKOK[XGPR.plotXdefMMXGP.setOptimizerXdefK|KXGP.plotData_1dXdefK[KfXGPC_FITCXclassM'MpXGP.fitXdefKKڇXGPXclassK6MXGPR_FITC.setNoiseXdefMMXGPR_FITCXclassMM'X GP.predictXdefKM+XGPCXclassMMXGPC.useInferenceXdefMMXGPMC.useLikelihoodXdefM2M6X GP.__init__XdefK8KOX GPR_FITC.plotXdefMM XGPR_FITC.useInferenceXdefM MXGPRXclassMMXGPC_FITC.setOptimizerXdefM2MHX GP.optimizeXdefKKX GPC.__init__XdefMMX GPR.__init__XdefMMX GPMC.setPriorXdefM$M.XGP_FITC.setDataXdefMMXGP_FITC.setPriorXdefMMXGP.plotData_2dXdefKfKrXGPMC.fitAndPredictXdefM?MbX GPMC.setDataXdefM6M?XGPR.useLikelihoodXdefMMXGPMC.useInferenceXdefM.M2X GPC_FITC.plotXdefMHMUXGP.predict_with_posteriorXdefM+MXGPR_FITC.__init__XdefMMu}rKMXGP.setOptimizerjDsj7MX"pyGPs.GraphExtensions.graphKernelsrLMrMM}rNMbXE#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ # Created on April, 2014 # @author: marion, shan import scipy.sparse as spsp import numpy as np import matplotlib.pyplot as plt def propagationKernel(A, l, gr_id, h_max, w, p, ktype=None, SUM=True, VIS=False, showEachStep=False): ''' Propagation kernel for graphs as described in: Neumann, M., Patricia, N., Garnett, R., Kersting, K.: Efficient Graph Kernels by Randomization. In: P.A. Flach, T.D. Bie, N. Cristianini (eds.) ECML/PKDD, Notes in Computer Science, vol. 7523, pp. 378-393. Springer (2012). :param A: adjacency matrix (num_nodes x num_nodes) :param l: label array (num_nodes x 1); values [1,...,k] or -1 for unlabeled nodes OR label array (num_nodes x num_labels); values [0,1], unlabeled nodes have only 0 entries :param gr_id: graph indicator array (num_nodes x 1); values [0,..,n] :param h_max: number of iterations :param w: bin widths parameter :param p: distance ('tv', 'hellinger', 'L1', 'L2') :param ktype: type of propagation kernel ['diffusion', 'label_propagation', 'label_spreading', 'belief_propagation'] :return: kernel matrix ''' # ToDO: SPLIT Varible label array and label probability # CAUTION: number of labels (k) > 1 !!! #=========================================================================== # ## Propagation Kernel COMPUTATION #=========================================================================== num_graphs = gr_id.max().astype(int) num_nodes = A.shape[0] if l.shape[1]==1: num_labels = l.max().astype(int) else: num_labels = l.shape[1] #=========================================================================== # ## INITIALIZATION #=========================================================================== ## CHECK and ADJUST sparseness if not spsp.issparse(A): A = spsp.csr_matrix(A) else: if not spsp.isspmatrix_csr(A): A = A.tocsr() # MAKE sure A is float! A = A.astype(float) ## INITIALIZE kernel matrix K = np.ndarray(shape=(num_graphs,num_graphs,h_max+1), dtype=float) ## INITIALIZE label probabilities of labeled nodes (num_nodes x num_labels) if l.shape[1]==1: lab_prob = np.zeros((num_nodes,num_labels),dtype=np.float64) for i in xrange(num_nodes): if l[i,0] > 0: lab_prob[i,l[i,0]-1] = 1 else: lab_prob = l if spsp.issparse(lab_prob): lab_prob = lab_prob.todense() lab_orig = lab_prob.copy() # lab_orig is the orignal label information ## GET indices of labeled and unlabeled nodes idx_unlab = np.where(np.max(lab_prob, axis=1)!=1)[0] idx_all = np.array(np.arange(0,A.shape[0]), ndmin=2) idx = np.array(np.setdiff1d(idx_all,idx_unlab), ndmin=2) ## INITIALIZE unlabeled/uninitialized nodes UNIFORMLY idx_unif = np.where(np.sum(lab_prob, axis=1)==0)[0] if idx_unif.shape[0] != 0: lab_prob[idx_unif,:] = 1./lab_prob.shape[1] ## row normalize A -> transition matrix T if (ktype=='label_propagation') or (ktype=='label_diffusion'): row_sums = np.array(A.sum(axis=1))[:,0] T = A.copy() T.data /= row_sums[A.nonzero()[0]] # so A and T are sparse matrix #=========================================================================== # ## PROPAGATION KERNEL ITERATIONS #=========================================================================== for h in xrange(h_max+1): print 'ITERATION: ', h if h > 0: ## LABEL UPDATE if showEachStep: print '...computing LABEL UPDATE' if ktype == 'label_propagation': lab_prob[idx,:] = lab_orig[idx,:] # PUSH BACK original LABELS lab_prob = T*lab_prob elif ktype == 'label_diffusion': lab_prob = T*lab_prob elif ktype == 'label_spreading': # S = D^(-1/2)*W*D^(-1/2) # y(t+1) = alpha*S*y(t)+(1-alpha)*y(0) alpha = 0.8 # compute S diag = np.array(A.sum(axis=1)).T**(-1/2) D = spsp.lil_matrix((A.shape[0], A.shape[1]), dtype=float) # D.setdiag(diag[0,:]) D = D.tocsr() S = D*A*D lab_prob = np.dot((alpha*S.todense()),lab_prob) + (1-alpha)*lab_orig ## COMPUTE hashvalues if showEachStep: print '...computing hashvalues' # determine path to take depending on chosen distance use_cauchy = (p =='L1') or (p =='tv') take_sqrt = (p =='hellinger') if take_sqrt: lab_prob = np.sqrt(lab_prob) # generate random projection vector v = np.random.normal(size=(lab_prob.shape[1], 1)) if use_cauchy: # if X, Y are N(0, 1), then X / Y has a standard Cauchy distribution v /= np.random.normal(size=(lab_prob.shape[1], 1)) # random offset b = w * np.random.rand() # compute hashes # hashLabels is a Vector with length: number of nodes (hashvalue for respective node) hashLabels = (np.dot(lab_prob,v) + b) / w hashLabels = np.floor(hashLabels) # take floor uniqueHash, hashLabels = np.unique(hashLabels, return_inverse=True) # map to consecutive integer from 0 ## COMPUTE kernel contribution if showEachStep: print '...computing KERNEL contribution' # aggregate counts on graphs # counts is a matrix: number of graphs x number of hashlabels num_bins = len(uniqueHash) counts = np.zeros((num_graphs, num_bins)) # init counts matrix # accumulate counts of hash labels for i in xrange(num_nodes): counts[ (gr_id[i,0]-1), hashLabels[i] ] +=1 # compute base kernel (here: LINEAR kernel) K_h = np.dot(counts,counts.T) ## SUM kernel contributions if SUM: if h == 0: K[:,:,h] = K_h else: K[:,:,h] = K[:,:,h-1] + K_h else: K[:,:,h] = K_h if showEachStep: print K[:,:,h] ## VISUALIZE KERNELS if VIS: for h in range(h_max+1): K_h = K[:,:,h] plt.figure() plt.title('height:'+str(h)) imgplot = plt.imshow(K_h) imgplot.set_interpolation('nearest') plt.colorbar() plt.show() return K }rOMXpropagationKernelXdefKKs}rPMXpropagationKernelhsj7MXpyGPs.Core.likrQMrRM}rSMbXqm# Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ # likelihood functions are provided to be used by the gp.py function: # # Erf (Error function, classification, probit regression) # Logistic [NOT IMPLEMENTED!] (Logistic, classification, logit regression) # Uni [NOT IMPLEMENTED!] (Uniform likelihood, classification) # # Gauss (Gaussian, regression) # Laplace (Laplacian or double exponential, regression) # Sech2 [NOT IMPLEMENTED!] (Sech-square, regression) # T [NOT IMPLEMENTED!] (Student's t, regression) # # Poisson [NOT IMPLEMENTED!] (Poisson regression, count data) # # Mix [NOT IMPLEMENTED!] (Mixture of individual covariance functions) # # See the documentation for the individual likelihood for the computations specific # to each likelihood function. # # # This is a object-oriented python implementation of gpml functionality # (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18). # based on the functional-version of python implementation # (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013) # # Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013 import numpy as np from scipy.special import erf import inf class Likelihood(object): """Base function for Likelihood function""" def __init__(self): self.hyp = [] def proceed(self, y=None, mu=None, s2=None, inffunc=None, der=None, nargout=1): ''' The likelihood functions have two possible modes, the mode being selected as follows (where "lik" stands for "proceed" method for any likelihood function): 1) With two or three input arguments: [PREDICTION MODE] lp = lik(y, mu) OR lp, ymu, ys2 = lik(y, mu, s2) This allows to evaluate the predictive distribution. Let p(y_*|f_*) be the likelihood of a test point and N(f_*|mu,s2) an approximation to the posterior marginal p(f_*|x_*,x,y) as returned by an inference method. The predictive distribution p(y_*|x_*,x,y) is approximated by: q(y_*) = \int N(f_*|mu,s2) p(y_*|f_*) df_* lp = log( q(y) ) for a particular value of y, if s2 is [] or 0, this corresponds to log( p(y|mu) ). ymu and ys2 are the mean and variance of the predictive marginal q(y) note that these two numbers do not depend on a particular value of y. All vectors have the same size. 2) With four or five input arguments, the fouth being an object of class "Inference" [INFERENCE MODE] lik(y, mu, s2, inf) OR lik(y, mu, s2, inf, i) There are two cases for inf, namely a) infLaplace, b) infEP The last input i, refers to derivatives w.r.t. the ith hyperparameter. | a1) | lp,dlp,d2lp,d3lp = lik(y, f, [], 'infLaplace'). | lp, dlp, d2lp and d3lp correspond to derivatives of the log likelihood. | log(p(y|f)) w.r.t. to the latent location f. | lp = log( p(y|f) ) | dlp = d log( p(y|f) ) / df | d2lp = d^2 log( p(y|f) ) / df^2 | d3lp = d^3 log( p(y|f) ) / df^3 | a2) | lp_dhyp,dlp_dhyp,d2lp_dhyp = lik(y, f, [], 'infLaplace', i) | returns derivatives w.r.t. to the ith hyperparameter | lp_dhyp = d log( p(y|f) ) / (dhyp_i) | dlp_dhyp = d^2 log( p(y|f) ) / (df dhyp_i) | d2lp_dhyp = d^3 log( p(y|f) ) / (df^2 dhyp_i) | b1) | lZ,dlZ,d2lZ = lik(y, mu, s2, 'infEP') | let Z = \int p(y|f) N(f|mu,s2) df then | lZ = log(Z) | dlZ = d log(Z) / dmu | d2lZ = d^2 log(Z) / dmu^2 | b2) | dlZhyp = lik(y, mu, s2, 'infEP', i) | returns derivatives w.r.t. to the ith hyperparameter | dlZhyp = d log(Z) / dhyp_i Cumulative likelihoods are designed for binary classification. Therefore, they only look at the sign of the targets y; zero values are treated as +1. Some examples for valid likelihood functions: | lik = likGauss([0.1]) | lik = likErf() ''' pass class Gauss(Likelihood): ''' Gaussian likelihood function for regression. :math:`Gauss(t)=\\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-\\frac{(t-y)^2}{2\\sigma^2}}`, where :math:`y` is the mean and :math:`\\sigma` is the standard deviation. hyp = [ log_sigma ] ''' def __init__(self, log_sigma=np.log(0.1) ): self.hyp = [log_sigma] def proceed(self, y=None, mu=None, s2=None, inffunc=None, der=None, nargout=1): sn2 = np.exp(2. * self.hyp[0]) if inffunc == None: # prediction mode if y == None: y = np.zeros_like(mu) s2zero = True if (not s2==None) and np.linalg.norm(s2) > 0: s2zero = False if s2zero: # log probability lp = -(y-mu)**2 /sn2/2 - np.log(2.*np.pi*sn2)/2. s2 = np.zeros_like(s2) else: inf_func = inf.EP() # prediction lp = self.proceed(y, mu, s2, inf_func) if nargout>1: ymu = mu # first y moment if nargout>2: ys2 = s2 + sn2 # second y moment return lp,ymu,ys2 else: return lp,ymu else: return lp else: if isinstance(inffunc, inf.EP): if der == None: # no derivative mode lZ = -(y-mu)**2/(sn2+s2)/2. - np.log(2*np.pi*(sn2+s2))/2. # log part function if nargout>1: dlZ = (y-mu)/(sn2+s2) # 1st derivative w.r.t. mean if nargout>2: d2lZ = -1/(sn2+s2) # 2nd derivative w.r.t. mean return lZ,dlZ,d2lZ else: return lZ,dlZ else: return lZ else: # derivative mode dlZhyp = ((y-mu)**2/(sn2+s2)-1) / (1+s2/sn2) # deriv. w.r.t. hyp.lik return dlZhyp elif isinstance(inffunc, inf.Laplace): if der == None: # no derivative mode if y == None: y=0 ymmu = y-mu lp = -ymmu**2/(2*sn2) - np.log(2*np.pi*sn2)/2. if nargout>1: dlp = ymmu/sn2 # dlp, derivative of log likelihood if nargout>2: # d2lp, 2nd derivative of log likelihood d2lp = -np.ones_like(ymmu)/sn2 if nargout>3: # d3lp, 3rd derivative of log likelihood d3lp = np.zeros_like(ymmu) return lp,dlp,d2lp,d3lp else: return lp,dlp,d2lp else: return lp,dlp else: return lp else: # derivative mode lp_dhyp = (y-mu)**2/sn2 - 1 # derivative of log likelihood w.r.t. hypers dlp_dhyp = 2*(mu-y)/sn2 # first derivative, d2lp_dhyp = 2*np.ones_like(mu)/sn2 # and also of the second mu derivative return lp_dhyp,dlp_dhyp,d2lp_dhyp ''' elif isinstance(inffunc, infVB): if der == None: # variational lower site bound # t(s) = exp(-(y-s)^2/2sn2)/sqrt(2*pi*sn2) # the bound has the form: b*s - s.^2/(2*ga) - h(ga)/2 with b=y/ga ga = s2 n = len(ga) b = y/ga y = y*np.ones((n,1)) db = -y/ga**2 d2b = 2*y/ga**3 h = np.zeros((n,1)) dh = h d2h = h # allocate memory for return args id = (ga <= sn2 + 1e-8) # OK below noise variance h[id] = y[id]**2/ga[id] + np.log(2*np.pi*sn2) h[np.logical_not(id)] = np.inf dh[id] = -y[id]**2/ga[id]**2 d2h[id] = 2*y[id]**2/ga[id]**3 id = ga < 0 h[id] = np.inf dh[id] = 0 d2h[id] = 0 # neg. var. treatment varargout = [h,b,dh,db,d2h,d2b] else: ga = s2 n = len(ga) dhhyp = np.zeros((n,1)) dhhyp[ga<=sn2] = 2 dhhyp[ga<0] = 0 # negative variances get a special treatment varargout = dhhyp # deriv. w.r.t. hyp.lik else: raise Exception('Incorrect inference in lik.Gauss\n') ''' class Erf(Likelihood): ''' Error function or cumulative Gaussian likelihood function for binary classification or probit regression. :math:`Erf(t)=\\frac{1}{2}(1+erf(\\frac{t}{\\sqrt{2}}))=normcdf(t)` ''' def __init__(self): self.hyp = [] def proceed(self, y=None, mu=None, s2=None, inffunc=None, der=None, nargout=1): if not y == None: y = np.sign(y) y[y==0] = 1 else: y = 1 # allow only +/- 1 values if inffunc == None: # prediction mode if inf is not present y = y*np.ones_like(mu) # make y a vector s2zero = True; if not s2 == None: if np.linalg.norm(s2)>0: s2zero = False # s2==0? if s2zero: # log probability evaluation p,lp = self.cumGauss(y,mu,2) else: # prediction lp = self.proceed(y, mu, s2, inf.EP()) p = np.exp(lp) if nargout>1: ymu = 2*p-1 # first y moment if nargout>2: ys2 = 4*p*(1-p) # second y moment return lp,ymu,ys2 else: return lp,ymu else: return lp else: # inference mode if isinstance(inffunc, inf.Laplace): if der == None: # no derivative mode f = mu; yf = y*f # product latents and labels p,lp = self.cumGauss(y,f,2) if nargout>1: # derivative of log likelihood n_p = self.gauOverCumGauss(yf,p) dlp = y*n_p # derivative of log likelihood if nargout>2: # 2nd derivative of log likelihood d2lp = -n_p**2 - yf*n_p if nargout>3: # 3rd derivative of log likelihood d3lp = 2*y*n_p**3 + 3*f*n_p**2 + y*(f**2-1)*n_p return lp,dlp,d2lp,d3lp else: return lp,dlp,d2lp else: return lp,dlp else: return lp else: # derivative mode return [] # derivative w.r.t. hypers elif isinstance(inffunc, inf.EP): if der == None: # no derivative mode z = mu/np.sqrt(1+s2) junk,lZ = self.cumGauss(y,z,2) # log part function if not y == None: z = z*y if nargout>1: if y == None: y = 1 n_p = self.gauOverCumGauss(z,np.exp(lZ)) dlZ = y*n_p/np.sqrt(1.+s2) # 1st derivative wrt mean if nargout>2: d2lZ = -n_p*(z+n_p)/(1.+s2) # 2nd derivative wrt mean return lZ,dlZ,d2lZ else: return lZ,dlZ else: return lZ else: # derivative mode return [] # deriv. wrt hyp.lik ''' if inffunc == 'inf.infVB': if der == None: # no derivative mode # naive variational lower bound based on asymptotical properties of lik # normcdf(t) -> -(t*A_hat^2-2dt+c)/2 for t->-np.inf (tight lower bound) d = 0.158482605320942; c = -1.785873318175113; ga = s2; n = len(ga); b = d*y*np.ones((n,1)); db = np.zeros((n,1)); d2b = db h = -2.*c*np.ones((n,1)); h[ga>1] = np.inf; dh = np.zeros((n,1)); d2h = dh varargout = [h,b,dh,db,d2h,d2b] else: # derivative mode varargout = [] # deriv. wrt hyp.lik ''' def cumGauss(self, y=None, f=None, nargout=1): # return [p,lp] = cumGauss(y,f) if not y == None: yf = y*f else: yf = f p = (1. + erf(yf/np.sqrt(2.)))/2. # likelihood if nargout>1: lp = self.logphi(yf,p) return p,lp else: return p def gauOverCumGauss(self,f,p): # return n_p = gauOverCumGauss(f,p) n_p = np.zeros_like(f) # safely compute Gaussian over cumulative Gaussian ok = f>-5 # naive evaluation for large values of f n_p[ok] = (np.exp(-f[ok]**2/2)/np.sqrt(2*np.pi)) / p[ok] bd = f<-6 # tight upper bound evaluation n_p[bd] = np.sqrt(f[bd]**2/4+1)-f[bd]/2 interp = np.logical_and(np.logical_not(ok),np.logical_not(bd)) # linearly interpolate between both of them tmp = f[interp] lam = -5. - f[interp] n_p[interp] = (1-lam)*(np.exp(-tmp**2/2)/np.sqrt(2*np.pi))/p[interp] + lam *(np.sqrt(tmp**2/4+1)-tmp/2); return n_p def logphi(self,z,p): # return lp = logphi(z,p) lp = np.zeros_like(z) # allocate memory zmin = -6.2; zmax = -5.5; ok = z>zmax # safe evaluation for large values bd = z0: s2zero = False # s2==0? if s2zero: # log probability evaluation lp = -np.abs(y-mu)/b -np.log(2*b); s2 = 0 else: # prediction lp = self.proceed(y, mu, s2, inf.EP()) if nargout>1: ymu = mu # first y moment if nargout>2: ys2 = s2 + sn**2 # second y moment return lp,ymu,ys2 else: return lp,ymu else: return lp else: # inference mode if isinstance(inffunc, inf.Laplace): if der == None: # no derivative mode if y == None: y = np.zeros_like(mu) ymmu = y-mu lp = np.abs(ymmu)/b - np.log(2*b) if nargout>1: # derivative of log likelihood dlp = np.sign(ymmu)/b if nargout>2: # 2nd derivative of log likelihood d2lp = np.zeros_like(ymmu) if nargout>3: # 3rd derivative of log likelihood d3lp = np.zeros_like(ymmu) return lp,dlp,d2lp,d3lp else: return lp,dlp,d2lp else: return lp,dlp else: return lp else: # derivative w.r.t. hypers lp_dhyp = np.abs(y-mu)/b - 1 # derivative of log likelihood w.r.t. hypers dlp_dhyp = np.sign(mu-y)/b # first derivative, d2lp_dhyp = np.zeros(mu.shape) # and also of the second mu derivative return lp_dhyp, dlp_dhyp, d2lp_dhyp elif isinstance(inffunc, inf.EP): n = np.max([len(y.flatten()),len(mu.flatten()),len(s2.flatten()),len(sn.flatten())]) on = np.ones((n,1)) y = y*on; mu = mu*on; s2 = s2*on; sn = sn*on; fac = 1e3; # factor between the widths of the two distributions ... # ... from when one considered a delta peak, we use 3 orders of magnitude #idlik = np.reshape( (fac*sn) < np.sqrt(s2) , (sn.shape[0],) ) # Likelihood is a delta peak #idgau = np.reshape( (fac*np.sqrt(s2)) < sn , (sn.shape[0],) ) # Gaussian is a delta peak idlik = (fac*sn) < np.sqrt(s2) idgau = (fac*np.sqrt(s2)) < sn id = np.logical_and(np.logical_not(idgau),np.logical_not(idlik)) # interesting case in between if der == None: # no derivative mode lZ = np.zeros((n,1)) dlZ = np.zeros((n,1)) d2lZ = np.zeros((n,1)) if np.any(idlik): l = Gauss(log_sigma=np.log(s2[idlik])/2) a = l.proceed(mu[idlik], y[idlik]) lZ[idlik] = a[0]; dlZ[idlik] = a[1]; d2lZ[idlik] = a[2] if np.any(idgau): l = Laplace(log_hyp=np.log(sn[idgau])) a = l.proceed(mu=mu[idgau], y=y[idgau]) lZ[idgau] = a[0]; dlZ[idgau] = a[1]; d2lZ[idgau] = a[2] if np.any(id): # substitution to obtain unit variance, zero mean Laplacian tvar = s2[id]/(sn[id]**2+1e-16) tmu = (mu[id]-y[id])/(sn[id]+1e-16) # an implementation based on logphi(t) = log(normcdf(t)) zp = (tmu+np.sqrt(2)*tvar)/np.sqrt(tvar) zm = (tmu-np.sqrt(2)*tvar)/np.sqrt(tvar) ap = self._logphi(-zp)+np.sqrt(2)*tmu am = self._logphi( zm)-np.sqrt(2)*tmu apam = np.vstack((ap,am)).T lZ[id] = self._logsum2exp(apam) + tvar - np.log(sn[id]*np.sqrt(2.)) if nargout>1: lqp = -0.5*zp**2 - 0.5*np.log(2*np.pi) - self._logphi(-zp); # log( N(z)/Phi(z) ) lqm = -0.5*zm**2 - 0.5*np.log(2*np.pi) - self._logphi( zm); dap = -np.exp(lqp-0.5*np.log(s2[id])) + np.sqrt(2)/sn[id] dam = np.exp(lqm-0.5*np.log(s2[id])) - np.sqrt(2)/sn[id] _z1 = np.vstack((ap,am)).T _z2 = np.vstack((dap,dam)).T _x = np.array([[1],[1]]) dlZ[id] = self._expABz_expAx(_z1, _x, _z2, _x) if nargout>2: a = np.sqrt(8.)/sn[id]/np.sqrt(s2[id]); bp = 2./sn[id]**2 - (a - zp/s2[id])*np.exp(lqp) bm = 2./sn[id]**2 - (a + zm/s2[id])*np.exp(lqm) _x = np.reshape(np.array([1,1]),(2,1)) _z1 = np.reshape(np.array([ap,am]),(1,2)) _z2 = np.reshape(np.array([bp,bm]),(1,2)) d2lZ[id] = self._expABz_expAx(_z1, _x, _z2, _x) - dlZ[id]**2 return lZ,dlZ,d2lZ else: return lZ,dlZ else: return lZ else: # derivative mode dlZhyp = np.zeros((n,1)) if np.any(idlik): dlZhyp[idlik] = 0 if np.any(idgau): l = Laplace(log_hyp=np.log(sn[idgau])) a = l.proceed(mu=mu[idgau], y=y[idgau], inffunc='inf.Laplace', nargout=1) dlZhyp[idgau] = a[0] if np.any(id): # substitution to obtain unit variance, zero mean Laplacian tmu = (mu[id]-y[id])/(sn[id]+1e-16); tvar = s2[id]/(sn[id]**2+1e-16) zp = (tvar+tmu/np.sqrt(2))/np.sqrt(tvar); vp = tvar+np.sqrt(2)*tmu zm = (tvar-tmu/np.sqrt(2))/np.sqrt(tvar); vm = tvar-np.sqrt(2)*tmu dzp = (-s2[id]/sn[id]+tmu*sn[id]/np.sqrt(2)) / np.sqrt(s2[id]) dvp = -2*tvar - np.sqrt(2)*tmu dzm = (-s2[id]/sn[id]-tmu*sn[id]/np.sqrt(2)) / np.sqrt(s2[id]) dvm = -2*tvar + np.sqrt(2)*tmu lezp = self._lerfc(zp); # ap = exp(vp).*ezp lezm = self._lerfc(zm); # am = exp(vm).*ezm vmax = np.max(np.array([vp+lezp,vm+lezm]),axis=0); # subtract max to avoid numerical pb ep = np.exp(vp+lezp-vmax) em = np.exp(vm+lezm-vmax) dap = ep*(dvp - 2/np.sqrt(np.pi)*np.exp(-zp**2-lezp)*dzp) dam = em*(dvm - 2/np.sqrt(np.pi)*np.exp(-zm**2-lezm)*dzm) dlZhyp[id] = (dap+dam)/(ep+em) - 1; return dlZhyp # deriv. wrt hyp.lik elif isinstance(inffunc, inf.VB): n = len(s2.flatten()); b = np.zeros((n,1)); y = y*np.ones((n,1)); z = y return b,z def _lerfc(self,t): ''' numerically safe implementation of f(t) = log(1-erf(t)) = log(erfc(t))''' from scipy.special import erfc f = np.zeros_like(t) tmin = 20; tmax = 25 ok = ttmax # evaluate tight bound nok = np.logical_not(ok) interp = np.logical_and(nok,np.logical_not(bd)) # interpolate between both of them f[nok] = np.log(2/np.sqrt(np.pi)) -t[nok]**2 -np.log(t[nok]+np.sqrt( t[nok]**2+4/np.pi )) lam = 1/(1+np.exp( 12*(0.5-(t[interp]-tmin)/(tmax-tmin)) )) # interp. weights f[interp] = lam*f[interp] + (1-lam)*np.log(erfc( t[interp] )) f[ok] += np.log(erfc( t[ok] )) # safe eval return f def _expABz_expAx(self,A,x,B,z): ''' Computes y = ( (exp(A).*B)*z ) ./ ( exp(A)*x ) in a numerically safe way The function is not general in the sense that it yields correct values for all types of inputs. 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_warnfuncrmSNub.pyGPs-1.2/doc/build/html/.doctrees/index.doctree000066400000000000000000000333701236346020200215760ustar00rootroot00000000000000cdocutils.nodes document q)q}q(U nametypesq}q(Xgetting startedqNXprocedual implementationqXabout the packageqNX(pygps - a package for gaussian processesq NuUsubstitution_defsq }q Uparse_messagesq ]q Ucurrent_sourceqNU decorationqNUautofootnote_startqKUnameidsq}q(hUgetting-startedqhUprocedual-implementationqhUabout-the-packageqh U&pygps-a-package-for-gaussian-processesquUchildrenq]q(cdocutils.nodes comment q)q}q(U rawsourceqXwObject-Oriented Gaussian Processes documentation master file, created by sphinx-quickstart on Thu Nov 14 19:54:52 2013.UparentqhUsourceqcdocutils.nodes reprunicode qX0/Users/hs/Program/git/pyGPs/doc/source/index.rstq q!}q"bUtagnameq#Ucommentq$U attributesq%}q&(U xml:spaceq'Upreserveq(Uidsq)]Ubackrefsq*]Udupnamesq+]Uclassesq,]Unamesq-]uUlineq.KUdocumentq/hh]q0cdocutils.nodes Text q1XwObject-Oriented Gaussian Processes documentation master file, created by sphinx-quickstart on Thu Nov 14 19:54:52 2013.q2q3}q4(hUhhubaubcdocutils.nodes section q5)q6}q7(hUhhhh!h#Usectionq8h%}q9(h+]h,]h*]h)]q:hah-]q;h auh.Kh/hh]q<(cdocutils.nodes title q=)q>}q?(hX(pyGPs - A Package for Gaussian Processesq@hh6hh!h#UtitleqAh%}qB(h+]h,]h*]h)]h-]uh.Kh/hh]qCh1X(pyGPs - A Package for Gaussian ProcessesqDqE}qF(hh@hh>ubaubh5)qG}qH(hUhh6hh!h#h8h%}qI(h+]h,]h*]h)]qJhah-]qKhauh.Kh/hh]qL(h=)qM}qN(hXAbout the packageqOhhGhh!h#hAh%}qP(h+]h,]h*]h)]h-]uh.Kh/hh]qQh1XAbout the packageqRqS}qT(hhOhhMubaubcdocutils.nodes paragraph qU)qV}qW(hXpyGPs is a library hosting Python implementations of Gaussian processes (GPs) for machine learning. pyGPs bridges the gap between systems designed primarily for users, who mainly want to apply GPs and need basic machine learning routines for model training, evaluation, and visualiztion, and expressive systems for developers, who focus on extending the core functionalities as covariance and likelihood functions, as well as inference techniques.qXhhGhh!h#U paragraphqYh%}qZ(h+]h,]h*]h)]h-]uh.Kh/hh]q[h1XpyGPs is a library hosting Python implementations of Gaussian processes (GPs) for machine learning. pyGPs bridges the gap between systems designed primarily for users, who mainly want to apply GPs and need basic machine learning routines for model training, evaluation, and visualiztion, and expressive systems for developers, who focus on extending the core functionalities as covariance and likelihood functions, as well as inference techniques.q\q]}q^(hhXhhVubaubhU)q_}q`(hXThe software package is released under the BSD 2-Clause (FreeBSD) License. :doc:`Copyright <_static/copyright>` (c) by Marion Neumann, Shan Huang, Daniel Marthaler, & Kristian Kersting, Feb.2014hhGhh!h#hYh%}qa(h+]h,]h*]h)]h-]uh.Kh/hh]qb(h1XKThe software package is released under the BSD 2-Clause (FreeBSD) License. qcqd}qe(hXKThe software package is released under the BSD 2-Clause (FreeBSD) License. hh_ubcsphinx.addnodes pending_xref qf)qg}qh(hX$:doc:`Copyright <_static/copyright>`qihh_hh!h#U pending_xrefqjh%}qk(UreftypeXdocqlUrefwarnqmU reftargetqnX_static/copyrightU refdomainUh)]h*]U refexplicith+]h,]h-]UrefdocqoXindexqpuh.Kh]qqcdocutils.nodes literal qr)qs}qt(hhih%}qu(h+]h,]qv(Uxrefqwhleh*]h)]h-]uhhgh]qxh1X Copyrightqyqz}q{(hUhhsubah#Uliteralq|ubaubh1XS (c) by Marion Neumann, Shan Huang, Daniel Marthaler, & Kristian Kersting, Feb.2014q}q~}q(hXS (c) by Marion Neumann, Shan Huang, Daniel Marthaler, & Kristian Kersting, Feb.2014hh_ubeubhU)q}q(hX'Further, it includes implementations ofqhhGhh!h#hYh%}q(h+]h,]h*]h)]h-]uh.Kh/hh]qh1X'Further, it includes implementations ofqq}q(hhhhubaubcdocutils.nodes bullet_list q)q}q(hUhhGhh!h#U bullet_listqh%}q(UbulletqX*h)]h*]h+]h,]h-]uh.Kh/hh]q(cdocutils.nodes list_item q)q}q(hXminimize.py implemented in python by Roland Memisevic 2008, following minimize.m (Copyright (c) Carl Edward Rasmussen (1999-2006)) hhhh!h#U 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You can download the developer's guide and manual containing an API here: qօq}q(hX for a list of changes to the previous release. If you observe problems or bugs, please contact us. You can download the developer's guide and manual containing an API here: hhubcsphinx.addnodes download_reference q)q}q(hX2:download:`API and manual <_static/pyGPs_API.pdf>`qhhhh!h#Udownload_referenceqh%}q(UreftypeXdownloadqhmhnX_static/pyGPs_API.pdfqU refdomainUh)]h*]U refexplicith+]h,]h-]hohpUfilenameqX pyGPs_API.pdfquh.Kh]qhr)q}q(hhh%}q(h+]h,]q(hwheh*]h)]h-]uhhh]qh1XAPI and manualq酁q}q(hUhhubah#h|ubaubh1X. You can also download a q셁q}q(hX. You can also download a hhubcdocutils.nodes reference q)q}q(hX`procedual implementation`_UresolvedqKhhh#U referenceqh%}q(UnameXprocedual implementationUrefuriqX&https://github.com/marionmari/pyGP_PR/qh)]h*]h+]h,]h-]uh]qh1Xprocedual implementationqq}q(hUhhubaubh1Xa of GP functionality from Github. However, the procedual version will not be supported in future.qq}q(hXa of GP functionality from Github. However, the procedual version will not be supported in future.hhubeubcdocutils.nodes target q)q}r(hXD.. _procedual implementation: https://github.com/marionmari/pyGP_PR/U referencedrKhhGhh!h#Utargetrh%}r(hhh)]rhah*]h+]h,]h-]rhauh.K#h/hh]ubcdocutils.nodes definition_list r)r}r(hUhhGhNh#Udefinition_listr h%}r (h+]h,]h*]h)]h-]uh.Nh/hh]r cdocutils.nodes definition_list_item r )r }r(hXAuthors: - Marion Neumann [marion dot neumann at uni-bonn dot de] - Shan Huang [shan dot huang at iais dot fraunhofer dot de] - Daniel Marthaler [dan dot marthaler at gmail dot com] - Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] hjhh!h#Udefinition_list_itemrh%}r(h+]h,]h*]h)]h-]uh.K+h]r(cdocutils.nodes term r)r}r(hXAuthors:rhj hh!h#Utermrh%}r(h+]h,]h*]h)]h-]uh.K+h]rh1XAuthors:rr}r(hjhjubaubcdocutils.nodes definition r)r}r(hUh%}r(h+]h,]h*]h)]h-]uhj h]r h)r!}r"(hUh%}r#(hX-h)]h*]h+]h,]h-]uhjh]r$(h)r%}r&(hX6Marion Neumann [marion dot neumann at uni-bonn dot de]r'h%}r((h+]h,]h*]h)]h-]uhj!h]r)hU)r*}r+(hj'hj%hh!h#hYh%}r,(h+]h,]h*]h)]h-]uh.K(h]r-h1X6Marion Neumann [marion dot neumann at uni-bonn dot de]r.r/}r0(hj'hj*ubaubah#hubh)r1}r2(hX9Shan Huang [shan dot huang at iais dot fraunhofer dot de]r3h%}r4(h+]h,]h*]h)]h-]uhj!h]r5hU)r6}r7(hj3hj1hh!h#hYh%}r8(h+]h,]h*]h)]h-]uh.K)h]r9h1X9Shan Huang [shan dot huang at iais dot fraunhofer dot de]r:r;}r<(hj3hj6ubaubah#hubh)r=}r>(hX5Daniel Marthaler [dan dot marthaler at gmail dot com]r?h%}r@(h+]h,]h*]h)]h-]uhj!h]rAhU)rB}rC(hj?hj=hh!h#hYh%}rD(h+]h,]h*]h)]h-]uh.K*h]rEh1X5Daniel Marthaler [dan dot marthaler at gmail dot com]rFrG}rH(hj?hjBubaubah#hubh)rI}rJ(hXGKristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] h%}rK(h+]h,]h*]h)]h-]uhj!h]rLhU)rM}rN(hXFKristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]rOhjIhh!h#hYh%}rP(h+]h,]h*]h)]h-]uh.K+h]rQh1XFKristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]rRrS}rT(hjOhjMubaubah#hubeh#hubah#U definitionrUubeubaubhU)rV}rW(hXThe following persons helped to improve this software: Roman Garnett, Maciej Kurek, Hannes Nickisch, Zhao Xu, and Alejandro Molina.rXhhGhh!h#hYh%}rY(h+]h,]h*]h)]h-]uh.K-h/hh]rZh1XThe following persons helped to improve this software: Roman Garnett, Maciej Kurek, Hannes Nickisch, Zhao Xu, and Alejandro Molina.r[r\}r](hjXhjVubaubhU)r^}r_(hXJThis work is partly supported by the Fraunhofer ATTRACT fellowship STREAM.r`hhGhh!h#hYh%}ra(h+]h,]h*]h)]h-]uh.K/h/hh]rbh1XJThis work is partly supported by the Fraunhofer ATTRACT fellowship STREAM.rcrd}re(hj`hj^ubaubeubh5)rf}rg(hUhh6hh!h#h8h%}rh(h+]h,]h*]h)]rihah-]rjhauh.K3h/hh]rk(h=)rl}rm(hXGetting startedrnhjfhh!h#hAh%}ro(h+]h,]h*]h)]h-]uh.K3h/hh]rph1XGetting startedrqrr}rs(hjnhjlubaubcdocutils.nodes compound rt)ru}rv(hUhjfhh!h#Ucompoundrwh%}rx(h+]h,]ryUtoctree-wrapperrzah*]h)]h-]uh.Nh/hh]r{csphinx.addnodes toctree r|)r}}r~(hUhjuhh!h#Utoctreerh%}r(UnumberedrKU includehiddenrhhpU titlesonlyrUglobrh)]h*]h+]h,]h-]Uentriesr]r(NXInstallrrNXTheoryrrNXExamplesrrNXKernelsrrNX LikelihoodsrrNXOptsrrNXGraphrreUhiddenrU includefilesr]r(jjjjjjjeUmaxdepthrKuh.K5h]ubaubeubeubehUU transformerrNU footnote_refsr}rUrefnamesr}rXprocedual implementation]rhasUsymbol_footnotesr]rUautofootnote_refsr]rUsymbol_footnote_refsr]rU citationsr]rh/hU current_linerNUtransform_messagesr]rUreporterrNUid_startrKU autofootnotesr]rU citation_refsr}rUindirect_targetsr]rUsettingsr(cdocutils.frontend Values ror}r(Ufootnote_backlinksrKUrecord_dependenciesrNU rfc_base_urlrUhttp://tools.ietf.org/html/rU tracebackrUpep_referencesrNUstrip_commentsrNU toc_backlinksrUentryrU language_coderUenrU datestamprNU report_levelrKU _destinationrNU halt_levelrKU strip_classesrNhANUerror_encoding_error_handlerrUbackslashreplacerUdebugrNUembed_stylesheetrUoutput_encoding_error_handlerrUstrictrU sectnum_xformrKUdump_transformsrNU docinfo_xformrKUwarning_streamrNUpep_file_url_templaterUpep-%04drUexit_status_levelrKUconfigrNUstrict_visitorrNUcloak_email_addressesrUtrim_footnote_reference_spacerUenvrNUdump_pseudo_xmlrNUexpose_internalsrNUsectsubtitle_xformrU source_linkrNUrfc_referencesrNUoutput_encodingrUutf-8rU source_urlrNUinput_encodingrU utf-8-sigrU_disable_configrNU id_prefixrUU tab_widthrKUerror_encodingrUUTF-8rU_sourcerU0/Users/hs/Program/git/pyGPs/doc/source/index.rstrUgettext_compactrU generatorrNUdump_internalsrNU smart_quotesrU pep_base_urlrUhttp://www.python.org/dev/peps/rUsyntax_highlightrUlongrUinput_encoding_error_handlerrjUauto_id_prefixrUidrUdoctitle_xformrUstrip_elements_with_classesrNU _config_filesr]Ufile_insertion_enabledrU raw_enabledrKU dump_settingsrNubUsymbol_footnote_startrKUidsr}r(hhhjfhh6hhGuUsubstitution_namesr}rh#h/h%}r(h+]h)]h*]Usourceh!h,]h-]uU footnotesr]rUrefidsr}rub.pyGPs-1.2/doc/build/html/CV.html000066400000000000000000000231371236346020200164300ustar00rootroot00000000000000 K-fold Cross-Validation — pyGPs v1.2 documentation

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      K-fold Cross-Validation

      In this demo, we’ll show you the typical process of using GP for machine learning from loading data, preprocessing, training, predicting to validation and evaluation.

      Load data

      We use the ionosphere dataset [1] from Johns Hopkins University Ionosphere database. It is available in UCI machine learning repository. Then we need to do some data cleaning. Here we deal with label in ionosphere data, change “b” to”-1”, and “g” to “+1”. These preprocessing implementation are availabe in the source code.

      Cross Validation

      Now, lets focus on the use of cross-validation.

      K = 10             # number of folds
for x_train, x_test, y_train, y_test in valid.k_fold_validation(x, y, K):
    # This is a binary classification problem
    model = pyGPs.GPC()
    # Since no prior knowldege, leave everything default
    model.optimize(x_train, y_train)
    # Predition
    ymu, ys2, fmu, fs2, lp = model.predict(x_test, ys=y_test)
    # ymu for classification is a continuous value over -1 to +1
    # If you want predicting result to either one of the classes, take a sign of ymu.
    ymu_class = np.sign(ymu)
    # Evluation
    acc = valid.ACC(ymu_class, y_test)       # accuracy
    rmse = valid.RMSE(ymu_class, y_test)     # root-mean-square error

      Evaluation measures

      We implemented some classical evaluation measures.
      • RMSE - root mean squared error
      • ACC - classification/regression accuracy
      • Prec - classification precision for class +1
      • Recall - classification recall for class +1
      • NLPD - negative log predictive density in transformed observation space
      [1]Sigillito, V. G., Wing, S. P., Hutton, L. V., & Baker, K. B. (1989). Classification of radar returns from the ionosphere using neural networks. Johns Hopkins APL Technical Digest, 10, 262-266.

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      Demos

      There are several demos exemplifying the use of pyGPs for various Gaussian process (GP) tasks. We recommend to first go through Basic GP Regression which introduces the GP regression model. Basic regression is the most intuitive and simplest learning task feasable with GPs. The other demos will then provide a general insight into more advanced functionalities of the package. You will also find the implementation of the demos in the source folder under pyGPs/Demo.

      The Demos give some theoretical explanations. Further, it is useful to have a look at our documentation on Kernels & Means and Optimizers.

      Regression

      Classification

      Some examples for real-world data

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      Basic Classification

      The demo shown in this tutorial can be obtained by running pyGPs/Demo/demo_GPC.py.

      Load data

      First, we import the data:

      # GPC target class are +1 and -1
demoData = np.load('data_for_demo/classification_data.npz')
x = demoData['x']            # training data
y = demoData['y']            # training target
z = demoData['xstar']        # test data

      The 120 data points were generated from two Gaussians with different means and covariances. One Gaussian is isotropic and contains 2/3 of the data (blue), the other is highly correlated and contains 1/3 of the points (red). Note, that the labels for the targets are specified to be \pm 1 (and not 0/1).

      In the plot, we superimpose the data points with the posterior equi-probability contour lines for the probability of the second class given complete information about the generating mechanism.

      _images/d2_1.png

      First example \rightarrow state default values

      Again, lets see the simplest use of gp classification at first

      model = pyGPs.gp.GPC()             # binary classification (default inference method: EP)
model.fit(x, y)              # fit default model (mean zero & rbf kernel) with data
model.optimize(x, y)            # optimize hyperparamters (default optimizer: single run minimize)
model.predict(z)             # predict test cases

      Note, that inference is done via expectation propagation (EP) approximation by deault. How to set inference to Laplace approximation, see A bit more things you can do.

      Second example \rightarrow GP classification

      So we first state the model to be GP classification now:

      model = pyGPs.GPC()

      The rest is similar to GPR:

      k = pyGPs.cov.RBFard(log_ell_list=[0.05,0.17], log_sigma=1.)
model.setPrior(kernel=k)

model.setData(x, y)
model.plotData_2d(x1,x2,t1,t2,p1,p2)

model.fit()
model.optimize()
model.predict(z, ys=np.ones((z.shape[0],1)))

      [Theory] In this example, we used an RBF kernel (squared exponential covariance function) with automatic relevance determination (ARD). This covariance function has one characteristic length-scale parameter for each dimension of the input space (here 2 in total), and a signal magnitude parameter, resulting in a total of 3 hyperparameters. ARD with separate length-scales for each input dimension is a very powerful tool to learn which inputs are important for the predictions: if length-scales are short, input dimensions are very important, and when they grow very large (compared to the spread of the data), the corresponding input dimensions will be mostly ignored.

      Note, pyGPs.GPC().plot() is a toy method for 2-d data:

      model.plot(x1,x2,t1,t2)

      The contour plot for the predictive distribution is shown below. Note, that the predictive probability is fairly close to the probabilities of the generating process in regions of high data density. Note also, that as you move away from the data, the probability approaches 1/3, the overall class probability.

      _images/d2_2.png

      Examining the two ARD characteristic length-scale parameters after learning, you will find that they are fairly similar, reflecting the fact that for this data set, both input dimensions are important.

      A bit more things you can do

      GPC uses expectation propagation (EP) inference and Error function likelihood by default, you can explictly change to other methods:

      model.useInference("Laplace")

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      Sparse Classification

      The demo in this tutorial can be obtained by running pyGPs/Demo/demo_GPC_FITC.py. This demo is more or less a repetition of the demo of FITC regression.

      First example \rightarrow default inducing points

      First load the same data as in the GPC demo.

      [Theory] In case the number of training inputs x exceeds a few hundred, approximate inference using Laplacian Approximation or Expectation Propagation takes too long. As in regression, we offer the FITC approximation based on a low-rank plus diagonal approximation to the exact covariance to deal with these cases. The general idea is to use inducing points u and to base the computations on cross-covariances between training, test and inducing points only.

      Okay, now the model is FITC classificiation:

      model = pyGPs.GPC_FITC()

      The difference between the usage of basic GP is that we will have to specify inducing points. In our first example, we will introduce how to perform sparse GPC with the default settings.

      The default inducing points form a grid (hypercube in higher dimension), where each dimension has 5 values in equidistant steps in [min, max], where min and max are the minimum and maximum values of the input data by default. In order to specify the dimension of input data, we HAVE TO set data first:

      model.setData(x, y)

      The number of inducing points per axis is 5 per default. How to change this, see A bit more things you can do.

      Then, the regular process follows:

      model.optimize()
model.predict(z, ys=np.ones((z.shape[0],1)))
model.plot(x1,x2,t1,t2)
      _images/d4_1.png

      The equispaced default inducing points u are shown as black circles in the plot.

      Second example \rightarrow user-defined inducing points

      Alternatively, a random subset of the training points can be used as inducing points. Note, that there are various different ways of how to set the inducing points. So, in the second example let us use a user-defined set of inducing points:

      u1,u2 = np.meshgrid(np.linspace(-2,2,5),np.linspace(-2,2,5))
u = np.array(zip(np.reshape(u2,(np.prod(u2.shape),)),np.reshape(u1,(np.prod(u1.shape),))))

      Here, we also use a grid euqually spaced, but without the values on the margin of the grid.(i.e. shrinking the grid) Then, we can just pass u when specifying prior:

      m = pyGPs.mean.Zero()
k = pyGPs.cov.RBFard(log_ell_list=[0.05,0.17], log_sigma=1.)
model.setPrior(mean=m, kernel=k, inducing_points=u)

      The prediction results for this set of inducing points are shown below:

      _images/d4_2.png

      A bit more things you can do

      As in standard GPC, it is possible to use other inference/likelihood in the FITC method:

      model.useInference("Laplace")

      Change the number of inducing points per axis:

      model.setData(x, y, value_per_axis=10)

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      Multi-class Classification

      GPMC is NOT based on multi-class Laplace approximation. It works as a one vs. one classification wrapper. In other words, GPMC trains a GPC model for each pair of two classes, and uses a majority voting scheme over all results to determine the final class. The method only returns the predictive class with highest rating; no other values (such as variance) are returned.

      Lets see a practical example to classify the 10 (0,1,2,...9) hand-written digits in the USPS digits dataset.

      Load data

      The USPS digits data were gathered at the Center of Excellence in Document Analysis and Recognition (CEDAR) at SUNY Buffalo, as part of a project sponsored by the US Postal Service. The dataset is described in [1].

      data = loadmat('data_for_demo/usps_resampled.mat')
x = data['train_patterns'].T   # train patterns
y = data['train_labels'].T     # train labels
xs = data['test_patterns'].T   # test patterns
ys = data['test_labels'].T     # test labels

      To be used in GPMC, labels should start from 0 to k (k = number of classes).

      GPMC example

      State model with 10-class classification problem:

      model = pyGPs.GPMC(10)

      Pass data to model:

      model.setData(x,y)

      Train default GPC model for each binary classification problem, and decide label for test patterns of hand-writen digits. The return value predictive_vote[i,j] is the probability of being class j for test pattern i.

      predictive_vote = model.optimizeAndPredict(xs)
predictive_class = np.argmax(predictive_vote, axis=1)

      Just like we did for GP classification, you can use specific settings (other than default) for all binary classificiation problem for example by:

      m = pyGPs.mean.Zero()
k = pyGPs.cov.RBF()
model.setPrior(mean=m,kernel=k)
model.useInference("Laplace")

      For more information on how to use non-default settings see demo_GPC and demo_GPR.

      Beside optimizeAndPredict(xs), there is also an option to perform prediction without hyperparameter optimization:

      model.fitAndPredict(xs)
      [1]A Database for Handwritten Text Recognition Research, J. J. Hull, IEEE PAMI 16(5) 550-554, 1994.

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      Basic Regression

      The code shown in this tutorial can be executed by running pyGPs/Demo/demo_GPR.py

      This demo will not only introduce the regression model, it also provides the general insight of how to use the package. This general information will not be repeated in the other demos.

      Import packages

      Once you installed pyGPs, the typical way to import it is:

      import pyGPs
import numpy as np

      Load data

      First, load the data for this demo. The data consists of n=20 1-d data points drawn from a unit Gaussian. This is the same data used in the GPML example (it is hardcoded in data/regression_data.npz).

      Note that target vector y can be in 2d matrix with shape (nn,1) or 1d vector with shape (nn,) where nn is number of test inputs. pyGPs can work with either format.

      demoData = np.load('data_for_demo/regression_data.npz')
x = demoData['x']      # training data
y = demoData['y']      # training target
z = demoData['xstar']  # test data

      A five-line toy example

      Now lets do regression with Gaussian processes. Using pyGPs for regression is really simple; here is the most basic example:

      model = pyGPs.GPR()    # specify model (GP regression)
model.fit(x, y)        # fit default model (mean zero & rbf kernel) with data
model.optimize(x, y)   # optimize hyperparamters (default optimizer: single run minimize)
model.predict(z)       # predict test cases
model.plot()           # and plot result

      By default, GPR uses a zero mean, the rbf kernel and a Gaussian likelihood. Default optimizer is a single run of Rasmussen’s minimize. You will see below how to set non-default values in another example.

      pyGPs.GPR().plot() will plot the result, where the dark line is the posterior mean and the green-shaded area is the posterior variance. Note, that plot() is not a general method as it is not trivial to visualize high dimensional data. Here, pyGPs.GPR().plot() works for 1-d data only, while pyGPs.GPC().plot() is a toy method visualising 2-d input data in a classification scenario.

      _images/d1_1.png

      A more complicated example

      Now lets do another example to get insight into more advanced features of the toolbox.

      You can specify non-default mean and covariance functions:

      m = pyGPs.mean.Linear( D=x.shape[1] ) + mean.Const()
k = pyGPs.cov.RBF()
model.setPrior(mean=m, kernel=k)

      Here, we use a composite mean as the sum of a linear and a constant function, and an rbf kernel. The initial hyperparameters are left to their default values. See Kernels & Means for a complete documentation of kernel/mean specification and custom kernel/mean construction. Once kernel and mean are specified, they are passed to the prior using setPrior().

      You can add the traning data to the model explicitly by using setData(). So, you avoid passing them into fit() or optimize() each time used. More importantly, the deafult mean will be adapted to the average value of the trainging labels y (if you do not specify mean function by your own).

      Further, you can plot the data in the 1-d case:

      model.plotData_1d()
      _images/d1_2.png

      You can specify a optimization method different from the default, which is a single run of Rasmussen’s minimize. For example, you can choose to rerun the optimization method several times with different random initializations:

      model.setOptimizer("Minimize", num_restarts=30)

      The optimized hyperparameters returned by optimize() are then set to be the ones obtained from the run with the best result. The whole functionality for optimization is introduced in detail in the documentation Optimizers.

      Instead of fit(), which only fits data using given hyperparameters, optimize() will optimize hyperparamters based on marginal likelihood:

      model.optimize()

      There are several properties you can get from the model:

      model.nlZ                   # negative log marginal likelihood
model.dnlZ.cov              # direvatives of negative log marginal likelihood
model.dnlZ.lik
model.dnlZ.mean
model.posterior.sW          # posterior structure
model.posterior.alpha
model.posterior.L
model.covfunc.hyp
model.meanfunc.hyp
model.likfunc.hyp
model.fm                    # latent mean
model.fs2                   # latent variance
model.ym                    # predictive mean
model.ys2                   # predictive variance
model.lp                    # log predictive probability

      For example, to get the log marginal likelihood use:

      print 'Optimized negative log marginal likelihood:', round(model.nlZ,3)

      Prediction on the test data will return five values, which are output mean (ymu) resp. variance (ys2), latent mean (fmu) resp. variance (fs2), and log predictive probabilities (lp)

      ym, ys2, fm, fs2, lp = model.predict(z)

      Plot data. Note that GPR.plot() is a toy method only for visualising 1-d data. Here we got a different posterior by using a different prior other than in the default example.

      model.plot()
      _images/d1_3.png

      A bit more things you can do

      [For all Models] Speed up computation time for prediction if you know posterior in advance. Posterior is passed as an object with three fields (attributes) post.alpha, post.sW and post.L. How to use these vectors to represent the posterior can be best seen from Algorithm 2.1 (page 19) in Chapeter 2 of the GPML book by Rasmussen and Williams, 2006.

      post = myPosterior()        # known in advance
ym, ys2, fm, fs2, lp = model.predict_with_posterior( post,z )

      [Only for Regression] Specify noise of data (with \sigma=0.1 by default):

      model.setNoise( log_sigma = np.log(0.1) )

      You do not need to specify the noise parameter if you are optimizing the hyperparamters later anyhow.

      All plotting methods have keyword axisvals. You can adjust plotting range if you want. For example:

      model.plot(axisvals = [-1.9, 1.9, -0.9, 3.9])

      Switch to other Inference and Likelihood functions.

      model.useInference("EP")
model.useLikelihood("Laplace")

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      Sparse Regression

      The code shown in this tutorial can be obtained by running pyGPs/Demo/demo_GPR_FITC.py This demo is more or less similar to the demo of FITC classification.

      First example \rightarrow default inducing points

      First load the same data as in the GPR demo.

      [Theory] In case the number of training inputs x exceeds a few hundred, approximate inference using Laplace approximation or expectation propagation takes too long. We offer the FITC approximation based on a low-rank plus diagonal approximation to the exact covariance to deal with these cases. The general idea is to use inducing points u and to base the computations on cross-covariances between training, test and inducing points only.

      Okay, now the model is FITC regression:

      model = pyGPs.GPR_FITC()

      The difference between the usage of basic GP regression is that we will have to specify inducing points. In the first example here, we will introduce you how to use the default settings.

      The default inducing points are a grid (hypercube for higher dimensions), where each dimension has 5 values in equidistant steps in [min, max], where min and max are the minimum and maximum values of the input data by default. In order to specify the dimension of input data, we HAVE TO set data first:

      model.setData(x, y)

      The number of inducing points per axis is 5 per default.

      Now, the regular training and prediction routines follow:

      model.optimize()
model.predict(z)
model.plot()
      _images/d3_1.png

      The equidistant default inducing points u that are shown in the figure as black x’s.

      To change the number of inducing points per axis just specify a different value per axis:

      model.setData(x, y, value_per_axis=10)

      Second example \rightarrow user-defined inducing points

      Alternatively, a random subset of the training points can be used as inducing points. Note, that there are plenty of methods to set these inducing points. So, in the second example let us use a user-defined set of inducing points.

      You can pick a set of fixed inducing points by hand:

      u = np.array([[-1], [-0.8], [-0.5], [0.3],[1.]])

      You can also use equidistant inducing points u, but without the values on the margin of the grid.(i.e. shrinking the range of values)

      num_u = np.fix(x.shape[0]/2)
u = np.linspace(-1.3,1.3,num_u).T
u = np.reshape(u,(num_u,1))

      Then pass u when specifying prior.

      m = pyGPs.mean.Zero()
k = pyGPs.cov.RBFard(log_ell_list=[0.05,0.17], log_sigma=1.)
model.setPrior(mean=m, kernel=k, inducing_points=u)

      The left figure below shows the result of fixed inducing points, and the right figure shows the result for equidistant u.

      _images/d3_2.png _images/d3_3.png

      [Theory] Note that the predictive variance is overestimated outside the support of the inducing inputs. In a multivariate example where densely sampled inducing inputs are infeasible, one can also try to simply use a random subset of the training points.

      A bit more things you can do

      Switch to other Inference and Likelihood functions.

      model.useInference("EP")
model.useLikelihood("Laplace")

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      Kernels for Graph Data

      You can refer to our demo for semi-supervised learning for a simple usage of kernels for graph data.

      And there is also a demo for graph kernel using an example of propagation kernel.

      Kernels on Graph

      pyGPs.GraphExtensions.nodeKernels.VNDKernel(A, alpha=0.5)[source]

      Von Neumann Diffusion Kernel on graph (Zhou et al., 2004) (also label spreading kernel)

      K = (I - alpha*S)^-1, where S = D^-1/2*A*D^-1/2

      Parameters:
      • A – adjacency matrix
      • alpha – hyperparameter alpha
      Returns:

      kernel matrix
      pyGPs.GraphExtensions.nodeKernels.cosKernel(A)[source]

      Cosine Kernel (also Inverse Cosine Kernel)

      K = cos (L*pi/4), where L is the normalized Laplacian

      Parameters:A – adjacency matrix
      Returns:kernel matrix
      pyGPs.GraphExtensions.nodeKernels.diffKernel(A, beta=0.5)[source]

      Diffusion Process Kernel

      K = exp(beta * H), where H = -L = A-D

      K = Q exp(beta * Lambda) Q.T

      Parameters:
      • A – adjacency matrix
      • beta – hyperparameter beta
      Returns:

      kernel matrix
      pyGPs.GraphExtensions.nodeKernels.normLap(A)[source]

      Normalized Laplacian

      Parameters:A – adjacency matrix
      Returns:kernel matrix
      pyGPs.GraphExtensions.nodeKernels.psInvLapKernel(A)[source]

      Pseudo inverse of the normalized Laplacian.

      Parameters:A – adjacency matrix
      Returns:kernel matrix
      pyGPs.GraphExtensions.nodeKernels.regLapKernel(A, sigma=1)[source]

      Regularized Laplacian Kernel

      Parameters:
      • A – adjacency matrix
      • sigma – hyperparameter sigma
      Returns:

      kernel matrix
      pyGPs.GraphExtensions.nodeKernels.rwKernel(A, p=1, a=2)[source]

      p-step Random Walk Kernel with a>1

      K = (aI-L)^p, p>1 and L is the normalized Laplacian

      Parameters:
      • A – adjacency matrix
      • p – step parameter
      • a – hyperparameter a
      Returns:

      kernel matrix

      Graph Kernels

      pyGPs.GraphExtensions.graphKernels.propagationKernel(A, l, gr_id, h_max, w, p, ktype=None, SUM=True, VIS=False, showEachStep=False)[source]

      Propagation kernel for graphs as described in: Neumann, M., Patricia, N., Garnett, R., Kersting, K.: Efficient Graph Kernels by Randomization. In: P.A. Flach, T.D. Bie, N. Cristianini (eds.) ECML/PKDD, Notes in Computer Science, vol. 7523, pp. 378-393. Springer (2012).

      Parameters:
      • A – adjacency matrix (num_nodes x num_nodes)
      • l – label array (num_nodes x 1); values [1,...,k] or -1 for unlabeled nodes OR label array (num_nodes x num_labels); values [0,1], unlabeled nodes have only 0 entries
      • gr_id – graph indicator array (num_nodes x 1); values [0,..,n]
      • h_max – number of iterations
      • w – bin widths parameter
      • p – distance (‘tv’, ‘hellinger’, ‘L1’, ‘L2’)
      • ktype – type of propagation kernel [‘diffusion’, ‘label_propagation’, ‘label_spreading’, ‘belief_propagation’]
      Returns:

      kernel matrix

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      Graph Kernels

      The code shown in this tutorial can be executed by running pyGPs/Demo/demo_GraphKernel.py

      Load data

      MUTAG [1] is a data set of 188 mutagenic aromatic and heteroaromatic nitro compounds labeled according to whether or not they have a mutagenic effect on the Gram-negative bacterium Salmonella typhimurium. MUTAG is a simple but popular dataset to benchmark graph kernels.

      data = np.load('MUTAG.npz')
A = csc_matrix( (data['adj_data'], data['adj_indice'], \
    data['adj_indptr']), shape=data['adj_shape'])     # n x n adjancy array (sparse matrix)
gr_id = data['graph_ind']                             # n x 1 graph id array
node_label = data['responses']                        # n x 1 node label array
graph_label = data['labels']                          # N x 1 graph label array

      Note that adjacency matrix for all graphs is usually too big to fit into memory when using GP. Therefore, we generate A using sparse matrix csc_matrix provided by scipy.

      Compute propagation kernel matrix

      Propagation kernels [2] are recently introduced fast and flexible graph kernels. Generate prapagation kernel based on given graph data.

      num_Iteration = 10
w = 1e-4
dist = 'tv'      # possible values: 'tv', 'hellinger'
K = graphKernels.propagationKernel(A, node_label, gr_id, num_Iteration, w, dist, 'label_diffusion')

      Adjacency matrix A can either be in format of standard numpy matrix or sparse matrix.

      Standard GP Classification

      K is a big square matrix with all rows and columns of the number of data points. By specifying the indice of training data and test data, we will form two matrix M1 and M2 with the exact format which pyGPs.Core.cov.Pre needed.

      M1,M2 = graphUtil.formKernelMatrix(Matrix, indice_train, indice_test)
k = pyGPs.cov.Pre(M1,M2)
      M1 is a matrix with shape number of training points plus 1 by number of test points
      • cross covariances matrix (train by test)
      • last row is self covariances (diagonal of test by test)
      M2 is a square matrix with number of training points for each dimension
      • training set covariance matrix (train by train)

      The following is the standard way to do GP classification

      model = pyGPs.GPC()
model.setPrior(kernel=k)
model.fit(x_train, y_train)
model.predict(x_test)
      [1]Debnath, A., Lopez de Compadre, R., Debnath, G., Shusterman, A., and Hansch, C.. Structure-activity relationship of mutagenic aromatic and heteroaromatic nitro compounds. Correlation with molecular orbital energies and hydrophobicity. J. Med. Chem., 34:786–797, 1991.
      [2]Neumann, M., Patricia, N., Garnett, R., Kersting, K.: Efficient Graph Kernels by Randomization. In: P.A. Flach, T.D. Bie, N. Cristianini (eds.) ECML/PKDD, Notes in Computer Science, vol. 7523, pp. 378-393. Springer (2012).

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      Installation

      1. First, download the archive from github and extract it to any local directory.

      1. You can either add the local directory to your PYTHONPATH

        export PYTHONPATH=$PYTHONPATH:/path/to/local/directory/../parent_folder_of_pyGPs

      2. Or install the package using setup.py:

        sudo python setup.py install

      Requirements

      Example installation on Ubuntu & Debian:

      sudo apt-get install python2.7 python-numpy python-scipy python-matplotlib

      Example installation on Mac via Macports (requires XCode and MacPorts):

      sudo port install python27 py27-numpy py27-scipy py27-matplotlib

      For other systems please check the installation instructions on the respective package web sites.

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      Kernels & Means

      Simple Kernel & Mean

      You may already seen, we can specify a kernel function like this(same for mean fucntions):

      k = pyGPs.cov.RBF( log_ell=-1., log_sigma=0. )

      There are several points need to be noticed:

      1. Most parameters are initilized in their logorithms. This is because we need to make sure they are positive during optimization. e.g. Here length scale and signal variance should always be positive.
      2. Most kernel functions have a scalar in front, namely signal variance(set by log_sigma)
      3. If you will do optimization later anyway, you can just leave parameters to be default

      Some Special Cases

      1. For some kernels/means, number of hyperparameters depends on the dimension of input data. You can either enter the dimension, which use default values:

        m = pyGPs.mean.Linear( D=x.shape[1] )

        or you can initialze with the exact hyperparameters, you should enter as a list, one element for each dimension

        m = pyGPs.mean.Linear( alpha_list=[0.2, 0.4, 0.3] )
        All these “hyp-dim-dependent” functions are:
        • pyGPs.mean.Linear
        • pyGPs.cov.RBFard
        • pyGPs.cov.LINard
        • pyGPs.cov.RQard

      2. For pyGPs.cov.RBFunit(), its signal variance is always 1 (because of unit magnitude). Therefore this function do not have a hyperparameter of “signal variance”.

      3. pyGPs.cov.Poly() has three parameters, where hyperparameters are:
        • c -> inhomogeneous offset
        • sigma -> signal deviation
        however,
        • d -> order of polynomial will be treated as normal parameter, i.e. will not be trained

      4. Explicitly set pyGPs.cov.Noise is not necessary, because noise are already added in likelihood.

      Composite Kernels & Meams

      Adding and muliplying Kernels(Means) is really simple:

      k = pyGPs.cov.Periodic() * pyGPs.cov.RBF()
k = 0.5*pyGPs.cov.LIN() + pyGPs.cov.Periodic()

      Scalar will also be treated as a hyperparameter. For example, k = s1 * k1 + s2 * k2, then the list of hyperparameters is hyp = [s1, k1.hyp, s2, k2.hyp]. Scalar is passed in logorithm domain such that it will always be positive during optimization.

      Beside + / * , there is also a power operator for mean functions:

      m = ( pyGPs.mean.One() + pyGPs.mean.Linear(alpha_list=[0.2]) )**2

      Precomputed Kernel Matrix

      In certain cases, you may have a precomputed kernel matrix, but its non-trivial to write down the exact formula of kernel functions. Then you can specify your kernel in the following way. A precomputed kernel also fits with other kernels. In other words, it can also be composited as the way other kernels functions do.

      k = pyGPs.cov.Pre(M1, M2)

      M1 and M2 are your precomputed kernel matrix,

      where,

      M1 is a matrix with shape number of training points plus 1 by number of test points
      • cross covariances matrix (train by test)
      • last row is self covariances (diagonal of test by test)
      M2 is a square matrix with number of training points for each dimension
      • training set covariance matrix (train by train)

      A precomputed kernel can also be composited with other kernels. You need to explictly add scalar for pyGPs.cov.Pre().

      k = 0.5*pyGPs.cov.Pre(M1, M2) + pyGPs.cov.RBF()

      Developing New Kernel & Mean Functions

      We also support the development of new kernel/mean classes, your customized kernel class need to follow the template as below:

      # Your kernel class needs to inherit base class Kernel,
# which is in the module of Core.cov
class MyKernel(Kernel):

  def __init__(self, hyp):
      '''
      Intialize hyperparameters for MyKernel.
      '''
      self.hyp = hyp

  def getCovMatrix(self,x=None,z=None,mode=None):
      '''
      Return the specific covariance matrix according to input mode

      :param x: training data
      :param z: test data
      :param str mode: 'self_test' return self covariance matrix of test data(test by 1).
                       'train' return training covariance matrix(train by train).
                       'cross' return cross covariance matrix between x and z(train by test)

      :return: the corresponding covariance matrix
      '''
      pass

  def getDerMatrix(self,x=None,z=None,mode=None,der=None):
      '''
      Compute derivatives wrt. hyperparameters according to input mode

      :param x: training data
      :param z: test data
      :param str mode: 'self_test' return self derivative matrix of test data(test by 1).
                       'train' return training derivative matrix(train by train).
                       'cross' return cross derivative matrix between x and z(train by test)
      :param int der: index of hyperparameter whose derivative to be computed

      :return: the corresponding derivative matrix
      '''
      pass

      and for customized mean class:

      # Your mean class needs to inherit base class Mean,
# which is in the module of Core.mean
class MyMean(Mean):

  def __init__(self, hyp):
      '''
      Intialize hyperparameters for MyMean.
      '''
      self.hyp = hyp

  def getMean(self, x=None):
      '''
      Get the mean vector.
      '''
      pass

  def getDerMatrix(self, x=None, der=None):
      '''
      Compute derivatives wrt. hyperparameters.

      :param x: training data
      :param int der: index of hyperparameter whose derivative to be computed

      :return: the corresponding derivative matrix
      '''
      pass

      You can test your customized mean/kernel function using our framework of unit test. Taking kernel test as an example, you can uncomment method test_cov_new in pyGPs.Testing.unit_test_cov.py to check the outputs of your kernel function.

      # Test your customized covariance function
def test_cov_new(self):
    k = myKernel()     # specify your covariance function
    self.checkCovariance(k)

      and testing mean function in pyGPs.Testing.unit_test_mean.py

      # Test your customized mean function
def test_mean_new(self):
    m = myMean         # specify your mean function
    self.checkMean(m)

      List of Kernels and Default Parameters

      class pyGPs.Core.cov.Const(log_sigma=0.0)[source]

      Constant kernel. hyp = [ log_sigma ]

      Parameters:log_sigma – signal deviation.
      class pyGPs.Core.cov.FITCOfKernel(cov, inducingInput)[source]

      Covariance function to be used together with the FITC approximation. The function allows for more than one output argument and does not respect the interface of a proper covariance function. Instead of outputing the full covariance, it returns cross-covariances between the inputs x, z and the inducing inputs xu as needed by infFITC

      class pyGPs.Core.cov.Kernel[source]

      This is a base class of Kernel functions there is no computation in this class, it just defines rules about a kernel class should have each covariance function will inherit it and implement its own behaviour

      fitc(inducingInput)[source]

      Covariance function to be used together with the FITC approximation. Setting FITC gp model will implicitly call this method.

      Returns:an instance of FITCOfKernel
      getCovMatrix(x=None, z=None, mode=None)[source]

      Return the specific covariance matrix according to input mode

      Parameters:
      • x – training data
      • z – test data
      • mode (str) – ‘self_test’ return self covariance matrix of test data(test by 1). ‘train’ return training covariance matrix(train by train). ‘cross’ return cross covariance matrix between x and z(train by test)
      Returns:

      the corresponding covariance matrix
      getDerMatrix(x=None, z=None, mode=None, der=None)[source]

      Compute derivatives wrt. hyperparameters according to input mode

      Parameters:
      • x – training data
      • z – test data
      • mode (str) – ‘self_test’ return self derivative matrix of test data(test by 1). ‘train’ return training derivative matrix(train by train). ‘cross’ return cross derivative matrix between x and z(train by test)
      • der (int) – index of hyperparameter whose derivative to be computed
      Returns:

      the corresponding derivative matrix
      class pyGPs.Core.cov.LINard(D=None, log_ell_list=None)[source]

      Linear covariance function with Automatic Relevance Detemination. hyp = log_ell_list

      Parameters:
      • D – dimension of training data. Set if you want default ell, which is 1 for each dimension.
      • log_ell_list – characteristic length scale for each dimension.
      class pyGPs.Core.cov.Linear(log_sigma=0.0)[source]

      Linear kernel. hyp = [ log_sigma ].

      Parameters:log_sigma – signal deviation.
      class pyGPs.Core.cov.Matern(log_ell=0.0, d=3, log_sigma=0.0)[source]

      Matern covariance function with nu = d/2 and isotropic distance measure. For d=1 the function is also known as the exponential covariance function or the Ornstein-Uhlenbeck covariance in 1d. d will be rounded to 1, 3, or 5. hyp = [ log_ell, log_sigma]

      Parameters:
      • d – d is 2 times nu. Can only be 1,3 or 5.
      • log_ell – characteristic length scale.
      • log_sigma – signal deviation.
      class pyGPs.Core.cov.Noise(log_sigma=0.0)[source]

      Independent covariance function, i.e “white noise”, with specified variance. Normally NOT used anymore since noise is now added in liklihood. hyp = [ log_sigma ]

      Parameters:log_sigma – signal deviation.
      class pyGPs.Core.cov.Periodic(log_ell=0.0, log_p=0.0, log_sigma=0.0)[source]

      Stationary kernel for a smooth periodic function. hyp = [ log_ell, log_p, log_sigma]

      Parameters:
      • log_p – period.
      • log_ell – characteristic length scale.
      • log_sigma – signal deviation.
      class pyGPs.Core.cov.PiecePoly(log_ell=0.0, v=2, log_sigma=0.0)[source]

      Piecewise polynomial kernel with compact support. hyp = [log_ell, log_sigma]

      Parameters:
      • log_ell – characteristic length scale.
      • log_sigma – signal deviation.
      • v – degree v will be rounded to 0,1,2,or 3. (not treated as hyperparameter, i.e. will not be trained).
      class pyGPs.Core.cov.Poly(log_c=0.0, d=2, log_sigma=0.0)[source]

      Polynomial covariance function. hyp = [ log_c, log_sigma ]

      Parameters:
      • log_c – inhomogeneous offset.
      • log_sigma – signal deviation.
      • d – degree of polynomial (not treated as hyperparameter, i.e. will not be trained).
      class pyGPs.Core.cov.Pre(M1, M2)[source]

      Precomputed kernel matrix. No hyperparameters and thus nothing will be optimised.

      Parameters:
      • M1 – cross covariances matrix(train+1 by test). last row is self covariances (diagonal of test by test)
      • M2 – training set covariance matrix (train by train)
      class pyGPs.Core.cov.ProductOfKernel(cov1, cov2)[source]

      Product of two kernel function.

      class pyGPs.Core.cov.RBF(log_ell=0.0, log_sigma=0.0)[source]

      Squared Exponential kernel with isotropic distance measure. hyp = [log_ell, log_sigma]

      Parameters:
      • log_ell – characteristic length scale.
      • log_sigma – signal deviation.
      class pyGPs.Core.cov.RBFard(D=None, log_ell_list=None, log_sigma=0.0)[source]

      Squared Exponential kernel with Automatic Relevance Determination. hyp = log_ell_list + [log_sigma]

      Parameters:
      • D – dimension of pattern. set if you want default ell, which is 1 for each dimension.
      • log_ell_list – characteristic length scale for each dimension.
      • log_sigma – signal deviation.
      class pyGPs.Core.cov.RBFunit(log_ell=0.0)[source]

      Squared Exponential kernel with isotropic distance measure with unit magnitude. i.e signal variance is always 1. hyp = [ log_ell ]

      Parameters:log_ell – characteristic length scale.
      class pyGPs.Core.cov.RQ(log_ell=0.0, log_sigma=0.0, log_alpha=0.0)[source]

      Rational Quadratic covariance function with isotropic distance measure. hyp = [ log_ell, log_sigma, log_alpha ]

      Parameters:
      • log_ell – characteristic length scale.
      • log_sigma – signal deviation.
      • log_alpha – shape parameter for the RQ covariance.
      class pyGPs.Core.cov.RQard(D=None, log_ell_list=None, log_sigma=0.0, log_alpha=0.0)[source]

      Rational Quadratic covariance function with Automatic Relevance Detemination (ARD) distance measure. hyp = log_ell_list + [ log_sigma, log_alpha ]

      Parameters:
      • D – dimension of pattern. set if you want default ell, which is 0.5 for each dimension.
      • log_ell_list – characteristic length scale for each dimension.
      • log_sigma – signal deviation.
      • log_alpha – shape parameter for the RQ covariance.
      class pyGPs.Core.cov.ScaleOfKernel(cov, scalar)[source]

      Scale of a kernel function.

      class pyGPs.Core.cov.SumOfKernel(cov1, cov2)[source]

      Sum of two kernel function.

      List of Means and Default Parameters

      class pyGPs.Core.mean.Const(c=5.0)[source]

      Constant mean function. hyp = [c]

      Parameters:c – constant value for mean
      class pyGPs.Core.mean.Linear(D=None, alpha_list=None)[source]

      Linear mean function. self.hyp = alpha_list

      Parameters:D – dimension of training data. Set if you want default alpha, which is 0.5 for each dimension.
      Alpha_list:scalar alpha for each dimension
      class pyGPs.Core.mean.Mean[source]

      The base function for mean function

      getDerMatrix(x=None, der=None)[source]

      Compute derivatives wrt. hyperparameters.

      Parameters:
      • x – training data
      • der (int) – index of hyperparameter whose derivative to be computed
      Returns:

      the corresponding derivative matrix
      getMean(x=None)[source]

      Get the mean vector.

      class pyGPs.Core.mean.One[source]

      One mean.

      class pyGPs.Core.mean.PowerOfMean(mean, d)[source]

      Power of a mean fucntion.

      class pyGPs.Core.mean.ProductOfMean(mean1, mean2)[source]

      Product of two mean fucntions.

      class pyGPs.Core.mean.ScaleOfMean(mean, scalar)[source]

      Scale of a mean function.

      class pyGPs.Core.mean.SumOfMean(mean1, mean2)[source]

      Sum of two mean functions.

      class pyGPs.Core.mean.Zero[source]

      Zero mean.

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      Likelihoods & Inference

      Changing Likelihood & Inference

      By default,
      • GPR uses Gaussian likelihood and exact inference.
      • GPC uses Error functionlikelihood and EP inference.
      • FITC model uses same default with corresponding FITC inference.
      • GPMC calls GPC and thus uses the default setting of GPC

      You can change to other likelihood or inference methods using:

      model.useLikelihood(newLik)
model.useInference(newInf)
      newLik and newInf are Strings. Currently the options are:
      1. Regression model
        • newLik: “Laplace”. Note this will force inference method to be EP.
        • newInf: “EP”, “Laplace”.
      2. Classification model (including GPMC)
        • newInf: “Laplace”

      Developing New Likelihood & Inference Functions

      We also support the development of new likelihood/inference classes, your customized inference class need to follow the template as below:

      # Your inference class needs to inherit base class Inference,
# which is in the module of Core.inf
class MyKernel(Kernel):

  def __init__(self):
      pass

  def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1):
      '''
      Inference computation based on inputs.
      post, nlZ, dnlZ = inf.proceed(mean, cov, lik, x, y)

      INPUT:
      cov: name of the covariance function (see covFunctions.m)
      lik: name of the likelihood function (see likFunctions.m)
      x: n by D matrix of training inputs
      y: 1d array (of size n) of targets

      OUTPUT:
      post(instance of postStruct): struct representation of the (approximate) posterior containing:
      nlZ: returned value of the negative log marginal likelihood
      dnlZ(instance of dnlZStruct): struct representation for derivatives of the negative log marginal likelihood
      w.r.t. each hyperparameter.

      Usually, the approximate posterior to be returned admits the form:
      N(m=K*alpha, V=inv(inv(K)+W)), where alpha is a vector and W is diagonal;
      if not, then L contains instead -inv(K+inv(W)), and sW is unused.

      For more information on the individual approximation methods and their
      implementations, see the respective inference function below. See also gp.py

      :param meanfunc: mean function
      :param covfunc: covariance function
      :param likfunc: likelihood function
      :param x: training data
      :param y: training labels
      :param nargout: specify the number of output(1,2 or 3)
      :return: posterior, negative-log-marginal-likelihood, derivative for negative-log-marginal-likelihood-likelihood
      '''
      pass

      where postStruct and dnlZStruct is also defined in Core.inf.

      class postStruct(object):
    '''
    Data structure for posterior

    post.alpha ->  1d array containing inv(K)*m,
                   where K is the prior covariance matrix and m the approx posterior mean
    post.sW:   ->  1d array containing diagonal of sqrt(W)
                   the approximate posterior covariance matrix is inv(inv(K)+W)
    post.L     ->  2d array, L = chol(sW*K*sW+identity(n))
    '''

class dnlZStruct(object):
    '''
    Data structure for the derivatives of mean, cov and lik functions.

    dnlZ.mean  ->  list of derivatives for each hyperparameters in mean function
    dnlZ.cov   ->  list of derivatives for each hyperparameters in covariance function
    dnlZ.lik   ->  list of derivatives for each hyperparameters in likelihood function
    '''

      Customizing likelihood function is more complicated. We will omit it here to keep this this page not too long. However, you can find detailed explaination either in the source code Core.lik or in coresponding section of manual.

      Just like testing kernel/mean fucntions, you can also find unit test module for likelihood and inference functions. To test your customized inference function, uncomment the following method in pyGPs.Testing.unit_test_inf.py.

      # Test your customized inference function
def test_inf_new(self):
    # specify your inf function
    # set mean/cov/lik functions
    post, nlZ, dnlZ = inffunc.proceed(meanfunc, covfunc, likfunc, self.x, self.y, nargout=3)
    self.checkFITCOutput(post, nlZ, dnlZ)

      and test customized likelihood function in pyGPs.Testing.unit_test_lik.py

      # Test your customized likelihood function
def test_cov_new(self):
    likelihood = myLikelihood()     # specify your likelihood function
    self.checkLikelihood(likelihood)

      List of Likelihoods

      class pyGPs.Core.lik.Erf[source]

      Error function or cumulative Gaussian likelihood function for binary classification or probit regression.

      Erf(t)=\frac{1}{2}(1+erf(\frac{t}{\sqrt{2}}))=normcdf(t)

      class pyGPs.Core.lik.Gauss(log_sigma=-2.3025850929940455)[source]

      Gaussian likelihood function for regression.

      Gauss(t)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(t-y)^2}{2\sigma^2}}, where y is the mean and \sigma is the standard deviation.

      hyp = [ log_sigma ]

      class pyGPs.Core.lik.Laplace(log_sigma=-2.3025850929940455)[source]

      Laplacian likelihood function for regression. ONLY works with EP inference!

      Laplace(t) = \frac{1}{2b}e^{-\frac{|t-y|}{b}} where b=\frac{\sigma}{\sqrt{2}}, y is the mean and \sigma is the standard deviation.

      hyp = [ log_sigma ]

      List of Inference

      class pyGPs.Core.inf.Exact[source]

      Exact inference for a GP with Gaussian likelihood. Compute a parametrization of the posterior, the negative log marginal likelihood and its derivatives w.r.t. the hyperparameters.

      class pyGPs.Core.inf.EP[source]

      Expectation Propagation approximation to the posterior Gaussian Process.

      class pyGPs.Core.inf.Laplace[source]

      Laplace’s Approximation to the posterior Gaussian process.

      class pyGPs.Core.inf.FITC_Exact[source]

      FITC approximation to the posterior Gaussian process. The function is equivalent to infExact with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku’ * inv(Quu) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu, snu2 = sn2/1e6 is the noise of the inducing inputs and Quu = Kuu + snu2*eye(nu).

      class pyGPs.Core.inf.FITC_EP[source]

      FITC-EP approximation to the posterior Gaussian process. The function is equivalent to infEP with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku’ * inv(Kuu + snu2 * eye(nu)) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu and snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard deviation of the inducing inputs snu to be a one per mil of the measurement noise’s standard deviation sn. In case of a likelihood without noise parameter sn2, we simply use snu2 = 1e-6. For details, see The Generalized FITC Approximation, Andrew Naish-Guzman and Sean Holden, NIPS, 2007.

      class pyGPs.Core.inf.FITC_Laplace[source]

      FITC-Laplace approximation to the posterior Gaussian process. The function is equivalent to infLaplace with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku’ * inv(Kuu + snu2 * eye(nu)) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu and snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard deviation of the inducing inputs snu to be a one per mil of the measurement noise’s standard deviation sn. In case of a likelihood without noise parameter sn2, we simply use snu2 = 1e-6.

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      pyGPs-1.2/doc/build/html/Opts.html000066400000000000000000000230531236346020200170420ustar00rootroot00000000000000 Optimizers — pyGPs v1.2 documentation

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      Optimizers

      Optimization Methods

      As you may have already seen in the demos, the optimizer is initialized in the following way:

      GP.setOptimizer(method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None)[source]

      This method is used to sepecify optimization configuration. By default, gp uses a single run “minimize”.

      Parameters:
      • method –

        Optimization methods. Possible values are:

        “Minimize” -> minimize by Carl Rasmussen (python implementation of “minimize” in GPML)

        “CG” -> conjugent gradient

        “BFGS” -> quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS)

        “SCG” -> scaled conjugent gradient (faster than CG)

      • num_restarts – Set if you want to run mulitiple times of optimization with different initial guess. It specifys the maximum number of runs/restarts/trials.
      • min_threshold – Set if you want to run mulitiple times of optimization with different initial guess. It specifys the threshold of objective function value. Stop optimization when this value is reached.
      • meanRange –

        The range of initial guess for mean hyperparameters. e.g. meanRange = [(-2,2), (-5,5), (0,1)]. Each tuple specifys the range (low, high) of this hyperparameter, This is only the range of initial guess, during optimization process, optimal hyperparameters may go out of this range.

        (-5,5) for each hyperparameter by default.

      • covRange – The range of initial guess for kernel hyperparameters. Usage see meanRange
      • likRange – The range of initial guess for likelihood hyperparameters. Usage see meanRange

      Developing Optimization Methods

      We also support the development of new optimizers.

      Your own optimizer should inherent base class Optimizer in pyGPs.Core.opt and follow the template as below:

      class MyOptimizer(Optimizer):
    def __init__(self, model=None, searchConfig = None):
        self.model = model

    def findMin(self, x, y):
        '''
        Find minimal value based on negative-log-marginal-likelihood.
        optimalHyp, funcValue = findMin(x, y)

        where funcValue is the minimal negative-log-marginal-likelihood during optimization,
        and optimalHyp is a flattened numpy array
        (in sequence of meanfunc.hyp, covfunc.hyp, likfunc.hyp)
        of the hyparameters to achieve such value.

        You can achieve advanced search strategy by initializing Optimizer with searchConfig,
        which is an instance of pyGPs.Optimization.conf.
        See more in pyGPs.Optimization.conf and pyGPs.Core.gp.GP.setOptimizer,
        as well as in online documentation of section Optimizers.
        '''
        pass

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      Semi-supervised Learning with Graphs

      The code shown in this tutorial can be executed by running pyGPs/Demo/demo_NodeKernel.py

      Import

      You may want to import some extensions we provide as follows:

      from pyGPs.GraphExtension import graphUtil, nodeKernels
from pyGPs.Validation import valid

      Load data

      We used the same dataset from GPMC example. i.e. The USPS digits dataset [1]. Each digit of 16*16 pixels is flattened into a 256 dimension vector. For the simplicity of demo, we only selected digits 1 s and 2 s such that we have a binary classification problem where digit 1 for class +1 and digit 2 for class -1. We also reduced the dataset into 100 samples per digit, where the original dataset consist of thousands of samples for each digit.

      Here are samples for two digits for 1

      _images/digit1_1.png _images/digit1_2.png

      and samples for two digits for 2

      _images/digit2_1.png _images/digit2_2.png

      Form a nearest neighbour graph

      We form a nearest-neighbor graph based on Euclidean distance of the vector representation of digits. Neighboring images have small Euclidean distance. Each digit is a node in the graph. There is an edge if digit i is the k-nearest neighbour of digit j. We form a symmetrized graph such that we connect nodes j, i if i is in j’s kNN and vice versa, and therefore a node can have more than k edges. You should import the corresponding module from pyGPs.GraphStuff

      x,y = load_binary(1,2,reduce=True)
A = graphUtil.formKnnGraph(x,2)

      A is the adjacency matrix of this 2-NN graph.

      Below shows an example of such symmetrized Euclidean 2-NN graph on some 1s and 2s taking from Xiaojin Zhu’s doctoral thesis [2].

      _images/2nnGraph.png

      Kernel on graph

      Several classical kernels on graph described in Structured Kernels can be built from adjacency matrix A. We use diffusion kernel for this example to get the precomputed kernel matrix.

      Matrix = nodeKernels.diffKernel(A)

      This a big square matrix with all rows and columns of the number of data points. By specifying the indice of training data and test data, we will form two matrix M1 and M2 with the exact format which pyGPs.Core.cov.Pre needed.

      M1,M2 = graphUtil.formKernelMatrix(Matrix, indice_train, indice_test)
      M1 is a matrix with shape number of training points plus 1 by number of test points
      • cross covariances matrix (train by test)
      • last row is self covariances (diagonal of test by test)
      M2 is a square matrix with number of training points for each dimension
      • training set covariance matrix (train by train)

      GP classification

      Every ingredients for a basic semi-supervised learning is prepared now. Lets see how to proceed for GP classification. First, the normal way with rbf kernel we have seen several times

      model = pyGPs.GPC()
k = pyGPs.cov.RBF()
model.setPrior(kernel=k)

      Then lets use our kernel precomputed matrix. If you only use precomputed kernel matrix, there is no training data. However you still need to specify x just to fit in the usage of pyGPs for generality reason. You can create any x as long as the dimension is correct.

      x = np.zeros((n,1))
k = pyGPs.cov.Pre(M1,M2) + pyGPs.cov.RBF()
model.setPrior(kernel=k)

      Moreover, you can composite a kernel for both precomputed matrix and regular kernel function if necessary.

      k = pyGPs.cov.Pre(M1,M2) + pyGPs.cov.RBFunit()
model.setPrior(kernel=k)

      The rest way of using pyGPs is exactly the same as the demo of GP classification.

      Result

      For our manually created graph data, an rbf kernel works better than a diffusion kernel on the graph (higher accuracy). The performance in general should depend on the application as well as features of data.

      The left image shows the digit that using diffusion kernel will predict the wrong result (should be 2), but rbf kernel does the job fine. The right image shows the digit that rbf kernel predicts the wrong class, diffusion kernel on the other hand, predicts correctly due to graph information! (should be 1).

      Interestingly, using a composite kernel with diffusion kernel on graph and an rbf kernel together. All test cases including the following are predicted correctly.

      _images/digitDiffwrong.png _images/digitRBFwrong.png
      [1]A Database for Handwritten Text Recognition Research, J. J. Hull, IEEE PAMI 16(5) 550-554, 1994.
      [2]Semi-Supervised Learning with Graphs, Xiaojin Zhu, CMU-LTI-05-192, 2005

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      Regression on UCI Housing data

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      pyGPs-1.2/doc/build/html/Theory.html000066400000000000000000000202231236346020200173630ustar00rootroot00000000000000 GPs & Functionality — pyGPs v1.2 documentation

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      GPs & Functionality

      Gaussian Processes (GPs) can conveniently be used for Bayesian supervised learning, such as regression and classification. In its simplest form, GP inference can be implemented in a few lines of code. However, in practice, things typically get a little more complicated: you might want to use expressive covariance and mean functions, learn good values for hyperparameters, use non-Gaussian likelihood functions (rendering exact inference intractable), use approximate inference algorithms, or combinations of many or all of the above.

      A comprehensive introduction to Gaussian Processes for Machine Learning is provided in the GPML book by Rasmussen and Williams, 2006.

      List of Functionality

      This table lists the functionality implemented in pyGPs.

      Functionality Kernel Mean Likelihood Inference Optimizer
      Simple Constant Constant Gaussian Exact Minimize
      Linear Linear Cumulative Gaussian (Erf) EP CG
      Linear ard One Laplace Laplace SCG
      Matérn (1,3,5) Zero     BFGS
      Periodic    
      Polynomial
      Piecewise Poly
      RBF iso
      RBF ard
      RBF unit
      RQ iso
      RQ ard
      Noise
      Composite Sum (+) Sum (+)      
      Product (*) Product (*)
      Scale (*) Scale (*)
        Power (**)
      Sparse GP       FITC Exact  
      FITC EP
      FITC Laplace
      Graphs Diffusion        
      VN Diffusion
      Pseudo Inv Laplace
      Regularized Laplace
      p-step Random Walk
      Inverse Cosine
      Propagation Kernel
      Other Customized Customized      
      Precomputed Kernel  

      pyGPs also provide cross-validation and some built-in evalation methods.

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      pyGPs-1.2/doc/build/html/_modules/pyGP_OO/000077500000000000000000000000001236346020200203075ustar00rootroot00000000000000pyGPs-1.2/doc/build/html/_modules/pyGP_OO/Core/000077500000000000000000000000001236346020200211775ustar00rootroot00000000000000pyGPs-1.2/doc/build/html/_modules/pyGP_OO/Core/cov.html000066400000000000000000004004751236346020200226660ustar00rootroot00000000000000 pyGP_OO.Core.cov — pyGPs 1.0.0 documentation

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      Source code for pyGP_OO.Core.cov

      #================================================================================
#    Marion Neumann [marion dot neumann at uni-bonn dot de]
#    Daniel Marthaler [marthaler at ge dot com]
#    Shan Huang [shan dot huang at iais dot fraunhofer dot de]
#    Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]
#
#    This file is part of pyGP_OO.
#    The software package is released under the BSD 2-Clause (FreeBSD) License.
#
#    Copyright (c) by
#    Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 30/09/2013
#================================================================================

# @author: Shan Huang (last update Sep.2013)
# This is a object-oriented python implementation of gpml functionality 
# (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18).
# based on the functional-version of python implementation
# (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013)
# 
#
# Copyright (c) by Marion Neumann and Shan Huang, 30/092013


import numpy as np
import math
import scipy.spatial.distance as spdist



      [docs]class Kernel(object):
    """
    This is a base class of Kernel functions
    there is no computation in this class, it just defines rules about a kernel class should have
    each covariance function will inherite it and implement its own behaviour
    """
    def __init__(self):
        self.hyp = []
        self.para = []
    def proceed(self):
        pass
    #overloading operators
    def __add__(self,cov):
        return SumOfKernel(self,cov)
    def __mul__(self,other):
        # using * for both scalar and production
        # depending on the types of two objects.
        if isinstance(other, int) or isinstance(other, float):
            return ScaleOfKernel(self,other)
        elif isinstance(other, Kernel):
            return ProductOfKernel(self,other)
        else:
            print "only numbers and Kernels are supported operand types for *"
    # FITC approximation
    def fitc(self,inducingInput):
        return FITCOfKernel(self,inducingInput)
    
    # not used anymore
    # replaced by spdist from scipy
    def sq_dist(self, a, b=None):
        # Compute a matrix of all pairwise squared distances
        # between two sets of vectors, stored in the row of the two matrices:
        # a (of size n by D) and b (of size m by D). 
        n = a.shape[0]
        D = a.shape[1]
        m = n    
        if b == None:
            b = a.transpose()
        else:
            m = b.shape[0]
            b = b.transpose()
        C = np.zeros((n,m))
        for d in range(0,D):
            tt  = a[:,d]
            tt  = tt.reshape(n,1)
            tem = np.kron(np.ones((1,m)), tt)
            tem = tem - np.kron(np.ones((n,1)), b[d,:])
            C   = C + tem * tem 
        return C

              
        
      
class ProductOfKernel(Kernel):
    def __init__(self,cov1,cov2):
        self.cov1 = cov1
        self.cov2 = cov2
        self._hyp = cov1.hyp + cov2.hyp

    def sethyp(self,hyp):
        assert len(hyp) == len(self._hyp)
        len1 = len(self.cov1.hyp)
        self._hyp = hyp 
        self.cov1.hyp = self._hyp[:len1]
        self.cov2.hyp = self._hyp[len1:]
    def gethyp(self):
        return self._hyp
    hyp = property(gethyp,sethyp)

    def proceed(self,x=None,z=None,der=None):
        n, D = x.shape
        AT = self.cov1.proceed(x,z)
        A = np.ones_like(AT)   
        if der == None:                          # compute cov vector
            A *= self.cov1.proceed(x,z)
            A *= self.cov2.proceed(x,z)
        elif isinstance(der, int):               # compute derivative vector  
            if der < len(self.cov1.hyp):
                A *= self.cov1.proceed(x, z, der)
                A *= self.cov2.proceed(x, z)
            elif der < len(self.hyp):
                der2 = der - len(self.cov1.hyp)
                A *= self.cov2.proceed(x, z, der2)
                A *= self.cov1.proceed(x,z) 
            else:
                raise Exception("Error: der out of range for covProduct")            
        return A
        

class SumOfKernel(Kernel):
    def __init__(self,cov1,cov2):
        self.cov1 = cov1
        self.cov2 = cov2
        self._hyp = cov1.hyp + cov2.hyp
    def sethyp(self,hyp):
        assert len(hyp) == len(self._hyp)
        len1 = len(self.cov1.hyp)
        self._hyp = hyp 
        self.cov1.hyp = self._hyp[:len1]
        self.cov2.hyp = self._hyp[len1:]
    def gethyp(self):
        return self._hyp
    hyp = property(gethyp,sethyp)

    def proceed(self,x=None,z=None,der=None):
        n, D = x.shape
        AT = self.cov1.proceed(x,z)
        A = np.zeros_like(AT)                   # Allocate covariance Matrix
        if der == None:                         # compute cov vector
            A += self.cov1.proceed(x,z)
            A += self.cov2.proceed(x,z)
        elif isinstance(der, int):              # compute derivative vector  
            if der < len(self.cov1.hyp):
                A += self.cov1.proceed(x, z, der)
            elif der < len(self.hyp):
                der2 = der - len(self.cov1.hyp)
                A += self.cov2.proceed(x, z, der2)
            else:
                raise Exception("Error: der out of range for covSum")            
        return A
    

class ScaleOfKernel(Kernel):
# Compose a covariance function as a scaled version of another one
# k(x^p,x^q) = sf2 * k0(x^p,x^q)
#
# The hyperparameter is :
# hyp = [ log(sf2) ]
    def __init__(self,cov,scalar):
        self.cov = cov
        if cov.hyp:
            self._hyp = [scalar] + cov.hyp 
        else:
            self._hyp = [scalar]
    def sethyp(self,hyp):
        assert len(hyp) == len(self._hyp)
        self._hyp = hyp 
        self.cov.hyp = self._hyp[1:]
    def gethyp(self):
        return self._hyp
    hyp = property(gethyp,sethyp)

    def proceed(self,x=None,z=None,der=None):
        sf2 = np.exp(2.* self.hyp[0])                 # scale parameter   
        if der == None:                               # compute cov vector
            A = sf2 * self.cov.proceed(x,z)           # accumulate cov
        elif isinstance(der, int) and der == 0:       # compute derivative w.r.t. sf2
            A = 2. * sf2 * self.cov.proceed(x,z)
        else:                                 
            A = sf2 * self.cov.proceed(x, z, der-1) 
        return A
     

class FITCOfKernel(Kernel):
    def __init__(self,cov,inducingInput):
        self.inducingInput = inducingInput
        self.covfunc = cov
        self._hyp = cov.hyp

    def getHyp(self):
        return self._hyp
    def setHyp(self, hyp):
        self._hyp = hyp
        self.covfunc.hyp = hyp
    hyp = property(getHyp,setHyp)

    def proceed(self,x=None,z=None,der=None):
    # Covariance function to be used together with the FITC approximation.
    # The function allows for more than one output argument and does not respect the
    # interface of a proper covariance function. 
    # Instead of outputing the full covariance, it returns cross-covariances between
    # the inputs x, z and the inducing inputs xu as needed by infFITC
        xu = self.inducingInput
        try:
            assert(xu.shape[1] == x.shape[1])
        except AssertionError:
            raise Exception('Dimensionality of inducing inputs must match training inputs')        
        if der == None:                        # compute covariance matrices for dataset x
            if z == None:
                K   = self.covfunc.proceed(x,'diag')
                Kuu = self.covfunc.proceed(xu)
                Ku  = self.covfunc.proceed(xu,x)
            elif z == 'diag':
                K = self.covfunc.proceed(x,z)
                return K
            else:
                K = self.covfunc.proceed(xu,z)
                return K
        else:                                  # compute derivative matrices
            if z == None:
                K   = self.covfunc.proceed(x,'diag',der)
                Kuu = self.covfunc.proceed(xu,None,der)
                Ku  = self.covfunc.proceed(xu,x,der)
            elif z == 'diag':
                K = self.covfunc.proceed(x,z,der)
                return K
            else:
                K = self.covfunc.proceed(xu,z,der)
                return K
        return K, Kuu, Ku



# an example of simple kernel function, the rest are similar
class Poly(Kernel):
    def __init__(self, log_c=0., log_d=np.log(2), log_sigma=0. ):
        self.hyp = [ log_c, log_sigma]
        self.para =  [log_d] 
        '''
        else:
            print "Polynomial covariance function is parameterized as:"
            print "k(x^p,x^q) = sf2 * ( c +  (x^p)'*(x^q) ) ** d"
            print "The number of parameters is 3:"
            print "hyp = [ log(c), log(sqrt(sf2)), log(d) ]"
            print "where,"
            print "      log(c), log(sqrt(sf2))  is hyperparameters"
            print "      log(d) is a special parameter which will not be optimized"
            print "------------------------------------------------------------------"
            raise Exception("Wrong number of hyperparameters.")
        '''
    def proceed(self, x=None, z=None, der=None):
        c   = np.exp(self.hyp[0])             # inhomogeneous offset
        sf2 = np.exp(2.*self.hyp[1])          # signal variance
        ord = np.exp(self.para[0])            # order of polynomial
        if np.abs(ord-np.round(ord)) < 1e-8:  # remove numerical error from format of parameter
            ord = int(round(ord))
        assert(ord >= 1.)                     # only nonzero integers for ord
        ord = int(ord)       
        n,D = x.shape
        if z == 'diag':
            A = np.reshape(np.sum(x*x,1), (n,1))
        elif z == None:
            A = np.dot(x,x.T)
        else:                              # compute covariance between data sets x and z
            A = np.dot(x,z.T)              # cross covariances
    
        if der == None:                    # compute covariance matix for dataset x
            A = sf2 * (c + A)**ord
        else:
            # compute derivative matrix wrt 1st parameter
            if der == 0:                   
                A = c * ord * sf2 * (c+A)**(ord-1)
            # compute derivative matrix wrt 2nd parameter
            elif der == 1:  
                A = 2. * sf2 * (c + A)**ord
            # Wants to compute derivative wrt order
            elif der == 2:  
                A = np.zeros_like(A)
            else:
                raise Exception("Wrong derivative entry in covPoly")
        return A
   

class RBF(Kernel):
    def __init__(self, log_ell=-1., log_sigma=0.):
        self.hyp = [log_ell, log_sigma]
        '''
            print "Squared Exponential covariance function with isotropic distance measure is parameterized as:"
            print "k(x^p,x^q) = sf2 * exp(-(x^p - x^q)'*inv(P)*(x^p - x^q)/2)"
            print "where the P matrix is ell^2 times the unit matrix and"
            print "sf2 is the signal variance"
            print "The number of hyperparameters is 2:"
            print "hyp = [ log(ell), log(sqrt(sf2)) ]"
            print "------------------------------------------------------------------"
            raise Exception("Wrong number of hyperparameters.")
        '''
    def proceed(self, x=None, z=None, der=None):

        ell = np.exp(self.hyp[0])         # characteristic length scale
        sf2 = np.exp(2.*self.hyp[1])      # signal variance
        n,D = x.shape
        if z == 'diag':
            A = np.zeros((n,1))
        elif z == None:
            A = spdist.cdist(x/ell,x/ell,'sqeuclidean')
        else:                              # compute covariance between data sets x and z
            A = spdist.cdist(x/ell,z/ell,'sqeuclidean') # self covariances      
        if der == None:                    # compute covariance matix for dataset x
            A = sf2 * np.exp(-0.5*A)
        else:
            if der == 0:    # compute derivative matrix wrt 1st parameter
                A = sf2 * np.exp(-0.5*A) * A
            elif der == 1:  # compute derivative matrix wrt 2nd parameter
                A = 2. * sf2 * np.exp(-0.5*A)
            else:
                raise Exception("Calling for a derivative in covSEiso that does not exist")
        return A


class RBFunit(Kernel):
    def __init__(self, log_ell=-1.):
        self.hyp = [log_ell]
        '''
            print "Squared Exponential covariance function with isotropic distance measure with"
            print "unit magnitude. The covariance function is parameterized as:"
            print "k(x^p,x^q) = exp( -(x^p - x^q)' * inv(P) * (x^p - x^q) / 2 )"
            print "where the P matrix is ell^2 times the unit matrix"
            print "The number of hyperparameters is 1:"
            print "hyp = [ log(ell) ]"
            print "------------------------------------------------------------------"
            raise Exception("Wrong number of hyperparameters.")
        '''
    def proceed(self, x=None, z=None, der=None):
        ell = np.exp(self.hyp[0])  # characteristic length scale
        n,D = x.shape
        if z == 'diag':
            A = np.zeros((n,1))
        elif z == None:
            A = spdist.cdist(x/ell, x/ell, 'sqeuclidean')
        else:                      # compute covariance between data sets x and z
            A = spdist.cdist(x/ell, z/ell, 'sqeuclidean')   # self covariances (needed for GPR)
        if der == None:            # compute covariance matix for dataset x
            A = np.exp(-0.5*A)
        else:
            if der == 0:           # compute derivative matrix wrt 1st parameter
                A = np.exp(-0.5*A) * A
            else:
                raise Exception("Wrong derivative index in covSEisoU")
        return A


class RBFard(Kernel):
    def __init__(self, D=None, log_ell_list=None, log_sigma=0.):
        if log_ell_list == None:
            self.hyp = [0.5 for i in xrange(D)] + [log_sigma]
        else:
            self.hyp = log_ell_list + [log_sigma]
    def proceed(self, x=None, z=None, der=None):
        n, D = x.shape
        '''
        if len(self.hyp) != D+1: 
            print "Squared Exponential covariance function with ARD is parameterized as:"
            print "k(x^p,x^q) = sf2 * exp(-(x^p - x^q)'*inv(P)*(x^p - x^q)/2)"
            print "where the P matrix is diagonal with ARD parameters ell_1^2,...,ell_D^2, where"
            print "D is the dimension of the input space and sf2 is the signal variance."
            print "The number of hyperparameters is %d (dimension of inputs +1):" % (D+1)
            print "hyp = [log(ell_1), log(ell_2), ..., log(ell_D), log(sqrt(sf2))]"
            print "------------------------------------------------------------------"
            raise Exception("Wrong number of hyperparameters.")
        else:
        '''    
        ell = 1./np.exp(self.hyp[0:D])    # characteristic length scale
        sf2 = np.exp(2.*self.hyp[D])      # signal variance
        if z == 'diag':
            A = np.zeros((n,1))
        elif z == None:
            tem = np.dot(np.diag(ell),x.T).T
            A = spdist.cdist(tem,tem,'sqeuclidean')
        else:                # compute covariance between data sets x and z
            A = spdist.cdist(np.dot(np.diag(ell),x.T).T,np.dot(np.diag(ell),z.T).T,'sqeuclidean')
        A = sf2*np.exp(-0.5*A)
        if der:
            if der < D:      # compute derivative matrix wrt length scale parameters
                if z == 'diag':
                    A = A*0
                elif z == None:
                    tem = np.atleast_2d(x[:,der])/ell[der]
                    A *= spdist.cdist(tem,tem,'sqeuclidean')
                else:
                    A *= spdist.cdist(np.atleast_2d(x[:,der]).T/ell[der],np.atleast_2d(z[:,der]).T/ell[der],'sqeuclidean')
            elif der==D:     # compute derivative matrix wrt magnitude parameter
                A = 2.*A
            else:
                raise Exception("Wrong derivative index in covSEard")   
        return A

            
class PPiso(Kernel):
    pass


class Const(Kernel):
    def __init__(self, log_sigma=0.):
        self.hyp = [log_sigma]
        '''
        if len(hyp) == 1:
            self.hyp = hyp
        else:
            print "Constant covariance function is parameterized as:"
            print "k(x^p,x^q) = sf2 "
            print "The number of hyperparameters is 1" 
            print "hyp = [ log(sqrt(sf2)) ]"
            print "------------------------------------------------------------------"
            raise Exception("Wrong number of hyperparameters.")
        '''
    def proceed(self, x=None, z=None, der=None):
        sf2 = np.exp(self.hyp[0])         # s2
        n,m = x.shape
        if z == 'diag':
            A = sf2*np.ones((n,1))
        elif z == None:
            A = sf2 * np.ones((n,n))
        else:
            A = sf2*np.ones((n,z.shape[0]))
        if der == 0:  # compute derivative matrix wrt sf2
            A = 2. * A
        elif der:
            raise Exception("Wrong derivative entry in covConst")
        return A


class LIN(Kernel):
# Linear covariance function is parameterized as:
# k(x^p,x^q) = x^p'*x^q
# There are no hyperparameters
#
# Note that there is no bias or scale term
# use covConst and * to add these
    def __init__(self):
        self.hyp = []
    def proceed(self, x=None, z=None, der=None):
        n,m = x.shape
        if z == 'diag':
            A = np.reshape(np.sum(x*x,1), (n,1))
        elif z == None:
            A = np.dot(x,x.T) + np.eye(n)*1e-16     # required for numerical accuracy
        else:                                       # compute covariance between data sets x and z
            A = np.dot(x,z.T)                       # cross covariances
        if der:
            raise Exception("No derivative available in covLIN")
        return A


class LINard(Kernel):
    def __init__(self, D=None, log_ell_list=None):
        if log_ell_list == None:
            self.hyp = [0.5 for i in xrange(D)]
        else:
            self.hyp = log_ell_list

    def proceed(self, x=None, z=None, der=None):
        n, D = x.shape
        '''
        if len(self.hyp) != D: 
            print "Linear covariance function with Automatic Relevance Detemination"
            print "(ARD) distance measure is parameterized as:"
            print "k(x^p,x^q) = x^p' * inv(P) * x^q"
            print "where the P matrix is diagonal with ARD parameters ell_1^2,...,ell_D^2, where"
            print "D is the dimension of the input space and sf2 is the signal variance."
            print "The number of hyperparameters is %d (dimension of inputs):" % D
            print "hyp = [log(ell_1), log(ell_2), ..., log(ell_D) ]"
            print "------------------------------------------------------------------"
            raise Exception("Wrong number of hyperparameters.")
        else:
        '''
        ell = np.exp(self.hyp)            # ARD parameters
        x = np.dot(x,np.diag(1./ell))
        if z == 'diag':
            A = np.reshape(np.sum(x*x,1), (n,1))
        elif z == None:
            A = np.dot(x,x.T)
        else:                             # compute covariance between data sets x and z
            z = np.dot(z,np.diag(1./ell))
            A = np.dot(x,z.T)             # cross covariances

        if not der == None and der < D:
            if z == 'diag':
                A = -2.*x[:,der]*x[:,der]
            elif z == None:
                A = -2.*np.dot(x[:,der],x[:,der].T)
            else:
                A = -2.*np.dot(x[:,der],z[:,der].T) # cross covariances
        elif der:
            raise Exception("Wrong derivative index in covLINard")
        return A


class Matern(Kernel):
    def __init__(self, log_ell=-1., log_d=0., log_sigma=0. ):
        self.hyp = [ log_ell, log_sigma, log_d ]
        '''
        if len(hyp) == 3:
            self.hyp = hyp
        else:
            print "Matern covariance function with nu = d/2 and isotropic distance measure."
            print "For d=1 the function is also known as the exponential covariance function or the "
            print "Ornstein-Uhlenbeck covariance in 1d. The covariance function is:"
            print "k(x^p,x^q) = s2f * f( sqrt(d)*r ) * exp(-sqrt(d)*r)"
            print "with f(t)=1 for d=1, f(t)=1+t for d=3 and f(t)=1+t+(t*t)/3 for d=5."
            print "Here, r is the distance sqrt( (x^p-x^q)'*inv(P)*(x^p-x^q)), "
            print "where P is ell times the unit matrix and sf2 is the signal variance."
            print "The number of hyperparameters is 3" 
            print "hyp = [ log(ell), log(sqrt(sf2)), log(d) ]"
            print "------------------------------------------------------------------"
            raise Exception("Wrong number of hyperparameters.")
        '''
    def func(self,d,t):
        if d == 1:
            return 1
        elif d == 3:
            return 1 + t
        elif d == 5:
            return 1 + t*(1+t/3.)
        else:
            raise Exception("Wrong value for d in covMatern")
    def dfunc(self,d,t):
        if d == 1:
            return 1
        elif d == 3:
            return t
        elif d == 5:
            return t*(1+t/3.)
        else:
            raise Exception("Wrong value for d in covMatern")
    def mfunc(self,d,t):
        return self.func(d,t)*np.exp(-1.*t)
    def dmfunc(self,d,t):
        return self.dfunc(d,t)*t*np.exp(-1.*t)
    def proceed(self, x=None, z=None, der=None):
        ell = np.exp(self.hyp[0])        # characteristic length scale
        sf2 = np.exp(2.* self.hyp[1])    # signal variance
        d   = np.exp(self.hyp[2])        # 2 times nu
        if np.abs(d-np.round(d)) < 1e-8: # remove numerical error from format of parameter
            d = int(round(d))
        d = int(d)
        try:
            assert(d in [1,3,5])         # Check for valid values of d
        except AssertionError:
            d = 3

        if z == 'diag':
            A = np.zeros((x.shape[0],1))
        elif z == None:
            x = np.sqrt(d)*x/ell   
            A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean'))
        else:
            x = np.sqrt(d)*x/ell
            z = np.sqrt(d)*z/ell
            A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean'))

        if der == None:                     # compute covariance matix for dataset x
            A = sf2 * self.mfunc(d,A)
        else:
            if der == 0:                    # compute derivative matrix wrt 1st parameter
                A = sf2 * self.dmfunc(d,A)
            elif der == 1:                  # compute derivative matrix wrt 2nd parameter
                A = 2 * sf2 * self.mfunc(d,A)
            elif der == 2:                  # Wants to compute derivative wrt nu
                A = np.zeros_like(A)        # Do nothing
            else:
                raise Exception("Wrong derivative value in covMatern")
        return A



class Periodic(Kernel):
    def __init__(self, log_ell=-1, log_p=0., log_sigma=0. ): 
        self.hyp = [ log_ell, log_p, log_sigma]
        '''
        if len(hyp) == 3:
            self.hyp = hyp
        else:
            print "Stationary covariance function for a smooth periodic function,"
            print "with period p:"
            print "k(x^p,x^q) = sf2 * exp( -2*sin^2( pi*||x^p - x^q)||/p )/ell**2 )"
            print "The number of hyperparameters is 3" 
            print "hyp = [ log(ell), log(p), log(sqrt(sf2)) ]"
            print "------------------------------------------------------------------"
            raise Exception("Wrong number of hyperparameters.")
        '''
    def proceed(self, x=None, z=None, der=None):
        ell = np.exp(self.hyp[0])        # characteristic length scale
        p   = np.exp(self.hyp[1])        # period
        sf2 = np.exp(2.*self.hyp[2])     # signal variance
        n,D = x.shape
        if z == 'diag':
            A = np.zeros((n,1))
        elif z == None:
            A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean'))
        else:
            A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean'))
        A = np.pi*A/p
        if der == None:             # compute covariance matix for dataset x
            A = np.sin(A)/ell
            A = A * A
            A = sf2 *np.exp(-2.*A)
        else:
            if der == 0:            # compute derivative matrix wrt 1st parameter
                A = np.sin(A)/ell
                A = A * A
                A = 4. *sf2 *np.exp(-2.*A) * A
            elif der == 1:          # compute derivative matrix wrt 2nd parameter
                R = np.sin(A)/ell
                A = 4 * sf2/ell * np.exp(-2.*R*R)*R*np.cos(A)*A
            elif der == 2:          # compute derivative matrix wrt 3rd parameter
                A = np.sin(A)/ell
                A = A * A
                A = 2. * sf2 * np.exp(-2.*A)
            else:
                raise Exception("Wrong derivative index in covPeriodic")            
        return A


class covNoise(Kernel):
# Independent covariance function, ie "white noise", with specified variance.
    def __init__(self, log_sigma=0.):
        self.hyp = [log_sigma]
        '''
        if len(hyp) == 1:
            self.hyp = hyp
        else:
            print "Noise covariance function is specified as:,"
            print "k(x^p,x^q) = s2 * \delta(p,q)"
            print "where s2 is the noise variance and \delta(p,q) is a Kronecker delta function"
            print "which is 1 iff p=q and zero otherwise."
            print "The number of hyperparameters is 1" 
            print "hyp = [ log(sqrt(s2)) ]"
            print "------------------------------------------------------------------"
            raise Exception("Wrong number of hyperparameters.")
        '''
    def proceed(self, x=None, z=None, der=None):
        tol = 1.e-9                 # Tolerance for declaring two vectors "equal"
        s2 = np.exp(2.*self.hyp[0]) # noise variance
        n,D = x.shape
        if z == 'diag':
            A = np.ones((n,1))
        elif z == None:
            A = np.eye(n)
        else:                       # compute covariance between data sets x and z
            M = spdist.cdist(x, z, 'sqeuclidean')
            A = np.zeros_like(M,dtype=np.float)
            A[M < tol] = 1.
        if der == None:
            A = s2*A
        else:                       # compute derivative matrix
            if der == 0:
                A = 2.*s2*A
            else:
                raise Exception("Wrong derivative index in covNoise")
        return A


class RQiso(Kernel):
    pass
class RQard(Kernel):
    pass


class Pre(Kernel):
# Precomputed kernel matrix
# M1: cross covariances train by test
#     last row: self covariances (diagonal of test by test)
# M2: training set covariance matrix (train by train)   
#-> no hyperparameters have to be optimised. 
    def __init__(self,M1,M2):
        self.M1 = M1
        self.M2 = M2
        self.hyp = []
    def proceed(self, x=None, z=None, der=None):
        if z == 'diag':             # diagonal covariance between test_test
            A = self.M1[-1,:]            # self covariances for the test cases (last row) 
            A = np.reshape(A, (A.shape[0],1))
        elif z == None:             # covariance between train_train
            A = self.M2
        else:                       # covariance between train_test
            A = self.M1[:-1,:]
        if der != None:
            raise Exception("Error: NO optimization in precomputed kernel matrix")
        return A

    

    
    

    
# You can make some test codes below:
    
# UPDATE USAGE:
# here I decided to get rid of the data structure hyp = np.array([1,2,3]), 
# a simple list hyp = [1,2,3] might be enough

if __name__ == '__main__':
    # test1: combinations of kernel functions
    k1 = covPoly([1,2,3])
    k2 = covPoly([2,2,2])
    k3 = covPoly([4,4,4])
    myCov = k1*k2*6 + k1*k2
    print myCov.hyp

    #########################################

    # test2: Does proceed() perform correctly compare to feval()?
    n = 20 # number of labeled/training data
    D = 1  # Dimension of input data
    x = np.array([2.083970427750732,  -0.821018066101379,  -0.617870699182597,  -1.183822608860694,\
              0.274087442277144,   0.599441729295593,   1.768897919204435,  -0.465645549031928,\
              0.588852784375935,  -0.832982214438054,  -0.512106527960363,   0.277883144210116,\
              -0.065870426922211,  -0.821412363806325,   0.185399443778088,  -0.858296174995998,\
               0.370786630037059,  -1.409869162416639,-0.144668412325022,-0.553299615220374]);
    x = np.reshape(x,(n,D))
    z = np.array([np.linspace(-1.9,1.9,101)]).T
    
    k = covSEard([0.1,0.2])
    print myCov.proceed(x,z,1)

    # have the same result if passing same inputs to feval() 
    # covPoly tested!

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      © Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting. Created using Sphinx 1.2.
      pyGPs-1.2/doc/build/html/_modules/pyGP_OO/Core/gp.html000066400000000000000000004413231236346020200225020ustar00rootroot00000000000000 pyGP_OO.Core.gp — pyGPs 1.0.0 documentation

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      Source code for pyGP_OO.Core.gp

      #================================================================================
#    Marion Neumann [marion dot neumann at uni-bonn dot de]
#    Daniel Marthaler [marthaler at ge dot com]
#    Shan Huang [shan dot huang at iais dot fraunhofer dot de]
#    Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]
#
#    This file is part of pyGP_OO.
#    The software package is released under the BSD 2-Clause (FreeBSD) License.
#
#    Copyright (c) by
#    Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 30/09/2013
#================================================================================
import itertools
import numpy as np
import matplotlib.pyplot as plt
import inf, mean, lik, cov, opt
from tools import unique
from copy import deepcopy
import pyGP_OO

#   MEANING OF NOTATION:
#  
#   inffunc      function specifying the inference method 
#   covfunc      prior covariance function (see below)
#   meanfunc     prior mean function
#   likfunc      likelihood function
#   x            n by D matrix of training inputs
#   y            column vector of length n of training targets
#   xs           ns by D matrix of test inputs
#   ys           column vector of length nn of test targets
#   der          flag for dnlZ computation determination (when xs == None also)
#
#   nlZ          returned value of the negative log marginal likelihood
#   dnlZ         column vector of partial derivatives of the negative
#                    log marginal likelihood w.r.t. each hyperparameter
#   ymu          column vector (of length ns) of predictive output means
#   ys2          column vector (of length ns) of predictive output variances
#   fmu          column vector (of length ns) of predictive latent means
#   fs2          column vector (of length ns) of predictive latent variances
#   lp           column vector (of length ns) of log predictive probabilities
#
#   post         struct representation of the (approximate) posterior
#                post is consist of post.alpha, post.L, post.sW
#
# @author: Shan Huang (last update Sep.2013)
# This is a object-oriented python implementation of gpml functionality 
# (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18).
# based on the functional-version of python implementation
# (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013)
# 
#
# Copyright (c) by Marion Neumann and Shan Huang, Sep.2013



SHADEDCOLOR = [0.75,0.98,0.75,1.0]

class GP(object):
    """Base class for GP model"""
    def __init__(self):
        super(GP, self).__init__()
        self.meanfunc = None
        self.covfunc = None
        self.likfunc = None
        self.inffunc = None
        self.optimizer = None
        self._neg_log_marginal_likelihood_ = None
        self._neg_log_marginal_likelihood_gradient_ = None  
        self._posterior_ = None  
        self.x = None
        self.y = None
        self.xs = None
        self.ys = None
        self.ym = None 
        self.ys2 = None
        self.lp = None

    def setData(self, x, y):
        self.x = x
        self.y = y

    def plotData_1d(self, axisvals=[-1.9, 1.9, -0.9, 3.9]):
        plt.figure()
        plt.plot(self.x, self.y,' b+', markersize=12)
        plt.axis(axisvals)
        plt.grid()
        plt.xlabel('input x')
        plt.ylabel('target y')
        plt.show()
    
    def plotData_2d(self,x1,x2,t1,t2,p1,p2,axisvals=[-4, 4, -4, 4]):
        fig = plt.figure()
        plt.plot(x1[:,0], x1[:,1], 'b+', markersize = 12)
        plt.plot(x2[:,0], x2[:,1], 'r+', markersize = 12)
        pc = plt.contour(t1, t2, np.reshape(p2/(p1+p2), (t1.shape[0],t1.shape[1]) ))
        fig.colorbar(pc)
        plt.grid()
        plt.axis(axisvals)
        plt.show()

    def setPrior(self, mean=None, kernel=None):
        """set prior mean and cov"""
        if mean != None:
            self.meanfunc = mean
        if kernel != None:
            self.covfunc = kernel
    

      [docs]    def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None):
        '''
        :param method: Optimizatin methods. Possible values are:
        "Minimize"   # minimize by Carl Rasmussen
        "CG"         # conjugent gradient
        "BFGS"       # quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS)
        "SCG"        # scaled conjugent gradient (faster than CG) 
        '''
        pass

      
    def train(self, x=None, y=None):
        '''
        train optimal hyperparameters 
        adjust to all mean/cov/lik functions
        '''
        if x != None:
            self.x = x  
        if y != None:
            self.y = y

        # optimize 
        optimalHyp, optimalNlZ = self.optimizer.findMin(self.x, self.y)
        self._neg_log_marginal_likelihood_ = optimalNlZ

        # apply optimal hyp to all mean/cov/lik functions here
        self.optimizer.apply_in_objects(optimalHyp)
        self.fit()

    def fit(self,x=None, y=None,der=True):
        '''
        fit the training data
        @return  [nlZ, post]        if der = False
        @return  [nlZ, dnlZ, post]  if der = True (default)
            
            where nlZ  is the negative log marginal likelihood
                  dnlZ is partial derivatives of nlZ w.r.t. each hyperparameter
                  post is struct representation of the (approximate) posterior
                  post is consist of post.alpha, post.L, post.sW
        '''
        if x != None:
            self.x = x  
        if y != None:
            self.y = y
        # call inference method
        if isinstance(self.likfunc, lik.Erf):  #or likLogistic)
            uy = unique(self.y)        
            ind = ( uy != 1 )
            if any( uy[ind] != -1):
                raise Exception('You attempt classification using labels different from {+1,-1}')
        if not der:
            post, nlZ = self.inffunc.proceed(self.meanfunc, self.covfunc, self.likfunc, self.x, self.y, 2)
            self._neg_log_marginal_likelihood_ = nlZ
            self._posterior_ = deepcopy(post)
            return nlZ, post          
        else:
            post, nlZ, dnlZ = self.inffunc.proceed(self.meanfunc, self.covfunc, self.likfunc, self.x, self.y, 3) 
            self._neg_log_marginal_likelihood_ = nlZ 
            self._neg_log_marginal_likelihood_gradient_ = deepcopy(dnlZ)
            self._posterior_ = deepcopy(post)
            return nlZ, dnlZ, post    

    def predict(self, xs, ys=None):
        '''
        prediction according to given inputs 

        @param xs           test input
        @param ys           test target(optional)

        @return ymu, ys2, fmu, fs2, lp
                where ymu is predictive output means
                      ys2 is predictive output variances
                      fm2 is predictive latent means
                      fs2 is predictive latent variances
                      lp  is log predictive probabilities(if ys is given, otherwise is None)
        '''
        meanfunc = self.meanfunc
        covfunc = self.covfunc
        likfunc = self.likfunc
        inffunc = self.inffunc
        x = self.x
        y = self.y
        self.xs = xs
        self.ys = ys
        
        if self._posterior_ == None:   
            self.fit()        
        alpha = self._posterior_.alpha
        L     = self._posterior_.L
        sW    = self._posterior_.sW
        
        nz = range(len(alpha[:,0]))         # non-sparse representation 
        if L == []:                         # in case L is not provided, we compute it
            K = covfunc.proceed(x[nz,:])
            L = np.linalg.cholesky( (np.eye(nz) + np.dot(sW,sW.T)*K).T )
        Ltril     = np.all( np.tril(L,-1) == 0 ) # is L an upper triangular matrix?
        ns        = xs.shape[0]                  # number of data points
        nperbatch = 1000                         # number of data points per mini batch
        nact      = 0                            # number of already processed test data points
        ymu = np.zeros((ns,1))
        ys2 = np.zeros((ns,1))
        fmu = np.zeros((ns,1))
        fs2 = np.zeros((ns,1))
        lp  = np.zeros((ns,1))
        while nact<=ns-1:                              # process minibatches of test cases to save memory
            id  = range(nact,min(nact+nperbatch,ns))   # data points to process
            kss = covfunc.proceed(xs[id,:], 'diag')    # self-variances
            Ks  = covfunc.proceed(x[nz,:], xs[id,:])   # cross-covariances
            ms  = meanfunc.proceed(xs[id,:])         
            N   = (alpha.shape)[1]                     # number of alphas (usually 1; more in case of sampling)
            Fmu = np.tile(ms,(1,N)) + np.dot(Ks.T,alpha[nz])          # conditional mean fs|f
            fmu[id] = np.reshape(Fmu.sum(axis=1)/N,(len(id),1))       # predictive means
            if Ltril: # L is triangular => use Cholesky parameters (alpha,sW,L)
                V       = np.linalg.solve(L.T,np.tile(sW,(1,len(id)))*Ks)
                fs2[id] = kss - np.array([(V*V).sum(axis=0)]).T             # predictive variances
            else:     # L is not triangular => use alternative parametrization
                fs2[id] = kss + np.array([(Ks*np.dot(L,Ks)).sum(axis=0)]).T # predictive variances
            fs2[id] = np.maximum(fs2[id],0)            # remove numerical noise i.e. negative variances
            Fs2 = np.tile(fs2[id],(1,N))               # we have multiple values in case of sampling
            if ys == None:
                [Lp, Ymu, Ys2] = likfunc.proceed(None,Fmu[:],Fs2[:],None,None,3)
            else:
                [Lp, Ymu, Ys2] = likfunc.proceed(np.tile(ys[id],(1,N)), Fmu[:], Fs2[:],None,None,3)
            lp[id]  = np.reshape( np.reshape(Lp,(np.prod(Lp.shape),N)).sum(axis=1)/N , (len(id),1) )   # log probability; sample averaging
            ymu[id] = np.reshape( np.reshape(Ymu,(np.prod(Ymu.shape),N)).sum(axis=1)/N ,(len(id),1) )  # predictive mean ys|y and ...
            ys2[id] = np.reshape( np.reshape(Ys2,(np.prod(Ys2.shape),N)).sum(axis=1)/N , (len(id),1) ) # .. variance
            nact = id[-1]+1                  # set counter to index of next data point

        self.ym = ymu
        self.ys2 = ys2
        self.lp = lp
        if ys == None:
            return ymu, ys2, fmu, fs2, None
        else:
            return ymu, ys2, fmu, fs2, lp 


    def predict_with_posterior(self, post, xs, ys=None):
        '''
        prediction with provided posterior
        (i.e. you already have the posterior and thus don't need fitting/training phases)
        
        @param post         posterior structcture
        @param xs           test input
        @param ys           test target(optional)

        @return ymu, ys2, fmu, fs2, lp
                where ymu is predictive output means
                      ys2 is predictive output variances
                      fm2 is predictive latent means
                      fs2 is predictive latent variances
                      lp  is log predictive probabilities(if ys is given, otherwise is None)
        '''
        meanfunc = self.meanfunc
        covfunc = self.covfunc
        likfunc = self.likfunc
        inffunc = self.inffunc
        x = self.x
        y = self.y
        
        self._posterior_ = deepcopy(post)
        alpha = post.alpha
        L     = post.L
        sW    = post.sW

        nz = range(len(alpha[:,0]))         # non-sparse representation 
        if L == []:                         # in case L is not provided, we compute it
            K = covfunc.proceed(x[nz,:])
            L = np.linalg.cholesky( (np.eye(nz) + np.dot(sW,sW.T)*K).T )
        Ltril     = np.all( np.tril(L,-1) == 0 ) # is L an upper triangular matrix?
        ns        = xs.shape[0]                  # number of data points
        nperbatch = 1000                         # number of data points per mini batch
        nact      = 0                            # number of already processed test data points
        ymu = np.zeros((ns,1))
        ys2 = np.zeros((ns,1))
        fmu = np.zeros((ns,1))
        fs2 = np.zeros((ns,1))
        lp  = np.zeros((ns,1))
        while nact<=ns-1:                              # process minibatches of test cases to save memory
            id  = range(nact,min(nact+nperbatch,ns))   # data points to process
            kss = covfunc.proceed(xs[id,:], 'diag')    # self-variances
            Ks  = covfunc.proceed(x[nz,:], xs[id,:])   # cross-covariances
            ms  = meanfunc.proceed(xs[id,:])         
            N   = (alpha.shape)[1]                     # number of alphas (usually 1; more in case of sampling)
            Fmu = np.tile(ms,(1,N)) + np.dot(Ks.T,alpha[nz])          # conditional mean fs|f
            fmu[id] = np.reshape(Fmu.sum(axis=1)/N,(len(id),1))       # predictive means
            if Ltril: # L is triangular => use Cholesky parameters (alpha,sW,L)
                V       = np.linalg.solve(L.T,np.tile(sW,(1,len(id)))*Ks)
                fs2[id] = kss - np.array([(V*V).sum(axis=0)]).T             # predictive variances
            else:     # L is not triangular => use alternative parametrization
                fs2[id] = kss + np.array([(Ks*np.dot(L,Ks)).sum(axis=0)]).T # predictive variances
            fs2[id] = np.maximum(fs2[id],0)            # remove numerical noise i.e. negative variances
            Fs2 = np.tile(fs2[id],(1,N))               # we have multiple values in case of sampling
            if ys == None:
                [Lp, Ymu, Ys2] = likfunc.proceed(None,Fmu[:],Fs2[:],None,None,3)
            else:
                [Lp, Ymu, Ys2] = likfunc.proceed(np.tile(ys[id],(1,N)), Fmu[:], Fs2[:],None,None,3)
            lp[id]  = np.reshape( np.reshape(Lp,(np.prod(Lp.shape),N)).sum(axis=1)/N , (len(id),1) )   # log probability; sample averaging
            ymu[id] = np.reshape( np.reshape(Ymu,(np.prod(Ymu.shape),N)).sum(axis=1)/N ,(len(id),1) )  # predictive mean ys|y and ...
            ys2[id] = np.reshape( np.reshape(Ys2,(np.prod(Ys2.shape),N)).sum(axis=1)/N , (len(id),1) ) # .. variance
            nact = id[-1]+1                  # set counter to index of next data point
        
        self.ym = ymu
        self.ys2 = ys2
        self.lp = lp
        if ys == None:
            return ymu, ys2, fmu, fs2, None
        else:
            return ymu, ys2, fmu, fs2, lp 






class GPR(GP):
    """Gaussian Process Regression"""
    def __init__(self):
        super(GPR, self).__init__()
        self.meanfunc = mean.Zero()                        # default prior mean 
        self.covfunc = cov.RBF()                           # default prior covariance
        self.likfunc = lik.Gauss()                         # likihood with default noise variance 0.1
        self.inffunc = inf.Exact()                         # inference method
        self.optimizer = opt.Minimize(self)                # default optimizer       

    def setNoise(self,log_sigma):
        """explicitly set noise variance other than default"""
        self.likfunc = lik.Gauss(log_sigma)

    def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None):
        conf = None
        if (num_restarts!=None) or (min_threshold!=None):
            conf = pyGP_OO.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc)
            conf.num_restarts = num_restarts
            conf.min_threshold = min_threshold
            if meanRange != None:
                conf.meanRange = meanRange
            if covRange != None:
                conf.covRange = covRange
            if likRange != None:
                conf.likRange = likRange   
        if method == "Minimize":
            self.optimizer = opt.Minimize(self,conf)            
        elif method == "SCG":
            self.optimizer = opt.SCG(self,conf)  
        elif method == "CG":
            self.optimizer = opt.CG(self,conf)  
        elif method == "BFGS":
            self.optimizer = opt.BFGS(self,conf)  
                       
    def plot(self,axisvals=[-1.9, 1.9, -0.9, 3.9]):
        xs = self.xs
        x = self.x
        y = self.y
        ym = self.ym
        ys2 = self.ys2
        plt.figure()
        xss  = np.reshape(xs,(xs.shape[0],))
        ymm  = np.reshape(ym,(ym.shape[0],))
        ys22 = np.reshape(ys2,(ys2.shape[0],))
        plt.plot(xs, ym, 'g-', x, y, 'b+', linewidth = 3.0, markersize = 10.0)
        plt.fill_between(xss,ymm + 2.*np.sqrt(ys22), ymm - 2.*np.sqrt(ys22), facecolor=SHADEDCOLOR,linewidths=0.0)
        plt.grid()
        if axisvals:
            plt.axis(axisvals)
        plt.xlabel('input x')
        plt.ylabel('target y')
        plt.show()






class GPC(GP):
    """Gaussian Process Classification"""
    def __init__(self):
        super(GPC, self).__init__()
        self.meanfunc = mean.Zero()                        # default prior mean 
        self.covfunc = cov.RBF()                           # default prior covariance
        self.likfunc = lik.Erf()                           # erf likihood 
        self.inffunc = inf.EP()                            # default inference method
        self.optimizer = opt.Minimize(self)                # default optimizer       

    def useLaplace(self):
        """use Laplace approxiamation other than EP"""
        self.inffunc = inf.Laplace() 

    def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None):
        conf = None
        if (num_restarts!=None) or (min_threshold!=None):
            conf = pyGP_OO.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc)
            conf.num_restarts = num_restarts
            conf.min_threshold = min_threshold
            if meanRange != None:
                conf.meanRange = meanRange
            if covRange != None:
                conf.covRange = covRange
            if likRange != None:
                conf.likRange = likRange   
        if method == "Minimize":
            self.optimizer = opt.Minimize(self,conf)            
        elif method == "SCG":
            self.optimizer = opt.SCG(self,conf)  
        elif method == "CG":
            self.optimizer = opt.CG(self,conf)  
        elif method == "BFGS":
            self.optimizer = opt.BFGS(self,conf)  
                       
    def plot(self,x1,x2,t1,t2,axisvals=[-4, 4, -4, 4]):
        fig = plt.figure()
        plt.plot(x1[:,0], x1[:,1], 'b+', markersize = 12)
        plt.plot(x2[:,0], x2[:,1], 'r+', markersize = 12)
        pc = plt.contour(t1, t2, np.reshape(np.exp(self.lp), (t1.shape[0],t1.shape[1]) ))
        fig.colorbar(pc)
        plt.grid()
        plt.axis(axisvals)
        plt.show()




class GPMC(object):
    """This is a one vs. one classification wrapper for GPC"""
    def __init__(self, n_class):
        self.meanfunc = mean.Zero()                # default prior mean 
        self.covfunc = cov.RBF()                   # default prior covariance
        self.n_class = n_class                     # number of different classes
        self.x_all = None
        self.y_all = None
        self.Laplace = False
        self.newPrior = False

    def useLaplace(self):
        """use Laplace approxiamation other than EP"""
        self.Laplace = True

    def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None):
        conf = None
        if (num_restarts!=None) or (min_threshold!=None):
            conf = pyGP_OO.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc)
            conf.num_restarts = num_restarts
            conf.min_threshold = min_threshold
            if meanRange != None:
                conf.meanRange = meanRange
            if covRange != None:
                conf.covRange = covRange
            if likRange != None:
                conf.likRange = likRange   
        if method == "Minimize":
            self.optimizer = opt.Minimize(self,conf)            
        elif method == "SCG":
            self.optimizer = opt.SCG(self,conf)  
        elif method == "CG":
            self.optimizer = opt.CG(self,conf)  
        elif method == "BFGS":
            self.optimizer = opt.BFGS(self,conf)     
    

    # for multi-class, data is x_all and y_all
    def setData(self,x,y):
        self.x_all = x
        self.y_all = y


    def fitAndPredict(self, xs):
        '''
        predictive_vote is a matrix where
        row i    -> each test point i
        column j -> probability for being eahc class j
        '''
        predictive_vote = np.zeros((xs.shape[0],self.n_class))
        for i in xrange(self.n_class):         # classifier for class i...
            for j in xrange(i+1,self.n_class): # ...and class j
                x,y = self.createBinaryClass(i,j)
                model = GPC()
                if self.newPrior:
                    model.setPrior(mean=self.meanfunc, kernel=self.covfunc)
                if self.Laplace:
                    model.useLaplace()
                model.fit(x,y)               # fitting
                ym = model.predict(xs)[0]
                ym += 1     # now scale into 0 to 2,  ym=0 is class j, ym=2 is class i 
                vote_i = np.zeros((xs.shape[0],self.n_class))
                vote_j = np.zeros((xs.shape[0],self.n_class))
                vote_i[:,i:i+1] = ym
                vote_j[:,j:j+1] = 2-ym
                predictive_vote += vote_i
                predictive_vote += vote_j
        predictive_vote /=  predictive_vote.sum(axis=1)[:,np.newaxis]
        return predictive_vote


    def trainAndPredict(self, xs):
        '''
        predictive_vote is a matrix where
        row i    -> each test point i
        column j -> probability for being eahc class j

        '''
        predictive_vote = np.zeros((xs.shape[0],self.n_class))
        for i in xrange(self.n_class):         # classifier for class i...
            for j in xrange(i+1,self.n_class): # ...and class j
                x,y = self.createBinaryClass(i,j)
                model = GPC()
                if self.newPrior:
                    model.setPrior(mean=self.meanfunc, kernel=self.covfunc)
                if self.Laplace:
                    model.useLaplace()
                model.train(x,y)               # training
                ym = model.predict(xs)[0]
                ym += 1     # now scale into 0 to 2,  ym=0 is class j, ym=2 is class i 
                vote_i = np.zeros((xs.shape[0],self.n_class))
                vote_j = np.zeros((xs.shape[0],self.n_class))
                vote_i[:,i:i+1] = ym
                vote_j[:,j:j+1] = 2-ym
                predictive_vote += vote_i
                predictive_vote += vote_j
        predictive_vote /=  predictive_vote.sum(axis=1)[:,np.newaxis]
        return predictive_vote



    def setPrior(self, mean=None, kernel=None):
        """set prior mean and cov"""
        if mean != None:
            self.meanfunc = mean
        if kernel != None:
            self.covfunc = kernel
        self.newPrior = True

    def createBinaryClass(self, i,j):
        ''' create data points x,y which only contains class i and j
        class_i is +1 and class_j is -1'''
        class_i = []
        class_j = []
        for index in xrange(len(self.y_all)):       # check all classes
            target = self.y_all[index]
            if target == i:
                class_i.append(index)
            elif target == j:
                class_j.append(index)
        n1 = len(class_i)
        n2 = len(class_j)
        class_i.extend(class_j)
        x = self.x_all[class_i,:]
        y = np.concatenate((np.ones((1,n1)),-np.ones((1,n2))),axis=1).T
        return x,y







class GP_FITC(GP):
    """GP_FITC base class"""
    def __init__(self):
        super(GP_FITC, self).__init__()
        self.u = None

    # override
    def setData(self, x, y, value_per_axis=5):
        '''set Data and derive deault inducing_points

        value_per_axis is number of value in each dimension...
        ...when using a default inducing point grid'''
        self.x = x
        self.y = y
        # get range of x in each dimension
        # 5 uniformally selected value for each dimension
        gridAxis=[]
        for d in xrange(x.shape[1]):
            column = x[:,d]
            mini = np.min(column)
            maxi = np.max(column)
            axis = np.linspace(mini,maxi,value_per_axis)
            gridAxis.append(axis)
        # default inducing points-> a grid
        if self.u == None:
            self.u = np.array(list(itertools.product(*gridAxis)))
            self.covfunc = self.covfunc.fitc(self.u)

    # override
    def setPrior(self, mean=None, kernel=None, inducing_points=None):
        """set prior and inducing_points"""
        if kernel != None:
            if inducing_points != None:
                self.covfunc = kernel.fitc(inducing_points)
                self.u = inducing_points
            else:
                if self.u != None:
                    self.covfunc = kernel.fitc(self.u)
                else:
                    raise error("To use default inducing points, please call setData() first!")
        if mean != None:
            self.meanfunc = mean


       




class GPR_FITC(GP_FITC):
    """Gaussian Process Regression FITC"""
    def __init__(self):
        super(GPR_FITC, self).__init__()
        self.meanfunc = mean.Zero()                        # default prior mean 
        self.covfunc = cov.RBF()                           # default prior covariance
        self.likfunc = lik.Gauss()                         # likihood with default noise variance 0.1
        self.inffunc = inf.FITC_Exact()                    # inference method
        self.optimizer = opt.Minimize(self)                # default optimizer 
        self.u = None                                      # no default inducing points

    def setNoise(self,log_sigma):
        """explicitly set noise variance other than default"""
        self.likfunc = lik.Gauss(log_sigma)

    def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None):
        conf = None
        if (num_restarts!=None) or (min_threshold!=None):
            conf = pyGP_OO.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc)
            conf.num_restarts = num_restarts
            conf.min_threshold = min_threshold
            if meanRange != None:
                conf.meanRange = meanRange
            if covRange != None:
                conf.covRange = covRange
            if likRange != None:
                conf.likRange = likRange   
        if method == "Minimize":
            self.optimizer = opt.Minimize(self,conf)            
        elif method == "SCG":
            self.optimizer = opt.SCG(self,conf)  
        elif method == "CG":
            self.optimizer = opt.CG(self,conf)  
        elif method == "BFGS":
            self.optimizer = opt.BFGS(self,conf)  
                       
    def plot(self,axisvals=[-1.9, 1.9, -0.9, 3.9]):
        plt.figure()
        xss  = np.reshape(self.xs,(self.xs.shape[0],))
        ymm  = np.reshape(self.ym,(self.ym.shape[0],))
        ys22 = np.reshape(self.ys2,(self.ys2.shape[0],))
        plt.plot(self.xs, self.ym, 'g-', self.x, self.y, 'b+', linewidth = 3.0, markersize = 10.0)
        plt.fill_between(xss,ymm + 2.*np.sqrt(ys22), ymm - 2.*np.sqrt(ys22), facecolor=SHADEDCOLOR,linewidths=0.0)
        plt.grid()
        if axisvals:
            plt.axis(axisvals)
        plt.xlabel('input x')
        plt.ylabel('output y')
        plt.plot(self.u,np.ones_like(self.u),'kx',markersize=12)
        plt.show()







class GPC_FITC(GP_FITC):
    """Gaussian Process Classification FITC"""
    def __init__(self):
        super(GPC_FITC, self).__init__()
        self.meanfunc = mean.Zero()                        # default prior mean 
        self.covfunc = cov.RBF()                           # default prior covariance
        self.likfunc = lik.Erf()                           # erf liklihood
        self.inffunc = inf.FITC_EP()                       # default inference method
        self.optimizer = opt.Minimize(self)                # default optimizer 
        self.u = None                                      # no default inducing points

    def useLaplace_FITC(self):
        """use Laplace approxiamation other than EP"""
        self.inffunc = inf.FITC_Laplace() 

    def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None):
        conf = None
        if (num_restarts!=None) or (min_threshold!=None):
            conf = pyGP_OO.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc)
            conf.num_restarts = num_restarts
            conf.min_threshold = min_threshold
            if meanRange != None:
                conf.meanRange = meanRange
            if covRange != None:
                conf.covRange = covRange
            if likRange != None:
                conf.likRange = likRange   
        if method == "Minimize":
            self.optimizer = opt.Minimize(self,conf)            
        elif method == "SCG":
            self.optimizer = opt.SCG(self,conf)  
        elif method == "CG":
            self.optimizer = opt.CG(self,conf)  
        elif method == "BFGS":
            self.optimizer = opt.BFGS(self,conf)  
                       
    def plot(self,x1,x2,t1,t2,axisvals=[-4, 4, -4, 4]):
        fig = plt.figure()
        plt.plot(x1[:,0], x1[:,1], 'b+', markersize = 12)
        plt.plot(x2[:,0], x2[:,1], 'r+', markersize = 12)
        plt.plot(self.u[:,0],self.u[:,1],'ko', markersize=12)
        pc = plt.contour(t1, t2, np.reshape(np.exp(self.lp), (t1.shape[0],t1.shape[1]) ))
        fig.colorbar(pc)
        plt.grid()
        plt.axis(axisvals)
        plt.show()

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      © Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting. Created using Sphinx 1.2.
      pyGPs-1.2/doc/build/html/_modules/pyGPs/000077500000000000000000000000001236346020200200755ustar00rootroot00000000000000pyGPs-1.2/doc/build/html/_modules/pyGPs/Core/000077500000000000000000000000001236346020200207655ustar00rootroot00000000000000pyGPs-1.2/doc/build/html/_modules/pyGPs/Core/cov.html000066400000000000000000007140371236346020200224560ustar00rootroot00000000000000 pyGPs.Core.cov — pyGPs v1.2 documentation

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      Source code for pyGPs.Core.cov

      #================================================================================
#    Marion Neumann [marion dot neumann at uni-bonn dot de]
#    Daniel Marthaler [marthaler at ge dot com]
#    Shan Huang [shan dot huang at iais dot fraunhofer dot de]
#    Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]
#
#    This file is part of pyGPs.
#    The software package is released under the BSD 2-Clause (FreeBSD) License.
#
#    Copyright (c) by
#    Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014
#================================================================================
# covariance functions to be use by Gaussian process functions. There are two
# different kinds of covariance functions: simple and composite:
#
# simple cov functions:
#
# Poly          - Polynomial covariance function
# PiecePoly     - Piecewise polynomial kernel with compact support.
# RBF           - Squared Exponential kernel.
# RBFunit       - Squared Exponential kernel with unit magnitude.
# RBFard        - Squared Exponential kernel with Automatic Relevance Determination.
# Const         - Constant kernel.
# Linear        - Linear kernel.
# LINard        - Linear covariance function with Automatic Relevance Detemination.
# Matern        - Matern covariance function.
# Periodic      - Stationary kernel for a smooth periodic function.
# Noise         - Independent covariance function, i.e "white noise".
# RQ            - Rational Quadratic covariance function with isotropic distance measure.
# RQard         - Rational Quadratic covariance function with ARD distance measure.
# Pre           - Precomputed kernel matrix.
# 
# composite covariance functions:
#
#   ScaleOfKernel     - scaled version of a covariance function
#   ProductOfKernel   - products of covariance functions
#   SumOfKernel       - sums of covariance functions
#   FITCOfKernel      - Covariance function to be used together with the FITC approximation
#
#
# This is a object-oriented python implementation of gpml functionality 
# (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18).
# based on the functional-version of python implementation
# (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013)
# 
# Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013


import numpy as np
import math
import scipy.spatial.distance as spdist



      [docs]class Kernel(object):
    """
    This is a base class of Kernel functions
    there is no computation in this class, it just defines rules about a kernel class should have
    each covariance function will inherit it and implement its own behaviour
    """
    def __init__(self):
        self.hyp = []
        self.para = []


      [docs]    def getCovMatrix(self,x=None,z=None,mode=None):
        '''
        Return the specific covariance matrix according to input mode

        :param x: training data
        :param z: test data
        :param str mode: 'self_test' return self covariance matrix of test data(test by 1). 
                         'train' return training covariance matrix(train by train).
                         'cross' return cross covariance matrix between x and z(train by test)

        :return: the corresponding covariance matrix
        '''
        pass

      

      [docs]    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        '''
        Compute derivatives wrt. hyperparameters according to input mode

        :param x: training data
        :param z: test data
        :param str mode: 'self_test' return self derivative matrix of test data(test by 1). 
                         'train' return training derivative matrix(train by train).
                         'cross' return cross derivative matrix between x and z(train by test)
        :param int der: index of hyperparameter whose derivative to be computed

        :return: the corresponding derivative matrix
        '''
        pass

    # overloading 
      
    def __add__(self,cov):
        '''
        Overloading + operator.
        
        :param cov: covariance function
        :return: an instance of SumOfKernel
        '''
        return SumOfKernel(self,cov)

    # overloading 
    def __mul__(self,other):
        '''
        Overloading * operator.
        Using * for both multiplication with scalar and product of kernels
        depending on the type of the two objects.
        
        :param other: covariance function as product or int/float as scalar
        :return: an instance of ScaleOfKernel or ProductOfKernel
        '''
        if isinstance(other, int) or isinstance(other, float):
            return ScaleOfKernel(self,other)
        elif isinstance(other, Kernel):
            return ProductOfKernel(self,other)
        else:
            print "only numbers and Kernels are supported operand types for *"

    # overloading 
    __rmul__ = __mul__


      [docs]    def fitc(self,inducingInput):
        '''
        Covariance function to be used together with the FITC approximation.
        Setting FITC gp model will implicitly call this method.

        :return: an instance of FITCOfKernel
        '''
        return FITCOfKernel(self,inducingInput)
    
    # can be replaced by spdist from scipy
      
    def _sq_dist(self, a, b=None):
        '''Compute a matrix of all pairwise squared distances
        between two sets of vectors, stored in the row of the two matrices:
        a (of size n by D) and b (of size m by D).'''
        n = a.shape[0]
        D = a.shape[1]
        m = n    
        if b == None:
            b = a.transpose()
        else:
            m = b.shape[0]
            b = b.transpose()
        C = np.zeros((n,m))
        for d in range(0,D):
            tt  = a[:,d]
            tt  = tt.reshape(n,1)
            tem = np.kron(np.ones((1,m)), tt)
            tem = tem - np.kron(np.ones((n,1)), b[d,:])
            C   = C + tem * tem 
        return C

              
        
      

      [docs]class ProductOfKernel(Kernel):
    '''Product of two kernel function.'''
    def __init__(self,cov1,cov2):
        self.cov1 = cov1
        self.cov2 = cov2
        self._hyp = cov1.hyp + cov2.hyp

    def _setHyp(self,hyp):
        assert len(hyp) == len(self._hyp)
        len1 = len(self.cov1.hyp)
        self._hyp = hyp 
        self.cov1.hyp = self._hyp[:len1]
        self.cov2.hyp = self._hyp[len1:]
    def _getHyp(self):
        return self._hyp
    hyp = property(_getHyp,_setHyp)

    def getCovMatrix(self,x=None,z=None,mode=None):
        A = self.cov1.getCovMatrix(x,z,mode) * self.cov2.getCovMatrix(x,z,mode)
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        if der < len(self.cov1.hyp):
            A = self.cov1.getDerMatrix(x,z,mode,der) * self.cov2.getCovMatrix(x,z,mode)
        elif der < len(self.hyp):
            der2 = der - len(self.cov1.hyp)
            A = self.cov2.getDerMatrix(x,z,mode,der2) * self.cov1.getCovMatrix(x,z,mode)
        else:
            raise Exception("Error: der out of range for covProduct") 
        return A



      

      [docs]class SumOfKernel(Kernel):
    '''Sum of two kernel function.'''
    def __init__(self,cov1,cov2):
        self.cov1 = cov1
        self.cov2 = cov2
        self._hyp = cov1.hyp + cov2.hyp
    def _setHyp(self,hyp):
        assert len(hyp) == len(self._hyp)
        len1 = len(self.cov1.hyp)
        self._hyp = hyp 
        self.cov1.hyp = self._hyp[:len1]
        self.cov2.hyp = self._hyp[len1:]
    def _getHyp(self):
        return self._hyp
    hyp = property(_getHyp,_setHyp)

    def getCovMatrix(self,x=None,z=None,mode=None):
        A = self.cov1.getCovMatrix(x,z,mode) + self.cov2.getCovMatrix(x,z,mode)
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        if der < len(self.cov1.hyp):
            A = self.cov1.getDerMatrix(x,z,mode,der) 
        elif der < len(self.hyp):
            der2 = der - len(self.cov1.hyp)
            A = self.cov2.getDerMatrix(x,z,mode,der2) 
        else:
            raise Exception("Error: der out of range for covSum") 
        return A



      

      [docs]class ScaleOfKernel(Kernel):
    '''Scale of a kernel function.'''
    def __init__(self,cov,scalar):
        self.cov = cov
        if cov.hyp:
            self._hyp = [scalar] + cov.hyp 
        else:
            self._hyp = [scalar]
    def _setHyp(self,hyp):
        assert len(hyp) == len(self._hyp)
        self._hyp = hyp 
        self.cov.hyp = self._hyp[1:]
    def _getHyp(self):
        return self._hyp
    hyp = property(_getHyp,_setHyp)

    def getCovMatrix(self,x=None,z=None,mode=None):
        sf2 = np.exp(self.hyp[0])                     # scale parameter  
        A = sf2 * self.cov.getCovMatrix(x,z,mode)     # accumulate cov 
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        sf2 = np.exp(self.hyp[0])                     # scale parameter  
        if der == 0:                                  # compute derivative w.r.t. sf2
            A = 2. * sf2 * self.cov.getCovMatrix(x,z,mode)
        else:                                 
            A = sf2 * self.cov.getDerMatrix(x,z,mode,der-1) 
        return A
     


      

      [docs]class FITCOfKernel(Kernel):
    '''
    Covariance function to be used together with the FITC approximation.
    The function allows for more than one output argument and does not respect the
    interface of a proper covariance function. 
    Instead of outputing the full covariance, it returns cross-covariances between
    the inputs x, z and the inducing inputs xu as needed by infFITC
    '''
    def __init__(self,cov,inducingInput):
        self.inducingInput = inducingInput
        self.covfunc = cov
        self._hyp = cov.hyp

    def _getHyp(self):
        return self._hyp
    def _setHyp(self, hyp):
        self._hyp = hyp
        self.covfunc.hyp = hyp
    hyp = property(_getHyp,_setHyp)

    def getCovMatrix(self,x=None,z=None,mode=None):
        xu = self.inducingInput
        if x != None:
            try:
                assert(xu.shape[1] == x.shape[1])
            except AssertionError:
                raise Exception('Dimensionality of inducing inputs must match training inputs') 
        if mode == 'self_test':           # self covariances for the test cases
            K = self.covfunc.getCovMatrix(z=z,mode='self_test')
            return K
        elif mode == 'train':             # compute covariance matix for training set
            K   = self.covfunc.getCovMatrix(z=x,mode='self_test')
            Kuu = self.covfunc.getCovMatrix(x=xu,mode='train')
            Ku  = self.covfunc.getCovMatrix(x=xu,z=x,mode='cross')
            return K, Kuu, Ku
        elif mode == 'cross':             # compute covariance between data sets x and z 
            K = self.covfunc.getCovMatrix(x=xu,z=z,mode='cross')
            return K 

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        xu = self.inducingInput
        if x!= None:
            try:
                assert(xu.shape[1] == x.shape[1])
            except AssertionError:
                raise Exception('Dimensionality of inducing inputs must match training inputs') 
        if mode == 'self_test':           # self covariances for the test cases
            K = self.covfunc.getDerMatrix(z=z,mode='self_test',der=der)
            return K
        elif mode == 'train':             # compute covariance matix for training set
            K   = self.covfunc.getDerMatrix(z=x,mode='self_test',der=der)
            Kuu = self.covfunc.getDerMatrix(x=xu,mode='train',der=der)
            Ku  = self.covfunc.getDerMatrix(x=xu,z=x,mode='cross',der=der)
            return K, Kuu, Ku
        elif mode == 'cross':             # compute covariance between data sets x and z 
            K = self.covfunc.getDerMatrix(x=xu,z=z,mode='cross',der=der)
            return K 
        


      

      [docs]class Poly(Kernel):
    '''
    Polynomial covariance function. hyp = [ log_c, log_sigma ]

    :param log_c: inhomogeneous offset. 
    :param log_sigma: signal deviation. 
    :param d: degree of polynomial (not treated as hyperparameter, i.e. will not be trained). 
    '''
    def __init__(self, log_c=0., d=2, log_sigma=0. ):
        self.hyp = [log_c, log_sigma]
        self.para = [d] 

    def getCovMatrix(self,x=None,z=None,mode=None):
        c   = np.exp(self.hyp[0])             # inhomogeneous offset
        sf2 = np.exp(2.*self.hyp[1])          # signal variance
        ord = self.para[0]                    # order of polynomial
        if np.abs(ord-np.round(ord)) < 1e-8:  # remove numerical error from format of parameter
            ord = int(round(ord))
        assert(ord >= 1.)                     # only nonzero integers for ord
        ord = int(ord) 
        if mode == 'self_test':               # self covariances for the test cases
            nn,D = z.shape
            A = np.reshape(np.sum(z*z,1), (nn,1))
        elif mode == 'train':                 # compute covariance matix for dataset x
            A = np.dot(x,x.T)
        elif mode == 'cross':                 # compute covariance between data sets x and z 
            A = np.dot(x,z.T) 
        A = sf2 * (c + A)**ord
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        c   = np.exp(self.hyp[0])             # inhomogeneous offset
        sf2 = np.exp(2.*self.hyp[1])          # signal variance
        ord = self.para[0]                    # order of polynomial
        if np.abs(ord-np.round(ord)) < 1e-8:  # remove numerical error from format of parameter
            ord = int(round(ord))
        assert(ord >= 1.)                     # only nonzero integers for ord
        ord = int(ord) 
        if mode == 'self_test':               # self covariances for the test cases
            nn,D = z.shape
            A = np.reshape(np.sum(z*z,1), (nn,1))
        elif mode == 'train':                 # compute covariance matix for dataset x
            A = np.dot(x,x.T)
        elif mode == 'cross':                 # compute covariance between data sets x and z 
            A = np.dot(x,z.T) 
        if der == 0:                          # compute derivative matrix wrt 1st parameter             
            A = c * ord * sf2 * (c+A)**(ord-1)
        elif der == 1:                        # compute derivative matrix wrt 2nd parameter
            A = 2. * sf2 * (c + A)**ord
        elif der == 2:                        # no derivative wrt 3rd parameter
            A = np.zeros_like(A)              # do nothing (d is not learned)
        else:
            raise Exception("Wrong derivative entry in Poly")
        return A
        


      

      [docs]class PiecePoly(Kernel):
    '''
    Piecewise polynomial kernel with compact support. 
    hyp = [log_ell, log_sigma]

    :param log_ell: characteristic length scale. 
    :param log_sigma: signal deviation. 
    :param v: degree v will be rounded to 0,1,2,or 3. (not treated as hyperparameter, i.e. will not be trained). 
    '''
    def __init__(self, log_ell=0., v=2, log_sigma=0. ):
        self.hyp = [log_ell, log_sigma]
        self.para = [v]

    def ppmax(self,A,B):
        return np.maximum(A,B*np.ones_like(A))

    def func(self,v,r,j):
        if v == 0:
            return 1
        elif v == 1:
            return ( 1. + (j+1) * r )
        elif v == 2:
            return ( 1. + (j+2)*r + (j*j + 4.*j+ 3)/3.*r*r )
        elif v == 3:
            return ( 1. + (j+3)*r + (6.*j*j+36.*j+45.)/15.*r*r + (j*j*j+9.*j*j+23.*j+15.)/15.*r*r*r )
        else:
             raise Exception (["Wrong degree in PiecePoly.  Should be 0,1,2 or 3, is " + str(v)])

    def dfunc(self,v,r,j):
        if v == 0:
            return 0
        elif v == 1:
            return ( j+1 )
        elif v == 2:
            return ( (j+2) + 2.*(j*j+ 4.*j+ 3.)/3.*r )
        elif v == 3:
            return ( (j+3) + 2.*(6.*j*j+36.*j+45.)/15.*r + (j*j*j+9.*j*j+23.*j+15.)/5.*r*r )
        else:
            raise Exception (["Wrong degree in PiecePoly.  Should be 0,1,2 or 3, is " + str(v)])

    def pp(self,r,j,v,func):
        return func(v,r,j)*(self.ppmax(1-r,0)**(j+v))

    def dpp(self,r,j,v,func,dfunc):
        return self.ppmax(1-r,0)**(j+v-1) * r * ( (j+v)*self.func(v,r,j) - self.ppmax(1-r,0) * self.dfunc(v,r,j) )

    def getCovMatrix(self,x=None,z=None,mode=None):
        if x != None:
            n, D = x.shape
        if z != None:
            nn, D = z.shape
        ell = np.exp(self.hyp[0])            # characteristic length scale
        sf2 = np.exp(2.*self.hyp[1])         # signal variance
        v   = self.para[0]                   # degree (v = 0,1,2 or 3 only)
        if np.abs(v-np.round(v)) < 1e-8:     # remove numerical error from format of parameter
            v = int(round(v))
        assert(int(v) in range(4))           # Only allowed degrees: 0,1,2 or 3
        v = int(v)
        j = np.floor(0.5*D) + v + 1
        if mode == 'self_test':              # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':                # compute covariance matix for dataset x
            A = np.sqrt( spdist.cdist(x/ell, x/ell, 'sqeuclidean') )
        elif mode == 'cross':                # compute covariance between data sets x and z 
            A = np.sqrt( spdist.cdist(x/ell, z/ell, 'sqeuclidean') )
        A = sf2 * self.pp(A,j,v,self.func)
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        if x != None:
            n, D = x.shape
        if z != None:
            nn, D = z.shape
        ell = np.exp(self.hyp[0])            # characteristic length scale
        sf2 = np.exp(2.*self.hyp[1])         # signal variance
        v   = self.para[0]                   # degree (v = 0,1,2 or 3 only)
        if np.abs(v-np.round(v)) < 1e-8:     # remove numerical error from format of parameter
            v = int(round(v))
        assert(int(v) in range(4))           # Only allowed degrees: 0,1,2 or 3
        v = int(v)
        j = np.floor(0.5*D) + v + 1
        if mode == 'self_test':              # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':                # compute covariance matix for dataset x
            A = np.sqrt( spdist.cdist(x/ell, x/ell, 'sqeuclidean') )
        elif mode == 'cross':                # compute covariance between data sets x and z 
            A = np.sqrt( spdist.cdist(x/ell, z/ell, 'sqeuclidean') )
        if der == 0:                            # compute derivative matrix wrt 1st parameter
            A = sf2 * self.dpp(A,j,v,self.func,self.dfunc)
        elif der == 1:                          # compute derivative matrix wrt 2nd parameter
            A = 2. * sf2 * self.pp(A,j,v,self.func)
        elif der == 2:                          # wants to compute derivative wrt order
            A = np.zeros_like(A)
        else:
            raise Exception("Wrong derivative entry in PiecePoly")
        return A



      

      [docs]class RBF(Kernel):
    '''
    Squared Exponential kernel with isotropic distance measure. hyp = [log_ell, log_sigma]

    :param log_ell: characteristic length scale. 
    :param log_sigma: signal deviation. 
    '''
    def __init__(self, log_ell=0., log_sigma=0.):
        self.hyp = [log_ell, log_sigma]

    def getCovMatrix(self,x=None,z=None,mode=None):
        ell = np.exp(self.hyp[0])         # characteristic length scale
        sf2 = np.exp(2.*self.hyp[1])      # signal variance
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for training set
            A = spdist.cdist(x/ell,x/ell,'sqeuclidean')
        elif mode == 'cross':             # compute covariance between data sets x and z 
            A = spdist.cdist(x/ell,z/ell,'sqeuclidean') 
        A = sf2 * np.exp(-0.5*A)
        return A


    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        ell = np.exp(self.hyp[0])         # characteristic length scale
        sf2 = np.exp(2.*self.hyp[1])      # signal variance
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            A = spdist.cdist(x/ell,x/ell,'sqeuclidean')
        elif mode == 'cross':             # compute covariance between data sets x and z 
            A = spdist.cdist(x/ell,z/ell,'sqeuclidean') 
        if der == 0:    # compute derivative matrix wrt 1st parameter
            A = sf2 * np.exp(-0.5*A) * A
        elif der == 1:  # compute derivative matrix wrt 2nd parameter
            A = 2. * sf2 * np.exp(-0.5*A)
        else:
            raise Exception("Calling for a derivative in RBF that does not exist")
        return A
        


      

      [docs]class RBFunit(Kernel):
    '''
    Squared Exponential kernel with isotropic distance measure with unit magnitude.
    i.e signal variance is always 1. hyp = [ log_ell ]

    :param log_ell: characteristic length scale. 
    '''
    def __init__(self, log_ell=0.):
        self.hyp = [log_ell]

    def getCovMatrix(self,x=None,z=None,mode=None):
        ell = np.exp(self.hyp[0])         # characteristic length scale
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            A = spdist.cdist(x/ell,x/ell,'sqeuclidean')
        elif mode == 'cross':             # compute covariance between data sets x and z 
            A = spdist.cdist(x/ell,z/ell,'sqeuclidean') 
        A = np.exp(-0.5*A)
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        ell = np.exp(self.hyp[0])         # characteristic length scale
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            A = spdist.cdist(x/ell,x/ell,'sqeuclidean')
        elif mode == 'cross':             # compute covariance between data sets x and z 
            A = spdist.cdist(x/ell,z/ell,'sqeuclidean') 
        if der == 0:           # compute derivative matrix wrt 1st parameter
            A = np.exp(-0.5*A) * A
        else:
            raise Exception("Wrong derivative index in RDFunit")
        return A



      

      [docs]class RBFard(Kernel):
    '''
    Squared Exponential kernel with Automatic Relevance Determination.
    hyp = log_ell_list + [log_sigma]

    :param D: dimension of pattern. set if you want default ell, which is 1 for each dimension.
    :param log_ell_list: characteristic length scale for each dimension.
    :param log_sigma: signal deviation. 
    '''
    def __init__(self, D=None, log_ell_list=None, log_sigma=0.):
        if log_ell_list == None:
            self.hyp = [0. for i in xrange(D)] + [log_sigma]
        else:
            self.hyp = log_ell_list + [log_sigma]

    def getCovMatrix(self,x=None,z=None,mode=None):
        if x != None:
            n, D = x.shape
        if z != None:
            nn, D = z.shape
        ell = 1./np.exp(self.hyp[0:D])    # characteristic length scale
        sf2 = np.exp(2.*self.hyp[D])      # signal variance
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            tem = np.dot(np.diag(ell),x.T).T
            A = spdist.cdist(tem,tem,'sqeuclidean')
        elif mode == 'cross':             # compute covariance between data sets x and z 
            A = spdist.cdist(np.dot(np.diag(ell),x.T).T,np.dot(np.diag(ell),z.T).T,'sqeuclidean')
        A = sf2*np.exp(-0.5*A)
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        if x != None:
            n, D = x.shape
        if z != None:
            nn, D = z.shape
        ell = 1./np.exp(self.hyp[0:D])    # characteristic length scale
        sf2 = np.exp(2.*self.hyp[D])      # signal variance
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            tem = np.dot(np.diag(ell),x.T).T
            A = spdist.cdist(tem,tem,'sqeuclidean')
        elif mode == 'cross':             # compute covariance between data sets x and z 
            A = spdist.cdist(np.dot(np.diag(ell),x.T).T,np.dot(np.diag(ell),z.T).T,'sqeuclidean')
        A = sf2*np.exp(-0.5*A)
        if der < D:                       # compute derivative matrix wrt length scale parameters
            if mode == 'self_test':
                A = A*0
            elif mode == 'train':
                tem = np.atleast_2d(x[:,der])/ell[der]
                A *= spdist.cdist(tem,tem,'sqeuclidean')
            elif mode == 'cross':
                A *= spdist.cdist(np.atleast_2d(x[:,der]).T/ell[der],np.atleast_2d(z[:,der]).T/ell[der],'sqeuclidean')
        elif der == D:                    # compute derivative matrix wrt magnitude parameter
            A = 2.*A
        else:
            raise Exception("Wrong derivative index in RDFard") 
        return A



      

      [docs]class Const(Kernel):
    '''
    Constant kernel. hyp = [ log_sigma ]

    :param log_sigma: signal deviation. 
    '''
    def __init__(self, log_sigma=0.):
        self.hyp = [log_sigma]

    def getCovMatrix(self,x=None,z=None,mode=None):
        sf2 = np.exp(self.hyp[0])         # s2
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = sf2 * np.ones((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            n,D = x.shape
            A = sf2 * np.ones((n,n))
        elif mode == 'cross':             # compute covariance between data sets x and z 
            n,D  = x.shape
            nn,D = z.shape
            A = sf2 * np.ones((n,nn))
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        sf2 = np.exp(self.hyp[0])         # s2
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = sf2 * np.ones((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            n,D = x.shape
            A = sf2 * np.ones((n,n))
        elif mode == 'cross':             # compute covariance between data sets x and z 
            n,D  = x.shape
            nn,D = z.shape
            A = sf2 * np.ones((n,nn))
        if der == 0:                      # compute derivative matrix wrt sf2
            A = 2. * A
        else:
            raise Exception("Wrong derivative entry in covConst")
        return A
        


      

      [docs]class Linear(Kernel):
    '''
    Linear kernel. hyp = [ log_sigma ].

    :param log_sigma: signal deviation.
    '''
    def __init__(self, log_sigma=0.):
        self.hyp = [ log_sigma ]

    def getCovMatrix(self,x=None,z=None,mode=None):
        sf2 = np.exp(self.hyp[0])         # s2
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.reshape(np.sum(z*z,1), (nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            n,D = x.shape
            A = np.dot(x,x.T) + np.eye(n)*1e-16    # required for numerical accuracy
        elif mode == 'cross':             # compute covariance between data sets x and z 
            A = np.dot(x,z.T)
        A = sf2 * A
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        sf2 = np.exp(self.hyp[0])         # s2
        if der == 0:
            if mode == 'self_test':           # self covariances for the test cases
                nn,D = z.shape
                A = np.reshape(np.sum(z*z,1), (nn,1))
            elif mode == 'train':             # compute covariance matix for dataset x
                n,D = x.shape
                A = np.dot(x,x.T) + np.eye(n)*1e-16    # required for numerical accuracy
            elif mode == 'cross':             # compute covariance between data sets x and z 
                A = np.dot(x,z.T)
            A = 2 * sf2 * A
        else:
            raise Exception("Wrong derivative index in covLinear")
        return A



      

      [docs]class LINard(Kernel):
    '''
    Linear covariance function with Automatic Relevance Detemination.
    hyp = log_ell_list 

    :param D: dimension of training data. Set if you want default ell, which is 1 for each dimension.
    :param log_ell_list: characteristic length scale for each dimension.
    '''
    def __init__(self, D=None, log_ell_list=None):
        if log_ell_list == None:
            self.hyp = [0. for i in xrange(D)]
        else:
            self.hyp = log_ell_list

    def getCovMatrix(self,x=None,z=None,mode=None):
        ell = np.exp(self.hyp)            # ARD parameters
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.reshape(np.sum(z*z,1), (nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            A = np.dot(x,x.T) 
        elif mode == 'cross':             # compute covariance between data sets x and z
            z = np.dot(z,np.diag(1./ell))
            A = np.dot(x,z.T) 
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        ell = np.exp(self.hyp)            # ARD parameters
        if x != None:
            n, D = x.shape
        if z != None:
            nn, D = z.shape
        if der < D:
            if mode == 'self_test':
                tem = np.atleast_2d(z[:,der]).T
                A = -2.* tem * tem
            elif mode == 'train':
                A = -2.*np.dot(np.atleast_2d(x[:,der]).T,np.atleast_2d(x[:,der]))
            elif mode == 'cross':
                z = np.dot(z,np.diag(1./ell))
                A = -2.*np.dot(np.atleast_2d(x[:,der]).T, np.atleast_2d(z[:,der])) # cross covariances
        else:
            raise Exception("Wrong derivative index in covLINard")
        return A



      

      [docs]class Matern(Kernel):
    '''
    Matern covariance function with nu = d/2 and isotropic distance measure. 
    For d=1 the function is also known as the exponential covariance function 
    or the Ornstein-Uhlenbeck covariance in 1d.
    d will be rounded to 1, 3, or 5.
    hyp = [ log_ell, log_sigma]
    
    :param d: d is 2 times nu. Can only be 1,3 or 5.
    :param log_ell: characteristic length scale. 
    :param log_sigma: signal deviation. 
    '''
    def __init__(self, log_ell=0., d=3, log_sigma=0. ):
        self.hyp = [ log_ell, log_sigma ]
        self.para = [d]

    def func(self,d,t):
        if d == 1:
            return 1
        elif d == 3:
            return 1 + t
        elif d == 5:
            return 1 + t*(1+t/3.)
        else:
            raise Exception("Wrong value for d in Matern")

    def dfunc(self,d,t):
        if d == 1:
            return 1
        elif d == 3:
            return t
        elif d == 5:
            return t*(1+t/3.)
        else:
            raise Exception("Wrong value for d in Matern")

    def mfunc(self,d,t):
        return self.func(d,t)*np.exp(-1.*t)

    def dmfunc(self,d,t):
        return self.dfunc(d,t)*t*np.exp(-1.*t)

    def getCovMatrix(self,x=None,z=None,mode=None):
        ell = np.exp(self.hyp[0])        # characteristic length scale
        sf2 = np.exp(2.* self.hyp[1])    # signal variance
        d   = self.para[0]               # 2 times nu
        if np.abs(d-np.round(d)) < 1e-8: # remove numerical error from format of parameter
            d = int(round(d))
        d = int(d)
        try:
            assert(d in [1,3,5])         # check for valid values of d
        except AssertionError:
            print "Warning: You specified d to be neither 1,3 nor 5. We set to d=3. "
            d = 3
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            x = np.sqrt(d)*x/ell   
            A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean'))
        elif mode == 'cross':             # compute covariance between data sets x and z
            x = np.sqrt(d)*x/ell
            z = np.sqrt(d)*z/ell
            A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean'))
        A = sf2 * self.mfunc(d,A)
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        ell = np.exp(self.hyp[0])        # characteristic length scale
        sf2 = np.exp(2.* self.hyp[1])    # signal variance
        d   = self.para[0]               # 2 times nu
        if np.abs(d-np.round(d)) < 1e-8: # remove numerical error from format of parameter
            d = int(round(d))
        d = int(d)
        try:
            assert(d in [1,3,5])         # check for valid values of d
        except AssertionError:
            print "Warning: You specified d to be neither 1,3 nor 5. We set to d=3. "
            d = 3
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            x = np.sqrt(d)*x/ell   
            A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean'))
        elif mode == 'cross':             # compute covariance between data sets x and z
            x = np.sqrt(d)*x/ell
            z = np.sqrt(d)*z/ell
            A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean'))
        A = sf2 * self.mfunc(d,A)
        if der == 0:                    # compute derivative matrix wrt 1st parameter
            A = sf2 * self.dmfunc(d,A)
        elif der == 1:                  # compute derivative matrix wrt 2nd parameter
            A = 2 * sf2 * self.mfunc(d,A)
        elif der == 2:                  # no derivative wrt 3rd parameter
            A = np.zeros_like(A)        # do nothing (d is not learned)
        else:
            raise Exception("Wrong derivative value in Matern")
        return A
        


      

      [docs]class Periodic(Kernel):
    '''
    Stationary kernel for a smooth periodic function. 
    hyp = [ log_ell, log_p, log_sigma]

    :param log_p: period.
    :param log_ell: characteristic length scale. 
    :param log_sigma: signal deviation. 
    '''
    def __init__(self, log_ell=0., log_p=0., log_sigma=0. ): 
        self.hyp = [ log_ell, log_p, log_sigma]

    def getCovMatrix(self,x=None,z=None,mode=None):
        ell = np.exp(self.hyp[0])        # characteristic length scale
        p   = np.exp(self.hyp[1])        # period
        sf2 = np.exp(2.*self.hyp[2])     # signal variance
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean'))
        elif mode == 'cross':             # compute covariance between data sets x and z
            A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean'))
        A = np.pi*A/p
        A = np.sin(A)/ell
        A = A * A
        A = sf2 *np.exp(-2.*A)
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        ell = np.exp(self.hyp[0])        # characteristic length scale
        p   = np.exp(self.hyp[1])        # period
        sf2 = np.exp(2.*self.hyp[2])     # signal variance
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean'))
        elif mode == 'cross':             # compute covariance between data sets x and z
            A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean'))
        A = np.pi*A/p
        if der == 0:            # compute derivative matrix wrt 1st parameter
            A = np.sin(A)/ell
            A = A * A
            A = 4. *sf2 *np.exp(-2.*A) * A
        elif der == 1:          # compute derivative matrix wrt 2nd parameter
            R = np.sin(A)/ell
            A = 4 * sf2/ell * np.exp(-2.*R*R)*R*np.cos(A)*A
        elif der == 2:          # compute derivative matrix wrt 3rd parameter
            A = np.sin(A)/ell
            A = A * A
            A = 2. * sf2 * np.exp(-2.*A)
        else:
            raise Exception("Wrong derivative index in covPeriodic")
        return A
        


      

      [docs]class Noise(Kernel):
    '''
    Independent covariance function, i.e "white noise", with specified variance.
    Normally NOT used anymore since noise is now added in liklihood.
    hyp = [ log_sigma ]

    :param log_sigma: signal deviation. 
    '''
    def __init__(self, log_sigma=0.):
        self.hyp = [log_sigma]

    def getCovMatrix(self,x=None,z=None,mode=None):
        tol = 1.e-9                       # Tolerance for declaring two vectors "equal"
        s2 = np.exp(2.*self.hyp[0])       # noise variance
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            n,D = x.shape
            A = np.eye(n)
        elif mode == 'cross':             # compute covariance between data sets x and z
            M = spdist.cdist(x, z, 'sqeuclidean')
            A = np.zeros_like(M,dtype=np.float)
            A[M < tol] = 1.
        A = s2*A
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        tol = 1.e-9                       # Tolerance for declaring two vectors "equal"
        s2 = np.exp(2.*self.hyp[0])       # noise variance
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            A = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            n,D = x.shape
            A = np.eye(n)
        elif mode == 'cross':             # compute covariance between data sets x and z
            M = spdist.cdist(x, z, 'sqeuclidean')
            A = np.zeros_like(M,dtype=np.float)
            A[M < tol] = 1.
        if der == 0:
            A = 2.*s2*A
        else:
            raise Exception("Wrong derivative index in covNoise")
        return A
        


      

      [docs]class RQ(Kernel):
    ''' 
    Rational Quadratic covariance function with isotropic distance measure.
    hyp = [ log_ell, log_sigma, log_alpha ]
    
    :param log_ell: characteristic length scale. 
    :param log_sigma: signal deviation.
    :param log_alpha: shape parameter for the RQ covariance.
    '''
    def __init__(self, log_ell=0., log_sigma=0., log_alpha=0.):
        self.hyp = [ log_ell, log_sigma, log_alpha ]

    def getCovMatrix(self,x=None,z=None,mode=None):
        ell   = np.exp(self.hyp[0])       # characteristic length scale
        sf2   = np.exp(2.*self.hyp[1])    # signal variance
        alpha = np.exp(self.hyp[2])
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            D2 = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            D2 = spdist.cdist(x/ell, x/ell, 'sqeuclidean')
        elif mode == 'cross':             # compute covariance between data sets x and z
            D2 = spdist.cdist(x/ell, z/ell, 'sqeuclidean')
        A = sf2 * ( ( 1.0 + 0.5*D2/alpha )**(-alpha) )
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        ell   = np.exp(self.hyp[0])       # characteristic length scale
        sf2   = np.exp(2.*self.hyp[1])    # signal variance
        alpha = np.exp(self.hyp[2])
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            D2 = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            D2 = spdist.cdist(x/ell, x/ell, 'sqeuclidean')
        elif mode == 'cross':             # compute covariance between data sets x and z
            D2 = spdist.cdist(x/ell, z/ell, 'sqeuclidean')
        if der == 0:                # compute derivative matrix wrt 1st parameter
            A = sf2 * ( 1.0 + 0.5*D2/alpha )**(-alpha-1) * D2
        elif der == 1:              # compute derivative matrix wrt 2nd parameter
            A = 2.* sf2 * ( ( 1.0 + 0.5*D2/alpha )**(-alpha) )
        elif der == 2:              # compute derivative matrix wrt 3rd parameter
            K = ( 1.0 + 0.5*D2/alpha )
            A = sf2 * K**(-alpha) * (0.5*D2/K - alpha*np.log(K) )
        else:
            raise Exception("Wrong derivative index in covRQ")
        return A



      

      [docs]class RQard(Kernel):
    '''
    Rational Quadratic covariance function with Automatic Relevance Detemination
    (ARD) distance measure.
    hyp = log_ell_list + [ log_sigma, log_alpha ]

    :param D: dimension of pattern. set if you want default ell, which is 0.5 for each dimension.
    :param log_ell_list: characteristic length scale for each dimension.
    :param log_sigma: signal deviation. 
    :param log_alpha: shape parameter for the RQ covariance.
    '''
    def __init__(self, D=None, log_ell_list=None, log_sigma=0., log_alpha=0.):
        if log_ell_list == None:
            self.hyp = [0. for i in xrange(D)] + [ log_sigma, log_alpha ]
        else:
            self.hyp = log_ell_list + [ log_sigma, log_alpha ]

    def getCovMatrix(self,x=None,z=None,mode=None):
        if x != None:
            n, D = x.shape
        if z != None:
            nn, D = z.shape
        ell = 1./np.exp(self.hyp[0:D])    # characteristic length scale
        sf2 = np.exp(2.*self.hyp[D])      # signal variance
        alpha = np.exp(self.hyp[D+1])
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            D2 = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            tmp = np.dot(np.diag(ell),x.T).T
            D2 = spdist.cdist(tmp, tmp, 'sqeuclidean')
        elif mode == 'cross':             # compute covariance between data sets x and z
            D2 = spdist.cdist(np.dot(np.diag(ell),x.T).T, np.dot(np.diag(ell),z.T).T, 'sqeuclidean')
        A = sf2 * ( ( 1.0 + 0.5*D2/alpha )**(-alpha) )
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        if x != None:
            n, D = x.shape
        if z != None:
            nn, D = z.shape
        ell = 1./np.exp(self.hyp[0:D])    # characteristic length scale
        sf2 = np.exp(2.*self.hyp[D])      # signal variance
        alpha = np.exp(self.hyp[D+1])
        if mode == 'self_test':           # self covariances for the test cases
            nn,D = z.shape
            D2 = np.zeros((nn,1))
        elif mode == 'train':             # compute covariance matix for dataset x
            tmp = np.dot(np.diag(ell),x.T).T
            D2 = spdist.cdist(tmp, tmp, 'sqeuclidean')
        elif mode == 'cross':             # compute covariance between data sets x and z
            D2 = spdist.cdist(np.dot(np.diag(ell),x.T).T, np.dot(np.diag(ell),z.T).T, 'sqeuclidean')
        if der < D:
            if mode == 'self_test':
                A = D2*0
            elif mode == 'train':
                tmp = np.atleast_2d(x[:,der])/ell[der]
                A = sf2 * ( 1.0 + 0.5*D2/alpha )**(-alpha-1) * spdist.cdist(tmp, tmp, 'sqeuclidean')
            elif mode == 'cross':
                A = sf2 * ( 1.0 + 0.5*D2/alpha )**(-alpha-1) * spdist.cdist(np.atleast_2d(x[:,der]).T/ell[der], np.atleast_2d(z[:,der]).T/ell[der], 'sqeuclidean')
        elif der==D:                # compute derivative matrix wrt magnitude parameter
            A = 2. * sf2 * ( ( 1.0 + 0.5*D2/alpha )**(-alpha) )
        elif der==(D+1):            # compute derivative matrix wrt magnitude parameter
            K = ( 1.0 + 0.5*D2/alpha )
            A = sf2 * K**(-alpha) * ( 0.5*D2/K - alpha*np.log(K) )
        else:
            raise Exception("Wrong derivative index in covRQard") 
        return A
        


      

      [docs]class Pre(Kernel):
    '''
    Precomputed kernel matrix. No hyperparameters and thus nothing will be optimised. 

    :param M1: cross covariances matrix(train+1 by test).
               last row is self covariances (diagonal of test by test)
    :param M2: training set covariance matrix (train by train)   
    '''
    def __init__(self,M1,M2):
        self.M1 = M1
        self.M2 = M2
        self.hyp = []

    def getCovMatrix(self,x=None,z=None,mode=None):
        if mode == 'self_test':           # diagonal covariance between test_test
            A = self.M1[-1,:]             # self covariances for the test cases (last row) 
            A = np.reshape(A, (A.shape[0],1))
        elif mode == 'train':             # covariance between train_train
            A = self.M2
        elif mode == 'cross':             # covariance between train_test
            A = self.M1[:-1,:]
        return A

    def getDerMatrix(self,x=None,z=None,mode=None,der=None):
        if der != None:
            raise Exception("Error: NO optimization in precomputed kernel matrix")
        return 0
        
    

      
if __name__ == '__main__':
    pass

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      © Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting. Created using Sphinx 1.2.2.
      pyGPs-1.2/doc/build/html/_modules/pyGPs/Core/gp.html000066400000000000000000005304631236346020200222740ustar00rootroot00000000000000 pyGPs.Core.gp — pyGPs v1.2 documentation

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      Source code for pyGPs.Core.gp

      #================================================================================
#    Marion Neumann [marion dot neumann at uni-bonn dot de]
#    Daniel Marthaler [marthaler at ge dot com]
#    Shan Huang [shan dot huang at iais dot fraunhofer dot de]
#    Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]
#
#    This file is part of pyGPs.
#    The software package is released under the BSD 2-Clause (FreeBSD) License.
#
#    Copyright (c) by
#    Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014
#================================================================================

#   MEANING OF NOTATION:
#  
#   inffunc      function specifying the inference method 
#   covfunc      prior covariance function (see below)
#   meanfunc     prior mean function
#   likfunc      likelihood function
#   x            n by D matrix of training inputs
#   y            column vector of length n of training targets
#   xs           n by D matrix of test inputs
#   ys           column vector of length nn of true test targets (optional)
#   nlZ          returned value of the negative log marginal likelihood
#   dnlZ         column vector of partial derivatives of the negative
#                    log marginal likelihood w.r.t. each hyperparameter
#   ym           column vector (of length ns) of predictive output means
#   ys2          column vector (of length ns) of predictive output variances
#   fm           column vector (of length ns) of predictive latent means
#   fs2          column vector (of length ns) of predictive latent variances
#   lp           column vector (of length ns) of log predictive probabilities
#   post         struct representation of the (approximate) posterior
#                post consists of post.alpha, post.L, post.sW
#
# This is a object-oriented python implementation of gpml functionality 
# (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18).
# based on the functional-version of python implementation
# (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013)
# 
# Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013

import itertools
import numpy as np
import matplotlib.pyplot as plt
import inf, mean, lik, cov, opt
from tools import unique
from copy import deepcopy
import pyGPs

SHADEDCOLOR = [0.7539, 0.89453125, 0.62890625, 1.0]
MEANCOLOR = [ 0.2109375, 0.63385, 0.1796875, 1.0]
DATACOLOR = [0.12109375, 0.46875, 1., 1.0]

class GP(object):
    '''Base class for GP model'''
    def __init__(self):
        super(GP, self).__init__()
        self.usingDefaultMean = True  # was using default mean function now?
        self.meanfunc = None
        self.covfunc = None
        self.likfunc = None
        self.inffunc = None
        self.optimizer = None 
        self.nlZ = None           # negative log marginal likelihood
        self.dnlZ = None          # column vector of partial derivatives of the negative
                                  # log marginal likelihood w.r.t. each hyperparameter
        self.posterior = None     # struct representation of the (approximate) posterior 
        self.x = None             # n by D matrix of training inputs
        self.y = None             # column vector of length n of training targets
        self.xs = None            # n by D matrix of test inputs
        self.ys = None            # column vector of length nn of true test targets (optional)
        self.ym = None            # column vector (of length ns) of predictive output means
        self.ys2 = None           # column vector (of length ns) of predictive output variances
        self.fm = None            # column vector (of length ns) of predictive latent means
        self.fs2 = None           # column vector (of length ns) of predictive latent variances
        self.lp = None            # column vector (of length ns) of log predictive probabilities


    def setData(self, x, y):
        '''Pass training data and traning labels to model.''' 
        if x.ndim == 1:
            x = np.reshape(x, (x.shape[0],1))
        if y.ndim == 1:
            y = np.reshape(y, (y.shape[0],1))  
        self.x = x
        self.y = y
        if self.usingDefaultMean:
            c = np.mean(y)
            self.meanfunc = mean.Const(c)    # adapt default prior mean wrt. training labels

    def plotData_1d(self, axisvals=None):
        '''Plot 1d (toy) data.'''
        plt.figure()
        plt.plot(self.x, self.y, ls='None', marker='+', color=DATACOLOR, ms=12, mew=2)
        if axisvals:
            plt.axis(axisvals)
        plt.grid()
        plt.xlabel('input x')
        plt.ylabel('target y')
        plt.show()
    
    def plotData_2d(self,x1,x2,t1,t2,p1,p2,axisvals=None):
        '''Plot 2d (toy) data.'''
        fig = plt.figure()
        plt.plot(x1[:,0], x1[:,1], 'b+', markersize = 12)
        plt.plot(x2[:,0], x2[:,1], 'r+', markersize = 12)
        pc = plt.contour(t1, t2, np.reshape(p2/(p1+p2), (t1.shape[0],t1.shape[1]) ))
        fig.colorbar(pc)
        plt.grid()
        if axisvals:
            plt.axis(axisvals)
        plt.show()

    def setPrior(self, mean=None, kernel=None):
        '''Set prior mean and cov'''
        if mean != None:
            self.meanfunc = mean
            self.usingDefaultMean = False
        if kernel != None:
            self.covfunc = kernel
            if type(kernel) is cov.Pre:
                self.usingDefaultMean = False
    

      [docs]    def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None):
        '''This method is used to sepecify optimization configuration. By default, gp uses a single run "minimize".

        :param method: Optimization methods. Possible values are:

                        "Minimize"   -> minimize by Carl Rasmussen (python implementation of "minimize" in GPML)

                        "CG"         -> conjugent gradient

                        "BFGS"       -> quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS)

                        "SCG"        -> scaled conjugent gradient (faster than CG) 
        :param num_restarts: Set if you want to run mulitiple times of optimization with different initial guess. 
                             It specifys the maximum number of runs/restarts/trials.
        :param min_threshold: Set if you want to run mulitiple times of optimization with different initial guess. 
                              It specifys the threshold of objective function value. Stop optimization when this value is reached.
        :param meanRange: The range of initial guess for mean hyperparameters. 
                          e.g. meanRange = [(-2,2), (-5,5), (0,1)].
                          Each tuple specifys the range (low, high) of this hyperparameter,
                          This is only the range of initial guess, during optimization process, optimal hyperparameters may go out of this range.

                          (-5,5) for each hyperparameter by default.
        :param covRange: The range of initial guess for kernel hyperparameters. Usage see meanRange
        :param likRange: The range of initial guess for likelihood hyperparameters. Usage see meanRange

        '''
        pass


      
    def optimize(self, x=None, y=None):
        '''
        Train optimal hyperparameters based on training data,
        adjust new hyperparameters to all mean/cov/lik functions
        '''
        if x != None:
            if x.ndim == 1:
                x = np.reshape(x, (x.shape[0],1))   
            self.x = x 

        if y != None:
            if y.ndim == 1:
                y = np.reshape(y, (y.shape[0],1))  
            self.y = y

        if self.usingDefaultMean and self.meanfunc == None:
            c = np.mean(y)
            self.meanfunc = mean.Const(c)    # adapt default prior mean wrt. training labels

        # optimize 
        optimalHyp, optimalNlZ = self.optimizer.findMin(self.x, self.y)
        self.nlZ= optimalNlZ

        # apply optimal hyp to all mean/cov/lik functions here
        self.optimizer._apply_in_objects(optimalHyp)
        self.fit()


    def fit(self, x=None, y=None, der=True):
        '''
        Fit the training data. Update negative log marginal likelihood(nlZ), 
        partial derivatives of nlZ w.r.t. each hyperparameter(dnlZ),
        and struct representation of the (approximate) posterior(post), 
        which consists of post.alpha, post.L, post.sW.
        '''
        if x != None:
            if x.ndim == 1:
                x = np.reshape(x, (x.shape[0],1))   
            self.x = x 
        if y != None:
            if y.ndim == 1:
                y = np.reshape(y, (y.shape[0],1))  
            self.y = y
        if self.usingDefaultMean and self.meanfunc == None:
            c = np.mean(y)
            self.meanfunc = mean.Const(c)    # adapt default prior mean wrt. training labels
        # call inference method
        if isinstance(self.likfunc, lik.Erf): #or is instance(self.likfunc, lik.Logistic):  
            uy = unique(self.y)        
            ind = ( uy != 1 )
            if any( uy[ind] != -1):
                raise Exception('You attempt classification using labels different from {+1,-1}')
        if not der:
            post, nlZ = self.inffunc.proceed(self.meanfunc, self.covfunc, self.likfunc, self.x, self.y, 2)
            self.nlZ = nlZ
            self.posterior = deepcopy(post)
            return nlZ, post          
        else:
            post, nlZ, dnlZ = self.inffunc.proceed(self.meanfunc, self.covfunc, self.likfunc, self.x, self.y, 3) 
            self.nlZ = nlZ 
            self.dnlZ = deepcopy(dnlZ)
            self.posterior = deepcopy(post)
            return nlZ, dnlZ, post    


    def predict(self, xs, ys=None):
        '''
        Prediction according to given inputs.
        Get predictive output means(ym), 
        predictive output variances(ys2),
        predictive latent means(fm),
        predictive latent variances(fs2),
        log predictive probabilities(lp).

        :param xs: test input
        :param ys: test target(optional)

        :return: ym, ys2, fm, fs2, lp
        '''
        meanfunc = self.meanfunc
        covfunc = self.covfunc
        likfunc = self.likfunc
        inffunc = self.inffunc
        x = self.x
        y = self.y
        if xs.ndim == 1:
            xs = np.reshape(xs, (xs.shape[0],1))   
        self.xs = xs
        if ys != None:
            if ys.ndim == 1:
                ys = np.reshape(ys, (ys.shape[0],1)) 
            self.ys = ys
        
        if self.posterior == None:   
            self.fit()        
        alpha = self.posterior.alpha
        L     = self.posterior.L
        sW    = self.posterior.sW
        
        nz = range(len(alpha[:,0]))         # non-sparse representation 
        if L == []:                         # in case L is not provided, we compute it
            K = covfunc.getCovMatrix(x=x[nz,:], mode='train')
            L = np.linalg.cholesky( (np.eye(nz) + np.dot(sW,sW.T)*K).T )
        Ltril     = np.all( np.tril(L,-1) == 0 ) # is L an upper triangular matrix?
        ns        = xs.shape[0]                  # number of data points
        nperbatch = 1000                         # number of data points per mini batch
        nact      = 0                            # number of already processed test data points
        ymu = np.zeros((ns,1))
        ys2 = np.zeros((ns,1))
        fmu = np.zeros((ns,1))
        fs2 = np.zeros((ns,1))
        lp  = np.zeros((ns,1))
        while nact<=ns-1:                              # process minibatches of test cases to save memory
            id  = range(nact,min(nact+nperbatch,ns))   # data points to process
            kss = covfunc.getCovMatrix(z=xs[id,:], mode='self_test')    # self-variances
            Ks  = covfunc.getCovMatrix(x=x[nz,:], z=xs[id,:], mode='cross')   # cross-covariances
            ms  = meanfunc.getMean(xs[id,:])         
            N   = (alpha.shape)[1]                     # number of alphas (usually 1; more in case of sampling)
            Fmu = np.tile(ms,(1,N)) + np.dot(Ks.T,alpha[nz])          # conditional mean fs|f
            fmu[id] = np.reshape(Fmu.sum(axis=1)/N,(len(id),1))       # predictive means
            if Ltril: # L is triangular => use Cholesky parameters (alpha,sW,L)
                V       = np.linalg.solve(L.T,np.tile(sW,(1,len(id)))*Ks)
                fs2[id] = kss - np.array([(V*V).sum(axis=0)]).T             # predictive variances
            else:     # L is not triangular => use alternative parametrization
                fs2[id] = kss + np.array([(Ks*np.dot(L,Ks)).sum(axis=0)]).T # predictive variances
            fs2[id] = np.maximum(fs2[id],0)            # remove numerical noise i.e. negative variances
            Fs2 = np.tile(fs2[id],(1,N))               # we have multiple values in case of sampling
            if ys == None:
                [Lp, Ymu, Ys2] = likfunc.proceed(None,Fmu[:],Fs2[:],None,None,3)
            else:
                [Lp, Ymu, Ys2] = likfunc.proceed(np.tile(ys[id],(1,N)), Fmu[:], Fs2[:],None,None,3)
            lp[id]  = np.reshape( np.reshape(Lp,(np.prod(Lp.shape),N)).sum(axis=1)/N , (len(id),1) )   # log probability; sample averaging
            ymu[id] = np.reshape( np.reshape(Ymu,(np.prod(Ymu.shape),N)).sum(axis=1)/N ,(len(id),1) )  # predictive mean ys|y and ...
            ys2[id] = np.reshape( np.reshape(Ys2,(np.prod(Ys2.shape),N)).sum(axis=1)/N , (len(id),1) ) # .. variance
            nact = id[-1]+1                  # set counter to index of next data point
        self.ym = ymu
        self.ys2 = ys2
        self.lp = lp
        self.fm = fmu
        self.fs2 = fs2
        if ys == None:
            return ymu, ys2, fmu, fs2, None
        else:
            return ymu, ys2, fmu, fs2, lp 


    def predict_with_posterior(self, post, xs, ys=None):
        '''
        Prediction with provided posterior
        (i.e. you already have the posterior and thus don't need fitting/training phases)
        Get predictive output means(ym), 
        predictive output variances(ys2),
        predictive latent means(fm),
        predictive latent variances(fs2),
        log predictive probabilities(lp).

        :param post: struct representation of posterior
        :param xs: test input
        :param ys: test target(optional)

        :return: ym, ys2, fm, fs2, lp
        '''
        meanfunc = self.meanfunc
        covfunc = self.covfunc
        likfunc = self.likfunc
        inffunc = self.inffunc
        x = self.x
        y = self.y
        if xs.ndim == 1:
            xs = np.reshape(xs, (xs.shape[0],1))   
        self.xs = xs
        if ys != None:
            if ys.ndim == 1:
                ys = np.reshape(ys, (ys.shape[0],1)) 
            self.ys = ys

        self.posterior = deepcopy(post)
        alpha = post.alpha
        L     = post.L
        sW    = post.sW

        nz = range(len(alpha[:,0]))         # non-sparse representation 
        if L == []:                         # in case L is not provided, we compute it
            K = covfunc.getCovMatrix(x=x[nz,:], mode='train')
            L = np.linalg.cholesky( (np.eye(nz) + np.dot(sW,sW.T)*K).T )
        Ltril     = np.all( np.tril(L,-1) == 0 ) # is L an upper triangular matrix?
        ns        = xs.shape[0]                  # number of data points
        nperbatch = 1000                         # number of data points per mini batch
        nact      = 0                            # number of already processed test data points
        ymu = np.zeros((ns,1))
        ys2 = np.zeros((ns,1))
        fmu = np.zeros((ns,1))
        fs2 = np.zeros((ns,1))
        lp  = np.zeros((ns,1))
        while nact<=ns-1:                              # process minibatches of test cases to save memory
            id  = range(nact,min(nact+nperbatch,ns))   # data points to process
            kss = covfunc.getCovMatrix(z=xs[id,:], mode='self_test')    # self-variances
            Ks  = covfunc.getCovMatrix(x=x[nz,:], z=xs[id,:], mode='cross')   # cross-covariances
            ms  = meanfunc.getMean(xs[id,:])         
            N   = (alpha.shape)[1]                     # number of alphas (usually 1; more in case of sampling)
            Fmu = np.tile(ms,(1,N)) + np.dot(Ks.T,alpha[nz])          # conditional mean fs|f
            fmu[id] = np.reshape(Fmu.sum(axis=1)/N,(len(id),1))       # predictive means
            if Ltril: # L is triangular => use Cholesky parameters (alpha,sW,L)
                V       = np.linalg.solve(L.T,np.tile(sW,(1,len(id)))*Ks)
                fs2[id] = kss - np.array([(V*V).sum(axis=0)]).T             # predictive variances
            else:     # L is not triangular => use alternative parametrization
                fs2[id] = kss + np.array([(Ks*np.dot(L,Ks)).sum(axis=0)]).T # predictive variances
            fs2[id] = np.maximum(fs2[id],0)            # remove numerical noise i.e. negative variances
            Fs2 = np.tile(fs2[id],(1,N))               # we have multiple values in case of sampling
            if ys == None:
                [Lp, Ymu, Ys2] = likfunc.proceed(None,Fmu[:],Fs2[:],None,None,3)
            else:
                [Lp, Ymu, Ys2] = likfunc.proceed(np.tile(ys[id],(1,N)), Fmu[:], Fs2[:],None,None,3)
            lp[id]  = np.reshape( np.reshape(Lp,(np.prod(Lp.shape),N)).sum(axis=1)/N , (len(id),1) )   # log probability; sample averaging
            ymu[id] = np.reshape( np.reshape(Ymu,(np.prod(Ymu.shape),N)).sum(axis=1)/N ,(len(id),1) )  # predictive mean ys|y and ...
            ys2[id] = np.reshape( np.reshape(Ys2,(np.prod(Ys2.shape),N)).sum(axis=1)/N , (len(id),1) ) # .. variance
            nact = id[-1]+1                  # set counter to index of next data point
        self.ym = ymu
        self.ys2 = ys2
        self.lp = lp
        self.fm = fmu
        self.fs2 = fs2
        if ys == None:
            return ymu, ys2, fmu, fs2, None
        else:
            return ymu, ys2, fmu, fs2, lp 






class GPR(GP):
    '''Gaussian Process Regression'''
    def __init__(self):
        super(GPR, self).__init__()
        self.meanfunc = mean.Zero()                        # default prior mean 
        self.covfunc = cov.RBF()                           # default prior covariance
        self.likfunc = lik.Gauss()                         # likihood with default noise variance 0.1
        self.inffunc = inf.Exact()                         # inference method
        self.optimizer = opt.Minimize(self)                # default optimizer       

    def setNoise(self,log_sigma):
        '''Set noise variance other than default'''
        self.likfunc = lik.Gauss(log_sigma)

    def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None):
        '''Set Optimizer. See base class.'''
        conf = None
        if (num_restarts!=None) or (min_threshold!=None):
            conf = pyGPs.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc)
            conf.num_restarts = num_restarts
            conf.min_threshold = min_threshold
            if meanRange != None:
                conf.meanRange = meanRange
            if covRange != None:
                conf.covRange = covRange
            if likRange != None:
                conf.likRange = likRange   
        if method == "Minimize":
            self.optimizer = opt.Minimize(self,conf)            
        elif method == "SCG":
            self.optimizer = opt.SCG(self,conf)  
        elif method == "CG":
            self.optimizer = opt.CG(self,conf)  
        elif method == "BFGS":
            self.optimizer = opt.BFGS(self,conf)  
                       
    def plot(self,axisvals=None):
        '''Plot 1d GP regression.'''
        xs = self.xs
        x = self.x
        y = self.y
        ym = self.ym
        ys2 = self.ys2
        plt.figure()
        xss  = np.reshape(xs,(xs.shape[0],))
        ymm  = np.reshape(ym,(ym.shape[0],))
        ys22 = np.reshape(ys2,(ys2.shape[0],))
        plt.plot(x, y, color=DATACOLOR, ls='None', marker='+',ms=12, mew=2)
        plt.plot(xs, ym, color=MEANCOLOR, ls='-', lw=3.)
        plt.fill_between(xss,ymm + 2.*np.sqrt(ys22), ymm - 2.*np.sqrt(ys22), facecolor=SHADEDCOLOR,linewidths=0.0)
        plt.grid()
        if axisvals != None:
            plt.axis(axisvals)
        plt.xlabel('input x')
        plt.ylabel('target y')
        plt.show()

    def useInference(self, newInf):
        '''
        Use another inference techinique other than default exact inference.
        
        :param str newInf: 'Laplace' or 'EP'
        '''

        if newInf == "Laplace":
            self.inffunc = inf.Laplace()
        elif newInf == "EP":
            self.inffunc = inf.EP()
        else:
            raise Exception('Possible inf values are "Laplace", "EP".')

    def useLikelihood(self,newLik):
        if newLik == "Laplace":
            self.likfunc = lik.Laplace()
            self.inffunc = inf.EP()
        else:
            raise Exception('Possible lik values are "Laplace".')



class GPC(GP):
    '''Gaussian Process Classification'''
    def __init__(self):
        super(GPC, self).__init__()
        self.meanfunc = mean.Zero()                        # default prior mean 
        self.covfunc = cov.RBF()                           # default prior covariance
        self.likfunc = lik.Erf()                           # erf likihood 
        self.inffunc = inf.EP()                            # default inference method
        self.optimizer = opt.Minimize(self)                # default optimizer       

    def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None):
        '''Set Optimizer. See base class.'''
        conf = None
        if (num_restarts!=None) or (min_threshold!=None):
            conf = pyGPs.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc)
            conf.num_restarts = num_restarts
            conf.min_threshold = min_threshold
            if meanRange != None:
                conf.meanRange = meanRange
            if covRange != None:
                conf.covRange = covRange
            if likRange != None:
                conf.likRange = likRange   
        if method == "Minimize":
            self.optimizer = opt.Minimize(self,conf)            
        elif method == "SCG":
            self.optimizer = opt.SCG(self,conf)  
        elif method == "CG":
            self.optimizer = opt.CG(self,conf)  
        elif method == "BFGS":
            self.optimizer = opt.BFGS(self,conf)  
                       
    def plot(self,x1,x2,t1,t2,axisvals=None): 
        '''Plot 2d gp classification.'''
        fig = plt.figure()
        plt.plot(x1[:,0], x1[:,1], 'b+', markersize = 12)
        plt.plot(x2[:,0], x2[:,1], 'r+', markersize = 12)
        pc = plt.contour(t1, t2, np.reshape(np.exp(self.lp), (t1.shape[0],t1.shape[1]) ))
        fig.colorbar(pc)
        plt.grid()
        if axisvals != None:
            plt.axis(axisvals)
        plt.show()

    def useInference(self, newInf):
        '''
        Use another inference techinique other than default EP inference.

        :param str newInf: 'Laplace'
        '''
        if newInf == "Laplace":
            self.inffunc = inf.Laplace()
        else:
            raise Exception('Possible inf values are "Laplace".')

    def useLikelihood(self,newLik):
        '''
        Use another likelihood function other than default Erf inference.
        (Not used in this version)

        :param str newLik: 'Logistic'
        '''
        if newLik == "Logistic":
            pass
            #self.likfunc = lik.Logistic()
        else:
            raise Exception('Possible lik values are "Logistic".')




class GPMC(object):
    '''This is a one vs. one classification wrapper for GPC'''
    def __init__(self, n_class):
        self.meanfunc = mean.Zero()                # default prior mean 
        self.covfunc = cov.RBF()                   # default prior covariance
        self.n_class = n_class                     # number of different classes
        self.x_all = None
        self.y_all = None
        self.newInf = None                         # new inference? -> call useInference 
        self.newLik = None                         # new likelihood? -> call useLikelihood
        self.newPrior = False

    def setPrior(self, mean=None, kernel=None):
        '''set prior mean function and covariance function'''
        if mean != None:
            self.meanfunc = mean
        if kernel != None:
            self.covfunc = kernel
            if type(kernel) is cov.Pre:
                self.usingDefaultMean = False
        self.newPrior = True

    def useInference(self, newInf):
        '''Use another inference techinique other than default EP inference.'''
        self.newInf = newInf

    def useLikelihood(self,newLik):
        '''Use another likelihood function other than default Erf inference.'''
        self.newLik = newLik    
    
    def setData(self,x,y):
        '''Set data for multi-class model'''
        if x.ndim == 1:
            x = np.reshape(x, (x.shape[0],1))  
        if y.ndim == 1:
            y = np.reshape(y, (y.shape[0],1))  
        self.x_all = x
        self.y_all = y

    def fitAndPredict(self, xs):
        '''
        Fitting model and prediction.
        predictive_vote is a matrix where
        row i is each test point i
        column j is the probability for being class j

        :return: predictive_vote
        '''
        if xs.ndim == 1:
            xs = np.reshape(xs, (xs.shape[0],1)) 
        predictive_vote = np.zeros((xs.shape[0],self.n_class))
        for i in xrange(self.n_class):         # classifier for class i...
            for j in xrange(i+1,self.n_class): # ...and class j
                x,y = self.createBinaryClass(i,j)
                model = GPC()
                if self.newPrior:
                    model.setPrior(mean=self.meanfunc, kernel=self.covfunc)
                if self.newInf:
                    model.useInference(self.newInf)
                if self.newLik:
                    model.useLikelihood(self.newLik)
                model.fit(x,y)               # fitting
                ym = model.predict(xs)[0]
                ym += 1     # now scale into 0 to 2,  ym=0 is class j, ym=2 is class i 
                vote_i = np.zeros((xs.shape[0],self.n_class))
                vote_j = np.zeros((xs.shape[0],self.n_class))
                vote_i[:,i:i+1] = ym
                vote_j[:,j:j+1] = 2-ym
                predictive_vote += vote_i
                predictive_vote += vote_j
        predictive_vote /=  predictive_vote.sum(axis=1)[:,np.newaxis]
        return predictive_vote


    def optimizeAndPredict(self, xs):
        '''
        Optimizing model and prediction.
        predictive_vote is a matrix where
        row i is each test point i
        column j is the probability for being class j

        :return: predictive_vote
        '''
        if xs.ndim == 1:
            xs = np.reshape(xs, (xs.shape[0],1)) 
        predictive_vote = np.zeros((xs.shape[0],self.n_class))
        for i in xrange(self.n_class):         # classifier for class i...
            for j in xrange(i+1,self.n_class): # ...and class j
                x,y = self.createBinaryClass(i,j)
                model = GPC()
                if self.newPrior:
                    model.setPrior(mean=self.meanfunc, kernel=self.covfunc)
                if self.newInf:
                    model.useInference(self.newInf)
                if self.newLik:
                    model.useLikelihood(self.newLik)
                model.optimize(x,y)               # training
                ym = model.predict(xs)[0]
                ym += 1     # now scale into 0 to 2,  ym=0 is class j, ym=2 is class i 
                vote_i = np.zeros((xs.shape[0],self.n_class))
                vote_j = np.zeros((xs.shape[0],self.n_class))
                vote_i[:,i:i+1] = ym
                vote_j[:,j:j+1] = 2-ym
                predictive_vote += vote_i
                predictive_vote += vote_j
        predictive_vote /=  predictive_vote.sum(axis=1)[:,np.newaxis]
        return predictive_vote

    def createBinaryClass(self, i,j):
        ''' Create dataset x(data) and y(label) which only contains class i and j.
        Relabel class i to +1 and class j to -1'''
        class_i = []
        class_j = []
        for index in xrange(len(self.y_all)):       # check all classes
            target = self.y_all[index]
            if target == i:
                class_i.append(index)
            elif target == j:
                class_j.append(index)
        n1 = len(class_i)
        n2 = len(class_j)
        class_i.extend(class_j)
        x = self.x_all[class_i,:]
        y = np.concatenate((np.ones((1,n1)),-np.ones((1,n2))),axis=1).T
        return x,y





class GP_FITC(GP):
    '''FITC GP base class'''
    def __init__(self):
        super(GP_FITC, self).__init__()
        self.u = None

    # overriding
    def setData(self, x, y, value_per_axis=5):
        '''Set training data and derive deault inducing_points.

        :param x: training data
        :param y: training target
        :param int value_per_axis: number of value in each dimension 
        when using a uni-distant default inducing points'''
        if x.ndim == 1:
            x = np.reshape(x, (x.shape[0],1))  
        if y.ndim == 1:
            y = np.reshape(y, (y.shape[0],1))  
        self.x = x
        self.y = y
        if self.usingDefaultMean:
            c = np.mean(y)
            self.meanfunc = mean.Const(c)    # adapt default prior mean wrt. training labels

        # get range of x in each dimension
        # 5 uniformally selected value for each dimension
        gridAxis=[]
        for d in xrange(x.shape[1]):
            column = x[:,d]
            mini = np.min(column)
            maxi = np.max(column)
            axis = np.linspace(mini,maxi,value_per_axis)
            gridAxis.append(axis)
        # default inducing points-> a grid
        if self.u == None:
            self.u = np.array(list(itertools.product(*gridAxis)))
            self.covfunc = self.covfunc.fitc(self.u)

    # overriding
    def setPrior(self, mean=None, kernel=None, inducing_points=None):
        '''Set GP prior and inducing points'''
        if kernel != None:
            if inducing_points != None:
                self.covfunc = kernel.fitc(inducing_points)
                self.u = inducing_points
            else:
                if self.u != None:
                    self.covfunc = kernel.fitc(self.u)
                else:
                    raise error("To use default inducing points, please call setData() first!")
            if type(kernel) is cov.Pre:
                self.usingDefaultMean = False
        if mean != None:
            self.meanfunc = mean
            self.usingDefaultMean = False




class GPR_FITC(GP_FITC):
    '''Gaussian Process Regression FITC'''
    def __init__(self):
        super(GPR_FITC, self).__init__()
        self.meanfunc = mean.Zero()                        # default prior mean 
        self.covfunc = cov.RBF()                           # default prior covariance
        self.likfunc = lik.Gauss()                         # likihood with default noise variance 0.1
        self.inffunc = inf.FITC_Exact()                    # inference method
        self.optimizer = opt.Minimize(self)                # default optimizer 
        self.u = None                                      # no default inducing points

    def setNoise(self,log_sigma):
        '''Set noise variance other than default value'''
        self.likfunc = lik.Gauss(log_sigma)

    def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None):
        '''Set Optimizer. See base class.'''
        conf = None
        if (num_restarts!=None) or (min_threshold!=None):
            conf = pyGPs.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc)
            conf.num_restarts = num_restarts
            conf.min_threshold = min_threshold
            if meanRange != None:
                conf.meanRange = meanRange
            if covRange != None:
                conf.covRange = covRange
            if likRange != None:
                conf.likRange = likRange   
        if method == "Minimize":
            self.optimizer = opt.Minimize(self,conf)            
        elif method == "SCG":
            self.optimizer = opt.SCG(self,conf)  
        elif method == "CG":
            self.optimizer = opt.CG(self,conf)  
        elif method == "BFGS":
            self.optimizer = opt.BFGS(self,conf)  
                       
    def plot(self,axisvals=None):
        '''Plot 1d GP FITC regression.'''
        plt.figure()
        xss  = np.reshape(self.xs,(self.xs.shape[0],))
        ymm  = np.reshape(self.ym,(self.ym.shape[0],))
        ys22 = np.reshape(self.ys2,(self.ys2.shape[0],))
        plt.plot(self.x, self.y, color=DATACOLOR, ls='None', marker='+',ms=12, mew=2)
        plt.plot(self.xs, self.ym, color=MEANCOLOR, ls='-', lw=3.)
        plt.fill_between(xss,ymm + 2.*np.sqrt(ys22), ymm - 2.*np.sqrt(ys22), facecolor=SHADEDCOLOR,linewidths=0.0)
        plt.grid()
        if axisvals != None:                
            plt.axis(axisvals)
        plt.xlabel('input x')
        plt.ylabel('output y')
        plt.plot(self.u,np.ones_like(self.u), ls='None', color='k',marker='x',markersize=12,mew=2)
        plt.show()

    def useInference(self, newInf):
        '''
        Use another inference techinique other than default exact inference.

        :param str newInf: 'Laplace' or 'EP'
        '''
        if newInf == "Laplace":
            self.inffunc = inf.FITC_Laplace()
        elif newInf == "EP":
            self.inffunc = inf.FITC_EP()
        else:
            raise Exception('Possible inf values are "Laplace", "EP".')

    def useLikelihood(self,newLik):
        '''
        Use another inference techinique other than default Gaussian likelihood.

        :param str newLik: 'Laplace'
        '''
        if newLik == "Laplace":
            self.likfunc = lik.Laplace()
            self.inffunc = inf.FITC_EP()
        else:
            raise Exception('Possible lik values are "Laplace".')   



   
class GPC_FITC(GP_FITC):
    '''Gaussian Process Classification FITC'''
    def __init__(self):
        super(GPC_FITC, self).__init__()
        self.meanfunc = mean.Zero()                        # default prior mean 
        self.covfunc = cov.RBF()                           # default prior covariance
        self.likfunc = lik.Erf()                           # erf liklihood
        self.inffunc = inf.FITC_EP()                       # default inference method
        self.optimizer = opt.Minimize(self)                # default optimizer 
        self.u = None                                      # no default inducing points

    def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None):
        '''Set optimizer. See base class.'''
        conf = None
        if (num_restarts!=None) or (min_threshold!=None):
            conf = pyGPs.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc)
            conf.num_restarts = num_restarts
            conf.min_threshold = min_threshold
            if meanRange != None:
                conf.meanRange = meanRange
            if covRange != None:
                conf.covRange = covRange
            if likRange != None:
                conf.likRange = likRange   
        if method == "Minimize":
            self.optimizer = opt.Minimize(self,conf)            
        elif method == "SCG":
            self.optimizer = opt.SCG(self,conf)  
        elif method == "CG":
            self.optimizer = opt.CG(self,conf)  
        elif method == "BFGS":
            self.optimizer = opt.BFGS(self,conf)  
                       
    def plot(self,x1,x2,t1,t2,axisvals=None): 
        '''Plot 2d GP FITC classification.'''
        fig = plt.figure()
        plt.plot(x1[:,0], x1[:,1], 'b+', markersize = 12)
        plt.plot(x2[:,0], x2[:,1], 'r+', markersize = 12)
        plt.plot(self.u[:,0],self.u[:,1],'ko', markersize=12)
        pc = plt.contour(t1, t2, np.reshape(np.exp(self.lp), (t1.shape[0],t1.shape[1]) ))
        fig.colorbar(pc)
        plt.grid()
        if axisvals != None:    
            plt.axis(axisvals)
        plt.show()  

    def useInference(self, newInf):
        '''
        Use another inference techinique other than default exact inference.

        :param str newInf: 'Laplace' or 'EP'
        '''
        if newInf == "Laplace":
            self.inffunc = inf.FITC_Laplace()
        else:
            raise Exception('Possible inf values are "Laplace".')

    def useLikelihood(self,newLik):
        '''
        Use another inference techinique other than default Erf likelihood.
        (Not used in this version)

        :param str newLik: 'Logistic'
        '''
        if newLik == "Logistic":
            pass
            #self.likfunc = lik.Logistic()
        else:
            raise Exception('Possible lik values are "Logistic".')

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      © Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting. Created using Sphinx 1.2.2.
      pyGPs-1.2/doc/build/html/_modules/pyGPs/Core/inf.html000066400000000000000000010201021236346020200224230ustar00rootroot00000000000000 pyGPs.Core.inf — pyGPs v1.2 documentation

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      Source code for pyGPs.Core.inf

      #================================================================================
#    Marion Neumann [marion dot neumann at uni-bonn dot de]
#    Daniel Marthaler [marthaler at ge dot com]
#    Shan Huang [shan dot huang at iais dot fraunhofer dot de]
#    Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]
#
#    This file is part of pyGPs.
#    The software package is released under the BSD 2-Clause (FreeBSD) License.
#
#    Copyright (c) by
#    Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014
#================================================================================

# Inference methods: Compute the (approximate) posterior for a Gaussian process.
# Methods currently implemented include:
#
#   infExact         Exact inference (only possible with Gaussian likelihood)
#   infLaplace       Laplace's Approximation
#   infEP            Expectation Propagation
#   infVB            [NOT IMPLEMENTED!] Variational Bayes Approximation 
#
#   infFITC          Large scale regression with approximate covariance matrix
#   infFITC_Laplace  Large scale inference  with approximate covariance matrix
#   infFITC_EP       Large scale inference  with approximate covariance matrix
#
#   infMCMC     [NOT IMPLEMENTED!]
#               Markov Chain Monte Carlo and Annealed Importance Sampling
#               We offer two samplers.
#                 - hmc: Hybrid Monte Carlo
#                 - ess: Elliptical Slice Sampling
#               No derivatives w.r.t. to hyperparameters are provided.
#
#   infLOO      [NOT IMPLEMENTED!]
#               Leave-One-Out predictive probability and Least-Squares Approxim.
#
#
# This is a object-oriented python implementation of gpml functionality 
# (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18).
# based on the functional-version of python implementation
# (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013)
# 
# Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013


import numpy as np
import lik, cov
from copy import copy,deepcopy
from tools import solve_chol,brentmin,cholupdate

np.seterr(all='ignore')

class postStruct(object):
    '''
    Data structure for posterior

    | post.alpha: 1d array containing inv(K)*m, 
    | where K is the prior covariance matrix and m the approx posterior mean
    | post.sW: 1d array containing diagonal of sqrt(W)
    | the approximate posterior covariance matrix is inv(inv(K)+W)
    | post.L : 2d array, L = chol(sW*K*sW+identity(n))
    '''
    def __init__(self):
        self.alpha = np.array([])
        self.L     = np.array([])
        self.sW    = np.array([])

class dnlZStruct(object):
    '''
    Data structure for the derivatives of mean, cov and lik functions.

    |dnlZ.mean: list of derivatives for each hyperparameters in mean function
    |dnlZ.cov: list of derivatives for each hyperparameters in covariance function
    |dnlZ.lik: list of derivatives for each hyperparameters in likelihood function
    '''
    def __init__(self, m, c, l):
        self.mean = []
        self.cov = []
        self.lik = []
        if m.hyp:
            self.mean = [0 for i in xrange(len(m.hyp))]
        if c.hyp:
            self.cov  = [0 for i in xrange(len(c.hyp))]
        if l.hyp:
            self.lik  = [0 for i in xrange(len(l.hyp))]        

class Inference(object):
    '''
    Base class for inference. Defined several tool methods in it. 
    '''
    def __init__(self):
        pass

    def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1):
        '''
        Inference computation based on inputs.
        post, nlZ, dnlZ = inf.proceed(mean, cov, lik, x, y)

            | INPUT:
            | cov: name of the covariance function (see covFunctions.m)
            | lik: name of the likelihood function (see likFunctions.m)
            | x: n by D matrix of training inputs 
            | y: 1d array (of size n) of targets

            | OUTPUT:
            | post(postStruct): struct representation of the (approximate) posterior containing: 
            | nlZ: returned value of the negative log marginal likelihood
            | dnlZ(dnlZStruct): struct representation for derivatives of the negative log marginal likelihood
            | w.r.t. each hyperparameter.

        Usually, the approximate posterior to be returned admits the form: 
        N(m=K*alpha, V=inv(inv(K)+W)), where alpha is a vector and W is diagonal;
        if not, then L contains instead -inv(K+inv(W)), and sW is unused.

        For more information on the individual approximation methods and their
        implementations, see the respective inference function below. See also gp.py

        :param meanfunc: mean function
        :param covfunc: covariance function
        :param likfunc: likelihood function
        :param x: training data
        :param y: training labels
        :param nargout: specify the number of output(1,2 or 3)
        :return: posterior, negative-log-marginal-likelihood, derivative for negative-log-marginal-likelihood-likelihood
        '''
        pass

    def _epComputeParams(self, K, y, ttau, tnu, likfunc, m, inffunc):
        n     = len(y)                                                # number of training cases
        ssi   = np.sqrt(ttau)                                         # compute Sigma and mu
        L     = np.linalg.cholesky(np.eye(n)+np.dot(ssi,ssi.T)*K).T   # L'*L=B=eye(n)+sW*K*sW
        V     = np.linalg.solve(L.T,np.tile(ssi,(1,n))*K)
        Sigma = K - np.dot(V.T,V)
        mu    = np.dot(Sigma,tnu)
        Dsigma = np.reshape(np.diag(Sigma),(np.diag(Sigma).shape[0],1)) 
        tau_n = 1/Dsigma - ttau               # compute the log marginal likelihood
        nu_n  = mu/Dsigma-tnu + m*tau_n       # vectors of cavity parameters
        lZ    = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc)
        nlZ   = np.log(np.diag(L)).sum() - lZ.sum() - np.dot(tnu.T,np.dot(Sigma,tnu))/2  \
                - np.dot((nu_n-m*tau_n).T,((ttau/tau_n*(nu_n-m*tau_n)-2*tnu) / (ttau+tau_n)))/2 \
                + (tnu**2/(tau_n+ttau)).sum()/2.- np.log(1.+ttau/tau_n).sum()/2.
        return Sigma, mu, nlZ[0], L
    
    def _logdetA(self,K,w,nargout):
        '''
        Compute the log determinant ldA and the inverse iA of a square nxn matrix
        A = eye(n) + K*diag(w) from its LU decomposition; for negative definite A, we 
        return ldA = Inf. We also return mwiA = -diag(w)*inv(A).
        [ldA,iA,mwiA] = _logdetA(K,w)'''
        n = K.shape[0]
        assert(K.shape[0] == K.shape[1])
        A = np.eye(n) + K*np.tile(w.T,(n,1))
        [L,U,P] = np.linalg.lu(A)    # compute LU decomposition, A = P'*L*U
        u = np.diag(U)           
        signU = np.prod(np.sign(u))  # sign of U
        detP = 1                     # compute sign (and det) of the permutation matrix P
        p = np.dot(P,np.array(range(n)).T)
        for ii in range(n):
            if ii != p[ii]: 
                detP = -detP
                j= [jj for jj,val in enumerate(p) if val == ii]
                p[ii,j[0]] = p[j[0],ii] 
        if signU != detP:            # log becomes complex for negative values, encoded by infinity
            ldA = np.inf
        else:                        # det(L) = 1 and U triangular => det(A) = det(P)*prod(diag(U))
            ldA = np.log(np.abs(u)).sum()
        if nargout>1:
            iA = inv(U)*inv(L)*P; 
            if nargout>2: 
                mwiA = -np.tile(w,(1,n))*iA
                return ldA,iA,mwiA
            else:
                return ldA,iA
        else:
            return ldA  
    
    def _Psi_line(self,s,dalpha,alpha,K,m,likfunc,y,inffunc):
        '''Criterion Psi at alpha + s*dalpha for line search
        [Psi,alpha,f,dlp,W] = _Psi_line(s,dalpha,alpha,hyp,K,m,lik,y,inf)'''
        alpha = alpha + s*dalpha
        f = np.dot(K,alpha) + m
        [lp,dlp,d2lp] = likfunc.proceed(y,f,None,inffunc,None,3) 
        W = -d2lp
        Psi = np.dot(alpha.T,(f-m))/2. - lp.sum()
        return Psi[0],alpha,f,dlp,W  
    
    def _epfitcZ(self,d,P,R,nn,gg,ttau,tnu,d0,R0,P0,y,likfunc,m,inffunc):
        '''Compute the marginal likelihood approximation
        effort is O(n*nu^2) provided that nu<n'''
        T = np.dot(np.dot(R,R0),P)              # temporary variable
        diag_sigma = d + np.array([(T*T).sum(axis=0)]).T 
        mu = nn + np.dot(P.T,gg)                # post moments O(n*nu^2)
        tau_n = 1./diag_sigma-ttau              # compute the log marginal likelihood
        nu_n  = mu/diag_sigma - tnu + m*tau_n   # vectors of cavity parameters
        lZ = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc, None, 1)
        nu = gg.shape[0]
        U = np.dot(R0,P0).T*np.tile(1/np.sqrt(d0+1/ttau),(1,nu))
        L = np.linalg.cholesky(np.eye(nu)+np.dot(U.T,U)).T
        ld = 2.*np.log(np.diag(L)).sum() + (np.log(d0+1/ttau)).sum() + (np.log(ttau)).sum()
        t = np.dot(T,tnu); tnu_Sigma_tnu = np.dot(tnu.T,(d*tnu)) + np.dot(t.T,t)
        nlZ = ld/2. - lZ.sum() -tnu_Sigma_tnu/2. \
            -np.dot((nu_n-m*tau_n).T,((ttau/tau_n*(nu_n-m*tau_n)-2.*tnu)/(ttau+tau_n)))/2. \
            + (tnu**2/(tau_n+ttau)).sum()/2. - (np.log(1+ttau/tau_n)).sum()/2.
        return nlZ,nu_n,tau_n
    
    def _epfitcRefresh(self,d0,P0,R0,R0P0,w,b):
        '''Refresh the representation of the posterior from initial and site parameters
        to prevent possible loss of numerical precision after many epfitcUpdates
        effort is O(n*nu^2) provided that nu<n
        Sigma = inv(inv(K)+diag(W)) = diag(d) + P'*R0'*R'*R*R0*P.'''
        nu = R0.shape[0]                                 # number of inducing points
        rot180   = lambda A: np.rot90(np.rot90(A))       # little helper functions
        chol_inv = lambda A: np.linalg.solve( rot180( np.linalg.cholesky(rot180(A)) ),np.eye(nu)) # chol(inv(A))
        t  = 1/(1+d0*w)                                  # temporary variable O(n)
        d  = d0*t                                        # O(n)
        P  = np.tile(t.T,(nu,1))*P0                      # O(n*nu)
        T  = np.tile((w*t).T,(nu,1))*R0P0                # temporary variable O(n*nu^2)
        R  = chol_inv(np.eye(nu)+np.dot(R0P0,T.T))       # O(n*nu^3)
        nn = d*b                                                  # O(n)
        gg = np.dot(R0.T,np.dot(R.T,np.dot(R,np.dot(R0P0,t*b))))  # O(n*nu)
        return d,P,R,nn,gg
    
    def _epfitcUpdate(self,d,P_i,R,nn,gg,w,b,ii,w_i,b_i,m,d0,P0,R0):
        dwi = w_i-w[ii]
        dbi = b_i-b[ii]
        hi = nn[ii] + m[ii] + np.dot(P_i.T,gg)           # posterior mean of site i O(nu)
        t = 1+dwi*d[ii]
        d[ii] = d[ii]/t                                  # O(1)
        nn[ii] = d[ii]*b_i                               # O(1)
        r = 1+d0[ii]*w[ii]
        r = (r*r)/dwi + r*d0[ii]
        v = np.dot(R,np.dot(R0,P0[:,ii]))
        v = np.reshape(v,(v.shape[0],1))
        r = 1/(r+np.dot(v.T,v))
        if r>0:
            R = cholupdate(R,np.sqrt(r)*np.dot(R.T,v),'-')
        else:
            R = cholupdate(R,np.sqrt(-r)*np.dot(R.T,v),'+')
        ttemp = np.dot(R0.T,np.dot(R.T,np.dot(R,np.dot(R0,P_i))))
        gg = gg + ((dbi-dwi*(hi-m[ii]))/t) * np.reshape(ttemp,(ttemp.shape[0],1)) # O(nu^2)
        w[ii] = w_i; b[ii] = b_i;                          # update site parameters O(1)
        P_i = P_i/t                                        # O(nu)
        return d,P_i,R,nn,gg,w,b
    
    def _mvmZ(self,x,RVdd,t):
        '''Matrix vector multiplication with Z=inv(K+inv(W))'''
        Zx = t*x - np.dot(RVdd.T,np.dot(RVdd,x))
        return Zx

    def _mvmK(self,al,V,d0):
        '''Matrix vector multiplication with approximate covariance matrix'''
        Kal = np.dot(V.T,np.dot(V,al)) + d0*al
        return Kal

    def _Psi_lineFITC(self,s,dalpha,alpha,V,d0,m,likfunc,y,inffunc):
        '''Criterion Psi at alpha + s*dalpha for line search'''
        alpha = alpha + s*dalpha
        f = self._mvmK(alpha,V,d0) + m
        vargout = likfunc.proceed(y,f,None,inffunc,None,3) 
        lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] 
        W = -d2lp
        Psi = np.dot(alpha.T,(f-m))/2. - lp.sum()
        return Psi[0],alpha,f,dlp,W

    def _fitcRefresh(self,d0,P0,R0,R0P0, w):
        '''Refresh the representation of the posterior from initial and site parameters
        to prevent possible loss of numerical precision after many epfitcUpdates
        effort is O(n*nu^2) provided that nu<n
        Sigma = inv(inv(K)+diag(W)) = diag(d) + P'*R0'*R'*R*R0*P.'''
        nu = R0.shape[0]                                  # number of inducing points
        rot180   = lambda A: np.rot90(np.rot90(A))        # little helper functions
        chol_inv = lambda A: np.linalg.solve( rot180( np.linalg.cholesky(rot180(A)) ),np.eye(nu)) # chol(inv(A))
        t  = 1/(1+d0*w)                                   # temporary variable O(n)
        d  = d0*t                                         # O(n)
        P  = np.tile(t.T,(nu,1))*P0;                      # O(n*nu)
        T  = np.tile((w*t).T,(nu,1))*R0P0;                # temporary variable O(n*nu^2)
        R  = chol_inv(np.eye(nu)+np.dot(R0P0,T.T))        # O(n*nu^3)
        return d,P,R



      [docs]class Exact(Inference):
    '''
    Exact inference for a GP with Gaussian likelihood. Compute a parametrization
    of the posterior, the negative log marginal likelihood and its derivatives
    w.r.t. the hyperparameters.
    '''
    def __init__(self):
        self.name = "Exact inference"
    def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1):
        if not isinstance(likfunc, lik.Gauss):
            raise Exception ('Exact inference only possible with Gaussian likelihood')
        n, D = x.shape
        K = covfunc.getCovMatrix(x=x, mode='train')            # evaluate covariance matrix
        m = meanfunc.getMean(x)                                # evaluate mean vector
        
        sn2   = np.exp(2*likfunc.hyp[0])                       # noise variance of likGauss
        L     = np.linalg.cholesky(K/sn2+np.eye(n)).T          # Cholesky factor of covariance with noise
        alpha = solve_chol(L,y-m)/sn2
        post = postStruct()
        post.alpha = alpha                                     # return the posterior parameters
        post.sW    = np.ones((n,1))/np.sqrt(sn2)               # sqrt of noise precision vector
        post.L     = L                                         # L = chol(eye(n)+sW*sW'.*K)

        if nargout>1:                                          # do we want the marginal likelihood?
            nlZ = np.dot((y-m).T,alpha)/2. + np.log(np.diag(L)).sum() + n*np.log(2*np.pi*sn2)/2. # -log marg lik
            if nargout>2:                                      # do we want derivatives?
                dnlZ = dnlZStruct(meanfunc, covfunc, likfunc)  # allocate space for derivatives
                Q = solve_chol(L,np.eye(n))/sn2 - np.dot(alpha,alpha.T) # precompute for convenience
                dnlZ.lik = [sn2*np.trace(Q)]
                if covfunc.hyp:
                    for ii in range(len(covfunc.hyp)):
                        dnlZ.cov[ii] = (Q*covfunc.getDerMatrix(x=x, mode='train', der=ii)).sum()/2.
                if meanfunc.hyp:
                    for ii in range(len(meanfunc.hyp)): 
                        dnlZ.mean[ii] = np.dot(-meanfunc.getDerMatrix(x, ii).T,alpha)
                        dnlZ.mean[ii] = dnlZ.mean[ii][0,0]
                return post, nlZ[0,0], dnlZ
            return post, nlZ[0,0]
        return post


      

      [docs]class FITC_Exact(Inference):
    '''
    FITC approximation to the posterior Gaussian process. The function is
    equivalent to infExact with the covariance function:
    Kt = Q + G; G = diag(g); g = diag(K-Q);  Q = Ku' * inv(Quu) * Ku; 
    where Ku and Kuu are covariances w.r.t. to inducing inputs xu, snu2 = sn2/1e6
    is the noise of the inducing inputs and Quu = Kuu + snu2*eye(nu).
    '''
    def __init__(self):
        self.name = 'FICT exact inference'

    def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1):
        if not isinstance(likfunc, lik.Gauss):                  # NOTE: no explicit call to likGauss
            raise Exception ('Exact inference only possible with Gaussian likelihood')
        if not isinstance(covfunc, cov.FITCOfKernel):
            raise Exception('Only covFITC supported.')          # check cov

        diagK,Kuu,Ku = covfunc.getCovMatrix(x=x, mode='train')  # evaluate covariance matrix
        m  = meanfunc.getMean(x)                                # evaluate mean vector
        n, D = x.shape
        nu = Kuu.shape[0]

        sn2   = np.exp(2*likfunc.hyp[0])                        # noise variance of likGauss
        snu2  = 1.e-6*sn2                                       # hard coded inducing inputs noise
        Luu   = np.linalg.cholesky(Kuu+snu2*np.eye(nu)).T       # Kuu + snu2*I = Luu'*Luu
        V     = np.linalg.solve(Luu.T,Ku)                       # V = inv(Luu')*Ku => V'*V = Q
        g_sn2 = diagK + sn2 - np.array([(V*V).sum(axis=0)]).T   # g + sn2 = diag(K) + sn2 - diag(Q)
        Lu    = np.linalg.cholesky(np.eye(nu) + np.dot(V/np.tile(g_sn2.T,(nu,1)),V.T)).T  # Lu'*Lu=I+V*diag(1/g_sn2)*V'
        r     = (y-m)/np.sqrt(g_sn2)
        be    = np.linalg.solve(Lu.T,np.dot(V,r/np.sqrt(g_sn2)))
        iKuu  = solve_chol(Luu,np.eye(nu))                      # inv(Kuu + snu2*I) = iKuu
        
        post = postStruct()
        post.alpha = np.linalg.solve(Luu,np.linalg.solve(Lu,be))# return the posterior parameters
        post.L  = solve_chol(np.dot(Lu,Luu),np.eye(nu)) - iKuu  # Sigma-inv(Kuu)
        post.sW = np.ones((n,1))/np.sqrt(sn2)                   # unused for FITC prediction  with gp.m

        if nargout>1:                                           # do we want the marginal likelihood
            nlZ = np.log(np.diag(Lu)).sum() + (np.log(g_sn2).sum() + n*np.log(2*np.pi) + np.dot(r.T,r) - np.dot(be.T,be))/2. 
            if nargout>2:                                       # do we want derivatives?
                dnlZ = dnlZStruct(meanfunc, covfunc, likfunc)   # allocate space for derivatives
                al = r/np.sqrt(g_sn2) - np.dot(V.T,np.linalg.solve(Lu,be))/g_sn2 # al = (Kt+sn2*eye(n))\y
                B = np.dot(iKuu,Ku)
                w = np.dot(B,al)
                W = np.linalg.solve(Lu.T,V/np.tile(g_sn2.T,(nu,1)))
                for ii in range(len(covfunc.hyp)):
                    [ddiagKi,dKuui,dKui] = covfunc.getDerMatrix(x=x, mode='train', der=ii)    # eval cov deriv
                    R = 2.*dKui-np.dot(dKuui,B)
                    v = ddiagKi - np.array([(R*B).sum(axis=0)]).T          # diag part of cov deriv
                    dnlZ.cov[ii] = ( np.dot(ddiagKi.T,1./g_sn2) + np.dot(w.T,(np.dot(dKuui,w)-2.*np.dot(dKui,al))) \
                                   - np.dot(al.T,(v*al)) - np.dot(np.array([(W*W).sum(axis=0)]),v) - (np.dot(R,W.T)*np.dot(B,W.T)).sum() )/2.
                    dnlZ.cov[ii] = dnlZ.cov[ii][0,0]
                dnlZ.lik = sn2*((1./g_sn2).sum() - (np.array([(W*W).sum(axis=0)])).sum() - np.dot(al.T,al))
                dKuui = 2*snu2
                R = -dKuui*B
                v = -np.array([(R*B).sum(axis=0)]).T     # diag part of cov deriv
                dnlZ.lik += (np.dot(w.T,np.dot(dKuui,w)) -np.dot(al.T,(v*al)) \
                                 - np.dot(np.array([(W*W).sum(axis=0)]),v) - (np.dot(R,W.T)*np.dot(B,W.T)).sum() )/2. 
                dnlZ.lik = list(dnlZ.lik[0])
                for ii in range(len(meanfunc.hyp)):
                    dnlZ.mean[ii] = np.dot(-meanfunc.getDerMatrix(x, ii).T, al) 
                    dnlZ.mean[ii] = dnlZ.mean[ii][0,0]

                return post, nlZ[0,0], dnlZ
            return post, nlZ[0,0]
        return post



      

      [docs]class Laplace(Inference): 
    '''
    Laplace's Approximation to the posterior Gaussian process.
    '''
    def __init__(self):
        self.last_alpha = None

    def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1):
        tol = 1e-6                           # tolerance for when to stop the Newton iterations
        smax = 2; Nline = 20; thr = 1e-4     # line search parameters
        maxit = 20                           # max number of Newton steps in f
        inffunc = self
        K = covfunc.getCovMatrix(x=x, mode='train')    # evaluate the covariance matrix
        m = meanfunc.getMean(x)              # evaluate the mean vector
        n, D = x.shape
        Psi_old = np.inf                     # make sure while loop starts by the largest old objective val
        if self.last_alpha == None:          # find a good starting point for alpha and f
            alpha = np.zeros((n,1))
            f = np.dot(K,alpha) + m          # start at mean if sizes not match 
            vargout = likfunc.proceed(y, f, None, inffunc, None, 3)
            lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2]
            W= -d2lp
            Psi_new = -lp.sum()
        else:
            alpha = self.last_alpha
            f = np.dot(K,alpha) + m                       # try last one
            vargout = likfunc.proceed(y, f, None, inffunc, None, 3)
            lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2]
            W= -d2lp
            Psi_new = np.dot(alpha.T,(f-m))/2. - lp.sum() # objective for last alpha
            vargout = - likfunc.proceed(y, m, None, inffunc, None, 1)
            Psi_def =  vargout[0]                         # objective for default init f==m
            if Psi_def < Psi_new:                         # if default is better, we use it
                alpha = np.zeros((n,1))
                f = np.dot(K,alpha) + m 
                vargout = likfunc.proceed(y, f, None, inffunc, None, 3)
                lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2]
                W=-d2lp; Psi_new = -lp.sum()
        isWneg = np.any(W<0)       # flag indicating whether we found negative values of W
        it = 0                     # this happens for the Student's t likelihood

        while (Psi_old - Psi_new > tol) and it<maxit:      # begin Newton
            Psi_old = Psi_new; it += 1
            if isWneg:                   # stabilise the Newton direction in case W has negative values
                W = np.maximum(W,0)      # stabilise the Hessian to guarantee postive definiteness
                tol = 1e-10              # increase accuracy to also get the derivatives right
            sW = np.sqrt(W); L = np.linalg.cholesky(np.eye(n) + np.dot(sW,sW.T)*K).T
            b = W*(f-m) + dlp; 
            dalpha = b - sW*solve_chol(L,sW*np.dot(K,b)) - alpha
            vargout = brentmin(0,smax,Nline,thr,self._Psi_line,4,dalpha,alpha,K,m,likfunc,y,inffunc)
            s = vargout[0]
            Psi_new = vargout[1]
            Nfun = vargout[2]
            alpha = vargout[3]
            f = vargout[4]
            dlp = vargout[5]
            W = vargout[6]
            isWneg = np.any(W<0)
        self.last_alpha = alpha                 # remember for next call
        vargout = likfunc.proceed(y,f,None,inffunc,None,4) 
        lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2]; d3lp = vargout[3] 
        W = -d2lp; isWneg = np.any(W<0)
        post = postStruct()
        post.alpha = alpha                      # return the posterior parameters
        post.sW = np.sqrt(np.abs(W))*np.sign(W) # preserve sign in case of negative
        if isWneg:
            [ldA,iA,post.L] = self._logdetA(K,W,3)
            nlZ = np.dot(alpha.T,(f-m))/2. - lp.sum() + ldA/2.
            nlZ = nlZ[0] 
        else:
            sW = post.sW
            post.L = np.linalg.cholesky(np.eye(n)+np.dot(sW,sW.T)*K).T 
            nlZ = np.dot(alpha.T,(f-m))/2. + (np.log(np.diag(post.L))-np.reshape(lp,(lp.shape[0],))).sum()
            nlZ = nlZ[0]
        if nargout>2:                                           # do we want derivatives?
            dnlZ = dnlZStruct(meanfunc, covfunc, likfunc)       # allocate space for derivatives
            if isWneg:                                          # switch between Cholesky and LU decomposition mode
                Z = -post.L                                     # inv(K+inv(W))
                g = np.atleast_2d((iA*K).sum(axis=1)).T /2      # deriv. of ln|B| wrt W; g = diag(inv(inv(K)+diag(W)))/2
            else:
                Z = np.tile(sW,(1,n))*solve_chol(post.L,np.diag(np.reshape(sW,(sW.shape[0],)))) #sW*inv(B)*sW=inv(K+inv(W))
                C = np.linalg.solve(post.L.T,np.tile(sW,(1,n))*K)            # deriv. of ln|B| wrt W
                g = np.atleast_2d((np.diag(K)-(C**2).sum(axis=0).T)).T /2.   # g = diag(inv(inv(K)+W))/2
            dfhat = g* d3lp                 # deriv. of nlZ wrt. fhat: dfhat=diag(inv(inv(K)+W)).*d3lp/2
            for ii in range(len(covfunc.hyp)):                               # covariance hypers
                dK = covfunc.getDerMatrix(x=x, mode='train', der=ii)
                dnlZ.cov[ii] = (Z*dK).sum()/2. - np.dot(alpha.T,np.dot(dK,alpha))/2.   # explicit part
                b = np.dot(dK,dlp)                                           # b-K*(Z*b) = inv(eye(n)+K*diag(W))*b
                dnlZ.cov[ii] -= np.dot(dfhat.T,b-np.dot(K,np.dot(Z,b)))      # implicit part
                dnlZ.cov[ii] = dnlZ.cov[ii][0,0]
            for ii in range(len(likfunc.hyp)):                               # likelihood hypers
                [lp_dhyp,dlp_dhyp,d2lp_dhyp] = likfunc.proceed(y,f,None,inffunc,ii,3)
                dnlZ.lik[ii] = -np.dot(g.T,d2lp_dhyp) - lp_dhyp.sum()        # explicit part
                b = np.dot(K,dlp_dhyp)                                       # b-K*(Z*b) = inv(eye(n)+K*diag(W))*b
                dnlZ.lik[ii] -= np.dot(dfhat.T,b-np.dot(K,np.dot(Z,b)))      # implicit part
                dnlZ.lik[ii] = dnlZ.lik[ii][0,0]
            for ii in range(len(meanfunc.hyp)):                              # mean hypers
                dm = meanfunc.getDerMatrix(x, ii)
                dnlZ.mean[ii] = -np.dot(alpha.T,dm)                          # explicit part
                dnlZ.mean[ii] -= np.dot(dfhat.T,dm-np.dot(K,np.dot(Z,dm)))   # implicit part
                dnlZ.mean[ii] = dnlZ.mean[ii][0,0]
            return post,nlZ[0],dnlZ
        else:
            return post, nlZ[0]



      

      [docs]class FITC_Laplace(Inference):
    '''
    FITC-Laplace approximation to the posterior Gaussian process. The function is
    equivalent to infLaplace with the covariance function:
    Kt = Q + G; G = diag(g); g = diag(K-Q);  Q = Ku' * inv(Kuu + snu2 * eye(nu)) * Ku;
    where Ku and Kuu are covariances w.r.t. to inducing inputs xu and
    snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard
    deviation of the inducing inputs snu to be a one per mil of the measurement 
    noise's standard deviation sn. In case of a likelihood without noise
    parameter sn2, we simply use snu2 = 1e-6.
    '''
    def __init__(self):
        self.last_alpha = None

    def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1):
        if not isinstance(covfunc, cov.FITCOfKernel):
            raise Exception('Only covFITC supported.')            
        tol = 1e-6                             # tolerance for when to stop the Newton iterations
        smax = 2; Nline = 100; thr = 1e-4      # line search parameters
        maxit = 20                             # max number of Newton steps in f
        inffunc = Laplace()
        diagK,Kuu,Ku = covfunc.getCovMatrix(x=x, mode='train')      # evaluate the covariance matrix
        m = meanfunc.getMean(x)                # evaluate the mean vector
        if likfunc.hyp:                        # hard coded inducing inputs noise
            sn2  = np.exp(2.*likfunc.hyp[-1]) 
            snu2 = 1.e-6*sn2                   # similar to infFITC
        else:
            snu2 = 1.e-6        
        
        n, D = x.shape
        nu = Kuu.shape[0]
        rot180   = lambda A: np.rot90(np.rot90(A))      # little helper functions
        chol_inv = lambda A: np.linalg.solve( rot180( np.linalg.cholesky(rot180(A)) ),np.eye(nu)) # chol(inv(A))
        R0 = chol_inv(Kuu+snu2*np.eye(nu))              # initial R, used for refresh O(nu^3)
        V  = np.dot(R0,Ku); d0 = diagK - np.array([(V*V).sum(axis=0)]).T     # initial d, needed
    
        Psi_old = np.inf                    # make sure while loop starts by the largest old objective val
        if self.last_alpha == None:         # find a good starting point for alpha and f
            alpha = np.zeros((n,1))
            f = self._mvmK(alpha,V,d0) + m   # start at mean if sizes not match 
            vargout = likfunc.proceed(y, f, None, inffunc, None, 3)
            lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2]
            W=-d2lp; Psi_new = -lp.sum()
        else:
            alpha = self.last_alpha
            f = self._mvmK(alpha,V,d0) + m                           # try last one
            vargout = likfunc.proceed(y, f, None, inffunc, None, 3)
            lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2]
            W=-d2lp
            Psi_new = np.dot(alpha.T,(f-m))/2. - lp.sum()           # objective for last alpha
            vargout = - likfunc.proceed(y, m, None, inffunc, None, 1)
            Psi_def =  vargout[0]                                   # objective for default init f==m
            if Psi_def < Psi_new:                                   # if default is better, we use it
                alpha = np.zeros((n,1))
                f = self._mvmK(alpha,V,d0) + m
                vargout = likfunc.proceed(y, f, None, inffunc, None, 3)
                lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2]
                W=-d2lp; Psi_new = -lp.sum()

        isWneg = np.any(W<0)                # flag indicating whether we found negative values of W
        it = 0                              # this happens for the Student's t likelihood

        while (Psi_old - Psi_new > tol) and it<maxit:               # begin Newton
            Psi_old = Psi_new
            it += 1
            if isWneg:                      # stabilise the Newton direction in case W has negative values
                W = np.maximum(W,0)         # stabilise the Hessian to guarantee postive definiteness
                tol = 1e-8                  # increase accuracy to also get the derivatives right
            b = W*(f-m) + dlp; dd = 1/(1+W*d0)
            RV = np.dot( chol_inv( np.eye(nu) + np.dot(V*np.tile((W*dd).T,(nu,1)),V.T)),V ) 
            dalpha = dd*b - (W*dd)*np.dot(RV.T,np.dot(RV,(dd*b))) - alpha # Newt dir + line search
            vargout = brentmin(0,smax,Nline,thr,self._Psi_lineFITC,4,dalpha,alpha,V,d0,m,likfunc,y,inffunc)
            s = vargout[0]; Psi_new = vargout[1]; Nfun = vargout[2]; alpha = vargout[3]
            f = vargout[4]; dlp = vargout[5]; W = vargout[6]
            isWneg = np.any(W<0)

        self.last_alpha = alpha                                     # remember for next call
        vargout = likfunc.proceed(y,f,None,inffunc,None,4) 
        lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2]; d3lp = vargout[3]  

        W=-d2lp; isWneg = np.any(W<0)
        post = postStruct()
        post.alpha = np.dot(R0.T,np.dot(V,alpha))                   # return the posterior parameters
        post.sW = np.sqrt(np.abs(W))*np.sign(W)                     # preserve sign in case of negative
        dd = 1/(1+d0*W)                                             # temporary variable O(n)
        A = np.eye(nu) + np.dot(V*np.tile((W*dd).T,(nu,1)),V.T)     # temporary variable O(n*nu^2)
        R0tV = np.dot(R0.T,V); B = R0tV*np.tile((W*dd).T,(nu,1))    # temporary variables O(n*nu^2)
        post.L = -np.dot(B,R0tV.T)                                  # L = -R0'*V*inv(Kt+diag(1./ttau))*V'*R0, first part
        if np.any(1+d0*W<0):
            raise Exception('W is too negative; nlZ and dnlZ cannot be computed.')
        nlZ = np.dot(alpha.T,(f-m))/2. - lp.sum() - np.log(dd).sum()/2. + \
            np.log(np.diag(np.linalg.cholesky(A).T)).sum()
        RV = np.dot(chol_inv(A),V)
        RVdd = RV * np.tile((W*dd).T,(nu,1))                        # RVdd needed for dnlZ
        B = np.dot(B,RV.T)
        post.L += np.dot(B,B.T)

        if nargout>2:                                               # do we want derivatives?
            dnlZ = dnlZStruct(meanfunc, covfunc, likfunc)           # allocate space for derivatives
            [d,P,R] = self._fitcRefresh(d0,Ku,R0,V,W)                # g = diag(inv(inv(K)+W))/2
            g = d/2 + 0.5*np.atleast_2d((np.dot(np.dot(R,R0),P)**2).sum(axis=0)).T
            t = W/(1+W*d0)
            
            dfhat = g*d3lp  # deriv. of nlZ wrt. fhat: dfhat=diag(inv(inv(K)+W)).*d3lp/2
            for ii in range(len(covfunc.hyp)):                      # covariance hypers
                ddiagK,dKuu,dKu = covfunc.getDerMatrix(x=x, mode='train', der=ii)      # eval cov derivatives
                dA = 2.*dKu.T-np.dot(R0tV.T,dKuu)                   # dQ = dA*R0tV
                w = np.atleast_2d((dA*R0tV.T).sum(axis=1)).T        # w = diag(dQ)
                v = ddiagK-w                                        # v = diag(dK)-diag(dQ);
                dnlZ.cov[ii] = np.dot(ddiagK.T,t) - np.dot((RVdd*RVdd).sum(axis=0),v)   # explicit part
                dnlZ.cov[ii] -= (np.dot(RVdd,dA)*np.dot(RVdd,R0tV.T)).sum()             # explicit part
                dnlZ.cov[ii] = 0.5*dnlZ.cov[ii] - np.dot(alpha.T,np.dot(dA,np.dot(R0tV,alpha))+v*alpha)/2. # explicit
                b = np.dot(dA,np.dot(R0tV,dlp)) + v*dlp             # b-K*(Z*b) = inv(eye(n)+K*diag(W))*b
                KZb = self._mvmK(self._mvmZ(b,RVdd,t),V,d0)
                dnlZ.cov[ii] -= np.dot(dfhat.T,(b-KZb))             # implicit part
                dnlZ.cov[ii] = dnlZ.cov[ii][0,0]
                
            for ii in range(len(likfunc.hyp)):                      # likelihood hypers
                vargout = likfunc.proceed(y,f,None,inffunc,ii,3)
                lp_dhyp = vargout[0]; dlp_dhyp = vargout[1]; d2lp_dhyp = vargout[2] 
                dnlZ.lik[ii] = -np.dot(g.T,d2lp_dhyp) - lp_dhyp.sum() # explicit part
                b = self._mvmK(dlp_dhyp,V,d0)                          # implicit part
                dnlZ.lik[ii] -= np.dot(dfhat.T,b-self._mvmK(self._mvmZ(b,RVdd,t),V,d0))
                if ii == len(likfunc.hyp)-1:
                    # since snu2 is a fixed fraction of sn2, there is a covariance-like term
                    # in the derivative as well
                    snu = np.sqrt(snu2);
                    T = chol_inv(Kuu + snu2*np.eye(nu)); 
                    T = np.dot(T.T,np.dot(T,snu*Ku)); 
                    t = np.array([(T*T).sum(axis=0)]).T 
                    z = np.dot(alpha.T,np.dot(T.T,np.dot(T,alpha))-t*alpha) - np.dot(np.array([(RVdd*RVdd).sum(axis=0)]),t)
                    z += (np.dot(RVdd,T.T)**2).sum()
                    b = (t*dlp-np.dot(T.T,np.dot(T,dlp)))/2.
                    KZb = self._mvmK(self._mvmZ(b,RVdd,t),V,d0)
                    z -= np.dot(dfhat.T,b-KZb)
                    dnlZ.lik[ii] += z
                    dnlZ.lik[ii] = dnlZ.lik[ii][0,0]

            for ii in range(len(meanfunc.hyp)):                           # mean hypers
                dm = meanfunc.getDerMatrix(x, ii)
                dnlZ.mean[ii] = -np.dot(alpha.T,dm)                       # explicit part
                Zdm = self._mvmZ(dm,RVdd,t)
                dnlZ.mean[ii] -= np.dot(dfhat.T,(dm-self._mvmK(Zdm,V,d0))) # implicit part
                dnlZ.mean[ii] = dnlZ.mean[ii][0,0]

            return post,nlZ[0,0],dnlZ
        else:
            return post, nlZ[0,0]



      

      [docs]class EP(Inference):
    '''
    Expectation Propagation approximation to the posterior Gaussian Process.
    '''
    def __init__(self):
        self.name = 'Expectation Propagation'
        self.last_ttau = None
        self.last_tnu = None
    def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1):
        tol = 1e-4; max_sweep = 10; min_sweep = 2 # tolerance to stop EP iterations
        n = x.shape[0]
        inffunc = self
        K = covfunc.getCovMatrix(x=x, mode='train') # evaluate the covariance matrix
        m = meanfunc.getMean(x)                   # evaluate the mean vector
        nlZ0 = -likfunc.proceed(y, m, np.reshape(np.diag(K),(np.diag(K).shape[0],1)), inffunc).sum()
        if self.last_ttau == None:                # find starting point for tilde parameters
            ttau  = np.zeros((n,1))               # initialize to zero if we have no better guess
            tnu   = np.zeros((n,1))
            Sigma = K                             # initialize Sigma and mu, the parameters of ..
            mu    = np.zeros((n,1))               # .. the Gaussian posterior approximation
            nlZ   = nlZ0
        else:
            ttau = self.last_ttau                 # try the tilde values from previous call
            tnu  = self.last_tnu
            Sigma, mu, nlZ, L = self._epComputeParams(K, y, ttau, tnu, likfunc, m, inffunc)
            if nlZ > nlZ0:                        # if zero is better ..
                ttau = np.zeros((n,1))            # .. then initialize with zero instead
                tnu  = np.zeros((n,1)) 
                Sigma = K                         # initialize Sigma and mu, the parameters of ..
                mu = np.zeros((n,1))              # .. the Gaussian posterior approximation
                nlZ = nlZ0
        nlZ_old = np.inf; sweep = 0               # converged, max. sweeps or min. sweeps?
        while (np.abs(nlZ-nlZ_old) > tol and sweep < max_sweep) or (sweep < min_sweep):
            nlZ_old = nlZ; sweep += 1
            rperm = xrange(n)                     # randperm(n)
            for ii in rperm:                      # iterate EP updates (in random order) over examples
                tau_ni = 1/Sigma[ii,ii] - ttau[ii]#  first find the cavity distribution ..
                nu_ni  = mu[ii]/Sigma[ii,ii] + m[ii]*tau_ni - tnu[ii]    # .. params tau_ni and nu_ni
                # compute the desired derivatives of the indivdual log partition function
                lZ,dlZ,d2lZ = likfunc.proceed(y[ii], nu_ni/tau_ni, 1/tau_ni, inffunc, None, 3)
                ttau_old = copy(ttau[ii])         # then find the new tilde parameters, keep copy of old
                ttau[ii] = -d2lZ  /(1.+d2lZ/tau_ni)
                ttau[ii] = max(ttau[ii],0)        # enforce positivity i.e. lower bound ttau by zero
                tnu[ii]  = ( dlZ + (m[ii]-nu_ni/tau_ni)*d2lZ )/(1.+d2lZ/tau_ni)
                ds2 = ttau[ii] - ttau_old         # finally rank-1 update Sigma ..
                si  = np.reshape(Sigma[:,ii],(Sigma.shape[0],1))
                Sigma = Sigma - ds2/(1.+ds2*si[ii])*np.dot(si,si.T)   # takes 70# of total time
                mu = np.dot(Sigma,tnu)                                # .. and recompute mu
            # recompute since repeated rank-one updates can destroy numerical precision
            Sigma, mu, nlZ, L = self._epComputeParams(K, y, ttau, tnu, likfunc, m, inffunc)
        if sweep == max_sweep:
            pass
            # print '[warning] maximum number of sweeps reached in function infEP'
        self.last_ttau = ttau; self.last_tnu = tnu          # remember for next call
        sW = np.sqrt(ttau); alpha = tnu-sW*solve_chol(L,sW*np.dot(K,tnu))
        post = postStruct()
        post.alpha = alpha                                  # return the posterior params
        post.sW    = sW
        post.L     = L
        if nargout>2:                                       # do we want derivatives?
            dnlZ = dnlZStruct(meanfunc, covfunc, likfunc)   # allocate space for derivatives
            ssi  = np.sqrt(ttau)
            V = np.linalg.solve(L.T,np.tile(ssi,(1,n))*K)
            Sigma = K - np.dot(V.T,V)
            mu = np.dot(Sigma,tnu)
            Dsigma = np.reshape(np.diag(Sigma),(np.diag(Sigma).shape[0],1))
            tau_n = 1/Dsigma-ttau                           # compute the log marginal likelihood
            nu_n  = mu/Dsigma-tnu                           # vectors of cavity parameters
            F = np.dot(alpha,alpha.T) - np.tile(sW,(1,n))* \
                solve_chol(L,np.diag(np.reshape(sW,(sW.shape[0],))))   # covariance hypers
            for jj in range(len(covfunc.hyp)):
                dK = covfunc.getDerMatrix(x=x, mode='train', der=jj)
                dnlZ.cov[jj] = -(F*dK).sum()/2.
            for ii in range(len(likfunc.hyp)):
                dlik = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc, ii)
                dnlZ.lik[ii] = -dlik.sum()    
            junk,dlZ = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc, None, 2) # mean hyps
            for ii in range(len(meanfunc.hyp)):
                dm = meanfunc.getDerMatrix(x, ii)
                dnlZ.mean[ii] = -np.dot(dlZ.T,dm)
                dnlZ.mean[ii] = dnlZ.mean[ii][0,0]
            return post, nlZ[0], dnlZ
        else:
            return post, nlZ[0]



      

      [docs]class FITC_EP(Inference):
    '''
    FITC-EP approximation to the posterior Gaussian process. The function is
    equivalent to infEP with the covariance function:
    Kt = Q + G; G = diag(g); g = diag(K-Q);  Q = Ku' * inv(Kuu + snu2 * eye(nu)) * Ku;
    where Ku and Kuu are covariances w.r.t. to inducing inputs xu and
    snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard
    deviation of the inducing inputs snu to be a one per mil of the measurement 
    noise's standard deviation sn. In case of a likelihood without noise
    parameter sn2, we simply use snu2 = 1e-6.
    For details, see The Generalized FITC Approximation, Andrew Naish-Guzman and
    Sean Holden, NIPS, 2007.
    '''
    def __init__(self):
        self.name = 'FITC Expectation Propagation'
        self.last_ttau = None
        self.last_tnu = None

    def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1):
        if not isinstance(covfunc, cov.FITCOfKernel):
            raise Exception('Only covFITC supported.')  # check cov
        tol = 1e-4; max_sweep = 10; min_sweep = 2       # tolerance to stop EP iterations
        inffunc = EP()

        diagK,Kuu,Ku = covfunc.getCovMatrix(x=x, mode='train')  # evaluate the covariance matrix
        m = meanfunc.getMean(x)                         # evaluate the mean vector

        if likfunc.hyp:                                 # hard coded inducing inputs noise
            sn2  = np.exp(2.*likfunc.hyp[-1]) 
            snu2 = 1.e-6*sn2                            # similar to infFITC
        else:
            snu2 = 1.e-6
    
        n, D = x.shape; nu = Kuu.shape[0]
        rot180   = lambda A: np.rot90(np.rot90(A))      # little helper functions
        chol_inv = lambda A: np.linalg.solve( rot180( np.linalg.cholesky(rot180(A)) ),np.eye(nu)) # chol(inv(A))

        R0 = chol_inv(Kuu+snu2*np.eye(nu))              # initial R, used for refresh O(nu^3)
        V  = np.dot(R0,Ku); d0 = diagK - np.array([(V*V).sum(axis=0)]).T # initial d, needed for refresh O(n*nu^2)

        # A note on naming: variables are given short but descriptive names in 
        # accordance with Rasmussen & Williams "GPs for Machine Learning" (2006): mu
        # and s2 are mean and variance, nu and tau are natural parameters. A leading t
        # means tilde, a subscript _ni means "not i" (for cavity parameters), or _n
        # for a vector of cavity parameters.

        # marginal likelihood for ttau = tnu = zeros(n,1); equals n*log(2) for likCum*
        nlZ0 = -1.* likfunc.proceed(y, m, np.reshape(diagK,(diagK.shape[0],1)), inffunc).sum()
        if self.last_ttau == None:                      # find starting point for tilde parameters
            ttau  = np.zeros((n,1))                     # initialize to zero if we have no better guess
            tnu   = np.zeros((n,1))
            [d,P,R,nn,gg] = self._epfitcRefresh(d0,Ku,R0,V,ttau,tnu)   # compute initial repres.
            nlZ = nlZ0
        else:
            ttau = self.last_ttau                       # try the tilde values from previous call
            tnu  = self.last_tnu
            [d,P,R,nn,gg] = self._epfitcRefresh(d0,Ku,R0,V,ttau,tnu) # compute initial repres.
            nlZ = self._epfitcZ(d,P,R,nn,gg,ttau,tnu,d0,R0,Ku,y,likfunc,m,inffunc)[0]
            if nlZ > nlZ0:                              # if zero is better ..
                ttau = np.zeros((n,1))                  # .. then initialize with zero instead
                tnu  = np.zeros((n,1))
                [d,P,R,nn,gg] = self._epfitcRefresh(d0,Ku,R0,V,ttau,tnu) # initial repres.
                nlZ = nlZ0

        nlZ_old = np.inf; sweep = 0                     # converged, max. sweeps or min. sweeps?
        while (np.abs(nlZ-nlZ_old) > tol and sweep < max_sweep) or (sweep < min_sweep):
            nlZ_old = nlZ
            sweep += 1
            rperm = range(n)                            # randperm(n)
            for ii in rperm:                            # iterate EP updates (in random order) over examples
                p_i = np.reshape(P[:,ii],(P.shape[0],1))
                t = np.dot(R,np.dot(R0,p_i))            # temporary variables
                sigma_i = d[ii] + np.dot(t.T,t); mu_i = nn[ii] + np.dot(p_i.T,gg) # post moments O(nu^2)    
                tau_ni = 1/sigma_i - ttau[ii]                   #  first find the cavity distribution ..
                nu_ni  = mu_i/sigma_i + m[ii]*tau_ni - tnu[ii]  # .. params tau_ni and nu_ni
                # compute the desired derivatives of the indivdual log partition function
                vargout = likfunc.proceed(y[ii], nu_ni/tau_ni, 1/tau_ni, inffunc, None, 3)
                lZ = vargout[0]; dlZ = vargout[1]; d2lZ = vargout[2]
                ttau_i = -d2lZ  /(1.+d2lZ/tau_ni)
                ttau_i = max(ttau_i,0)                  # enforce positivity i.e. lower bound ttau by zero
                tnu_i  = ( dlZ + (m[ii]-nu_ni/tau_ni)*d2lZ )/(1.+d2lZ/tau_ni)
                [d,P[:,ii],R,nn,gg,ttau,tnu] = self._epfitcUpdate(d,P[:,ii],R,nn,gg,ttau,tnu,ii,ttau_i,tnu_i,m,d0,Ku,R0)# update representation
      
            # recompute since repeated rank-one updates can destroy numerical precision
            [d,P,R,nn,gg] = self._epfitcRefresh(d0,Ku,R0,V,ttau,tnu)
            [nlZ,nu_n,tau_n] = self._epfitcZ(d,P,R,nn,gg,ttau,tnu,d0,R0,Ku,y,likfunc,m,inffunc)
        if sweep == max_sweep:
            pass
            # print '[warning] maximum number of sweeps reached in function infEP'
        
        self.last_ttau = ttau
        self.last_tnu = tnu       # remember for next call
        post = postStruct()
        post.sW = np.sqrt(ttau)   # unused for FITC_EP prediction with gp.m
        dd = 1/(d0+1/ttau)
        alpha = tnu/ttau*dd
        RV = np.dot(R,V)
        R0tV = np.dot(R0.T,V)
        alpha = alpha - np.dot(RV.T,np.dot(RV,alpha))*dd      # long alpha vector for ordinary infEP
        post.alpha = np.dot(R0tV,alpha)                       # alpha = R0'*V*inv(Kt+diag(1./ttau))*(tnu./ttau)
        B = R0tV*np.tile(dd.T,(nu,1)); L = np.dot(B,R0tV.T); B = np.dot(B,RV.T)
        post.L = np.dot(B,B.T) - L                            # L = -R0'*V*inv(Kt+diag(1./ttau))*V'*R0

        if nargout>2:                                         # do we want derivatives?
            dnlZ = dnlZStruct(meanfunc, covfunc, likfunc)     # allocate space for derivatives
            RVdd = RV*np.tile(dd.T,(nu,1))
            for ii in range(len(covfunc.hyp)):
                ddiagK,dKuu,dKu = covfunc.getDerMatrix(x=x, mode='train', der=ii)
                dA = 2*dKu.T - np.dot(R0tV.T,dKuu)            # dQ = dA*R0tV
                w = np.atleast_2d((dA*R0tV.T).sum(axis=1)).T  # w = diag(dQ)
                v = ddiagK - w                                # v = diag(dK)-diag(dQ)
                z = np.dot(dd.T,(v+w)) - np.dot(np.atleast_2d((RVdd*RVdd).sum(axis=0)), v) \
                       - (np.dot(RVdd,dA).T * np.dot(R0tV,RVdd.T)).sum()
                dnlZ.cov[ii] = (z - np.dot(alpha.T,(alpha*v)) - np.dot(np.dot(alpha.T,dA),np.dot(R0tV,alpha)))/2.
                dnlZ.cov[ii] = dnlZ.cov[ii][0,0]
            for ii in range(len(likfunc.hyp)):                # likelihood hypers
                dlik = likfunc.proceed(y, nu_n/tau_n+m, 1/tau_n, inffunc, ii, 1)
                dnlZ.lik[ii] = -dlik.sum()                                 
                if ii == len(likfunc.hyp)-1:
                    # since snu2 is a fixed fraction of sn2, there is a covariance-like term
                    # in the derivative as well
                    v = np.atleast_2d((R0tV*R0tV).sum(axis=0)).T
                    z = (np.dot(RVdd,R0tV.T)**2).sum()  - np.dot(np.atleast_2d((RVdd*RVdd).sum(axis=0)),v)
                    z = z + np.dot(post.alpha.T,post.alpha) - np.dot(alpha.T,(v*alpha))
                    dnlZ.lik[ii] += snu2*z
                    dnlZ.lik[ii] = dnlZ.lik[ii][0,0]
            [junk,dlZ] = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc, None, 2) # mean hyps
            for ii in range(len(meanfunc.hyp)):
                dm = meanfunc.getDerMatrix(x, ii)
                dnlZ.mean[ii] = -np.dot(dlZ.T,dm)
                dnlZ.mean[ii] = dnlZ.mean[ii][0,0]
        
            return post, nlZ[0,0], dnlZ
        else:
            return post, nlZ[0,0]
    



      
if __name__ == '__main__':
    pass

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      © Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting. Created using Sphinx 1.2.2.
      pyGPs-1.2/doc/build/html/_modules/pyGPs/Core/lik.html000066400000000000000000003636201236346020200224440ustar00rootroot00000000000000 pyGPs.Core.lik — pyGPs v1.2 documentation

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      Source code for pyGPs.Core.lik

      #    Marion Neumann [marion dot neumann at uni-bonn dot de]
#    Daniel Marthaler [marthaler at ge dot com]
#    Shan Huang [shan dot huang at iais dot fraunhofer dot de]
#    Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]
#
#    This file is part of pyGPs.
#    The software package is released under the BSD 2-Clause (FreeBSD) License.
#
#    Copyright (c) by
#    Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014
#================================================================================

# likelihood functions are provided to be used by the gp.py function:
#
#   Erf         (Error function, classification, probit regression)
#   Logistic    [NOT IMPLEMENTED!] (Logistic, classification, logit regression)
#   Uni         [NOT IMPLEMENTED!] (Uniform likelihood, classification)
#
#   Gauss       (Gaussian, regression)
#   Laplace     (Laplacian or double exponential, regression)
#   Sech2       [NOT IMPLEMENTED!] (Sech-square, regression)
#   T           [NOT IMPLEMENTED!] (Student's t, regression)
#
#   Poisson     [NOT IMPLEMENTED!] (Poisson regression, count data)
#
#   Mix         [NOT IMPLEMENTED!] (Mixture of individual covariance functions)
#
# See the documentation for the individual likelihood for the computations specific 
# to each likelihood function.
#
#
# This is a object-oriented python implementation of gpml functionality 
# (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18).
# based on the functional-version of python implementation
# (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013)
# 
# Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013


import numpy as np
from scipy.special import erf
import inf

class Likelihood(object):
    """Base function for Likelihood function"""
    def __init__(self):
        self.hyp = []
    def proceed(self, y=None, mu=None, s2=None, inffunc=None, der=None, nargout=1):
        '''
        The likelihood functions have two possible modes, the mode being selected
        as follows (where "lik" stands for "proceed" method for any likelihood function):


        1) With two or three input arguments:                       [PREDICTION MODE]

         lp = lik(y, mu) OR lp, ymu, ys2 = lik(y, mu, s2)

            This allows to evaluate the predictive distribution. Let p(y_*|f_*) be the
            likelihood of a test point and N(f_*|mu,s2) an approximation to the posterior
            marginal p(f_*|x_*,x,y) as returned by an inference method. The predictive
            distribution p(y_*|x_*,x,y) is approximated by:
            q(y_*) = \int N(f_*|mu,s2) p(y_*|f_*) df_*

            lp = log( q(y) ) for a particular value of y, if s2 is [] or 0, this
            corresponds to log( p(y|mu) ).
            
            ymu and ys2 are the mean and variance of the predictive marginal q(y)
            note that these two numbers do not depend on a particular 
            value of y.
            All vectors have the same size.


        2) With four or five input arguments, the fouth being an object of class "Inference" [INFERENCE MODE]

         lik(y, mu, s2, inf) OR lik(y, mu, s2, inf, i)

         There are two cases for inf, namely a) infLaplace, b) infEP 
         The last input i, refers to derivatives w.r.t. the ith hyperparameter. 

         | a1) 
         | lp,dlp,d2lp,d3lp = lik(y, f, [], 'infLaplace'). 
         | lp, dlp, d2lp and d3lp correspond to derivatives of the log likelihood. 
         | log(p(y|f)) w.r.t. to the latent location f.
         | lp = log( p(y|f) ) 
         | dlp = d log( p(y|f) ) / df 
         | d2lp = d^2 log( p(y|f) ) / df^2
         | d3lp = d^3 log( p(y|f) ) / df^3 

         | a2)
         | lp_dhyp,dlp_dhyp,d2lp_dhyp = lik(y, f, [], 'infLaplace', i) 
         | returns derivatives w.r.t. to the ith hyperparameter 
         | lp_dhyp = d log( p(y|f) ) / (dhyp_i) 
         | dlp_dhyp = d^2 log( p(y|f) ) / (df   dhyp_i) 
         | d2lp_dhyp = d^3 log( p(y|f) ) / (df^2 dhyp_i) 


         | b1)
         | lZ,dlZ,d2lZ = lik(y, mu, s2, 'infEP') 
         | let Z = \int p(y|f) N(f|mu,s2) df then 
         | lZ = log(Z) 
         | dlZ = d log(Z) / dmu 
         | d2lZ = d^2 log(Z) / dmu^2 

         | b2)
         | dlZhyp = lik(y, mu, s2, 'infEP', i)
         | returns derivatives w.r.t. to the ith hyperparameter 
         | dlZhyp = d log(Z) / dhyp_i 

        Cumulative likelihoods are designed for binary classification. Therefore, they
        only look at the sign of the targets y; zero values are treated as +1.

        Some examples for valid likelihood functions:
         | lik = likGauss([0.1])
         | lik = likErf()
        '''
        pass



      [docs]class Gauss(Likelihood):
    '''
    Gaussian likelihood function for regression. 

    :math:`Gauss(t)=\\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-\\frac{(t-y)^2}{2\\sigma^2}}`,
    where :math:`y` is the mean and :math:`\\sigma` is the standard deviation.

    hyp = [ log_sigma ]
    '''
    def __init__(self, log_sigma=np.log(0.1) ):
        self.hyp = [log_sigma]

    def proceed(self, y=None, mu=None, s2=None, inffunc=None, der=None, nargout=1):
        sn2 = np.exp(2. * self.hyp[0])
        if inffunc == None:              # prediction mode 
            if y == None:
                y = np.zeros_like(mu)
            s2zero = True
            if (not s2==None) and np.linalg.norm(s2) > 0:
                s2zero = False     
            if s2zero:                   # log probability
                lp = -(y-mu)**2 /sn2/2 - np.log(2.*np.pi*sn2)/2. 
                s2 = np.zeros_like(s2)
            else:
                inf_func = inf.EP()   # prediction
                lp = self.proceed(y, mu, s2, inf_func)
            if nargout>1:
                ymu = mu                 # first y moment
                if nargout>2:
                    ys2 = s2 + sn2       # second y moment
                    return lp,ymu,ys2
                else:
                    return lp,ymu
            else:
                return lp  
        else:
            if isinstance(inffunc, inf.EP):
                if der == None:                                  # no derivative mode
                    lZ = -(y-mu)**2/(sn2+s2)/2. - np.log(2*np.pi*(sn2+s2))/2. # log part function
                    if nargout>1:
                        dlZ  = (y-mu)/(sn2+s2)                   # 1st derivative w.r.t. mean
                        if nargout>2:
                            d2lZ = -1/(sn2+s2)                   # 2nd derivative w.r.t. mean
                            return lZ,dlZ,d2lZ
                        else:
                           return lZ,dlZ
                    else:
                        return lZ
                else:                                            # derivative mode
                    dlZhyp = ((y-mu)**2/(sn2+s2)-1) / (1+s2/sn2) # deriv. w.r.t. hyp.lik
                    return dlZhyp
            elif isinstance(inffunc, inf.Laplace):
                if der == None:                                  # no derivative mode
                    if y == None: 
                        y=0 
                    ymmu = y-mu
                    lp = -ymmu**2/(2*sn2) - np.log(2*np.pi*sn2)/2. 
                    if nargout>1:
                        dlp = ymmu/sn2                           # dlp, derivative of log likelihood
                        if nargout>2:                            # d2lp, 2nd derivative of log likelihood
                            d2lp = -np.ones_like(ymmu)/sn2
                            if nargout>3:                        # d3lp, 3rd derivative of log likelihood
                                d3lp = np.zeros_like(ymmu)
                                return lp,dlp,d2lp,d3lp
                            else:
                                return lp,dlp,d2lp
                        else:
                            return lp,dlp
                    else:
                        return lp
                else:                                            # derivative mode
                    lp_dhyp   = (y-mu)**2/sn2 - 1                # derivative of log likelihood w.r.t. hypers
                    dlp_dhyp  = 2*(mu-y)/sn2                     # first derivative,
                    d2lp_dhyp = 2*np.ones_like(mu)/sn2           # and also of the second mu derivative
                    return lp_dhyp,dlp_dhyp,d2lp_dhyp
            '''
            elif isinstance(inffunc, infVB):
                if der == None:
                    # variational lower site bound
                    # t(s) = exp(-(y-s)^2/2sn2)/sqrt(2*pi*sn2)
                    # the bound has the form: b*s - s.^2/(2*ga) - h(ga)/2 with b=y/ga
                    ga  = s2
                    n   = len(ga)
                    b   = y/ga
                    y   = y*np.ones((n,1))
                    db  = -y/ga**2 
                    d2b = 2*y/ga**3
                    h   = np.zeros((n,1))
                    dh  = h
                    d2h = h                           # allocate memory for return args
                    id  = (ga <= sn2 + 1e-8)          # OK below noise variance
                    h[id]   = y[id]**2/ga[id] + np.log(2*np.pi*sn2)
                    h[np.logical_not(id)] = np.inf
                    dh[id]  = -y[id]**2/ga[id]**2
                    d2h[id] = 2*y[id]**2/ga[id]**3
                    id = ga < 0
                    h[id] = np.inf
                    dh[id] = 0
                    d2h[id] = 0                       # neg. var. treatment
                    varargout = [h,b,dh,db,d2h,d2b]
                else:
                    ga = s2 
                    n  = len(ga)
                    dhhyp = np.zeros((n,1))
                    dhhyp[ga<=sn2] = 2
                    dhhyp[ga<0] = 0                   # negative variances get a special treatment
                    varargout = dhhyp                 # deriv. w.r.t. hyp.lik
            else:
                raise Exception('Incorrect inference in lik.Gauss\n')
        '''


      

      [docs]class Erf(Likelihood):
    '''
    Error function or cumulative Gaussian likelihood function for binary
    classification or probit regression. 

    :math:`Erf(t)=\\frac{1}{2}(1+erf(\\frac{t}{\\sqrt{2}}))=normcdf(t)`
    '''
    def __init__(self):
        self.hyp = []

    def proceed(self, y=None, mu=None, s2=None, inffunc=None, der=None, nargout=1):
        if not y == None:
            y = np.sign(y)
            y[y==0] = 1
        else:
            y = 1                                        # allow only +/- 1 values
        if inffunc == None:                              # prediction mode if inf is not present
            y = y*np.ones_like(mu)                       # make y a vector
            s2zero = True; 
            if not s2 == None: 
                if np.linalg.norm(s2)>0:
                    s2zero = False                       # s2==0?       
            if s2zero:                                   # log probability evaluation
                p,lp = self.cumGauss(y,mu,2)
            else:                                        # prediction
                lp = self.proceed(y, mu, s2, inf.EP())
                p = np.exp(lp)
            if nargout>1:
                ymu = 2*p-1                              # first y moment
                if nargout>2:
                    ys2 = 4*p*(1-p)                      # second y moment
                    return lp,ymu,ys2
                else:
                    return lp,ymu
            else:
                return lp
        else:                                            # inference mode
            if isinstance(inffunc, inf.Laplace):
                if der == None:                          # no derivative mode
                    f = mu; yf = y*f                     # product latents and labels
                    p,lp = self.cumGauss(y,f,2)
                    if nargout>1:                        # derivative of log likelihood
                        n_p = self.gauOverCumGauss(yf,p)
                        dlp = y*n_p                      # derivative of log likelihood
                        if nargout>2:                    # 2nd derivative of log likelihood
                            d2lp = -n_p**2 - yf*n_p
                            if nargout>3:                # 3rd derivative of log likelihood
                                d3lp = 2*y*n_p**3 + 3*f*n_p**2 + y*(f**2-1)*n_p 
                                return lp,dlp,d2lp,d3lp
                            else:
                                return lp,dlp,d2lp
                        else:
                            return lp,dlp
                    else:
                        return lp
                else:                                    # derivative mode
                    return []                            # derivative w.r.t. hypers

            elif isinstance(inffunc, inf.EP):
                if der == None:                          # no derivative mode
                    z = mu/np.sqrt(1+s2) 
                    junk,lZ = self.cumGauss(y,z,2)       # log part function
                    if not y == None:
                         z = z*y
                    if nargout>1:
                        if y == None: y = 1
                        n_p = self.gauOverCumGauss(z,np.exp(lZ))
                        dlZ = y*n_p/np.sqrt(1.+s2)       # 1st derivative wrt mean
                        if nargout>2:
                            d2lZ = -n_p*(z+n_p)/(1.+s2)  # 2nd derivative wrt mean
                            return lZ,dlZ,d2lZ
                        else:
                            return lZ,dlZ
                    else:
                        return lZ
                else:                                    # derivative mode
                    return []                       # deriv. wrt hyp.lik
        '''
        if inffunc == 'inf.infVB':
            if der == None:                              # no derivative mode
                # naive variational lower bound based on asymptotical properties of lik
                # normcdf(t) -> -(t*A_hat^2-2dt+c)/2 for t->-np.inf (tight lower bound)
                d =  0.158482605320942;
                c = -1.785873318175113;
                ga = s2; n = len(ga); b = d*y*np.ones((n,1)); db = np.zeros((n,1)); d2b = db
                h = -2.*c*np.ones((n,1)); h[ga>1] = np.inf; dh = np.zeros((n,1)); d2h = dh   
                varargout = [h,b,dh,db,d2h,d2b]
            else:                                        # derivative mode
                varargout = []                           # deriv. wrt hyp.lik
        '''

    def cumGauss(self, y=None, f=None, nargout=1):
        # return [p,lp] = cumGauss(y,f)
        if not y == None: 
            yf = y*f 
        else:
            yf = f 
        p = (1. + erf(yf/np.sqrt(2.)))/2. # likelihood
        if nargout>1: 
            lp = self.logphi(yf,p)
            return p,lp 
        else:
            return p

    def gauOverCumGauss(self,f,p):
        # return n_p = gauOverCumGauss(f,p)
        n_p = np.zeros_like(f)       # safely compute Gaussian over cumulative Gaussian
        ok = f>-5                    # naive evaluation for large values of f
        n_p[ok] = (np.exp(-f[ok]**2/2)/np.sqrt(2*np.pi)) / p[ok] 
        bd = f<-6                    # tight upper bound evaluation
        n_p[bd] = np.sqrt(f[bd]**2/4+1)-f[bd]/2
        interp = np.logical_and(np.logical_not(ok),np.logical_not(bd)) # linearly interpolate between both of them
        tmp = f[interp]
        lam = -5. - f[interp]
        n_p[interp] = (1-lam)*(np.exp(-tmp**2/2)/np.sqrt(2*np.pi))/p[interp] + lam *(np.sqrt(tmp**2/4+1)-tmp/2);
        return n_p

    def logphi(self,z,p):
        # return lp = logphi(z,p)
        lp = np.zeros_like(z)                       # allocate memory
        zmin = -6.2; zmax = -5.5;
        ok = z>zmax                                 # safe evaluation for large values
        bd = z<zmin                                 # use asymptotics
        nok = np.logical_not(ok)
        ip = np.logical_and(nok,np.logical_not(bd)) # interpolate between both of them
        lam = 1/(1.+np.exp( 25.*(0.5-(z[ip]-zmin)/(zmax-zmin)) ))  # interp. weights
        lp[ok] = np.log(p[ok])
        lp[nok] = -np.log(np.pi)/2. -z[nok]**2/2. - np.log( np.sqrt(z[nok]**2/2.+2.) - z[nok]/np.sqrt(2.) )
        lp[ip] = (1-lam)*lp[ip] + lam*np.log( p[ip] )
        return lp



      

      [docs]class Laplace(Likelihood):
    ''' 
    Laplacian likelihood function for regression. ONLY works with EP inference!

    :math:`Laplace(t) = \\frac{1}{2b}e^{-\\frac{|t-y|}{b}}` where :math:`b=\\frac{\\sigma}{\\sqrt{2}}`,
    :math:`y` is the mean and :math:`\\sigma` is the standard deviation.

    hyp = [ log_sigma ]
    '''
    def __init__(self, log_sigma=np.log(0.1) ):
        self.hyp = [ log_sigma ]

    def proceed(self, y=None, mu=None, s2=None, inffunc=None, der=None, nargout=1):
        sn = np.exp(self.hyp); b = sn/np.sqrt(2);
        if y == None:
            y = np.zeros_like(mu) 
        if inffunc == None:                              # prediction mode if inf is not present
            if y == None:
                y = np.zeros_like(mu)
            s2zero = True; 
            if not s2 == None: 
                if np.linalg.norm(s2)>0:
                    s2zero = False                       # s2==0?       
            if s2zero:                                   # log probability evaluation
                lp = -np.abs(y-mu)/b -np.log(2*b); s2 = 0
            else:                                        # prediction
                lp = self.proceed(y, mu, s2, inf.EP())
            if nargout>1:
                ymu = mu                              # first y moment
                if nargout>2:
                    ys2 = s2 + sn**2                  # second y moment
                    return lp,ymu,ys2
                else:
                    return lp,ymu
            else:
                return lp
        else:                                            # inference mode
            if isinstance(inffunc, inf.Laplace):
                if der == None:                          # no derivative mode
                    if y == None:
                        y = np.zeros_like(mu) 
                    ymmu = y-mu
                    lp = np.abs(ymmu)/b - np.log(2*b)
                    if nargout>1:                        # derivative of log likelihood
                        dlp = np.sign(ymmu)/b
                        if nargout>2:                    # 2nd derivative of log likelihood
                            d2lp = np.zeros_like(ymmu)
                            if nargout>3:                # 3rd derivative of log likelihood
                                d3lp = np.zeros_like(ymmu)
                                return lp,dlp,d2lp,d3lp
                            else:
                                return lp,dlp,d2lp
                        else:
                            return lp,dlp
                    else:
                        return lp
                else:                                    # derivative w.r.t. hypers
                    lp_dhyp = np.abs(y-mu)/b - 1           # derivative of log likelihood w.r.t. hypers
                    dlp_dhyp = np.sign(mu-y)/b              # first derivative,
                    d2lp_dhyp = np.zeros(mu.shape)         # and also of the second mu derivative
                    return lp_dhyp, dlp_dhyp, d2lp_dhyp     
            elif isinstance(inffunc, inf.EP):
                n = np.max([len(y.flatten()),len(mu.flatten()),len(s2.flatten()),len(sn.flatten())])
                on = np.ones((n,1))
                y = y*on; mu = mu*on; s2 = s2*on; sn = sn*on; 
                fac = 1e3;          # factor between the widths of the two distributions ...
                                    # ... from when one considered a delta peak, we use 3 orders of magnitude
                #idlik = np.reshape( (fac*sn) < np.sqrt(s2) , (sn.shape[0],) ) # Likelihood is a delta peak
                #idgau = np.reshape( (fac*np.sqrt(s2)) < sn , (sn.shape[0],) ) # Gaussian is a delta peak
                idlik = (fac*sn) < np.sqrt(s2) 
                idgau = (fac*np.sqrt(s2)) < sn 
                id    = np.logical_and(np.logical_not(idgau),np.logical_not(idlik)) # interesting case in between

                if der == None:                          # no derivative mode
                    lZ = np.zeros((n,1))
                    dlZ = np.zeros((n,1))
                    d2lZ = np.zeros((n,1))
                    if np.any(idlik):
                        l = Gauss(log_sigma=np.log(s2[idlik])/2)
                        a = l.proceed(mu[idlik], y[idlik])
                        lZ[idlik] = a[0]; dlZ[idlik] = a[1]; d2lZ[idlik] = a[2]
                    if np.any(idgau):
                        l = Laplace(log_hyp=np.log(sn[idgau]))
                        a = l.proceed(mu=mu[idgau], y=y[idgau])
                        lZ[idgau] = a[0]; dlZ[idgau] = a[1]; d2lZ[idgau] = a[2] 
                    if np.any(id):
                        # substitution to obtain unit variance, zero mean Laplacian
                        tvar = s2[id]/(sn[id]**2+1e-16)
                        tmu = (mu[id]-y[id])/(sn[id]+1e-16)
                        # an implementation based on logphi(t) = log(normcdf(t))
                        zp = (tmu+np.sqrt(2)*tvar)/np.sqrt(tvar)
                        zm = (tmu-np.sqrt(2)*tvar)/np.sqrt(tvar)
                        ap =  self._logphi(-zp)+np.sqrt(2)*tmu
                        am =  self._logphi( zm)-np.sqrt(2)*tmu
                        apam = np.vstack((ap,am)).T
                        lZ[id] = self._logsum2exp(apam) + tvar - np.log(sn[id]*np.sqrt(2.))

                    if nargout>1:
                        lqp = -0.5*zp**2 - 0.5*np.log(2*np.pi) - self._logphi(-zp);       # log( N(z)/Phi(z) )
                        lqm = -0.5*zm**2 - 0.5*np.log(2*np.pi) - self._logphi( zm);
                        dap = -np.exp(lqp-0.5*np.log(s2[id])) + np.sqrt(2)/sn[id]
                        dam =  np.exp(lqm-0.5*np.log(s2[id])) - np.sqrt(2)/sn[id]
                        _z1 = np.vstack((ap,am)).T
                        _z2 = np.vstack((dap,dam)).T
                        _x = np.array([[1],[1]])
                        dlZ[id] = self._expABz_expAx(_z1, _x, _z2, _x)
                        if nargout>2:
                            a = np.sqrt(8.)/sn[id]/np.sqrt(s2[id]);
                            bp = 2./sn[id]**2 - (a - zp/s2[id])*np.exp(lqp)
                            bm = 2./sn[id]**2 - (a + zm/s2[id])*np.exp(lqm)
                            _x = np.reshape(np.array([1,1]),(2,1))
                            _z1 = np.reshape(np.array([ap,am]),(1,2))
                            _z2 = np.reshape(np.array([bp,bm]),(1,2))
                            d2lZ[id] = self._expABz_expAx(_z1, _x, _z2, _x) - dlZ[id]**2
                            return lZ,dlZ,d2lZ
                        else:
                            return lZ,dlZ
                    else:
                        return lZ
                else:                                    # derivative mode
                    dlZhyp = np.zeros((n,1))
                    if np.any(idlik):
                        dlZhyp[idlik] = 0
                    if np.any(idgau):
                        l = Laplace(log_hyp=np.log(sn[idgau]))
                        a =  l.proceed(mu=mu[idgau], y=y[idgau], inffunc='inf.Laplace', nargout=1)
                        dlZhyp[idgau] = a[0]

                    if np.any(id):
                        # substitution to obtain unit variance, zero mean Laplacian
                        tmu = (mu[id]-y[id])/(sn[id]+1e-16);        tvar = s2[id]/(sn[id]**2+1e-16)
                        zp  = (tvar+tmu/np.sqrt(2))/np.sqrt(tvar);  vp = tvar+np.sqrt(2)*tmu
                        zm  = (tvar-tmu/np.sqrt(2))/np.sqrt(tvar);  vm = tvar-np.sqrt(2)*tmu
                        dzp = (-s2[id]/sn[id]+tmu*sn[id]/np.sqrt(2)) / np.sqrt(s2[id])
                        dvp = -2*tvar - np.sqrt(2)*tmu
                        dzm = (-s2[id]/sn[id]-tmu*sn[id]/np.sqrt(2)) / np.sqrt(s2[id])
                        dvm = -2*tvar + np.sqrt(2)*tmu
                        lezp = self._lerfc(zp); # ap = exp(vp).*ezp
                        lezm = self._lerfc(zm); # am = exp(vm).*ezm
                        vmax = np.max(np.array([vp+lezp,vm+lezm]),axis=0); # subtract max to avoid numerical pb
                        ep  = np.exp(vp+lezp-vmax)
                        em  = np.exp(vm+lezm-vmax)
                        dap = ep*(dvp - 2/np.sqrt(np.pi)*np.exp(-zp**2-lezp)*dzp)
                        dam = em*(dvm - 2/np.sqrt(np.pi)*np.exp(-zm**2-lezm)*dzm)        
                        dlZhyp[id] = (dap+dam)/(ep+em) - 1;       
                    return dlZhyp               # deriv. wrt hyp.lik
            elif isinstance(inffunc, inf.VB):
                n = len(s2.flatten()); b = np.zeros((n,1)); y = y*np.ones((n,1)); z = y
                return b,z

    def _lerfc(self,t):
        ''' numerically safe implementation of f(t) = log(1-erf(t)) = log(erfc(t))'''
        from scipy.special import erfc
        f  = np.zeros_like(t)
        tmin = 20; tmax = 25
        ok = t<tmin                              # log(1-erf(t)) is safe to evaluate
        bd = t>tmax                              # evaluate tight bound
        nok = np.logical_not(ok)
        interp = np.logical_and(nok,np.logical_not(bd)) # interpolate between both of them
        f[nok] = np.log(2/np.sqrt(np.pi)) -t[nok]**2 -np.log(t[nok]+np.sqrt( t[nok]**2+4/np.pi ))
        lam = 1/(1+np.exp( 12*(0.5-(t[interp]-tmin)/(tmax-tmin)) ))   # interp. weights
        f[interp] = lam*f[interp] + (1-lam)*np.log(erfc( t[interp] ))
        f[ok] += np.log(erfc( t[ok] ))             # safe eval
        return f

    def _expABz_expAx(self,A,x,B,z):
        ''' 
        Computes y = ( (exp(A).*B)*z ) ./ ( exp(A)*x ) in a numerically safe way
        The function is not general in the sense that it yields correct values for
        all types of inputs. We assume that the values are close together.
        '''
        N = A.shape[1]
        maxA = np.max(A,axis=1)                    # number of columns, max over columns
        maxA = np.array([maxA]).T
        A = A - np.dot(maxA, np.ones((1,N)))       # subtract maximum value
        y = ( np.dot((np.exp(A)*B),z) ) / ( np.dot(np.exp(A),x) )
        return y[0]

    def _logphi(self,z):
        ''' Safe implementation of the log of phi(x) = \int_{-\infty}^x N(f|0,1) df
         returns lp = log(normcdf(z))
        '''
        lp = np.zeros_like(z)                       # allocate memory
        zmin = -6.2; zmax = -5.5;
        ok = z>zmax                                 # safe evaluation for large values
        bd = z<zmin                                 # use asymptotics
        nok = np.logical_not(ok)
        ip = np.logical_and(nok,np.logical_not(bd)) # interpolate between both of them
        lam = 1./(1.+np.exp( 25.*(0.5-(z[ip]-zmin)/(zmax-zmin)) ))  # interp. weights
        lp[ok] = np.log( 0.5*( 1.+erf(z[ok]/np.sqrt(2.)) ) )
        lp[nok] = -0.5*(np.log(np.pi) + z[nok]**2) - np.log( np.sqrt(2.+0.5*(z[nok]**2)) - z[nok]/np.sqrt(2)) 
        lp[ip] = (1-lam)*lp[ip] + lam*np.log( 0.5*( 1.+erf(z[ip]/np.sqrt(2.)) ) )
        return lp

    def _logsum2exp(self,logx):
        '''computes y = log( sum(exp(x),2) ) in a numerically safe way 
        by subtracting the row maximum to avoid cancelation after taking 
        the exp the sum is done along the rows'''
        N = logx.shape[1]
        max_logx = logx.max(1)
        max_logx = np.array([max_logx]).T
        # we have all values in the log domain, and want to calculate a sum
        x = np.exp(logx - np.dot(max_logx,np.ones((1,N))))
        y = np.log(np.array([np.sum(x,1)]).T) + max_logx
        return list(y.flatten())


 

      
if __name__ == '__main__':
    pass

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      © Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting. Created using Sphinx 1.2.2.
      pyGPs-1.2/doc/build/html/_modules/pyGPs/Core/mean.html000066400000000000000000001505371236346020200226060ustar00rootroot00000000000000 pyGPs.Core.mean — pyGPs v1.2 documentation

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      Source code for pyGPs.Core.mean

      #================================================================================
#    Marion Neumann [marion dot neumann at uni-bonn dot de]
#    Daniel Marthaler [marthaler at ge dot com]
#    Shan Huang [shan dot huang at iais dot fraunhofer dot de]
#    Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]
#
#    This file is part of pyGPs.
#    The software package is released under the BSD 2-Clause (FreeBSD) License.
#
#    Copyright (c) by
#    Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014
#================================================================================

# mean functions to be use by Gaussian process functions. There are two
# different kinds of mean functions: simple and composite:
#
# simple mean functions:
#
#   Zero      - zero mean function
#   One       - one mean function
#   Const     - constant mean function
#   Linear    - linear mean function
# 
# composite mean functions:
#
#   ScaleOfMean     - scaled version of a mean function
#   PowerOfMean     - power of a mean function
#   ProductOfMean   - products of mean functions
#   SumOfMean       - sums of mean functions
#
#
# This is a object-oriented python implementation of gpml functionality 
# (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18).
# based on the functional-version of python implementation
# (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013)
# 
#
# Copyright (c) by Marion Neumann and Shan Huang, Sep.2013

import numpy as np
import math


      [docs]class Mean(object):
    """
    The base function for mean function
    """
    def __init__(self):
        super(Mean, self).__init__()
        self.hyp = []
        self.para = []

    # overloading 
    def __add__(self,mean):
        '''
        Overloading + operator.
        
        :param mean: mean function
        :return: an instance of SumOfMean
        '''
        return SumOfMean(self,mean)

    # overloading
    def __mul__(self,other):
        '''
        Overloading * operator.
        Using * for both multiplication with scalar and product of means
        depending on the type of the two objects.
        
        :param other: mean function as product or int/float as scalar
        :return: an instance of ScaleOfMean or ProductOfMean
        '''
        if isinstance(other, int) or isinstance(other, float):
            return ScaleOfMean(self,other)
        elif isinstance(other, Mean):
            return ProductOfMean(self,other)
        else:
            print "only numbers and Means are allowed for *"

    # overloading
    __rmul__ = __mul__

    # overloading
    def __pow__(self,number):
        '''
        Overloading ** operator.
        
        :param int number: power of the mean function
        :return: an instance of PowerOfMean
        '''
        if isinstance(number, int) and number > 0:
            return PowerOfMean(self,number)
        else:
            print "only non-zero integers are supported for **"


      [docs]    def getMean(self, x=None):
        '''
        Get the mean vector.
        '''
        pass

      

      [docs]    def getDerMatrix(self, x=None, der=None):
        '''
        Compute derivatives wrt. hyperparameters.

        :param x: training data
        :param int der: index of hyperparameter whose derivative to be computed

        :return: the corresponding derivative matrix
        '''
        pass
    


      

      [docs]class ProductOfMean(Mean):
    '''Product of two mean fucntions.'''
    def __init__(self,mean1,mean2):
        self.mean1 = mean1
        self.mean2 = mean2
        if mean1.hyp and mean2.hyp:
            self._hyp = mean1.hyp + mean2.hyp
        elif not mean1.hyp:
            self._hyp = mean2.hyp
        elif not mean2.hyp:
            self._hyp = mean1.hyp
    def _setHyp(self,hyp):
        assert len(hyp) == len(self._hyp)
        len1 = len(self.mean1.hyp)
        self._hyp = hyp 
        self.mean1.hyp = self._hyp[:len1]
        self.mean2.hyp = self._hyp[len1:]
    def _getHyp(self):
        return self._hyp
    hyp = property(_getHyp,_setHyp)
    
    def getMean(self, x=None):
        A = self.mean1.getMean(x) * self.mean2.getMean(x)
        return A

    def getDerMatrix(self, x=None, der=None):
        if der < len(self.mean1.hyp):
            A = self.mean1.getDerMatrix(x, der) * self.mean2.getMean(x)
        elif der < len(self.hyp):
            der2 = der - len(self.mean1.hyp)
            A = self.mean2.getDerMatrix(x, der2) * self.mean1.getMean(x) 
        else:
            raise Exception("Error: der out of range for meanProduct")
        return A



      

      [docs]class SumOfMean(Mean):
    '''Sum of two mean functions.'''
    def __init__(self,mean1,mean2):
        self.mean1 = mean1
        self.mean2 = mean2
        if mean1.hyp and mean2.hyp:
            self._hyp = mean1.hyp + mean2.hyp
        elif not mean1.hyp:
            self._hyp = mean2.hyp
        elif not mean2.hyp:
            self._hyp = mean1.hyp
    def _setHyp(self,hyp):
        assert len(hyp) == len(self._hyp)
        len1 = len(self.mean1.hyp)
        self._hyp = hyp 
        self.mean1.hyp = self._hyp[:len1]
        self.mean2.hyp = self._hyp[len1:]
    def _getHyp(self):
        return self._hyp
    hyp = property(_getHyp,_setHyp)

    def getMean(self, x=None):
        A = self.mean1.getMean(x) + self.mean2.getMean(x)
        return A

    def getDerMatrix(self, x=None, der=None):
        if der < len(self.mean1.hyp):
            A = self.mean1.getDerMatrix(x, der)
        elif der < len(self.hyp):
            der2 = der - len(self.mean1.hyp)
            A = self.mean2.getDerMatrix(x, der2) 
        else:
            raise Exception("Error: der out of range for meanSum")
        return A



      

      [docs]class ScaleOfMean(Mean):
    '''Scale of a mean function.'''
    def __init__(self,mean,scalar):
        self.mean = mean
        if mean.hyp:
            self._hyp = [scalar] + mean.hyp 
        else:
            self._hyp = [scalar]
    def _setHyp(self,hyp):
        assert len(hyp) == len(self._hyp)
        self._hyp = hyp 
        self.mean.hyp = self._hyp[1:]
    def _getHyp(self):
        return self._hyp
    hyp = property(_getHyp,_setHyp)

    def getMean(self, x=None):
        c = self.hyp[0]                          # scale parameter
        A = c * self.mean.getMean(x)             # accumulate means
        return A

    def getDerMatrix(self, x=None, der=None):
        c = self.hyp[0]                          # scale parameter
        if der == 0:                             # compute derivative w.r.t. c
            A = self.mean.getMean(x)
        else:
            A = c * self.mean.getDerMatrix(x,der-1)
        return A



      

      [docs]class PowerOfMean(Mean):
    '''Power of a mean fucntion.'''
    def __init__(self, mean, d):
        self.mean = mean
        if mean.hyp:
            self._hyp = [d] + mean.hyp 
        else:
            self._hyp = [d]
    def _setHyp(self,hyp):
        assert len(hyp) == len(self._hyp)
        self._hyp = hyp 
        self.mean.hyp = self._hyp[1:]
    def _getHyp(self):
        return self._hyp
    hyp = property(_getHyp,_setHyp)

    def getMean(self, x=None):
        d = np.abs(np.floor(self.hyp[0])) 
        d = max(d,1)
        A = self.mean.getMean(x) **d              # accumulate means
        return A

    def getDerMatrix(self, x=None, der=None):
        d = np.abs(np.floor(self.hyp[0])) 
        d = max(d,1)
        if der == 0:                             # compute derivative w.r.t. c
            a = self.mean.getMean(x)
            A = a**d * np.log(a) 
        else:
            A = d * self.mean.getMean(x) ** (d-1) * self.mean.getDerMatrix(x, der-1)
        return A



      

      [docs]class Zero(Mean):
    '''Zero mean.'''
    def __init__(self):
        self.hyp = []
        self.name = '0'

    def getMean(self, x=None):
        n, D = x.shape
        A = np.zeros((n,1))
        return A

    def getDerMatrix(self, x=None, der=None):
        n, D = x.shape
        A = np.zeros((n,1))
        return A



      

      [docs]class One(Mean):
    ''' One mean.'''
    def __init__(self):
        self.hyp = []
        self.name = '1'

    def getMean(self, x=None):
        n, D = x.shape
        A = np.ones((n,1))
        return A

    def getDerMatrix(self, x=None, der=None):
        n, D = x.shape
        A = np.zeros((n,1))
        return A



      

      [docs]class Const(Mean):
    '''
    Constant mean function. hyp = [c]

    :param c: constant value for mean
    '''
    def __init__(self, c=5.):
        self.hyp = [c]

    def getMean(self, x=None):
        n,D = x.shape
        A = self.hyp[0] * np.ones((n,1))
        return A

    def getDerMatrix(self, x=None, der=None):
        n,D = x.shape
        if der == 0:                  # compute derivative vector wrt c
            A = np.ones((n,1))
        else:
            A = np.zeros((n,1)) 
        return A          



      

      [docs]class Linear(Mean):
    '''
    Linear mean function. self.hyp = alpha_list

    :param D: dimension of training data. Set if you want default alpha, which is 0.5 for each dimension.
    :alpha_list: scalar alpha for each dimension
    '''
    def __init__(self, D=None, alpha_list=None):
        if alpha_list == None:
            self.hyp = [0.5 for i in xrange(D)]
        else:
            self.hyp = alpha_list

    def getMean(self, x=None):
        n, D = x.shape
        c = np.array(self.hyp)
        c = np.reshape(c,(len(c),1))
        A = np.dot(x,c)
        return A

    def getDerMatrix(self, x=None, der=None):
        n, D = x.shape
        c = np.array(self.hyp)
        c = np.reshape(c,(len(c),1))
        if isinstance(der, int) and der < D:     # compute derivative vector wrt meanparameters
            A = np.reshape(x[:,der], (len(x[:,der]),1) ) 
        else:   
            A = np.zeros((n,1)) 
        return A




      
if __name__ == '__main__':
    pass

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      © Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting. Created using Sphinx 1.2.2.
      pyGPs-1.2/doc/build/html/_modules/pyGPs/GraphExtensions/000077500000000000000000000000001236346020200232165ustar00rootroot00000000000000pyGPs-1.2/doc/build/html/_modules/pyGPs/GraphExtensions/graphKernels.html000066400000000000000000000763421236346020200265450ustar00rootroot00000000000000 pyGPs.GraphExtensions.graphKernels — pyGPs v1.2 documentation

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      Source code for pyGPs.GraphExtensions.graphKernels

      #================================================================================
#    Marion Neumann [marion dot neumann at uni-bonn dot de]
#    Daniel Marthaler [marthaler at ge dot com]
#    Shan Huang [shan dot huang at iais dot fraunhofer dot de]
#    Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]
#
#    This file is part of pyGPs.
#    The software package is released under the BSD 2-Clause (FreeBSD) License.
#
#    Copyright (c) by
#    Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014
#================================================================================


# Created on April, 2014
# @author: marion, shan

import scipy.sparse as spsp
import numpy as np
import matplotlib.pyplot as plt
    

      [docs]def propagationKernel(A, l, gr_id, h_max, w, p, ktype=None, SUM=True, VIS=False, showEachStep=False):
    '''
    Propagation kernel for graphs as described in: 
    Neumann, M., Patricia, N., Garnett, R., Kersting, K.: Efficient Graph Kernels by 
    Randomization. In: P.A. Flach, T.D. Bie, N. Cristianini (eds.) ECML/PKDD, Notes in 
    Computer Science, vol. 7523, pp. 378-393. Springer (2012).

    :param A: adjacency matrix (num_nodes x num_nodes)
    :param l: label array (num_nodes x 1); values [1,...,k] or -1 for unlabeled nodes 
              OR label array (num_nodes x num_labels); values [0,1], unlabeled nodes have only 0 entries
    :param gr_id: graph indicator array (num_nodes x 1); values [0,..,n]
    :param h_max: number of iterations
    :param w: bin widths parameter
    :param p: distance ('tv', 'hellinger', 'L1', 'L2')
    :param ktype: type of propagation kernel ['diffusion', 'label_propagation', 'label_spreading', 'belief_propagation']
    
    :return: kernel matrix 
    '''
    # ToDO: SPLIT Varible label array and label probability
        
    # CAUTION: number of labels (k) > 1 !!! 
    
    #===========================================================================
    # ## Propagation Kernel COMPUTATION
    #===========================================================================
    num_graphs = gr_id.max().astype(int)
    num_nodes = A.shape[0]
    if l.shape[1]==1:
        num_labels = l.max().astype(int)
    else:
        num_labels = l.shape[1]
 
    #===========================================================================
    # ## INITIALIZATION
    #===========================================================================
    
    ## CHECK and ADJUST sparseness
    if not spsp.issparse(A): 
        A = spsp.csr_matrix(A)
    else:
        if not spsp.isspmatrix_csr(A):     
            A = A.tocsr()
    # MAKE sure A is float!        
    A = A.astype(float)                  
    ## INITIALIZE kernel matrix
    K = np.ndarray(shape=(num_graphs,num_graphs,h_max+1), dtype=float)
    ## INITIALIZE label probabilities of labeled nodes (num_nodes x num_labels)
    if l.shape[1]==1:
        lab_prob = np.zeros((num_nodes,num_labels),dtype=np.float64)            
        for i in xrange(num_nodes):
            if l[i,0] > 0:
                lab_prob[i,l[i,0]-1] = 1
    else: 
        lab_prob = l
        if spsp.issparse(lab_prob):
            lab_prob = lab_prob.todense()
    lab_orig  = lab_prob.copy()      # lab_orig is the orignal label information       
    ## GET indices of labeled and unlabeled nodes
    idx_unlab = np.where(np.max(lab_prob, axis=1)!=1)[0]
    idx_all   = np.array(np.arange(0,A.shape[0]), ndmin=2)
    idx       = np.array(np.setdiff1d(idx_all,idx_unlab), ndmin=2)     
    ## INITIALIZE unlabeled/uninitialized nodes UNIFORMLY
    idx_unif = np.where(np.sum(lab_prob, axis=1)==0)[0]         
    if idx_unif.shape[0] != 0:
        lab_prob[idx_unif,:] = 1./lab_prob.shape[1] 
    ## row normalize A -> transition matrix T
    if (ktype=='label_propagation') or (ktype=='label_diffusion'):
        row_sums = np.array(A.sum(axis=1))[:,0] 
        T = A.copy()
        T.data /= row_sums[A.nonzero()[0]]    # so A and T are sparse matrix

    #===========================================================================
    # ## PROPAGATION KERNEL ITERATIONS
    #===========================================================================
    for h in xrange(h_max+1):  
        print 'ITERATION: ', h
        if h > 0:
            ## LABEL UPDATE 
            if showEachStep:
                print '...computing LABEL UPDATE'
            
            if ktype == 'label_propagation':
                lab_prob[idx,:] = lab_orig[idx,:]   # PUSH BACK original LABELS
                lab_prob = T*lab_prob
                
            elif ktype == 'label_diffusion': 
                lab_prob = T*lab_prob
            
            elif ktype == 'label_spreading':
                # S = D^(-1/2)*W*D^(-1/2)
                # y(t+1) = alpha*S*y(t)+(1-alpha)*y(0)
                alpha = 0.8
                # compute S
                diag = np.array(A.sum(axis=1)).T**(-1/2)
                D = spsp.lil_matrix((A.shape[0], A.shape[1]), dtype=float) #
                D.setdiag(diag[0,:])
                D = D.tocsr()
                S = D*A*D   
                lab_prob = np.dot((alpha*S.todense()),lab_prob) + (1-alpha)*lab_orig
        
        ## COMPUTE hashvalues 
        if showEachStep:
            print '...computing hashvalues'
        # determine path to take depending on chosen distance
        use_cauchy = (p =='L1') or (p =='tv')
        take_sqrt  = (p =='hellinger') 
        if take_sqrt:
            lab_prob = np.sqrt(lab_prob)      
        # generate random projection vector
        v = np.random.normal(size=(lab_prob.shape[1], 1))
        if use_cauchy:
            # if X, Y are N(0, 1), then X / Y has a standard Cauchy distribution
            v /= np.random.normal(size=(lab_prob.shape[1], 1))           
        # random offset
        b = w * np.random.rand()         
        # compute hashes
        # hashLabels is a Vector with length: number of nodes (hashvalue for respective node)
        hashLabels = (np.dot(lab_prob,v) + b) / w
        hashLabels = np.floor(hashLabels)                           # take floor
        uniqueHash, hashLabels = np.unique(hashLabels, return_inverse=True)  # map to consecutive integer from 0        
        ## COMPUTE kernel contribution 
        if showEachStep:
            print '...computing KERNEL contribution'                     
        # aggregate counts on graphs
        # counts is a matrix: number of graphs x number of hashlabels
        num_bins = len(uniqueHash)
        counts = np.zeros((num_graphs, num_bins))      # init counts matrix 
        # accumulate counts of hash labels                              
        for i in xrange(num_nodes):
            counts[ (gr_id[i,0]-1), hashLabels[i] ] +=1
            
        # compute base kernel (here: LINEAR kernel)
        K_h = np.dot(counts,counts.T)      
        
        ## SUM kernel contributions
        if SUM:
            if h == 0:
                K[:,:,h] = K_h 
            else:
                K[:,:,h] = K[:,:,h-1] + K_h 
        else:
            K[:,:,h] = K_h
        
        if showEachStep:
            print K[:,:,h] 

    ## VISUALIZE KERNELS
    if VIS:  
        for h in range(h_max+1): 
            K_h = K[:,:,h]
            plt.figure()
            plt.title('height:'+str(h))
            imgplot = plt.imshow(K_h)
            imgplot.set_interpolation('nearest')
            plt.colorbar()
        plt.show()
    
    return K

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      © Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting. Created using Sphinx 1.2.2.
      pyGPs-1.2/doc/build/html/_modules/pyGPs/GraphExtensions/nodeKernels.html000066400000000000000000000455041236346020200263650ustar00rootroot00000000000000 pyGPs.GraphExtensions.nodeKernels — pyGPs v1.2 documentation

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      Source code for pyGPs.GraphExtensions.nodeKernels

      #================================================================================
#    Marion Neumann [marion dot neumann at uni-bonn dot de]
#    Daniel Marthaler [marthaler at ge dot com]
#    Shan Huang [shan dot huang at iais dot fraunhofer dot de]
#    Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]
#
#    This file is part of pyGPs.
#    The software package is released under the BSD 2-Clause (FreeBSD) License.
#
#    Copyright (c) by
#    Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014
#================================================================================


import numpy as np
import scipy.linalg as spla
import scipy.sparse as spsp



      [docs]def normLap(A):
    '''
    Normalized Laplacian

    :param A: adjacency matrix
    :return: kernel matrix
    '''
    I = np.identity(A.shape[0])
    d = A.sum(axis=0)
    d = np.sqrt(1./d)    
    D = np.diag(d)
    L = I - np.dot( np.dot(D,A),D )
    return L


      

      [docs]def regLapKernel(A, sigma=1):
    '''
    Regularized Laplacian Kernel 
        
    :param A: adjacency matrix
    :param sigma: hyperparameter sigma
    :return: kernel matrix
    '''
    I = np.identity(A.shape[0])
    L = normLap(A)
    K = np.linalg.inv( I+(sigma**2)*L )
    return K


      

      [docs]def psInvLapKernel(A):
    '''
    Pseudo inverse of the normalized Laplacian.

    :param A: adjacency matrix
    :return: kernel matrix
    '''
    L = normLap(A)
    K = np.linalg.pinv(L)
    return K


      

      [docs]def diffKernel(A, beta=0.5):
    '''
    Diffusion Process Kernel 
        
    K = exp(beta * H), where H = -L = A-D
    
    K = Q exp(beta * Lambda) Q.T
        
    :param A: adjacency matrix
    :param beta: hyperparameter beta
    :return: kernel matrix
    '''
    A = np.array(A) # make sure that A is a numpy array!!
    H = A - np.diag(np.sum(A, axis=1))
    w, Q = np.linalg.eigh(H)
    Lambda = np.diag(np.exp(beta*w))
    K = np.dot(np.dot(Q, Lambda), Q.T)
    return K


      

      [docs]def VNDKernel(A, alpha=0.5):
    '''
    Von Neumann Diffusion Kernel on graph (Zhou et al., 2004)
    (also label spreading kernel)
        
    K = (I - alpha*S)^-1, where S = D^-1/2*A*D^-1/2
        
    :param A: adjacency matrix
    :param alpha: hyperparameter alpha
    :return: kernel matrix
    ''' 
    I = np.identity(A.shape[0])
    d = A.sum(axis=0)
    d = np.sqrt(1./d)    
    D = np.diag(d)
    S = np.dot( np.dot(D,A),D )
    K = np.linalg.inv( I - alpha*S )
    return K


      

      [docs]def rwKernel(A, p=1, a=2):
    '''
    p-step Random Walk Kernel with a>1
       
    K = (aI-L)^p, p>1 and L is the normalized Laplacian 
     
    :param A: adjacency matrix
    :param p: step parameter
    :param a: hyperparameter a
    :return: kernel matrix
    '''
    if type(p) != int:
	p = int(p)
    if p < 1:
	raise Exception('Step parameter p needs to be larger than 0.') 	
    if a <= 1:	
	a=1.0001
    I = np.identity(A.shape[0])
    L = normLap(A)
    K = np.linalg.matrix_power( a*I - L, p)	
    return K



      

      [docs]def cosKernel(A):
    '''
    Cosine Kernel (also Inverse Cosine Kernel)
    
    K = cos (L*pi/4), where L is the normalized Laplacian
                
    :param A: adjacency matrix
    :return: kernel matrix
    '''
    L = normLap(A)
    K = np.cos(L*np.pi/4)   
    return K

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      © Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting. Created using Sphinx 1.2.2.
      pyGPs-1.2/doc/build/html/_modules/pyGPs/GraphStuff/000077500000000000000000000000001236346020200221465ustar00rootroot00000000000000pyGPs-1.2/doc/build/html/_modules/pyGPs/GraphStuff/kernels_on_graph.html000066400000000000000000000477031236346020200263670ustar00rootroot00000000000000 pyGPs.GraphStuff.kernels_on_graph — pyGPs v1.2 documentation

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      Source code for pyGPs.GraphStuff.kernels_on_graph

      #================================================================================
#    Marion Neumann [marion dot neumann at uni-bonn dot de]
#    Daniel Marthaler [marthaler at ge dot com]
#    Shan Huang [shan dot huang at iais dot fraunhofer dot de]
#    Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]
#
#    This file is part of pyGPs.
#    The software package is released under the BSD 2-Clause (FreeBSD) License.
#
#    Copyright (c) by
#    Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 30/09/2013
#================================================================================


import numpy as np
import scipy.linalg as spla
import scipy.sparse as spsp


def normalizeKernel(K):
    K_min = np.float(np.min(K))
    K_max = np.float(np.max(K))  
    #K_norm = (K - K_min)/(np.float(K_max-K_min))
    K_norm = K/K_max
    
    if np.isnan(np.sum(K_norm)):
        print K_max 
        print K_min
        print K_norm
        print K
        print '======================'
        exit()
    return K_norm



      [docs]def normLap(A):
    ''' normalized Laplacian '''
    I = np.identity(A.shape[0])
    d = A.sum(axis=0)
    
    d = np.sqrt(1./d)    
    D = np.diag(d)
    L = I - np.dot( np.dot(D,A),D )
    return L



      

      [docs]def regLapKernel(A, sigma=1):
    ''' Regularized Laplacian Kernel 
        
    :param A:  adjacency matrix
    :param sigma: (hyper)parameter(s)
    '''

    I = np.identity(A.shape[0])
    L = normLap(A)
    K = np.linalg.inv( I+(sigma**2)*L )
    return K


      

      [docs]def psInvLapKernel(A):
    ''' Pseudo inverse of the normalized Laplacian.

    :param A:  adjacency matrix
    '''

    L = normLap(A)
    K = np.linalg.pinv(L)
    return K


      

      [docs]def diffKernel(A, beta=0.5):
    ''' Diffusion Process Kernel 
        
    K = exp(beta * H), where H = -L = A-D
    
    K = Q exp(beta * Lambda) Q.T
        
    :param A:  adjacency matrix
    :param beta: (hyper)parameter(s)
    '''

    A = np.array(A) # make sure that A is a numpy array!!

    H = A - np.diag(np.sum(A, axis=1))
    w, Q = np.linalg.eigh(H)
    Lambda = np.diag(np.exp(beta*w))
    K = np.dot(np.dot(Q, Lambda), Q.T)

    return K


      

      [docs]def VNDKernel(A, alpha=0.5):
    '''  Von Neumann Diffusion Kernel on graph (Zhou et al., 2004)
    (also label spreading kernel)
        
    K = (I - alpha*S)^-1, where S = D^-1/2*A*D^-1/2
        
    :param A:  adjacency matrix
    :param alpha: (hyper)parameter(s)
    '''
        
    I = np.identity(A.shape[0])
    d = A.sum(axis=0)
    d = np.sqrt(1./d)    
    D = np.diag(d)
    S = np.dot( np.dot(D,A),D )

    K = np.linalg.inv( I - alpha*S )
    return K



      

      [docs]def rwKernel(A, p=1, a=2):
    ''' p-step Random Walk Kernel with a>1
       
    K = (aI-L)^p, p>1 and L is the normalized Laplacian 
     
    :param A:  adjacency matrix
    :param p:  step parameter
    :param a:  (hyper)parameter(s)
    '''
    if type(p) != int:
	p = int(p)
	
    if p < 1:
	raise Exception('Step parameter p needs to be larger than 0.') 	

    if a <= 1:	
	a=1.0001

    I = np.identity(A.shape[0])
    L = normLap(A)
    K = np.linalg.matrix_power( a*I - L, p)	
    return K



      

      [docs]def cosKernel(A):
    ''' Cosine Kernel (also Inverse Cosine Kernel)
    
    K = cos (L*pi/4), where L is the normalized Laplacian
                
    :param A:  adjacency matrix

    '''
       
    L = normLap(A)
    K = np.cos(L*np.pi/4)   
    return K

      Navigation

      © Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting. Created using Sphinx 1.1.3.
      pyGPs-1.2/doc/build/html/_sources/000077500000000000000000000000001236346020200170465ustar00rootroot00000000000000pyGPs-1.2/doc/build/html/_sources/CV.txt000066400000000000000000000036401236346020200201220ustar00rootroot00000000000000K-fold Cross-Validation ================================== In this demo, we'll show you the typical process of using GP for machine learning from loading data, preprocessing, training, predicting to validation and evaluation. Load data -------------------- We use the ionosphere dataset [1]_ from Johns Hopkins University Ionosphere database. It is available in UCI machine learning repository. Then we need to do some data cleaning. Here we deal with label in ionosphere data, change "b" to"-1", and "g" to "+1". These preprocessing implementation are availabe in the source code. Cross Validation ---------------------------- Now, lets focus on the use of cross-validation. :: K = 10 # number of folds for x_train, x_test, y_train, y_test in valid.k_fold_validation(x, y, K): # This is a binary classification problem model = pyGPs.GPC() # Since no prior knowldege, leave everything default model.optimize(x_train, y_train) # Predition ymu, ys2, fmu, fs2, lp = model.predict(x_test, ys=y_test) # ymu for classification is a continuous value over -1 to +1 # If you want predicting result to either one of the classes, take a sign of ymu. ymu_class = np.sign(ymu) # Evluation acc = valid.ACC(ymu_class, y_test) # accuracy rmse = valid.RMSE(ymu_class, y_test) # root-mean-square error Evaluation measures ----------------------------- We implemented some classical evaluation measures. - RMSE - root mean squared error - ACC - classification/regression accuracy - Prec - classification precision for class +1 - Recall - classification recall for class +1 - NLPD - negative log predictive density in transformed observation space .. [1] Sigillito, V. G., Wing, S. P., Hutton, L. V., \& Baker, K. B. (1989). Classification of radar returns from the ionosphere using neural networks. Johns Hopkins APL Technical Digest, 10, 262-266. pyGPs-1.2/doc/build/html/_sources/Examples.txt000066400000000000000000000021701236346020200213650ustar00rootroot00000000000000Demos ========================= There are several demos exemplifying the use of pyGPs for various Gaussian process (:math:`GP`) tasks. We recommend to first go through *Basic GP Regression* which introduces the :math:`GP` regression model. Basic regression is the most intuitive and simplest learning task feasable with :math:`GPs`. The other demos will then provide a general insight into more advanced functionalities of the package. You will also find the implementation of the demos in the source_ folder under `pyGPs/Demo`_. .. _pyGPs/Demo: https://github.com/marionmari/pyGPs/tree/master/pyGPs/Demo The Demos give some theoretical explanations. Further, it is useful to have a look at our documentation on `Kernels & Means`_ and `Optimizers`_. .. _Kernels & Means: Kernels.html .. _Optimizers: Opts.html .. _source: https://github.com/marionmari/pyGPs Regression .. toctree:: :maxdepth: 1 GPR GPR_FITC Classification .. toctree:: :maxdepth: 1 GPC GPC_FITC GPMC Some examples for real-world data .. toctree:: :maxdepth: 1 CV demoMaunaLoa demoHousing SemiSupervised GraphKernel pyGPs-1.2/doc/build/html/_sources/GPC.txt000066400000000000000000000074671236346020200202360ustar00rootroot00000000000000Basic Classification ========================= The demo shown in this tutorial can be obtained by running *pyGPs/Demo/demo_GPC.py*. Load data -------------------- First, we import the data:: # GPC target class are +1 and -1 demoData = np.load('data_for_demo/classification_data.npz') x = demoData['x'] # training data y = demoData['y'] # training target z = demoData['xstar'] # test data The :math:`120` data points were generated from two Gaussians with different means and covariances. One Gaussian is isotropic and contains :math:`2/3` of the data (blue), the other is highly correlated and contains :math:`1/3` of the points (red). Note, that the labels for the targets are specified to be :math:`\pm 1` (and not :math:`0/1`). In the plot, we superimpose the data points with the posterior equi-probability contour lines for the probability of the second class given complete information about the generating mechanism. .. figure:: _images/d2_1.png :height: 600 px :width: 800 px :align: center :scale: 70 % First example :math:`\rightarrow` state default values ---------------------------------------------------------------- Again, lets see the simplest use of gp classification at first :: model = pyGPs.gp.GPC() # binary classification (default inference method: EP) model.fit(x, y) # fit default model (mean zero & rbf kernel) with data model.optimize(x, y) # optimize hyperparamters (default optimizer: single run minimize) model.predict(z) # predict test cases Note, that inference is done via expectation propagation (EP) approximation by deault. How to set inference to Laplace approximation, see :ref:`more_on_GPC`. Second example :math:`\rightarrow` GP classification ------------------------------------------------------------ So we first state the model to be :math:`GP` classification now:: model = pyGPs.GPC() The rest is similar to GPR:: k = pyGPs.cov.RBFard(log_ell_list=[0.05,0.17], log_sigma=1.) model.setPrior(kernel=k) model.setData(x, y) model.plotData_2d(x1,x2,t1,t2,p1,p2) model.fit() model.optimize() model.predict(z, ys=np.ones((z.shape[0],1))) **[Theory]** In this example, we used an RBF kernel (squared exponential covariance function) with automatic relevance determination (ARD). This covariance function has one characteristic length-scale parameter for each dimension of the input space (here :math:`2` in total), and a signal magnitude parameter, resulting in a total of :math:`3` hyperparameters. ARD with separate length-scales for each input dimension is a very powerful tool to learn which inputs are important for the predictions: if length-scales are short, input dimensions are very important, and when they grow very large (compared to the spread of the data), the corresponding input dimensions will be mostly ignored. Note, *pyGPs.GPC().plot()* is a toy method for 2-d data:: model.plot(x1,x2,t1,t2) The contour plot for the predictive distribution is shown below. Note, that the predictive probability is fairly close to the probabilities of the generating process in regions of high data density. Note also, that as you move away from the data, the probability approaches :math:`1/3`, the overall class probability. .. figure:: _images/d2_2.png :height: 600 px :width: 800 px :align: center :scale: 70 % Examining the two ARD characteristic length-scale parameters after learning, you will find that they are fairly similar, reflecting the fact that for this data set, both input dimensions are important. .. _more_on_GPC: A bit more things you can do ---------------------------------------------------- GPC uses expectation propagation (EP) inference and Error function likelihood by default, you can explictly change to other methods: :: model.useInference("Laplace") pyGPs-1.2/doc/build/html/_sources/GPC_FITC.txt000066400000000000000000000063401236346020200210300ustar00rootroot00000000000000Sparse Classification ======================== The demo in this tutorial can be obtained by running *pyGPs/Demo/demo_GPC_FITC.py*. This demo is more or less a repetition of the demo of `FITC regression`_. .. _FITC regression: GPR_FITC.html First example :math:`\rightarrow` default inducing points ------------------------------------------------------------- First load the same data as in the GPC demo. **[Theory]** In case the number of training inputs :math:`x` exceeds a few hundred, approximate inference using Laplacian Approximation or Expectation Propagation takes too long. As in regression, we offer the FITC approximation based on a low-rank plus diagonal approximation to the exact covariance to deal with these cases. The general idea is to use inducing points :math:`u` and to base the computations on cross-covariances between training, test and inducing points only. Okay, now the model is FITC classificiation:: model = pyGPs.GPC_FITC() The difference between the usage of basic :math:`GP` is that we will have to specify inducing points. In our first example, we will introduce how to perform sparse GPC with the default settings. The default inducing points form a grid (hypercube in higher dimension), where each dimension has :math:`5` values in equidistant steps in :math:`[min, max]`, where :math:`min` and :math:`max` are the minimum and maximum values of the input data by default. In order to specify the dimension of input data, we HAVE TO set data first:: model.setData(x, y) The number of inducing points per axis is :math:`5` per default. How to change this, see :ref:`more_on_GPC_FITC`. Then, the regular process follows:: model.optimize() model.predict(z, ys=np.ones((z.shape[0],1))) model.plot(x1,x2,t1,t2) .. figure:: _images/d4_1.png :height: 600 px :width: 800 px :align: center :scale: 70 % The equispaced default inducing points :math:`u` are shown as black circles in the plot. Second example :math:`\rightarrow` user-defined inducing points -------------------------------------------------------------------- Alternatively, a random subset of the training points can be used as inducing points. Note, that there are various different ways of how to set the inducing points. So, in the second example let us use a user-defined set of inducing points:: u1,u2 = np.meshgrid(np.linspace(-2,2,5),np.linspace(-2,2,5)) u = np.array(zip(np.reshape(u2,(np.prod(u2.shape),)),np.reshape(u1,(np.prod(u1.shape),)))) Here, we also use a grid euqually spaced, but without the values on the margin of the grid.(i.e. shrinking the grid) Then, we can just pass :math:`u` when specifying prior:: m = pyGPs.mean.Zero() k = pyGPs.cov.RBFard(log_ell_list=[0.05,0.17], log_sigma=1.) model.setPrior(mean=m, kernel=k, inducing_points=u) The prediction results for this set of inducing points are shown below: .. figure:: _images/d4_2.png :height: 600 px :width: 800 px :align: center :scale: 70 % .. _more_on_GPC_FITC: A bit more things you can do ------------------------------ As in standard GPC, it is possible to use other inference/likelihood in the FITC method:: model.useInference("Laplace") Change the number of inducing points per axis:: model.setData(x, y, value_per_axis=10) pyGPs-1.2/doc/build/html/_sources/GPMC.txt000066400000000000000000000043321236346020200203370ustar00rootroot00000000000000Multi-class Classification =========================== GPMC is NOT based on multi-class Laplace approximation. It works as a one vs. one classification wrapper. In other words, GPMC trains a GPC model for each pair of two classes, and uses a majority voting scheme over all results to determine the final class. The method only returns the predictive class with highest rating; no other values (such as variance) are returned. Lets see a practical example to classify the 10 (0,1,2,...9) hand-written digits in the USPS digits dataset. Load data -------------------- The USPS digits data were gathered at the Center of Excellence in Document Analysis and Recognition (CEDAR) at SUNY Buffalo, as part of a project sponsored by the US Postal Service. The dataset is described in [1]_. :: data = loadmat('data_for_demo/usps_resampled.mat') x = data['train_patterns'].T # train patterns y = data['train_labels'].T # train labels xs = data['test_patterns'].T # test patterns ys = data['test_labels'].T # test labels To be used in GPMC, labels should start from 0 to k (k = number of classes). GPMC example --------------------- State model with 10-class classification problem: :: model = pyGPs.GPMC(10) Pass data to model: :: model.setData(x,y) Train default GPC model for each binary classification problem, and decide label for test patterns of hand-writen digits. The return value *predictive_vote[i,j]* is the probability of being class *j* for test pattern *i*. :: predictive_vote = model.optimizeAndPredict(xs) predictive_class = np.argmax(predictive_vote, axis=1) Just like we did for GP classification, you can use specific settings (other than default) for all binary classificiation problem for example by: :: m = pyGPs.mean.Zero() k = pyGPs.cov.RBF() model.setPrior(mean=m,kernel=k) model.useInference("Laplace") For more information on how to use non-default settings see `demo_GPC`_ and `demo_GPR`_. .. _demo_GPC: GPC.html .. _demo_GPR: GPR.html Beside *optimizeAndPredict(xs)*, there is also an option to perform prediction without hyperparameter optimization: :: model.fitAndPredict(xs) .. [1] A Database for Handwritten Text Recognition Research, J. J. Hull, IEEE PAMI 16(5) 550-554, 1994. pyGPs-1.2/doc/build/html/_sources/GPR.txt000066400000000000000000000150431236346020200202420ustar00rootroot00000000000000Basic Regression ========================= The code shown in this tutorial can be executed by running *pyGPs/Demo/demo_GPR.py* This demo will not only introduce the regression model, it also provides the general insight of how to use the package. This general information will not be repeated in the other demos. Import packages -------------------- Once you installed pyGPs, the typical way to import it is: :: import pyGPs import numpy as np Load data -------------------- First, load the data for this demo. The data consists of :math:`n=20` 1-d data points drawn from a unit Gaussian. This is the same data used in the GPML example (it is hardcoded in *data/regression_data.npz*). Note that target vector :math:`y` can be in 2d matrix with shape (nn,1) or 1d vector with shape (nn,) where :math:`nn` is number of test inputs. pyGPs can work with either format. :: demoData = np.load('data_for_demo/regression_data.npz') x = demoData['x'] # training data y = demoData['y'] # training target z = demoData['xstar'] # test data A five-line toy example --------------------------- Now lets do regression with Gaussian processes. Using pyGPs for regression is really simple; here is the most basic example: :: model = pyGPs.GPR() # specify model (GP regression) model.fit(x, y) # fit default model (mean zero & rbf kernel) with data model.optimize(x, y) # optimize hyperparamters (default optimizer: single run minimize) model.predict(z) # predict test cases model.plot() # and plot result By default, GPR uses a zero mean, the rbf kernel and a Gaussian likelihood. Default optimizer is a single run of Rasmussen's minimize. You will see below how to set non-default values in another example. *pyGPs.GPR().plot()* will plot the result, where the dark line is the posterior mean and the green-shaded area is the posterior variance. Note, that *plot()* is not a general method as it is not trivial to visualize high dimensional data. Here, *pyGPs.GPR().plot()* works for 1-d data only, while *pyGPs.GPC().plot()* is a toy method visualising 2-d input data in a classification scenario. .. figure:: _images/d1_1.png :height: 600 px :width: 800 px :align: center :scale: 70 % A more complicated example --------------------------- Now lets do another example to get insight into more advanced features of the toolbox. You can specify non-default mean and covariance functions: :: m = pyGPs.mean.Linear( D=x.shape[1] ) + mean.Const() k = pyGPs.cov.RBF() model.setPrior(mean=m, kernel=k) Here, we use a composite mean as the sum of a linear and a constant function, and an rbf kernel. The initial hyperparameters are left to their default values. See `Kernels & Means`_ for a complete documentation of kernel/mean specification and custom kernel/mean construction. Once kernel and mean are specified, they are passed to the prior using *setPrior()*. .. _Kernels & Means: Kernels.html You can add the traning data to the model explicitly by using *setData()*. So, you avoid passing them into *fit()* or *optimize()* each time used. More importantly, the deafult mean will be adapted to the average value of the trainging labels :math:`y` (if you do not specify mean function by your own). Further, you can plot the data in the 1-d case: :: model.plotData_1d() .. figure:: _images/d1_2.png :height: 600 px :width: 800 px :align: center :scale: 70 % You can specify a optimization method different from the default, which is a single run of Rasmussen's minimize. For example, you can choose to rerun the optimization method several times with different random initializations: :: model.setOptimizer("Minimize", num_restarts=30) The optimized hyperparameters returned by *optimize()* are then set to be the ones obtained from the run with the best result. The whole functionality for optimization is introduced in detail in the documentation `Optimizers`_. .. _Optimizers: Opts.html Instead of *fit()*, which only fits data using given hyperparameters, *optimize()* will optimize hyperparamters based on marginal likelihood: :: model.optimize() There are several properties you can get from the model: :: model.nlZ # negative log marginal likelihood model.dnlZ.cov # direvatives of negative log marginal likelihood model.dnlZ.lik model.dnlZ.mean model.posterior.sW # posterior structure model.posterior.alpha model.posterior.L model.covfunc.hyp model.meanfunc.hyp model.likfunc.hyp model.fm # latent mean model.fs2 # latent variance model.ym # predictive mean model.ys2 # predictive variance model.lp # log predictive probability For example, to get the log marginal likelihood use: :: print 'Optimized negative log marginal likelihood:', round(model.nlZ,3) Prediction on the test data will return five values, which are output mean (ymu) resp. variance (ys2), latent mean (fmu) resp. variance (fs2), and log predictive probabilities (lp) :: ym, ys2, fm, fs2, lp = model.predict(z) Plot data. Note that *GPR.plot()* is a toy method only for visualising 1-d data. Here we got a different posterior by using a different prior other than in the default example. :: model.plot() .. figure:: _images/d1_3.png :height: 600 px :width: 800 px :align: center :scale: 70 % A bit more things you can do ------------------------------- **[For all Models]** Speed up computation time for prediction if you know posterior in advance. Posterior is passed as an object with three fields (attributes) post.alpha, post.sW and post.L. How to use these vectors to represent the posterior can be best seen from Algorithm 2.1 (page 19) in Chapeter 2 of the `GPML`_ book by Rasmussen and Williams, 2006. :: post = myPosterior() # known in advance ym, ys2, fm, fs2, lp = model.predict_with_posterior( post,z ) .. _GPML: http://www.gaussianprocess.org/gpml/chapters/RW2.pdf **[Only for Regression]** Specify noise of data (with :math:`\sigma=0.1` by default): :: model.setNoise( log_sigma = np.log(0.1) ) You do not need to specify the noise parameter if you are optimizing the hyperparamters later anyhow. All plotting methods have keyword axisvals. You can adjust plotting range if you want. For example: :: model.plot(axisvals = [-1.9, 1.9, -0.9, 3.9]) Switch to other Inference and Likelihood functions. :: model.useInference("EP") model.useLikelihood("Laplace") pyGPs-1.2/doc/build/html/_sources/GPR_FITC.txt000066400000000000000000000070011236346020200210420ustar00rootroot00000000000000Sparse Regression ========================= The code shown in this tutorial can be obtained by running *pyGPs/Demo/demo_GPR_FITC.py* This demo is more or less similar to the demo of `FITC classification`_. .. _FITC classification: GPC_FITC.html First example :math:`\rightarrow` default inducing points -------------------------------------------------------------- First load the same data as in the GPR demo. **[Theory]** In case the number of training inputs :math:`x` exceeds a few hundred, approximate inference using Laplace approximation or expectation propagation takes too long. We offer the FITC approximation based on a low-rank plus diagonal approximation to the exact covariance to deal with these cases. The general idea is to use inducing points :math:`u` and to base the computations on cross-covariances between training, test and inducing points only. Okay, now the model is FITC regression:: model = pyGPs.GPR_FITC() The difference between the usage of basic :math:`GP` regression is that we will have to specify inducing points. In the first example here, we will introduce you how to use the default settings. The default inducing points are a grid (hypercube for higher dimensions), where each dimension has 5 values in equidistant steps in :math:`[min, max]`, where :math:`min` and :math:`max` are the minimum and maximum values of the input data by default. In order to specify the dimension of input data, we HAVE TO set data first:: model.setData(x, y) The number of inducing points per axis is :math:`5` per default. Now, the regular training and prediction routines follow: :: model.optimize() model.predict(z) model.plot() .. figure:: _images/d3_1.png :height: 600 px :width: 800 px :align: center :scale: 70 % The equidistant default inducing points :math:`u` that are shown in the figure as black x's. To change the number of inducing points per axis just specify a different value per axis:: model.setData(x, y, value_per_axis=10) Second example :math:`\rightarrow` user-defined inducing points ------------------------------------------------------------------ Alternatively, a random subset of the training points can be used as inducing points. Note, that there are plenty of methods to set these inducing points. So, in the second example let us use a user-defined set of inducing points. You can pick a set of fixed inducing points by hand: :: u = np.array([[-1], [-0.8], [-0.5], [0.3],[1.]]) You can also use equidistant inducing points :math:`u`, but without the values on the margin of the grid.(i.e. shrinking the range of values) :: num_u = np.fix(x.shape[0]/2) u = np.linspace(-1.3,1.3,num_u).T u = np.reshape(u,(num_u,1)) Then pass :math:`u` when specifying prior. :: m = pyGPs.mean.Zero() k = pyGPs.cov.RBFard(log_ell_list=[0.05,0.17], log_sigma=1.) model.setPrior(mean=m, kernel=k, inducing_points=u) The left figure below shows the result of fixed inducing points, and the right figure shows the result for equidistant :math:`u`. .. image:: _images/d3_2.png :width: 45% .. image:: _images/d3_3.png :width: 45% **[Theory]** Note that the predictive variance is overestimated outside the support of the inducing inputs. In a multivariate example where densely sampled inducing inputs are infeasible, one can also try to simply use a random subset of the training points. A bit more things you can do ----------------------------- Switch to other Inference and Likelihood functions. :: model.useInference("EP") model.useLikelihood("Laplace") pyGPs-1.2/doc/build/html/_sources/Graph.txt000066400000000000000000000010761236346020200206540ustar00rootroot00000000000000Kernels for Graph Data ============================ You can refer to our demo for `semi-supervised learning`_ for a simple usage of kernels for graph data. And there is also a demo for `graph kernel`_ using an example of propagation kernel. .. _`semi-supervised learning`: SemiSupervised.html .. _`graph kernel`: GraphKernel.html Kernels on Graph --------------------------------------- .. automodule:: pyGPs.GraphExtensions.nodeKernels :members: Graph Kernels --------------------------------------- .. automodule:: pyGPs.GraphExtensions.graphKernels :members: pyGPs-1.2/doc/build/html/_sources/GraphKernel.txt000066400000000000000000000057461236346020200220250ustar00rootroot00000000000000Graph Kernels ========================= The code shown in this tutorial can be executed by running *pyGPs/Demo/demo_GraphKernel.py* Load data -------------------- MUTAG [1]_ is a data set of 188 mutagenic aromatic and heteroaromatic nitro compounds labeled according to whether or not they have a mutagenic effect on the Gram-negative bacterium Salmonella typhimurium. MUTAG is a simple but popular dataset to benchmark graph kernels. :: data = np.load('MUTAG.npz') A = csc_matrix( (data['adj_data'], data['adj_indice'], \ data['adj_indptr']), shape=data['adj_shape']) # n x n adjancy array (sparse matrix) gr_id = data['graph_ind'] # n x 1 graph id array node_label = data['responses'] # n x 1 node label array graph_label = data['labels'] # N x 1 graph label array Note that adjacency matrix for all graphs is usually too big to fit into memory when using :math:`GP`. Therefore, we generate :math:`A` using sparse matrix *csc_matrix* provided by *scipy*. Compute propagation kernel matrix ---------------------------------------- Propagation kernels [2]_ are recently introduced fast and flexible graph kernels. Generate prapagation kernel based on given graph data. :: num_Iteration = 10 w = 1e-4 dist = 'tv' # possible values: 'tv', 'hellinger' K = graphKernels.propagationKernel(A, node_label, gr_id, num_Iteration, w, dist, 'label_diffusion') Adjacency matrix :math:`A` can either be in format of standard numpy matrix or sparse matrix. Standard GP Classification ---------------------------------------- :math:`K` is a big square matrix with all rows and columns of the number of data points. By specifying the indice of training data and test data, we will form two matrix M1 and M2 with the exact format which *pyGPs.Core.cov.Pre* needed. :: M1,M2 = graphUtil.formKernelMatrix(Matrix, indice_train, indice_test) k = pyGPs.cov.Pre(M1,M2) M1 is a matrix with shape **number of training points plus 1** by **number of test points** - cross covariances matrix (train by test) - last row is self covariances (diagonal of test by test) M2 is a square matrix with **number of training points** for each dimension - training set covariance matrix (train by train) The following is the standard way to do GP classification :: model = pyGPs.GPC() model.setPrior(kernel=k) model.fit(x_train, y_train) model.predict(x_test) .. [1] Debnath, A., Lopez de Compadre, R., Debnath, G., Shusterman, A., and Hansch, C.. Structure-activity relationship of mutagenic aromatic and heteroaromatic nitro compounds. Correlation with molecular orbital energies and hydrophobicity. J. Med. Chem., 34:786–797, 1991. .. [2] Neumann, M., Patricia, N., Garnett, R., Kersting, K.: Efficient Graph Kernels by Randomization. In: P.A. Flach, T.D. Bie, N. Cristianini (eds.) ECML/PKDD, Notes in Computer Science, vol. 7523, pp. 378-393. Springer (2012). pyGPs-1.2/doc/build/html/_sources/Install.txt000066400000000000000000000021111236346020200212100ustar00rootroot00000000000000Installation ============================ 1. First, download_ the archive from github and extract it to any local directory. .. _download: https://github.com/marionmari/pyGPs/releases/tag/v1.1 2. You can either add the local directory to your PYTHONPATH :: export PYTHONPATH=$PYTHONPATH:/path/to/local/directory/../parent_folder_of_pyGPs 3. Or install the package using setup.py:: sudo python setup.py install Requirements ------------ * `python 2.6 or 2.7`_ * `scipy`_, `numpy`_, and `matplotlib`_: open-source packages for scientific computing in Python. .. _python 2.6 or 2.7: http://www.python.org/ .. _scipy: http://www.scipy.org/ .. _numpy: http://www.numpy.org/ .. _matplotlib: http://matplotlib.org/ Example installation on Ubuntu & Debian:: sudo apt-get install python2.7 python-numpy python-scipy python-matplotlib Example installation on Mac via Macports (requires XCode and MacPorts):: sudo port install python27 py27-numpy py27-scipy py27-matplotlib For other systems please check the installation instructions on the respective package web sites. pyGPs-1.2/doc/build/html/_sources/Kernels.txt000066400000000000000000000150401236346020200212120ustar00rootroot00000000000000Kernels & Means ============================ Simple Kernel & Mean --------------------- You may already seen, we can specify a kernel function like this(same for mean fucntions): :: k = pyGPs.cov.RBF( log_ell=-1., log_sigma=0. ) There are several points need to be noticed: 1. Most parameters are initilized in their logorithms. This is because we need to make sure they are positive during optimization. e.g. Here length scale and signal variance should always be positive. 2. Most kernel functions have a scalar in front, namely signal variance(set by log_sigma) 3. If you will do optimization later anyway, you can just leave parameters to be default Some Special Cases --------------------- 1. For some kernels/means, number of hyperparameters depends on the dimension of input data. You can either enter the dimension, which use default values: :: m = pyGPs.mean.Linear( D=x.shape[1] ) or you can initialze with the exact hyperparameters, you should enter as a list, one element for each dimension :: m = pyGPs.mean.Linear( alpha_list=[0.2, 0.4, 0.3] ) All these "hyp-dim-dependent" functions are: * *pyGPs.mean.Linear* * *pyGPs.cov.RBFard* * *pyGPs.cov.LINard* * *pyGPs.cov.RQard* 2. For *pyGPs.cov.RBFunit()*, its signal variance is always 1 (because of unit magnitude). Therefore this function do not have a hyperparameter of "signal variance". 3. *pyGPs.cov.Poly()* has three parameters, where hyperparameters are: * c -> inhomogeneous offset * sigma -> signal deviation however, * d -> order of polynomial will be treated as normal parameter, i.e. will not be trained 4. Explicitly set *pyGPs.cov.Noise* is not necessary, because noise are already added in likelihood. Composite Kernels & Meams ---------------------------- Adding and muliplying Kernels(Means) is really simple: :: k = pyGPs.cov.Periodic() * pyGPs.cov.RBF() k = 0.5*pyGPs.cov.LIN() + pyGPs.cov.Periodic() Scalar will also be treated as a hyperparameter. For example, k = s1 * k1 + s2 * k2, then the list of hyperparameters is hyp = [s1, k1.hyp, s2, k2.hyp]. Scalar is passed in logorithm domain such that it will always be positive during optimization. Beside + / * , there is also a power operator for mean functions: :: m = ( pyGPs.mean.One() + pyGPs.mean.Linear(alpha_list=[0.2]) )**2 Precomputed Kernel Matrix ----------------------------- In certain cases, you may have a precomputed kernel matrix, but its non-trivial to write down the exact formula of kernel functions. Then you can specify your kernel in the following way. A precomputed kernel also fits with other kernels. In other words, it can also be composited as the way other kernels functions do. :: k = pyGPs.cov.Pre(M1, M2) M1 and M2 are your precomputed kernel matrix, where, M1 is a matrix with shape **number of training points plus 1** by **number of test points** - cross covariances matrix (train by test) - last row is self covariances (diagonal of test by test) M2 is a square matrix with **number of training points** for each dimension - training set covariance matrix (train by train) A precomputed kernel can also be composited with other kernels. You need to explictly add scalar for *pyGPs.cov.Pre()*. :: k = 0.5*pyGPs.cov.Pre(M1, M2) + pyGPs.cov.RBF() Developing New Kernel & Mean Functions ----------------------------------------- We also support the development of new kernel/mean classes, your customized kernel class need to follow the template as below: :: # Your kernel class needs to inherit base class Kernel, # which is in the module of Core.cov class MyKernel(Kernel): def __init__(self, hyp): ''' Intialize hyperparameters for MyKernel. ''' self.hyp = hyp def getCovMatrix(self,x=None,z=None,mode=None): ''' Return the specific covariance matrix according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self covariance matrix of test data(test by 1). 'train' return training covariance matrix(train by train). 'cross' return cross covariance matrix between x and z(train by test) :return: the corresponding covariance matrix ''' pass def getDerMatrix(self,x=None,z=None,mode=None,der=None): ''' Compute derivatives wrt. hyperparameters according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self derivative matrix of test data(test by 1). 'train' return training derivative matrix(train by train). 'cross' return cross derivative matrix between x and z(train by test) :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' pass and for customized mean class: :: # Your mean class needs to inherit base class Mean, # which is in the module of Core.mean class MyMean(Mean): def __init__(self, hyp): ''' Intialize hyperparameters for MyMean. ''' self.hyp = hyp def getMean(self, x=None): ''' Get the mean vector. ''' pass def getDerMatrix(self, x=None, der=None): ''' Compute derivatives wrt. hyperparameters. :param x: training data :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' pass You can test your customized mean/kernel function using our framework of unit test. Taking kernel test as an example, you can uncomment method *test_cov_new* in *pyGPs.Testing.unit_test_cov.py* to check the outputs of your kernel function. :: # Test your customized covariance function def test_cov_new(self): k = myKernel() # specify your covariance function self.checkCovariance(k) and testing mean function in *pyGPs.Testing.unit_test_mean.py* :: # Test your customized mean function def test_mean_new(self): m = myMean # specify your mean function self.checkMean(m) List of Kernels and Default Parameters --------------------------------------- .. automodule:: pyGPs.Core.cov :members: List of Means and Default Parameters ----------------------------------------- .. automodule:: pyGPs.Core.mean :members: pyGPs-1.2/doc/build/html/_sources/Likelihoods.txt000066400000000000000000000124211236346020200220550ustar00rootroot00000000000000Likelihoods & Inference ============================ Changing Likelihood & Inference ------------------------------------ By default, * GPR uses Gaussian likelihood and exact inference. * GPC uses Error functionlikelihood and EP inference. * FITC model uses same default with corresponding FITC inference. * GPMC calls GPC and thus uses the default setting of GPC You can change to other likelihood or inference methods using: :: model.useLikelihood(newLik) model.useInference(newInf) *newLik* and *newInf* are **Strings**. Currently the options are: 1. Regression model * newLik: **"Laplace"**. Note this will force inference method to be EP. * newInf: **"EP"**, **"Laplace"**. 2. Classification model (including GPMC) * newInf: **"Laplace"** Developing New Likelihood & Inference Functions ------------------------------------------------- We also support the development of new likelihood/inference classes, your customized inference class need to follow the template as below: :: # Your inference class needs to inherit base class Inference, # which is in the module of Core.inf class MyKernel(Kernel): def __init__(self): pass def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): ''' Inference computation based on inputs. post, nlZ, dnlZ = inf.proceed(mean, cov, lik, x, y) INPUT: cov: name of the covariance function (see covFunctions.m) lik: name of the likelihood function (see likFunctions.m) x: n by D matrix of training inputs y: 1d array (of size n) of targets OUTPUT: post(instance of postStruct): struct representation of the (approximate) posterior containing: nlZ: returned value of the negative log marginal likelihood dnlZ(instance of dnlZStruct): struct representation for derivatives of the negative log marginal likelihood w.r.t. each hyperparameter. Usually, the approximate posterior to be returned admits the form: N(m=K*alpha, V=inv(inv(K)+W)), where alpha is a vector and W is diagonal; if not, then L contains instead -inv(K+inv(W)), and sW is unused. For more information on the individual approximation methods and their implementations, see the respective inference function below. See also gp.py :param meanfunc: mean function :param covfunc: covariance function :param likfunc: likelihood function :param x: training data :param y: training labels :param nargout: specify the number of output(1,2 or 3) :return: posterior, negative-log-marginal-likelihood, derivative for negative-log-marginal-likelihood-likelihood ''' pass where **postStruct** and **dnlZStruct** is also defined in *Core.inf*. :: class postStruct(object): ''' Data structure for posterior post.alpha -> 1d array containing inv(K)*m, where K is the prior covariance matrix and m the approx posterior mean post.sW: -> 1d array containing diagonal of sqrt(W) the approximate posterior covariance matrix is inv(inv(K)+W) post.L -> 2d array, L = chol(sW*K*sW+identity(n)) ''' class dnlZStruct(object): ''' Data structure for the derivatives of mean, cov and lik functions. dnlZ.mean -> list of derivatives for each hyperparameters in mean function dnlZ.cov -> list of derivatives for each hyperparameters in covariance function dnlZ.lik -> list of derivatives for each hyperparameters in likelihood function ''' Customizing likelihood function is more complicated. We will omit it here to keep this this page not too long. However, you can find detailed explaination either in the **source code** *Core.lik* or in coresponding section of **manual**. Just like testing kernel/mean fucntions, you can also find unit test module for likelihood and inference functions. To test your customized inference function, uncomment the following method in *pyGPs.Testing.unit_test_inf.py*. :: # Test your customized inference function def test_inf_new(self): # specify your inf function # set mean/cov/lik functions post, nlZ, dnlZ = inffunc.proceed(meanfunc, covfunc, likfunc, self.x, self.y, nargout=3) self.checkFITCOutput(post, nlZ, dnlZ) and test customized likelihood function in *pyGPs.Testing.unit_test_lik.py* :: # Test your customized likelihood function def test_cov_new(self): likelihood = myLikelihood() # specify your likelihood function self.checkLikelihood(likelihood) List of Likelihoods --------------------------------------- .. autoclass:: pyGPs.Core.lik.Erf :members: .. autoclass:: pyGPs.Core.lik.Gauss :members: .. autoclass:: pyGPs.Core.lik.Laplace :members: List of Inference ----------------------------------------- .. autoclass:: pyGPs.Core.inf.Exact :members: .. autoclass:: pyGPs.Core.inf.EP :members: .. autoclass:: pyGPs.Core.inf.Laplace :members: .. autoclass:: pyGPs.Core.inf.FITC_Exact :members: .. autoclass:: pyGPs.Core.inf.FITC_EP :members: .. autoclass:: pyGPs.Core.inf.FITC_Laplace :members: pyGPs-1.2/doc/build/html/_sources/Opts.txt000066400000000000000000000025711236346020200205410ustar00rootroot00000000000000Optimizers ============================ Optimization Methods ---------------------------- As you may have already seen in the demos, the optimizer is initialized in the following way: .. automethod:: pyGPs.Core.gp.GP.setOptimizer Developing Optimization Methods ------------------------------------- We also support the development of new optimizers. Your own optimizer should inherent base class Optimizer in *pyGPs.Core.opt* and follow the template as below: :: class MyOptimizer(Optimizer): def __init__(self, model=None, searchConfig = None): self.model = model def findMin(self, x, y): ''' Find minimal value based on negative-log-marginal-likelihood. optimalHyp, funcValue = findMin(x, y) where funcValue is the minimal negative-log-marginal-likelihood during optimization, and optimalHyp is a flattened numpy array (in sequence of meanfunc.hyp, covfunc.hyp, likfunc.hyp) of the hyparameters to achieve such value. You can achieve advanced search strategy by initializing Optimizer with searchConfig, which is an instance of pyGPs.Optimization.conf. See more in pyGPs.Optimization.conf and pyGPs.Core.gp.GP.setOptimizer, as well as in online documentation of section Optimizers. ''' pass pyGPs-1.2/doc/build/html/_sources/SemiSupervised.txt000066400000000000000000000116771236346020200225720ustar00rootroot00000000000000Semi-supervised Learning with Graphs ======================================= The code shown in this tutorial can be executed by running *pyGPs/Demo/demo_NodeKernel.py* Import ------------------- You may want to import some extensions we provide as follows: :: from pyGPs.GraphExtension import graphUtil, nodeKernels from pyGPs.Validation import valid Load data -------------------- We used the same dataset from GPMC example. i.e. The USPS digits dataset [1]_. Each digit of :math:`16*16` pixels is flattened into a :math:`256` dimension vector. For the simplicity of demo, we only selected digits :math:`1` s and :math:`2` s such that we have a binary classification problem where digit :math:`1` for class +1 and digit :math:`2` for class -1. We also reduced the dataset into :math:`100` samples per digit, where the original dataset consist of thousands of samples for each digit. Here are samples for two digits for :math:`1` .. image:: _images/digit1_1.png :width: 30% .. image:: _images/digit1_2.png :width: 30% and samples for two digits for :math:`2` .. image:: _images/digit2_1.png :width: 30% .. image:: _images/digit2_2.png :width: 30% Form a nearest neighbour graph -------------------------------- We form a nearest-neighbor graph based on Euclidean distance of the vector representation of digits. Neighboring images have small Euclidean distance. Each digit is a node in the graph. There is an edge if digit :math:`i` is the k-nearest neighbour of digit :math:`j`. We form a symmetrized graph such that we connect nodes :math:`j`, :math:`i` if i is in j’s kNN and vice versa, and therefore a node can have more than k edges. You should import the corresponding module from *pyGPs.GraphStuff* :: x,y = load_binary(1,2,reduce=True) A = graphUtil.formKnnGraph(x,2) A is the adjacency matrix of this :math:`2-NN` graph. Below shows an example of such symmetrized Euclidean :math:`2-NN` graph on some 1s and 2s taking from Xiaojin Zhu's doctoral thesis [2]_. .. figure:: _images/2nnGraph.png :align: center Kernel on graph ------------------ Several classical kernels on graph described in `Structured Kernels`_ can be built from adjacency matrix :math:`A`. We use diffusion kernel for this example to get the precomputed kernel matrix. :: Matrix = nodeKernels.diffKernel(A) .. _Structured Kernels: Graph.html This a big square matrix with all rows and columns of the number of data points. By specifying the indice of training data and test data, we will form two matrix M1 and M2 with the exact format which *pyGPs.Core.cov.Pre* needed. :: M1,M2 = graphUtil.formKernelMatrix(Matrix, indice_train, indice_test) M1 is a matrix with shape **number of training points plus 1** by **number of test points** - cross covariances matrix (train by test) - last row is self covariances (diagonal of test by test) M2 is a square matrix with **number of training points** for each dimension - training set covariance matrix (train by train) GP classification ----------------------- Every ingredients for a basic semi-supervised learning is prepared now. Lets see how to proceed for :math:`GP` classification. First, the normal way with rbf kernel we have seen several times :: model = pyGPs.GPC() k = pyGPs.cov.RBF() model.setPrior(kernel=k) Then lets use our kernel precomputed matrix. If you only use precomputed kernel matrix, there is no training data. However you still need to specify :math:`x` just to fit in the usage of pyGPs for generality reason. You can create any :math:`x` as long as the dimension is correct. :: x = np.zeros((n,1)) k = pyGPs.cov.Pre(M1,M2) + pyGPs.cov.RBF() model.setPrior(kernel=k) Moreover, you can composite a kernel for both precomputed matrix and regular kernel function if necessary. :: k = pyGPs.cov.Pre(M1,M2) + pyGPs.cov.RBFunit() model.setPrior(kernel=k) The rest way of using pyGPs is exactly the same as the demo of GP classification. Result ----------------------- For our manually created graph data, an rbf kernel works better than a diffusion kernel on the graph (higher accuracy). The performance in general should depend on the application as well as features of data. The left image shows the digit that using diffusion kernel will predict the wrong result (should be :math:`2`), but rbf kernel does the job fine. The right image shows the digit that rbf kernel predicts the wrong class, diffusion kernel on the other hand, predicts correctly due to graph information! (should be :math:`1`). Interestingly, using a composite kernel with diffusion kernel on graph and an rbf kernel together. All test cases including the following are predicted correctly. .. image:: _images/digitDiffwrong.png :width: 50% .. image:: _images/digitRBFwrong.png :width: 50% .. [1] A Database for Handwritten Text Recognition Research, J. J. Hull, IEEE PAMI 16(5) 550-554, 1994. .. [2] Semi-Supervised Learning with Graphs, Xiaojin Zhu, CMU-LTI-05-192, 2005 pyGPs-1.2/doc/build/html/_sources/Theory.txt000066400000000000000000000167561236346020200211000ustar00rootroot00000000000000GPs & Functionality ======================== Gaussian Processes (GPs) can conveniently be used for Bayesian supervised learning, such as regression and classification. In its simplest form, GP inference can be implemented in a few lines of code. However, in practice, things typically get a little more complicated: you might want to use expressive covariance and mean functions, learn good values for hyperparameters, use non-Gaussian likelihood functions (rendering exact inference intractable), use approximate inference algorithms, or combinations of many or all of the above. A comprehensive introduction to Gaussian Processes for Machine Learning is provided in the `GPML`_ book by Rasmussen and Williams, 2006. List of Functionality ------------------------ This table lists the functionality implemented in pyGPs. +-------------+-------------------+------------+-------------------------+---------------+------------------+ |Functionality| Kernel | Mean | Likelihood | Inference | Optimizer | +=============+===================+============+=========================+===============+==================+ | Simple | Constant | Constant | Gaussian | Exact | Minimize | | +-------------------+------------+-------------------------+---------------+------------------+ | | Linear | Linear |Cumulative Gaussian (Erf)| EP | CG | | +-------------------+------------+-------------------------+---------------+------------------+ | | Linear ard | One | Laplace | Laplace | SCG | | +-------------------+------------+-------------------------+---------------+------------------+ | | Matérn (1,3,5) | Zero | | | BFGS | | +-------------------+------------+ | +------------------+ | | Periodic | | | | | | +-------------------+ | | | | | | Polynomial | | | | | | +-------------------+ | | | | | | Piecewise Poly | | | | | | +-------------------+ | | | | | | RBF iso | | | | | | +-------------------+ | | | | | | RBF ard | | | | | | +-------------------+ | | | | | | RBF unit | | | | | | +-------------------+ | | | | | | RQ iso | | | | | | +-------------------+ | | | | | | RQ ard | | | | | | +-------------------+ | | | | | | Noise | | | | | +-------------+-------------------+------------+-------------------------+---------------+------------------+ | Composite | Sum (+) | Sum (+) | | | | | +-------------------+------------+ | | | | | Product (*) | Product (*)| | | | | +-------------------+------------+ | | | | | Scale (*) | Scale (*) | | | | | +-------------------+------------+ | | | | | | Power (**) | | | | +-------------+-------------------+------------+-------------------------+---------------+------------------+ | Sparse GP | | | | FITC Exact | | | | | | +---------------+ | | | | | | FITC EP | | | | | | +---------------+ | | | | | | FITC Laplace | | +-------------+-------------------+------------+-------------------------+---------------+------------------+ | Graphs | Diffusion | | | | | | +-------------------+ | | | | | | VN Diffusion | | | | | | +-------------------+ | | | | | |Pseudo Inv Laplace | | | | | | +-------------------+ | | | | | |Regularized Laplace| | | | | | +-------------------+ | | | | | |p-step Random Walk | | | | | | +-------------------+ | | | | | |Inverse Cosine | | | | | | +-------------------+ | | | | | |Propagation Kernel | | | | | +-------------+-------------------+------------+-------------------------+---------------+------------------+ | Other | Customized | Customized | | | | | +-------------------+------------+ | | | | |Precomputed Kernel | | | | | +-------------+-------------------+------------+-------------------------+---------------+------------------+ pyGPs also provide cross-validation and some built-in evalation methods. .. _GPML: http://www.gaussianprocess.org/gpml pyGPs-1.2/doc/build/html/_sources/_static/000077500000000000000000000000001236346020200204745ustar00rootroot00000000000000pyGPs-1.2/doc/build/html/_sources/_static/changelog.txt000066400000000000000000000030461236346020200231670ustar00rootroot00000000000000 Changelog pyGPs v1.2 ===================== June 30th 2014 ---------------- structural updates: - input target now can either be in 2-d array with size (n,1) or in 1-d array with size (n,) - setup.py updated - "import pyGPs" instead of "from pyGPs.Core import gp" - rename ".train()" to ".optimize()" - rename "Graph-stuff" to "graphExtension" - rename kernelOnGraph to "nodeKernels" and graphKernel to "graphKernels" - redundancy removed for model.setData(x,y) - rewrite "mean.proceed()" to "getMean()" and "getDerMatrix()" - rewrite "cov.proceed()" to "getCovMatrix()" and "getDerMatrix()" - rename cov.LIN to cov.Linear (to be consistent with mean.Linear) - rename module "valid" to "validation" - add graph dataset Mutag in python file. (.npz and .mat) - add graphUtil.nomalizeKernel() - fix number of iteration problem in graphKernels "PropagationKernel" - add unit testing for covariance, mean functions bug fixes: - derivatives for cov.LINard - derivative of the scalar for cov.covScale - demo_GPR_FITC.py missing pyGPs.mean July 8th 2014 ---------------- structural updates: - add hyperparameter(signal variance s2) for linear covariance - add unit testing for inference,likelihood functions as well as models - NOT show(print) "maximum number of sweep warning in inference EP" any more - documentation updated bug fixes: - typos in lik.Laplace - derivative in lik.Laplace July 14th 2014 ---------------- documentation updates: - online docs updated - API file updated structural updates: - made private for methods that users don't need to call pyGPs-1.2/doc/build/html/_sources/_static/copyright.txt000066400000000000000000000035241236346020200232510ustar00rootroot00000000000000Copyright ========= Authors ------- * Marion Neumann [marion dot neumann at uni-bonn dot de] * Shan Huang [shan dot huang at iais dot fraunhofer dot de] * Daniel Marthaler [marthaler at ge dot com] * Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] This file is part of pyGPs. License ------- The code is released under the BSD 2-Clause (FreeBSD) License. Copyright (c) 2014 Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: - Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. - Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. pyGPs-1.2/doc/build/html/_sources/demoHousing.txt000066400000000000000000000043421236346020200220730ustar00rootroot00000000000000Regression on UCI Housing data ------------------------------ Boston Housing is a fairly standard dataset used for testing regression problems. It contains 506 data points with 12 numeric attributes, and one binary categorical attribute. The goal is to determine median home values, based on various census attributes. This dataset is available at the `UCI Repository`_. The demo follows that in [1]_. The data set was preprocessed as follows: each continuous feature was transformed to zero mean and unit variance (The categorical variable was dropped). The data was partitioned into :math:`481` points for training and :math:`25` points for testing. The mean function used was :func:`src.Core.means.meanZero` and the covariance (using the :func:`src.Core.kernels.covSum` function) was a composite of :func:`src.Core.kernels.covSEiso` and :func:`src.Core.kernels.covNoise`. The initial values of the hyperparameters were selected randomly from a zero-mean, unit-variance normal distribtion. The actual values were: :math:`[ -0.75, \; 0.59, \; -0.45 ]`. The initial likelihood hyperparameter was :math:`-2.30`. The regression started with initial negative log marginal likelihood of :math:` 752.46`. Note the initial zero mean and the variance that is uniform over the test set. :: model = pyGPs.GPR() model.optimize(x,y) ym, ys2, fm, fs2, lp = model.predict(xs) xa = np.concatenate((data[:,:4],data[:,5:-1]),axis=1) xa = (xa - np.mean(xa,axis=0))/(np.std(xa,axis=0)+1.e-16) ya, ys2a, fma, fs2a, lpa = model.predict(xa) .. figure:: _images/demoH1.png :align: center :width: 600pt :height: 300pt After hyperparameter optimization, the covariance hyperparameters were :math:`[ 1.17, \; 0.45, \; -1.41 ]` and the likelihood hyperparameter was :math:`-2.27`. The final negative log marginal likelihood (optimized) was :math:`214.46`. .. figure:: _images/demoH2.png :align: center :width: 600pt :height: 300pt .. _UCI Repository: http://archive.ics.uci.edu/ml/datasets/Housing .. [1] T. Suttorp and C. Igel, Approximation of Gaussian process regression models after training. In M. Verleysen (Hrsg.), Proceedings of the 16th European Symposium on Artificial Neural Networks (ESANN 2008) , pp. 427–432 (2008). pyGPs-1.2/doc/build/html/_sources/demoMaunaLoa.txt000066400000000000000000000147501236346020200221600ustar00rootroot00000000000000Regression on Mauna Loa data ---------------------------- This example does regression on the Hawaiian Mauna Loa data (example taken from chapter :math:`5` of the `GPML`_ book by Rasmussen and Williams, 2006) We will use a modelling problem concerning the concentration of :math:`CO_2` in the atmosphere to illustrate how the marginal likelihood can be used to set multiple hyperparameters in hierarchical Gaussian process models. A complex covariance function is derived by combining several different kinds of simple covariance functions, and the resulting model provides an excellent fit to the data as well as insight into its properties by interpretation of the adapted hyperparameters. Although the data is one-dimensional, and therefore easy to visualize, a total of :math:`11` hyperparameters are used, which in practice rules out the use of cross-validation for setting parameters, except for the gradient-based LOO-CV procedure. The data [1]_ consists of monthly average atmospheric :math:`CO_2` concentrations (in parts per million by volume (ppmv)) derived from *in-situ* air samples collected at the Mauna Loa Observatory, Hawaii, between :math:`1958` and :math:`2003` (with some missing values) `[2]`_. .. figure:: _images/demoML1.png :height: 600 px :width: 800 px :align: center :scale: 70 % The data is shown in the above plot. Our goal is to model the :math:`CO_2` concentration as a function of time :math:`t`. Several features are immediately apparent: a long term rising trend, a pronounced seasonal variation and some smaller irregularities. In the following, contributions to a combined covariance function which takes care of these individual properties are suggessted. This is meant primarily to illustrate the power and flexibility of the Gaussian process framework—it is possible that other choices would be more appropriate for this data set. To model the long term smooth rising trend, a squared exponential (SE) covariance term with two hyperparameters controlling the amplitude :math:`\theta_1` and characteristic length-scale :math:`\theta_2` is used: .. math:: k_1(x,x') = \theta_1^2 \exp \left(-\frac{(x-x')^2}{2\theta_2^2}\right). Note that we just use a smooth trend; actually enforcing the trend *a priori* to be increasing is probably not so simple and (hopefully) not desirable. We can use the periodic covariance function with a period of one year to model the seasonal variation. However, it is not clear that the seasonal trend is exactly periodic, so we modify it by taking the product with a squared exponential component to allow a decay away from exact periodicity: .. math:: k_2(x,x') = \theta_3^2 \exp\left(-\frac{(x-x')^2}{2\theta_4^2} \frac{2\sin^2(\pi(x-x'))}{\theta_5^2}\right). where :math:`\theta_3` gives the magnitude, :math:`\theta_4` the decay-time for the periodic component, and :math:`\theta_5` the smoothness of the periodic component; the period has been fixed to one (year). The seasonal component in the data is caused primarily by different rates of :math:`CO_2` uptake for plants depending on the season, and it is probably reasonable to assume that this pattern may itself change slowly over time, partially due to the elevation of the :math:`CO_2` level itself; if this effect turns out not to be relevant, then it can be effectively removed at the fitting stage by allowing :math:`\theta_4` to become very large. To model the (small) medium term irregularities, a rational quadratic term is used: .. math:: k_3(x,x') = \theta_6^2\left(1+\frac{(x-x')^2}{2\theta_8\theta_7^2}\right)^{\theta_8}. where :math:`\theta_6` is the magnitude, :math:`\theta_7` is the typical length-scale and :math:`\theta_8` is the shape parameter determining diffuseness of the length-scales. One could also have used a squared exponential form for this component, but it turns out that the rational quadratic works better (gives higher marginal likelihood), probably because it can accommodate several length-scales simultaneously. Finally we specify a noise model as the sum of a squared exponential contrubition and an independent component: .. math:: k_4(x_p,x_q) = \theta_9^2\exp\left(-\frac{(x_p - x_q)^2}{2\theta_{10}^2}\right) + \theta_{11}^2\delta_{pq}. where :math:`\theta_9` is the magnitude of the correlated noise component, :math:`\theta_{10}` is its length scale and :math:`\theta_{11}` is the magnitude of the independent noise component. Noise in the series could be caused by measurement inaccuracies, and by local short-term weather phenomena, so it is probably reasonable to assume at least a modest amount of correlation in time. Notice that the correlated noise component, the first term has an identical expression to the long term component in the trend covariance. When optimizing the hyperparameters, we will see that one of these components becomes large with a long length-scale (the long term trend), while the other remains small with a short length-scale (noise). The fact that we have chosen to call one of these components ‘signal’ and the other one ‘noise’ is only a question of interpretation. Presumably, we are less interested in very short-term effect, and thus call it noise; if on the other hand we were interested in this effect, we would call it signal. The final covariance function is: .. math:: k(x,x') = k_1(x,x') + k_2(x,x') + k_3(x,x') + k_4(x,x') with hyperparameters :math:`\theta = (\theta_1,\ldots,\theta_{11})` :: # DEFINE parameterized covariance function k1 = pyGPs.cov.RBF(np.log(67.), np.log(66.)) k2 = pyGPs.cov.Periodic(np.log(1.3), np.log(1.0), np.log(2.4)) * cov.RBF(np.log(90.), np.log(2.4)) k3 = pyGPs.cov.RQ(np.log(1.2), np.log(0.66), np.log(0.78)) k4 = pyGPs.cov.RBF(np.log(1.6/12.), np.log(0.18)) + cov.Noise(np.log(0.19)) k = k1 + k2 + k3 + k4 After running the minimization, :: t0 = clock() model.optimize(x,y) t1 = clock() model.predict(xs) The extrapolated data looks like: .. figure:: _images/demoML2.png :height: 600 px :width: 800 px :align: center :scale: 70 % and the optimized values of the hyperparameters allow for a principled analysis of different components driving the model. .. [1] Keeling, C. D. and Whorf, T. P. (2004). Atmospheric :math:`CO_2` Records from Sites in the SIO Air Sampling Network. In Trends: A Compendium of Data on Global Change. Carbon Dioxide Information Analysis Center, Oak Ridge National Laboratory, Oak Ridge, Tenn., U.S.A. .. _[2]: http://cdiac.esd.ornl.gov/ftp/trends/co2/maunaloa.co2 .. _GPML: http://www.gaussianprocess.org/gpml/chapters/RW5.pdf pyGPs-1.2/doc/build/html/_sources/index.txt000066400000000000000000000047321236346020200207240ustar00rootroot00000000000000.. Object-Oriented Gaussian Processes documentation master file, created by sphinx-quickstart on Thu Nov 14 19:54:52 2013. pyGPs - A Package for Gaussian Processes =========================================== About the package --------------------- pyGPs is a library hosting Python implementations of Gaussian processes (GPs) for machine learning. pyGPs bridges the gap between systems designed primarily for users, who mainly want to apply GPs and need basic machine learning routines for model training, evaluation, and visualiztion, and expressive systems for developers, who focus on extending the core functionalities as covariance and likelihood functions, as well as inference techniques. The software package is released under the BSD 2-Clause (FreeBSD) License. :doc:`Copyright <_static/copyright>` (c) by Marion Neumann, Shan Huang, Daniel Marthaler, & Kristian Kersting, Feb.2014 Further, it includes implementations of * minimize.py implemented in python by Roland Memisevic 2008, following minimize.m (Copyright (c) Carl Edward Rasmussen (1999-2006)) * scg.py (Copyright (c) Ian T Nabney (1996-2001)) * brentmin.py (Copyright (c) Hannes Nickisch 2010-01-10) * FITC functionality (following matlab implementations under Copyright (c) by Ed Snelson, Carl Edward Rasmussen and Hannes Nickisch, 2011-11-02) The most recent stable release is pyGPs v1.2. See :doc:`changelog <_static/changelog>` for a list of changes to the previous release. If you observe problems or bugs, please contact us. You can download the developer's guide and manual containing an API here: :download:`API and manual <_static/pyGPs_API.pdf>`. You can also download a `procedual implementation`_ of GP functionality from Github. However, the procedual version will not be supported in future. .. _procedual implementation: https://github.com/marionmari/pyGP_PR/ Authors: - Marion Neumann [marion dot neumann at uni-bonn dot de] - Shan Huang [shan dot huang at iais dot fraunhofer dot de] - Daniel Marthaler [dan dot marthaler at gmail dot com] - Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] The following persons helped to improve this software: Roman Garnett, Maciej Kurek, Hannes Nickisch, Zhao Xu, and Alejandro Molina. This work is partly supported by the Fraunhofer ATTRACT fellowship STREAM. Getting started ------------------------------ .. toctree:: :maxdepth: 1 Install Theory Examples Kernels Likelihoods Opts Graph pyGPs-1.2/doc/build/html/_static/000077500000000000000000000000001236346020200166525ustar00rootroot00000000000000pyGPs-1.2/doc/build/html/_static/agogo.css000066400000000000000000000155671236346020200204760ustar00rootroot00000000000000/* * agogo.css_t * ~~~~~~~~~~~ * * Sphinx stylesheet -- agogo theme. * * :copyright: Copyright 2007-2013 by the Sphinx team, see AUTHORS. * :license: BSD, see LICENSE for details. * */ * { margin: 0px; padding: 0px; } body { font-family: "Verdana", Arial, sans-serif; line-height: 1.4em; color: black; background-color: #eeeeec; } /* Page layout */ div.header, div.content, div.footer { width: 70em; margin-left: auto; margin-right: auto; } div.header-wrapper { background: #555573 url(bgtop.png) top left repeat-x; border-bottom: 3px solid #2e3436; } /* Default body styles */ a { color: #ce5c00; } div.bodywrapper a, div.footer a { text-decoration: underline; 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pHYs  tIME +3IDATHб W6`#_Vz#  &>AklrIENDB`pyGPs-1.2/doc/build/html/_static/bgfooter.png000066400000000000000000000006621236346020200211730ustar00rootroot00000000000000PNG  IHDRP2sRGBbKGDϴ Bb pHYs  tIME4\tEXtCommentCreated with GIMPW IDAT8m;%13jk1a >{1mC(_>Z7  zfH[v|BT:m882GK##>k׎9!ҾK8s(igISjl;a͇e]iw  "HVyb_ \i #l*喈j:hv0qePzD=ٸ@Bd&dt3ܰ|ǞSϳ6 亓pwF/O\҈k Hy3IENDB`pyGPs-1.2/doc/build/html/_static/bgtop.png000066400000000000000000000006561236346020200205020ustar00rootroot00000000000000PNG  IHDRP2sRGBbKGDUWS!ÿ pHYs  tIME-tEXtCommentCreated with GIMPW IDAT8]ˎ@!C_d2mgAQy#|qI w X/`ᶮqdϏϪfH=C&H9M Cv"QV1T^ΜF#lvTت&j0PGyV!.6`HSsVfCxAƎ/]MyTږ9vne`cSLFUs%J'h v|m5ku|*l>*6D夻zdDX?IENDB`pyGPs-1.2/doc/build/html/_static/bullet_orange.png000066400000000000000000000005551236346020200222070ustar00rootroot00000000000000PNG  IHDRz`PLTE~*̓m7޵ث*ŋzn˅*<ņt⟄bKGD  pHYsHHFk> vpAg/IDAT u? .r/P5z=,%&;%tEXtdate:create2010-01-07T12:49:06+01:00:y%tEXtdate:modify2010-01-07T12:49:06+01:00KtIENDB`pyGPs-1.2/doc/build/html/_static/changelog.html000066400000000000000000000140731236346020200214740ustar00rootroot00000000000000 Changelog pyGPs v1.2 — pyGPs v1.2 documentation

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      Changelog pyGPs v1.2

      June 30th 2014

      structural updates:

      • input target now can either be in 2-d array with size (n,1) or in 1-d array with size (n,)
      • setup.py updated
      • “import pyGPs” instead of “from pyGPs.Core import gp”
      • rename ”.train()” to ”.optimize()”
      • rename “Graph-stuff” to “graphExtension”
      • rename kernelOnGraph to “nodeKernels” and graphKernel to “graphKernels”
      • redundancy removed for model.setData(x,y)
      • rewrite “mean.proceed()” to “getMean()” and “getDerMatrix()”
      • rewrite “cov.proceed()” to “getCovMatrix()” and “getDerMatrix()”
      • rename cov.LIN to cov.Linear (to be consistent with mean.Linear)
      • rename module “valid” to “validation”
      • add graph dataset Mutag in python file. (.npz and .mat)
      • add graphUtil.nomalizeKernel()
      • fix number of iteration problem in graphKernels “PropagationKernel”
      • add unit testing for covariance, mean functions

      bug fixes:

      • derivatives for cov.LINard
      • derivative of the scalar for cov.covScale
      • demo_GPR_FITC.py missing pyGPs.mean

      July 8th 2014

      structural updates:

      • add hyperparameter(signal variance s2) for linear covariance
      • add unit testing for inference,likelihood functions as well as models
      • NOT show(print) “maximum number of sweep warning in inference EP” any more
      • documentation updated

      bug fixes:

      • typos in lik.Laplace
      • derivative in lik.Laplace

      July 14th 2014

      documentation updates:

      • online docs updated
      • API file updated

      structural updates:

      • made private for methods that users don’t need to call

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      © Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting. Created using Sphinx 1.2.2.
      pyGPs-1.2/doc/build/html/_static/changelog.rst000066400000000000000000000030461236346020200213360ustar00rootroot00000000000000 Changelog pyGPs v1.2 ===================== June 30th 2014 ---------------- structural updates: - input target now can either be in 2-d array with size (n,1) or in 1-d array with size (n,) - setup.py updated - "import pyGPs" instead of "from pyGPs.Core import gp" - rename ".train()" to ".optimize()" - rename "Graph-stuff" to "graphExtension" - rename kernelOnGraph to "nodeKernels" and graphKernel to "graphKernels" - redundancy removed for model.setData(x,y) - rewrite "mean.proceed()" to "getMean()" and "getDerMatrix()" - rewrite "cov.proceed()" to "getCovMatrix()" and "getDerMatrix()" - rename cov.LIN to cov.Linear (to be consistent with mean.Linear) - rename module "valid" to "validation" - add graph dataset Mutag in python file. (.npz and .mat) - add graphUtil.nomalizeKernel() - fix number of iteration problem in graphKernels "PropagationKernel" - add unit testing for covariance, mean functions bug fixes: - derivatives for cov.LINard - derivative of the scalar for cov.covScale - demo_GPR_FITC.py missing pyGPs.mean July 8th 2014 ---------------- structural updates: - add hyperparameter(signal variance s2) for linear covariance - add unit testing for inference,likelihood functions as well as models - NOT show(print) "maximum number of sweep warning in inference EP" any more - documentation updated bug fixes: - typos in lik.Laplace - derivative in lik.Laplace July 14th 2014 ---------------- documentation updates: - online docs updated - API file updated structural updates: - made private for methods that users don't need to call pyGPs-1.2/doc/build/html/_static/comment-bright.png000066400000000000000000000066541236346020200223120ustar00rootroot00000000000000PNG  IHDRa OiCCPPhotoshop ICC profilexڝSgTS=BKKoR RB&*! 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      © Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting. Created using Sphinx 1.2.2.
      pyGPs-1.2/doc/build/html/_static/copyright.rst000066400000000000000000000035241236346020200214200ustar00rootroot00000000000000Copyright ========= Authors ------- * Marion Neumann [marion dot neumann at uni-bonn dot de] * Shan Huang [shan dot huang at iais dot fraunhofer dot de] * Daniel Marthaler [marthaler at ge dot com] * Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] This file is part of pyGPs. License ------- The code is released under the BSD 2-Clause (FreeBSD) License. Copyright (c) 2014 Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: - Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. - Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. pyGPs-1.2/doc/build/html/_static/darkmetal.png000066400000000000000000001265111236346020200213320ustar00rootroot00000000000000PNG  IHDR,brtEXtSoftwareAdobe ImageReadyqe<IDATxڬ} I8?ݶ2H뽙2#H? ϯ_?~_ 0~qϏW?ߵ֯_|_?sX?_>p|_G s3}k|] r}K<ϗ|&~ѯ_ /]k[Y3_^s~ j?G>,ςy]ϟi[#|.ճX?9qϲ~ܟU9*s~^Y؟!w-~~_~ݧ>?+pUvB,_~)|~k{\5ose>}~~Q#=پryO~J?.·~4r,g>k[JGfbӑ]XĽFbO/F/b{iFhM5b,@in(^f{mJX!a#`si>=VcŰ},o;O;.Ai4׈t`404B"ĴL-]ew@ >G<A?)wd[&93;r˨uyWZ?!#]YWn-ݸys^L!mo}DwI?cƓ|>`HFyɫ~E= dz3V ̾+]ġGb wcu~}|.AH CqC,01~vl?|ʜ:\z}N.MΔ^ԀL=ϩ,S 5G 0XfSmk!OlJyGMTt=!w$7` AvTKRGEgDRP3C,gdh2.r>Oj$|3'E,vj;Lu˶\_X]yC9\("4{)r$V gx%d-$gߝOSR4gV cwQ]*TDSe ʭ%+_KSF(.rL iPCM1UNͼ? ?N" ^kcۃJ8tFGwҹuwgŖƣ۝`\2491ʼn#Du%00lH> |rq.`^́{ " %!%$=L>`[ptQ w-{kBuBזgi[LCƴе}=CZ&שa r?.$A0F1kH(Kpj6 ۠! {X%Q8zK.?4 e~l{6p14^/"T5dps\9~82ET 9mf#-Iؖ;`~ b9xocBx?'-#.ޗJIo3*,􉗑X2q<6BTpM"t XxAxr`ruh ~"qY/R@G@m)Q$)5ܛr\lbXs]0XՅɈsT:~ԛS) +)E.TYiF4QA^ 32ȡ=x`˲Iݙ3]͉Q= KWR, jc ='eA3< &Kwj\AIMI#8%k~9D\U BI Z˦Χ3zvx>ci:\4O +z4rB>@ϺkޕtD:g#8}l7R3b6977=oXq6H<<(İSsDW@L2@SeIŐ:yNC•1CwS5$d`X2CPcE=@>>!$BK-D#ij)}ShRرoÎsЌ[xzil xd|cy㸰z+:Z.#(q!.wk6_\zDMY0SD?W,t|4!s VLV훆0CoF1!, C3z"(EH?ʌD_172^Kghf7[PNkb ]̨ 8K炣#y+Jyjx'%N89>)aߥfL*$[X6 V1!?0|fnBcb6p|vfl0egӅtV=7A3[]}f3I[hbC!s_[B{!@w"lD2-.d,yp1|cq&X]c0]gX&fc(egndI<ے$g1Ab5%J49mx5%r8{o{$&]a!Sb\YPBuP3I>[eu _Rkx? ]I$2cZS`9 /y{鱆K4G`GG uYdZHơ1Ki6 xMuHo+[n^5 ̹fh(mzO YQ8@Kz5lu64#UFM@EH$I 8jFF `2pb%sαNm [kc>٫b?ǐ9SҡaT{4a Ć5NhfGA86jB1o~UpД$pl#"{`$p*%=s9ԴwG*ݖB"T%7Mɥg4m6~)G賁4EN~"SRY܂0$ F"2)h:rMby$!P,_}\iGTFTI &k`Q+H!i_*E|h! 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/** * make the code below compatible with browsers without * an installed firebug like debugger if (!window.console || !console.firebug) { var names = ["log", "debug", "info", "warn", "error", "assert", "dir", "dirxml", "group", "groupEnd", "time", "timeEnd", "count", "trace", "profile", "profileEnd"]; window.console = {}; for (var i = 0; i < names.length; ++i) window.console[names[i]] = function() {}; } */ /** * small helper function to urldecode strings */ jQuery.urldecode = function(x) { return decodeURIComponent(x).replace(/\+/g, ' '); }; /** * small helper function to urlencode strings */ jQuery.urlencode = encodeURIComponent; /** * This function returns the parsed url parameters of the * current request. Multiple values per key are supported, * it will always return arrays of strings for the value parts. */ jQuery.getQueryParameters = function(s) { if (typeof s == 'undefined') s = document.location.search; var parts = s.substr(s.indexOf('?') + 1).split('&'); var result = {}; for (var i = 0; i < parts.length; i++) { var tmp = parts[i].split('=', 2); var key = jQuery.urldecode(tmp[0]); var value = jQuery.urldecode(tmp[1]); if (key in result) result[key].push(value); else result[key] = [value]; } return result; }; /** * highlight a given string on a jquery object by wrapping it in * span elements with the given class name. */ jQuery.fn.highlightText = function(text, className) { function highlight(node) { if (node.nodeType == 3) { var val = node.nodeValue; var pos = val.toLowerCase().indexOf(text); if (pos >= 0 && !jQuery(node.parentNode).hasClass(className)) { var span = document.createElement("span"); span.className = className; span.appendChild(document.createTextNode(val.substr(pos, text.length))); node.parentNode.insertBefore(span, node.parentNode.insertBefore( document.createTextNode(val.substr(pos + text.length)), node.nextSibling)); node.nodeValue = val.substr(0, pos); } } else if (!jQuery(node).is("button, select, textarea")) { jQuery.each(node.childNodes, function() { highlight(this); }); } } return this.each(function() { highlight(this); }); }; /** * Small JavaScript module for the documentation. */ var Documentation = { init : function() { this.fixFirefoxAnchorBug(); this.highlightSearchWords(); this.initIndexTable(); }, /** * i18n support */ TRANSLATIONS : {}, PLURAL_EXPR : function(n) { return n == 1 ? 0 : 1; }, LOCALE : 'unknown', // gettext and ngettext don't access this so that the functions // can safely bound to a different name (_ = Documentation.gettext) gettext : function(string) { var translated = Documentation.TRANSLATIONS[string]; if (typeof translated == 'undefined') return string; return (typeof translated == 'string') ? translated : translated[0]; }, ngettext : function(singular, plural, n) { var translated = Documentation.TRANSLATIONS[singular]; if (typeof translated == 'undefined') return (n == 1) ? singular : plural; return translated[Documentation.PLURALEXPR(n)]; }, addTranslations : function(catalog) { for (var key in catalog.messages) this.TRANSLATIONS[key] = catalog.messages[key]; this.PLURAL_EXPR = new Function('n', 'return +(' + catalog.plural_expr + ')'); this.LOCALE = catalog.locale; }, /** * add context elements like header anchor links */ addContextElements : function() { $('div[id] > :header:first').each(function() { $('\u00B6'). attr('href', '#' + this.id). attr('title', _('Permalink to this headline')). appendTo(this); }); $('dt[id]').each(function() { $('\u00B6'). attr('href', '#' + this.id). attr('title', _('Permalink to this definition')). appendTo(this); }); }, /** * workaround a firefox stupidity */ fixFirefoxAnchorBug : function() { if (document.location.hash && $.browser.mozilla) window.setTimeout(function() { document.location.href += ''; }, 10); }, /** * highlight the search words provided in the url in the text */ highlightSearchWords : function() { var params = $.getQueryParameters(); var terms = (params.highlight) ? params.highlight[0].split(/\s+/) : []; if (terms.length) { var body = $('div.body'); if (!body.length) { body = $('body'); } window.setTimeout(function() { $.each(terms, function() { body.highlightText(this.toLowerCase(), 'highlighted'); }); }, 10); $('

      ' + _('Hide Search Matches') + '

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}); var url = parts.join('/'); return path.substring(url.lastIndexOf('/') + 1, path.length - 1); } }; // quick alias for translations _ = Documentation.gettext; $(document).ready(function() { Documentation.init(); }); pyGPs-1.2/doc/build/html/_static/down-pressed.png000066400000000000000000000005601236346020200217730ustar00rootroot00000000000000PNG  IHDRasRGBbKGDC pHYs B(xtIME -vF#IDAT8!OAJ, ++@I vbÿ@W7F HN#48646TMvv޼7Dsax1U q;< E-f)j%po4xF78G>)- EYm4%7YTk-Qa"NWAo-yeq,) Ypt\hqmszG]Nar߶s^l vh\2%0EeRvIENDB`pyGPs-1.2/doc/build/html/_static/down.png000066400000000000000000000005531236346020200203320ustar00rootroot00000000000000PNG  IHDRasRGBbKGDC pHYs B(xtIME"U{IDAT8ҡNCAJ, ++@4>/U^,~T&3M^^^PM6ٹs*RJa)eG*W<"F Fg78G>q OIp:sAj5GنyD^+yU:p_%G@D|aOs(yM,"msx:.b@D|`Vٟ۲иeKſ/G!IENDB`pyGPs-1.2/doc/build/html/_static/epub.css000066400000000000000000000127341236346020200203260ustar00rootroot00000000000000/* * default.css_t * ~~~~~~~~~~~~~ * * Sphinx stylesheet -- default theme. * * :copyright: Copyright 2007-2013 by the Sphinx team, see AUTHORS. * :license: BSD, see LICENSE for details. * */ @import url("basic.css"); /* -- page layout ----------------------------------------------------------- */ body { font-family: {{ theme_bodyfont }}; font-size: 100%; background-color: {{ theme_footerbgcolor }}; color: #000; margin: 0; padding: 0; } div.document { background-color: {{ theme_sidebarbgcolor }}; } div.documentwrapper { float: left; width: 100%; } div.bodywrapper { margin: 0 0 0 230px; } div.body { background-color: {{ theme_bgcolor }}; color: {{ theme_textcolor }}; padding: 0 20px 30px 20px; } {%- if theme_rightsidebar|tobool %} div.bodywrapper { margin: 0 230px 0 0; } {%- endif %} div.footer { color: {{ theme_footertextcolor }}; width: 100%; padding: 9px 0 9px 0; text-align: center; font-size: 75%; } div.footer a { color: {{ theme_footertextcolor }}; text-decoration: underline; } div.related { background-color: {{ theme_relbarbgcolor }}; line-height: 30px; color: {{ theme_relbartextcolor }}; } div.related a { color: {{ theme_relbarlinkcolor }}; } div.sphinxsidebar { {%- if theme_stickysidebar|tobool %} top: 30px; bottom: 0; margin: 0; position: fixed; overflow: auto; height: auto; {%- endif %} {%- if theme_rightsidebar|tobool %} float: right; {%- if theme_stickysidebar|tobool %} right: 0; {%- endif %} {%- endif %} } {%- if theme_stickysidebar|tobool %} /* this is nice, but it it leads to hidden headings when jumping to an anchor */ /* div.related { position: fixed; } div.documentwrapper { margin-top: 30px; } */ {%- endif %} div.sphinxsidebar h3 { font-family: {{ theme_headfont }}; color: {{ theme_sidebartextcolor }}; font-size: 1.4em; font-weight: normal; margin: 0; padding: 0; } div.sphinxsidebar h3 a { color: {{ theme_sidebartextcolor }}; } div.sphinxsidebar h4 { font-family: {{ theme_headfont }}; color: {{ theme_sidebartextcolor }}; font-size: 1.3em; font-weight: normal; margin: 5px 0 0 0; padding: 0; } div.sphinxsidebar p { color: {{ theme_sidebartextcolor }}; } div.sphinxsidebar p.topless { margin: 5px 10px 10px 10px; } div.sphinxsidebar ul { margin: 10px; padding: 0; color: {{ theme_sidebartextcolor }}; } div.sphinxsidebar a { color: {{ theme_sidebarlinkcolor }}; } div.sphinxsidebar input { border: 1px solid {{ theme_sidebarlinkcolor }}; font-family: sans-serif; font-size: 1em; } {% if theme_collapsiblesidebar|tobool %} /* for collapsible sidebar */ div#sidebarbutton { background-color: {{ theme_sidebarbtncolor }}; } {% endif %} /* -- hyperlink styles ------------------------------------------------------ */ a { color: {{ theme_linkcolor }}; text-decoration: none; } a:visited { color: {{ theme_visitedlinkcolor }}; text-decoration: none; } a:hover { text-decoration: underline; } {% if theme_externalrefs|tobool %} a.external { text-decoration: none; border-bottom: 1px dashed {{ theme_linkcolor }}; } a.external:hover { text-decoration: none; border-bottom: none; } a.external:visited { text-decoration: none; border-bottom: 1px dashed {{ theme_visitedlinkcolor }}; } {% endif %} /* -- body styles ----------------------------------------------------------- */ div.body h1, div.body h2, div.body h3, div.body h4, div.body h5, div.body h6 { font-family: {{ theme_headfont }}; background-color: {{ theme_headbgcolor }}; font-weight: normal; color: {{ theme_headtextcolor }}; border-bottom: 1px solid #ccc; margin: 20px -20px 10px -20px; padding: 3px 0 3px 10px; } div.body h1 { margin-top: 0; font-size: 200%; } div.body h2 { font-size: 160%; } div.body h3 { font-size: 140%; } div.body h4 { font-size: 120%; } div.body h5 { font-size: 110%; } div.body h6 { font-size: 100%; } a.headerlink { color: {{ theme_headlinkcolor }}; font-size: 0.8em; padding: 0 4px 0 4px; text-decoration: none; } a.headerlink:hover { background-color: {{ theme_headlinkcolor }}; color: white; } div.body p, div.body dd, div.body li { text-align: justify; line-height: 130%; } div.admonition p.admonition-title + p { display: inline; } div.admonition p { margin-bottom: 5px; } div.admonition pre { margin-bottom: 5px; } div.admonition ul, div.admonition ol { margin-bottom: 5px; } div.note { background-color: #eee; border: 1px solid #ccc; } div.seealso { background-color: #ffc; border: 1px solid #ff6; } div.topic { background-color: #eee; } div.warning { background-color: #ffe4e4; border: 1px solid #f66; } p.admonition-title { display: inline; } p.admonition-title:after { content: ":"; } pre { padding: 5px; background-color: {{ theme_codebgcolor }}; color: {{ theme_codetextcolor }}; line-height: 120%; border: 1px solid #ac9; border-left: none; border-right: none; } tt { background-color: #ecf0f3; padding: 0 1px 0 1px; font-size: 0.95em; } th { background-color: #ede; } .warning tt { background: #efc2c2; } .note tt { background: #d6d6d6; } .viewcode-back { font-family: {{ theme_bodyfont }}; } div.viewcode-block:target { background-color: #f4debf; border-top: 1px solid #ac9; border-bottom: 1px solid #ac9; } pyGPs-1.2/doc/build/html/_static/file.png000066400000000000000000000006101236346020200202740ustar00rootroot00000000000000PNG  IHDRabKGD pHYs  tIME  )TIDAT8˭J@Ir('[ "&xYZ X0!i|_@tD] #xjv YNaEi(əy@D&`6PZk$)5%"z.NA#Aba`Vs_3c,2mj [klvy|!Iմy;v "߮a?A7`c^nk?Bg}TЙD# "RD1yER*6MJ3K_Ut8F~IENDB`pyGPs-1.2/doc/build/html/_static/footerbg.png000066400000000000000000000005151236346020200211700ustar00rootroot00000000000000PNG  IHDRfIDAT8ɑ0Z샔%[ڇ58| X0`oqNG@ř;mH#8Kk8FO-&19Z#AdF\g/8MLGp8dk躓}}tse.u9d=|9MӍ}ջ晍k>;t.k~!;?۟ZqY:wumM|<+3ݝ9p-nV=F|IENDB`pyGPs-1.2/doc/build/html/_static/haiku.css000066400000000000000000000150341236346020200204700ustar00rootroot00000000000000/* * haiku.css_t * ~~~~~~~~~~~ * * Sphinx stylesheet -- haiku theme. * * Adapted from http://haiku-os.org/docs/Haiku-doc.css. * Original copyright message: * * Copyright 2008-2009, Haiku. All rights reserved. * Distributed under the terms of the MIT License. * * Authors: * Francois Revol * Stephan Assmus * Braden Ewing * Humdinger * * :copyright: Copyright 2007-2013 by the Sphinx team, see AUTHORS. * :license: BSD, see LICENSE for details. * */ @import url("basic.css"); html { margin: 0px; padding: 0px; background: #FFF url(bg-page.png) top left repeat-x; } body { line-height: 1.5; margin: auto; padding: 0px; font-family: "DejaVu Sans", Arial, Helvetica, sans-serif; min-width: 59em; max-width: 70em; color: #333333; } div.footer { padding: 8px; font-size: 11px; text-align: center; letter-spacing: 0.5px; } /* link colors and text decoration */ a:link { font-weight: bold; text-decoration: none; color: #dc3c01; } a:visited { font-weight: bold; text-decoration: none; color: #892601; } a:hover, a:active { text-decoration: underline; color: #ff4500; } /* Some headers act as anchors, don't give them a hover effect */ h1 a:hover, a:active { text-decoration: none; color: #0c3762; } h2 a:hover, a:active { text-decoration: none; color: #0c3762; } h3 a:hover, a:active { text-decoration: none; color: #0c3762; } h4 a:hover, a:active { text-decoration: none; color: #0c3762; } a.headerlink { color: #a7ce38; padding-left: 5px; } a.headerlink:hover { color: #a7ce38; } /* basic text elements */ div.content { margin-top: 20px; margin-left: 40px; margin-right: 40px; margin-bottom: 50px; font-size: 0.9em; } /* heading and navigation */ div.header { position: relative; left: 0px; top: 0px; height: 85px; /* background: #eeeeee; */ padding: 0 40px; } div.header h1 { font-size: 1.6em; font-weight: normal; letter-spacing: 1px; color: #0c3762; border: 0; margin: 0; padding-top: 15px; } div.header h1 a { font-weight: normal; color: #0c3762; } div.header h2 { font-size: 1.3em; font-weight: normal; letter-spacing: 1px; text-transform: uppercase; color: #aaa; border: 0; margin-top: -3px; padding: 0; } div.header img.rightlogo { float: right; } div.title { font-size: 1.3em; 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font-size: 100%; background-color: #393939; color: #ffffff; margin: 0; padding: 0; } div.documentwrapper { float: left; width: 100%; } div.bodywrapper { margin: 0 0 0 230px; } hr { border: 1px solid #B1B4B6; } div.document { background-color: #eee; } div.header { width:100%; background: #f4ad32 url(headerbg.png) repeat-x 0 top; border-bottom: 2px solid #ffffff; } div.logo { text-align: center; padding-top: 10px; } div.body { background-color: #ffffff; color: #3E4349; padding: 0 30px 30px 30px; font-size: 1em; border: 2px solid #ddd; border-right-style: none; overflow: auto; } div.footer { color: #ffffff; width: 100%; padding: 13px 0; text-align: center; font-size: 75%; background: transparent; clear:both; } div.footer a { color: #ffffff; text-decoration: none; } div.footer a:hover { color: #e88f00; text-decoration: underline; } div.related { line-height: 30px; color: #373839; font-size: 0.8em; background-color: #eee; } div.related a { color: #1b61d6; } div.related ul { padding-left: 240px; 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} p.sidebar-title { font-weight: bold; } /* -- body styles ----------------------------------------------------------- */ a, a .pre { color: #1b61d6; text-decoration: none; } a:hover, a:hover .pre { text-decoration: underline; } div.body h1, div.body h2, div.body h3, div.body h4, div.body h5, div.body h6 { font-family: "Neuton", sans-serif; background-color: #ffffff; font-weight: normal; color: #373839; margin: 30px 0px 10px 0px; padding: 5px 0; } div.body h1 { border-top: 20px solid white; margin-top: 0; font-size: 200%; } div.body h2 { font-size: 150%; background-color: #ffffff; } div.body h3 { font-size: 120%; background-color: #ffffff; } div.body h4 { font-size: 110%; background-color: #ffffff; } div.body h5 { font-size: 100%; background-color: #ffffff; } div.body h6 { font-size: 100%; background-color: #ffffff; } a.headerlink { color: #1b61d6; font-size: 0.8em; padding: 0 4px 0 4px; text-decoration: none; } a.headerlink:hover { text-decoration: underline; } div.body p, div.body dd, div.body li { line-height: 1.5em; } div.admonition p.admonition-title + p { display: inline; } div.admonition { background: #eeeeec; border: 2px solid #babdb6; border-right-style: none; border-left-style: none; padding: 10px 20px 10px 60px; } div.highlight{ background-color: white; } div.note { border: 2px solid #7a9eec; border-right-style: none; border-left-style: none; padding: 10px 20px 10px 60px; background: #e1ecfe url(dialog-note.png) no-repeat 10px 8px; } div.seealso { background: #fff6bf url(dialog-seealso.png) no-repeat 10px 8px; border: 2px solid #ffd324; border-left-style: none; border-right-style: none; padding: 10px 20px 10px 60px; } div.topic { background: #eeeeee; border: 2px solid #C6C9CB; padding: 10px 20px; border-right-style: none; border-left-style: none; } div.warning { background: #fbe3e4 url(dialog-warning.png) no-repeat 10px 8px; border: 2px solid #fbc2c4; border-right-style: none; border-left-style: none; padding: 10px 20px 10px 60px; } div.admonition-todo { background: #f2d9b4 url(dialog-todo.png) no-repeat 10px 8px; 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font-weight: normal; font-style: normal; } tt.xref { font-weight: normal; font-style: normal; }pyGPs-1.2/doc/build/html/_static/scrolls.css000066400000000000000000000151011236346020200210430ustar00rootroot00000000000000/* * scrolls.css_t * ~~~~~~~~~~~~~ * * Sphinx stylesheet -- scrolls theme. * * :copyright: Copyright 2007-2013 by the Sphinx team, see AUTHORS. * :license: BSD, see LICENSE for details. * */ body { background-color: #222; margin: 0; padding: 0; font-family: 'Georgia', serif; font-size: 15px; color: #eee; } div.footer { border-top: 1px solid #111; padding: 8px; font-size: 11px; text-align: center; letter-spacing: 0.5px; } div.footer a { color: #eee; } div.header { margin: 0 -15px 0 -15px; background: url(headerbg.png) repeat-x; border-top: 6px solid #1752b4; } div.relnav { border-bottom: 1px solid #111; background: url(navigation.png); margin: 0 -15px 0 -15px; padding: 2px 20px 0 28px; line-height: 25px; color: #aaa; font-size: 12px; text-align: center; } div.relnav a { color: #eee; font-weight: bold; text-decoration: none; } div.relnav a:hover { text-decoration: underline; } #content { background-color: white; color: #111; border-bottom: 1px solid black; background: url(watermark.png) center 0; padding: 0 15px 0 15px; margin: 0; } h1 { margin: 0; padding: 15px 0 0 0; } h1.heading { margin: 0; padding: 0; height: 80px; } h1.heading:hover { background: #222; } h1.heading a { background: url(logo.png) no-repeat center 0; display: block; width: 100%; height: 80px; } h1.heading a:focus { -moz-outline: none; outline: none; } h1.heading span { display: none; } #contentwrapper { max-width: 680px; padding: 0 18px 20px 18px; margin: 0 auto 0 auto; border-right: 1px solid #eee; border-left: 1px solid #eee; background: url(watermark_blur.png) center -114px; } #contentwrapper h2, #contentwrapper h2 a { color: #222; font-size: 24px; margin: 20px 0 0 0; } #contentwrapper h3, #contentwrapper h3 a { color: #0d306b; font-size: 20px; margin: 20px 0 0 0; } table.docutils { border-collapse: collapse; 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} a.reference:hover tt { border-bottom-color: #aaa; } cite { /* abusing , it's generated by ReST for `x` */ font-size: 13px; font-family: 'Bitstream Vera Sans Mono', 'Monaco', monospace; font-weight: bold; font-style: normal; } div.admonition { margin: 10px 0 10px 0; padding: 10px; border: 1px solid #ccc; } div.admonition p.admonition-title { background-color: #28437f; color: white; margin: -10px -10px 10px -10px; padding: 4px 10px 4px 10px; font-weight: bold; font-size: 15px; } div.admonition p.admonition-title a { color: white!important; } a.headerlink { color: #B4B4B4!important; font-size: 0.8em; padding: 0 4px 0 4px; text-decoration: none!important; visibility: hidden; } h1:hover > a.headerlink, h2:hover > a.headerlink, h3:hover > a.headerlink, h4:hover > a.headerlink, h5:hover > a.headerlink, h6:hover > a.headerlink, dt:hover > a.headerlink, dt:hover > a.headerlink { visibility: visible; } a.headerlink:hover { background-color: #B4B4B4; color: #F0F0F0!important; } table.indextable { width: 100%; } table.genindextable td { vertical-align: top; width: 50%; } table.indextable dl dd { font-size: 11px; } table.indextable dl dd a { color: #000; } div.modindex-jumpbox { border-top: 1px solid #ddd; border-bottom: 1px solid #ddd; margin: 1em 0 1em 0; padding: 0.4em; } table.modindextable { width: 100%; border: none; } table.modindextable img.toggler { margin-right: 10px; } dl.function dt, dl.class dt, dl.exception dt, dl.method dt, dl.attribute dt { font-weight: normal; } dt .descname { font-weight: bold; margin-right: 4px; } dt .descname, dt .descclassname { padding: 0; background: transparent; border-bottom: 1px solid #111; } dt .descclassname { margin-left: 2px; } dl dt big { font-size: 100%; } ul.search { margin: 10px 0 0 30px; padding: 0; } ul.search li { margin: 10px 0 0 0; padding: 0; } ul.search div.context { font-size: 12px; padding: 4px 0 0 20px; color: #888; } span.highlight { background-color: #eee; border: 1px solid #ccc; } #toc { margin: 0 -17px 0 -17px; display: none; } #toc h3 { float: right; 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}pyGPs-1.2/doc/build/html/_static/searchtools.js000066400000000000000000000427021236346020200215430ustar00rootroot00000000000000/* * searchtools.js_t * ~~~~~~~~~~~~~~~~ * * Sphinx JavaScript utilties for the full-text search. * * :copyright: Copyright 2007-2014 by the Sphinx team, see AUTHORS. * :license: BSD, see LICENSE for details. * */ /** * Porter Stemmer */ var Stemmer = function() { var step2list = { ational: 'ate', tional: 'tion', enci: 'ence', anci: 'ance', izer: 'ize', bli: 'ble', alli: 'al', entli: 'ent', eli: 'e', ousli: 'ous', ization: 'ize', ation: 'ate', ator: 'ate', alism: 'al', iveness: 'ive', fulness: 'ful', ousness: 'ous', aliti: 'al', iviti: 'ive', biliti: 'ble', logi: 'log' }; var step3list = { icate: 'ic', ative: '', alize: 'al', iciti: 'ic', ical: 'ic', ful: '', ness: '' }; var c = "[^aeiou]"; // consonant var v = "[aeiouy]"; // vowel var C = c + "[^aeiouy]*"; // consonant sequence var V = v + "[aeiou]*"; // vowel sequence var mgr0 = "^(" + C + ")?" + V + C; // [C]VC... is m>0 var meq1 = "^(" + C + ")?" + V + C + "(" + V + ")?$"; // [C]VC[V] is m=1 var mgr1 = "^(" + C + ")?" + V + C + V + C; // [C]VCVC... is m>1 var s_v = "^(" + C + ")?" + v; // vowel in stem this.stemWord = function (w) { var stem; var suffix; var firstch; var origword = w; if (w.length < 3) return w; var re; var re2; var re3; var re4; firstch = w.substr(0,1); if (firstch == "y") w = firstch.toUpperCase() + w.substr(1); // Step 1a re = /^(.+?)(ss|i)es$/; re2 = /^(.+?)([^s])s$/; if (re.test(w)) w = w.replace(re,"$1$2"); else if (re2.test(w)) w = w.replace(re2,"$1$2"); // Step 1b re = /^(.+?)eed$/; re2 = /^(.+?)(ed|ing)$/; if (re.test(w)) { var fp = re.exec(w); re = new RegExp(mgr0); if (re.test(fp[1])) { re = /.$/; w = w.replace(re,""); } } else if (re2.test(w)) { var fp = re2.exec(w); stem = fp[1]; re2 = new RegExp(s_v); if (re2.test(stem)) { w = stem; re2 = /(at|bl|iz)$/; re3 = new RegExp("([^aeiouylsz])\\1$"); re4 = new RegExp("^" + C + v + "[^aeiouwxy]$"); if (re2.test(w)) w = w + "e"; else if (re3.test(w)) { re = /.$/; w = w.replace(re,""); } else if (re4.test(w)) w = w + "e"; } } // Step 1c re = /^(.+?)y$/; if (re.test(w)) { var fp = re.exec(w); stem = fp[1]; re = new RegExp(s_v); if (re.test(stem)) w = stem + "i"; } // Step 2 re = /^(.+?)(ational|tional|enci|anci|izer|bli|alli|entli|eli|ousli|ization|ation|ator|alism|iveness|fulness|ousness|aliti|iviti|biliti|logi)$/; if (re.test(w)) { var fp = re.exec(w); stem = fp[1]; suffix = fp[2]; re = new RegExp(mgr0); if (re.test(stem)) w = stem + step2list[suffix]; } // Step 3 re = /^(.+?)(icate|ative|alize|iciti|ical|ful|ness)$/; if (re.test(w)) { var fp = re.exec(w); stem = fp[1]; suffix = fp[2]; re = new RegExp(mgr0); if (re.test(stem)) w = stem + step3list[suffix]; } // Step 4 re = /^(.+?)(al|ance|ence|er|ic|able|ible|ant|ement|ment|ent|ou|ism|ate|iti|ous|ive|ize)$/; re2 = /^(.+?)(s|t)(ion)$/; if (re.test(w)) { var fp = re.exec(w); stem = fp[1]; re = new RegExp(mgr1); if (re.test(stem)) w = stem; } else if (re2.test(w)) { var fp = re2.exec(w); stem = fp[1] + fp[2]; re2 = new RegExp(mgr1); if (re2.test(stem)) w = stem; } // Step 5 re = /^(.+?)e$/; if (re.test(w)) { var fp = re.exec(w); stem = fp[1]; re = new RegExp(mgr1); re2 = new RegExp(meq1); re3 = new RegExp("^" + C + v + "[^aeiouwxy]$"); if (re.test(stem) || (re2.test(stem) && !(re3.test(stem)))) w = stem; } re = /ll$/; re2 = new RegExp(mgr1); if (re.test(w) && re2.test(w)) { re = /.$/; w = w.replace(re,""); } // and turn initial Y back to y if (firstch == "y") w = firstch.toLowerCase() + w.substr(1); return w; } } /** * Simple result scoring code. */ var Scorer = { // Implement the following function to further tweak the score for each result // The function takes a result array [filename, title, anchor, descr, score] // and returns the new score. /* score: function(result) { return result[4]; }, */ // query matches the full name of an object objNameMatch: 11, // or matches in the last dotted part of the object name objPartialMatch: 6, // Additive scores depending on the priority of the object objPrio: {0: 15, // used to be importantResults 1: 5, // used to be objectResults 2: -5}, // used to be unimportantResults // Used when the priority is not in the mapping. objPrioDefault: 0, // query found in title title: 15, // query found in terms term: 5 }; /** * Search Module */ var Search = { _index : null, _queued_query : null, _pulse_status : -1, init : function() { var params = $.getQueryParameters(); if (params.q) { var query = params.q[0]; $('input[name="q"]')[0].value = query; this.performSearch(query); } }, loadIndex : function(url) { $.ajax({type: "GET", url: url, data: null, dataType: "script", cache: true, complete: function(jqxhr, textstatus) { if (textstatus != "success") { document.getElementById("searchindexloader").src = url; } }}); }, setIndex : function(index) { var q; this._index = index; if ((q = this._queued_query) !== null) { this._queued_query = null; Search.query(q); } }, hasIndex : function() { return this._index !== null; }, deferQuery : function(query) { this._queued_query = query; }, stopPulse : function() { this._pulse_status = 0; }, startPulse : function() { if (this._pulse_status >= 0) return; function pulse() { var i; Search._pulse_status = (Search._pulse_status + 1) % 4; var dotString = ''; for (i = 0; i < Search._pulse_status; i++) dotString += '.'; Search.dots.text(dotString); if (Search._pulse_status > -1) window.setTimeout(pulse, 500); } pulse(); }, /** * perform a search for something (or wait until index is loaded) */ performSearch : function(query) { // create the required interface elements this.out = $('#search-results'); this.title = $('

      ' + _('Searching') + '

      ').appendTo(this.out); this.dots = $('').appendTo(this.title); this.status = $('

      ').appendTo(this.out); this.output = $('
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          Regression on UCI Housing data

          Boston Housing is a fairly standard dataset used for testing regression problems. It contains 506 data points with 12 numeric attributes, and one binary categorical attribute. The goal is to determine median home values, based on various census attributes. This dataset is available at the UCI Repository.

          The demo follows that in [1]. The data set was preprocessed as follows: each continuous feature was transformed to zero mean and unit variance (The categorical variable was dropped). The data was partitioned into 481 points for training and 25 points for testing.

          The mean function used was src.Core.means.meanZero() and the covariance (using the src.Core.kernels.covSum() function) was a composite of src.Core.kernels.covSEiso() and src.Core.kernels.covNoise(). The initial values of the hyperparameters were selected randomly from a zero-mean, unit-variance normal distribtion. The actual values were: [ -0.75, \; 0.59, \; -0.45 ]. The initial likelihood hyperparameter was -2.30. The regression started with initial negative log marginal likelihood of :math:` 752.46`. Note the initial zero mean and the variance that is uniform over the test set.

          model = pyGPs.GPR()
model.optimize(x,y)
ym, ys2, fm, fs2, lp = model.predict(xs)
xa  = np.concatenate((data[:,:4],data[:,5:-1]),axis=1)
xa = (xa - np.mean(xa,axis=0))/(np.std(xa,axis=0)+1.e-16)
ya, ys2a, fma, fs2a, lpa = model.predict(xa)
          _images/demoH1.png

          After hyperparameter optimization, the covariance hyperparameters were [ 1.17, \; 0.45, \; -1.41 ] and the likelihood hyperparameter was -2.27. The final negative log marginal likelihood (optimized) was 214.46.

          _images/demoH2.png
          [1]
          1. Suttorp and C. Igel, Approximation of Gaussian process regression models after training. In M. Verleysen (Hrsg.), Proceedings of the 16th European Symposium on Artificial Neural Networks (ESANN 2008) , pp. 427–432 (2008).

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          Regression on Mauna Loa data

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          Regression on Mauna Loa data

          This example does regression on the Hawaiian Mauna Loa data (example taken from chapter 5 of the GPML book by Rasmussen and Williams, 2006)

          We will use a modelling problem concerning the concentration of CO_2 in the atmosphere to illustrate how the marginal likelihood can be used to set multiple hyperparameters in hierarchical Gaussian process models. A complex covariance function is derived by combining several different kinds of simple covariance functions, and the resulting model provides an excellent fit to the data as well as insight into its properties by interpretation of the adapted hyperparameters. Although the data is one-dimensional, and therefore easy to visualize, a total of 11 hyperparameters are used, which in practice rules out the use of cross-validation for setting parameters, except for the gradient-based LOO-CV procedure.

          The data [1] consists of monthly average atmospheric CO_2 concentrations (in parts per million by volume (ppmv)) derived from in-situ air samples collected at the Mauna Loa Observatory, Hawaii, between 1958 and 2003 (with some missing values) [2].

          _images/demoML1.png

          The data is shown in the above plot. Our goal is to model the CO_2 concentration as a function of time t. Several features are immediately apparent: a long term rising trend, a pronounced seasonal variation and some smaller irregularities. In the following, contributions to a combined covariance function which takes care of these individual properties are suggessted. This is meant primarily to illustrate the power and flexibility of the Gaussian process framework—it is possible that other choices would be more appropriate for this data set.

          To model the long term smooth rising trend, a squared exponential (SE) covariance term with two hyperparameters controlling the amplitude \theta_1 and characteristic length-scale \theta_2 is used:

          k_1(x,x') = \theta_1^2 \exp \left(-\frac{(x-x')^2}{2\theta_2^2}\right).

          Note that we just use a smooth trend; actually enforcing the trend a priori to be increasing is probably not so simple and (hopefully) not desirable. We can use the periodic covariance function with a period of one year to model the seasonal variation. However, it is not clear that the seasonal trend is exactly periodic, so we modify it by taking the product with a squared exponential component to allow a decay away from exact periodicity:

          k_2(x,x') = \theta_3^2 \exp\left(-\frac{(x-x')^2}{2\theta_4^2} \frac{2\sin^2(\pi(x-x'))}{\theta_5^2}\right).

          where \theta_3 gives the magnitude, \theta_4 the decay-time for the periodic component, and \theta_5 the smoothness of the periodic component; the period has been fixed to one (year). The seasonal component in the data is caused primarily by different rates of CO_2 uptake for plants depending on the season, and it is probably reasonable to assume that this pattern may itself change slowly over time, partially due to the elevation of the CO_2 level itself; if this effect turns out not to be relevant, then it can be effectively removed at the fitting stage by allowing \theta_4 to become very large.

          To model the (small) medium term irregularities, a rational quadratic term is used:

          k_3(x,x') = \theta_6^2\left(1+\frac{(x-x')^2}{2\theta_8\theta_7^2}\right)^{\theta_8}.

          where \theta_6 is the magnitude, \theta_7 is the typical length-scale and \theta_8 is the shape parameter determining diffuseness of the length-scales.

          One could also have used a squared exponential form for this component, but it turns out that the rational quadratic works better (gives higher marginal likelihood), probably because it can accommodate several length-scales simultaneously.

          Finally we specify a noise model as the sum of a squared exponential contrubition and an independent component:

          k_4(x_p,x_q) = \theta_9^2\exp\left(-\frac{(x_p - x_q)^2}{2\theta_{10}^2}\right) + \theta_{11}^2\delta_{pq}.

          where \theta_9 is the magnitude of the correlated noise component, \theta_{10} is its length scale and \theta_{11} is the magnitude of the independent noise component. Noise in the series could be caused by measurement inaccuracies, and by local short-term weather phenomena, so it is probably reasonable to assume at least a modest amount of correlation in time. Notice that the correlated noise component, the first term has an identical expression to the long term component in the trend covariance. When optimizing the hyperparameters, we will see that one of these components becomes large with a long length-scale (the long term trend), while the other remains small with a short length-scale (noise). The fact that we have chosen to call one of these components ‘signal’ and the other one ‘noise’ is only a question of interpretation. Presumably, we are less interested in very short-term effect, and thus call it noise; if on the other hand we were interested in this effect, we would call it signal.

          The final covariance function is:

          k(x,x') = k_1(x,x') + k_2(x,x') + k_3(x,x') + k_4(x,x')

          with hyperparameters \theta = (\theta_1,\ldots,\theta_{11})

          # DEFINE parameterized covariance function
k1 = pyGPs.cov.RBF(np.log(67.), np.log(66.))
k2 = pyGPs.cov.Periodic(np.log(1.3), np.log(1.0), np.log(2.4)) * cov.RBF(np.log(90.), np.log(2.4))
k3 = pyGPs.cov.RQ(np.log(1.2), np.log(0.66), np.log(0.78))
k4 = pyGPs.cov.RBF(np.log(1.6/12.), np.log(0.18)) + cov.Noise(np.log(0.19))
k  = k1 + k2 + k3 + k4

          After running the minimization,

          t0 = clock()
model.optimize(x,y)
t1 = clock()
model.predict(xs)

          The extrapolated data looks like:

          _images/demoML2.png

          and the optimized values of the hyperparameters allow for a principled analysis of different components driving the model.

          [1]Keeling, C. D. and Whorf, T. P. (2004). Atmospheric CO_2 Records from Sites in the SIO Air Sampling Network. In Trends: A Compendium of Data on Global Change. Carbon Dioxide Information Analysis Center, Oak Ridge National Laboratory, Oak Ridge, Tenn., U.S.A.

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          K-fold Cross-Validation

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          pyGPs-1.2/doc/build/html/genindex.html000066400000000000000000000312411236346020200177140ustar00rootroot00000000000000 Index — pyGPs v1.2 documentation

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          Index

          C | D | E | F | G | K | L | M | N | O | P | R | S | V | Z

          C

          Const (class in pyGPs.Core.cov)
          (class in pyGPs.Core.mean)
          cosKernel() (in module pyGPs.GraphExtensions.nodeKernels)

          D

          diffKernel() (in module pyGPs.GraphExtensions.nodeKernels)

          E

          EP (class in pyGPs.Core.inf)
          Erf (class in pyGPs.Core.lik)
          Exact (class in pyGPs.Core.inf)

          F

          fitc() (pyGPs.Core.cov.Kernel method)
          FITC_EP (class in pyGPs.Core.inf)
          FITC_Exact (class in pyGPs.Core.inf)
          FITC_Laplace (class in pyGPs.Core.inf)
          FITCOfKernel (class in pyGPs.Core.cov)

          G

          Gauss (class in pyGPs.Core.lik)
          getCovMatrix() (pyGPs.Core.cov.Kernel method)
          getDerMatrix() (pyGPs.Core.cov.Kernel method)
          (pyGPs.Core.mean.Mean method)
          getMean() (pyGPs.Core.mean.Mean method)

          K

          Kernel (class in pyGPs.Core.cov)

          L

          Laplace (class in pyGPs.Core.inf)
          (class in pyGPs.Core.lik)
          LINard (class in pyGPs.Core.cov)
          Linear (class in pyGPs.Core.cov)
          (class in pyGPs.Core.mean)

          M

          Matern (class in pyGPs.Core.cov)
          Mean (class in pyGPs.Core.mean)

          N

          Noise (class in pyGPs.Core.cov)
          normLap() (in module pyGPs.GraphExtensions.nodeKernels)

          O

          One (class in pyGPs.Core.mean)

          P

          Periodic (class in pyGPs.Core.cov)
          PiecePoly (class in pyGPs.Core.cov)
          Poly (class in pyGPs.Core.cov)
          PowerOfMean (class in pyGPs.Core.mean)
          Pre (class in pyGPs.Core.cov)
          ProductOfKernel (class in pyGPs.Core.cov)
          ProductOfMean (class in pyGPs.Core.mean)
          propagationKernel() (in module pyGPs.GraphExtensions.graphKernels)
          psInvLapKernel() (in module pyGPs.GraphExtensions.nodeKernels)
          pyGPs.Core.cov (module)
          pyGPs.Core.mean (module)
          pyGPs.GraphExtensions.graphKernels (module)
          pyGPs.GraphExtensions.nodeKernels (module)

          R

          RBF (class in pyGPs.Core.cov)
          RBFard (class in pyGPs.Core.cov)
          RBFunit (class in pyGPs.Core.cov)
          regLapKernel() (in module pyGPs.GraphExtensions.nodeKernels)
          RQ (class in pyGPs.Core.cov)
          RQard (class in pyGPs.Core.cov)
          rwKernel() (in module pyGPs.GraphExtensions.nodeKernels)

          S

          ScaleOfKernel (class in pyGPs.Core.cov)
          ScaleOfMean (class in pyGPs.Core.mean)
          setOptimizer() (pyGPs.Core.gp.GP method)
          SumOfKernel (class in pyGPs.Core.cov)
          SumOfMean (class in pyGPs.Core.mean)

          V

          VNDKernel() (in module pyGPs.GraphExtensions.nodeKernels)

          Z

          Zero (class in pyGPs.Core.mean)

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          pyGPs-1.2/doc/build/html/index.html000066400000000000000000000171061236346020200172260ustar00rootroot00000000000000 pyGPs - A Package for Gaussian Processes — pyGPs v1.2 documentation

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          pyGPs - A Package for Gaussian Processes

          About the package

          pyGPs is a library hosting Python implementations of Gaussian processes (GPs) for machine learning. pyGPs bridges the gap between systems designed primarily for users, who mainly want to apply GPs and need basic machine learning routines for model training, evaluation, and visualiztion, and expressive systems for developers, who focus on extending the core functionalities as covariance and likelihood functions, as well as inference techniques.

          The software package is released under the BSD 2-Clause (FreeBSD) License. Copyright (c) by Marion Neumann, Shan Huang, Daniel Marthaler, & Kristian Kersting, Feb.2014

          Further, it includes implementations of

          • minimize.py implemented in python by Roland Memisevic 2008, following minimize.m (Copyright (c) Carl Edward Rasmussen (1999-2006))
          • scg.py (Copyright (c) Ian T Nabney (1996-2001))
          • brentmin.py (Copyright (c) Hannes Nickisch 2010-01-10)
          • FITC functionality (following matlab implementations under Copyright (c) by Ed Snelson, Carl Edward Rasmussen and Hannes Nickisch, 2011-11-02)

          The most recent stable release is pyGPs v1.2. See changelog for a list of changes to the previous release. If you observe problems or bugs, please contact us. You can download the developer’s guide and manual containing an API here: API and manual. You can also download a procedual implementation of GP functionality from Github. However, the procedual version will not be supported in future.

          Authors:
          • Marion Neumann [marion dot neumann at uni-bonn dot de]
          • Shan Huang [shan dot huang at iais dot fraunhofer dot de]
          • Daniel Marthaler [dan dot marthaler at gmail dot com]
          • Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de]

          The following persons helped to improve this software: Roman Garnett, Maciej Kurek, Hannes Nickisch, Zhao Xu, and Alejandro Molina.

          This work is partly supported by the Fraunhofer ATTRACT fellowship STREAM.

          Table Of Contents

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          Python Module Index

          p
           
          p
          pyGPs
              pyGPs.Core.cov
              pyGPs.Core.mean
              pyGPs.GraphExtensions.graphKernels
              pyGPs.GraphExtensions.nodeKernels

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          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OFF REM Command file for Sphinx documentation if "%SPHINXBUILD%" == "" ( set SPHINXBUILD=sphinx-build ) set BUILDDIR=build set ALLSPHINXOPTS=-d %BUILDDIR%/doctrees %SPHINXOPTS% source set I18NSPHINXOPTS=%SPHINXOPTS% source if NOT "%PAPER%" == "" ( set ALLSPHINXOPTS=-D latex_paper_size=%PAPER% %ALLSPHINXOPTS% set I18NSPHINXOPTS=-D latex_paper_size=%PAPER% %I18NSPHINXOPTS% ) if "%1" == "" goto help if "%1" == "help" ( :help echo.Please use `make ^` where ^ is one of echo. html to make standalone HTML files echo. dirhtml to make HTML files named index.html in directories echo. singlehtml to make a single large HTML file echo. pickle to make pickle files echo. json to make JSON files echo. htmlhelp to make HTML files and a HTML help project echo. qthelp to make HTML files and a qthelp project echo. devhelp to make HTML files and a Devhelp project echo. epub to make an epub echo. latex to make LaTeX files, you can set PAPER=a4 or PAPER=letter echo. text to make text files echo. man to make 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The XML files are in %BUILDDIR%/xml. goto end ) if "%1" == "pseudoxml" ( %SPHINXBUILD% -b pseudoxml %ALLSPHINXOPTS% %BUILDDIR%/pseudoxml if errorlevel 1 exit /b 1 echo. echo.Build finished. The pseudo-XML files are in %BUILDDIR%/pseudoxml. goto end ) :end pyGPs-1.2/doc/source/000077500000000000000000000000001236346020200144615ustar00rootroot00000000000000pyGPs-1.2/doc/source/CV.rst000066400000000000000000000036401236346020200155260ustar00rootroot00000000000000K-fold Cross-Validation ================================== In this demo, we'll show you the typical process of using GP for machine learning from loading data, preprocessing, training, predicting to validation and evaluation. Load data -------------------- We use the ionosphere dataset [1]_ from Johns Hopkins University Ionosphere database. It is available in UCI machine learning repository. Then we need to do some data cleaning. Here we deal with label in ionosphere data, change "b" to"-1", and "g" to "+1". These preprocessing implementation are availabe in the source code. Cross Validation ---------------------------- Now, lets focus on the use of cross-validation. :: K = 10 # number of folds for x_train, x_test, y_train, y_test in valid.k_fold_validation(x, y, K): # This is a binary classification problem model = pyGPs.GPC() # Since no prior knowldege, leave everything default model.optimize(x_train, y_train) # Predition ymu, ys2, fmu, fs2, lp = model.predict(x_test, ys=y_test) # ymu for classification is a continuous value over -1 to +1 # If you want predicting result to either one of the classes, take a sign of ymu. ymu_class = np.sign(ymu) # Evluation acc = valid.ACC(ymu_class, y_test) # accuracy rmse = valid.RMSE(ymu_class, y_test) # root-mean-square error Evaluation measures ----------------------------- We implemented some classical evaluation measures. - RMSE - root mean squared error - ACC - classification/regression accuracy - Prec - classification precision for class +1 - Recall - classification recall for class +1 - NLPD - negative log predictive density in transformed observation space .. [1] Sigillito, V. G., Wing, S. P., Hutton, L. V., \& Baker, K. B. (1989). Classification of radar returns from the ionosphere using neural networks. Johns Hopkins APL Technical Digest, 10, 262-266. pyGPs-1.2/doc/source/Examples.rst000066400000000000000000000021701236346020200167710ustar00rootroot00000000000000Demos ========================= There are several demos exemplifying the use of pyGPs for various Gaussian process (:math:`GP`) tasks. We recommend to first go through *Basic GP Regression* which introduces the :math:`GP` regression model. Basic regression is the most intuitive and simplest learning task feasable with :math:`GPs`. The other demos will then provide a general insight into more advanced functionalities of the package. You will also find the implementation of the demos in the source_ folder under `pyGPs/Demo`_. .. _pyGPs/Demo: https://github.com/marionmari/pyGPs/tree/master/pyGPs/Demo The Demos give some theoretical explanations. Further, it is useful to have a look at our documentation on `Kernels & Means`_ and `Optimizers`_. .. _Kernels & Means: Kernels.html .. _Optimizers: Opts.html .. _source: https://github.com/marionmari/pyGPs Regression .. toctree:: :maxdepth: 1 GPR GPR_FITC Classification .. toctree:: :maxdepth: 1 GPC GPC_FITC GPMC Some examples for real-world data .. toctree:: :maxdepth: 1 CV demoMaunaLoa demoHousing SemiSupervised GraphKernel pyGPs-1.2/doc/source/GPC.rst000066400000000000000000000074671236346020200156420ustar00rootroot00000000000000Basic Classification ========================= The demo shown in this tutorial can be obtained by running *pyGPs/Demo/demo_GPC.py*. Load data -------------------- First, we import the data:: # GPC target class are +1 and -1 demoData = np.load('data_for_demo/classification_data.npz') x = demoData['x'] # training data y = demoData['y'] # training target z = demoData['xstar'] # test data The :math:`120` data points were generated from two Gaussians with different means and covariances. One Gaussian is isotropic and contains :math:`2/3` of the data (blue), the other is highly correlated and contains :math:`1/3` of the points (red). Note, that the labels for the targets are specified to be :math:`\pm 1` (and not :math:`0/1`). In the plot, we superimpose the data points with the posterior equi-probability contour lines for the probability of the second class given complete information about the generating mechanism. .. figure:: _images/d2_1.png :height: 600 px :width: 800 px :align: center :scale: 70 % First example :math:`\rightarrow` state default values ---------------------------------------------------------------- Again, lets see the simplest use of gp classification at first :: model = pyGPs.gp.GPC() # binary classification (default inference method: EP) model.fit(x, y) # fit default model (mean zero & rbf kernel) with data model.optimize(x, y) # optimize hyperparamters (default optimizer: single run minimize) model.predict(z) # predict test cases Note, that inference is done via expectation propagation (EP) approximation by deault. How to set inference to Laplace approximation, see :ref:`more_on_GPC`. Second example :math:`\rightarrow` GP classification ------------------------------------------------------------ So we first state the model to be :math:`GP` classification now:: model = pyGPs.GPC() The rest is similar to GPR:: k = pyGPs.cov.RBFard(log_ell_list=[0.05,0.17], log_sigma=1.) model.setPrior(kernel=k) model.setData(x, y) model.plotData_2d(x1,x2,t1,t2,p1,p2) model.fit() model.optimize() model.predict(z, ys=np.ones((z.shape[0],1))) **[Theory]** In this example, we used an RBF kernel (squared exponential covariance function) with automatic relevance determination (ARD). This covariance function has one characteristic length-scale parameter for each dimension of the input space (here :math:`2` in total), and a signal magnitude parameter, resulting in a total of :math:`3` hyperparameters. ARD with separate length-scales for each input dimension is a very powerful tool to learn which inputs are important for the predictions: if length-scales are short, input dimensions are very important, and when they grow very large (compared to the spread of the data), the corresponding input dimensions will be mostly ignored. Note, *pyGPs.GPC().plot()* is a toy method for 2-d data:: model.plot(x1,x2,t1,t2) The contour plot for the predictive distribution is shown below. Note, that the predictive probability is fairly close to the probabilities of the generating process in regions of high data density. Note also, that as you move away from the data, the probability approaches :math:`1/3`, the overall class probability. .. figure:: _images/d2_2.png :height: 600 px :width: 800 px :align: center :scale: 70 % Examining the two ARD characteristic length-scale parameters after learning, you will find that they are fairly similar, reflecting the fact that for this data set, both input dimensions are important. .. _more_on_GPC: A bit more things you can do ---------------------------------------------------- GPC uses expectation propagation (EP) inference and Error function likelihood by default, you can explictly change to other methods: :: model.useInference("Laplace") pyGPs-1.2/doc/source/GPC_FITC.rst000066400000000000000000000063401236346020200164340ustar00rootroot00000000000000Sparse Classification ======================== The demo in this tutorial can be obtained by running *pyGPs/Demo/demo_GPC_FITC.py*. This demo is more or less a repetition of the demo of `FITC regression`_. .. _FITC regression: GPR_FITC.html First example :math:`\rightarrow` default inducing points ------------------------------------------------------------- First load the same data as in the GPC demo. **[Theory]** In case the number of training inputs :math:`x` exceeds a few hundred, approximate inference using Laplacian Approximation or Expectation Propagation takes too long. As in regression, we offer the FITC approximation based on a low-rank plus diagonal approximation to the exact covariance to deal with these cases. The general idea is to use inducing points :math:`u` and to base the computations on cross-covariances between training, test and inducing points only. Okay, now the model is FITC classificiation:: model = pyGPs.GPC_FITC() The difference between the usage of basic :math:`GP` is that we will have to specify inducing points. In our first example, we will introduce how to perform sparse GPC with the default settings. The default inducing points form a grid (hypercube in higher dimension), where each dimension has :math:`5` values in equidistant steps in :math:`[min, max]`, where :math:`min` and :math:`max` are the minimum and maximum values of the input data by default. In order to specify the dimension of input data, we HAVE TO set data first:: model.setData(x, y) The number of inducing points per axis is :math:`5` per default. How to change this, see :ref:`more_on_GPC_FITC`. Then, the regular process follows:: model.optimize() model.predict(z, ys=np.ones((z.shape[0],1))) model.plot(x1,x2,t1,t2) .. figure:: _images/d4_1.png :height: 600 px :width: 800 px :align: center :scale: 70 % The equispaced default inducing points :math:`u` are shown as black circles in the plot. Second example :math:`\rightarrow` user-defined inducing points -------------------------------------------------------------------- Alternatively, a random subset of the training points can be used as inducing points. Note, that there are various different ways of how to set the inducing points. So, in the second example let us use a user-defined set of inducing points:: u1,u2 = np.meshgrid(np.linspace(-2,2,5),np.linspace(-2,2,5)) u = np.array(zip(np.reshape(u2,(np.prod(u2.shape),)),np.reshape(u1,(np.prod(u1.shape),)))) Here, we also use a grid euqually spaced, but without the values on the margin of the grid.(i.e. shrinking the grid) Then, we can just pass :math:`u` when specifying prior:: m = pyGPs.mean.Zero() k = pyGPs.cov.RBFard(log_ell_list=[0.05,0.17], log_sigma=1.) model.setPrior(mean=m, kernel=k, inducing_points=u) The prediction results for this set of inducing points are shown below: .. figure:: _images/d4_2.png :height: 600 px :width: 800 px :align: center :scale: 70 % .. _more_on_GPC_FITC: A bit more things you can do ------------------------------ As in standard GPC, it is possible to use other inference/likelihood in the FITC method:: model.useInference("Laplace") Change the number of inducing points per axis:: model.setData(x, y, value_per_axis=10) pyGPs-1.2/doc/source/GPMC.rst000066400000000000000000000043321236346020200157430ustar00rootroot00000000000000Multi-class Classification =========================== GPMC is NOT based on multi-class Laplace approximation. It works as a one vs. one classification wrapper. In other words, GPMC trains a GPC model for each pair of two classes, and uses a majority voting scheme over all results to determine the final class. The method only returns the predictive class with highest rating; no other values (such as variance) are returned. Lets see a practical example to classify the 10 (0,1,2,...9) hand-written digits in the USPS digits dataset. Load data -------------------- The USPS digits data were gathered at the Center of Excellence in Document Analysis and Recognition (CEDAR) at SUNY Buffalo, as part of a project sponsored by the US Postal Service. The dataset is described in [1]_. :: data = loadmat('data_for_demo/usps_resampled.mat') x = data['train_patterns'].T # train patterns y = data['train_labels'].T # train labels xs = data['test_patterns'].T # test patterns ys = data['test_labels'].T # test labels To be used in GPMC, labels should start from 0 to k (k = number of classes). GPMC example --------------------- State model with 10-class classification problem: :: model = pyGPs.GPMC(10) Pass data to model: :: model.setData(x,y) Train default GPC model for each binary classification problem, and decide label for test patterns of hand-writen digits. The return value *predictive_vote[i,j]* is the probability of being class *j* for test pattern *i*. :: predictive_vote = model.optimizeAndPredict(xs) predictive_class = np.argmax(predictive_vote, axis=1) Just like we did for GP classification, you can use specific settings (other than default) for all binary classificiation problem for example by: :: m = pyGPs.mean.Zero() k = pyGPs.cov.RBF() model.setPrior(mean=m,kernel=k) model.useInference("Laplace") For more information on how to use non-default settings see `demo_GPC`_ and `demo_GPR`_. .. _demo_GPC: GPC.html .. _demo_GPR: GPR.html Beside *optimizeAndPredict(xs)*, there is also an option to perform prediction without hyperparameter optimization: :: model.fitAndPredict(xs) .. [1] A Database for Handwritten Text Recognition Research, J. J. Hull, IEEE PAMI 16(5) 550-554, 1994. pyGPs-1.2/doc/source/GPR.rst000066400000000000000000000150431236346020200156460ustar00rootroot00000000000000Basic Regression ========================= The code shown in this tutorial can be executed by running *pyGPs/Demo/demo_GPR.py* This demo will not only introduce the regression model, it also provides the general insight of how to use the package. This general information will not be repeated in the other demos. Import packages -------------------- Once you installed pyGPs, the typical way to import it is: :: import pyGPs import numpy as np Load data -------------------- First, load the data for this demo. The data consists of :math:`n=20` 1-d data points drawn from a unit Gaussian. This is the same data used in the GPML example (it is hardcoded in *data/regression_data.npz*). Note that target vector :math:`y` can be in 2d matrix with shape (nn,1) or 1d vector with shape (nn,) where :math:`nn` is number of test inputs. pyGPs can work with either format. :: demoData = np.load('data_for_demo/regression_data.npz') x = demoData['x'] # training data y = demoData['y'] # training target z = demoData['xstar'] # test data A five-line toy example --------------------------- Now lets do regression with Gaussian processes. Using pyGPs for regression is really simple; here is the most basic example: :: model = pyGPs.GPR() # specify model (GP regression) model.fit(x, y) # fit default model (mean zero & rbf kernel) with data model.optimize(x, y) # optimize hyperparamters (default optimizer: single run minimize) model.predict(z) # predict test cases model.plot() # and plot result By default, GPR uses a zero mean, the rbf kernel and a Gaussian likelihood. Default optimizer is a single run of Rasmussen's minimize. You will see below how to set non-default values in another example. *pyGPs.GPR().plot()* will plot the result, where the dark line is the posterior mean and the green-shaded area is the posterior variance. Note, that *plot()* is not a general method as it is not trivial to visualize high dimensional data. Here, *pyGPs.GPR().plot()* works for 1-d data only, while *pyGPs.GPC().plot()* is a toy method visualising 2-d input data in a classification scenario. .. figure:: _images/d1_1.png :height: 600 px :width: 800 px :align: center :scale: 70 % A more complicated example --------------------------- Now lets do another example to get insight into more advanced features of the toolbox. You can specify non-default mean and covariance functions: :: m = pyGPs.mean.Linear( D=x.shape[1] ) + mean.Const() k = pyGPs.cov.RBF() model.setPrior(mean=m, kernel=k) Here, we use a composite mean as the sum of a linear and a constant function, and an rbf kernel. The initial hyperparameters are left to their default values. See `Kernels & Means`_ for a complete documentation of kernel/mean specification and custom kernel/mean construction. Once kernel and mean are specified, they are passed to the prior using *setPrior()*. .. _Kernels & Means: Kernels.html You can add the traning data to the model explicitly by using *setData()*. So, you avoid passing them into *fit()* or *optimize()* each time used. More importantly, the deafult mean will be adapted to the average value of the trainging labels :math:`y` (if you do not specify mean function by your own). Further, you can plot the data in the 1-d case: :: model.plotData_1d() .. figure:: _images/d1_2.png :height: 600 px :width: 800 px :align: center :scale: 70 % You can specify a optimization method different from the default, which is a single run of Rasmussen's minimize. For example, you can choose to rerun the optimization method several times with different random initializations: :: model.setOptimizer("Minimize", num_restarts=30) The optimized hyperparameters returned by *optimize()* are then set to be the ones obtained from the run with the best result. The whole functionality for optimization is introduced in detail in the documentation `Optimizers`_. .. _Optimizers: Opts.html Instead of *fit()*, which only fits data using given hyperparameters, *optimize()* will optimize hyperparamters based on marginal likelihood: :: model.optimize() There are several properties you can get from the model: :: model.nlZ # negative log marginal likelihood model.dnlZ.cov # direvatives of negative log marginal likelihood model.dnlZ.lik model.dnlZ.mean model.posterior.sW # posterior structure model.posterior.alpha model.posterior.L model.covfunc.hyp model.meanfunc.hyp model.likfunc.hyp model.fm # latent mean model.fs2 # latent variance model.ym # predictive mean model.ys2 # predictive variance model.lp # log predictive probability For example, to get the log marginal likelihood use: :: print 'Optimized negative log marginal likelihood:', round(model.nlZ,3) Prediction on the test data will return five values, which are output mean (ymu) resp. variance (ys2), latent mean (fmu) resp. variance (fs2), and log predictive probabilities (lp) :: ym, ys2, fm, fs2, lp = model.predict(z) Plot data. Note that *GPR.plot()* is a toy method only for visualising 1-d data. Here we got a different posterior by using a different prior other than in the default example. :: model.plot() .. figure:: _images/d1_3.png :height: 600 px :width: 800 px :align: center :scale: 70 % A bit more things you can do ------------------------------- **[For all Models]** Speed up computation time for prediction if you know posterior in advance. Posterior is passed as an object with three fields (attributes) post.alpha, post.sW and post.L. How to use these vectors to represent the posterior can be best seen from Algorithm 2.1 (page 19) in Chapeter 2 of the `GPML`_ book by Rasmussen and Williams, 2006. :: post = myPosterior() # known in advance ym, ys2, fm, fs2, lp = model.predict_with_posterior( post,z ) .. _GPML: http://www.gaussianprocess.org/gpml/chapters/RW2.pdf **[Only for Regression]** Specify noise of data (with :math:`\sigma=0.1` by default): :: model.setNoise( log_sigma = np.log(0.1) ) You do not need to specify the noise parameter if you are optimizing the hyperparamters later anyhow. All plotting methods have keyword axisvals. You can adjust plotting range if you want. For example: :: model.plot(axisvals = [-1.9, 1.9, -0.9, 3.9]) Switch to other Inference and Likelihood functions. :: model.useInference("EP") model.useLikelihood("Laplace") pyGPs-1.2/doc/source/GPR_FITC.rst000066400000000000000000000070011236346020200164460ustar00rootroot00000000000000Sparse Regression ========================= The code shown in this tutorial can be obtained by running *pyGPs/Demo/demo_GPR_FITC.py* This demo is more or less similar to the demo of `FITC classification`_. .. _FITC classification: GPC_FITC.html First example :math:`\rightarrow` default inducing points -------------------------------------------------------------- First load the same data as in the GPR demo. **[Theory]** In case the number of training inputs :math:`x` exceeds a few hundred, approximate inference using Laplace approximation or expectation propagation takes too long. We offer the FITC approximation based on a low-rank plus diagonal approximation to the exact covariance to deal with these cases. The general idea is to use inducing points :math:`u` and to base the computations on cross-covariances between training, test and inducing points only. Okay, now the model is FITC regression:: model = pyGPs.GPR_FITC() The difference between the usage of basic :math:`GP` regression is that we will have to specify inducing points. In the first example here, we will introduce you how to use the default settings. The default inducing points are a grid (hypercube for higher dimensions), where each dimension has 5 values in equidistant steps in :math:`[min, max]`, where :math:`min` and :math:`max` are the minimum and maximum values of the input data by default. In order to specify the dimension of input data, we HAVE TO set data first:: model.setData(x, y) The number of inducing points per axis is :math:`5` per default. Now, the regular training and prediction routines follow: :: model.optimize() model.predict(z) model.plot() .. figure:: _images/d3_1.png :height: 600 px :width: 800 px :align: center :scale: 70 % The equidistant default inducing points :math:`u` that are shown in the figure as black x's. To change the number of inducing points per axis just specify a different value per axis:: model.setData(x, y, value_per_axis=10) Second example :math:`\rightarrow` user-defined inducing points ------------------------------------------------------------------ Alternatively, a random subset of the training points can be used as inducing points. Note, that there are plenty of methods to set these inducing points. So, in the second example let us use a user-defined set of inducing points. You can pick a set of fixed inducing points by hand: :: u = np.array([[-1], [-0.8], [-0.5], [0.3],[1.]]) You can also use equidistant inducing points :math:`u`, but without the values on the margin of the grid.(i.e. shrinking the range of values) :: num_u = np.fix(x.shape[0]/2) u = np.linspace(-1.3,1.3,num_u).T u = np.reshape(u,(num_u,1)) Then pass :math:`u` when specifying prior. :: m = pyGPs.mean.Zero() k = pyGPs.cov.RBFard(log_ell_list=[0.05,0.17], log_sigma=1.) model.setPrior(mean=m, kernel=k, inducing_points=u) The left figure below shows the result of fixed inducing points, and the right figure shows the result for equidistant :math:`u`. .. image:: _images/d3_2.png :width: 45% .. image:: _images/d3_3.png :width: 45% **[Theory]** Note that the predictive variance is overestimated outside the support of the inducing inputs. In a multivariate example where densely sampled inducing inputs are infeasible, one can also try to simply use a random subset of the training points. A bit more things you can do ----------------------------- Switch to other Inference and Likelihood functions. :: model.useInference("EP") model.useLikelihood("Laplace") pyGPs-1.2/doc/source/Graph.rst000066400000000000000000000010761236346020200162600ustar00rootroot00000000000000Kernels for Graph Data ============================ You can refer to our demo for `semi-supervised learning`_ for a simple usage of kernels for graph data. And there is also a demo for `graph kernel`_ using an example of propagation kernel. .. _`semi-supervised learning`: SemiSupervised.html .. _`graph kernel`: GraphKernel.html Kernels on Graph --------------------------------------- .. automodule:: pyGPs.GraphExtensions.nodeKernels :members: Graph Kernels --------------------------------------- .. automodule:: pyGPs.GraphExtensions.graphKernels :members: pyGPs-1.2/doc/source/GraphKernel.rst000066400000000000000000000057461236346020200174310ustar00rootroot00000000000000Graph Kernels ========================= The code shown in this tutorial can be executed by running *pyGPs/Demo/demo_GraphKernel.py* Load data -------------------- MUTAG [1]_ is a data set of 188 mutagenic aromatic and heteroaromatic nitro compounds labeled according to whether or not they have a mutagenic effect on the Gram-negative bacterium Salmonella typhimurium. MUTAG is a simple but popular dataset to benchmark graph kernels. :: data = np.load('MUTAG.npz') A = csc_matrix( (data['adj_data'], data['adj_indice'], \ data['adj_indptr']), shape=data['adj_shape']) # n x n adjancy array (sparse matrix) gr_id = data['graph_ind'] # n x 1 graph id array node_label = data['responses'] # n x 1 node label array graph_label = data['labels'] # N x 1 graph label array Note that adjacency matrix for all graphs is usually too big to fit into memory when using :math:`GP`. Therefore, we generate :math:`A` using sparse matrix *csc_matrix* provided by *scipy*. Compute propagation kernel matrix ---------------------------------------- Propagation kernels [2]_ are recently introduced fast and flexible graph kernels. Generate prapagation kernel based on given graph data. :: num_Iteration = 10 w = 1e-4 dist = 'tv' # possible values: 'tv', 'hellinger' K = graphKernels.propagationKernel(A, node_label, gr_id, num_Iteration, w, dist, 'label_diffusion') Adjacency matrix :math:`A` can either be in format of standard numpy matrix or sparse matrix. Standard GP Classification ---------------------------------------- :math:`K` is a big square matrix with all rows and columns of the number of data points. By specifying the indice of training data and test data, we will form two matrix M1 and M2 with the exact format which *pyGPs.Core.cov.Pre* needed. :: M1,M2 = graphUtil.formKernelMatrix(Matrix, indice_train, indice_test) k = pyGPs.cov.Pre(M1,M2) M1 is a matrix with shape **number of training points plus 1** by **number of test points** - cross covariances matrix (train by test) - last row is self covariances (diagonal of test by test) M2 is a square matrix with **number of training points** for each dimension - training set covariance matrix (train by train) The following is the standard way to do GP classification :: model = pyGPs.GPC() model.setPrior(kernel=k) model.fit(x_train, y_train) model.predict(x_test) .. [1] Debnath, A., Lopez de Compadre, R., Debnath, G., Shusterman, A., and Hansch, C.. Structure-activity relationship of mutagenic aromatic and heteroaromatic nitro compounds. Correlation with molecular orbital energies and hydrophobicity. J. Med. Chem., 34:786–797, 1991. .. [2] Neumann, M., Patricia, N., Garnett, R., Kersting, K.: Efficient Graph Kernels by Randomization. In: P.A. Flach, T.D. Bie, N. Cristianini (eds.) ECML/PKDD, Notes in Computer Science, vol. 7523, pp. 378-393. Springer (2012). pyGPs-1.2/doc/source/Install.rst000066400000000000000000000021111236346020200166140ustar00rootroot00000000000000Installation ============================ 1. First, download_ the archive from github and extract it to any local directory. .. _download: https://github.com/marionmari/pyGPs/releases/tag/v1.1 2. You can either add the local directory to your PYTHONPATH :: export PYTHONPATH=$PYTHONPATH:/path/to/local/directory/../parent_folder_of_pyGPs 3. Or install the package using setup.py:: sudo python setup.py install Requirements ------------ * `python 2.6 or 2.7`_ * `scipy`_, `numpy`_, and `matplotlib`_: open-source packages for scientific computing in Python. .. _python 2.6 or 2.7: http://www.python.org/ .. _scipy: http://www.scipy.org/ .. _numpy: http://www.numpy.org/ .. _matplotlib: http://matplotlib.org/ Example installation on Ubuntu & Debian:: sudo apt-get install python2.7 python-numpy python-scipy python-matplotlib Example installation on Mac via Macports (requires XCode and MacPorts):: sudo port install python27 py27-numpy py27-scipy py27-matplotlib For other systems please check the installation instructions on the respective package web sites. pyGPs-1.2/doc/source/Kernels.rst000066400000000000000000000150401236346020200166160ustar00rootroot00000000000000Kernels & Means ============================ Simple Kernel & Mean --------------------- You may already seen, we can specify a kernel function like this(same for mean fucntions): :: k = pyGPs.cov.RBF( log_ell=-1., log_sigma=0. ) There are several points need to be noticed: 1. Most parameters are initilized in their logorithms. This is because we need to make sure they are positive during optimization. e.g. Here length scale and signal variance should always be positive. 2. Most kernel functions have a scalar in front, namely signal variance(set by log_sigma) 3. If you will do optimization later anyway, you can just leave parameters to be default Some Special Cases --------------------- 1. For some kernels/means, number of hyperparameters depends on the dimension of input data. You can either enter the dimension, which use default values: :: m = pyGPs.mean.Linear( D=x.shape[1] ) or you can initialze with the exact hyperparameters, you should enter as a list, one element for each dimension :: m = pyGPs.mean.Linear( alpha_list=[0.2, 0.4, 0.3] ) All these "hyp-dim-dependent" functions are: * *pyGPs.mean.Linear* * *pyGPs.cov.RBFard* * *pyGPs.cov.LINard* * *pyGPs.cov.RQard* 2. For *pyGPs.cov.RBFunit()*, its signal variance is always 1 (because of unit magnitude). Therefore this function do not have a hyperparameter of "signal variance". 3. *pyGPs.cov.Poly()* has three parameters, where hyperparameters are: * c -> inhomogeneous offset * sigma -> signal deviation however, * d -> order of polynomial will be treated as normal parameter, i.e. will not be trained 4. Explicitly set *pyGPs.cov.Noise* is not necessary, because noise are already added in likelihood. Composite Kernels & Meams ---------------------------- Adding and muliplying Kernels(Means) is really simple: :: k = pyGPs.cov.Periodic() * pyGPs.cov.RBF() k = 0.5*pyGPs.cov.LIN() + pyGPs.cov.Periodic() Scalar will also be treated as a hyperparameter. For example, k = s1 * k1 + s2 * k2, then the list of hyperparameters is hyp = [s1, k1.hyp, s2, k2.hyp]. Scalar is passed in logorithm domain such that it will always be positive during optimization. Beside + / * , there is also a power operator for mean functions: :: m = ( pyGPs.mean.One() + pyGPs.mean.Linear(alpha_list=[0.2]) )**2 Precomputed Kernel Matrix ----------------------------- In certain cases, you may have a precomputed kernel matrix, but its non-trivial to write down the exact formula of kernel functions. Then you can specify your kernel in the following way. A precomputed kernel also fits with other kernels. In other words, it can also be composited as the way other kernels functions do. :: k = pyGPs.cov.Pre(M1, M2) M1 and M2 are your precomputed kernel matrix, where, M1 is a matrix with shape **number of training points plus 1** by **number of test points** - cross covariances matrix (train by test) - last row is self covariances (diagonal of test by test) M2 is a square matrix with **number of training points** for each dimension - training set covariance matrix (train by train) A precomputed kernel can also be composited with other kernels. You need to explictly add scalar for *pyGPs.cov.Pre()*. :: k = 0.5*pyGPs.cov.Pre(M1, M2) + pyGPs.cov.RBF() Developing New Kernel & Mean Functions ----------------------------------------- We also support the development of new kernel/mean classes, your customized kernel class need to follow the template as below: :: # Your kernel class needs to inherit base class Kernel, # which is in the module of Core.cov class MyKernel(Kernel): def __init__(self, hyp): ''' Intialize hyperparameters for MyKernel. ''' self.hyp = hyp def getCovMatrix(self,x=None,z=None,mode=None): ''' Return the specific covariance matrix according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self covariance matrix of test data(test by 1). 'train' return training covariance matrix(train by train). 'cross' return cross covariance matrix between x and z(train by test) :return: the corresponding covariance matrix ''' pass def getDerMatrix(self,x=None,z=None,mode=None,der=None): ''' Compute derivatives wrt. hyperparameters according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self derivative matrix of test data(test by 1). 'train' return training derivative matrix(train by train). 'cross' return cross derivative matrix between x and z(train by test) :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' pass and for customized mean class: :: # Your mean class needs to inherit base class Mean, # which is in the module of Core.mean class MyMean(Mean): def __init__(self, hyp): ''' Intialize hyperparameters for MyMean. ''' self.hyp = hyp def getMean(self, x=None): ''' Get the mean vector. ''' pass def getDerMatrix(self, x=None, der=None): ''' Compute derivatives wrt. hyperparameters. :param x: training data :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' pass You can test your customized mean/kernel function using our framework of unit test. Taking kernel test as an example, you can uncomment method *test_cov_new* in *pyGPs.Testing.unit_test_cov.py* to check the outputs of your kernel function. :: # Test your customized covariance function def test_cov_new(self): k = myKernel() # specify your covariance function self.checkCovariance(k) and testing mean function in *pyGPs.Testing.unit_test_mean.py* :: # Test your customized mean function def test_mean_new(self): m = myMean # specify your mean function self.checkMean(m) List of Kernels and Default Parameters --------------------------------------- .. automodule:: pyGPs.Core.cov :members: List of Means and Default Parameters ----------------------------------------- .. automodule:: pyGPs.Core.mean :members: pyGPs-1.2/doc/source/Likelihoods.rst000066400000000000000000000124211236346020200174610ustar00rootroot00000000000000Likelihoods & Inference ============================ Changing Likelihood & Inference ------------------------------------ By default, * GPR uses Gaussian likelihood and exact inference. * GPC uses Error functionlikelihood and EP inference. * FITC model uses same default with corresponding FITC inference. * GPMC calls GPC and thus uses the default setting of GPC You can change to other likelihood or inference methods using: :: model.useLikelihood(newLik) model.useInference(newInf) *newLik* and *newInf* are **Strings**. Currently the options are: 1. Regression model * newLik: **"Laplace"**. Note this will force inference method to be EP. * newInf: **"EP"**, **"Laplace"**. 2. Classification model (including GPMC) * newInf: **"Laplace"** Developing New Likelihood & Inference Functions ------------------------------------------------- We also support the development of new likelihood/inference classes, your customized inference class need to follow the template as below: :: # Your inference class needs to inherit base class Inference, # which is in the module of Core.inf class MyKernel(Kernel): def __init__(self): pass def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): ''' Inference computation based on inputs. post, nlZ, dnlZ = inf.proceed(mean, cov, lik, x, y) INPUT: cov: name of the covariance function (see covFunctions.m) lik: name of the likelihood function (see likFunctions.m) x: n by D matrix of training inputs y: 1d array (of size n) of targets OUTPUT: post(instance of postStruct): struct representation of the (approximate) posterior containing: nlZ: returned value of the negative log marginal likelihood dnlZ(instance of dnlZStruct): struct representation for derivatives of the negative log marginal likelihood w.r.t. each hyperparameter. Usually, the approximate posterior to be returned admits the form: N(m=K*alpha, V=inv(inv(K)+W)), where alpha is a vector and W is diagonal; if not, then L contains instead -inv(K+inv(W)), and sW is unused. For more information on the individual approximation methods and their implementations, see the respective inference function below. See also gp.py :param meanfunc: mean function :param covfunc: covariance function :param likfunc: likelihood function :param x: training data :param y: training labels :param nargout: specify the number of output(1,2 or 3) :return: posterior, negative-log-marginal-likelihood, derivative for negative-log-marginal-likelihood-likelihood ''' pass where **postStruct** and **dnlZStruct** is also defined in *Core.inf*. :: class postStruct(object): ''' Data structure for posterior post.alpha -> 1d array containing inv(K)*m, where K is the prior covariance matrix and m the approx posterior mean post.sW: -> 1d array containing diagonal of sqrt(W) the approximate posterior covariance matrix is inv(inv(K)+W) post.L -> 2d array, L = chol(sW*K*sW+identity(n)) ''' class dnlZStruct(object): ''' Data structure for the derivatives of mean, cov and lik functions. dnlZ.mean -> list of derivatives for each hyperparameters in mean function dnlZ.cov -> list of derivatives for each hyperparameters in covariance function dnlZ.lik -> list of derivatives for each hyperparameters in likelihood function ''' Customizing likelihood function is more complicated. We will omit it here to keep this this page not too long. However, you can find detailed explaination either in the **source code** *Core.lik* or in coresponding section of **manual**. Just like testing kernel/mean fucntions, you can also find unit test module for likelihood and inference functions. To test your customized inference function, uncomment the following method in *pyGPs.Testing.unit_test_inf.py*. :: # Test your customized inference function def test_inf_new(self): # specify your inf function # set mean/cov/lik functions post, nlZ, dnlZ = inffunc.proceed(meanfunc, covfunc, likfunc, self.x, self.y, nargout=3) self.checkFITCOutput(post, nlZ, dnlZ) and test customized likelihood function in *pyGPs.Testing.unit_test_lik.py* :: # Test your customized likelihood function def test_cov_new(self): likelihood = myLikelihood() # specify your likelihood function self.checkLikelihood(likelihood) List of Likelihoods --------------------------------------- .. autoclass:: pyGPs.Core.lik.Erf :members: .. autoclass:: pyGPs.Core.lik.Gauss :members: .. autoclass:: pyGPs.Core.lik.Laplace :members: List of Inference ----------------------------------------- .. autoclass:: pyGPs.Core.inf.Exact :members: .. autoclass:: pyGPs.Core.inf.EP :members: .. autoclass:: pyGPs.Core.inf.Laplace :members: .. autoclass:: pyGPs.Core.inf.FITC_Exact :members: .. autoclass:: pyGPs.Core.inf.FITC_EP :members: .. autoclass:: pyGPs.Core.inf.FITC_Laplace :members: pyGPs-1.2/doc/source/Opts.rst000066400000000000000000000025711236346020200161450ustar00rootroot00000000000000Optimizers ============================ Optimization Methods ---------------------------- As you may have already seen in the demos, the optimizer is initialized in the following way: .. automethod:: pyGPs.Core.gp.GP.setOptimizer Developing Optimization Methods ------------------------------------- We also support the development of new optimizers. Your own optimizer should inherent base class Optimizer in *pyGPs.Core.opt* and follow the template as below: :: class MyOptimizer(Optimizer): def __init__(self, model=None, searchConfig = None): self.model = model def findMin(self, x, y): ''' Find minimal value based on negative-log-marginal-likelihood. optimalHyp, funcValue = findMin(x, y) where funcValue is the minimal negative-log-marginal-likelihood during optimization, and optimalHyp is a flattened numpy array (in sequence of meanfunc.hyp, covfunc.hyp, likfunc.hyp) of the hyparameters to achieve such value. You can achieve advanced search strategy by initializing Optimizer with searchConfig, which is an instance of pyGPs.Optimization.conf. See more in pyGPs.Optimization.conf and pyGPs.Core.gp.GP.setOptimizer, as well as in online documentation of section Optimizers. ''' pass pyGPs-1.2/doc/source/SemiSupervised.rst000066400000000000000000000116771236346020200201760ustar00rootroot00000000000000Semi-supervised Learning with Graphs ======================================= The code shown in this tutorial can be executed by running *pyGPs/Demo/demo_NodeKernel.py* Import ------------------- You may want to import some extensions we provide as follows: :: from pyGPs.GraphExtension import graphUtil, nodeKernels from pyGPs.Validation import valid Load data -------------------- We used the same dataset from GPMC example. i.e. The USPS digits dataset [1]_. Each digit of :math:`16*16` pixels is flattened into a :math:`256` dimension vector. For the simplicity of demo, we only selected digits :math:`1` s and :math:`2` s such that we have a binary classification problem where digit :math:`1` for class +1 and digit :math:`2` for class -1. We also reduced the dataset into :math:`100` samples per digit, where the original dataset consist of thousands of samples for each digit. Here are samples for two digits for :math:`1` .. image:: _images/digit1_1.png :width: 30% .. image:: _images/digit1_2.png :width: 30% and samples for two digits for :math:`2` .. image:: _images/digit2_1.png :width: 30% .. image:: _images/digit2_2.png :width: 30% Form a nearest neighbour graph -------------------------------- We form a nearest-neighbor graph based on Euclidean distance of the vector representation of digits. Neighboring images have small Euclidean distance. Each digit is a node in the graph. There is an edge if digit :math:`i` is the k-nearest neighbour of digit :math:`j`. We form a symmetrized graph such that we connect nodes :math:`j`, :math:`i` if i is in j’s kNN and vice versa, and therefore a node can have more than k edges. You should import the corresponding module from *pyGPs.GraphStuff* :: x,y = load_binary(1,2,reduce=True) A = graphUtil.formKnnGraph(x,2) A is the adjacency matrix of this :math:`2-NN` graph. Below shows an example of such symmetrized Euclidean :math:`2-NN` graph on some 1s and 2s taking from Xiaojin Zhu's doctoral thesis [2]_. .. figure:: _images/2nnGraph.png :align: center Kernel on graph ------------------ Several classical kernels on graph described in `Structured Kernels`_ can be built from adjacency matrix :math:`A`. We use diffusion kernel for this example to get the precomputed kernel matrix. :: Matrix = nodeKernels.diffKernel(A) .. _Structured Kernels: Graph.html This a big square matrix with all rows and columns of the number of data points. By specifying the indice of training data and test data, we will form two matrix M1 and M2 with the exact format which *pyGPs.Core.cov.Pre* needed. :: M1,M2 = graphUtil.formKernelMatrix(Matrix, indice_train, indice_test) M1 is a matrix with shape **number of training points plus 1** by **number of test points** - cross covariances matrix (train by test) - last row is self covariances (diagonal of test by test) M2 is a square matrix with **number of training points** for each dimension - training set covariance matrix (train by train) GP classification ----------------------- Every ingredients for a basic semi-supervised learning is prepared now. Lets see how to proceed for :math:`GP` classification. First, the normal way with rbf kernel we have seen several times :: model = pyGPs.GPC() k = pyGPs.cov.RBF() model.setPrior(kernel=k) Then lets use our kernel precomputed matrix. If you only use precomputed kernel matrix, there is no training data. However you still need to specify :math:`x` just to fit in the usage of pyGPs for generality reason. You can create any :math:`x` as long as the dimension is correct. :: x = np.zeros((n,1)) k = pyGPs.cov.Pre(M1,M2) + pyGPs.cov.RBF() model.setPrior(kernel=k) Moreover, you can composite a kernel for both precomputed matrix and regular kernel function if necessary. :: k = pyGPs.cov.Pre(M1,M2) + pyGPs.cov.RBFunit() model.setPrior(kernel=k) The rest way of using pyGPs is exactly the same as the demo of GP classification. Result ----------------------- For our manually created graph data, an rbf kernel works better than a diffusion kernel on the graph (higher accuracy). The performance in general should depend on the application as well as features of data. The left image shows the digit that using diffusion kernel will predict the wrong result (should be :math:`2`), but rbf kernel does the job fine. The right image shows the digit that rbf kernel predicts the wrong class, diffusion kernel on the other hand, predicts correctly due to graph information! (should be :math:`1`). Interestingly, using a composite kernel with diffusion kernel on graph and an rbf kernel together. All test cases including the following are predicted correctly. .. image:: _images/digitDiffwrong.png :width: 50% .. image:: _images/digitRBFwrong.png :width: 50% .. [1] A Database for Handwritten Text Recognition Research, J. J. Hull, IEEE PAMI 16(5) 550-554, 1994. .. [2] Semi-Supervised Learning with Graphs, Xiaojin Zhu, CMU-LTI-05-192, 2005 pyGPs-1.2/doc/source/Theory.rst000077500000000000000000000167561236346020200165070ustar00rootroot00000000000000GPs & Functionality ======================== Gaussian Processes (GPs) can conveniently be used for Bayesian supervised learning, such as regression and classification. In its simplest form, GP inference can be implemented in a few lines of code. However, in practice, things typically get a little more complicated: you might want to use expressive covariance and mean functions, learn good values for hyperparameters, use non-Gaussian likelihood functions (rendering exact inference intractable), use approximate inference algorithms, or combinations of many or all of the above. A comprehensive introduction to Gaussian Processes for Machine Learning is provided in the `GPML`_ book by Rasmussen and Williams, 2006. List of Functionality ------------------------ This table lists the functionality implemented in pyGPs. +-------------+-------------------+------------+-------------------------+---------------+------------------+ |Functionality| Kernel | Mean | Likelihood | Inference | Optimizer | +=============+===================+============+=========================+===============+==================+ | Simple | Constant | Constant | Gaussian | Exact | Minimize | | +-------------------+------------+-------------------------+---------------+------------------+ | | Linear | Linear |Cumulative Gaussian (Erf)| EP | CG | | +-------------------+------------+-------------------------+---------------+------------------+ | | Linear ard | One | Laplace | Laplace | SCG | | +-------------------+------------+-------------------------+---------------+------------------+ | | Matérn (1,3,5) | Zero | | | BFGS | | +-------------------+------------+ | +------------------+ | | Periodic | | | | | | +-------------------+ | | | | | | Polynomial | | | | | | +-------------------+ | | | | | | Piecewise Poly | | | | | | +-------------------+ | | | | | | RBF iso | | | | | | +-------------------+ | | | | | | RBF ard | | | | | | +-------------------+ | | | | | | RBF unit | | | | | | +-------------------+ | | | | | | RQ iso | | | | | | +-------------------+ | | | | | | RQ ard | | | | | | +-------------------+ | | | | | | Noise | | | | | +-------------+-------------------+------------+-------------------------+---------------+------------------+ | Composite | Sum (+) | Sum (+) | | | | | +-------------------+------------+ | | | | | Product (*) | Product (*)| | | | | +-------------------+------------+ | | | | | Scale (*) | Scale (*) | | | | | +-------------------+------------+ | | | | | | Power (**) | | | | +-------------+-------------------+------------+-------------------------+---------------+------------------+ | Sparse GP | | | | FITC Exact | | | | | | +---------------+ | | | | | | FITC EP | | | | | | +---------------+ | | | | | | FITC Laplace | | +-------------+-------------------+------------+-------------------------+---------------+------------------+ | Graphs | Diffusion | | | | | | +-------------------+ | | | | | | VN Diffusion | | | | | | +-------------------+ | | | | | |Pseudo Inv Laplace | | | | | | +-------------------+ | | | | | |Regularized Laplace| | | | | | +-------------------+ | | | | | |p-step Random Walk | | | | | | +-------------------+ | | | | | |Inverse Cosine | | | | | | +-------------------+ | | | | | |Propagation Kernel | | | | | +-------------+-------------------+------------+-------------------------+---------------+------------------+ | Other | Customized | Customized | | | | | +-------------------+------------+ | | | | |Precomputed Kernel | | | | | +-------------+-------------------+------------+-------------------------+---------------+------------------+ pyGPs also provide cross-validation and some built-in evalation methods. .. _GPML: http://www.gaussianprocess.org/gpml pyGPs-1.2/doc/source/_images/000077500000000000000000000000001236346020200160655ustar00rootroot00000000000000pyGPs-1.2/doc/source/_images/2nnGraph.png000066400000000000000000001616031236346020200202610ustar00rootroot00000000000000JFIF XICC_PROFILE HLinomntrRGB XYZ  1acspMSFTIEC sRGB-HP cprtP3desclwtptbkptrXYZgXYZ,bXYZ@dmndTpdmddvuedLview$lumimeas $tech0 rTRC< gTRC< bTRC< textCopyright (c) 1998 Hewlett-Packard CompanydescsRGB IEC61966-2.1sRGB IEC61966-2.1XYZ QXYZ XYZ o8XYZ bXYZ $descIEC http://www.iec.chIEC http://www.iec.chdesc.IEC 61966-2.1 Default RGB colour space - sRGB.IEC 61966-2.1 Default RGB colour space - sRGBdesc,Reference Viewing Condition in IEC61966-2.1,Reference Viewing Condition in IEC61966-2.1view_. \XYZ L VPWmeassig CRT curv #(-27;@EJOTY^chmrw| %+28>ELRY`gnu| &/8AKT]gqz !-8COZfr~ -;HUcq~ +:IXgw'7HYj{+=Oat 2FZn  % : O d y  ' = T j " 9 Q i  * C \ u & @ Z t .Id %A^z &Ca~1Om&Ed#Cc'Ij4Vx&IlAe@e Ek*Qw;c*R{Gp@j>i  A l !!H!u!!!"'"U"""# #8#f###$$M$|$$% %8%h%%%&'&W&&&''I'z''( (?(q(())8)k))**5*h**++6+i++,,9,n,,- -A-v--..L.../$/Z///050l0011J1112*2c223 3F3334+4e4455M555676r667$7`7788P8899B999:6:t::;-;k;;<' >`>>?!?a??@#@d@@A)AjAAB0BrBBC:C}CDDGDDEEUEEF"FgFFG5G{GHHKHHIIcIIJ7J}JK KSKKL*LrLMMJMMN%NnNOOIOOP'PqPQQPQQR1R|RSS_SSTBTTU(UuUVV\VVWDWWX/X}XYYiYZZVZZ[E[[\5\\]']x]^^l^__a_``W``aOaabIbbcCccd@dde=eef=ffg=ggh?hhiCiijHjjkOkklWlmm`mnnknooxop+ppq:qqrKrss]sttptu(uuv>vvwVwxxnxy*yyzFz{{c{|!||}A}~~b~#G k͂0WGrׇ;iΉ3dʋ0cʍ1fΏ6n֑?zM _ɖ4 uL$h՛BdҞ@iءG&vVǥ8nRĩ7u\ЭD-u`ֲK³8%yhYѹJº;.! 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G}t򽿪x C >[?=*PJ7|s+VWWO=K/{9rdX(v,zɌ98xkO~СCUUUſWXսűc{]c=Gydqܫ}m}ֿeeee˖m_zIENDB`pyGPs-1.2/doc/source/_static/000077500000000000000000000000001236346020200161075ustar00rootroot00000000000000pyGPs-1.2/doc/source/_static/changelog.rst000066400000000000000000000030461236346020200205730ustar00rootroot00000000000000 Changelog pyGPs v1.2 ===================== June 30th 2014 ---------------- structural updates: - input target now can either be in 2-d array with size (n,1) or in 1-d array with size (n,) - setup.py updated - "import pyGPs" instead of "from pyGPs.Core import gp" - rename ".train()" to ".optimize()" - rename "Graph-stuff" to "graphExtension" - rename kernelOnGraph to "nodeKernels" and graphKernel to "graphKernels" - redundancy removed for model.setData(x,y) - rewrite "mean.proceed()" to "getMean()" and "getDerMatrix()" - rewrite "cov.proceed()" to "getCovMatrix()" and "getDerMatrix()" - rename cov.LIN to cov.Linear (to be consistent with mean.Linear) - rename module "valid" to "validation" - add graph dataset Mutag in python file. (.npz and .mat) - add graphUtil.nomalizeKernel() - fix number of iteration problem in graphKernels "PropagationKernel" - add unit testing for covariance, mean functions bug fixes: - derivatives for cov.LINard - derivative of the scalar for cov.covScale - demo_GPR_FITC.py missing pyGPs.mean July 8th 2014 ---------------- structural updates: - add hyperparameter(signal variance s2) for linear covariance - add unit testing for inference,likelihood functions as well as models - NOT show(print) "maximum number of sweep warning in inference EP" any more - documentation updated bug fixes: - typos in lik.Laplace - derivative in lik.Laplace July 14th 2014 ---------------- documentation updates: - online docs updated - API file updated structural updates: - made private for methods that users don't need to call pyGPs-1.2/doc/source/_static/copyright.rst000066400000000000000000000035241236346020200206550ustar00rootroot00000000000000Copyright ========= Authors ------- * Marion Neumann [marion dot neumann at uni-bonn dot de] * Shan Huang [shan dot huang at iais dot fraunhofer dot de] * Daniel Marthaler [marthaler at ge dot com] * Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] This file is part of pyGPs. License ------- The code is released under the BSD 2-Clause (FreeBSD) License. Copyright (c) 2014 Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: - Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. - Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. pyGPs-1.2/doc/source/_static/pyGPs_API.pdf000077500000000000000000045161371236346020200203610ustar00rootroot00000000000000%PDF-1.5 % 97 0 obj << /Length 306 /Filter /FlateDecode >> stream xڽPN0+H3>BVi}lVpGc=Yaq'^N0"o Jp~:?5BqX"1=_Gyo`<>YU$ځ2瞠xCZ:9%y(V)YqNi|..yYI"s6uTΪ*U$Н 3Xdd4zDj.vПۗmTpbR6&u("i . ݽutHkUHk'ϻ,NzÌ́{Xd}Lw7⇰ endstream endobj 106 0 obj << /Length 19 /Filter /FlateDecode >> stream x3PHW0Pp2Ac( endstream endobj 133 0 obj << /Length 915 /Filter /FlateDecode >> stream xIs0:u.8ԓI=9#L0g鯯ddI'bޓ>M2AMip+

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If the directory is relative to the # documentation root, use os.path.abspath to make it absolute, like shown here. sys.path.insert(0, os.path.abspath("../../")) # -- General configuration ------------------------------------------------ # If your documentation needs a minimal Sphinx version, state it here. #needs_sphinx = '1.0' # Add any Sphinx extension module names here, as strings. They can be # extensions coming with Sphinx (named 'sphinx.ext.*') or your custom # ones. extensions = [ 'sphinx.ext.autodoc', 'sphinx.ext.doctest', 'sphinx.ext.intersphinx', 'sphinx.ext.todo', 'sphinx.ext.coverage', 'sphinx.ext.pngmath', 'sphinx.ext.ifconfig', 'sphinx.ext.viewcode', ] # Add any paths that contain templates here, relative to this directory. templates_path = ['_templates'] # The suffix of source filenames. source_suffix = '.rst' # The encoding of source files. #source_encoding = 'utf-8-sig' # The master toctree document. master_doc = 'index' # General information about the project. project = u'pyGPs' copyright = u'2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting' # The version info for the project you're documenting, acts as replacement for # |version| and |release|, also used in various other places throughout the # built documents. # # The short X.Y version. version = 'v1.2' # The full version, including alpha/beta/rc tags. release = 'v1.2' # The language for content autogenerated by Sphinx. Refer to documentation # for a list of supported languages. #language = None # There are two options for replacing |today|: either, you set today to some # non-false value, then it is used: #today = '' # Else, today_fmt is used as the format for a strftime call. #today_fmt = '%B %d, %Y' # List of patterns, relative to source directory, that match files and # directories to ignore when looking for source files. exclude_patterns = [] # The reST default role (used for this markup: `text`) to use for all # documents. #default_role = None # If true, '()' will be appended to :func: etc. cross-reference text. #add_function_parentheses = True # If true, the current module name will be prepended to all description # unit titles (such as .. function::). #add_module_names = True # If true, sectionauthor and moduleauthor directives will be shown in the # output. They are ignored by default. #show_authors = False # The name of the Pygments (syntax highlighting) style to use. pygments_style = 'sphinx' # A list of ignored prefixes for module index sorting. #modindex_common_prefix = [] # If true, keep warnings as "system message" paragraphs in the built documents. #keep_warnings = False # -- Options for HTML output ---------------------------------------------- # The theme to use for HTML and HTML Help pages. See the documentation for # a list of builtin themes. html_theme = 'nature' # Theme options are theme-specific and customize the look and feel of a theme # further. For a list of options available for each theme, see the # documentation. #html_theme_options = {} # Add any paths that contain custom themes here, relative to this directory. #html_theme_path = [] # The name for this set of Sphinx documents. If None, it defaults to # " v documentation". #html_title = None # A shorter title for the navigation bar. Default is the same as html_title. #html_short_title = None # The name of an image file (relative to this directory) to place at the top # of the sidebar. #html_logo = None # The name of an image file (within the static path) to use as favicon of the # docs. This file should be a Windows icon file (.ico) being 16x16 or 32x32 # pixels large. #html_favicon = None # Add any paths that contain custom static files (such as style sheets) here, # relative to this directory. They are copied after the builtin static files, # so a file named "default.css" will overwrite the builtin "default.css". html_static_path = ['_static'] # Add any extra paths that contain custom files (such as robots.txt or # .htaccess) here, relative to this directory. These files are copied # directly to the root of the documentation. #html_extra_path = [] # If not '', a 'Last updated on:' timestamp is inserted at every page bottom, # using the given strftime format. #html_last_updated_fmt = '%b %d, %Y' # If true, SmartyPants will be used to convert quotes and dashes to # typographically correct entities. #html_use_smartypants = True # Custom sidebar templates, maps document names to template names. #html_sidebars = {} # Additional templates that should be rendered to pages, maps page names to # template names. #html_additional_pages = {} # If false, no module index is generated. #html_domain_indices = True # If false, no index is generated. #html_use_index = True # If true, the index is split into individual pages for each letter. #html_split_index = False # If true, links to the reST sources are added to the pages. #html_show_sourcelink = True # If true, "Created using Sphinx" is shown in the HTML footer. Default is True. #html_show_sphinx = True # If true, "(C) Copyright ..." is shown in the HTML footer. Default is True. #html_show_copyright = True # If true, an OpenSearch description file will be output, and all pages will # contain a tag referring to it. The value of this option must be the # base URL from which the finished HTML is served. #html_use_opensearch = '' # This is the file name suffix for HTML files (e.g. ".xhtml"). #html_file_suffix = None # Output file base name for HTML help builder. htmlhelp_basename = 'pyGP_OOdoc' # -- Options for LaTeX output --------------------------------------------- latex_elements = { # The paper size ('letterpaper' or 'a4paper'). #'papersize': 'letterpaper', # The font size ('10pt', '11pt' or '12pt'). #'pointsize': '10pt', # Additional stuff for the LaTeX preamble. #'preamble': '', } # Grouping the document tree into LaTeX files. List of tuples # (source start file, target name, title, # author, documentclass [howto, manual, or own class]). latex_documents = [ ('index', 'pyGP_OO.tex', u'pyGP\\_OO Documentation', u'Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting', 'manual'), ] # The name of an image file (relative to this directory) to place at the top of # the title page. #latex_logo = None # For "manual" documents, if this is true, then toplevel headings are parts, # not chapters. #latex_use_parts = False # If true, show page references after internal links. #latex_show_pagerefs = False # If true, show URL addresses after external links. #latex_show_urls = False # Documents to append as an appendix to all manuals. #latex_appendices = [] # If false, no module index is generated. #latex_domain_indices = True # -- Options for manual page output --------------------------------------- # One entry per manual page. List of tuples # (source start file, name, description, authors, manual section). man_pages = [ ('index', 'pygp_oo', u'pyGP_OO Documentation', [u'Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting'], 1) ] # If true, show URL addresses after external links. #man_show_urls = False # -- Options for Texinfo output ------------------------------------------- # Grouping the document tree into Texinfo files. List of tuples # (source start file, target name, title, author, # dir menu entry, description, category) texinfo_documents = [ ('index', 'pyGP_OO', u'pyGP_OO Documentation', u'Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting', 'pyGP_OO', 'One line description of project.', 'Miscellaneous'), ] # Documents to append as an appendix to all manuals. #texinfo_appendices = [] # If false, no module index is generated. #texinfo_domain_indices = True # How to display URL addresses: 'footnote', 'no', or 'inline'. #texinfo_show_urls = 'footnote' # If true, do not generate a @detailmenu in the "Top" node's menu. #texinfo_no_detailmenu = False # Example configuration for intersphinx: refer to the Python standard library. intersphinx_mapping = {'http://docs.python.org/': None} pyGPs-1.2/doc/source/demoHousing.rst000066400000000000000000000043421236346020200174770ustar00rootroot00000000000000Regression on UCI Housing data ------------------------------ Boston Housing is a fairly standard dataset used for testing regression problems. It contains 506 data points with 12 numeric attributes, and one binary categorical attribute. The goal is to determine median home values, based on various census attributes. This dataset is available at the `UCI Repository`_. The demo follows that in [1]_. The data set was preprocessed as follows: each continuous feature was transformed to zero mean and unit variance (The categorical variable was dropped). The data was partitioned into :math:`481` points for training and :math:`25` points for testing. The mean function used was :func:`src.Core.means.meanZero` and the covariance (using the :func:`src.Core.kernels.covSum` function) was a composite of :func:`src.Core.kernels.covSEiso` and :func:`src.Core.kernels.covNoise`. The initial values of the hyperparameters were selected randomly from a zero-mean, unit-variance normal distribtion. The actual values were: :math:`[ -0.75, \; 0.59, \; -0.45 ]`. The initial likelihood hyperparameter was :math:`-2.30`. The regression started with initial negative log marginal likelihood of :math:` 752.46`. Note the initial zero mean and the variance that is uniform over the test set. :: model = pyGPs.GPR() model.optimize(x,y) ym, ys2, fm, fs2, lp = model.predict(xs) xa = np.concatenate((data[:,:4],data[:,5:-1]),axis=1) xa = (xa - np.mean(xa,axis=0))/(np.std(xa,axis=0)+1.e-16) ya, ys2a, fma, fs2a, lpa = model.predict(xa) .. figure:: _images/demoH1.png :align: center :width: 600pt :height: 300pt After hyperparameter optimization, the covariance hyperparameters were :math:`[ 1.17, \; 0.45, \; -1.41 ]` and the likelihood hyperparameter was :math:`-2.27`. The final negative log marginal likelihood (optimized) was :math:`214.46`. .. figure:: _images/demoH2.png :align: center :width: 600pt :height: 300pt .. _UCI Repository: http://archive.ics.uci.edu/ml/datasets/Housing .. [1] T. Suttorp and C. Igel, Approximation of Gaussian process regression models after training. In M. Verleysen (Hrsg.), Proceedings of the 16th European Symposium on Artificial Neural Networks (ESANN 2008) , pp. 427–432 (2008). pyGPs-1.2/doc/source/demoMaunaLoa.rst000077500000000000000000000147501236346020200175670ustar00rootroot00000000000000Regression on Mauna Loa data ---------------------------- This example does regression on the Hawaiian Mauna Loa data (example taken from chapter :math:`5` of the `GPML`_ book by Rasmussen and Williams, 2006) We will use a modelling problem concerning the concentration of :math:`CO_2` in the atmosphere to illustrate how the marginal likelihood can be used to set multiple hyperparameters in hierarchical Gaussian process models. A complex covariance function is derived by combining several different kinds of simple covariance functions, and the resulting model provides an excellent fit to the data as well as insight into its properties by interpretation of the adapted hyperparameters. Although the data is one-dimensional, and therefore easy to visualize, a total of :math:`11` hyperparameters are used, which in practice rules out the use of cross-validation for setting parameters, except for the gradient-based LOO-CV procedure. The data [1]_ consists of monthly average atmospheric :math:`CO_2` concentrations (in parts per million by volume (ppmv)) derived from *in-situ* air samples collected at the Mauna Loa Observatory, Hawaii, between :math:`1958` and :math:`2003` (with some missing values) `[2]`_. .. figure:: _images/demoML1.png :height: 600 px :width: 800 px :align: center :scale: 70 % The data is shown in the above plot. Our goal is to model the :math:`CO_2` concentration as a function of time :math:`t`. Several features are immediately apparent: a long term rising trend, a pronounced seasonal variation and some smaller irregularities. In the following, contributions to a combined covariance function which takes care of these individual properties are suggessted. This is meant primarily to illustrate the power and flexibility of the Gaussian process framework—it is possible that other choices would be more appropriate for this data set. To model the long term smooth rising trend, a squared exponential (SE) covariance term with two hyperparameters controlling the amplitude :math:`\theta_1` and characteristic length-scale :math:`\theta_2` is used: .. math:: k_1(x,x') = \theta_1^2 \exp \left(-\frac{(x-x')^2}{2\theta_2^2}\right). Note that we just use a smooth trend; actually enforcing the trend *a priori* to be increasing is probably not so simple and (hopefully) not desirable. We can use the periodic covariance function with a period of one year to model the seasonal variation. However, it is not clear that the seasonal trend is exactly periodic, so we modify it by taking the product with a squared exponential component to allow a decay away from exact periodicity: .. math:: k_2(x,x') = \theta_3^2 \exp\left(-\frac{(x-x')^2}{2\theta_4^2} \frac{2\sin^2(\pi(x-x'))}{\theta_5^2}\right). where :math:`\theta_3` gives the magnitude, :math:`\theta_4` the decay-time for the periodic component, and :math:`\theta_5` the smoothness of the periodic component; the period has been fixed to one (year). The seasonal component in the data is caused primarily by different rates of :math:`CO_2` uptake for plants depending on the season, and it is probably reasonable to assume that this pattern may itself change slowly over time, partially due to the elevation of the :math:`CO_2` level itself; if this effect turns out not to be relevant, then it can be effectively removed at the fitting stage by allowing :math:`\theta_4` to become very large. To model the (small) medium term irregularities, a rational quadratic term is used: .. math:: k_3(x,x') = \theta_6^2\left(1+\frac{(x-x')^2}{2\theta_8\theta_7^2}\right)^{\theta_8}. where :math:`\theta_6` is the magnitude, :math:`\theta_7` is the typical length-scale and :math:`\theta_8` is the shape parameter determining diffuseness of the length-scales. One could also have used a squared exponential form for this component, but it turns out that the rational quadratic works better (gives higher marginal likelihood), probably because it can accommodate several length-scales simultaneously. Finally we specify a noise model as the sum of a squared exponential contrubition and an independent component: .. math:: k_4(x_p,x_q) = \theta_9^2\exp\left(-\frac{(x_p - x_q)^2}{2\theta_{10}^2}\right) + \theta_{11}^2\delta_{pq}. where :math:`\theta_9` is the magnitude of the correlated noise component, :math:`\theta_{10}` is its length scale and :math:`\theta_{11}` is the magnitude of the independent noise component. Noise in the series could be caused by measurement inaccuracies, and by local short-term weather phenomena, so it is probably reasonable to assume at least a modest amount of correlation in time. Notice that the correlated noise component, the first term has an identical expression to the long term component in the trend covariance. When optimizing the hyperparameters, we will see that one of these components becomes large with a long length-scale (the long term trend), while the other remains small with a short length-scale (noise). The fact that we have chosen to call one of these components ‘signal’ and the other one ‘noise’ is only a question of interpretation. Presumably, we are less interested in very short-term effect, and thus call it noise; if on the other hand we were interested in this effect, we would call it signal. The final covariance function is: .. math:: k(x,x') = k_1(x,x') + k_2(x,x') + k_3(x,x') + k_4(x,x') with hyperparameters :math:`\theta = (\theta_1,\ldots,\theta_{11})` :: # DEFINE parameterized covariance function k1 = pyGPs.cov.RBF(np.log(67.), np.log(66.)) k2 = pyGPs.cov.Periodic(np.log(1.3), np.log(1.0), np.log(2.4)) * cov.RBF(np.log(90.), np.log(2.4)) k3 = pyGPs.cov.RQ(np.log(1.2), np.log(0.66), np.log(0.78)) k4 = pyGPs.cov.RBF(np.log(1.6/12.), np.log(0.18)) + cov.Noise(np.log(0.19)) k = k1 + k2 + k3 + k4 After running the minimization, :: t0 = clock() model.optimize(x,y) t1 = clock() model.predict(xs) The extrapolated data looks like: .. figure:: _images/demoML2.png :height: 600 px :width: 800 px :align: center :scale: 70 % and the optimized values of the hyperparameters allow for a principled analysis of different components driving the model. .. [1] Keeling, C. D. and Whorf, T. P. (2004). Atmospheric :math:`CO_2` Records from Sites in the SIO Air Sampling Network. In Trends: A Compendium of Data on Global Change. Carbon Dioxide Information Analysis Center, Oak Ridge National Laboratory, Oak Ridge, Tenn., U.S.A. .. _[2]: http://cdiac.esd.ornl.gov/ftp/trends/co2/maunaloa.co2 .. _GPML: http://www.gaussianprocess.org/gpml/chapters/RW5.pdf pyGPs-1.2/doc/source/index.rst000066400000000000000000000047321236346020200163300ustar00rootroot00000000000000.. Object-Oriented Gaussian Processes documentation master file, created by sphinx-quickstart on Thu Nov 14 19:54:52 2013. pyGPs - A Package for Gaussian Processes =========================================== About the package --------------------- pyGPs is a library hosting Python implementations of Gaussian processes (GPs) for machine learning. pyGPs bridges the gap between systems designed primarily for users, who mainly want to apply GPs and need basic machine learning routines for model training, evaluation, and visualiztion, and expressive systems for developers, who focus on extending the core functionalities as covariance and likelihood functions, as well as inference techniques. The software package is released under the BSD 2-Clause (FreeBSD) License. :doc:`Copyright <_static/copyright>` (c) by Marion Neumann, Shan Huang, Daniel Marthaler, & Kristian Kersting, Feb.2014 Further, it includes implementations of * minimize.py implemented in python by Roland Memisevic 2008, following minimize.m (Copyright (c) Carl Edward Rasmussen (1999-2006)) * scg.py (Copyright (c) Ian T Nabney (1996-2001)) * brentmin.py (Copyright (c) Hannes Nickisch 2010-01-10) * FITC functionality (following matlab implementations under Copyright (c) by Ed Snelson, Carl Edward Rasmussen and Hannes Nickisch, 2011-11-02) The most recent stable release is pyGPs v1.2. See :doc:`changelog <_static/changelog>` for a list of changes to the previous release. If you observe problems or bugs, please contact us. You can download the developer's guide and manual containing an API here: :download:`API and manual <_static/pyGPs_API.pdf>`. You can also download a `procedual implementation`_ of GP functionality from Github. However, the procedual version will not be supported in future. .. _procedual implementation: https://github.com/marionmari/pyGP_PR/ Authors: - Marion Neumann [marion dot neumann at uni-bonn dot de] - Shan Huang [shan dot huang at iais dot fraunhofer dot de] - Daniel Marthaler [dan dot marthaler at gmail dot com] - Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] The following persons helped to improve this software: Roman Garnett, Maciej Kurek, Hannes Nickisch, Zhao Xu, and Alejandro Molina. This work is partly supported by the Fraunhofer ATTRACT fellowship STREAM. Getting started ------------------------------ .. toctree:: :maxdepth: 1 Install Theory Examples Kernels Likelihoods Opts Graph pyGPs-1.2/pyGPs/000077500000000000000000000000001236346020200134565ustar00rootroot00000000000000pyGPs-1.2/pyGPs/Core/000077500000000000000000000000001236346020200143465ustar00rootroot00000000000000pyGPs-1.2/pyGPs/Core/__init__.py000066400000000000000000000001701236346020200164550ustar00rootroot00000000000000import inf import cov import mean import lik import gp import opt __all__ = ['inf', 'cov', 'mean', 'lik','gp','opt'] pyGPs-1.2/pyGPs/Core/cov.py000066400000000000000000001313031236346020200155100ustar00rootroot00000000000000#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ # covariance functions to be use by Gaussian process functions. There are two # different kinds of covariance functions: simple and composite: # # simple cov functions: # # Poly - Polynomial covariance function # PiecePoly - Piecewise polynomial kernel with compact support. # RBF - Squared Exponential kernel. # RBFunit - Squared Exponential kernel with unit magnitude. # RBFard - Squared Exponential kernel with Automatic Relevance Determination. # Const - Constant kernel. # Linear - Linear kernel. # LINard - Linear covariance function with Automatic Relevance Detemination. # Matern - Matern covariance function. # Periodic - Stationary kernel for a smooth periodic function. # Noise - Independent covariance function, i.e "white noise". # RQ - Rational Quadratic covariance function with isotropic distance measure. # RQard - Rational Quadratic covariance function with ARD distance measure. # Pre - Precomputed kernel matrix. # # composite covariance functions: # # ScaleOfKernel - scaled version of a covariance function # ProductOfKernel - products of covariance functions # SumOfKernel - sums of covariance functions # FITCOfKernel - Covariance function to be used together with the FITC approximation # # # This is a object-oriented python implementation of gpml functionality # (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18). # based on the functional-version of python implementation # (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013) # # Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013 import numpy as np import math import scipy.spatial.distance as spdist class Kernel(object): """ This is a base class of Kernel functions there is no computation in this class, it just defines rules about a kernel class should have each covariance function will inherit it and implement its own behaviour """ def __init__(self): self.hyp = [] self.para = [] def getCovMatrix(self,x=None,z=None,mode=None): ''' Return the specific covariance matrix according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self covariance matrix of test data(test by 1). 'train' return training covariance matrix(train by train). 'cross' return cross covariance matrix between x and z(train by test) :return: the corresponding covariance matrix ''' pass def getDerMatrix(self,x=None,z=None,mode=None,der=None): ''' Compute derivatives wrt. hyperparameters according to input mode :param x: training data :param z: test data :param str mode: 'self_test' return self derivative matrix of test data(test by 1). 'train' return training derivative matrix(train by train). 'cross' return cross derivative matrix between x and z(train by test) :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' pass # overloading def __add__(self,cov): ''' Overloading + operator. :param cov: covariance function :return: an instance of SumOfKernel ''' return SumOfKernel(self,cov) # overloading def __mul__(self,other): ''' Overloading * operator. Using * for both multiplication with scalar and product of kernels depending on the type of the two objects. :param other: covariance function as product or int/float as scalar :return: an instance of ScaleOfKernel or ProductOfKernel ''' if isinstance(other, int) or isinstance(other, float): return ScaleOfKernel(self,other) elif isinstance(other, Kernel): return ProductOfKernel(self,other) else: print "only numbers and Kernels are supported operand types for *" # overloading __rmul__ = __mul__ def fitc(self,inducingInput): ''' Covariance function to be used together with the FITC approximation. Setting FITC gp model will implicitly call this method. :return: an instance of FITCOfKernel ''' return FITCOfKernel(self,inducingInput) # can be replaced by spdist from scipy def _sq_dist(self, a, b=None): '''Compute a matrix of all pairwise squared distances between two sets of vectors, stored in the row of the two matrices: a (of size n by D) and b (of size m by D).''' n = a.shape[0] D = a.shape[1] m = n if b == None: b = a.transpose() else: m = b.shape[0] b = b.transpose() C = np.zeros((n,m)) for d in range(0,D): tt = a[:,d] tt = tt.reshape(n,1) tem = np.kron(np.ones((1,m)), tt) tem = tem - np.kron(np.ones((n,1)), b[d,:]) C = C + tem * tem return C class ProductOfKernel(Kernel): '''Product of two kernel function.''' def __init__(self,cov1,cov2): self.cov1 = cov1 self.cov2 = cov2 self._hyp = cov1.hyp + cov2.hyp def _setHyp(self,hyp): assert len(hyp) == len(self._hyp) len1 = len(self.cov1.hyp) self._hyp = hyp self.cov1.hyp = self._hyp[:len1] self.cov2.hyp = self._hyp[len1:] def _getHyp(self): return self._hyp hyp = property(_getHyp,_setHyp) def getCovMatrix(self,x=None,z=None,mode=None): A = self.cov1.getCovMatrix(x,z,mode) * self.cov2.getCovMatrix(x,z,mode) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): if der < len(self.cov1.hyp): A = self.cov1.getDerMatrix(x,z,mode,der) * self.cov2.getCovMatrix(x,z,mode) elif der < len(self.hyp): der2 = der - len(self.cov1.hyp) A = self.cov2.getDerMatrix(x,z,mode,der2) * self.cov1.getCovMatrix(x,z,mode) else: raise Exception("Error: der out of range for covProduct") return A class SumOfKernel(Kernel): '''Sum of two kernel function.''' def __init__(self,cov1,cov2): self.cov1 = cov1 self.cov2 = cov2 self._hyp = cov1.hyp + cov2.hyp def _setHyp(self,hyp): assert len(hyp) == len(self._hyp) len1 = len(self.cov1.hyp) self._hyp = hyp self.cov1.hyp = self._hyp[:len1] self.cov2.hyp = self._hyp[len1:] def _getHyp(self): return self._hyp hyp = property(_getHyp,_setHyp) def getCovMatrix(self,x=None,z=None,mode=None): A = self.cov1.getCovMatrix(x,z,mode) + self.cov2.getCovMatrix(x,z,mode) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): if der < len(self.cov1.hyp): A = self.cov1.getDerMatrix(x,z,mode,der) elif der < len(self.hyp): der2 = der - len(self.cov1.hyp) A = self.cov2.getDerMatrix(x,z,mode,der2) else: raise Exception("Error: der out of range for covSum") return A class ScaleOfKernel(Kernel): '''Scale of a kernel function.''' def __init__(self,cov,scalar): self.cov = cov if cov.hyp: self._hyp = [scalar] + cov.hyp else: self._hyp = [scalar] def _setHyp(self,hyp): assert len(hyp) == len(self._hyp) self._hyp = hyp self.cov.hyp = self._hyp[1:] def _getHyp(self): return self._hyp hyp = property(_getHyp,_setHyp) def getCovMatrix(self,x=None,z=None,mode=None): sf2 = np.exp(self.hyp[0]) # scale parameter A = sf2 * self.cov.getCovMatrix(x,z,mode) # accumulate cov return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): sf2 = np.exp(self.hyp[0]) # scale parameter if der == 0: # compute derivative w.r.t. sf2 A = 2. * sf2 * self.cov.getCovMatrix(x,z,mode) else: A = sf2 * self.cov.getDerMatrix(x,z,mode,der-1) return A class FITCOfKernel(Kernel): ''' Covariance function to be used together with the FITC approximation. The function allows for more than one output argument and does not respect the interface of a proper covariance function. Instead of outputing the full covariance, it returns cross-covariances between the inputs x, z and the inducing inputs xu as needed by infFITC ''' def __init__(self,cov,inducingInput): self.inducingInput = inducingInput self.covfunc = cov self._hyp = cov.hyp def _getHyp(self): return self._hyp def _setHyp(self, hyp): self._hyp = hyp self.covfunc.hyp = hyp hyp = property(_getHyp,_setHyp) def getCovMatrix(self,x=None,z=None,mode=None): xu = self.inducingInput if x != None: try: assert(xu.shape[1] == x.shape[1]) except AssertionError: raise Exception('Dimensionality of inducing inputs must match training inputs') if mode == 'self_test': # self covariances for the test cases K = self.covfunc.getCovMatrix(z=z,mode='self_test') return K elif mode == 'train': # compute covariance matix for training set K = self.covfunc.getCovMatrix(z=x,mode='self_test') Kuu = self.covfunc.getCovMatrix(x=xu,mode='train') Ku = self.covfunc.getCovMatrix(x=xu,z=x,mode='cross') return K, Kuu, Ku elif mode == 'cross': # compute covariance between data sets x and z K = self.covfunc.getCovMatrix(x=xu,z=z,mode='cross') return K def getDerMatrix(self,x=None,z=None,mode=None,der=None): xu = self.inducingInput if x!= None: try: assert(xu.shape[1] == x.shape[1]) except AssertionError: raise Exception('Dimensionality of inducing inputs must match training inputs') if mode == 'self_test': # self covariances for the test cases K = self.covfunc.getDerMatrix(z=z,mode='self_test',der=der) return K elif mode == 'train': # compute covariance matix for training set K = self.covfunc.getDerMatrix(z=x,mode='self_test',der=der) Kuu = self.covfunc.getDerMatrix(x=xu,mode='train',der=der) Ku = self.covfunc.getDerMatrix(x=xu,z=x,mode='cross',der=der) return K, Kuu, Ku elif mode == 'cross': # compute covariance between data sets x and z K = self.covfunc.getDerMatrix(x=xu,z=z,mode='cross',der=der) return K class Poly(Kernel): ''' Polynomial covariance function. hyp = [ log_c, log_sigma ] :param log_c: inhomogeneous offset. :param log_sigma: signal deviation. :param d: degree of polynomial (not treated as hyperparameter, i.e. will not be trained). ''' def __init__(self, log_c=0., d=2, log_sigma=0. ): self.hyp = [log_c, log_sigma] self.para = [d] def getCovMatrix(self,x=None,z=None,mode=None): c = np.exp(self.hyp[0]) # inhomogeneous offset sf2 = np.exp(2.*self.hyp[1]) # signal variance ord = self.para[0] # order of polynomial if np.abs(ord-np.round(ord)) < 1e-8: # remove numerical error from format of parameter ord = int(round(ord)) assert(ord >= 1.) # only nonzero integers for ord ord = int(ord) if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.reshape(np.sum(z*z,1), (nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.dot(x,x.T) elif mode == 'cross': # compute covariance between data sets x and z A = np.dot(x,z.T) A = sf2 * (c + A)**ord return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): c = np.exp(self.hyp[0]) # inhomogeneous offset sf2 = np.exp(2.*self.hyp[1]) # signal variance ord = self.para[0] # order of polynomial if np.abs(ord-np.round(ord)) < 1e-8: # remove numerical error from format of parameter ord = int(round(ord)) assert(ord >= 1.) # only nonzero integers for ord ord = int(ord) if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.reshape(np.sum(z*z,1), (nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.dot(x,x.T) elif mode == 'cross': # compute covariance between data sets x and z A = np.dot(x,z.T) if der == 0: # compute derivative matrix wrt 1st parameter A = c * ord * sf2 * (c+A)**(ord-1) elif der == 1: # compute derivative matrix wrt 2nd parameter A = 2. * sf2 * (c + A)**ord elif der == 2: # no derivative wrt 3rd parameter A = np.zeros_like(A) # do nothing (d is not learned) else: raise Exception("Wrong derivative entry in Poly") return A class PiecePoly(Kernel): ''' Piecewise polynomial kernel with compact support. hyp = [log_ell, log_sigma] :param log_ell: characteristic length scale. :param log_sigma: signal deviation. :param v: degree v will be rounded to 0,1,2,or 3. (not treated as hyperparameter, i.e. will not be trained). ''' def __init__(self, log_ell=0., v=2, log_sigma=0. ): self.hyp = [log_ell, log_sigma] self.para = [v] def ppmax(self,A,B): return np.maximum(A,B*np.ones_like(A)) def func(self,v,r,j): if v == 0: return 1 elif v == 1: return ( 1. + (j+1) * r ) elif v == 2: return ( 1. + (j+2)*r + (j*j + 4.*j+ 3)/3.*r*r ) elif v == 3: return ( 1. + (j+3)*r + (6.*j*j+36.*j+45.)/15.*r*r + (j*j*j+9.*j*j+23.*j+15.)/15.*r*r*r ) else: raise Exception (["Wrong degree in PiecePoly. Should be 0,1,2 or 3, is " + str(v)]) def dfunc(self,v,r,j): if v == 0: return 0 elif v == 1: return ( j+1 ) elif v == 2: return ( (j+2) + 2.*(j*j+ 4.*j+ 3.)/3.*r ) elif v == 3: return ( (j+3) + 2.*(6.*j*j+36.*j+45.)/15.*r + (j*j*j+9.*j*j+23.*j+15.)/5.*r*r ) else: raise Exception (["Wrong degree in PiecePoly. Should be 0,1,2 or 3, is " + str(v)]) def pp(self,r,j,v,func): return func(v,r,j)*(self.ppmax(1-r,0)**(j+v)) def dpp(self,r,j,v,func,dfunc): return self.ppmax(1-r,0)**(j+v-1) * r * ( (j+v)*self.func(v,r,j) - self.ppmax(1-r,0) * self.dfunc(v,r,j) ) def getCovMatrix(self,x=None,z=None,mode=None): if x != None: n, D = x.shape if z != None: nn, D = z.shape ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.*self.hyp[1]) # signal variance v = self.para[0] # degree (v = 0,1,2 or 3 only) if np.abs(v-np.round(v)) < 1e-8: # remove numerical error from format of parameter v = int(round(v)) assert(int(v) in range(4)) # Only allowed degrees: 0,1,2 or 3 v = int(v) j = np.floor(0.5*D) + v + 1 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.sqrt( spdist.cdist(x/ell, x/ell, 'sqeuclidean') ) elif mode == 'cross': # compute covariance between data sets x and z A = np.sqrt( spdist.cdist(x/ell, z/ell, 'sqeuclidean') ) A = sf2 * self.pp(A,j,v,self.func) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): if x != None: n, D = x.shape if z != None: nn, D = z.shape ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.*self.hyp[1]) # signal variance v = self.para[0] # degree (v = 0,1,2 or 3 only) if np.abs(v-np.round(v)) < 1e-8: # remove numerical error from format of parameter v = int(round(v)) assert(int(v) in range(4)) # Only allowed degrees: 0,1,2 or 3 v = int(v) j = np.floor(0.5*D) + v + 1 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.sqrt( spdist.cdist(x/ell, x/ell, 'sqeuclidean') ) elif mode == 'cross': # compute covariance between data sets x and z A = np.sqrt( spdist.cdist(x/ell, z/ell, 'sqeuclidean') ) if der == 0: # compute derivative matrix wrt 1st parameter A = sf2 * self.dpp(A,j,v,self.func,self.dfunc) elif der == 1: # compute derivative matrix wrt 2nd parameter A = 2. * sf2 * self.pp(A,j,v,self.func) elif der == 2: # wants to compute derivative wrt order A = np.zeros_like(A) else: raise Exception("Wrong derivative entry in PiecePoly") return A class RBF(Kernel): ''' Squared Exponential kernel with isotropic distance measure. hyp = [log_ell, log_sigma] :param log_ell: characteristic length scale. :param log_sigma: signal deviation. ''' def __init__(self, log_ell=0., log_sigma=0.): self.hyp = [log_ell, log_sigma] def getCovMatrix(self,x=None,z=None,mode=None): ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.*self.hyp[1]) # signal variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for training set A = spdist.cdist(x/ell,x/ell,'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z A = spdist.cdist(x/ell,z/ell,'sqeuclidean') A = sf2 * np.exp(-0.5*A) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.*self.hyp[1]) # signal variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = spdist.cdist(x/ell,x/ell,'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z A = spdist.cdist(x/ell,z/ell,'sqeuclidean') if der == 0: # compute derivative matrix wrt 1st parameter A = sf2 * np.exp(-0.5*A) * A elif der == 1: # compute derivative matrix wrt 2nd parameter A = 2. * sf2 * np.exp(-0.5*A) else: raise Exception("Calling for a derivative in RBF that does not exist") return A class RBFunit(Kernel): ''' Squared Exponential kernel with isotropic distance measure with unit magnitude. i.e signal variance is always 1. hyp = [ log_ell ] :param log_ell: characteristic length scale. ''' def __init__(self, log_ell=0.): self.hyp = [log_ell] def getCovMatrix(self,x=None,z=None,mode=None): ell = np.exp(self.hyp[0]) # characteristic length scale if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = spdist.cdist(x/ell,x/ell,'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z A = spdist.cdist(x/ell,z/ell,'sqeuclidean') A = np.exp(-0.5*A) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): ell = np.exp(self.hyp[0]) # characteristic length scale if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = spdist.cdist(x/ell,x/ell,'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z A = spdist.cdist(x/ell,z/ell,'sqeuclidean') if der == 0: # compute derivative matrix wrt 1st parameter A = np.exp(-0.5*A) * A else: raise Exception("Wrong derivative index in RDFunit") return A class RBFard(Kernel): ''' Squared Exponential kernel with Automatic Relevance Determination. hyp = log_ell_list + [log_sigma] :param D: dimension of pattern. set if you want default ell, which is 1 for each dimension. :param log_ell_list: characteristic length scale for each dimension. :param log_sigma: signal deviation. ''' def __init__(self, D=None, log_ell_list=None, log_sigma=0.): if log_ell_list == None: self.hyp = [0. for i in xrange(D)] + [log_sigma] else: self.hyp = log_ell_list + [log_sigma] def getCovMatrix(self,x=None,z=None,mode=None): if x != None: n, D = x.shape if z != None: nn, D = z.shape ell = 1./np.exp(self.hyp[0:D]) # characteristic length scale sf2 = np.exp(2.*self.hyp[D]) # signal variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x tem = np.dot(np.diag(ell),x.T).T A = spdist.cdist(tem,tem,'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z A = spdist.cdist(np.dot(np.diag(ell),x.T).T,np.dot(np.diag(ell),z.T).T,'sqeuclidean') A = sf2*np.exp(-0.5*A) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): if x != None: n, D = x.shape if z != None: nn, D = z.shape ell = 1./np.exp(self.hyp[0:D]) # characteristic length scale sf2 = np.exp(2.*self.hyp[D]) # signal variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x tem = np.dot(np.diag(ell),x.T).T A = spdist.cdist(tem,tem,'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z A = spdist.cdist(np.dot(np.diag(ell),x.T).T,np.dot(np.diag(ell),z.T).T,'sqeuclidean') A = sf2*np.exp(-0.5*A) if der < D: # compute derivative matrix wrt length scale parameters if mode == 'self_test': A = A*0 elif mode == 'train': tem = np.atleast_2d(x[:,der])/ell[der] A *= spdist.cdist(tem,tem,'sqeuclidean') elif mode == 'cross': A *= spdist.cdist(np.atleast_2d(x[:,der]).T/ell[der],np.atleast_2d(z[:,der]).T/ell[der],'sqeuclidean') elif der == D: # compute derivative matrix wrt magnitude parameter A = 2.*A else: raise Exception("Wrong derivative index in RDFard") return A class Const(Kernel): ''' Constant kernel. hyp = [ log_sigma ] :param log_sigma: signal deviation. ''' def __init__(self, log_sigma=0.): self.hyp = [log_sigma] def getCovMatrix(self,x=None,z=None,mode=None): sf2 = np.exp(self.hyp[0]) # s2 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = sf2 * np.ones((nn,1)) elif mode == 'train': # compute covariance matix for dataset x n,D = x.shape A = sf2 * np.ones((n,n)) elif mode == 'cross': # compute covariance between data sets x and z n,D = x.shape nn,D = z.shape A = sf2 * np.ones((n,nn)) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): sf2 = np.exp(self.hyp[0]) # s2 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = sf2 * np.ones((nn,1)) elif mode == 'train': # compute covariance matix for dataset x n,D = x.shape A = sf2 * np.ones((n,n)) elif mode == 'cross': # compute covariance between data sets x and z n,D = x.shape nn,D = z.shape A = sf2 * np.ones((n,nn)) if der == 0: # compute derivative matrix wrt sf2 A = 2. * A else: raise Exception("Wrong derivative entry in covConst") return A class Linear(Kernel): ''' Linear kernel. hyp = [ log_sigma ]. :param log_sigma: signal deviation. ''' def __init__(self, log_sigma=0.): self.hyp = [ log_sigma ] def getCovMatrix(self,x=None,z=None,mode=None): sf2 = np.exp(self.hyp[0]) # s2 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.reshape(np.sum(z*z,1), (nn,1)) elif mode == 'train': # compute covariance matix for dataset x n,D = x.shape A = np.dot(x,x.T) + np.eye(n)*1e-16 # required for numerical accuracy elif mode == 'cross': # compute covariance between data sets x and z A = np.dot(x,z.T) A = sf2 * A return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): sf2 = np.exp(self.hyp[0]) # s2 if der == 0: if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.reshape(np.sum(z*z,1), (nn,1)) elif mode == 'train': # compute covariance matix for dataset x n,D = x.shape A = np.dot(x,x.T) + np.eye(n)*1e-16 # required for numerical accuracy elif mode == 'cross': # compute covariance between data sets x and z A = np.dot(x,z.T) A = 2 * sf2 * A else: raise Exception("Wrong derivative index in covLinear") return A class LINard(Kernel): ''' Linear covariance function with Automatic Relevance Detemination. hyp = log_ell_list :param D: dimension of training data. Set if you want default ell, which is 1 for each dimension. :param log_ell_list: characteristic length scale for each dimension. ''' def __init__(self, D=None, log_ell_list=None): if log_ell_list == None: self.hyp = [0. for i in xrange(D)] else: self.hyp = log_ell_list def getCovMatrix(self,x=None,z=None,mode=None): ell = np.exp(self.hyp) # ARD parameters if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.reshape(np.sum(z*z,1), (nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.dot(x,x.T) elif mode == 'cross': # compute covariance between data sets x and z z = np.dot(z,np.diag(1./ell)) A = np.dot(x,z.T) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): ell = np.exp(self.hyp) # ARD parameters if x != None: n, D = x.shape if z != None: nn, D = z.shape if der < D: if mode == 'self_test': tem = np.atleast_2d(z[:,der]).T A = -2.* tem * tem elif mode == 'train': A = -2.*np.dot(np.atleast_2d(x[:,der]).T,np.atleast_2d(x[:,der])) elif mode == 'cross': z = np.dot(z,np.diag(1./ell)) A = -2.*np.dot(np.atleast_2d(x[:,der]).T, np.atleast_2d(z[:,der])) # cross covariances else: raise Exception("Wrong derivative index in covLINard") return A class Matern(Kernel): ''' Matern covariance function with nu = d/2 and isotropic distance measure. For d=1 the function is also known as the exponential covariance function or the Ornstein-Uhlenbeck covariance in 1d. d will be rounded to 1, 3, or 5. hyp = [ log_ell, log_sigma] :param d: d is 2 times nu. Can only be 1,3 or 5. :param log_ell: characteristic length scale. :param log_sigma: signal deviation. ''' def __init__(self, log_ell=0., d=3, log_sigma=0. ): self.hyp = [ log_ell, log_sigma ] self.para = [d] def func(self,d,t): if d == 1: return 1 elif d == 3: return 1 + t elif d == 5: return 1 + t*(1+t/3.) else: raise Exception("Wrong value for d in Matern") def dfunc(self,d,t): if d == 1: return 1 elif d == 3: return t elif d == 5: return t*(1+t/3.) else: raise Exception("Wrong value for d in Matern") def mfunc(self,d,t): return self.func(d,t)*np.exp(-1.*t) def dmfunc(self,d,t): return self.dfunc(d,t)*t*np.exp(-1.*t) def getCovMatrix(self,x=None,z=None,mode=None): ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.* self.hyp[1]) # signal variance d = self.para[0] # 2 times nu if np.abs(d-np.round(d)) < 1e-8: # remove numerical error from format of parameter d = int(round(d)) d = int(d) try: assert(d in [1,3,5]) # check for valid values of d except AssertionError: print "Warning: You specified d to be neither 1,3 nor 5. We set to d=3. " d = 3 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x x = np.sqrt(d)*x/ell A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean')) elif mode == 'cross': # compute covariance between data sets x and z x = np.sqrt(d)*x/ell z = np.sqrt(d)*z/ell A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean')) A = sf2 * self.mfunc(d,A) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.* self.hyp[1]) # signal variance d = self.para[0] # 2 times nu if np.abs(d-np.round(d)) < 1e-8: # remove numerical error from format of parameter d = int(round(d)) d = int(d) try: assert(d in [1,3,5]) # check for valid values of d except AssertionError: print "Warning: You specified d to be neither 1,3 nor 5. We set to d=3. " d = 3 if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x x = np.sqrt(d)*x/ell A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean')) elif mode == 'cross': # compute covariance between data sets x and z x = np.sqrt(d)*x/ell z = np.sqrt(d)*z/ell A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean')) A = sf2 * self.mfunc(d,A) if der == 0: # compute derivative matrix wrt 1st parameter A = sf2 * self.dmfunc(d,A) elif der == 1: # compute derivative matrix wrt 2nd parameter A = 2 * sf2 * self.mfunc(d,A) elif der == 2: # no derivative wrt 3rd parameter A = np.zeros_like(A) # do nothing (d is not learned) else: raise Exception("Wrong derivative value in Matern") return A class Periodic(Kernel): ''' Stationary kernel for a smooth periodic function. hyp = [ log_ell, log_p, log_sigma] :param log_p: period. :param log_ell: characteristic length scale. :param log_sigma: signal deviation. ''' def __init__(self, log_ell=0., log_p=0., log_sigma=0. ): self.hyp = [ log_ell, log_p, log_sigma] def getCovMatrix(self,x=None,z=None,mode=None): ell = np.exp(self.hyp[0]) # characteristic length scale p = np.exp(self.hyp[1]) # period sf2 = np.exp(2.*self.hyp[2]) # signal variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean')) elif mode == 'cross': # compute covariance between data sets x and z A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean')) A = np.pi*A/p A = np.sin(A)/ell A = A * A A = sf2 *np.exp(-2.*A) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): ell = np.exp(self.hyp[0]) # characteristic length scale p = np.exp(self.hyp[1]) # period sf2 = np.exp(2.*self.hyp[2]) # signal variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x A = np.sqrt(spdist.cdist(x, x, 'sqeuclidean')) elif mode == 'cross': # compute covariance between data sets x and z A = np.sqrt(spdist.cdist(x, z, 'sqeuclidean')) A = np.pi*A/p if der == 0: # compute derivative matrix wrt 1st parameter A = np.sin(A)/ell A = A * A A = 4. *sf2 *np.exp(-2.*A) * A elif der == 1: # compute derivative matrix wrt 2nd parameter R = np.sin(A)/ell A = 4 * sf2/ell * np.exp(-2.*R*R)*R*np.cos(A)*A elif der == 2: # compute derivative matrix wrt 3rd parameter A = np.sin(A)/ell A = A * A A = 2. * sf2 * np.exp(-2.*A) else: raise Exception("Wrong derivative index in covPeriodic") return A class Noise(Kernel): ''' Independent covariance function, i.e "white noise", with specified variance. Normally NOT used anymore since noise is now added in liklihood. hyp = [ log_sigma ] :param log_sigma: signal deviation. ''' def __init__(self, log_sigma=0.): self.hyp = [log_sigma] def getCovMatrix(self,x=None,z=None,mode=None): tol = 1.e-9 # Tolerance for declaring two vectors "equal" s2 = np.exp(2.*self.hyp[0]) # noise variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x n,D = x.shape A = np.eye(n) elif mode == 'cross': # compute covariance between data sets x and z M = spdist.cdist(x, z, 'sqeuclidean') A = np.zeros_like(M,dtype=np.float) A[M < tol] = 1. A = s2*A return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): tol = 1.e-9 # Tolerance for declaring two vectors "equal" s2 = np.exp(2.*self.hyp[0]) # noise variance if mode == 'self_test': # self covariances for the test cases nn,D = z.shape A = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x n,D = x.shape A = np.eye(n) elif mode == 'cross': # compute covariance between data sets x and z M = spdist.cdist(x, z, 'sqeuclidean') A = np.zeros_like(M,dtype=np.float) A[M < tol] = 1. if der == 0: A = 2.*s2*A else: raise Exception("Wrong derivative index in covNoise") return A class RQ(Kernel): ''' Rational Quadratic covariance function with isotropic distance measure. hyp = [ log_ell, log_sigma, log_alpha ] :param log_ell: characteristic length scale. :param log_sigma: signal deviation. :param log_alpha: shape parameter for the RQ covariance. ''' def __init__(self, log_ell=0., log_sigma=0., log_alpha=0.): self.hyp = [ log_ell, log_sigma, log_alpha ] def getCovMatrix(self,x=None,z=None,mode=None): ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.*self.hyp[1]) # signal variance alpha = np.exp(self.hyp[2]) if mode == 'self_test': # self covariances for the test cases nn,D = z.shape D2 = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x D2 = spdist.cdist(x/ell, x/ell, 'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z D2 = spdist.cdist(x/ell, z/ell, 'sqeuclidean') A = sf2 * ( ( 1.0 + 0.5*D2/alpha )**(-alpha) ) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): ell = np.exp(self.hyp[0]) # characteristic length scale sf2 = np.exp(2.*self.hyp[1]) # signal variance alpha = np.exp(self.hyp[2]) if mode == 'self_test': # self covariances for the test cases nn,D = z.shape D2 = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x D2 = spdist.cdist(x/ell, x/ell, 'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z D2 = spdist.cdist(x/ell, z/ell, 'sqeuclidean') if der == 0: # compute derivative matrix wrt 1st parameter A = sf2 * ( 1.0 + 0.5*D2/alpha )**(-alpha-1) * D2 elif der == 1: # compute derivative matrix wrt 2nd parameter A = 2.* sf2 * ( ( 1.0 + 0.5*D2/alpha )**(-alpha) ) elif der == 2: # compute derivative matrix wrt 3rd parameter K = ( 1.0 + 0.5*D2/alpha ) A = sf2 * K**(-alpha) * (0.5*D2/K - alpha*np.log(K) ) else: raise Exception("Wrong derivative index in covRQ") return A class RQard(Kernel): ''' Rational Quadratic covariance function with Automatic Relevance Detemination (ARD) distance measure. hyp = log_ell_list + [ log_sigma, log_alpha ] :param D: dimension of pattern. set if you want default ell, which is 0.5 for each dimension. :param log_ell_list: characteristic length scale for each dimension. :param log_sigma: signal deviation. :param log_alpha: shape parameter for the RQ covariance. ''' def __init__(self, D=None, log_ell_list=None, log_sigma=0., log_alpha=0.): if log_ell_list == None: self.hyp = [0. for i in xrange(D)] + [ log_sigma, log_alpha ] else: self.hyp = log_ell_list + [ log_sigma, log_alpha ] def getCovMatrix(self,x=None,z=None,mode=None): if x != None: n, D = x.shape if z != None: nn, D = z.shape ell = 1./np.exp(self.hyp[0:D]) # characteristic length scale sf2 = np.exp(2.*self.hyp[D]) # signal variance alpha = np.exp(self.hyp[D+1]) if mode == 'self_test': # self covariances for the test cases nn,D = z.shape D2 = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x tmp = np.dot(np.diag(ell),x.T).T D2 = spdist.cdist(tmp, tmp, 'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z D2 = spdist.cdist(np.dot(np.diag(ell),x.T).T, np.dot(np.diag(ell),z.T).T, 'sqeuclidean') A = sf2 * ( ( 1.0 + 0.5*D2/alpha )**(-alpha) ) return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): if x != None: n, D = x.shape if z != None: nn, D = z.shape ell = 1./np.exp(self.hyp[0:D]) # characteristic length scale sf2 = np.exp(2.*self.hyp[D]) # signal variance alpha = np.exp(self.hyp[D+1]) if mode == 'self_test': # self covariances for the test cases nn,D = z.shape D2 = np.zeros((nn,1)) elif mode == 'train': # compute covariance matix for dataset x tmp = np.dot(np.diag(ell),x.T).T D2 = spdist.cdist(tmp, tmp, 'sqeuclidean') elif mode == 'cross': # compute covariance between data sets x and z D2 = spdist.cdist(np.dot(np.diag(ell),x.T).T, np.dot(np.diag(ell),z.T).T, 'sqeuclidean') if der < D: if mode == 'self_test': A = D2*0 elif mode == 'train': tmp = np.atleast_2d(x[:,der])/ell[der] A = sf2 * ( 1.0 + 0.5*D2/alpha )**(-alpha-1) * spdist.cdist(tmp, tmp, 'sqeuclidean') elif mode == 'cross': A = sf2 * ( 1.0 + 0.5*D2/alpha )**(-alpha-1) * spdist.cdist(np.atleast_2d(x[:,der]).T/ell[der], np.atleast_2d(z[:,der]).T/ell[der], 'sqeuclidean') elif der==D: # compute derivative matrix wrt magnitude parameter A = 2. * sf2 * ( ( 1.0 + 0.5*D2/alpha )**(-alpha) ) elif der==(D+1): # compute derivative matrix wrt magnitude parameter K = ( 1.0 + 0.5*D2/alpha ) A = sf2 * K**(-alpha) * ( 0.5*D2/K - alpha*np.log(K) ) else: raise Exception("Wrong derivative index in covRQard") return A class Pre(Kernel): ''' Precomputed kernel matrix. No hyperparameters and thus nothing will be optimised. :param M1: cross covariances matrix(train+1 by test). last row is self covariances (diagonal of test by test) :param M2: training set covariance matrix (train by train) ''' def __init__(self,M1,M2): self.M1 = M1 self.M2 = M2 self.hyp = [] def getCovMatrix(self,x=None,z=None,mode=None): if mode == 'self_test': # diagonal covariance between test_test A = self.M1[-1,:] # self covariances for the test cases (last row) A = np.reshape(A, (A.shape[0],1)) elif mode == 'train': # covariance between train_train A = self.M2 elif mode == 'cross': # covariance between train_test A = self.M1[:-1,:] return A def getDerMatrix(self,x=None,z=None,mode=None,der=None): if der != None: raise Exception("Error: NO optimization in precomputed kernel matrix") return 0 if __name__ == '__main__': pass pyGPs-1.2/pyGPs/Core/gp.py000066400000000000000000001110111236346020200153210ustar00rootroot00000000000000#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ # MEANING OF NOTATION: # # inffunc function specifying the inference method # covfunc prior covariance function (see below) # meanfunc prior mean function # likfunc likelihood function # x n by D matrix of training inputs # y column vector of length n of training targets # xs n by D matrix of test inputs # ys column vector of length nn of true test targets (optional) # nlZ returned value of the negative log marginal likelihood # dnlZ column vector of partial derivatives of the negative # log marginal likelihood w.r.t. each hyperparameter # ym column vector (of length ns) of predictive output means # ys2 column vector (of length ns) of predictive output variances # fm column vector (of length ns) of predictive latent means # fs2 column vector (of length ns) of predictive latent variances # lp column vector (of length ns) of log predictive probabilities # post struct representation of the (approximate) posterior # post consists of post.alpha, post.L, post.sW # # This is a object-oriented python implementation of gpml functionality # (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18). # based on the functional-version of python implementation # (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013) # # Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013 import itertools import numpy as np import matplotlib.pyplot as plt import inf, mean, lik, cov, opt from tools import unique from copy import deepcopy import pyGPs SHADEDCOLOR = [0.7539, 0.89453125, 0.62890625, 1.0] MEANCOLOR = [ 0.2109375, 0.63385, 0.1796875, 1.0] DATACOLOR = [0.12109375, 0.46875, 1., 1.0] class GP(object): '''Base class for GP model''' def __init__(self): super(GP, self).__init__() self.usingDefaultMean = True # was using default mean function now? self.meanfunc = None self.covfunc = None self.likfunc = None self.inffunc = None self.optimizer = None self.nlZ = None # negative log marginal likelihood self.dnlZ = None # column vector of partial derivatives of the negative # log marginal likelihood w.r.t. each hyperparameter self.posterior = None # struct representation of the (approximate) posterior self.x = None # n by D matrix of training inputs self.y = None # column vector of length n of training targets self.xs = None # n by D matrix of test inputs self.ys = None # column vector of length nn of true test targets (optional) self.ym = None # column vector (of length ns) of predictive output means self.ys2 = None # column vector (of length ns) of predictive output variances self.fm = None # column vector (of length ns) of predictive latent means self.fs2 = None # column vector (of length ns) of predictive latent variances self.lp = None # column vector (of length ns) of log predictive probabilities def setData(self, x, y): '''Pass training data and traning labels to model.''' if x.ndim == 1: x = np.reshape(x, (x.shape[0],1)) if y.ndim == 1: y = np.reshape(y, (y.shape[0],1)) self.x = x self.y = y if self.usingDefaultMean: c = np.mean(y) self.meanfunc = mean.Const(c) # adapt default prior mean wrt. training labels def plotData_1d(self, axisvals=None): '''Plot 1d (toy) data.''' plt.figure() plt.plot(self.x, self.y, ls='None', marker='+', color=DATACOLOR, ms=12, mew=2) if axisvals: plt.axis(axisvals) plt.grid() plt.xlabel('input x') plt.ylabel('target y') plt.show() def plotData_2d(self,x1,x2,t1,t2,p1,p2,axisvals=None): '''Plot 2d (toy) data.''' fig = plt.figure() plt.plot(x1[:,0], x1[:,1], 'b+', markersize = 12) plt.plot(x2[:,0], x2[:,1], 'r+', markersize = 12) pc = plt.contour(t1, t2, np.reshape(p2/(p1+p2), (t1.shape[0],t1.shape[1]) )) fig.colorbar(pc) plt.grid() if axisvals: plt.axis(axisvals) plt.show() def setPrior(self, mean=None, kernel=None): '''Set prior mean and cov''' if mean != None: self.meanfunc = mean self.usingDefaultMean = False if kernel != None: self.covfunc = kernel if type(kernel) is cov.Pre: self.usingDefaultMean = False def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None): '''This method is used to sepecify optimization configuration. By default, gp uses a single run "minimize". :param method: Optimization methods. Possible values are: "Minimize" -> minimize by Carl Rasmussen (python implementation of "minimize" in GPML) "CG" -> conjugent gradient "BFGS" -> quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS) "SCG" -> scaled conjugent gradient (faster than CG) :param num_restarts: Set if you want to run mulitiple times of optimization with different initial guess. It specifys the maximum number of runs/restarts/trials. :param min_threshold: Set if you want to run mulitiple times of optimization with different initial guess. It specifys the threshold of objective function value. Stop optimization when this value is reached. :param meanRange: The range of initial guess for mean hyperparameters. e.g. meanRange = [(-2,2), (-5,5), (0,1)]. Each tuple specifys the range (low, high) of this hyperparameter, This is only the range of initial guess, during optimization process, optimal hyperparameters may go out of this range. (-5,5) for each hyperparameter by default. :param covRange: The range of initial guess for kernel hyperparameters. Usage see meanRange :param likRange: The range of initial guess for likelihood hyperparameters. Usage see meanRange ''' pass def optimize(self, x=None, y=None): ''' Train optimal hyperparameters based on training data, adjust new hyperparameters to all mean/cov/lik functions ''' if x != None: if x.ndim == 1: x = np.reshape(x, (x.shape[0],1)) self.x = x if y != None: if y.ndim == 1: y = np.reshape(y, (y.shape[0],1)) self.y = y if self.usingDefaultMean and self.meanfunc == None: c = np.mean(y) self.meanfunc = mean.Const(c) # adapt default prior mean wrt. training labels # optimize optimalHyp, optimalNlZ = self.optimizer.findMin(self.x, self.y) self.nlZ= optimalNlZ # apply optimal hyp to all mean/cov/lik functions here self.optimizer._apply_in_objects(optimalHyp) self.fit() def fit(self, x=None, y=None, der=True): ''' Fit the training data. Update negative log marginal likelihood(nlZ), partial derivatives of nlZ w.r.t. each hyperparameter(dnlZ), and struct representation of the (approximate) posterior(post), which consists of post.alpha, post.L, post.sW. ''' if x != None: if x.ndim == 1: x = np.reshape(x, (x.shape[0],1)) self.x = x if y != None: if y.ndim == 1: y = np.reshape(y, (y.shape[0],1)) self.y = y if self.usingDefaultMean and self.meanfunc == None: c = np.mean(y) self.meanfunc = mean.Const(c) # adapt default prior mean wrt. training labels # call inference method if isinstance(self.likfunc, lik.Erf): #or is instance(self.likfunc, lik.Logistic): uy = unique(self.y) ind = ( uy != 1 ) if any( uy[ind] != -1): raise Exception('You attempt classification using labels different from {+1,-1}') if not der: post, nlZ = self.inffunc.proceed(self.meanfunc, self.covfunc, self.likfunc, self.x, self.y, 2) self.nlZ = nlZ self.posterior = deepcopy(post) return nlZ, post else: post, nlZ, dnlZ = self.inffunc.proceed(self.meanfunc, self.covfunc, self.likfunc, self.x, self.y, 3) self.nlZ = nlZ self.dnlZ = deepcopy(dnlZ) self.posterior = deepcopy(post) return nlZ, dnlZ, post def predict(self, xs, ys=None): ''' Prediction according to given inputs. Get predictive output means(ym), predictive output variances(ys2), predictive latent means(fm), predictive latent variances(fs2), log predictive probabilities(lp). :param xs: test input :param ys: test target(optional) :return: ym, ys2, fm, fs2, lp ''' meanfunc = self.meanfunc covfunc = self.covfunc likfunc = self.likfunc inffunc = self.inffunc x = self.x y = self.y if xs.ndim == 1: xs = np.reshape(xs, (xs.shape[0],1)) self.xs = xs if ys != None: if ys.ndim == 1: ys = np.reshape(ys, (ys.shape[0],1)) self.ys = ys if self.posterior == None: self.fit() alpha = self.posterior.alpha L = self.posterior.L sW = self.posterior.sW nz = range(len(alpha[:,0])) # non-sparse representation if L == []: # in case L is not provided, we compute it K = covfunc.getCovMatrix(x=x[nz,:], mode='train') L = np.linalg.cholesky( (np.eye(nz) + np.dot(sW,sW.T)*K).T ) Ltril = np.all( np.tril(L,-1) == 0 ) # is L an upper triangular matrix? ns = xs.shape[0] # number of data points nperbatch = 1000 # number of data points per mini batch nact = 0 # number of already processed test data points ymu = np.zeros((ns,1)) ys2 = np.zeros((ns,1)) fmu = np.zeros((ns,1)) fs2 = np.zeros((ns,1)) lp = np.zeros((ns,1)) while nact<=ns-1: # process minibatches of test cases to save memory id = range(nact,min(nact+nperbatch,ns)) # data points to process kss = covfunc.getCovMatrix(z=xs[id,:], mode='self_test') # self-variances Ks = covfunc.getCovMatrix(x=x[nz,:], z=xs[id,:], mode='cross') # cross-covariances ms = meanfunc.getMean(xs[id,:]) N = (alpha.shape)[1] # number of alphas (usually 1; more in case of sampling) Fmu = np.tile(ms,(1,N)) + np.dot(Ks.T,alpha[nz]) # conditional mean fs|f fmu[id] = np.reshape(Fmu.sum(axis=1)/N,(len(id),1)) # predictive means if Ltril: # L is triangular => use Cholesky parameters (alpha,sW,L) V = np.linalg.solve(L.T,np.tile(sW,(1,len(id)))*Ks) fs2[id] = kss - np.array([(V*V).sum(axis=0)]).T # predictive variances else: # L is not triangular => use alternative parametrization fs2[id] = kss + np.array([(Ks*np.dot(L,Ks)).sum(axis=0)]).T # predictive variances fs2[id] = np.maximum(fs2[id],0) # remove numerical noise i.e. negative variances Fs2 = np.tile(fs2[id],(1,N)) # we have multiple values in case of sampling if ys == None: [Lp, Ymu, Ys2] = likfunc.proceed(None,Fmu[:],Fs2[:],None,None,3) else: [Lp, Ymu, Ys2] = likfunc.proceed(np.tile(ys[id],(1,N)), Fmu[:], Fs2[:],None,None,3) lp[id] = np.reshape( np.reshape(Lp,(np.prod(Lp.shape),N)).sum(axis=1)/N , (len(id),1) ) # log probability; sample averaging ymu[id] = np.reshape( np.reshape(Ymu,(np.prod(Ymu.shape),N)).sum(axis=1)/N ,(len(id),1) ) # predictive mean ys|y and ... ys2[id] = np.reshape( np.reshape(Ys2,(np.prod(Ys2.shape),N)).sum(axis=1)/N , (len(id),1) ) # .. variance nact = id[-1]+1 # set counter to index of next data point self.ym = ymu self.ys2 = ys2 self.lp = lp self.fm = fmu self.fs2 = fs2 if ys == None: return ymu, ys2, fmu, fs2, None else: return ymu, ys2, fmu, fs2, lp def predict_with_posterior(self, post, xs, ys=None): ''' Prediction with provided posterior (i.e. you already have the posterior and thus don't need fitting/training phases) Get predictive output means(ym), predictive output variances(ys2), predictive latent means(fm), predictive latent variances(fs2), log predictive probabilities(lp). :param post: struct representation of posterior :param xs: test input :param ys: test target(optional) :return: ym, ys2, fm, fs2, lp ''' meanfunc = self.meanfunc covfunc = self.covfunc likfunc = self.likfunc inffunc = self.inffunc x = self.x y = self.y if xs.ndim == 1: xs = np.reshape(xs, (xs.shape[0],1)) self.xs = xs if ys != None: if ys.ndim == 1: ys = np.reshape(ys, (ys.shape[0],1)) self.ys = ys self.posterior = deepcopy(post) alpha = post.alpha L = post.L sW = post.sW nz = range(len(alpha[:,0])) # non-sparse representation if L == []: # in case L is not provided, we compute it K = covfunc.getCovMatrix(x=x[nz,:], mode='train') L = np.linalg.cholesky( (np.eye(nz) + np.dot(sW,sW.T)*K).T ) Ltril = np.all( np.tril(L,-1) == 0 ) # is L an upper triangular matrix? ns = xs.shape[0] # number of data points nperbatch = 1000 # number of data points per mini batch nact = 0 # number of already processed test data points ymu = np.zeros((ns,1)) ys2 = np.zeros((ns,1)) fmu = np.zeros((ns,1)) fs2 = np.zeros((ns,1)) lp = np.zeros((ns,1)) while nact<=ns-1: # process minibatches of test cases to save memory id = range(nact,min(nact+nperbatch,ns)) # data points to process kss = covfunc.getCovMatrix(z=xs[id,:], mode='self_test') # self-variances Ks = covfunc.getCovMatrix(x=x[nz,:], z=xs[id,:], mode='cross') # cross-covariances ms = meanfunc.getMean(xs[id,:]) N = (alpha.shape)[1] # number of alphas (usually 1; more in case of sampling) Fmu = np.tile(ms,(1,N)) + np.dot(Ks.T,alpha[nz]) # conditional mean fs|f fmu[id] = np.reshape(Fmu.sum(axis=1)/N,(len(id),1)) # predictive means if Ltril: # L is triangular => use Cholesky parameters (alpha,sW,L) V = np.linalg.solve(L.T,np.tile(sW,(1,len(id)))*Ks) fs2[id] = kss - np.array([(V*V).sum(axis=0)]).T # predictive variances else: # L is not triangular => use alternative parametrization fs2[id] = kss + np.array([(Ks*np.dot(L,Ks)).sum(axis=0)]).T # predictive variances fs2[id] = np.maximum(fs2[id],0) # remove numerical noise i.e. negative variances Fs2 = np.tile(fs2[id],(1,N)) # we have multiple values in case of sampling if ys == None: [Lp, Ymu, Ys2] = likfunc.proceed(None,Fmu[:],Fs2[:],None,None,3) else: [Lp, Ymu, Ys2] = likfunc.proceed(np.tile(ys[id],(1,N)), Fmu[:], Fs2[:],None,None,3) lp[id] = np.reshape( np.reshape(Lp,(np.prod(Lp.shape),N)).sum(axis=1)/N , (len(id),1) ) # log probability; sample averaging ymu[id] = np.reshape( np.reshape(Ymu,(np.prod(Ymu.shape),N)).sum(axis=1)/N ,(len(id),1) ) # predictive mean ys|y and ... ys2[id] = np.reshape( np.reshape(Ys2,(np.prod(Ys2.shape),N)).sum(axis=1)/N , (len(id),1) ) # .. variance nact = id[-1]+1 # set counter to index of next data point self.ym = ymu self.ys2 = ys2 self.lp = lp self.fm = fmu self.fs2 = fs2 if ys == None: return ymu, ys2, fmu, fs2, None else: return ymu, ys2, fmu, fs2, lp class GPR(GP): '''Gaussian Process Regression''' def __init__(self): super(GPR, self).__init__() self.meanfunc = mean.Zero() # default prior mean self.covfunc = cov.RBF() # default prior covariance self.likfunc = lik.Gauss() # likihood with default noise variance 0.1 self.inffunc = inf.Exact() # inference method self.optimizer = opt.Minimize(self) # default optimizer def setNoise(self,log_sigma): '''Set noise variance other than default''' self.likfunc = lik.Gauss(log_sigma) def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None): '''Set Optimizer. See base class.''' conf = None if (num_restarts!=None) or (min_threshold!=None): conf = pyGPs.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc) conf.num_restarts = num_restarts conf.min_threshold = min_threshold if meanRange != None: conf.meanRange = meanRange if covRange != None: conf.covRange = covRange if likRange != None: conf.likRange = likRange if method == "Minimize": self.optimizer = opt.Minimize(self,conf) elif method == "SCG": self.optimizer = opt.SCG(self,conf) elif method == "CG": self.optimizer = opt.CG(self,conf) elif method == "BFGS": self.optimizer = opt.BFGS(self,conf) def plot(self,axisvals=None): '''Plot 1d GP regression.''' xs = self.xs x = self.x y = self.y ym = self.ym ys2 = self.ys2 plt.figure() xss = np.reshape(xs,(xs.shape[0],)) ymm = np.reshape(ym,(ym.shape[0],)) ys22 = np.reshape(ys2,(ys2.shape[0],)) plt.plot(x, y, color=DATACOLOR, ls='None', marker='+',ms=12, mew=2) plt.plot(xs, ym, color=MEANCOLOR, ls='-', lw=3.) plt.fill_between(xss,ymm + 2.*np.sqrt(ys22), ymm - 2.*np.sqrt(ys22), facecolor=SHADEDCOLOR,linewidths=0.0) plt.grid() if axisvals != None: plt.axis(axisvals) plt.xlabel('input x') plt.ylabel('target y') plt.show() def useInference(self, newInf): ''' Use another inference techinique other than default exact inference. :param str newInf: 'Laplace' or 'EP' ''' if newInf == "Laplace": self.inffunc = inf.Laplace() elif newInf == "EP": self.inffunc = inf.EP() else: raise Exception('Possible inf values are "Laplace", "EP".') def useLikelihood(self,newLik): if newLik == "Laplace": self.likfunc = lik.Laplace() self.inffunc = inf.EP() else: raise Exception('Possible lik values are "Laplace".') class GPC(GP): '''Gaussian Process Classification''' def __init__(self): super(GPC, self).__init__() self.meanfunc = mean.Zero() # default prior mean self.covfunc = cov.RBF() # default prior covariance self.likfunc = lik.Erf() # erf likihood self.inffunc = inf.EP() # default inference method self.optimizer = opt.Minimize(self) # default optimizer def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None): '''Set Optimizer. See base class.''' conf = None if (num_restarts!=None) or (min_threshold!=None): conf = pyGPs.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc) conf.num_restarts = num_restarts conf.min_threshold = min_threshold if meanRange != None: conf.meanRange = meanRange if covRange != None: conf.covRange = covRange if likRange != None: conf.likRange = likRange if method == "Minimize": self.optimizer = opt.Minimize(self,conf) elif method == "SCG": self.optimizer = opt.SCG(self,conf) elif method == "CG": self.optimizer = opt.CG(self,conf) elif method == "BFGS": self.optimizer = opt.BFGS(self,conf) def plot(self,x1,x2,t1,t2,axisvals=None): '''Plot 2d gp classification.''' fig = plt.figure() plt.plot(x1[:,0], x1[:,1], 'b+', markersize = 12) plt.plot(x2[:,0], x2[:,1], 'r+', markersize = 12) pc = plt.contour(t1, t2, np.reshape(np.exp(self.lp), (t1.shape[0],t1.shape[1]) )) fig.colorbar(pc) plt.grid() if axisvals != None: plt.axis(axisvals) plt.show() def useInference(self, newInf): ''' Use another inference techinique other than default EP inference. :param str newInf: 'Laplace' ''' if newInf == "Laplace": self.inffunc = inf.Laplace() else: raise Exception('Possible inf values are "Laplace".') def useLikelihood(self,newLik): ''' Use another likelihood function other than default Erf inference. (Not used in this version) :param str newLik: 'Logistic' ''' if newLik == "Logistic": pass #self.likfunc = lik.Logistic() else: raise Exception('Possible lik values are "Logistic".') class GPMC(object): '''This is a one vs. one classification wrapper for GPC''' def __init__(self, n_class): self.meanfunc = mean.Zero() # default prior mean self.covfunc = cov.RBF() # default prior covariance self.n_class = n_class # number of different classes self.x_all = None self.y_all = None self.newInf = None # new inference? -> call useInference self.newLik = None # new likelihood? -> call useLikelihood self.newPrior = False def setPrior(self, mean=None, kernel=None): '''set prior mean function and covariance function''' if mean != None: self.meanfunc = mean if kernel != None: self.covfunc = kernel if type(kernel) is cov.Pre: self.usingDefaultMean = False self.newPrior = True def useInference(self, newInf): '''Use another inference techinique other than default EP inference.''' self.newInf = newInf def useLikelihood(self,newLik): '''Use another likelihood function other than default Erf inference.''' self.newLik = newLik def setData(self,x,y): '''Set data for multi-class model''' if x.ndim == 1: x = np.reshape(x, (x.shape[0],1)) if y.ndim == 1: y = np.reshape(y, (y.shape[0],1)) self.x_all = x self.y_all = y def fitAndPredict(self, xs): ''' Fitting model and prediction. predictive_vote is a matrix where row i is each test point i column j is the probability for being class j :return: predictive_vote ''' if xs.ndim == 1: xs = np.reshape(xs, (xs.shape[0],1)) predictive_vote = np.zeros((xs.shape[0],self.n_class)) for i in xrange(self.n_class): # classifier for class i... for j in xrange(i+1,self.n_class): # ...and class j x,y = self.createBinaryClass(i,j) model = GPC() if self.newPrior: model.setPrior(mean=self.meanfunc, kernel=self.covfunc) if self.newInf: model.useInference(self.newInf) if self.newLik: model.useLikelihood(self.newLik) model.fit(x,y) # fitting ym = model.predict(xs)[0] ym += 1 # now scale into 0 to 2, ym=0 is class j, ym=2 is class i vote_i = np.zeros((xs.shape[0],self.n_class)) vote_j = np.zeros((xs.shape[0],self.n_class)) vote_i[:,i:i+1] = ym vote_j[:,j:j+1] = 2-ym predictive_vote += vote_i predictive_vote += vote_j predictive_vote /= predictive_vote.sum(axis=1)[:,np.newaxis] return predictive_vote def optimizeAndPredict(self, xs): ''' Optimizing model and prediction. predictive_vote is a matrix where row i is each test point i column j is the probability for being class j :return: predictive_vote ''' if xs.ndim == 1: xs = np.reshape(xs, (xs.shape[0],1)) predictive_vote = np.zeros((xs.shape[0],self.n_class)) for i in xrange(self.n_class): # classifier for class i... for j in xrange(i+1,self.n_class): # ...and class j x,y = self.createBinaryClass(i,j) model = GPC() if self.newPrior: model.setPrior(mean=self.meanfunc, kernel=self.covfunc) if self.newInf: model.useInference(self.newInf) if self.newLik: model.useLikelihood(self.newLik) model.optimize(x,y) # training ym = model.predict(xs)[0] ym += 1 # now scale into 0 to 2, ym=0 is class j, ym=2 is class i vote_i = np.zeros((xs.shape[0],self.n_class)) vote_j = np.zeros((xs.shape[0],self.n_class)) vote_i[:,i:i+1] = ym vote_j[:,j:j+1] = 2-ym predictive_vote += vote_i predictive_vote += vote_j predictive_vote /= predictive_vote.sum(axis=1)[:,np.newaxis] return predictive_vote def createBinaryClass(self, i,j): ''' Create dataset x(data) and y(label) which only contains class i and j. Relabel class i to +1 and class j to -1''' class_i = [] class_j = [] for index in xrange(len(self.y_all)): # check all classes target = self.y_all[index] if target == i: class_i.append(index) elif target == j: class_j.append(index) n1 = len(class_i) n2 = len(class_j) class_i.extend(class_j) x = self.x_all[class_i,:] y = np.concatenate((np.ones((1,n1)),-np.ones((1,n2))),axis=1).T return x,y class GP_FITC(GP): '''FITC GP base class''' def __init__(self): super(GP_FITC, self).__init__() self.u = None # overriding def setData(self, x, y, value_per_axis=5): '''Set training data and derive deault inducing_points. :param x: training data :param y: training target :param int value_per_axis: number of value in each dimension when using a uni-distant default inducing points''' if x.ndim == 1: x = np.reshape(x, (x.shape[0],1)) if y.ndim == 1: y = np.reshape(y, (y.shape[0],1)) self.x = x self.y = y if self.usingDefaultMean: c = np.mean(y) self.meanfunc = mean.Const(c) # adapt default prior mean wrt. training labels # get range of x in each dimension # 5 uniformally selected value for each dimension gridAxis=[] for d in xrange(x.shape[1]): column = x[:,d] mini = np.min(column) maxi = np.max(column) axis = np.linspace(mini,maxi,value_per_axis) gridAxis.append(axis) # default inducing points-> a grid if self.u == None: self.u = np.array(list(itertools.product(*gridAxis))) self.covfunc = self.covfunc.fitc(self.u) # overriding def setPrior(self, mean=None, kernel=None, inducing_points=None): '''Set GP prior and inducing points''' if kernel != None: if inducing_points != None: self.covfunc = kernel.fitc(inducing_points) self.u = inducing_points else: if self.u != None: self.covfunc = kernel.fitc(self.u) else: raise error("To use default inducing points, please call setData() first!") if type(kernel) is cov.Pre: self.usingDefaultMean = False if mean != None: self.meanfunc = mean self.usingDefaultMean = False class GPR_FITC(GP_FITC): '''Gaussian Process Regression FITC''' def __init__(self): super(GPR_FITC, self).__init__() self.meanfunc = mean.Zero() # default prior mean self.covfunc = cov.RBF() # default prior covariance self.likfunc = lik.Gauss() # likihood with default noise variance 0.1 self.inffunc = inf.FITC_Exact() # inference method self.optimizer = opt.Minimize(self) # default optimizer self.u = None # no default inducing points def setNoise(self,log_sigma): '''Set noise variance other than default value''' self.likfunc = lik.Gauss(log_sigma) def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None): '''Set Optimizer. See base class.''' conf = None if (num_restarts!=None) or (min_threshold!=None): conf = pyGPs.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc) conf.num_restarts = num_restarts conf.min_threshold = min_threshold if meanRange != None: conf.meanRange = meanRange if covRange != None: conf.covRange = covRange if likRange != None: conf.likRange = likRange if method == "Minimize": self.optimizer = opt.Minimize(self,conf) elif method == "SCG": self.optimizer = opt.SCG(self,conf) elif method == "CG": self.optimizer = opt.CG(self,conf) elif method == "BFGS": self.optimizer = opt.BFGS(self,conf) def plot(self,axisvals=None): '''Plot 1d GP FITC regression.''' plt.figure() xss = np.reshape(self.xs,(self.xs.shape[0],)) ymm = np.reshape(self.ym,(self.ym.shape[0],)) ys22 = np.reshape(self.ys2,(self.ys2.shape[0],)) plt.plot(self.x, self.y, color=DATACOLOR, ls='None', marker='+',ms=12, mew=2) plt.plot(self.xs, self.ym, color=MEANCOLOR, ls='-', lw=3.) plt.fill_between(xss,ymm + 2.*np.sqrt(ys22), ymm - 2.*np.sqrt(ys22), facecolor=SHADEDCOLOR,linewidths=0.0) plt.grid() if axisvals != None: plt.axis(axisvals) plt.xlabel('input x') plt.ylabel('output y') plt.plot(self.u,np.ones_like(self.u), ls='None', color='k',marker='x',markersize=12,mew=2) plt.show() def useInference(self, newInf): ''' Use another inference techinique other than default exact inference. :param str newInf: 'Laplace' or 'EP' ''' if newInf == "Laplace": self.inffunc = inf.FITC_Laplace() elif newInf == "EP": self.inffunc = inf.FITC_EP() else: raise Exception('Possible inf values are "Laplace", "EP".') def useLikelihood(self,newLik): ''' Use another inference techinique other than default Gaussian likelihood. :param str newLik: 'Laplace' ''' if newLik == "Laplace": self.likfunc = lik.Laplace() self.inffunc = inf.FITC_EP() else: raise Exception('Possible lik values are "Laplace".') class GPC_FITC(GP_FITC): '''Gaussian Process Classification FITC''' def __init__(self): super(GPC_FITC, self).__init__() self.meanfunc = mean.Zero() # default prior mean self.covfunc = cov.RBF() # default prior covariance self.likfunc = lik.Erf() # erf liklihood self.inffunc = inf.FITC_EP() # default inference method self.optimizer = opt.Minimize(self) # default optimizer self.u = None # no default inducing points def setOptimizer(self, method, num_restarts=None, min_threshold=None, meanRange=None, covRange=None, likRange=None): '''Set optimizer. See base class.''' conf = None if (num_restarts!=None) or (min_threshold!=None): conf = pyGPs.Optimization.conf.random_init_conf(self.meanfunc,self.covfunc,self.likfunc) conf.num_restarts = num_restarts conf.min_threshold = min_threshold if meanRange != None: conf.meanRange = meanRange if covRange != None: conf.covRange = covRange if likRange != None: conf.likRange = likRange if method == "Minimize": self.optimizer = opt.Minimize(self,conf) elif method == "SCG": self.optimizer = opt.SCG(self,conf) elif method == "CG": self.optimizer = opt.CG(self,conf) elif method == "BFGS": self.optimizer = opt.BFGS(self,conf) def plot(self,x1,x2,t1,t2,axisvals=None): '''Plot 2d GP FITC classification.''' fig = plt.figure() plt.plot(x1[:,0], x1[:,1], 'b+', markersize = 12) plt.plot(x2[:,0], x2[:,1], 'r+', markersize = 12) plt.plot(self.u[:,0],self.u[:,1],'ko', markersize=12) pc = plt.contour(t1, t2, np.reshape(np.exp(self.lp), (t1.shape[0],t1.shape[1]) )) fig.colorbar(pc) plt.grid() if axisvals != None: plt.axis(axisvals) plt.show() def useInference(self, newInf): ''' Use another inference techinique other than default exact inference. :param str newInf: 'Laplace' or 'EP' ''' if newInf == "Laplace": self.inffunc = inf.FITC_Laplace() else: raise Exception('Possible inf values are "Laplace".') def useLikelihood(self,newLik): ''' Use another inference techinique other than default Erf likelihood. (Not used in this version) :param str newLik: 'Logistic' ''' if newLik == "Logistic": pass #self.likfunc = lik.Logistic() else: raise Exception('Possible lik values are "Logistic".') pyGPs-1.2/pyGPs/Core/inf.py000066400000000000000000001404151236346020200155010ustar00rootroot00000000000000#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ # Inference methods: Compute the (approximate) posterior for a Gaussian process. # Methods currently implemented include: # # infExact Exact inference (only possible with Gaussian likelihood) # infLaplace Laplace's Approximation # infEP Expectation Propagation # infVB [NOT IMPLEMENTED!] Variational Bayes Approximation # # infFITC Large scale regression with approximate covariance matrix # infFITC_Laplace Large scale inference with approximate covariance matrix # infFITC_EP Large scale inference with approximate covariance matrix # # infMCMC [NOT IMPLEMENTED!] # Markov Chain Monte Carlo and Annealed Importance Sampling # We offer two samplers. # - hmc: Hybrid Monte Carlo # - ess: Elliptical Slice Sampling # No derivatives w.r.t. to hyperparameters are provided. # # infLOO [NOT IMPLEMENTED!] # Leave-One-Out predictive probability and Least-Squares Approxim. # # # This is a object-oriented python implementation of gpml functionality # (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18). # based on the functional-version of python implementation # (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013) # # Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013 import numpy as np import lik, cov from copy import copy,deepcopy from tools import solve_chol,brentmin,cholupdate np.seterr(all='ignore') class postStruct(object): ''' Data structure for posterior | post.alpha: 1d array containing inv(K)*m, | where K is the prior covariance matrix and m the approx posterior mean | post.sW: 1d array containing diagonal of sqrt(W) | the approximate posterior covariance matrix is inv(inv(K)+W) | post.L : 2d array, L = chol(sW*K*sW+identity(n)) ''' def __init__(self): self.alpha = np.array([]) self.L = np.array([]) self.sW = np.array([]) class dnlZStruct(object): ''' Data structure for the derivatives of mean, cov and lik functions. |dnlZ.mean: list of derivatives for each hyperparameters in mean function |dnlZ.cov: list of derivatives for each hyperparameters in covariance function |dnlZ.lik: list of derivatives for each hyperparameters in likelihood function ''' def __init__(self, m, c, l): self.mean = [] self.cov = [] self.lik = [] if m.hyp: self.mean = [0 for i in xrange(len(m.hyp))] if c.hyp: self.cov = [0 for i in xrange(len(c.hyp))] if l.hyp: self.lik = [0 for i in xrange(len(l.hyp))] class Inference(object): ''' Base class for inference. Defined several tool methods in it. ''' def __init__(self): pass def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): ''' Inference computation based on inputs. post, nlZ, dnlZ = inf.proceed(mean, cov, lik, x, y) | INPUT: | cov: name of the covariance function (see covFunctions.m) | lik: name of the likelihood function (see likFunctions.m) | x: n by D matrix of training inputs | y: 1d array (of size n) of targets | OUTPUT: | post(postStruct): struct representation of the (approximate) posterior containing: | nlZ: returned value of the negative log marginal likelihood | dnlZ(dnlZStruct): struct representation for derivatives of the negative log marginal likelihood | w.r.t. each hyperparameter. Usually, the approximate posterior to be returned admits the form: N(m=K*alpha, V=inv(inv(K)+W)), where alpha is a vector and W is diagonal; if not, then L contains instead -inv(K+inv(W)), and sW is unused. For more information on the individual approximation methods and their implementations, see the respective inference function below. See also gp.py :param meanfunc: mean function :param covfunc: covariance function :param likfunc: likelihood function :param x: training data :param y: training labels :param nargout: specify the number of output(1,2 or 3) :return: posterior, negative-log-marginal-likelihood, derivative for negative-log-marginal-likelihood-likelihood ''' pass def _epComputeParams(self, K, y, ttau, tnu, likfunc, m, inffunc): n = len(y) # number of training cases ssi = np.sqrt(ttau) # compute Sigma and mu L = np.linalg.cholesky(np.eye(n)+np.dot(ssi,ssi.T)*K).T # L'*L=B=eye(n)+sW*K*sW V = np.linalg.solve(L.T,np.tile(ssi,(1,n))*K) Sigma = K - np.dot(V.T,V) mu = np.dot(Sigma,tnu) Dsigma = np.reshape(np.diag(Sigma),(np.diag(Sigma).shape[0],1)) tau_n = 1/Dsigma - ttau # compute the log marginal likelihood nu_n = mu/Dsigma-tnu + m*tau_n # vectors of cavity parameters lZ = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc) nlZ = np.log(np.diag(L)).sum() - lZ.sum() - np.dot(tnu.T,np.dot(Sigma,tnu))/2 \ - np.dot((nu_n-m*tau_n).T,((ttau/tau_n*(nu_n-m*tau_n)-2*tnu) / (ttau+tau_n)))/2 \ + (tnu**2/(tau_n+ttau)).sum()/2.- np.log(1.+ttau/tau_n).sum()/2. return Sigma, mu, nlZ[0], L def _logdetA(self,K,w,nargout): ''' Compute the log determinant ldA and the inverse iA of a square nxn matrix A = eye(n) + K*diag(w) from its LU decomposition; for negative definite A, we return ldA = Inf. We also return mwiA = -diag(w)*inv(A). [ldA,iA,mwiA] = _logdetA(K,w)''' n = K.shape[0] assert(K.shape[0] == K.shape[1]) A = np.eye(n) + K*np.tile(w.T,(n,1)) [L,U,P] = np.linalg.lu(A) # compute LU decomposition, A = P'*L*U u = np.diag(U) signU = np.prod(np.sign(u)) # sign of U detP = 1 # compute sign (and det) of the permutation matrix P p = np.dot(P,np.array(range(n)).T) for ii in range(n): if ii != p[ii]: detP = -detP j= [jj for jj,val in enumerate(p) if val == ii] p[ii,j[0]] = p[j[0],ii] if signU != detP: # log becomes complex for negative values, encoded by infinity ldA = np.inf else: # det(L) = 1 and U triangular => det(A) = det(P)*prod(diag(U)) ldA = np.log(np.abs(u)).sum() if nargout>1: iA = inv(U)*inv(L)*P; if nargout>2: mwiA = -np.tile(w,(1,n))*iA return ldA,iA,mwiA else: return ldA,iA else: return ldA def _Psi_line(self,s,dalpha,alpha,K,m,likfunc,y,inffunc): '''Criterion Psi at alpha + s*dalpha for line search [Psi,alpha,f,dlp,W] = _Psi_line(s,dalpha,alpha,hyp,K,m,lik,y,inf)''' alpha = alpha + s*dalpha f = np.dot(K,alpha) + m [lp,dlp,d2lp] = likfunc.proceed(y,f,None,inffunc,None,3) W = -d2lp Psi = np.dot(alpha.T,(f-m))/2. - lp.sum() return Psi[0],alpha,f,dlp,W def _epfitcZ(self,d,P,R,nn,gg,ttau,tnu,d0,R0,P0,y,likfunc,m,inffunc): '''Compute the marginal likelihood approximation effort is O(n*nu^2) provided that nu0: R = cholupdate(R,np.sqrt(r)*np.dot(R.T,v),'-') else: R = cholupdate(R,np.sqrt(-r)*np.dot(R.T,v),'+') ttemp = np.dot(R0.T,np.dot(R.T,np.dot(R,np.dot(R0,P_i)))) gg = gg + ((dbi-dwi*(hi-m[ii]))/t) * np.reshape(ttemp,(ttemp.shape[0],1)) # O(nu^2) w[ii] = w_i; b[ii] = b_i; # update site parameters O(1) P_i = P_i/t # O(nu) return d,P_i,R,nn,gg,w,b def _mvmZ(self,x,RVdd,t): '''Matrix vector multiplication with Z=inv(K+inv(W))''' Zx = t*x - np.dot(RVdd.T,np.dot(RVdd,x)) return Zx def _mvmK(self,al,V,d0): '''Matrix vector multiplication with approximate covariance matrix''' Kal = np.dot(V.T,np.dot(V,al)) + d0*al return Kal def _Psi_lineFITC(self,s,dalpha,alpha,V,d0,m,likfunc,y,inffunc): '''Criterion Psi at alpha + s*dalpha for line search''' alpha = alpha + s*dalpha f = self._mvmK(alpha,V,d0) + m vargout = likfunc.proceed(y,f,None,inffunc,None,3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W = -d2lp Psi = np.dot(alpha.T,(f-m))/2. - lp.sum() return Psi[0],alpha,f,dlp,W def _fitcRefresh(self,d0,P0,R0,R0P0, w): '''Refresh the representation of the posterior from initial and site parameters to prevent possible loss of numerical precision after many epfitcUpdates effort is O(n*nu^2) provided that nu1: # do we want the marginal likelihood? nlZ = np.dot((y-m).T,alpha)/2. + np.log(np.diag(L)).sum() + n*np.log(2*np.pi*sn2)/2. # -log marg lik if nargout>2: # do we want derivatives? dnlZ = dnlZStruct(meanfunc, covfunc, likfunc) # allocate space for derivatives Q = solve_chol(L,np.eye(n))/sn2 - np.dot(alpha,alpha.T) # precompute for convenience dnlZ.lik = [sn2*np.trace(Q)] if covfunc.hyp: for ii in range(len(covfunc.hyp)): dnlZ.cov[ii] = (Q*covfunc.getDerMatrix(x=x, mode='train', der=ii)).sum()/2. if meanfunc.hyp: for ii in range(len(meanfunc.hyp)): dnlZ.mean[ii] = np.dot(-meanfunc.getDerMatrix(x, ii).T,alpha) dnlZ.mean[ii] = dnlZ.mean[ii][0,0] return post, nlZ[0,0], dnlZ return post, nlZ[0,0] return post class FITC_Exact(Inference): ''' FITC approximation to the posterior Gaussian process. The function is equivalent to infExact with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Quu) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu, snu2 = sn2/1e6 is the noise of the inducing inputs and Quu = Kuu + snu2*eye(nu). ''' def __init__(self): self.name = 'FICT exact inference' def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): if not isinstance(likfunc, lik.Gauss): # NOTE: no explicit call to likGauss raise Exception ('Exact inference only possible with Gaussian likelihood') if not isinstance(covfunc, cov.FITCOfKernel): raise Exception('Only covFITC supported.') # check cov diagK,Kuu,Ku = covfunc.getCovMatrix(x=x, mode='train') # evaluate covariance matrix m = meanfunc.getMean(x) # evaluate mean vector n, D = x.shape nu = Kuu.shape[0] sn2 = np.exp(2*likfunc.hyp[0]) # noise variance of likGauss snu2 = 1.e-6*sn2 # hard coded inducing inputs noise Luu = np.linalg.cholesky(Kuu+snu2*np.eye(nu)).T # Kuu + snu2*I = Luu'*Luu V = np.linalg.solve(Luu.T,Ku) # V = inv(Luu')*Ku => V'*V = Q g_sn2 = diagK + sn2 - np.array([(V*V).sum(axis=0)]).T # g + sn2 = diag(K) + sn2 - diag(Q) Lu = np.linalg.cholesky(np.eye(nu) + np.dot(V/np.tile(g_sn2.T,(nu,1)),V.T)).T # Lu'*Lu=I+V*diag(1/g_sn2)*V' r = (y-m)/np.sqrt(g_sn2) be = np.linalg.solve(Lu.T,np.dot(V,r/np.sqrt(g_sn2))) iKuu = solve_chol(Luu,np.eye(nu)) # inv(Kuu + snu2*I) = iKuu post = postStruct() post.alpha = np.linalg.solve(Luu,np.linalg.solve(Lu,be))# return the posterior parameters post.L = solve_chol(np.dot(Lu,Luu),np.eye(nu)) - iKuu # Sigma-inv(Kuu) post.sW = np.ones((n,1))/np.sqrt(sn2) # unused for FITC prediction with gp.m if nargout>1: # do we want the marginal likelihood nlZ = np.log(np.diag(Lu)).sum() + (np.log(g_sn2).sum() + n*np.log(2*np.pi) + np.dot(r.T,r) - np.dot(be.T,be))/2. if nargout>2: # do we want derivatives? dnlZ = dnlZStruct(meanfunc, covfunc, likfunc) # allocate space for derivatives al = r/np.sqrt(g_sn2) - np.dot(V.T,np.linalg.solve(Lu,be))/g_sn2 # al = (Kt+sn2*eye(n))\y B = np.dot(iKuu,Ku) w = np.dot(B,al) W = np.linalg.solve(Lu.T,V/np.tile(g_sn2.T,(nu,1))) for ii in range(len(covfunc.hyp)): [ddiagKi,dKuui,dKui] = covfunc.getDerMatrix(x=x, mode='train', der=ii) # eval cov deriv R = 2.*dKui-np.dot(dKuui,B) v = ddiagKi - np.array([(R*B).sum(axis=0)]).T # diag part of cov deriv dnlZ.cov[ii] = ( np.dot(ddiagKi.T,1./g_sn2) + np.dot(w.T,(np.dot(dKuui,w)-2.*np.dot(dKui,al))) \ - np.dot(al.T,(v*al)) - np.dot(np.array([(W*W).sum(axis=0)]),v) - (np.dot(R,W.T)*np.dot(B,W.T)).sum() )/2. dnlZ.cov[ii] = dnlZ.cov[ii][0,0] dnlZ.lik = sn2*((1./g_sn2).sum() - (np.array([(W*W).sum(axis=0)])).sum() - np.dot(al.T,al)) dKuui = 2*snu2 R = -dKuui*B v = -np.array([(R*B).sum(axis=0)]).T # diag part of cov deriv dnlZ.lik += (np.dot(w.T,np.dot(dKuui,w)) -np.dot(al.T,(v*al)) \ - np.dot(np.array([(W*W).sum(axis=0)]),v) - (np.dot(R,W.T)*np.dot(B,W.T)).sum() )/2. dnlZ.lik = list(dnlZ.lik[0]) for ii in range(len(meanfunc.hyp)): dnlZ.mean[ii] = np.dot(-meanfunc.getDerMatrix(x, ii).T, al) dnlZ.mean[ii] = dnlZ.mean[ii][0,0] return post, nlZ[0,0], dnlZ return post, nlZ[0,0] return post class Laplace(Inference): ''' Laplace's Approximation to the posterior Gaussian process. ''' def __init__(self): self.last_alpha = None def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): tol = 1e-6 # tolerance for when to stop the Newton iterations smax = 2; Nline = 20; thr = 1e-4 # line search parameters maxit = 20 # max number of Newton steps in f inffunc = self K = covfunc.getCovMatrix(x=x, mode='train') # evaluate the covariance matrix m = meanfunc.getMean(x) # evaluate the mean vector n, D = x.shape Psi_old = np.inf # make sure while loop starts by the largest old objective val if self.last_alpha == None: # find a good starting point for alpha and f alpha = np.zeros((n,1)) f = np.dot(K,alpha) + m # start at mean if sizes not match vargout = likfunc.proceed(y, f, None, inffunc, None, 3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W= -d2lp Psi_new = -lp.sum() else: alpha = self.last_alpha f = np.dot(K,alpha) + m # try last one vargout = likfunc.proceed(y, f, None, inffunc, None, 3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W= -d2lp Psi_new = np.dot(alpha.T,(f-m))/2. - lp.sum() # objective for last alpha vargout = - likfunc.proceed(y, m, None, inffunc, None, 1) Psi_def = vargout[0] # objective for default init f==m if Psi_def < Psi_new: # if default is better, we use it alpha = np.zeros((n,1)) f = np.dot(K,alpha) + m vargout = likfunc.proceed(y, f, None, inffunc, None, 3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W=-d2lp; Psi_new = -lp.sum() isWneg = np.any(W<0) # flag indicating whether we found negative values of W it = 0 # this happens for the Student's t likelihood while (Psi_old - Psi_new > tol) and it2: # do we want derivatives? dnlZ = dnlZStruct(meanfunc, covfunc, likfunc) # allocate space for derivatives if isWneg: # switch between Cholesky and LU decomposition mode Z = -post.L # inv(K+inv(W)) g = np.atleast_2d((iA*K).sum(axis=1)).T /2 # deriv. of ln|B| wrt W; g = diag(inv(inv(K)+diag(W)))/2 else: Z = np.tile(sW,(1,n))*solve_chol(post.L,np.diag(np.reshape(sW,(sW.shape[0],)))) #sW*inv(B)*sW=inv(K+inv(W)) C = np.linalg.solve(post.L.T,np.tile(sW,(1,n))*K) # deriv. of ln|B| wrt W g = np.atleast_2d((np.diag(K)-(C**2).sum(axis=0).T)).T /2. # g = diag(inv(inv(K)+W))/2 dfhat = g* d3lp # deriv. of nlZ wrt. fhat: dfhat=diag(inv(inv(K)+W)).*d3lp/2 for ii in range(len(covfunc.hyp)): # covariance hypers dK = covfunc.getDerMatrix(x=x, mode='train', der=ii) dnlZ.cov[ii] = (Z*dK).sum()/2. - np.dot(alpha.T,np.dot(dK,alpha))/2. # explicit part b = np.dot(dK,dlp) # b-K*(Z*b) = inv(eye(n)+K*diag(W))*b dnlZ.cov[ii] -= np.dot(dfhat.T,b-np.dot(K,np.dot(Z,b))) # implicit part dnlZ.cov[ii] = dnlZ.cov[ii][0,0] for ii in range(len(likfunc.hyp)): # likelihood hypers [lp_dhyp,dlp_dhyp,d2lp_dhyp] = likfunc.proceed(y,f,None,inffunc,ii,3) dnlZ.lik[ii] = -np.dot(g.T,d2lp_dhyp) - lp_dhyp.sum() # explicit part b = np.dot(K,dlp_dhyp) # b-K*(Z*b) = inv(eye(n)+K*diag(W))*b dnlZ.lik[ii] -= np.dot(dfhat.T,b-np.dot(K,np.dot(Z,b))) # implicit part dnlZ.lik[ii] = dnlZ.lik[ii][0,0] for ii in range(len(meanfunc.hyp)): # mean hypers dm = meanfunc.getDerMatrix(x, ii) dnlZ.mean[ii] = -np.dot(alpha.T,dm) # explicit part dnlZ.mean[ii] -= np.dot(dfhat.T,dm-np.dot(K,np.dot(Z,dm))) # implicit part dnlZ.mean[ii] = dnlZ.mean[ii][0,0] return post,nlZ[0],dnlZ else: return post, nlZ[0] class FITC_Laplace(Inference): ''' FITC-Laplace approximation to the posterior Gaussian process. The function is equivalent to infLaplace with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Kuu + snu2 * eye(nu)) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu and snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard deviation of the inducing inputs snu to be a one per mil of the measurement noise's standard deviation sn. In case of a likelihood without noise parameter sn2, we simply use snu2 = 1e-6. ''' def __init__(self): self.last_alpha = None def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): if not isinstance(covfunc, cov.FITCOfKernel): raise Exception('Only covFITC supported.') tol = 1e-6 # tolerance for when to stop the Newton iterations smax = 2; Nline = 100; thr = 1e-4 # line search parameters maxit = 20 # max number of Newton steps in f inffunc = Laplace() diagK,Kuu,Ku = covfunc.getCovMatrix(x=x, mode='train') # evaluate the covariance matrix m = meanfunc.getMean(x) # evaluate the mean vector if likfunc.hyp: # hard coded inducing inputs noise sn2 = np.exp(2.*likfunc.hyp[-1]) snu2 = 1.e-6*sn2 # similar to infFITC else: snu2 = 1.e-6 n, D = x.shape nu = Kuu.shape[0] rot180 = lambda A: np.rot90(np.rot90(A)) # little helper functions chol_inv = lambda A: np.linalg.solve( rot180( np.linalg.cholesky(rot180(A)) ),np.eye(nu)) # chol(inv(A)) R0 = chol_inv(Kuu+snu2*np.eye(nu)) # initial R, used for refresh O(nu^3) V = np.dot(R0,Ku); d0 = diagK - np.array([(V*V).sum(axis=0)]).T # initial d, needed Psi_old = np.inf # make sure while loop starts by the largest old objective val if self.last_alpha == None: # find a good starting point for alpha and f alpha = np.zeros((n,1)) f = self._mvmK(alpha,V,d0) + m # start at mean if sizes not match vargout = likfunc.proceed(y, f, None, inffunc, None, 3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W=-d2lp; Psi_new = -lp.sum() else: alpha = self.last_alpha f = self._mvmK(alpha,V,d0) + m # try last one vargout = likfunc.proceed(y, f, None, inffunc, None, 3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W=-d2lp Psi_new = np.dot(alpha.T,(f-m))/2. - lp.sum() # objective for last alpha vargout = - likfunc.proceed(y, m, None, inffunc, None, 1) Psi_def = vargout[0] # objective for default init f==m if Psi_def < Psi_new: # if default is better, we use it alpha = np.zeros((n,1)) f = self._mvmK(alpha,V,d0) + m vargout = likfunc.proceed(y, f, None, inffunc, None, 3) lp = vargout[0]; dlp = vargout[1]; d2lp = vargout[2] W=-d2lp; Psi_new = -lp.sum() isWneg = np.any(W<0) # flag indicating whether we found negative values of W it = 0 # this happens for the Student's t likelihood while (Psi_old - Psi_new > tol) and it2: # do we want derivatives? dnlZ = dnlZStruct(meanfunc, covfunc, likfunc) # allocate space for derivatives [d,P,R] = self._fitcRefresh(d0,Ku,R0,V,W) # g = diag(inv(inv(K)+W))/2 g = d/2 + 0.5*np.atleast_2d((np.dot(np.dot(R,R0),P)**2).sum(axis=0)).T t = W/(1+W*d0) dfhat = g*d3lp # deriv. of nlZ wrt. fhat: dfhat=diag(inv(inv(K)+W)).*d3lp/2 for ii in range(len(covfunc.hyp)): # covariance hypers ddiagK,dKuu,dKu = covfunc.getDerMatrix(x=x, mode='train', der=ii) # eval cov derivatives dA = 2.*dKu.T-np.dot(R0tV.T,dKuu) # dQ = dA*R0tV w = np.atleast_2d((dA*R0tV.T).sum(axis=1)).T # w = diag(dQ) v = ddiagK-w # v = diag(dK)-diag(dQ); dnlZ.cov[ii] = np.dot(ddiagK.T,t) - np.dot((RVdd*RVdd).sum(axis=0),v) # explicit part dnlZ.cov[ii] -= (np.dot(RVdd,dA)*np.dot(RVdd,R0tV.T)).sum() # explicit part dnlZ.cov[ii] = 0.5*dnlZ.cov[ii] - np.dot(alpha.T,np.dot(dA,np.dot(R0tV,alpha))+v*alpha)/2. # explicit b = np.dot(dA,np.dot(R0tV,dlp)) + v*dlp # b-K*(Z*b) = inv(eye(n)+K*diag(W))*b KZb = self._mvmK(self._mvmZ(b,RVdd,t),V,d0) dnlZ.cov[ii] -= np.dot(dfhat.T,(b-KZb)) # implicit part dnlZ.cov[ii] = dnlZ.cov[ii][0,0] for ii in range(len(likfunc.hyp)): # likelihood hypers vargout = likfunc.proceed(y,f,None,inffunc,ii,3) lp_dhyp = vargout[0]; dlp_dhyp = vargout[1]; d2lp_dhyp = vargout[2] dnlZ.lik[ii] = -np.dot(g.T,d2lp_dhyp) - lp_dhyp.sum() # explicit part b = self._mvmK(dlp_dhyp,V,d0) # implicit part dnlZ.lik[ii] -= np.dot(dfhat.T,b-self._mvmK(self._mvmZ(b,RVdd,t),V,d0)) if ii == len(likfunc.hyp)-1: # since snu2 is a fixed fraction of sn2, there is a covariance-like term # in the derivative as well snu = np.sqrt(snu2); T = chol_inv(Kuu + snu2*np.eye(nu)); T = np.dot(T.T,np.dot(T,snu*Ku)); t = np.array([(T*T).sum(axis=0)]).T z = np.dot(alpha.T,np.dot(T.T,np.dot(T,alpha))-t*alpha) - np.dot(np.array([(RVdd*RVdd).sum(axis=0)]),t) z += (np.dot(RVdd,T.T)**2).sum() b = (t*dlp-np.dot(T.T,np.dot(T,dlp)))/2. KZb = self._mvmK(self._mvmZ(b,RVdd,t),V,d0) z -= np.dot(dfhat.T,b-KZb) dnlZ.lik[ii] += z dnlZ.lik[ii] = dnlZ.lik[ii][0,0] for ii in range(len(meanfunc.hyp)): # mean hypers dm = meanfunc.getDerMatrix(x, ii) dnlZ.mean[ii] = -np.dot(alpha.T,dm) # explicit part Zdm = self._mvmZ(dm,RVdd,t) dnlZ.mean[ii] -= np.dot(dfhat.T,(dm-self._mvmK(Zdm,V,d0))) # implicit part dnlZ.mean[ii] = dnlZ.mean[ii][0,0] return post,nlZ[0,0],dnlZ else: return post, nlZ[0,0] class EP(Inference): ''' Expectation Propagation approximation to the posterior Gaussian Process. ''' def __init__(self): self.name = 'Expectation Propagation' self.last_ttau = None self.last_tnu = None def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): tol = 1e-4; max_sweep = 10; min_sweep = 2 # tolerance to stop EP iterations n = x.shape[0] inffunc = self K = covfunc.getCovMatrix(x=x, mode='train') # evaluate the covariance matrix m = meanfunc.getMean(x) # evaluate the mean vector nlZ0 = -likfunc.proceed(y, m, np.reshape(np.diag(K),(np.diag(K).shape[0],1)), inffunc).sum() if self.last_ttau == None: # find starting point for tilde parameters ttau = np.zeros((n,1)) # initialize to zero if we have no better guess tnu = np.zeros((n,1)) Sigma = K # initialize Sigma and mu, the parameters of .. mu = np.zeros((n,1)) # .. the Gaussian posterior approximation nlZ = nlZ0 else: ttau = self.last_ttau # try the tilde values from previous call tnu = self.last_tnu Sigma, mu, nlZ, L = self._epComputeParams(K, y, ttau, tnu, likfunc, m, inffunc) if nlZ > nlZ0: # if zero is better .. ttau = np.zeros((n,1)) # .. then initialize with zero instead tnu = np.zeros((n,1)) Sigma = K # initialize Sigma and mu, the parameters of .. mu = np.zeros((n,1)) # .. the Gaussian posterior approximation nlZ = nlZ0 nlZ_old = np.inf; sweep = 0 # converged, max. sweeps or min. sweeps? while (np.abs(nlZ-nlZ_old) > tol and sweep < max_sweep) or (sweep < min_sweep): nlZ_old = nlZ; sweep += 1 rperm = xrange(n) # randperm(n) for ii in rperm: # iterate EP updates (in random order) over examples tau_ni = 1/Sigma[ii,ii] - ttau[ii]# first find the cavity distribution .. nu_ni = mu[ii]/Sigma[ii,ii] + m[ii]*tau_ni - tnu[ii] # .. params tau_ni and nu_ni # compute the desired derivatives of the indivdual log partition function lZ,dlZ,d2lZ = likfunc.proceed(y[ii], nu_ni/tau_ni, 1/tau_ni, inffunc, None, 3) ttau_old = copy(ttau[ii]) # then find the new tilde parameters, keep copy of old ttau[ii] = -d2lZ /(1.+d2lZ/tau_ni) ttau[ii] = max(ttau[ii],0) # enforce positivity i.e. lower bound ttau by zero tnu[ii] = ( dlZ + (m[ii]-nu_ni/tau_ni)*d2lZ )/(1.+d2lZ/tau_ni) ds2 = ttau[ii] - ttau_old # finally rank-1 update Sigma .. si = np.reshape(Sigma[:,ii],(Sigma.shape[0],1)) Sigma = Sigma - ds2/(1.+ds2*si[ii])*np.dot(si,si.T) # takes 70# of total time mu = np.dot(Sigma,tnu) # .. and recompute mu # recompute since repeated rank-one updates can destroy numerical precision Sigma, mu, nlZ, L = self._epComputeParams(K, y, ttau, tnu, likfunc, m, inffunc) if sweep == max_sweep: pass # print '[warning] maximum number of sweeps reached in function infEP' self.last_ttau = ttau; self.last_tnu = tnu # remember for next call sW = np.sqrt(ttau); alpha = tnu-sW*solve_chol(L,sW*np.dot(K,tnu)) post = postStruct() post.alpha = alpha # return the posterior params post.sW = sW post.L = L if nargout>2: # do we want derivatives? dnlZ = dnlZStruct(meanfunc, covfunc, likfunc) # allocate space for derivatives ssi = np.sqrt(ttau) V = np.linalg.solve(L.T,np.tile(ssi,(1,n))*K) Sigma = K - np.dot(V.T,V) mu = np.dot(Sigma,tnu) Dsigma = np.reshape(np.diag(Sigma),(np.diag(Sigma).shape[0],1)) tau_n = 1/Dsigma-ttau # compute the log marginal likelihood nu_n = mu/Dsigma-tnu # vectors of cavity parameters F = np.dot(alpha,alpha.T) - np.tile(sW,(1,n))* \ solve_chol(L,np.diag(np.reshape(sW,(sW.shape[0],)))) # covariance hypers for jj in range(len(covfunc.hyp)): dK = covfunc.getDerMatrix(x=x, mode='train', der=jj) dnlZ.cov[jj] = -(F*dK).sum()/2. for ii in range(len(likfunc.hyp)): dlik = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc, ii) dnlZ.lik[ii] = -dlik.sum() junk,dlZ = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc, None, 2) # mean hyps for ii in range(len(meanfunc.hyp)): dm = meanfunc.getDerMatrix(x, ii) dnlZ.mean[ii] = -np.dot(dlZ.T,dm) dnlZ.mean[ii] = dnlZ.mean[ii][0,0] return post, nlZ[0], dnlZ else: return post, nlZ[0] class FITC_EP(Inference): ''' FITC-EP approximation to the posterior Gaussian process. The function is equivalent to infEP with the covariance function: Kt = Q + G; G = diag(g); g = diag(K-Q); Q = Ku' * inv(Kuu + snu2 * eye(nu)) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu and snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard deviation of the inducing inputs snu to be a one per mil of the measurement noise's standard deviation sn. In case of a likelihood without noise parameter sn2, we simply use snu2 = 1e-6. For details, see The Generalized FITC Approximation, Andrew Naish-Guzman and Sean Holden, NIPS, 2007. ''' def __init__(self): self.name = 'FITC Expectation Propagation' self.last_ttau = None self.last_tnu = None def proceed(self, meanfunc, covfunc, likfunc, x, y, nargout=1): if not isinstance(covfunc, cov.FITCOfKernel): raise Exception('Only covFITC supported.') # check cov tol = 1e-4; max_sweep = 10; min_sweep = 2 # tolerance to stop EP iterations inffunc = EP() diagK,Kuu,Ku = covfunc.getCovMatrix(x=x, mode='train') # evaluate the covariance matrix m = meanfunc.getMean(x) # evaluate the mean vector if likfunc.hyp: # hard coded inducing inputs noise sn2 = np.exp(2.*likfunc.hyp[-1]) snu2 = 1.e-6*sn2 # similar to infFITC else: snu2 = 1.e-6 n, D = x.shape; nu = Kuu.shape[0] rot180 = lambda A: np.rot90(np.rot90(A)) # little helper functions chol_inv = lambda A: np.linalg.solve( rot180( np.linalg.cholesky(rot180(A)) ),np.eye(nu)) # chol(inv(A)) R0 = chol_inv(Kuu+snu2*np.eye(nu)) # initial R, used for refresh O(nu^3) V = np.dot(R0,Ku); d0 = diagK - np.array([(V*V).sum(axis=0)]).T # initial d, needed for refresh O(n*nu^2) # A note on naming: variables are given short but descriptive names in # accordance with Rasmussen & Williams "GPs for Machine Learning" (2006): mu # and s2 are mean and variance, nu and tau are natural parameters. A leading t # means tilde, a subscript _ni means "not i" (for cavity parameters), or _n # for a vector of cavity parameters. # marginal likelihood for ttau = tnu = zeros(n,1); equals n*log(2) for likCum* nlZ0 = -1.* likfunc.proceed(y, m, np.reshape(diagK,(diagK.shape[0],1)), inffunc).sum() if self.last_ttau == None: # find starting point for tilde parameters ttau = np.zeros((n,1)) # initialize to zero if we have no better guess tnu = np.zeros((n,1)) [d,P,R,nn,gg] = self._epfitcRefresh(d0,Ku,R0,V,ttau,tnu) # compute initial repres. nlZ = nlZ0 else: ttau = self.last_ttau # try the tilde values from previous call tnu = self.last_tnu [d,P,R,nn,gg] = self._epfitcRefresh(d0,Ku,R0,V,ttau,tnu) # compute initial repres. nlZ = self._epfitcZ(d,P,R,nn,gg,ttau,tnu,d0,R0,Ku,y,likfunc,m,inffunc)[0] if nlZ > nlZ0: # if zero is better .. ttau = np.zeros((n,1)) # .. then initialize with zero instead tnu = np.zeros((n,1)) [d,P,R,nn,gg] = self._epfitcRefresh(d0,Ku,R0,V,ttau,tnu) # initial repres. nlZ = nlZ0 nlZ_old = np.inf; sweep = 0 # converged, max. sweeps or min. sweeps? while (np.abs(nlZ-nlZ_old) > tol and sweep < max_sweep) or (sweep < min_sweep): nlZ_old = nlZ sweep += 1 rperm = range(n) # randperm(n) for ii in rperm: # iterate EP updates (in random order) over examples p_i = np.reshape(P[:,ii],(P.shape[0],1)) t = np.dot(R,np.dot(R0,p_i)) # temporary variables sigma_i = d[ii] + np.dot(t.T,t); mu_i = nn[ii] + np.dot(p_i.T,gg) # post moments O(nu^2) tau_ni = 1/sigma_i - ttau[ii] # first find the cavity distribution .. nu_ni = mu_i/sigma_i + m[ii]*tau_ni - tnu[ii] # .. params tau_ni and nu_ni # compute the desired derivatives of the indivdual log partition function vargout = likfunc.proceed(y[ii], nu_ni/tau_ni, 1/tau_ni, inffunc, None, 3) lZ = vargout[0]; dlZ = vargout[1]; d2lZ = vargout[2] ttau_i = -d2lZ /(1.+d2lZ/tau_ni) ttau_i = max(ttau_i,0) # enforce positivity i.e. lower bound ttau by zero tnu_i = ( dlZ + (m[ii]-nu_ni/tau_ni)*d2lZ )/(1.+d2lZ/tau_ni) [d,P[:,ii],R,nn,gg,ttau,tnu] = self._epfitcUpdate(d,P[:,ii],R,nn,gg,ttau,tnu,ii,ttau_i,tnu_i,m,d0,Ku,R0)# update representation # recompute since repeated rank-one updates can destroy numerical precision [d,P,R,nn,gg] = self._epfitcRefresh(d0,Ku,R0,V,ttau,tnu) [nlZ,nu_n,tau_n] = self._epfitcZ(d,P,R,nn,gg,ttau,tnu,d0,R0,Ku,y,likfunc,m,inffunc) if sweep == max_sweep: pass # print '[warning] maximum number of sweeps reached in function infEP' self.last_ttau = ttau self.last_tnu = tnu # remember for next call post = postStruct() post.sW = np.sqrt(ttau) # unused for FITC_EP prediction with gp.m dd = 1/(d0+1/ttau) alpha = tnu/ttau*dd RV = np.dot(R,V) R0tV = np.dot(R0.T,V) alpha = alpha - np.dot(RV.T,np.dot(RV,alpha))*dd # long alpha vector for ordinary infEP post.alpha = np.dot(R0tV,alpha) # alpha = R0'*V*inv(Kt+diag(1./ttau))*(tnu./ttau) B = R0tV*np.tile(dd.T,(nu,1)); L = np.dot(B,R0tV.T); B = np.dot(B,RV.T) post.L = np.dot(B,B.T) - L # L = -R0'*V*inv(Kt+diag(1./ttau))*V'*R0 if nargout>2: # do we want derivatives? dnlZ = dnlZStruct(meanfunc, covfunc, likfunc) # allocate space for derivatives RVdd = RV*np.tile(dd.T,(nu,1)) for ii in range(len(covfunc.hyp)): ddiagK,dKuu,dKu = covfunc.getDerMatrix(x=x, mode='train', der=ii) dA = 2*dKu.T - np.dot(R0tV.T,dKuu) # dQ = dA*R0tV w = np.atleast_2d((dA*R0tV.T).sum(axis=1)).T # w = diag(dQ) v = ddiagK - w # v = diag(dK)-diag(dQ) z = np.dot(dd.T,(v+w)) - np.dot(np.atleast_2d((RVdd*RVdd).sum(axis=0)), v) \ - (np.dot(RVdd,dA).T * np.dot(R0tV,RVdd.T)).sum() dnlZ.cov[ii] = (z - np.dot(alpha.T,(alpha*v)) - np.dot(np.dot(alpha.T,dA),np.dot(R0tV,alpha)))/2. dnlZ.cov[ii] = dnlZ.cov[ii][0,0] for ii in range(len(likfunc.hyp)): # likelihood hypers dlik = likfunc.proceed(y, nu_n/tau_n+m, 1/tau_n, inffunc, ii, 1) dnlZ.lik[ii] = -dlik.sum() if ii == len(likfunc.hyp)-1: # since snu2 is a fixed fraction of sn2, there is a covariance-like term # in the derivative as well v = np.atleast_2d((R0tV*R0tV).sum(axis=0)).T z = (np.dot(RVdd,R0tV.T)**2).sum() - np.dot(np.atleast_2d((RVdd*RVdd).sum(axis=0)),v) z = z + np.dot(post.alpha.T,post.alpha) - np.dot(alpha.T,(v*alpha)) dnlZ.lik[ii] += snu2*z dnlZ.lik[ii] = dnlZ.lik[ii][0,0] [junk,dlZ] = likfunc.proceed(y, nu_n/tau_n, 1/tau_n, inffunc, None, 2) # mean hyps for ii in range(len(meanfunc.hyp)): dm = meanfunc.getDerMatrix(x, ii) dnlZ.mean[ii] = -np.dot(dlZ.T,dm) dnlZ.mean[ii] = dnlZ.mean[ii][0,0] return post, nlZ[0,0], dnlZ else: return post, nlZ[0,0] if __name__ == '__main__': pass pyGPs-1.2/pyGPs/Core/lik.py000066400000000000000000000665611236346020200155150ustar00rootroot00000000000000# Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ # likelihood functions are provided to be used by the gp.py function: # # Erf (Error function, classification, probit regression) # Logistic [NOT IMPLEMENTED!] (Logistic, classification, logit regression) # Uni [NOT IMPLEMENTED!] (Uniform likelihood, classification) # # Gauss (Gaussian, regression) # Laplace (Laplacian or double exponential, regression) # Sech2 [NOT IMPLEMENTED!] (Sech-square, regression) # T [NOT IMPLEMENTED!] (Student's t, regression) # # Poisson [NOT IMPLEMENTED!] (Poisson regression, count data) # # Mix [NOT IMPLEMENTED!] (Mixture of individual covariance functions) # # See the documentation for the individual likelihood for the computations specific # to each likelihood function. # # # This is a object-oriented python implementation of gpml functionality # (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18). # based on the functional-version of python implementation # (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013) # # Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013 import numpy as np from scipy.special import erf import inf class Likelihood(object): """Base function for Likelihood function""" def __init__(self): self.hyp = [] def proceed(self, y=None, mu=None, s2=None, inffunc=None, der=None, nargout=1): ''' The likelihood functions have two possible modes, the mode being selected as follows (where "lik" stands for "proceed" method for any likelihood function): 1) With two or three input arguments: [PREDICTION MODE] lp = lik(y, mu) OR lp, ymu, ys2 = lik(y, mu, s2) This allows to evaluate the predictive distribution. Let p(y_*|f_*) be the likelihood of a test point and N(f_*|mu,s2) an approximation to the posterior marginal p(f_*|x_*,x,y) as returned by an inference method. The predictive distribution p(y_*|x_*,x,y) is approximated by: q(y_*) = \int N(f_*|mu,s2) p(y_*|f_*) df_* lp = log( q(y) ) for a particular value of y, if s2 is [] or 0, this corresponds to log( p(y|mu) ). ymu and ys2 are the mean and variance of the predictive marginal q(y) note that these two numbers do not depend on a particular value of y. All vectors have the same size. 2) With four or five input arguments, the fouth being an object of class "Inference" [INFERENCE MODE] lik(y, mu, s2, inf) OR lik(y, mu, s2, inf, i) There are two cases for inf, namely a) infLaplace, b) infEP The last input i, refers to derivatives w.r.t. the ith hyperparameter. | a1) | lp,dlp,d2lp,d3lp = lik(y, f, [], 'infLaplace'). | lp, dlp, d2lp and d3lp correspond to derivatives of the log likelihood. | log(p(y|f)) w.r.t. to the latent location f. | lp = log( p(y|f) ) | dlp = d log( p(y|f) ) / df | d2lp = d^2 log( p(y|f) ) / df^2 | d3lp = d^3 log( p(y|f) ) / df^3 | a2) | lp_dhyp,dlp_dhyp,d2lp_dhyp = lik(y, f, [], 'infLaplace', i) | returns derivatives w.r.t. to the ith hyperparameter | lp_dhyp = d log( p(y|f) ) / (dhyp_i) | dlp_dhyp = d^2 log( p(y|f) ) / (df dhyp_i) | d2lp_dhyp = d^3 log( p(y|f) ) / (df^2 dhyp_i) | b1) | lZ,dlZ,d2lZ = lik(y, mu, s2, 'infEP') | let Z = \int p(y|f) N(f|mu,s2) df then | lZ = log(Z) | dlZ = d log(Z) / dmu | d2lZ = d^2 log(Z) / dmu^2 | b2) | dlZhyp = lik(y, mu, s2, 'infEP', i) | returns derivatives w.r.t. to the ith hyperparameter | dlZhyp = d log(Z) / dhyp_i Cumulative likelihoods are designed for binary classification. Therefore, they only look at the sign of the targets y; zero values are treated as +1. Some examples for valid likelihood functions: | lik = likGauss([0.1]) | lik = likErf() ''' pass class Gauss(Likelihood): ''' Gaussian likelihood function for regression. :math:`Gauss(t)=\\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-\\frac{(t-y)^2}{2\\sigma^2}}`, where :math:`y` is the mean and :math:`\\sigma` is the standard deviation. hyp = [ log_sigma ] ''' def __init__(self, log_sigma=np.log(0.1) ): self.hyp = [log_sigma] def proceed(self, y=None, mu=None, s2=None, inffunc=None, der=None, nargout=1): sn2 = np.exp(2. * self.hyp[0]) if inffunc == None: # prediction mode if y == None: y = np.zeros_like(mu) s2zero = True if (not s2==None) and np.linalg.norm(s2) > 0: s2zero = False if s2zero: # log probability lp = -(y-mu)**2 /sn2/2 - np.log(2.*np.pi*sn2)/2. s2 = np.zeros_like(s2) else: inf_func = inf.EP() # prediction lp = self.proceed(y, mu, s2, inf_func) if nargout>1: ymu = mu # first y moment if nargout>2: ys2 = s2 + sn2 # second y moment return lp,ymu,ys2 else: return lp,ymu else: return lp else: if isinstance(inffunc, inf.EP): if der == None: # no derivative mode lZ = -(y-mu)**2/(sn2+s2)/2. - np.log(2*np.pi*(sn2+s2))/2. # log part function if nargout>1: dlZ = (y-mu)/(sn2+s2) # 1st derivative w.r.t. mean if nargout>2: d2lZ = -1/(sn2+s2) # 2nd derivative w.r.t. mean return lZ,dlZ,d2lZ else: return lZ,dlZ else: return lZ else: # derivative mode dlZhyp = ((y-mu)**2/(sn2+s2)-1) / (1+s2/sn2) # deriv. w.r.t. hyp.lik return dlZhyp elif isinstance(inffunc, inf.Laplace): if der == None: # no derivative mode if y == None: y=0 ymmu = y-mu lp = -ymmu**2/(2*sn2) - np.log(2*np.pi*sn2)/2. if nargout>1: dlp = ymmu/sn2 # dlp, derivative of log likelihood if nargout>2: # d2lp, 2nd derivative of log likelihood d2lp = -np.ones_like(ymmu)/sn2 if nargout>3: # d3lp, 3rd derivative of log likelihood d3lp = np.zeros_like(ymmu) return lp,dlp,d2lp,d3lp else: return lp,dlp,d2lp else: return lp,dlp else: return lp else: # derivative mode lp_dhyp = (y-mu)**2/sn2 - 1 # derivative of log likelihood w.r.t. hypers dlp_dhyp = 2*(mu-y)/sn2 # first derivative, d2lp_dhyp = 2*np.ones_like(mu)/sn2 # and also of the second mu derivative return lp_dhyp,dlp_dhyp,d2lp_dhyp ''' elif isinstance(inffunc, infVB): if der == None: # variational lower site bound # t(s) = exp(-(y-s)^2/2sn2)/sqrt(2*pi*sn2) # the bound has the form: b*s - s.^2/(2*ga) - h(ga)/2 with b=y/ga ga = s2 n = len(ga) b = y/ga y = y*np.ones((n,1)) db = -y/ga**2 d2b = 2*y/ga**3 h = np.zeros((n,1)) dh = h d2h = h # allocate memory for return args id = (ga <= sn2 + 1e-8) # OK below noise variance h[id] = y[id]**2/ga[id] + np.log(2*np.pi*sn2) h[np.logical_not(id)] = np.inf dh[id] = -y[id]**2/ga[id]**2 d2h[id] = 2*y[id]**2/ga[id]**3 id = ga < 0 h[id] = np.inf dh[id] = 0 d2h[id] = 0 # neg. var. treatment varargout = [h,b,dh,db,d2h,d2b] else: ga = s2 n = len(ga) dhhyp = np.zeros((n,1)) dhhyp[ga<=sn2] = 2 dhhyp[ga<0] = 0 # negative variances get a special treatment varargout = dhhyp # deriv. w.r.t. hyp.lik else: raise Exception('Incorrect inference in lik.Gauss\n') ''' class Erf(Likelihood): ''' Error function or cumulative Gaussian likelihood function for binary classification or probit regression. :math:`Erf(t)=\\frac{1}{2}(1+erf(\\frac{t}{\\sqrt{2}}))=normcdf(t)` ''' def __init__(self): self.hyp = [] def proceed(self, y=None, mu=None, s2=None, inffunc=None, der=None, nargout=1): if not y == None: y = np.sign(y) y[y==0] = 1 else: y = 1 # allow only +/- 1 values if inffunc == None: # prediction mode if inf is not present y = y*np.ones_like(mu) # make y a vector s2zero = True; if not s2 == None: if np.linalg.norm(s2)>0: s2zero = False # s2==0? if s2zero: # log probability evaluation p,lp = self.cumGauss(y,mu,2) else: # prediction lp = self.proceed(y, mu, s2, inf.EP()) p = np.exp(lp) if nargout>1: ymu = 2*p-1 # first y moment if nargout>2: ys2 = 4*p*(1-p) # second y moment return lp,ymu,ys2 else: return lp,ymu else: return lp else: # inference mode if isinstance(inffunc, inf.Laplace): if der == None: # no derivative mode f = mu; yf = y*f # product latents and labels p,lp = self.cumGauss(y,f,2) if nargout>1: # derivative of log likelihood n_p = self.gauOverCumGauss(yf,p) dlp = y*n_p # derivative of log likelihood if nargout>2: # 2nd derivative of log likelihood d2lp = -n_p**2 - yf*n_p if nargout>3: # 3rd derivative of log likelihood d3lp = 2*y*n_p**3 + 3*f*n_p**2 + y*(f**2-1)*n_p return lp,dlp,d2lp,d3lp else: return lp,dlp,d2lp else: return lp,dlp else: return lp else: # derivative mode return [] # derivative w.r.t. hypers elif isinstance(inffunc, inf.EP): if der == None: # no derivative mode z = mu/np.sqrt(1+s2) junk,lZ = self.cumGauss(y,z,2) # log part function if not y == None: z = z*y if nargout>1: if y == None: y = 1 n_p = self.gauOverCumGauss(z,np.exp(lZ)) dlZ = y*n_p/np.sqrt(1.+s2) # 1st derivative wrt mean if nargout>2: d2lZ = -n_p*(z+n_p)/(1.+s2) # 2nd derivative wrt mean return lZ,dlZ,d2lZ else: return lZ,dlZ else: return lZ else: # derivative mode return [] # deriv. wrt hyp.lik ''' if inffunc == 'inf.infVB': if der == None: # no derivative mode # naive variational lower bound based on asymptotical properties of lik # normcdf(t) -> -(t*A_hat^2-2dt+c)/2 for t->-np.inf (tight lower bound) d = 0.158482605320942; c = -1.785873318175113; ga = s2; n = len(ga); b = d*y*np.ones((n,1)); db = np.zeros((n,1)); d2b = db h = -2.*c*np.ones((n,1)); h[ga>1] = np.inf; dh = np.zeros((n,1)); d2h = dh varargout = [h,b,dh,db,d2h,d2b] else: # derivative mode varargout = [] # deriv. wrt hyp.lik ''' def cumGauss(self, y=None, f=None, nargout=1): # return [p,lp] = cumGauss(y,f) if not y == None: yf = y*f else: yf = f p = (1. + erf(yf/np.sqrt(2.)))/2. # likelihood if nargout>1: lp = self.logphi(yf,p) return p,lp else: return p def gauOverCumGauss(self,f,p): # return n_p = gauOverCumGauss(f,p) n_p = np.zeros_like(f) # safely compute Gaussian over cumulative Gaussian ok = f>-5 # naive evaluation for large values of f n_p[ok] = (np.exp(-f[ok]**2/2)/np.sqrt(2*np.pi)) / p[ok] bd = f<-6 # tight upper bound evaluation n_p[bd] = np.sqrt(f[bd]**2/4+1)-f[bd]/2 interp = np.logical_and(np.logical_not(ok),np.logical_not(bd)) # linearly interpolate between both of them tmp = f[interp] lam = -5. - f[interp] n_p[interp] = (1-lam)*(np.exp(-tmp**2/2)/np.sqrt(2*np.pi))/p[interp] + lam *(np.sqrt(tmp**2/4+1)-tmp/2); return n_p def logphi(self,z,p): # return lp = logphi(z,p) lp = np.zeros_like(z) # allocate memory zmin = -6.2; zmax = -5.5; ok = z>zmax # safe evaluation for large values bd = z0: s2zero = False # s2==0? if s2zero: # log probability evaluation lp = -np.abs(y-mu)/b -np.log(2*b); s2 = 0 else: # prediction lp = self.proceed(y, mu, s2, inf.EP()) if nargout>1: ymu = mu # first y moment if nargout>2: ys2 = s2 + sn**2 # second y moment return lp,ymu,ys2 else: return lp,ymu else: return lp else: # inference mode if isinstance(inffunc, inf.Laplace): if der == None: # no derivative mode if y == None: y = np.zeros_like(mu) ymmu = y-mu lp = np.abs(ymmu)/b - np.log(2*b) if nargout>1: # derivative of log likelihood dlp = np.sign(ymmu)/b if nargout>2: # 2nd derivative of log likelihood d2lp = np.zeros_like(ymmu) if nargout>3: # 3rd derivative of log likelihood d3lp = np.zeros_like(ymmu) return lp,dlp,d2lp,d3lp else: return lp,dlp,d2lp else: return lp,dlp else: return lp else: # derivative w.r.t. hypers lp_dhyp = np.abs(y-mu)/b - 1 # derivative of log likelihood w.r.t. hypers dlp_dhyp = np.sign(mu-y)/b # first derivative, d2lp_dhyp = np.zeros(mu.shape) # and also of the second mu derivative return lp_dhyp, dlp_dhyp, d2lp_dhyp elif isinstance(inffunc, inf.EP): n = np.max([len(y.flatten()),len(mu.flatten()),len(s2.flatten()),len(sn.flatten())]) on = np.ones((n,1)) y = y*on; mu = mu*on; s2 = s2*on; sn = sn*on; fac = 1e3; # factor between the widths of the two distributions ... # ... from when one considered a delta peak, we use 3 orders of magnitude #idlik = np.reshape( (fac*sn) < np.sqrt(s2) , (sn.shape[0],) ) # Likelihood is a delta peak #idgau = np.reshape( (fac*np.sqrt(s2)) < sn , (sn.shape[0],) ) # Gaussian is a delta peak idlik = (fac*sn) < np.sqrt(s2) idgau = (fac*np.sqrt(s2)) < sn id = np.logical_and(np.logical_not(idgau),np.logical_not(idlik)) # interesting case in between if der == None: # no derivative mode lZ = np.zeros((n,1)) dlZ = np.zeros((n,1)) d2lZ = np.zeros((n,1)) if np.any(idlik): l = Gauss(log_sigma=np.log(s2[idlik])/2) a = l.proceed(mu[idlik], y[idlik]) lZ[idlik] = a[0]; dlZ[idlik] = a[1]; d2lZ[idlik] = a[2] if np.any(idgau): l = Laplace(log_hyp=np.log(sn[idgau])) a = l.proceed(mu=mu[idgau], y=y[idgau]) lZ[idgau] = a[0]; dlZ[idgau] = a[1]; d2lZ[idgau] = a[2] if np.any(id): # substitution to obtain unit variance, zero mean Laplacian tvar = s2[id]/(sn[id]**2+1e-16) tmu = (mu[id]-y[id])/(sn[id]+1e-16) # an implementation based on logphi(t) = log(normcdf(t)) zp = (tmu+np.sqrt(2)*tvar)/np.sqrt(tvar) zm = (tmu-np.sqrt(2)*tvar)/np.sqrt(tvar) ap = self._logphi(-zp)+np.sqrt(2)*tmu am = self._logphi( zm)-np.sqrt(2)*tmu apam = np.vstack((ap,am)).T lZ[id] = self._logsum2exp(apam) + tvar - np.log(sn[id]*np.sqrt(2.)) if nargout>1: lqp = -0.5*zp**2 - 0.5*np.log(2*np.pi) - self._logphi(-zp); # log( N(z)/Phi(z) ) lqm = -0.5*zm**2 - 0.5*np.log(2*np.pi) - self._logphi( zm); dap = -np.exp(lqp-0.5*np.log(s2[id])) + np.sqrt(2)/sn[id] dam = np.exp(lqm-0.5*np.log(s2[id])) - np.sqrt(2)/sn[id] _z1 = np.vstack((ap,am)).T _z2 = np.vstack((dap,dam)).T _x = np.array([[1],[1]]) dlZ[id] = self._expABz_expAx(_z1, _x, _z2, _x) if nargout>2: a = np.sqrt(8.)/sn[id]/np.sqrt(s2[id]); bp = 2./sn[id]**2 - (a - zp/s2[id])*np.exp(lqp) bm = 2./sn[id]**2 - (a + zm/s2[id])*np.exp(lqm) _x = np.reshape(np.array([1,1]),(2,1)) _z1 = np.reshape(np.array([ap,am]),(1,2)) _z2 = np.reshape(np.array([bp,bm]),(1,2)) d2lZ[id] = self._expABz_expAx(_z1, _x, _z2, _x) - dlZ[id]**2 return lZ,dlZ,d2lZ else: return lZ,dlZ else: return lZ else: # derivative mode dlZhyp = np.zeros((n,1)) if np.any(idlik): dlZhyp[idlik] = 0 if np.any(idgau): l = Laplace(log_hyp=np.log(sn[idgau])) a = l.proceed(mu=mu[idgau], y=y[idgau], inffunc='inf.Laplace', nargout=1) dlZhyp[idgau] = a[0] if np.any(id): # substitution to obtain unit variance, zero mean Laplacian tmu = (mu[id]-y[id])/(sn[id]+1e-16); tvar = s2[id]/(sn[id]**2+1e-16) zp = (tvar+tmu/np.sqrt(2))/np.sqrt(tvar); vp = tvar+np.sqrt(2)*tmu zm = (tvar-tmu/np.sqrt(2))/np.sqrt(tvar); vm = tvar-np.sqrt(2)*tmu dzp = (-s2[id]/sn[id]+tmu*sn[id]/np.sqrt(2)) / np.sqrt(s2[id]) dvp = -2*tvar - np.sqrt(2)*tmu dzm = (-s2[id]/sn[id]-tmu*sn[id]/np.sqrt(2)) / np.sqrt(s2[id]) dvm = -2*tvar + np.sqrt(2)*tmu lezp = self._lerfc(zp); # ap = exp(vp).*ezp lezm = self._lerfc(zm); # am = exp(vm).*ezm vmax = np.max(np.array([vp+lezp,vm+lezm]),axis=0); # subtract max to avoid numerical pb ep = np.exp(vp+lezp-vmax) em = np.exp(vm+lezm-vmax) dap = ep*(dvp - 2/np.sqrt(np.pi)*np.exp(-zp**2-lezp)*dzp) dam = em*(dvm - 2/np.sqrt(np.pi)*np.exp(-zm**2-lezm)*dzm) dlZhyp[id] = (dap+dam)/(ep+em) - 1; return dlZhyp # deriv. wrt hyp.lik elif isinstance(inffunc, inf.VB): n = len(s2.flatten()); b = np.zeros((n,1)); y = y*np.ones((n,1)); z = y return b,z def _lerfc(self,t): ''' numerically safe implementation of f(t) = log(1-erf(t)) = log(erfc(t))''' from scipy.special import erfc f = np.zeros_like(t) tmin = 20; tmax = 25 ok = ttmax # evaluate tight bound nok = np.logical_not(ok) interp = np.logical_and(nok,np.logical_not(bd)) # interpolate between both of them f[nok] = np.log(2/np.sqrt(np.pi)) -t[nok]**2 -np.log(t[nok]+np.sqrt( t[nok]**2+4/np.pi )) lam = 1/(1+np.exp( 12*(0.5-(t[interp]-tmin)/(tmax-tmin)) )) # interp. weights f[interp] = lam*f[interp] + (1-lam)*np.log(erfc( t[interp] )) f[ok] += np.log(erfc( t[ok] )) # safe eval return f def _expABz_expAx(self,A,x,B,z): ''' Computes y = ( (exp(A).*B)*z ) ./ ( exp(A)*x ) in a numerically safe way The function is not general in the sense that it yields correct values for all types of inputs. We assume that the values are close together. ''' N = A.shape[1] maxA = np.max(A,axis=1) # number of columns, max over columns maxA = np.array([maxA]).T A = A - np.dot(maxA, np.ones((1,N))) # subtract maximum value y = ( np.dot((np.exp(A)*B),z) ) / ( np.dot(np.exp(A),x) ) return y[0] def _logphi(self,z): ''' Safe implementation of the log of phi(x) = \int_{-\infty}^x N(f|0,1) df returns lp = log(normcdf(z)) ''' lp = np.zeros_like(z) # allocate memory zmin = -6.2; zmax = -5.5; ok = z>zmax # safe evaluation for large values bd = z 0: return PowerOfMean(self,number) else: print "only non-zero integers are supported for **" def getMean(self, x=None): ''' Get the mean vector. ''' pass def getDerMatrix(self, x=None, der=None): ''' Compute derivatives wrt. hyperparameters. :param x: training data :param int der: index of hyperparameter whose derivative to be computed :return: the corresponding derivative matrix ''' pass class ProductOfMean(Mean): '''Product of two mean fucntions.''' def __init__(self,mean1,mean2): self.mean1 = mean1 self.mean2 = mean2 if mean1.hyp and mean2.hyp: self._hyp = mean1.hyp + mean2.hyp elif not mean1.hyp: self._hyp = mean2.hyp elif not mean2.hyp: self._hyp = mean1.hyp def _setHyp(self,hyp): assert len(hyp) == len(self._hyp) len1 = len(self.mean1.hyp) self._hyp = hyp self.mean1.hyp = self._hyp[:len1] self.mean2.hyp = self._hyp[len1:] def _getHyp(self): return self._hyp hyp = property(_getHyp,_setHyp) def getMean(self, x=None): A = self.mean1.getMean(x) * self.mean2.getMean(x) return A def getDerMatrix(self, x=None, der=None): if der < len(self.mean1.hyp): A = self.mean1.getDerMatrix(x, der) * self.mean2.getMean(x) elif der < len(self.hyp): der2 = der - len(self.mean1.hyp) A = self.mean2.getDerMatrix(x, der2) * self.mean1.getMean(x) else: raise Exception("Error: der out of range for meanProduct") return A class SumOfMean(Mean): '''Sum of two mean functions.''' def __init__(self,mean1,mean2): self.mean1 = mean1 self.mean2 = mean2 if mean1.hyp and mean2.hyp: self._hyp = mean1.hyp + mean2.hyp elif not mean1.hyp: self._hyp = mean2.hyp elif not mean2.hyp: self._hyp = mean1.hyp def _setHyp(self,hyp): assert len(hyp) == len(self._hyp) len1 = len(self.mean1.hyp) self._hyp = hyp self.mean1.hyp = self._hyp[:len1] self.mean2.hyp = self._hyp[len1:] def _getHyp(self): return self._hyp hyp = property(_getHyp,_setHyp) def getMean(self, x=None): A = self.mean1.getMean(x) + self.mean2.getMean(x) return A def getDerMatrix(self, x=None, der=None): if der < len(self.mean1.hyp): A = self.mean1.getDerMatrix(x, der) elif der < len(self.hyp): der2 = der - len(self.mean1.hyp) A = self.mean2.getDerMatrix(x, der2) else: raise Exception("Error: der out of range for meanSum") return A class ScaleOfMean(Mean): '''Scale of a mean function.''' def __init__(self,mean,scalar): self.mean = mean if mean.hyp: self._hyp = [scalar] + mean.hyp else: self._hyp = [scalar] def _setHyp(self,hyp): assert len(hyp) == len(self._hyp) self._hyp = hyp self.mean.hyp = self._hyp[1:] def _getHyp(self): return self._hyp hyp = property(_getHyp,_setHyp) def getMean(self, x=None): c = self.hyp[0] # scale parameter A = c * self.mean.getMean(x) # accumulate means return A def getDerMatrix(self, x=None, der=None): c = self.hyp[0] # scale parameter if der == 0: # compute derivative w.r.t. c A = self.mean.getMean(x) else: A = c * self.mean.getDerMatrix(x,der-1) return A class PowerOfMean(Mean): '''Power of a mean fucntion.''' def __init__(self, mean, d): self.mean = mean if mean.hyp: self._hyp = [d] + mean.hyp else: self._hyp = [d] def _setHyp(self,hyp): assert len(hyp) == len(self._hyp) self._hyp = hyp self.mean.hyp = self._hyp[1:] def _getHyp(self): return self._hyp hyp = property(_getHyp,_setHyp) def getMean(self, x=None): d = np.abs(np.floor(self.hyp[0])) d = max(d,1) A = self.mean.getMean(x) **d # accumulate means return A def getDerMatrix(self, x=None, der=None): d = np.abs(np.floor(self.hyp[0])) d = max(d,1) if der == 0: # compute derivative w.r.t. c a = self.mean.getMean(x) A = a**d * np.log(a) else: A = d * self.mean.getMean(x) ** (d-1) * self.mean.getDerMatrix(x, der-1) return A class Zero(Mean): '''Zero mean.''' def __init__(self): self.hyp = [] self.name = '0' def getMean(self, x=None): n, D = x.shape A = np.zeros((n,1)) return A def getDerMatrix(self, x=None, der=None): n, D = x.shape A = np.zeros((n,1)) return A class One(Mean): ''' One mean.''' def __init__(self): self.hyp = [] self.name = '1' def getMean(self, x=None): n, D = x.shape A = np.ones((n,1)) return A def getDerMatrix(self, x=None, der=None): n, D = x.shape A = np.zeros((n,1)) return A class Const(Mean): ''' Constant mean function. hyp = [c] :param c: constant value for mean ''' def __init__(self, c=5.): self.hyp = [c] def getMean(self, x=None): n,D = x.shape A = self.hyp[0] * np.ones((n,1)) return A def getDerMatrix(self, x=None, der=None): n,D = x.shape if der == 0: # compute derivative vector wrt c A = np.ones((n,1)) else: A = np.zeros((n,1)) return A class Linear(Mean): ''' Linear mean function. self.hyp = alpha_list :param D: dimension of training data. Set if you want default alpha, which is 0.5 for each dimension. :alpha_list: scalar alpha for each dimension ''' def __init__(self, D=None, alpha_list=None): if alpha_list == None: self.hyp = [0.5 for i in xrange(D)] else: self.hyp = alpha_list def getMean(self, x=None): n, D = x.shape c = np.array(self.hyp) c = np.reshape(c,(len(c),1)) A = np.dot(x,c) return A def getDerMatrix(self, x=None, der=None): n, D = x.shape c = np.array(self.hyp) c = np.reshape(c,(len(c),1)) if isinstance(der, int) and der < D: # compute derivative vector wrt meanparameters A = np.reshape(x[:,der], (len(x[:,der]),1) ) else: A = np.zeros((n,1)) return A if __name__ == '__main__': pass pyGPs-1.2/pyGPs/Core/opt.py000066400000000000000000000361451236346020200155330ustar00rootroot00000000000000#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ # This is a object-oriented python implementation of gpml functionality # (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18). # based on the functional-version of python implementation # (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013) # # Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013 import numpy as np import gp from scipy.optimize import fmin_bfgs as bfgs from scipy.optimize import fmin_cg as cg from ..Optimization import minimize, scg from copy import deepcopy class Optimizer(object): def __init__(self, model=None, searchConfig = None): self.model = model def findMin(self, x, y): ''' Find minimal value based on negative-log-marginal-likelihood. optimalHyp, funcValue = findMin(x, y) where funcValue is the minimal negative-log-marginal-likelihood during optimization, and optimalHyp is a flattened numpy array (in sequence of meanfunc.hyp, covfunc.hyp, likfunc.hyp) of the hyparameters to achieve such value. You can achieve advanced search strategy by initializing Optimizer with searchConfig, which is an instance of pyGPs.Optimization.conf. See more in pyGPs.Optimization.conf and pyGPs.Core.gp.GP.setOptimizer, as well as in online documentation of section Optimizers. ''' pass def _nlml(self, hypInArray): '''Find negative-log-marginal-likelihood''' self._apply_in_objects(hypInArray) nlZ, dnlZ = self.model.fit(der=False) return nlZ def _dnlml(self, hypInArray): '''Find derivatives wrt. negative-log-marginal-likelihood''' self._apply_in_objects(hypInArray) nlZ, dnlZ, post = self.model.fit() dnlml_List = dnlZ.mean + dnlZ.cov + dnlZ.lik return np.array(dnlml_List) def _nlzAnddnlz(self, hypInArray): '''Find negative-log-marginal-likelihood and derivatives in one pass(faster)''' self._apply_in_objects(hypInArray) nlZ, dnlZ, post = self.model.fit() dnlml_List = dnlZ.mean + dnlZ.cov + dnlZ.lik return nlZ, np.array(dnlml_List) def _convert_to_array(self): '''Convert all hyparameters in the model to an array''' hyplist = self.model.meanfunc.hyp + self.model.covfunc.hyp + self.model.likfunc.hyp return np.array(hyplist) def _apply_in_objects(self, hypInArray): '''Apply the values in the input array to hyparameters of model.''' Lm = len(self.model.meanfunc.hyp) Lc = len(self.model.covfunc.hyp) hypInList = hypInArray.tolist() self.model.meanfunc.hyp = hypInList[:Lm] self.model.covfunc.hyp = hypInList[Lm:(Lm+Lc)] self.model.likfunc.hyp = hypInList[(Lm+Lc):] class CG(Optimizer): '''Conjugent gradient''' def __init__(self, model, searchConfig = None): super(CG, self).__init__() self.model = model self.searchConfig = searchConfig self.trailsCounter = 0 self.errorCounter = 0 def findMin(self, x, y): meanfunc = self.model.meanfunc covfunc = self.model.covfunc likfunc = self.model.likfunc inffunc = self.model.inffunc hypInArray = self._convert_to_array() try: opt = cg(self._nlml, hypInArray, self._dnlml, maxiter=100, disp=False, full_output=True) optimalHyp = deepcopy(opt[0]) funcValue = opt[1] warnFlag = opt[4] if warnFlag == 1: print "Maximum number of iterations exceeded." elif warnFlag == 2: print "Gradient and/or function calls not changing." except: self.errorCounter += 1 if not self.searchConfig: raise Exception("Can not use conjugate gradient. Try other hyparameters") self.trailsCounter += 1 if self.searchConfig: searchRange = self.searchConfig.meanRange + self.searchConfig.covRange + self.searchConfig.likRange if not (self.searchConfig.num_restarts or self.searchConfig.min_threshold): raise Exception('Specify at least one of the stop conditions') while True: self.trailsCounter += 1 # increase counter for i in xrange(hypInArray.shape[0]): # random init of hyp hypInArray[i]= np.random.uniform(low=searchRange[i][0], high=searchRange[i][1]) # value this time is better than optiaml min value try: thisopt = cg(self._nlml, hypInArray, self._dnlml, maxiter=100, disp=False, full_output=True) if thisopt[1] < funcValue: funcValue = thisopt[1] optimalHyp = thisopt[0] except: self.errorCounter += 1 if self.searchConfig.num_restarts and self.errorCounter > self.searchConfig.num_restarts/2: print "[CG] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter) raise Exception("Over half of the trails failed for conjugate gradient") if self.searchConfig.num_restarts and self.trailsCounter > self.searchConfig.num_restarts-1: # if exceed num_restarts print "[CG] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter) return optimalHyp, funcValue if self.searchConfig.min_threshold and funcValue <= self.searchConfig.min_threshold: # reach provided mininal print "[CG] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter) return optimalHyp, funcValue return optimalHyp, funcValue class BFGS(Optimizer): '''quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS)''' def __init__(self, model, searchConfig = None): super(BFGS, self).__init__() self.model = model self.searchConfig = searchConfig self.trailsCounter = 0 self.errorCounter = 0 def findMin(self, x, y): meanfunc = self.model.meanfunc covfunc = self.model.covfunc likfunc = self.model.likfunc inffunc = self.model.inffunc hypInArray = self._convert_to_array() try: opt = bfgs(self._nlml, hypInArray, self._dnlml, maxiter=100, disp=False, full_output=True) optimalHyp = deepcopy(opt[0]) funcValue = opt[1] warnFlag = opt[6] if warnFlag == 1: print "Maximum number of iterations exceeded." elif warnFlag == 2: print "Gradient and/or function calls not changing." except: self.errorCounter += 1 if not self.searchConfig: raise Exception("Can not use BFGS. Try other hyparameters") self.trailsCounter += 1 if self.searchConfig: searchRange = self.searchConfig.meanRange + self.searchConfig.covRange + self.searchConfig.likRange if not (self.searchConfig.num_restarts or self.searchConfig.min_threshold): raise Exception('Specify at least one of the stop conditions') while True: self.trailsCounter += 1 # increase counter for i in xrange(hypInArray.shape[0]): # random init of hyp hypInArray[i]= np.random.uniform(low=searchRange[i][0], high=searchRange[i][1]) # value this time is better than optiaml min value try: thisopt = bfgs(self._nlml, hypInArray, self._dnlml, maxiter=100, disp=False, full_output=True) if thisopt[1] < funcValue: funcValue = thisopt[1] optimalHyp = thisopt[0] except: self.errorCounter += 1 if self.searchConfig.num_restarts and self.errorCounter > self.searchConfig.num_restarts/2: print "[BFGS] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter) raise Exception("Over half of the trails failed for BFGS") if self.searchConfig.num_restarts and self.trailsCounter > self.searchConfig.num_restarts-1: # if exceed num_restarts print "[BFGS] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter) return optimalHyp, funcValue if self.searchConfig.min_threshold and funcValue <= self.searchConfig.min_threshold: # reach provided mininal print "[BFGS] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter) return optimalHyp, funcValue return optimalHyp, funcValue class Minimize(Optimizer): '''minimize by Carl Rasmussen (python implementation of "minimize" in GPML)''' def __init__(self, model, searchConfig = None): super(Minimize, self).__init__() self.model = model self.searchConfig = searchConfig self.trailsCounter = 0 self.errorCounter = 0 def findMin(self, x, y): meanfunc = self.model.meanfunc covfunc = self.model.covfunc likfunc = self.model.likfunc inffunc = self.model.inffunc hypInArray = self._convert_to_array() try: opt = minimize.run(self._nlzAnddnlz, hypInArray, length=-40) optimalHyp = deepcopy(opt[0]) funcValue = opt[1][-1] except: self.errorCounter += 1 if not self.searchConfig: raise Exception("Can not use minimize. Try other hyparameters") self.trailsCounter += 1 if self.searchConfig: searchRange = self.searchConfig.meanRange + self.searchConfig.covRange + self.searchConfig.likRange if not (self.searchConfig.num_restarts or self.searchConfig.min_threshold): raise Exception('Specify at least one of the stop conditions') while True: self.trailsCounter += 1 # increase counter for i in xrange(hypInArray.shape[0]): # random init of hyp hypInArray[i]= np.random.uniform(low=searchRange[i][0], high=searchRange[i][1]) # value this time is better than optiaml min value try: thisopt = minimize.run(self._nlzAnddnlz, hypInArray, length=-40) if thisopt[1][-1] < funcValue: funcValue = thisopt[1][-1] optimalHyp = thisopt[0] except: self.errorCounter += 1 if self.searchConfig.num_restarts and self.errorCounter > self.searchConfig.num_restarts/2: print "[Minimize] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter) raise Exception("Over half of the trails failed for minimize") if self.searchConfig.num_restarts and self.trailsCounter > self.searchConfig.num_restarts-1: # if exceed num_restarts print "[Minimize] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter) return optimalHyp, funcValue if self.searchConfig.min_threshold and funcValue <= self.searchConfig.min_threshold: # reach provided mininal print "[Minimize] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter) return optimalHyp, funcValue return optimalHyp, funcValue class SCG(Optimizer): '''Scaled conjugent gradient (faster than CG)''' def __init__(self, model, searchConfig = None): super(SCG, self).__init__() self.model = model self.searchConfig = searchConfig self.trailsCounter = 0 self.errorCounter = 0 def findMin(self, x, y): meanfunc = self.model.meanfunc covfunc = self.model.covfunc likfunc = self.model.likfunc inffunc = self.model.inffunc hypInArray = self._convert_to_array() try: opt = scg.run(self._nlzAnddnlz, hypInArray) optimalHyp = deepcopy(opt[0]) funcValue = opt[1][-1] except: self.errorCounter += 1 if not self.searchConfig: raise Exception("Can not use Scaled conjugate gradient. Try other hyparameters") self.trailsCounter += 1 if self.searchConfig: searchRange = self.searchConfig.meanRange + self.searchConfig.covRange + self.searchConfig.likRange if not (self.searchConfig.num_restarts or self.searchConfig.min_threshold): raise Exception('Specify at least one of the stop conditions') while True: self.trailsCounter += 1 # increase counter for i in xrange(hypInArray.shape[0]): # random init of hyp hypInArray[i]= np.random.uniform(low=searchRange[i][0], high=searchRange[i][1]) # value this time is better than optiaml min value try: thisopt = scg.run(self._nlzAnddnlz, hypInArray) if thisopt[1][-1] < funcValue: funcValue = thisopt[1][-1] optimalHyp = thisopt[0] except: self.errorCounter += 1 if self.searchConfig.num_restarts and self.errorCounter > self.searchConfig.num_restarts/2: print "[SCG] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter) raise Exception("Over half of the trails failed for Scaled conjugate gradient") if self.searchConfig.num_restarts and self.trailsCounter > self.searchConfig.num_restarts-1: # if exceed num_restarts print "[SCG] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter) return optimalHyp, funcValue if self.searchConfig.min_threshold and funcValue <= self.searchConfig.min_threshold: # reach provided mininal print "[SCG] %d out of %d trails failed during optimization" % (self.errorCounter, self.trailsCounter) return optimalHyp, funcValue return optimalHyp, funcValue pyGPs-1.2/pyGPs/Core/tools.py000066400000000000000000000176741236346020200160770ustar00rootroot00000000000000#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ # This is a object-oriented python implementation of gpml functionality # (Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18). # based on the functional-version of python implementation # (Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013) # # Copyright (c) by Marion Neumann and Shan Huang, 30/09/2013 import numpy as np import matplotlib.pyplot as plt import sys from math import sqrt def solve_chol(L, B): ''' Solve linear equations from the Cholesky factorization. Solve A*X = B for X, where A is square, symmetric, positive definite. The input to the function is R the Cholesky decomposition of A and the matrix B. Example: X = solve_chol(chol(A),B); ''' try: assert(L.shape[0] == L.shape[1] and L.shape[0] == B.shape[0]) except AssertionError: raise Exception('Wrong sizes of matrix arguments in solve_chol.py'); X = np.linalg.solve(L,np.linalg.solve(L.T,B)) return X def unique(x): ''' Return a list with unique elements. ''' # First flatten x y = [item for sublist in x for item in sublist] if isinstance(x,np.ndarray): n,D = x.shape assert(D == 1) y = np.array( list(set(x[:,0])) ) y = np.reshape(y, (len(y),1)) else: y = list(set(y)) return y def brentmin(xlow,xupp,Nitmax,tol,f,nout=None,*args): ''' Brent's minimization method in one dimension. Given a function f, and given a search interval this routine isolates the minimum of fractional precision of about tol using Brent's method. Reference: Section 10.2 Parabolic Interpolation and Brent's Method in One Dimension Press, Teukolsky, Vetterling & Flannery Numerical Recipes in C, Cambridge University Press, 2002 This is a python implementation of gpml functionality (Copyright (c) by Hannes Nickisch 2010-01-10). xmin,fmin,funccout,varargout = BRENTMIN(xlow,xupp,Nit,tol,f,nout,varargin) :param xlow: lower bound. i.e. search interval such that xlow<=xmin<=xupp :param xupp: uppper bound. i.e. search interval such that xlow<=xmin<=xupp :param Nitmax: maximum number of function evaluations made by the routine :param tol: fractional precision :param f: [y,varargout{:}] = f(x,varargin{:}) is the function :param nout: no. of outputs of f (in varargout) in addition to the y value :return: fmin is minimal function value. xmin is corresponding abscissa-value funccount is the number of function evaluations made. varargout is additional outputs of f at optimum. ''' # code taken from # Section 10.2 Parabolic Interpolation and Brent's Method in One Dimension # Press, Teukolsky, Vetterling & Flannery # Numerical Recipes in C, Cambridge University Press, 2002 # # [xmin,fmin,funccout,varargout] = BRENTMIN(xlow,xupp,Nit,tol,f,nout,varargin) # Given a function f, and given a search interval this routine isolates # the minimum of fractional precision of about tol using Brent's method. # # INPUT # ----- # xlow,xupp: search interval such that xlow<=xmin<=xupp # Nitmax: maximum number of function evaluations made by the routine # tol: fractional precision # f: [y,varargout{:}] = f(x,varargin{:}) is the function # nout: no. of outputs of f (in varargout) in addition to the y value # # OUTPUT # ------ # fmin: minimal function value # xmin: corresponding abscissa-value # funccount: number of function evaluations made # varargout: additional outputs of f at optimum # # This is a python implementation of gpml functionality (Copyright (c) by # Hannes Nickisch 2010-01-10). if nout == None: nout = 0 eps = sys.float_info.epsilon # tolerance is no smaller than machine's floating point precision tol = max(tol,eps) # Evaluate endpoints vargout = f(xlow,*args); fa = vargout[0][0] vargout = f(xupp,*args); fb = vargout[0][0] funccount = 2 # number of function evaluations # Compute the start point seps = sqrt(eps) c = 0.5*(3.0 - sqrt(5.0)) # golden ratio a = xlow b = xupp v = a + c*(b-a) w = v xf = v d = 0. e = 0. x = xf vargout = f(x,*args) fx = vargout[0][0] varargout = vargout[1:] funccount += 1 fv = fx; fw = fx xm = 0.5*(a+b) tol1 = seps*abs(xf) + tol/3.0; tol2 = 2.0*tol1 # Main loop while ( abs(xf-xm) > (tol2 - 0.5*(b-a)) ): gs = True # Is a parabolic fit possible if abs(e) > tol1: # Yes, so fit parabola gs = False r = (xf-w)*(fx-fv) q = (xf-v)*(fx-fw) p = (xf-v)*q-(xf-w)*r q = 2.0*(q-r) if q > 0.0: p = -p q = abs(q) r = e; e = d # Is the parabola acceptable if ( (abs(p)q*(a-xf)) and (p= xm: e = a-xf else: e = b-xf d = c*e # The function must not be evaluated too close to xf si = cmp(d,0) if (d == 0): si += 1 x = xf + si * max(abs(d),tol1) vargout = f(x,*args); fu = vargout[0][0]; varargout = vargout[1:] funccount += 1 # Update a, b, v, w, x, xm, tol1, tol2 if fu <= fx: if x >= xf: a = xf else: b = xf v = w; fv = fw w = xf; fw = fx xf = x; fx = fu else: # fu > fx if x < xf: a = x else: b = x if ( (fu <= fw) or (w == xf) ): v = w; fv = fw w = x; fw = fu elif ( (fu <= fv) or ((v == xf) or (v == w)) ): v = x; fv = fu xm = 0.5*(a+b) tol1 = seps*abs(xf) + tol/3.0; tol2 = 2.0*tol1 if funccount >= Nitmax: # typically we should not get here # print 'Warning: Specified number of function evaluation reached (brentmin)' break # check that endpoints are less than the minimum found if ( (fa < fx) and (fa <= fb) ): xf = xlow; fx = fa elif fb < fx: xf = xupp; fx = fb fmin = fx xmin = xf vargout = [xmin,fmin,funccount] for vv in varargout: vargout.append(vv) return vargout def cholupdate(R,x,sgn='+'): '''Placeholder for a python version of MATLAB's cholupdate. Now it is O(n^3)''' if len(x.shape) == 1: # Reshape x so that the dot product below is correct x = np.reshape(x,(x.shape[0],1)) assert(R.shape[0] == x.shape[0]) A = np.dot(R.T,R) if sgn == '+': R1 = A + np.dot(x,x.T) elif sgn == '-': R1 = A - np.dot(x,x.T) else: raise Exception('Sign needs to be + or - in cholupdate') return np.linalg.cholesky(R1).T pyGPs-1.2/pyGPs/Demo/000077500000000000000000000000001236346020200143425ustar00rootroot00000000000000pyGPs-1.2/pyGPs/Demo/data_for_demo/000077500000000000000000000000001236346020200171255ustar00rootroot00000000000000pyGPs-1.2/pyGPs/Demo/data_for_demo/README.txt000066400000000000000000000035461236346020200206330ustar00rootroot00000000000000 ---------------------------------------------------- Johns Hopkins University Ionosphere database Source Information: -- Donor: Vince Sigillito (vgs@aplcen.apl.jhu.edu) -- Date: 1989 -- Source: Space Physics Group Applied Physics Laboratory Johns Hopkins University Johns Hopkins Road Laurel, MD 20723 The relevant paper is: Sigillito, V. G., Wing, S. P., Hutton, L. V., \& Baker, K. B. (1989). Classification of radar returns from the ionosphere using neural networks. Johns Hopkins APL Technical Digest, 10, 262-266. ---------------------------------------------------- For more detail attribute description and source information, see "ionosphere.names.txt" ========================================================================= The USPS digits data were gathered at the Center of Excellence in Document Analysis and Recognition (CEDAR) at SUNY Buffalo, as part of a project sponsored by the US Postal Service. The dataset is described in A Database for Handwritten Text Recognition Research, J. J. Hull, IEEE PAMI 16(5) 550-554, 1994. ========================================================================= There are two datasets available: - usps_all.mat is the original sups dataset - usps_resampled.mat is collected in a different way of original dataset. See http://www.gaussianprocess.org/gpml/data/ ========================================================================= If you want to test on binary classification tasks, there are also two scripts for conveniently load databases with only selecting two classes: - [Matlab]loadBinaryUSPS.m Copyright (C) 2005 and 2006, Carl Edward Rasmussen - [Python]loadBinaryUSPS.py Copyright (C) 2013, Shan Huang With slight changes, you can use these scripts to load any subsets of dataset you want to evaluate on. pyGPs-1.2/pyGPs/Demo/data_for_demo/build.py000066400000000000000000000221321236346020200205760ustar00rootroot00000000000000import numpy as np def save_data_regresssion(): # n = 20 # number of labeled/training data # D = 1 # dimension of input data x = np.array([[2.083970427750732, -0.821018066101379, -0.617870699182597, -1.183822608860694,\ 0.274087442277144, 0.599441729295593, 1.768897919204435, -0.465645549031928,\ 0.588852784375935, -0.832982214438054, -0.512106527960363, 0.277883144210116,\ -0.065870426922211, -0.821412363806325, 0.185399443778088, -0.858296174995998,\ 0.370786630037059, -1.409869162416639,-0.144668412325022,-0.553299615220374]]).T y = np.array([[4.549203746331698, 0.371985574437271, 0.711307965514790, -0.013212893618430, 2.255473255338191,\ 1.009915749295733, 3.744675937965029, 0.424592771793202, 1.322833652295811, 0.278298293510020,\ 0.267229130945574, 2.200112286723833, 1.200609983308969, 0.439971697236094, 2.628580433511255,\ 0.503774817336353, 1.942525313820564, 0.579133950013327, 0.670874423968554, 0.377353755100965]]).T # TEST points # test points evenly distributed in the interval [-2, 2.5] xstar = np.array(range(-200,250,4), dtype=np.float64, ndmin=2).T xstar /= 100 np.savez('regression_data', x=x, y=y, xstar=xstar) def save_data_classification(): # Synthetic data for binary classification: two partially overlapping # Gaussians in two dimensions. 120 data points are generated from two # Gaussians with different means and covariances. One Gaussian is # isotropic and contains 2/3 of the data (blue), the other is highly # correlated and contains 1/3 of the points (red). Note, that the # labels for the targets are -1/+1 (and not 0/1). n1 = 80; n2 = 40 x1 = np.array([[0.089450165731417, -0.000700765006939],\ [ 1.171605560541542, 1.177765337635947],\ [ 1.404722675089394, -0.017417915887421],\ [ 0.556096196907929, -1.489370243839215],\ [ 1.213163445267992, 0.044545401368647],\ [ 0.173404742510759, -0.675668036759603],\ [ 2.225008556585363, 0.469803193769368],\ [ 1.470329290331445, 0.887642323697526],\ [ 2.715199208821485, 0.621044646503113],\ [ 0.173640760494328, -0.936054178730056],\ [ 2.038152815025167, 0.262587298316711],\ [ 1.670218375320427, -2.633186886994263],\ [ 0.270098501389591, -0.948779657473203],\ [ 1.396339236138275, -1.114992287201776],\ [-1.482070589718501, -0.654590652482805],\ [-1.493788226272929, 0.382017940248275],\ [ 1.025083846875763, -0.860344923788873],\ [ 0.750316336734172, -0.101864205602753],\ [ 0.184311310148912, -0.258523866245887],\ [ 0.221868667121623, -1.393954437105630],\ [ 2.258881477897777, -0.786806071526136],\ [ 1.211362530151533, -0.423431246029886],\ [ 1.525307406741207, -0.097975367602030],\ [ 0.978930232706465, 0.476154349549524],\ [ 1.347884229346280, -0.248408186838667],\ [ 1.205779546204216, -0.090878327349907],\ [ 0.124388644862000, 0.599612645000285],\ [ 0.784044356662233, 0.356596736271853],\ [ 1.060216683845210, -0.318474838087900],\ [ 1.678114484474938, 0.678735373910422],\ [ 0.973851135005570, 0.024880700382574],\ [ 0.016237746864886, -0.480899874254564],\ [ 0.979406721923196, 0.697708815321128],\ [ 2.217307638531248, -0.956931847027775],\ [ 2.150475558834153, 1.059031573329512],\ [ 1.050502393215048, 0.532141747419667],\ [ 1.210593098269218, -0.318123542280113],\ [ 0.426309208807901, -0.571727978045793],\ [ 0.742552105732714, -0.122112766396886],\ [ 0.757210723588679, 0.862002000781123],\ [-0.431639130160791, -0.763118261936640],\ [-0.748398486307095, -0.603667649379360],\ [ 0.975086541108249, -1.525297946453790],\ [ 0.074503762788667, -0.092155036190678],\ [-0.668889572018935, 1.305400680048752],\ [ 0.725632503186580, 0.096286255882168],\ [-1.042270707136463, 1.297009698531055],\ [ 1.943144890398260, -1.051176922438962],\ [ 1.191448645802597, 0.261349747400059],\ [ 0.778004017505022, -1.046301123377022],\ [ 0.628873970760607, 1.103926629619643],\ [ 1.295113890591403, -0.479519217798997],\ [ 1.522065175744686, 0.993476032742058],\ [ 1.100255776045601, 0.961069161713818],\ [-0.593243832838153, -0.479418953496258],\ [ 2.023196521366462, -0.275055494808503],\ [-0.788103134597041, -1.090707985778480],\ [-0.085168420896236, 1.226858390046108],\ [ 1.691706923196703, -1.153144804780540],\ [ 1.989279380395157, 1.974704317386435],\ [ 0.398799861652602, 3.051291814188982],\ [-0.707217210772927, 0.185505264874794],\ [ 0.697550136765320, 0.222287208720035],\ [ 2.186126058382323, -0.327829143438683],\ [ 1.368068331060010, 1.708138258453435],\ [ 0.883049126818189, -1.334269372314072],\ [ 1.737643116893527, 0.618452933813739],\ [ 2.002228743955222, 0.103381966018445],\ [-0.202638622737115, 0.495024938090909],\ [ 0.543309203560769, -0.802120609128192],\ [-1.796161599703804, -0.054795478648902],\ [ 1.460693782000059, 0.750052171180825],\ [ 0.133277872804608, -1.154891068006907],\ [ 0.203670382700157, -0.480336687666025],\ [-0.278985011909341, 0.030578590108392],\ [ 2.070490237052893, 2.420782751903098],\ [ 0.599023881366768, -1.673208560658818],\ [ 0.140506592147238, 0.804938444757444],\ [-0.980799204108985, -1.847987723222053],\ [-0.102350006007740, -0.822093851434857]]) x2 = np.array([[1.160257057434194, 1.544111720606185],\ [-0.458434595629321, 0.205667827100987],\ [-1.053562345687376, -0.614938261650010],\ [-1.687901005751336, -0.780028275457715],\ [-0.467035854712698, 0.561692074343868],\ [-0.703391186121452, 0.281301267639200],\ [-1.568557779993616, -0.629129013661319],\ [-2.176478596101226, -1.176211396013793],\ [ 0.768109265900499, 1.376893437232103],\ [-0.514772970064353, 0.474264363701950],\ [-1.301924381487904, -0.525179228127957],\ [-1.312024947004566, -0.049469442305628],\ [-0.623417800418214, 0.226456899059445],\ [ 0.020290591370131, 0.374055846421580],\ [-1.002901826023476, 0.076597486786743],\ [-2.553713136283273, -1.731788289864902],\ [-1.788156378743716, -0.742460481943494],\ [-1.119582270077321, -0.256154464598782],\ [-0.423084091988017, 0.395108309297119],\ [-1.645945345460644, -1.216319293733455],\ [ 0.227805611684674, 0.925948003854262],\ [-1.298719171366801, -0.965511301629466],\ [-0.618292817021891, 0.140045887498202],\ [ 0.794935039731655, 1.917830760420081],\ [-0.213709179946402, 0.617751634356751],\ [-0.474251035850546, -0.054854432018974],\ [ 0.056077816960464, 1.046282980014428],\ [ 0.887136693467512, 1.536490289895764],\ [ 1.377161915854166, 1.764872700787871],\ [-0.901195709427863, -0.340855547886558],\ [-0.783104424735034, -0.330927422324566],\ [-1.507139570543989, 0.137504213149820],\ [-0.348999111724700, 0.235931187612453],\ [-0.367309385513174, 0.655996377722041],\ [-0.050622309620072, 0.410969334468070],\ [ 1.734919039047271, 2.611080177877894],\ [-0.567413078682755, -0.458249564234885],\ [-0.622230797920433, 0.258401595566888],\ [-1.642146761593230, -1.138579130251617],\ [-0.285298076847255, 0.085451489400687]]) x = np.concatenate((x1,x2),axis=0) y = np.concatenate((-np.ones((1,n1)),np.ones((1,n2))),axis=1).T # For plotting, we superimpose the data points with the posterior equi-probability contour # lines for the probability of class two given complete information about the generating mechanism. t1,t2 = np.meshgrid(np.arange(-4,4.1,0.1),np.arange(-4,4.1,0.1)) t = np.array(zip(np.reshape(t1,(np.prod(t1.shape),)),np.reshape(t2,(np.prod(t2.shape),)))) # these are the test inputs n = t.shape[0] tmm = np.zeros_like(t) S1 = np.eye(2); S2 = np.array([[1, 0.95], [0.95, 1]]) m1 = np.array([0.75, 0]); m2 = np.array([-0.75, 0]) tmm[:,0] = t[:,0] - m1[0]; tmm[:,1] = t[:,1] - m1[1] p1 = n1*np.exp( (-np.dot(tmm,np.linalg.inv(S1))*tmm/2).sum(axis=1) ) tmm[:,0] = t[:,0] - m2[0]; 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Oۿ`s-?`ƕ? ? :?Ԁ?`Zۿ>*M y#ٿ A?7?`ʿ@.ҿ7?ҿ=@o?`5?׿|@P|ݿ T?` }?x?e?_?ֿ}ؿ`?@`?>1@ ?@? ?`[?)?ڠ?e?Q ?h?f63_Xݿ? p??`c?@4S? v??5޶?`<8kk duҿ,?1?@h#? }U?8cֿP@ou5@xk ) test_labels4kpyGPs-1.2/pyGPs/Demo/demo_GPC.py000066400000000000000000000060611236346020200163340ustar00rootroot00000000000000#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ import pyGPs import numpy as np # To have a gerneral idea, # you may want to read demo_GPR, demo_kernel and demo_optimization first! # Here, the focus is on the difference of classification model. print '' print '---------------------GPC DEMO-------------------------' #---------------------------------------------------------------------- # Load demo data (generated from Gaussians) #---------------------------------------------------------------------- # GPC target class are +1 and -1 demoData = np.load('data_for_demo/classification_data.npz') x = demoData['x'] # training data y = demoData['y'] # training target z = demoData['xstar'] # test data # only needed for 2-d contour plotting x1 = demoData['x1'] # x for class 1 (with label -1) x2 = demoData['x2'] # x for class 2 (with label +1) t1 = demoData['t1'] # y for class 1 (with label -1) t2 = demoData['t2'] # y for class 2 (with label +1) p1 = demoData['p1'] # prior for class 1 (with label -1) p2 = demoData['p2'] # prior for class 2 (with label +1) #---------------------------------------------------------------------- # First example -> state default values #---------------------------------------------------------------------- print 'Basic Example' model = pyGPs.GPC() # binary classification (default inference method: EP) model.fit(x, y) # fit default model (mean zero & rbf kernel) with data model.optimize(x, y) # optimize hyperparamters (default optimizer: single run minimize) model.predict(z) # predict test cases #---------------------------------------------------------------------- # GP classification example #---------------------------------------------------------------------- print 'More Advanced Example' # Start from a new model model = pyGPs.GPC() # Analogously to GPR k = pyGPs.cov.RBFard(log_ell_list=[0.05,0.17], log_sigma=1.) model.setPrior(kernel=k) model.plotData_2d(x1,x2,t1,t2,p1,p2) model.fit(x, y) print "Negative log marginal liklihood before:", round(model.nlZ,3) model.optimize(x, y) print "Negative log marginal liklihood optimized:", round(model.nlZ,3) # Prediction n = z.shape[0] ymu, ys2, fmu, fs2, lp = model.predict(z, ys=np.ones((n,1))) # GPC.plot() is a toy method for 2-d data # plot log probability distribution for class +1 model.plot(x1,x2,t1,t2) print '--------------------END OF DEMO-----------------------' pyGPs-1.2/pyGPs/Demo/demo_GPC_FITC.py000066400000000000000000000070511236346020200171410ustar00rootroot00000000000000#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ import pyGPs import numpy as np # To have a gerneral idea, # you may want to read demo_GPR, demo_kernel and demo_optimization first! # Here, the focus is on the difference of FITC classification. print '' print '-------------------GPC_FITC DEMO----------------------' #---------------------------------------------------------------------- # Load demo data (generated from Gaussians) #---------------------------------------------------------------------- # GPC_FITC target class are +1 and -1 demoData = np.load('data_for_demo/classification_data.npz') x = demoData['x'] # training data y = demoData['y'] # training target z = demoData['xstar'] # test data # only needed for 2-d contour plotting x1 = demoData['x1'] # x for class 1 (with label -1) x2 = demoData['x2'] # x for class 2 (with label +1) t1 = demoData['t1'] # y for class 1 (with label -1) t2 = demoData['t2'] # y for class 2 (with label +1) p1 = demoData['p1'] # prior for class 1 (with label -1) p2 = demoData['p2'] # prior for class 2 (with label +1) #---------------------------------------------------------------------- # Sparse GP classification (FITC) example #---------------------------------------------------------------------- print "Example 1: default inducing points" # Start from a new model model = pyGPs.GPC_FITC() # Notice if you want to use default inducing points: # You MUST call setData(x,y) FIRST! # The default inducing points is a grid(hypercube in higher dimension), where # each dimension has 5 values in same step between min and max value of data by default. model.setData(x, y) # To set value per dimension use: # model.setData(x, y, value_per_axis=10) model.optimize() print "Negative log marginal liklihood optimized:", round(model.nlZ,3) # Prediction n = z.shape[0] model.predict(z, ys=np.ones((n,1))) # Again, plot() is a toy method for 2-d data model.plot(x1,x2,t1,t2) print '------------------------------------------------------' print "Example 2: user-defined inducing points" model = pyGPs.GPC_FITC() # You can define inducing points yourself. # u = np.array([]) u1,u2 = np.meshgrid(np.linspace(-2,2,5),np.linspace(-2,2,5)) u = np.array(zip(np.reshape(u2,(np.prod(u2.shape),)),np.reshape(u1,(np.prod(u1.shape),)))) # and specify inducing point when seting prior m = pyGPs.mean.Zero() k = pyGPs.cov.RBFard(log_ell_list=[0.05,0.17], log_sigma=1.) model.setPrior(mean=m, kernel=k, inducing_points=u) # The rest is analogous to what we have done before. model.setData(x, y) model.fit() print "Negative log marginal liklihood before optimization:", round(model.nlZ,3) model.optimize() print "Negative log marginal liklihood optimized:", round(model.nlZ,3) # predict n = z.shape[0] ymu, ys2, fmu, fs2, lp = model.predict(z, ys=np.ones((n,1))) model.plot(x1,x2,t1,t2) print '--------------------END OF DEMO-----------------------' pyGPs-1.2/pyGPs/Demo/demo_GPMC.py000066400000000000000000000070341236346020200164520ustar00rootroot00000000000000#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ import pyGPs from pyGPs.Validation import valid import numpy as np from scipy.io import loadmat # To have a gerneral idea, # you may want to read demo_GPR, demo_kernel and demo_optimization first! # Here, the focus is on multi-class classification. print '' print '---------------------GPMC DEMO-----------------------' # GPMC is NOT based on multi-class Laplace Approximation. # It works as a one vs. one classification wrapper # i.e. GPMC trains GPC model for each combinations of two classes # and uses voting scheme over all results to determine the final class # It only returns the predictive class with highest rating, # but no other values (such as variance) returned. # Lets see a practical example to classify 10(0~9) hand-writen digits, # using USPS digits dataset. #---------------------------------------------------------------------- # Load USPS digits dataset #---------------------------------------------------------------------- data = loadmat('data_for_demo/usps_resampled.mat') x = data['train_patterns'].T # train patterns y = data['train_labels'].T # train labels xs = data['test_patterns'].T # test patterns ys = data['test_labels'].T # test labels # To be used in GPMC, we need to change label to integer from 0 to n(number of classes) # here class value should be 0,1,...,9. y = np.argmax(y, axis=1) y = np.reshape(y, (y.shape[0],1)) ys = np.argmax(ys, axis=1) ys = np.reshape(ys, (ys.shape[0],1)) # To save some time for demo, # lets reduce the number of training and testing patterns x = x[:100,:] y = y[:100,:] xs = xs[:20,:] ys = ys[:20,:] #---------------------------------------------------------------------- # GPMC example #---------------------------------------------------------------------- # State model with 10 classes model = pyGPs.GPMC(10) # Set data to model model.setData(x,y) # optimize default GPC model (see demo_GPC) for each binary classification problem, # and decide label for test patterns of hand-writen digits # prdictive_vote[i,j] is the probability of being class j for test pattern i predictive_vote = model.optimizeAndPredict(xs) predictive_class = np.argmax(predictive_vote, axis=1) predictive_class = np.reshape(predictive_class, (predictive_class.shape[0],1)) # Accuracy of recognized digit acc = valid.ACC(predictive_class, ys) print "Accuracy of recognizing hand-writen digits:", round(acc,2) #---------------------------------------------------------------------- # A bit more things you can do #---------------------------------------------------------------------- # Just like we did for GP classification # You can use specify the setting for all binary classificiation problem by: m = pyGPs.mean.Zero() k = pyGPs.cov.RBF() model.setPrior(mean=m,kernel=k) model.useInference("Laplace") # Beside optimizeAndPredict(xs), # there is also an option to predict without optimization # model.fitAndPredict(xs) print '--------------------END OF DEMO-----------------------' pyGPs-1.2/pyGPs/Demo/demo_GPR.py000066400000000000000000000105501236346020200163510ustar00rootroot00000000000000#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ import pyGPs import numpy as np # This demo will not only introduce GP regression model, # but provides a gerneral insight of our tourbox. # You may want to read it before reading other models. # current possible models are: # pyGPs.GPR -> Regression # pyGPs.GPC -> Classification # pyGPs.GPR_FITC -> Sparse GP Regression # pyGPs.GPC_FITC -> Sparse GP Classification # pyGPs.GPMC -> Muli-class Classification print '' print '---------------------GPR DEMO-------------------------' #---------------------------------------------------------------------- # Load demo data (generated from Gaussians) #---------------------------------------------------------------------- demoData = np.load('data_for_demo/regression_data.npz') x = demoData['x'] # training data y = demoData['y'] # training target z = demoData['xstar'] # test data #---------------------------------------------------------------------- # A five-line example #---------------------------------------------------------------------- print 'Basic Example' model = pyGPs.GPR() # model model.optimize(x, y) # optimize hyperparamters (default optimizer: single run minimize) model.predict(z) # predict test cases model.plot() # and plot result #---------------------------------------------------------------------- # Now lets do another example to get more insight to the toolbox #---------------------------------------------------------------------- print 'More Advanced Example' model = pyGPs.GPR() # start from a new model # Specify non-default mean and covariance functions # @SEE doc_kernel_mean for documentation of all kernels/means m = pyGPs.mean.Zero() k = pyGPs.cov.RBF() model.setPrior(kernel=k) # Specify optimization method (single run "Minimize" by default) # @SEE doc_optimization for documentation of optimization methods model.setOptimizer("Minimize", num_restarts=30) # Instead of fit(), which only fits data using given hyperparameters, # optimize() will optimize hyperparamters based on marginal likelihood # the deafult mean will be adapted to the average value of the training labels.. # ..if you do not specify mean function by your own. model.optimize(x, y) model.plotData_1d() # There are several propertys you can get from the model # For example: # model.nlZ # model.dnlZ.cov # model.dnlZ.lik # model.dnlZ.mean # model.posterior.sW # model.posterior.alpha # model.posterior.L # model.covfunc.hyp # model.meanfunc.hyp # model.likfunc.hyp # model.ym (predictive means) # model.ys2 (predictive variances) # model.fm (predictive latent means) # model.fs2 (predictive latent variances) # model.lp (log predictive probability) print 'Optimized negative log marginal likelihood:', round(model.nlZ,3) # Predict test data # output mean(ymu)/variance(ys2), latent mean(fmu)/variance(fs2), and log predictive probabilities(lp) ym, ys2, fmu, fs2, lp = model.predict(z) # Set range of axis for plotting # NOTE: plot() is a toy method only for 1-d data model.plot() # model.plot(axisvals=[-1.9, 1.9, -0.9, 3.9])) #---------------------------------------------------------------------- # A bit more things you can do #---------------------------------------------------------------------- # [For all model] Speed up prediction time if you know posterior in advance post = model.posterior # already known before ym, ys2, fmu, fs2, lp = model.predict_with_posterior(post,z) # ...other than model.predict(z) # [Only for Regresstion] Specify noise of data (sigma=0.1 by default) # You don't need it if you optimize it later anyway model.setNoise( log_sigma=np.log(0.1) ) print '--------------------END OF DEMO-----------------------' pyGPs-1.2/pyGPs/Demo/demo_GPR_FITC.py000066400000000000000000000061471236346020200171650ustar00rootroot00000000000000#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ import pyGPs import numpy as np # To have a gerneral idea, # you may want to read demo_GPR, demo_kernel and demo_optimization first! # Here, the focus is on the difference of FITC model. print '' print '-------------------GPR_FITC DEMO----------------------' #---------------------------------------------------------------------- # Load demo data (generated from Gaussians) #---------------------------------------------------------------------- demoData = np.load('data_for_demo/regression_data.npz') x = demoData['x'] # training data y = demoData['y'] # training target z = demoData['xstar'] # test data #---------------------------------------------------------------------- # Sparse GP regression (FITC) example #---------------------------------------------------------------------- print "Example 1: default inducing points" # Start from a new model model = pyGPs.GPR_FITC() # Notice if you want to use default inducing points: # You MUST call setData(x,y) FIRST! # The default inducing points are a grid (hypercube in higher dimension), where # each dimension has 5 values in equidistant steps between min and max value of the input data by default. model.setData(x, y) # To set value per dimension use: # model.setData(x, y, value_per_axis=10) model.optimize() print "Negative log marginal liklihood optimized:", round(model.nlZ,3) # Prediction model.predict(z) # Again, plot() is a toy method for 1-d data model.plot() print '------------------------------------------------------' print "Example 2: user-defined inducing points" # Start from a new model model = pyGPs.GPR_FITC() # You can define inducing points yourself. # You can pick some points by hand u = np.array([[-1], [-0.8], [-0.5], [0.3],[1.]]) # or equally-spaced inducing points num_u = np.fix(x.shape[0]/2) u = np.linspace(-1.3,1.3,num_u).T u = np.reshape(u,(num_u,1)) # and specify inducing point when seting prior m = pyGPs.mean.Linear( D=x.shape[1] ) + pyGPs.mean.Const() k = pyGPs.cov.RBF() model.setPrior(mean=m, kernel=k, inducing_points=u) # The rest is analogous to what we have done before model.setData(x, y) model.fit() print "Negative log marginal liklihood before optimization:", round(model.nlZ,3) model.optimize() print "Negative log marginal liklihood optimized:", round(model.nlZ,3) # Prediction ymu, ys2, fmu, fs2, lp = model.predict(z) # Again, plot() is a toy method for 1-d data model.plot() print '--------------------END OF DEMO-----------------------' pyGPs-1.2/pyGPs/Demo/demo_GraphKernel.py000066400000000000000000000053621236346020200201300ustar00rootroot00000000000000#! /usr/bin/env python #coding=utf-8 import numpy as np from scipy.sparse.csc import csc_matrix import pyGPs from pyGPs.Validation import valid from pyGPs.GraphExtensions import graphUtil,graphKernels location = 'data_for_demo/graphData/' data = np.load(location+'MUTAG.npz') # n = num of nodes # N = num of graphs # p = num of labels A = csc_matrix( (data['adj_data'], data['adj_indice'], \ data['adj_indptr']), shape=data['adj_shape']) # n x n adjancy array (sparse matrix) gr_id = data['graph_ind'] # n x 1 graph id array node_label = data['responses'] # n x 1 node label array graph_label = data['labels'] # N x 1 graph label array N = graph_label.shape[0] # number of graphs) graph_label = np.int8(graph_label) for i in xrange(N): if graph_label[i,0] == 0: graph_label[i,0] -= 1 #=========================================================================== # COMPUTE PROPAGATION KERNELS #=========================================================================== num_Iteration = 10 w = 1e-4 dist = 'tv' # possible values: 'tv', 'hellinger' np.random.seed(1) # set random seed to get reproducible kernel matrices (to account for randomness in kernel average resutls over several returns of the experiment) K = graphKernels.propagationKernel(A, node_label, gr_id, num_Iteration, w, dist, 'label_diffusion', SUM=True, VIS=False, showEachStep=False) #---------------------------------------------------------------------- # Cross Validation #---------------------------------------------------------------------- print '...GP prediction (10-fold CV)' for t in xrange(num_Iteration+1): ACC = [] # accuracy print 'number of kernel iterations =', t Matrix = K[:,:,t] # normalize kernel matrix (not useful for MUTAG) # Matrix = graphUtil.normalizeKernel(Matrix) # start cross-validation for this t for index_train, index_test in valid.k_fold_index(N, K=10): y_train = graph_label[index_train,:] y_test = graph_label[index_test,:] n1 = len(index_train) n2 = len(index_test) model = pyGPs.GPC() M1,M2 = graphUtil.formKernelMatrix(Matrix, index_train, index_test) k = pyGPs.cov.Pre(M1,M2) model.setPrior(kernel=k) # gp x_train = np.zeros((n1,1)) x_test = np.zeros((n2,1)) model.fit(x_train, y_train) model.predict(x_test) predictive_class = np.sign(model.ym) # evaluation acc = valid.ACC(predictive_class, y_test) ACC.append(acc) print 'Accuracy: ', np.round(np.mean(ACC),2), '('+str(np.round(np.std(ACC),2))+')' pyGPs-1.2/pyGPs/Demo/demo_Housing.py000077500000000000000000000061041236346020200173400ustar00rootroot00000000000000#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGP_PR. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ import pyGPs import numpy as np import matplotlib.pyplot as plt if __name__ == '__main__': infile = 'data_for_demo/housing.txt' data = np.genfromtxt(infile) DN, DD = data.shape N = 25 # Get all data (exclude the 4th column which is binary) except the last 50 points for training x = np.concatenate((data[:-N,:4],data[:-N,5:-1]),axis=1) x = (x - np.mean(x,axis=0))/(np.std(x,axis=0)+1.e-16) # The function we will perform regression on: Median Value of owner occupied homes y = np.reshape(data[:-N,-1],(len(data[:-N,-1]),1)) y = (y-np.mean(y))/(np.std(y)+1.e-16) # Test on the last 50 points xs = np.concatenate((data[-N:,:4],data[-N:,5:-1]),axis=1) xs = (xs - np.mean(xs,axis=0))/(np.std(xs,axis=0)+1.e-16) ys = np.reshape(data[-N:,-1],(N,1)) ys = (ys-np.mean(ys))/(np.std(ys)+1.e-16) N,D = x.shape model = pyGPs.GPR() model.fit(x, y) print 'Initial negative log marginal likelihood = ', round(model.nlZ,3) # train and predict from time import clock t0 = clock() model.optimize(x,y) t1 = clock() ym, ys2, fm, fs2, lp = model.predict(xs) xa = np.concatenate((data[:,:4],data[:,5:-1]),axis=1) xa = (xa - np.mean(xa,axis=0))/(np.std(xa,axis=0)+1.e-16) ya, ys2a, fma, fs2a, lpa = model.predict(xa) print 'Time to optimize = ', t1-t0 print 'Optimized mean = ', model.meanfunc.hyp print 'Optimized covariance = ', model.covfunc.hyp print 'Optimized liklihood = ', model.likfunc.hyp print 'Final negative log marginal likelihood = ', round(model.nlZ,3) #HousingPlotter(range(len(y)),y,range(len(ym)),ym,ys2,range(len(y),len(y)+len(ys)),ys) xm = np.array(range(len(y),len(y)+ym.shape[0])) ym = np.reshape(ym,(ym.shape[0],)) zm = np.reshape(ys2,(ym.shape[0],)) plt.plot(ya,'g') plt.fill_between(xm, ym + 1.*np.sqrt(zm), ym - 1.*np.sqrt(zm), facecolor=[0.,1.0,0.0,0.9],linewidths=0.0) plt.fill_between(xm, ym + 2.*np.sqrt(zm), ym - 2.*np.sqrt(zm), facecolor=[0.,1.0,0.0,0.7],linewidths=0.0) plt.fill_between(xm, ym + 3.*np.sqrt(zm), ym - 3.*np.sqrt(zm), facecolor=[0.,1.0,0.0,0.5],linewidths=0.0) plt.plot(y,'r.',linewidth = 3.0, markersize = 5.0) plt.plot(xm,ym[-N:], 'bx', linewidth = 3.0, markersize = 5.0) plt.grid() plt.xlabel('Index') plt.ylabel('Median Home Values (normalized)') plt.axis([0.,510.,-3.5,3.5]) plt.show() pyGPs-1.2/pyGPs/Demo/demo_MaunaLoa.py000077500000000000000000000057601236346020200174300ustar00rootroot00000000000000#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGP_PR. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ import pyGPs import numpy as np # Example demo for pyGP prediction of carbon dioxide concentration using # the Mauna Loa CO2 data [Pieter Tans, Aug 2012]. # # The data is constantly updated and publically available under the link: # ftp://ftp.cmdl.noaa.gov/ccg/co2/trends/co2_mm_mlo.txt # # The used covariance function was proposed in [Gaussian Processes for # Machine Learning,Carl Edward Rasmussen and Christopher K. I. Williams, # The MIT Press, 2006. ISBN 0-262-18253-X]. # # Copyright (c) by Marion Neumann and Daniel Marthaler, 20/05/2013 if __name__ == '__main__': # LOAD data infile = 'data_for_demo/mauna.txt' # Note: Samples with value -99.99 were dropped. f = open(infile,'r') year = [] co2 = [] for line in f: z = line.split(' ') z1 = z[1].split('\n') if float(z1[0]) != -99.99: year.append(float(z[0])) co2.append(float(z1[0])) X = [i for (i,j) in zip(year,co2) if i < 2004] y = [j for (i,j) in zip(year,co2) if i < 2004] xx = [i for (i,j) in zip(year,co2) if i >= 2004] yy = [j for (i,j) in zip(year,co2) if i >= 2004] x = np.array(X) y = np.array(y) x = x.reshape((len(x),1)) y = y.reshape((len(y),1)) n,D = x.shape # TEST POINTS xs = np.arange(2004+1./24.,2024-1./24.,1./12.) xs = xs.reshape(len(xs),1) # DEFINE parameterized covariance function k1 = pyGPs.cov.RBF(np.log(67.), np.log(66.)) k2 = pyGPs.cov.Periodic(np.log(1.3), np.log(1.0), np.log(2.4)) * cov.RBF(np.log(90.), np.log(2.4)) k3 = pyGPs.cov.RQ(np.log(1.2), np.log(0.66), np.log(0.78)) k4 = pyGPs.cov.RBF(np.log(1.6/12.), np.log(0.18)) + cov.Noise(np.log(0.19)) k = k1 + k2 + k3 + k4 # STANDARD GP (prediction) model = pyGPs.GPR() model.setData(x,y) model.plotData_1d() model.setPrior(kernel=k) model.predict(xs) # STANDARD GP (training) from time import clock t0 = clock() model.optimize(x,y) t1 = clock() model.predict(xs) print 'Time to optimize = ', t1-t0 print 'Optimized mean = ', model.meanfunc.hyp print 'Optimized covariance = ', model.covfunc.hyp print 'Optimized liklihood = ', model.likfunc.hyp print 'Final negative log marginal likelihood = ', round(model.nlZ,3) model.plot() pyGPs-1.2/pyGPs/Demo/demo_NodeKernel.py000066400000000000000000000134451236346020200177550ustar00rootroot00000000000000#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ import numpy as np from scipy.io import loadmat import matplotlib.pyplot as plt import pyGPs from pyGPs.GraphExtensions import graphUtil, nodeKernels from pyGPs.Validation import valid def load_binary(D1,D2,reduce=False): data = loadmat('data_for_demo/usps_resampled.mat') x = data['train_patterns'].T # train patterns y = data['train_labels'].T # train_labels xx = data['test_patterns'].T # test patterns yy = data['test_labels'].T # test labels D1_list = [] D2_list = [] n,D = x.shape for i in xrange(n): if y[i,D1] == 1: D1_list.append(i) elif y[i,D2] == 1: D2_list.append(i) if reduce == True: D1_list = D1_list[:100] D2_list = D2_list[:100] n1 = len(D1_list) n2 = len(D2_list) x_binary = np.concatenate((x[D1_list,:], x[D2_list,:])) y_binary = np.concatenate((np.ones((n1,1)),-np.ones((n2,1)))) return x_binary,y_binary def plotDigit(digit, title_str=''): fig, ax = plt.subplots() ax.imshow(digit, cmap=plt.cm.gray, interpolation='nearest') ax.spines['right'].set_visible(False) ax.spines['top'].set_visible(False) ax.spines['left'].set_visible(False) ax.spines['bottom'].set_visible(False) plt.title(title_str) plt.show() if __name__ == "__main__": # load small reduced dataset with 2 classes # digit 1 for +1 and digit 2 for -1 x,y = load_binary(1,2,reduce=True) # plot example digit exampleDigit1_1 = x[38] # sample digit 1 exampleDigit1_1 = np.reshape(exampleDigit1_1,(16,16)) plotDigit(exampleDigit1_1, 'This is a 1.') exampleDigit1_2 = x[81] # another sample digit 1 exampleDigit1_2 = np.reshape(exampleDigit1_2,(16,16)) plotDigit(exampleDigit1_2, 'This is another 1.') exampleDigit7_1 = x[124] # sample digit 2 exampleDigit7_1 = np.reshape(exampleDigit7_1,(16,16)) plotDigit(exampleDigit7_1, 'This is a 2.') exampleDigit7_2 = x[129] # another sample digit 2 exampleDigit7_2 = np.reshape(exampleDigit7_2,(16,16)) plotDigit(exampleDigit7_2, 'This is another 2.') # true class 1 exampleDigitBad = x[70] # digit that predicts wrong for rbf exampleDigitBad = np.reshape(exampleDigitBad,(16,16)) plotDigit(exampleDigitBad, 'This digit is an example where the rbf kernel predicts the wrong class (2). \nDiffusion kernel predicts correctly due to graph information!') # true class 2 exampleDigitBad = x[190] # digit that predicts wrong for diff exampleDigitBad = np.reshape(exampleDigitBad,(16,16)) plotDigit(exampleDigitBad, 'This digit is an example where the diff kernel predicts the wrong class (1). \nrbf kernel, however, predicts correctly!') # form a 2-nearest neighbour graph A = graphUtil.formKnnGraph(x,2) # use diffusion kernel to get precomputed matrix Matrix = nodeKernels.diffKernel(A) N = Matrix.shape[0] # set training and test set index_train = xrange(N) index_test = [38, 81, 124, 129, 70, 190] index_train = np.setdiff1d(index_train, index_test) ## RBF kernel # initialize Gaussian process model = pyGPs.GPC() k = pyGPs.cov.RBF() model.setPrior(kernel=k) # split training and test data x_train = x[index_train,:] y_train = y[index_train,:] x_test = x[index_test,:] y_test = y[index_test,:] # gp model.optimize(x_train, y_train) model.predict(x_test) # evaluation predictive_class_rbf = np.sign(model.ym) ACC_rbf = valid.ACC(predictive_class_rbf, y_test) ## DIFFUSION Kernel # compute kernel matrix and initalize GP with precomputed kernel model = pyGPs.GPC() M1,M2 = graphUtil.formKernelMatrix(Matrix, index_train, index_test) k = pyGPs.cov.Pre(M1,M2) model.setPrior(kernel=k) # if you only use precomputed kernel matrix, there is no training data needed, # but you still need to specify x_train (due to general structure of pyGPs) # e.g. you can use the following: n = len(index_train) x_train = np.zeros((n,1)) # gp model.optimize(x_train, y_train) model.predict(x_test) # evaluation predictive_class_diff = np.sign(model.ym) ACC_diff = valid.ACC(predictive_class_diff, y_test) ## SUM of DIFFUSION and RBF Kernel # compute kernel matrix and initalize GP with precomputed kernel model = pyGPs.gp.GPC() M1,M2 = graphUtil.formKernelMatrix(Matrix, index_train, index_test) k = pyGPs.cov.Pre(M1,M2) + pyGPs.cov.RBFunit() model.setPrior(kernel=k) # if you use combination of precomputed matrix and other kernel function, # you can pass traning data in the normal way: x_train = x[index_train,:] x_train = x[index_train,:] # gp model.optimize(x_train, y_train) model.predict(x_test) # evaluation predictive_class_sum = np.sign(model.ym) ACC_sum = valid.ACC(predictive_class_sum, y_test) print np.hstack((np.array(index_test, ndmin=2).T, y_test, predictive_class_rbf, predictive_class_diff, predictive_class_sum)) print 'accuracy (RBF): ' , ACC_rbf print 'accuracy (DIFF): ' , ACC_diff print 'accuracy (SUM): ' , ACC_sum pyGPs-1.2/pyGPs/Demo/demo_Validation.py000066400000000000000000000070031236346020200200120ustar00rootroot00000000000000#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ import pyGPs from pyGPs.Validation import valid import numpy as np # To have a gerneral idea, # you may want to read demo_GPR, demo_kernel and demo_optimization first! # This example shows the process of cross-validation using real dataset. print '' print '---------------Cross-Validation DEMO------------------' #---------------------------------------------------------------------- # Load raw data (ionosphere dataset from UCI) #---------------------------------------------------------------------- data_source = "data_for_demo/ionosphere.data.txt" x = [] y = [] with open(data_source) as f: for index,line in enumerate(f): feature = line.split(',') attr = feature[:-1] attr = [float(i) for i in attr] target = [feature[-1]] x.append(attr) y.append(target) x = np.array(x) y = np.array(y) #---------------------------------------------------------------------- # Data cleaning/preprocessing #---------------------------------------------------------------------- # Here we deal with label in ionosphere data, # change "b" to"-1", and "g" to "+1" n,D = x.shape for i in xrange(n): if y[i,0][0] == 'g': y[i,0] = 1 else: y[i,0] = -1 y = np.int8(y) #---------------------------------------------------------------------- # Cross Validation #---------------------------------------------------------------------- K = 10 # number of fold ACC = [] # accuracy RMSE = [] # root-mean-square error cv_run = 0 for x_train, x_test, y_train, y_test in valid.k_fold_validation(x, y, K): print 'Run:', cv_run # This is a binary classification problem model = pyGPs.GPC() # Since no prior knowldege, leave everything default model.optimize(x_train, y_train) # Predit ymu, ys2, fmu, fs2, lp = model.predict(x_test, ys=y_test) # ymu for classification is a continuous value over -1 to +1 # If you want predicting result to either one of the classes, take a sign of ymu. ymu_class = np.sign(ymu) # Evluation acc = valid.ACC(ymu_class, y_test) print ' accuracy =', round(acc,2) rmse = valid.RMSE(ymu_class, y_test) print ' rmse =', round(rmse,2) ACC.append(acc) RMSE.append(rmse) # Toward next run cv_run += 1 print '\nAccuracy: ', np.round(np.mean(ACC),2), '('+str(np.round(np.std(ACC),2))+')' print 'Root-Mean-Square Error: ', np.round(np.mean(RMSE),2) #---------------------------------------------------------------------- # Evaluation measures #---------------------------------------------------------------------- ''' We defined the following measures(@SEE pyGPs.Valid.valid): RMSE - Root mean squared error ACC - Classification accuracy Prec - Precision for class +1 Recall - Recall for class +1 NLPD - Negative log predictive density in transformed observation space ''' print '--------------------END OF DEMO-----------------------' pyGPs-1.2/pyGPs/GraphExtensions/000077500000000000000000000000001236346020200165775ustar00rootroot00000000000000pyGPs-1.2/pyGPs/GraphExtensions/__init__.py000066400000000000000000000000701236346020200207050ustar00rootroot00000000000000import graphKernels import graphUtil import nodeKernels pyGPs-1.2/pyGPs/GraphExtensions/graphKernels.py000066400000000000000000000165051236346020200216050ustar00rootroot00000000000000#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ # Created on April, 2014 # @author: marion, shan import scipy.sparse as spsp import numpy as np import matplotlib.pyplot as plt def propagationKernel(A, l, gr_id, h_max, w, p, ktype=None, SUM=True, VIS=False, showEachStep=False): ''' Propagation kernel for graphs as described in: Neumann, M., Patricia, N., Garnett, R., Kersting, K.: Efficient Graph Kernels by Randomization. In: P.A. Flach, T.D. Bie, N. Cristianini (eds.) ECML/PKDD, Notes in Computer Science, vol. 7523, pp. 378-393. Springer (2012). :param A: adjacency matrix (num_nodes x num_nodes) :param l: label array (num_nodes x 1); values [1,...,k] or -1 for unlabeled nodes OR label array (num_nodes x num_labels); values [0,1], unlabeled nodes have only 0 entries :param gr_id: graph indicator array (num_nodes x 1); values [0,..,n] :param h_max: number of iterations :param w: bin widths parameter :param p: distance ('tv', 'hellinger', 'L1', 'L2') :param ktype: type of propagation kernel ['diffusion', 'label_propagation', 'label_spreading', 'belief_propagation'] :return: kernel matrix ''' # ToDO: SPLIT Varible label array and label probability # CAUTION: number of labels (k) > 1 !!! #=========================================================================== # ## Propagation Kernel COMPUTATION #=========================================================================== num_graphs = gr_id.max().astype(int) num_nodes = A.shape[0] if l.shape[1]==1: num_labels = l.max().astype(int) else: num_labels = l.shape[1] #=========================================================================== # ## INITIALIZATION #=========================================================================== ## CHECK and ADJUST sparseness if not spsp.issparse(A): A = spsp.csr_matrix(A) else: if not spsp.isspmatrix_csr(A): A = A.tocsr() # MAKE sure A is float! A = A.astype(float) ## INITIALIZE kernel matrix K = np.ndarray(shape=(num_graphs,num_graphs,h_max+1), dtype=float) ## INITIALIZE label probabilities of labeled nodes (num_nodes x num_labels) if l.shape[1]==1: lab_prob = np.zeros((num_nodes,num_labels),dtype=np.float64) for i in xrange(num_nodes): if l[i,0] > 0: lab_prob[i,l[i,0]-1] = 1 else: lab_prob = l if spsp.issparse(lab_prob): lab_prob = lab_prob.todense() lab_orig = lab_prob.copy() # lab_orig is the orignal label information ## GET indices of labeled and unlabeled nodes idx_unlab = np.where(np.max(lab_prob, axis=1)!=1)[0] idx_all = np.array(np.arange(0,A.shape[0]), ndmin=2) idx = np.array(np.setdiff1d(idx_all,idx_unlab), ndmin=2) ## INITIALIZE unlabeled/uninitialized nodes UNIFORMLY idx_unif = np.where(np.sum(lab_prob, axis=1)==0)[0] if idx_unif.shape[0] != 0: lab_prob[idx_unif,:] = 1./lab_prob.shape[1] ## row normalize A -> transition matrix T if (ktype=='label_propagation') or (ktype=='label_diffusion'): row_sums = np.array(A.sum(axis=1))[:,0] T = A.copy() T.data /= row_sums[A.nonzero()[0]] # so A and T are sparse matrix #=========================================================================== # ## PROPAGATION KERNEL ITERATIONS #=========================================================================== for h in xrange(h_max+1): print 'ITERATION: ', h if h > 0: ## LABEL UPDATE if showEachStep: print '...computing LABEL UPDATE' if ktype == 'label_propagation': lab_prob[idx,:] = lab_orig[idx,:] # PUSH BACK original LABELS lab_prob = T*lab_prob elif ktype == 'label_diffusion': lab_prob = T*lab_prob elif ktype == 'label_spreading': # S = D^(-1/2)*W*D^(-1/2) # y(t+1) = alpha*S*y(t)+(1-alpha)*y(0) alpha = 0.8 # compute S diag = np.array(A.sum(axis=1)).T**(-1/2) D = spsp.lil_matrix((A.shape[0], A.shape[1]), dtype=float) # D.setdiag(diag[0,:]) D = D.tocsr() S = D*A*D lab_prob = np.dot((alpha*S.todense()),lab_prob) + (1-alpha)*lab_orig ## COMPUTE hashvalues if showEachStep: print '...computing hashvalues' # determine path to take depending on chosen distance use_cauchy = (p =='L1') or (p =='tv') take_sqrt = (p =='hellinger') if take_sqrt: lab_prob = np.sqrt(lab_prob) # generate random projection vector v = np.random.normal(size=(lab_prob.shape[1], 1)) if use_cauchy: # if X, Y are N(0, 1), then X / Y has a standard Cauchy distribution v /= np.random.normal(size=(lab_prob.shape[1], 1)) # random offset b = w * np.random.rand() # compute hashes # hashLabels is a Vector with length: number of nodes (hashvalue for respective node) hashLabels = (np.dot(lab_prob,v) + b) / w hashLabels = np.floor(hashLabels) # take floor uniqueHash, hashLabels = np.unique(hashLabels, return_inverse=True) # map to consecutive integer from 0 ## COMPUTE kernel contribution if showEachStep: print '...computing KERNEL contribution' # aggregate counts on graphs # counts is a matrix: number of graphs x number of hashlabels num_bins = len(uniqueHash) counts = np.zeros((num_graphs, num_bins)) # init counts matrix # accumulate counts of hash labels for i in xrange(num_nodes): counts[ (gr_id[i,0]-1), hashLabels[i] ] +=1 # compute base kernel (here: LINEAR kernel) K_h = np.dot(counts,counts.T) ## SUM kernel contributions if SUM: if h == 0: K[:,:,h] = K_h else: K[:,:,h] = K[:,:,h-1] + K_h else: K[:,:,h] = K_h if showEachStep: print K[:,:,h] ## VISUALIZE KERNELS if VIS: for h in range(h_max+1): K_h = K[:,:,h] plt.figure() plt.title('height:'+str(h)) imgplot = plt.imshow(K_h) imgplot.set_interpolation('nearest') plt.colorbar() plt.show() return K pyGPs-1.2/pyGPs/GraphExtensions/graphUtil.py000066400000000000000000000054231236346020200211140ustar00rootroot00000000000000#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ import scipy.spatial as spspatial import scipy.sparse as spsp import numpy as np def formKnnGraph(pc,k): ''' Form a k-nearest-neighbour graph from data points :param pc: n by D data matrix :param k: number of neighbours for each node :return: adjacency matrix ''' num_points = pc.shape[0] tree = spspatial.KDTree(pc) [dist, nearest_neighbours] = tree.query(pc, k+1) nearest_neighbours = nearest_neighbours[:,1:] # no self edges weights = np.ones((dist.shape[0],dist.shape[1]-1)) # same weights for edges row_idx = np.kron(np.arange(0,num_points),np.ones((k,1))).T num_edges = row_idx.shape[0]*row_idx.shape[1] A = spsp.coo_matrix((weights.reshape(num_edges), (row_idx.reshape(num_edges),nearest_neighbours.reshape(num_edges))), shape=(num_points,num_points)) A = np.array(A.todense()) A = np.maximum(A,A.T) # make symmetric return A def formKernelMatrix(M, indice_train, indice_test): ''' Format precomputed kernel matrix into two matrix, which fit the structure to be used in cov.Pre() in pyGP :param M: n by n precomputed kernel matrix :param indice_train: list of indice of training examples :param indice_test: list of indice of test examples :return: M1 is a train+1 by test matrix, where the last row is the diagonal of test-test covariance. and M2 is a train by train matrix. ''' train_test = M[indice_train,:][:,indice_test] test_test = M[indice_test,:][:,indice_test] dia = np.diag(test_test) test_dia = np.reshape(dia, (1,dia.shape[0])) M1 = np.concatenate((train_test,test_dia)) M2 = M[indice_train,:][:,indice_train] return M1,M2 def normalizeKernel(K): ''' Normalize the given kernel matrix. Each entry[i,j] is normalized by square root of entry[i,i] * entry[j,j]. (i.e. compute the correlation matrix from covariance matrix). :param K: n by D kernel matrix(covariance matrix) :return: n by D normalized kernel matrix(correlation matrix) ''' Kdiag = np.atleast_2d(np.diag(K)) normalizers = np.sqrt(Kdiag*Kdiag.T) K_norm = K/normalizers return K_norm pyGPs-1.2/pyGPs/GraphExtensions/nodeKernels.py000066400000000000000000000062741236346020200214330ustar00rootroot00000000000000#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ import numpy as np import scipy.linalg as spla import scipy.sparse as spsp def normLap(A): ''' Normalized Laplacian :param A: adjacency matrix :return: kernel matrix ''' I = np.identity(A.shape[0]) d = A.sum(axis=0) d = np.sqrt(1./d) D = np.diag(d) L = I - np.dot( np.dot(D,A),D ) return L def regLapKernel(A, sigma=1): ''' Regularized Laplacian Kernel :param A: adjacency matrix :param sigma: hyperparameter sigma :return: kernel matrix ''' I = np.identity(A.shape[0]) L = normLap(A) K = np.linalg.inv( I+(sigma**2)*L ) return K def psInvLapKernel(A): ''' Pseudo inverse of the normalized Laplacian. :param A: adjacency matrix :return: kernel matrix ''' L = normLap(A) K = np.linalg.pinv(L) return K def diffKernel(A, beta=0.5): ''' Diffusion Process Kernel K = exp(beta * H), where H = -L = A-D K = Q exp(beta * Lambda) Q.T :param A: adjacency matrix :param beta: hyperparameter beta :return: kernel matrix ''' A = np.array(A) # make sure that A is a numpy array!! H = A - np.diag(np.sum(A, axis=1)) w, Q = np.linalg.eigh(H) Lambda = np.diag(np.exp(beta*w)) K = np.dot(np.dot(Q, Lambda), Q.T) return K def VNDKernel(A, alpha=0.5): ''' Von Neumann Diffusion Kernel on graph (Zhou et al., 2004) (also label spreading kernel) K = (I - alpha*S)^-1, where S = D^-1/2*A*D^-1/2 :param A: adjacency matrix :param alpha: hyperparameter alpha :return: kernel matrix ''' I = np.identity(A.shape[0]) d = A.sum(axis=0) d = np.sqrt(1./d) D = np.diag(d) S = np.dot( np.dot(D,A),D ) K = np.linalg.inv( I - alpha*S ) return K def rwKernel(A, p=1, a=2): ''' p-step Random Walk Kernel with a>1 K = (aI-L)^p, p>1 and L is the normalized Laplacian :param A: adjacency matrix :param p: step parameter :param a: hyperparameter a :return: kernel matrix ''' if type(p) != int: p = int(p) if p < 1: raise Exception('Step parameter p needs to be larger than 0.') if a <= 1: a=1.0001 I = np.identity(A.shape[0]) L = normLap(A) K = np.linalg.matrix_power( a*I - L, p) return K def cosKernel(A): ''' Cosine Kernel (also Inverse Cosine Kernel) K = cos (L*pi/4), where L is the normalized Laplacian :param A: adjacency matrix :return: kernel matrix ''' L = normLap(A) K = np.cos(L*np.pi/4) return K pyGPs-1.2/pyGPs/Optimization/000077500000000000000000000000001236346020200161445ustar00rootroot00000000000000pyGPs-1.2/pyGPs/Optimization/__init__.py000066400000000000000000000000461236346020200202550ustar00rootroot00000000000000import scg import minimize import confpyGPs-1.2/pyGPs/Optimization/conf.py000066400000000000000000000037401236346020200174470ustar00rootroot00000000000000#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ import numpy as np class random_init_conf(object): def __init__(self, mean, cov, lik): self.num_restarts = None self.min_threshold = None self.mean = mean self.cov = cov self.lik = lik self._meanRange = [(-5,5) for i in mean.hyp] self._covRange = [(-5,5) for i in cov.hyp] self._likRange = [(-5,5) for i in lik.hyp] def _getmr(self): return self._meanRange def _setmr(self, value): if len(value) == len(self.mean.hyp): self._meanRange = value else: raise Exception('The length of meanRange is not consistent with number of mean hyparameters') meanRange = property(_getmr,_setmr) def _getcr(self): return self._covRange def _setcr(self, value): if len(value) == len(self.cov.hyp): self._covRange = value else: raise Exception('The length of covRange is not consistent with number of covariance hyparameters') covRange = property(_getcr,_setcr) def _getlr(self): return self._likRange def _setlr(self, value): if len(value) == len(self.lik.hyp): self._likRange = value else: raise Exception('The length of likRange is not consistent with number of liklihood hyparameters') likRange = property(_getlr,_setlr) pyGPs-1.2/pyGPs/Optimization/minimize.py000066400000000000000000000207541236346020200203470ustar00rootroot00000000000000 #=============================================================================== # This program is distributed WITHOUT ANY WARRANTY; without even the implied # warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. # # # This file contains a Python version of Carl Rasmussen's Matlab-function # minimize.m # # minimize.m is copyright (C) 1999 - 2006, Carl Edward Rasmussen. # Python adaptation by Roland Memisevic 2008. # updates by Shan Huang 2013 # # # The following is the original copyright notice that comes with the # function minimize.m # (from http://www.kyb.tuebingen.mpg.de/bs/people/carl/code/minimize/Copyright): # Rasmussen # # "(C) Copyright 1999 - 2006, Carl Edward Rasmussen # # Permission is granted for anyone to copy, use, or modify these # programs and accompanying documents for purposes of research or # education, provided this copyright notice is retained, and note is # made of any changes that have been made. # # These programs and documents are distributed without any warranty, # express or implied. As the programs were written for research # purposes only, they have not been tested to the degree that would be # advisable in any important application. All use of these programs is # entirely at the user's own risk." #=============================================================================== from numpy import dot, isinf, isnan, any, sqrt, isreal, real, nan, inf def run(f, X, args=(), length=None, red=1.0, verbose=False): ''' This is a function that performs unconstrained gradient based optimization using nonlinear conjugate gradients. The function is a straightforward Python-translation of Carl Rasmussen's Matlab-function minimize.m ''' INT = 0.1;# don't reevaluate within 0.1 of the limit of the current bracket EXT = 3.0; # extrapolate maximum 3 times the current step-size MAX = 20; # max 20 function evaluations per line search RATIO = 10; # maximum allowed slope ratio SIG = 0.1;RHO = SIG/2;# SIG and RHO are the constants controlling the Wolfe- #Powell conditions. SIG is the maximum allowed absolute ratio between #previous and new slopes (derivatives in the search direction), thus setting #SIG to low (positive) values forces higher precision in the line-searches. #RHO is the minimum allowed fraction of the expected (from the slope at the #initial point in the linesearch). Constants must satisfy 0 < RHO < SIG < 1. #Tuning of SIG (depending on the nature of the function to be optimized) may #speed up the minimization; it is probably not worth playing much with RHO. SMALL = 10.**-16 #minimize.m uses matlab's realmin i = 0 # zero the run length counter ls_failed = 0 # no previous line search has failed result = f(X, *args) f0 = result[0] # get function value and gradient df0 = result[1] fX = [f0] i = i + (length<0) # count epochs?! s = -df0 d0 = -dot(s,s) # initial search direction (steepest) and slope x3 = red/(1.0-d0) # initial step is red/(|s|+1) while i < abs(length): # while not finished i = i + (length>0) # count iterations?! X0 = X; F0 = f0; dF0 = df0 # make a copy of current values if length>0: M = MAX else: M = min(MAX, -length-i) while 1: # keep extrapolating as long as necessary x2 = 0; f2 = f0; d2 = d0; f3 = f0; df3 = df0 success = 0 while (not success) and (M > 0): try: M = M - 1; i = i + (length<0) # count epochs?! result3 = f(X+x3*s, *args) f3 = result3[0] df3 = result3[1] if isnan(f3) or isinf(f3) or any(isnan(df3)+isinf(df3)): return success = 1 except: # catch any error which occured in f x3 = (x2+x3)/2 # bisect and try again if f3 < F0: X0 = X+x3*s; F0 = f3; dF0 = df3 # keep best values d3 = dot(df3,s) # new slope if d3 > SIG*d0 or f3 > f0+x3*RHO*d0 or M == 0: # are we done extrapolating? break x1 = x2; f1 = f2; d1 = d2 # move point 2 to point 1 x2 = x3; f2 = f3; d2 = d3 # move point 3 to point 2 A = 6*(f1-f2)+3*(d2+d1)*(x2-x1) # make cubic extrapolation B = 3*(f2-f1)-(2*d1+d2)*(x2-x1) Z = B+sqrt(complex(B*B-A*d1*(x2-x1))) if Z != 0.0: x3 = x1-d1*(x2-x1)**2/Z # num. error possible, ok! else: x3 = inf if (not isreal(x3)) or isnan(x3) or isinf(x3) or (x3 < 0): # num prob | wrong sign? x3 = x2*EXT # extrapolate maximum amount elif x3 > x2*EXT: # new point beyond extrapolation limit? x3 = x2*EXT # extrapolate maximum amount elif x3 < x2+INT*(x2-x1): # new point too close to previous point? x3 = x2+INT*(x2-x1) x3 = real(x3) while (abs(d3) > -SIG*d0 or f3 > f0+x3*RHO*d0) and M > 0: # keep interpolating if (d3 > 0) or (f3 > f0+x3*RHO*d0): # choose subinterval x4 = x3; f4 = f3; d4 = d3 # move point 3 to point 4 else: x2 = x3; f2 = f3; d2 = d3 # move point 3 to point 2 if f4 > f0: x3 = x2-(0.5*d2*(x4-x2)**2)/(f4-f2-d2*(x4-x2)) # quadratic interpolation else: A = 6*(f2-f4)/(x4-x2)+3*(d4+d2) # cubic interpolation B = 3*(f4-f2)-(2*d2+d4)*(x4-x2) if A != 0: x3=x2+(sqrt(B*B-A*d2*(x4-x2)**2)-B)/A # num. error possible, ok! else: x3 = inf if isnan(x3) or isinf(x3): x3 = (x2+x4)/2 # if we had a numerical problem then bisect x3 = max(min(x3, x4-INT*(x4-x2)),x2+INT*(x4-x2)) # don't accept too close result3 = f(X+x3*s, *args) f3 = result3[0] df3 = result3[1] if f3 < F0: X0 = X+x3*s; F0 = f3; dF0 = df3 # keep best values M = M - 1; i = i + (length<0) # count epochs?! d3 = dot(df3,s) # new slope if abs(d3) < -SIG*d0 and f3 < f0+x3*RHO*d0: # if line search succeeded X = X+x3*s; f0 = f3; fX.append(f0) # update variables s = (dot(df3,df3)-dot(df0,df3))/dot(df0,df0)*s - df3 # Polack-Ribiere CG direction df0 = df3 # swap derivatives d3 = d0; d0 = dot(df0,s) if d0 > 0: # new slope must be negative s = -df0; d0 = -dot(s,s) # otherwise use steepest direction x3 = x3 * min(RATIO, d3/(d0-SMALL)) # slope ratio but max RATIO ls_failed = 0 # this line search did not fail else: X = X0; f0 = F0; df0 = dF0 # restore best point so far if ls_failed or (i>abs(length)): # line search failed twice in a row break # or we ran out of time, so we give up s = -df0; d0 = -dot(s,s) # try steepest x3 = 1/(1-d0) ls_failed = 1 # this line search failed if verbose: print "\n" #print fX return X, fX, i pyGPs-1.2/pyGPs/Optimization/scg.py000066400000000000000000000141201236346020200172700ustar00rootroot00000000000000#=============================================================================== # SCG Scaled conjugate gradient optimization. # # Copyright (c) Ian T Nabney (1996-2001) # updates by Shan Huang 2013 # # Permission is granted for anyone to copy, use, or modify these # programs and accompanying documents for purposes of research or # education, provided this copyright notice is retained, and note is # made of any changes that have been made. # # These programs and documents are distributed without any warranty, # express or implied. As the programs were written for research # purposes only, they have not been tested to the degree that would be # advisable in any important application. All use of these programs is # entirely at the user's own risk." #=============================================================================== from math import sqrt import numpy as np def run(f, x, args=(), niters = 100, gradcheck = False, display = 0, flog = False, pointlog = False, scalelog = False, tolX = 1.0e-8, tolO = 1.0e-8, eval = None): '''Scaled conjugate gradient optimization. ''' if display: print '\n***** starting optimization (SCG) *****\n' nparams = len(x); # Check gradients if gradcheck: pass eps = 1.0e-4 sigma0 = 1.0e-4 result = f(x, *args) fold = result[0] # Initial function value. fnow = fold funcCount = 1 # Increment function evaluation counter. gradnew = result[1] # Initial gradient. gradold = gradnew gradCount = 1 # Increment gradient evaluation counter. d = -gradnew # Initial search direction. success = 1 # Force calculation of directional derivs. nsuccess = 0 # nsuccess counts number of successes. beta = 1.0 # Initial scale parameter. betamin = 1.0e-15 # Lower bound on scale. betamax = 1.0e50 # Upper bound on scale. j = 1 # j counts number of iterations. if flog: pass #flog(j, :) = fold; if pointlog: pass #pointlog(j, :) = x; # Main optimization loop. listF = [fold] if eval is not None: evalue, timevalue = eval(x, *args) evalList = [evalue] time = [timevalue] while (j <= niters): # Calculate first and second directional derivatives. if (success == 1): mu = np.dot(d, gradnew) if (mu >= 0): d = - gradnew mu = np.dot(d, gradnew) kappa = np.dot(d, d) if (kappa < eps): #print "FNEW: " , fnow #options(8) = fnow if eval is not None: return x, listF, evalList, time else: return x, listF sigma = sigma0/sqrt(kappa) xplus = x + sigma*d gplus = f(xplus, *args)[1] gradCount += 1 theta = (np.dot(d, (gplus - gradnew)))/sigma; # Increase effective curvature and evaluate step size alpha. delta = theta + beta*kappa if (delta <= 0): delta = beta*kappa beta = beta - theta/kappa alpha = - mu/delta # Calculate the comparison ratio. xnew = x + alpha*d fnew = f(xnew, *args)[0] funcCount += 1; Delta = 2*(fnew - fold)/(alpha*mu) if (Delta >= 0): success = 1; nsuccess += 1; x = xnew; fnow = fnew; listF.append(fnow) if eval is not None: evalue, timevalue = eval(x, *args) evalList.append(evalue) time.append(timevalue) else: success = 0; fnow = fold; if flog: # Store relevant variables #flog(j) = fnow; # Current function value pass if pointlog: #pointlog(j,:) = x; # Current position pass if scalelog: #scalelog(j) = beta; # Current scale parameter pass if display > 0: print('***** Cycle %4d Error %11.6f Scale %e' %( j, fnow, beta)) if (success == 1): # Test for termination # print type (alpha), type(d), type(tolX), type(fnew), type(fold) if ((max(abs(alpha*d)) < tolX) & (abs(fnew-fold) < tolO)): # options(8) = fnew; # print "FNEW: " , fnew if eval is not None: return x, listF, evalList, time else: return x, listF else: # Update variables for new position fold = fnew gradold = gradnew gradnew = f(x, *args)[1] gradCount += 1 # If the gradient is zero then we are done. if (np.dot(gradnew, gradnew) == 0): # print "FNEW: " , fnew # options(8) = fnew; if eval is not None: return x, listF, evalList, time else: return x, listF # Adjust beta according to comparison ratio. if (Delta < 0.25): beta = min(4.0*beta, betamax); if (Delta > 0.75): beta = max(0.5*beta, betamin); # Update search direction using Polak-Ribiere formula, or re-start # in direction of negative gradient after nparams steps. if (nsuccess == nparams): d = -gradnew; nsuccess = 0; else: if (success == 1): gamma = np.dot((gradold - gradnew), gradnew)/(mu) d = gamma*d - gradnew; j += 1 # If we get here, then we haven't terminated in the given number of # iterations. # options(8) = fold; if (display): print "maximum number of iterations reached" if eval is not None: return x, listF, evalList, time else: return x, listF pyGPs-1.2/pyGPs/Testing/000077500000000000000000000000001236346020200150735ustar00rootroot00000000000000pyGPs-1.2/pyGPs/Testing/unit_test_cov.py000066400000000000000000000146471236346020200203460ustar00rootroot00000000000000#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ import unittest import numpy as np import pyGPs class CovarianceTests(unittest.TestCase): def setUp(self): # random 2d data for testing self.x = np.random.normal(loc=0.0, scale=1.0, size=(20,2)) self.z = np.random.normal(loc=0.0, scale=1.0, size=(10,2)) # random precomputed kernel matrix self.M1 = np.random.random(size=(21,10)) self.M2 = np.random.random(size=(20,20)) def checkCovOutput(self, train_train, train_test, self_test): n,D = self.x.shape nn, D = self.z.shape self.assertTrue(train_train.shape == (n,n)) self.assertTrue(train_test.shape == (n,nn)) self.assertTrue(self_test.shape == (nn,1)) def checkDerOutput(self, train_train, train_test, self_test): n,D = self.x.shape nn, D = self.z.shape self.assertTrue(train_train.shape == (n,n)) self.assertTrue(train_test.shape == (n,nn)) self.assertTrue(self_test.shape == (nn,1)) def checkCovariance(self, k): k1 = k.getCovMatrix(x=self.x, mode='train') # test train by train covariance k2 = k.getCovMatrix(x=self.x, z=self.z, mode='cross') # test train by test covariance k3 = k.getCovMatrix(z=self.z, mode='self_test') # test test by test self covariance self.checkCovOutput(k1,k2,k3) for der in xrange(len(k.hyp)): kd1 = k.getDerMatrix(x=self.x, mode='train',der=der) # test train by train derivative kd2 = k.getDerMatrix(x=self.x, z=self.z, mode='cross',der=der) # test train by test derivative kd3 = k.getDerMatrix(z=self.z, mode='self_test',der=der) # test test by test self derivative self.checkDerOutput(kd1, kd2, kd3) def test_covRBF(self): print "testing covRBF..." k = pyGPs.cov.RBF() self.checkCovariance(k) def test_covRBFunit(self): print "testing covRBFunit..." k = pyGPs.cov.RBFunit() self.checkCovariance(k) def test_covRBFard(self): print "testing covRBFard..." k = pyGPs.cov.RBFard(D=self.x.shape[1]) self.checkCovariance(k) def test_covConst(self): print "testing covConst..." k = pyGPs.cov.Const() self.checkCovariance(k) def test_covLinear(self): print "testing covLinear..." k = pyGPs.cov.Linear() self.checkCovariance(k) def test_covLINard(self): print "testing covLINard..." k = pyGPs.cov.LINard(D=self.x.shape[1]) self.checkCovariance(k) def test_covMatern(self): print "testing covMatern..." k = pyGPs.cov.Matern() self.checkCovariance(k) def test_covPeriodic(self): print "testing covPeriodic..." k = pyGPs.cov.Periodic() self.checkCovariance(k) def test_covNoise(self): print "testing covNoise..." k = pyGPs.cov.Noise() self.checkCovariance(k) def test_covRQ(self): print "testing covRQ..." k = pyGPs.cov.RQ() self.checkCovariance(k) def test_covRQard(self): print "testing covRQard..." k = pyGPs.cov.RQard(D=self.x.shape[1]) self.checkCovariance(k) def test_covPre(self): print "testing covPre..." k = pyGPs.cov.Pre(self.M1, self.M2) # load precomputed kernel matrix self.checkCovariance(k) def test_covPiecePoly(self): print "testing covPiecePoly..." k = pyGPs.cov.PiecePoly() self.checkCovariance(k) def test_covPoly(self): print "testing covPoly..." k = pyGPs.cov.Poly() self.checkCovariance(k) def test_covScale(self): print "testing (compositing kernel) muliply by a scalar..." k = pyGPs.cov.RBF()*5 self.checkCovariance(k) def test_covSum(self): print "testing (compositing kernel) sum of two kernels..." k = pyGPs.cov.RBF() + pyGPs.cov.PiecePoly() self.checkCovariance(k) def test_covProduct(self): print "testing (compositing kernel) product of two kernels..." k = pyGPs.cov.RBF() * pyGPs.cov.PiecePoly() self.checkCovariance(k) def test_covFITC(self): print "testing FITC kernel to be used with sparse GP..." n,D = self.x.shape nn,D = self.z.shape u = np.random.random(size=(5,2)) # random inducing points nu,D = u.shape k = pyGPs.cov.RBF().fitc(u) K, Kuu, Ku = k.getCovMatrix(x=self.x, mode='train') # test train by train covariance k2 = k.getCovMatrix(x=self.x, z=self.z, mode='cross') # test train by test covariance k3 = k.getCovMatrix(z=self.z, mode='self_test') # test test by test self covariance self.assertTrue(K.shape == (n,1)) self.assertTrue(Kuu.shape == (nu,nu)) self.assertTrue(Ku.shape == (nu,n)) self.assertTrue(k2.shape == (nu,nn)) self.assertTrue(k3.shape == (nn,1)) for der in xrange(len(k.hyp)): Kd, Kuud, Kud = k.getDerMatrix(x=self.x, mode='train',der=der) # test train by train derivative kd2 = k.getDerMatrix(x=self.x, z=self.z, mode='cross',der=der) # test train by test derivative kd3 = k.getDerMatrix(z=self.z, mode='self_test',der=der) # test test by test self derivative self.assertTrue(Kd.shape == (n,1)) self.assertTrue(Kuud.shape == (nu,nu)) self.assertTrue(Kud.shape == (nu,n)) self.assertTrue(kd2.shape == (nu,nn)) self.assertTrue(kd3.shape == (nn,1)) # Test your customized covariance function ''' def test_cov_new(self): k = myKernel() # specify your covariance function self.checkCovariance(k) ''' if __name__ == "__main__": print "Running unit tests..." unittest.main() pyGPs-1.2/pyGPs/Testing/unit_test_inf.py000066400000000000000000000113401236346020200203160ustar00rootroot00000000000000#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ import unittest import numpy as np import pyGPs class InfTests(unittest.TestCase): # here focus on testing inference function. # therefore only use one example set of covariance/mean functions def setUp(self): # random 2d data for testing self.x = np.random.normal(loc=0.0, scale=1.0, size=(20,2)) self.y = np.random.normal(loc=0.0, scale=1.0, size=(20,1)) self.u = np.random.random(size=(5,2)) # random inducing points def checkInferenceOutput(self, post, nlZ, dnlZ): n,D = self.x.shape self.assertTrue(post.alpha.shape[0] == n) self.assertTrue(post.L.shape == (n,n)) self.assertTrue(post.sW.shape == (n,1)) self.assertTrue(type(nlZ) is np.float64) for entry in dnlZ.mean: self.assertTrue(type(entry) is np.float64) for entry in dnlZ.cov: self.assertTrue(type(entry) is np.float64) for entry in dnlZ.lik: self.assertTrue(type(entry) is np.float64) def checkFITCOutput(self, post, nlZ, dnlZ): n,D = self.x.shape nu,D = self.u.shape self.assertTrue(post.alpha.shape[0] == nu) self.assertTrue(post.L.shape == (nu,nu)) self.assertTrue(post.sW.shape == (n,1)) self.assertTrue(type(nlZ) is np.float64) for entry in dnlZ.mean: self.assertTrue(type(entry) is np.float64) for entry in dnlZ.cov: self.assertTrue(type(entry) is np.float64) for entry in dnlZ.lik: self.assertTrue(type(entry) is np.float64) def test_infExact(self): print "testing exact inference..." inffunc = pyGPs.inf.Exact() meanfunc = pyGPs.mean.Zero() covfunc = pyGPs.cov.RBF() likfunc = pyGPs.lik.Gauss() post, nlZ, dnlZ = inffunc.proceed(meanfunc, covfunc, likfunc, self.x, self.y, nargout=3) self.checkInferenceOutput(post, nlZ, dnlZ) def test_infFITC_Exact(self): print "testing FITC inference..." inffunc = pyGPs.inf.FITC_Exact() meanfunc = pyGPs.mean.Zero() covfunc = pyGPs.cov.RBF().fitc(self.u) likfunc = pyGPs.lik.Gauss() post, nlZ, dnlZ = inffunc.proceed(meanfunc, covfunc, likfunc, self.x, self.y, nargout=3) self.checkFITCOutput(post, nlZ, dnlZ) def test_infEP(self): print "testing EP inference..." inffunc = pyGPs.inf.EP() meanfunc = pyGPs.mean.Zero() covfunc = pyGPs.cov.RBF() likfunc = pyGPs.lik.Gauss() post, nlZ, dnlZ = inffunc.proceed(meanfunc, covfunc, likfunc, self.x, self.y, nargout=3) self.checkInferenceOutput(post, nlZ, dnlZ) def test_infFITC_EP(self): print "testing FITC EP inference..." inffunc = pyGPs.inf.FITC_EP() meanfunc = pyGPs.mean.Zero() covfunc = pyGPs.cov.RBF().fitc(self.u) likfunc = pyGPs.lik.Gauss() post, nlZ, dnlZ = inffunc.proceed(meanfunc, covfunc, likfunc, self.x, self.y, nargout=3) self.checkFITCOutput(post, nlZ, dnlZ) def test_infLaplace(self): print "testing Laplace inference..." inffunc = pyGPs.inf.Laplace() meanfunc = pyGPs.mean.Zero() covfunc = pyGPs.cov.RBF() likfunc = pyGPs.lik.Gauss() post, nlZ, dnlZ = inffunc.proceed(meanfunc, covfunc, likfunc, self.x, self.y, nargout=3) self.checkInferenceOutput(post, nlZ, dnlZ) def test_infFITC_Laplace(self): print "testing FITC EP inference..." inffunc = pyGPs.inf.FITC_Laplace() meanfunc = pyGPs.mean.Zero() covfunc = pyGPs.cov.RBF().fitc(self.u) likfunc = pyGPs.lik.Gauss() post, nlZ, dnlZ = inffunc.proceed(meanfunc, covfunc, likfunc, self.x, self.y, nargout=3) self.checkFITCOutput(post, nlZ, dnlZ) # Test your customized inference function ''' def test_inf_new(self): # specify your inf function # set mean/cov/lik functions post, nlZ, dnlZ = inffunc.proceed(meanfunc, covfunc, likfunc, self.x, self.y, nargout=3) self.checkFITCOutput(post, nlZ, dnlZ) ''' if __name__ == "__main__": print "Running unit tests..." unittest.main() pyGPs-1.2/pyGPs/Testing/unit_test_lik.py000066400000000000000000000075211236346020200203270ustar00rootroot00000000000000#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ import unittest import numpy as np import pyGPs class LikTests(unittest.TestCase): def setUp(self): # random data for testing self.y = np.random.normal(loc=0.0, scale=1.0, size=(1,1)) # tiled target self.mu = np.random.normal(loc=0.0, scale=1.0, size=(1,1)) # tiled conditional mean self.s2 = np.random.normal(loc=0.0, scale=1.0, size=(1,1)) # tiled conditional variance def checkPrediction(self, lp, ymu, ys2): n,D = self.y.shape self.assertTrue(lp.shape == (n,D)) self.assertTrue(ymu.shape == (n,D)) self.assertTrue(ys2.shape == (n,D)) def checkInferenceLaplace(self, lp, dlp, d2lp, d3lp): n,D = self.y.shape self.assertTrue(lp.shape == (n,D)) self.assertTrue(dlp.shape == (n,D)) self.assertTrue(d2lp.shape == (n,D)) self.assertTrue(d3lp.shape == (n,D)) def checkDerivativeLaplace(self, lp, dlp, d2lp): n,D = self.y.shape self.assertTrue(lp.shape == (n,D)) self.assertTrue(dlp.shape == (n,D)) self.assertTrue(d2lp.shape == (n,D)) def checkInferenceEP(self, lp, dlp, d2lp): n,D = self.y.shape self.assertTrue(lp.shape == (n,D)) self.assertTrue(dlp.shape == (n,D)) self.assertTrue(d2lp.shape == (n,D)) def checkDerivativeEP(self, dlZhyp): n,D = self.y.shape self.assertTrue(dlZhyp.shape == (n,D)) def checkLikelihood(self, likelihood): print "predictive mode" lp, ymu, ys2 = likelihood.proceed(y=self.y, mu=self.mu, s2=self.s2, nargout=3) self.checkPrediction(lp, ymu, ys2) print "inference mode(Laplace inference)" lp,dlp,d2lp,d3lp = likelihood.proceed(y=self.y, mu=self.mu, s2=self.s2, inffunc=pyGPs.inf.Laplace(), nargout=4) self.checkInferenceLaplace(lp,dlp,d2lp,d3lp) for der in xrange(len(likelihood.hyp)): lp_dhyp,dlp_dhyp,d2lp_dhyp = likelihood.proceed(y=self.y, mu=self.mu, s2=self.s2, inffunc=pyGPs.inf.Laplace(), der=der, nargout=4) self.checkDerivativeLaplace(lp_dhyp,dlp_dhyp,d2lp_dhyp) print "inference mode(EP inference)" lp,dlp,d2lp = likelihood.proceed(y=self.y, mu=self.mu, s2=self.s2, inffunc=pyGPs.inf.EP(), nargout=3) self.checkInferenceEP(lp,dlp,d2lp) for der in xrange(len(likelihood.hyp)): dlZhyp = likelihood.proceed(y=self.y, mu=self.mu, s2=self.s2, inffunc=pyGPs.inf.EP(), der=der, nargout=3) self.checkDerivativeEP(dlZhyp) def test_likGauss(self): print "testing Gaussian likelihood..." likelihood = pyGPs.lik.Gauss() self.checkLikelihood(likelihood) def test_likErf(self): print "testing error function(cumulative Gaussian) likelihood..." likelihood = pyGPs.lik.Erf() self.checkLikelihood(likelihood) def test_likLaplace(self): print "testing Laplacian likelihood..." likelihood = pyGPs.lik.Laplace() self.checkLikelihood(likelihood) # Test your customized likelihood function ''' def test_cov_new(self): # specify your likelihood function self.checkLikelihood(likelihood) ''' if __name__ == "__main__": print "Running unit tests..." unittest.main() pyGPs-1.2/pyGPs/Testing/unit_test_mean.py000066400000000000000000000053441236346020200204710ustar00rootroot00000000000000#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ import unittest import numpy as np import pyGPs class MeanTests(unittest.TestCase): def setUp(self): # random 2d data for testing self.x = np.random.normal(loc=0.0, scale=1.0, size=(20,2)) def checkMeanOutput(self, mean): n,D = self.x.shape self.assertTrue(mean.shape == (n,1)) def checkDerOutput(self, derivative): n,D = self.x.shape self.assertTrue(derivative.shape == (n,1)) def checkMean(self, m): mean = m.getMean(x=self.x) # get mean self.checkMeanOutput(mean) for der in xrange(len(m.hyp)): # get derivatives derivative = m.getDerMatrix(x=self.x, der=der) self.checkDerOutput(derivative) def test_meanZero(self): print "testing meanZero..." m = pyGPs.mean.Zero() self.checkMean(m) def test_meanLinear(self): print "testing meanLinear..." m = pyGPs.mean.Linear(D=self.x.shape[1]) self.checkMean(m) def test_meanOne(self): print "testing meanOne..." m = pyGPs.mean.One() self.checkMean(m) def test_meanConst(self): print "testing meanConst..." m = pyGPs.mean.Const() self.checkMean(m) def test_meanScale(self): print "testing (compositing mean) muliply by a scalar..." m = pyGPs.mean.One() * 6 self.checkMean(m) def test_meanSum(self): print "testing (compositing mean) sum of two means..." m = pyGPs.mean.One() + pyGPs.mean.Const() self.checkMean(m) def test_meanProduct(self): print "testing (compositing mean) product of two means..." m = pyGPs.mean.One() * pyGPs.mean.Const() self.checkMean(m) def test_meanPower(self): print "testing (compositing mean) power of a mean..." m = pyGPs.mean.Const() ** 2 self.checkMean(m) # Test your customized mean function ''' def test_mean_new(self): # specify your mean function self.checkMean(m) ''' if __name__ == "__main__": print "Running unit tests..." unittest.main() pyGPs-1.2/pyGPs/Testing/unit_test_model.py000066400000000000000000000107401236346020200206450ustar00rootroot00000000000000#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ import unittest import numpy as np import pyGPs class ModelTests(unittest.TestCase): # note!! # due to completely random data(which may not make sense for gp in some cases), # it will rasis error sometimes(failed too much times in multiple-start optimizations) # therefore we construct toy data reasonably as memtioned in GPML book def setUp(self): regData = np.load('../Demo/data_for_demo/regression_data.npz') self.xr = regData['x'] # training data self.yr = regData['y'] # training target self.zr = regData['xstar'] # test data self.ur = np.array([[-1], [-0.8], [-0.5], [0.3],[1.]]) # inducing points clsData = np.load('../Demo/data_for_demo/classification_data.npz') self.xc = clsData['x'] # training data self.yc = clsData['y'] # training target self.zc = clsData['xstar'] # test data u1,u2 = np.meshgrid(np.linspace(-2,2,5),np.linspace(-2,2,5)) self.uc = np.array(zip(np.reshape(u2,(np.prod(u2.shape),)),np.reshape(u1,(np.prod(u1.shape),)))) # inducing points def checkRegressionOutput(self, model): nn,D = self.zr.shape self.assertTrue(model.ym.shape == (nn,1)) self.assertTrue(model.ys2.shape == (nn,1)) self.assertTrue(model.fm.shape == (nn,1)) self.assertTrue(model.fs2.shape == (nn,1)) def checkClassificationOutput(self, model): nn,D = self.zc.shape self.assertTrue(model.ym.shape == (nn,1)) self.assertTrue(model.ys2.shape == (nn,1)) self.assertTrue(model.fm.shape == (nn,1)) self.assertTrue(model.fs2.shape == (nn,1)) def test_GPR(self): print "testing GP regression..." model = pyGPs.GPR() m = pyGPs.mean.Zero() k = pyGPs.cov.RBF() model.setPrior(mean=m, kernel=k) model.setOptimizer("Minimize", num_restarts=10) model.optimize(self.xr, self.yr) model.predict(self.zr) self.checkRegressionOutput(model) def test_GPC(self): print "testing GP classification..." model = pyGPs.GPC() m = pyGPs.mean.Zero() k = pyGPs.cov.RBF() model.setPrior(mean=m, kernel=k) model.optimize(self.xc, self.yc) model.predict(self.zc) self.checkClassificationOutput(model) model = pyGPs.GPC() m = pyGPs.mean.Zero() k = pyGPs.cov.RBF() model.setPrior(mean=m, kernel=k) model.useInference("Laplace") model.optimize(self.xc, self.yc) model.predict(self.zc) self.checkClassificationOutput(model) def test_GPR_FITC(self): print "testing GP sparse regression..." model = pyGPs.GPR_FITC() m = pyGPs.mean.Zero() k = pyGPs.cov.RBF() model.setPrior(mean=m, kernel=k, inducing_points=self.ur) model.setOptimizer("Minimize", num_restarts=10) model.optimize(self.xr, self.yr) model.predict(self.zr) self.checkRegressionOutput(model) def test_GPC_FITC(self): print "testing GP sparse classification..." model = pyGPs.GPC_FITC() m = pyGPs.mean.Zero() k = pyGPs.cov.RBF() model.setPrior(mean=m, kernel=k, inducing_points=self.uc) model.setOptimizer("Minimize", num_restarts=10) model.optimize(self.xc, self.yc) model.predict(self.zc) self.checkClassificationOutput(model) model = pyGPs.GPC_FITC() m = pyGPs.mean.Zero() k = pyGPs.cov.RBF() model.setPrior(mean=m, kernel=k, inducing_points=self.uc) model.setOptimizer("Minimize", num_restarts=10) model.useInference("Laplace") model.optimize(self.xc, self.yc) model.predict(self.zc) self.checkClassificationOutput(model) if __name__ == "__main__": print "Running unit tests(about 5 min)..." unittest.main() pyGPs-1.2/pyGPs/Validation/000077500000000000000000000000001236346020200155505ustar00rootroot00000000000000pyGPs-1.2/pyGPs/Validation/__init__.py000066400000000000000000000000201236346020200176510ustar00rootroot00000000000000import valid pyGPs-1.2/pyGPs/Validation/valid.py000066400000000000000000000077671236346020200172420ustar00rootroot00000000000000#================================================================================ # Marion Neumann [marion dot neumann at uni-bonn dot de] # Daniel Marthaler [marthaler at ge dot com] # Shan Huang [shan dot huang at iais dot fraunhofer dot de] # Kristian Kersting [kristian dot kersting at cs dot tu-dortmund dot de] # # This file is part of pyGPs. # The software package is released under the BSD 2-Clause (FreeBSD) License. # # Copyright (c) by # Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014 #================================================================================ import numpy as np def k_fold_validation(x, y, K=10, randomise=False): ''' Generates K (training, validation) pairs from the items in X. The validation iterables are a partition of X, and each validation iterable is of length len(X)/K. Each training iterable is the complement (within X) of the validation iterable, and so each training iterable is of length (K-1)*len(X)/K. :param x: training data :param y: training targets :param K: number of folds :param randomise: boolean flag. Shuffle data first if it is true.s ''' # whether needed to shuffle data first if randomise: data = np.append(x, y, axis=1) np.random.shuffle(data) x = data[:,:-1] y = data[:,-1] n, D = x.shape assert n > K for k in xrange(K): x_train = np.array([e for i, e in enumerate(x) if i % K != k]) x_test = np.array([e for i, e in enumerate(x) if i % K == k]) y_train = np.array([e for i, e in enumerate(y) if i % K != k]) y_test = np.array([e for i, e in enumerate(y) if i % K == k]) yield x_train, x_test, y_train, y_test def k_fold_index(n, K=10): ''' Similar to k_fold_validation, but only yields indice of folds instead of data in each iteration :param n: size of data (number of instances) :param K: number of folds ''' for k in xrange(K): indice_train = [] indice_test = [] for i in xrange(n): if i % K == k: indice_test.append(i) else: indice_train.append(i) yield indice_train,indice_test def RMSE(predict,target): ''' Root mean squared error :param predict: vector of predicted means :param target: vector of true means :return: root mean squared error ''' error = predict - target return np.sqrt(np.mean(error**2)) def ACC(predict,target): ''' Classification accuracy :param predict: vector of predicted labels(+/- 1) :param target: vector of true labels :return: accuracy ''' n,D = target.shape count = 0. for i in xrange(n): if predict[i,0] == target[i,0]: count += 1 return count/n def Prec(predict,target): ''' Precision for class +1 :param predict: vector of predicted labels(+/- 1) :param target: vector of true labels :return: precision ''' n,D = target.shape count_1 = 0. count_2 = 0. for i in xrange(n): if predict[i,0] == 1: count_1 += 1 if target[i,0] == 1: count_2 += 1 return count_2 / count_1 def Recall(predict,target): ''' Recall for class +1 :param predict: vector of predicted labels(+/- 1) :param target: vector of true labels :return: recall ''' n,D = target.shape count_1 = 0. count_2 = 0. for i in xrange(n): if target[i,0] == 1: count_1 += 1 if predict[i,0] == 1: count_2 += 1 return count_2 / count_1 def NLPD(y, MU, S2): ''' Calculate evaluation measure NLPD in transformed observation space. :param y: observed targets :param MU: vector of predictions/predicted means :param S2: vector of 'self' variances :return: Negative Log Predictive Density. 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Kersting) /Title (pyGPs API) /Author()/Title()/Subject()/Creator(LaTeX with hyperref package)/Producer(pdfTeX-1.40.14)/Keywords() /CreationDate (D:20140722135759+02'00') /ModDate (D:20140722135759+02'00') /Trapped /False /PTEX.Fullbanner (This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/MacPorts 2013_5) kpathsea version 6.1.1) >> endobj 1236 0 obj << /Type /ObjStm /N 12 /First 108 /Length 443 /Filter /FlateDecode >> stream xڅOo0| H=1`iTuoMINmo"m NEl ELP* @ ZFZM)))ܼgbBBB6aiVNp c8T K<cbL|=#D4iYYfNhy±t 2X~9ȧ;-6NOB6uu'dm]_i(݇mɇqZc^x^MXӶ}1 =,jCVÏ)9H endstream endobj 1249 0 obj << /Type /XRef /Index [0 1250] /Size 1250 /W [1 3 1] /Root 1247 0 R /Info 1248 0 R /ID [ ] /Length 2934 /Filter /FlateDecode >> stream x%Yl]Gwk;v$Ŏxwk;,؎8*Rڇ0G%RHtR:[zj#q>sгcbbKH:F#ih-Uwq?1 ~>]^Ej ^ְ8?ojKvq7> Nq%oWuk hy~YMY-$z;sx:@$:6. $/^sV7ZzqMj t3v%?t4`Ivv4`c<&/<`b<*vQ4>'iwR ߘ:OHOqV 絋n1ƳR`mYK{Zc}6,5i<פ8Q=Y)5,tVm׋3f^?b OaHoI#:g2=ޓ@JrJok7 8ZG2 ;U튤>SÁ$뾭R'ndA(%$%RR iPt>tή:S7ZI N'xʛtV>jM<5O6Iδv c3}[ pa/ɕz !0*϶~@d@#v@L`d%sǾ[oF3ڽ.K`-% $Ֆ{RO=U0/]I}!ߛ :$}ݰ*:& %nvr-Ӥ%^wZ![ 6Yld&"YDdɖlcH,&Z>,YҷU&M/?!\VݘcWTmr_ p!a8dqbU8FvX|Y\, >ԋO/_,YRv`{0bEvKmNDC%L} ⁽'B ZpGKXT+Š$()4(Uu]J؜K@hMViO0 :%` !0 F(Ȁ p,)0 fp\&XY t̃p\W5eV ?%s&o6$ԭ;`{6 o3 < [u0ha A - !0aҤ%\ֳ A AqK'Z`Hc # F 73#A A N@U 0ap!7sZ½E=|1bPŠA*U T1bPŠAGb/_ 2d0` !-3'ݬސn$4b Al$4XbĨ%ha ck~aA 7<<*U.x,X#gQ'Z_%ᓯA^W/#ώV,q 7a O>H[pȻuȻuȻuȻuȻuȻuȻuʧr'v endstream endobj startxref 1216483 %%EOF pyGPs-1.2/setup.py000066400000000000000000000010751236346020200141310ustar00rootroot00000000000000 from setuptools import setup, find_packages setup( name='pyGPs', version='1.2', author=['Marion Neumann', 'Shan Huang','Daniel Marthaler', 'Kristian Kersting'], author_email=['marion.neumann@uni-bonn.de.com', 'shan.huang@iais.fraunhofer.de', 'marthaler@ge.com', 'kristian.kersting@cs.tu-dortmund.de'], packages=find_packages(), url='https://github.com/marionmari/pyGPs', license='COPYRIGHT.txt', description='Gaussian Processes for Regression and Classification', long_description=open('README.md').read(), install_requires=[], )

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