gpml-matlab-v3.2-2013-01-15/000755 000765 000024 00000000000 12077253603 015056 5ustar00carlstaff000000 000000 gpml-matlab-v3.2-2013-01-15/.octaverc000644 000765 000024 00000000010 11511036276 016651 0ustar00carlstaff000000 000000 startup gpml-matlab-v3.2-2013-01-15/gp.m000644 000765 000024 00000021004 12077253574 015646 0ustar00carlstaff000000 000000 function [varargout] = gp(hyp, inf, mean, cov, lik, x, y, xs, ys) % Gaussian Process inference and prediction. The gp function provides a % flexible framework for Bayesian inference and prediction with Gaussian % processes for scalar targets, i.e. both regression and binary % classification. The prior is Gaussian process, defined through specification % of its mean and covariance function. The likelihood function is also % specified. Both the prior and the likelihood may have hyperparameters % associated with them. % % Two modes are possible: training or prediction: if no test cases are % supplied, then the negative log marginal likelihood and its partial % derivatives w.r.t. the hyperparameters is computed; this mode is used to fit % the hyperparameters. If test cases are given, then the test set predictive % probabilities are returned. Usage: % % training: [nlZ dnlZ ] = gp(hyp, inf, mean, cov, lik, x, y); % prediction: [ymu ys2 fmu fs2 ] = gp(hyp, inf, mean, cov, lik, x, y, xs); % or: [ymu ys2 fmu fs2 lp] = gp(hyp, inf, mean, cov, lik, x, y, xs, ys); % % where: % % hyp column vector of hyperparameters % inf function specifying the inference method % cov prior covariance function (see below) % mean prior mean function % lik 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 % % 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 % 3rd output in training mode or 6th output in prediction mode % can be reused in prediction mode gp(.., cov, lik, x, post, xs,..) % % See also covFunctions.m, infMethods.m, likFunctions.m, meanFunctions.m. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2013-01-21 if nargin<7 || nargin>9 disp('Usage: [nlZ dnlZ ] = gp(hyp, inf, mean, cov, lik, x, y);') disp(' or: [ymu ys2 fmu fs2 ] = gp(hyp, inf, mean, cov, lik, x, y, xs);') disp(' or: [ymu ys2 fmu fs2 lp] = gp(hyp, inf, mean, cov, lik, x, y, xs, ys);') return end if isempty(mean), mean = {@meanZero}; end % set default mean if ischar(mean) || isa(mean, 'function_handle'), mean = {mean}; end % make cell if isempty(cov), error('Covariance function cannot be empty'); end % no default if ischar(cov) || isa(cov, 'function_handle'), cov = {cov}; end % make cell cov1 = cov{1}; if isa(cov1, 'function_handle'), cov1 = func2str(cov1); end if isempty(inf) % set default inference method if strcmp(cov1,'covFITC'), inf = @infFITC; else inf = @infExact; end else if iscell(inf), inf = inf{1}; end % cell input is allowed if ischar(inf), inf = str2func(inf); end % convert into function handle end if strcmp(cov1,'covFITC') % only infFITC* are possible if isempty(strfind(func2str(inf),'infFITC')==1) error('Only infFITC* are possible inference algorithms') end end % only one possible class of inference algorithms if isempty(lik), lik = {@likGauss}; end % set default lik if ischar(lik) || isa(lik, 'function_handle'), lik = {lik}; end % make cell if iscell(lik), likstr = lik{1}; else likstr = lik; end if ~ischar(likstr), likstr = func2str(likstr); end D = size(x,2); if ~isfield(hyp,'mean'), hyp.mean = []; end % check the hyp specification if eval(feval(mean{:})) ~= numel(hyp.mean) error('Number of mean function hyperparameters disagree with mean function') end if ~isfield(hyp,'cov'), hyp.cov = []; end if eval(feval(cov{:})) ~= numel(hyp.cov) error('Number of cov function hyperparameters disagree with cov function') end if ~isfield(hyp,'lik'), hyp.lik = []; end if eval(feval(lik{:})) ~= numel(hyp.lik) error('Number of lik function hyperparameters disagree with lik function') end try % call the inference method % issue a warning if a classification likelihood is used in conjunction with % labels different from +1 and -1 if strcmp(likstr,'likErf') || strcmp(likstr,'likLogistic') if ~isstruct(y) uy = unique(y); if any( uy~=+1 & uy~=-1 ) warning('You try classification with labels different from {+1,-1}') end end end if nargin>7 % compute marginal likelihood and its derivatives only if needed if isstruct(y) post = y; % reuse a previously computed posterior approximation else post = inf(hyp, mean, cov, lik, x, y); end else if nargout==1 [post nlZ] = inf(hyp, mean, cov, lik, x, y); dnlZ = {}; else [post nlZ dnlZ] = inf(hyp, mean, cov, lik, x, y); end end catch msgstr = lasterr; if nargin > 7, error('Inference method failed [%s]', msgstr); else warning('Inference method failed [%s] .. attempting to continue',msgstr) dnlZ = struct('cov',0*hyp.cov, 'mean',0*hyp.mean, 'lik',0*hyp.lik); varargout = {NaN, dnlZ}; return % continue with a warning end end if nargin==7 % if no test cases are provided varargout = {nlZ, dnlZ, post}; % report -log marg lik, derivatives and post else alpha = post.alpha; L = post.L; sW = post.sW; if issparse(alpha) % handle things for sparse representations nz = alpha ~= 0; % determine nonzero indices if issparse(L), L = full(L(nz,nz)); end % convert L and sW if necessary if issparse(sW), sW = full(sW(nz)); end else nz = true(size(alpha,1),1); end % non-sparse representation if numel(L)==0 % in case L is not provided, we compute it K = feval(cov{:}, hyp.cov, x(nz,:)); L = chol(eye(sum(nz))+sW*sW'.*K); end Ltril = all(all(tril(L,-1)==0)); % is L an upper triangular matrix? ns = size(xs,1); % number of data points nperbatch = 1000; % number of data points per mini batch nact = 0; % number of already processed test data points ymu = zeros(ns,1); ys2 = ymu; fmu = ymu; fs2 = ymu; lp = ymu; % allocate mem while nact use Cholesky parameters (alpha,sW,L) V = L'\(repmat(sW,1,length(id)).*Ks); fs2(id) = kss - sum(V.*V,1)'; % predictive variances else % L is not triangular => use alternative parametrisation fs2(id) = kss + sum(Ks.*(L*Ks),1)'; % predictive variances end fs2(id) = max(fs2(id),0); % remove numerical noise i.e. negative variances Fs2 = repmat(fs2(id),1,N); % we have multiple values in case of sampling if nargin<9 [Lp, Ymu, Ys2] = feval(lik{:},hyp.lik,[],Fmu(:),Fs2(:)); else [Lp, Ymu, Ys2] = feval(lik{:},hyp.lik,repmat(ys(id),1,N),Fmu(:),Fs2(:)); end lp(id) = sum(reshape(Lp, [],N),2)/N; % log probability; sample averaging ymu(id) = sum(reshape(Ymu,[],N),2)/N; % predictive mean ys|y and .. ys2(id) = sum(reshape(Ys2,[],N),2)/N; % .. variance nact = id(end); % set counter to index of last processed data point end if nargin<9 varargout = {ymu, ys2, fmu, fs2, [], post}; % assign output arguments else varargout = {ymu, ys2, fmu, fs2, lp, post}; end end gpml-matlab-v3.2-2013-01-15/mean/000755 000765 000024 00000000000 12077253603 015776 5ustar00carlstaff000000 000000 gpml-matlab-v3.2-2013-01-15/util/000755 000765 000024 00000000000 12077253603 016033 5ustar00carlstaff000000 000000 gpml-matlab-v3.2-2013-01-15/likFunctions.m000644 000765 000024 00000010016 12077253575 017712 0ustar00carlstaff000000 000000 % likelihood functions are provided to be used by the gp.m function: % % likErf (Error function, classification, probit regression) % likLogistic (Logistic, classification, logit regression) % likUni (Uniform likelihood, classification) % % likGauss (Gaussian, regression) % likLaplace (Laplacian or double exponential, regression) % likSech2 (Sech-square, regression) % likT (Student's t, regression) % % likPoisson (Poisson regression, count data) % % likMix (Mixture of individual covariance functions) % % The likelihood functions have three possible modes, the mode being selected % as follows (where "lik" stands for any likelihood function in "lik/lik*.m".): % % 1) With one or no input arguments: [REPORT NUMBER OF HYPERPARAMETERS] % % s = lik OR s = lik(hyp) % % The likelihood function returns a string telling how many hyperparameters it % expects, using the convention that "D" is the dimension of the input space. % For example, calling "likLogistic" returns the string '0'. % % % 2) With three or four input arguments: [PREDICTION MODE] % % lp = lik(hyp, y, mu) OR [lp, ymu, ys2] = lik(hyp, 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. % % % 3) With five or six input arguments, the fifth being a string [INFERENCE MODE] % % [varargout] = lik(hyp, y, mu, s2, inf) OR % [varargout] = lik(hyp, y, mu, s2, inf, i) % % There are three cases for inf, namely a) infLaplace, b) infEP and c) infVB. % The last input i, refers to derivatives w.r.t. the ith hyperparameter. % % a1) [lp,dlp,d2lp,d3lp] = lik(hyp, 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(hyp, 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(hyp, 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(hyp, y, mu, s2, 'infEP', i) % returns derivatives w.r.t. to the ith hyperparameter % dlZhyp = d log(Z) / dhyp_i % % % c1) [h,b,dh,db,d2h,d2b] = lik(hyp, y, [], ga, 'infVB') % ga is the variance of a Gaussian lower bound to the likelihood p(y|f). % p(y|f) \ge exp( b*f - f.^2/(2*ga) - h(ga)/2 ) \propto N(f|b*ga,ga) % The function returns the linear part b and the "scaling function" h(ga) and % derivatives dh = d h/dga, db = d b/dga, d2h = d^2 h/dga and d2b = d^2 b/dga. % % c2) [dhhyp] = lik(hyp, y, [], ga, 'infVB', i) % dhhyp = dh / dhyp_i, the derivative w.r.t. the ith hyperparameter % % 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 = @likLogistic; % lik = {'likMix',{'likUni',@likErf}} % lik = {@likPoisson,'logistic'}; % % See the help for the individual likelihood for the computations specific to % each likelihood function. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2013-01-21 gpml-matlab-v3.2-2013-01-15/covFunctions.m000644 000765 000024 00000011346 12077253575 017731 0ustar00carlstaff000000 000000 % covariance functions to be use by Gaussian process functions. There are two % different kinds of covariance functions: simple and composite: % % simple covariance functions: % covConst - covariance for constant functions % covLIN - linear covariance function % covLINard - linear covariance function with ARD % covLINone - linear covariance function with bias % covMaterniso - Matern covariance function with nu=1/2, 3/2 or 5/2 % covNNone - neural network covariance function % covNoise - independent covariance function (i.e. white noise) % covPeriodic - smooth periodic covariance function (1d) with unit period % covPoly - polynomial covariance function % covPPiso - piecewise polynomial covariance function (compact support) % covRQard - rational quadratic covariance function with ARD % covRQiso - isotropic rational quadratic covariance function % covSEard - squared exponential covariance function with ARD % covSEiso - isotropic squared exponential covariance function % covSEisoU - as above but without latent scale % % composite (meta) covariance functions (see explanation at the bottom): % covScale - scaled version of a covariance function % covProd - products of covariance functions % covSum - sums of covariance functions % covADD - additive covariance function % covMask - mask some dimensions of the data % % special purpose (wrapper) covariance functions % covFITC - to be used in conjunction with infFITC for large scale % regression problems; any covariance can be wrapped by % covFITC such that the FITC approximation is applicable % % Naming convention: all covariance functions are named "cov/cov*.m". A trailing % "iso" means isotropic, "ard" means Automatic Relevance Determination, and % "one" means that the distance measure is parameterized by a single parameter. % % The covariance functions are written according to a special convention where % the exact behaviour depends on the number of input and output arguments % passed to the function. If you want to add new covariance functions, you % should follow this convention if you want them to work with the function gp. % There are four different ways of calling the covariance functions: % % 1) With no (or one) input argument(s): % % s = cov % % The covariance function returns a string s telling how many hyperparameters it % expects, using the convention that "D" is the dimension of the input space. % For example, calling "covRQard" returns the string '(D+2)'. % % 2) With two input arguments: % % K = cov(hyp, x) equivalent to K = cov(hyp, x, []) % % The function computes and returns the covariance matrix where hyp are % the hyperparameters and x is an n by D matrix of cases, where % D is the dimension of the input space. The returned covariance matrix is of % size n by n. % % 3) With three input arguments: % % Ks = cov(hyp, x, xs) % kss = cov(hyp, xs, 'diag') % % The function computes test set covariances; kss is a vector of self covariances % for the test cases in xs (of length ns) and Ks is an (n by ns) matrix of cross % covariances between training cases x and test cases xs. % % 4) With four input arguments: % % dKi = cov(hyp, x, [], i) % dKsi = cov(hyp, x, xs, i) % dkssi = cov(hyp, xs, 'diag', i) % % The function computes and returns the partial derivatives of the % covariance matrices with respect to hyp(i), i.e. with % respect to the hyperparameter number i. % % Covariance functions can be specified in two ways: either as a string % containing the name of the covariance function or using a cell array. For % example: % % cov = 'covRQard'; % cov = {'covRQard'}; % cov = {@covRQard}; % % are supported. Only the second and third form using the cell array can be used % for specifying composite covariance functions, made up of several % contributions. For example: % % cov = {'covScale', {'covRQiso'}}; % cov = {'covSum', {'covRQiso','covSEard','covNoise'}}; % cov = {'covProd',{'covRQiso','covSEard','covNoise'}}; % cov = {'covMask',{mask,'covSEiso'}} % q=1; cov = {'covPPiso',q}; % d=3; cov = {'covPoly',d}; % cov = {'covADD',{[1,2],'covSEiso'}}; % cov = {@covFITC, {@covSEiso}, u}; where u are the inducing inputs % % specifies a covariance function which is the sum of three contributions. To % find out how many hyperparameters this covariance function requires, we do: % % feval(cov{:}) % % which returns the string '3+(D+1)+1' (i.e. the 'covRQiso' contribution uses % 3 parameters, the 'covSEard' uses D+1 and 'covNoise' a single parameter). % % See also doc/usageCov.m. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2013-01-21 gpml-matlab-v3.2-2013-01-15/README000644 000765 000024 00000026333 12077253602 015744 0ustar00carlstaff000000 000000 GAUSSIAN PROCESS REGRESSION AND CLASSIFICATION Toolbox version 3.2 for GNU Octave 3.2.x and Matlab 7.x Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2013-01-15. 0) HOW TO READ ============== If you want to get started without further delay, then read section 1) below and jump right to the examples in doc/index.html. 1) ABOUT THESE PROGRAMS ======================= This collection of matlab programs implements and demonstrates some of the algorithms described in a) the book by Rasmussen and Williams: "Gaussian Processes for Machine Learning", the MIT Press 2006, in b) the article by Nickisch and Rasmussen: "Approximations for Binary Gaussian Process Classification", JMLR 2008, in c) the article by Candela and Rasmussen: "A Unifying View of Sparse Approximate Gaussian Process Regression", JMLR 2005, in d) the paper by Murray, Adams and Mackay: "Elliptical slice sampling", AISTATS 2010, in e) the report by Neal: "Annealed Importance Sampling", UToronto 1998, in f) the paper by Naish-Guzman and Holden: "The Generalized FITC Approximation", NIPS, 2007, and in g) the paper by Duvenaud, Nickisch and Rasmussen: "Additive Gaussian Processes", NIPS, 2011. There are 6 code subdirectories: cov, doc, inf, lik, mean and util. cov/ contains covariance functions cov*.m => see covFunctions.m doc/ contains an index.html file providing documentation. This information is also available from http://www.GaussianProcess.org/gpml/code. Usage of mean, cov, classification and regression is demonstrated in usage*.m. Further details can be found in the developer documentation manual.pdf. inf/ contains the inference algorithms inf*.m => see infMethods.m lik/ contains the likelihood functions lik*.m => see likFunctions.m mean/ contains the mean functions mean*.m => see meanFunctions.m util/ contains optimisation routines, backward compatibility programs and small auxiliary scripts Before running the demos or any other part of the code, you should execute startup.m to add all necessary directories to your path. In Octave, you can also use the .octaverc file. 2) ABOUT MEX FILES ================== Some of the programs make use of the MEX facility in matlab for more efficient implementation. However, if you don't know about how to compile MEX files, you do not need to worry about this -- the code runs anyway. If you do compile the MEX files, this is automatically detected, and the program will run more efficiently. If you do not have a compiler installed on your system, you might want to consult [2]. Two components of the toolbox can be accelerated by MEX. First, there is an auxiliary functions solve_chol.m having a MEX equivalent. This can be compiled by either executing util/make.m from the Matlab/Octave command line. In addition to that, we provide a Makefile for Matlab in util/ that you can run by simply typing make in your shell after having supplied your Matlab path and your operating system. We recommend the make.m script because it works both under Matlab and Octave. Second, in order to use the L-BFGS minimiser for hyperparameter optimisation as an equivalent to minimize.m, you have to compile Peter Carbonetto's "Matlab interface for L-BFGS-B" [1]. The challenge here is the Fortran 77 code. We provide a Makefile suitable for Linux 32/64 bit and Mac whenever you have 1) g77 or 2) gfortran properly set up. Under Ubuntu, you can achieve this by installing the packages fort77 and gfortran, respectively. Compilation is done by editing util/lbfgsb/Makefile. In any case, you need to provide $MATLAB_HOME which can be found by the commands 'locate matlab' or 'find / -name "matlab"'. You can choose between two compilation modes: a) using the mex utility by Matlab [default] provide $MEX, then type 'make mex' b) without mex utility by Matlab provide $MEX_SUFFIX and $MATLAB_LIB, then type 'make nomex' In Ubuntu 10.04 LTS, the libg2c library needed for both 1)+a) and 1)+b) is not included per default. If 'ls /usr/lib/libg2c.*' does not list anything this is the case on your machine. You then whant to install the packages gcc-3.4-base and libg2c0 e.g. from http://packages.ubuntu.com/hardy/gcc-3.4-base and http://packages.ubuntu.com/hardy/libg2c0. After installation, you have to create a symbolic link by 'cd /usr/lib' and 'ln -s libg2c.so.0 libg2c.so'. [1] http://www.cs.ubc.ca/~pcarbo/lbfgsb-for-matlab.html [2] http://www.mathworks.com/support/compilers/R2010a 3) CURRENT VERSION ================== The current version of the programs is 3.2. Previous versions of the code are available at http://www.gaussianprocess.org/gpml/code/oldcode.html. 4) DIFFERENCES TO PREVIOUS VERSIONS =================================== NEW in version 3.2, 2013-01-15 ------------------------------ - added inf/infFITC_Laplace.m and inf/infFITC_EP.m for large scale approximate inference - added inf/infMCMC.m to allow for Markov Chain Monte Carlo sampling and Annealed Importance Sampling; we currently support two samplers: Elliptical Slice Sampling (ESS) and Hybrid Monte Carlo (HMC) - improved accuracy of the digamma function lik/likT.m/dloggamma from three to ten digits precision (thanks to Edward Challis) - added gfortran support to util/lbfgsb/Makefile (thanks to Ernst Kloppenburg) - changed parametrisation in cov/covPeriodic.m (thanks to Philipp Hennig) - fixed a stupid bug in cov/covFITC.m (thanks to Krzysztof Chalupka) - fixed a severe stability bug in the derivatives of inf/infFITC.m (thanks to Andrew McHutchon) - changed scalars in util/solve_chol.c to long to provide 64bit compatibility (thanks to Ernst Kloppenburg) - added infLOO to allow pseudo-likelihood optimisation instead of the marginal likelihood - fixed a performance issue in util/sq_dist.m (thanks to Krzysztof Chalupka) - fixed a bug causing numerical problems in infFITC and covFITC whenever Kuu is close to singular (thanks to Joris M. Mooij) - fixed an issue in covNoise (thanks to Daniel Marthaler) - fixed implicit derivative w.r.t. likelihood in infLaplace* - added two more likelihood functions likMix.m for mixtures and likUni for classification noise - slightly changed the interface for the likelihood functions - added likPoisson for Poisson regression of count data NEW in version 3.1, 2010-10-10 ------------------------------ - following a suggestion by Ed Snelson we now support FITC regression - cov/covFITC.m and inf/infFITC.m have been added with Ed Snelson's help - the covariance interface was slightly changed to make that possible NEW in version 3.0, 2010-07-23 ------------------------------ A major code reorganisation effort did take place in the current release. First, classification and regression are now done by a single file gp.m which is completely generic in the likelihood. The previous regression program gpr.m corresponds to gp.m with Gaussian likelihood. Several other likelihoods for robust regresssion were added. Further, the code now supports mean functions with a similar specification mechanism as already used by the covariance functions. Previous implementations correspond to using meanZero.m. We merged the covariance functions covMatern3iso.m and covMatern5.iso into a single covMaterniso.m and added a covariance function for additive functions as well as the possibility to use only certain components of the data. Finally, we included covPPiso.m a piecewise polynomial covariance function with compact support. New likelihood functions in lik/: likGauss.m, likLaplace.m, likLogistic.m and likT.m New mean functions in mean/: meanConst.m, meanLinear.m, meanPow.m, meanProd.m, meanSum.m and meanZero.m New covariance functions in cov/: covADD.m, covMask.m, covPPiso.m The gprSRPP.m function which previously provided "Subset of Regressors" and the "Projected Process" approximation has now been removed. NEW in version 2.1, 2007-07-25 ------------------------------ covConst.m: fixed a bug which caused an error in the derivative of the log marginal likelihood for certain combinations of covariance functions and approximation methods. (Thanks to Antonio Eleuteri for reporting the problem) gauher.m: added the function "gauher.m" which was mistakenly missing from the previous release. This caused an error for certain combinations of approximation method and likelihood function. logistic.m: modified the approximation of moments calculation to use a mixture of cumulative Gaussian, rather than Gauss-Hermite quadrature, as the former turns out to be more accurate. NEW in version 2.0, 2007-06-25 ------------------------------ Some code restructuring has taken place for the classification code to make it more modular, to facilitate addition of new likelihood functions and approximations methods. Now, all classification is done using the binaryGP function, which (among other things) takes an approximation method and a likelihood function as an arguments. Thus, binaryGP replaces both binaryEPGP and binaryLapaceGP, although wrapper functions are still provided for backward compatibility. This gives added flexibility: now EP can also be used wth the logistic likelihood function (implemented using Gauss-Hermite quadrature). approxEP.m: New file, containing the Expectation Propagation approximation method, which was previously contained in binaryEPGP.m approxLA.m: New file, containing Laplaces approximation method, which was previously contained in binaryLaplace.m approximations.m: New file, help for the approximation methods. binaryEPGP.m: This file has been replaced by a wrapper (for backward compatibility) which calls the more general binaryGP function. binaryGP.m: New general function to do binary classification. binaryLaplaceGP.m: This file has been replaced by a wrapper (for backward compatibility) which calls the more general binaryGP function. covMatern3iso.m, covMatern5iso.m, covNNone.m, covRQard.m, covRQiso.m, cosSEard, covSEiso: now check more carefully, that persistent variables have the correct sizes, and some variable names have been modified. cumGauss.m: New file, containing code for the cumulative Gaussian likelihood function likelihoods.m: New file, help for likelihood functions logistic.m: New file, logistic likelihood NEW in version 1.3, 2006-09-08 ------------------------------ covRQard.m: bugfix: replaced x with x' and z with z' in line 36 covRQiso.m: bugfix: replaced x with x' and z with z' in line 28 minimize.m: correction: replaced "error()" with "error('')", and made a few cosmetic changes binaryEPGP.m: added the line "lml = -n*log(2);" in line 77. This change should be largely inconsequential, but occationally may save things when the covariance matrix is exceptionally badly conditioned. NEW in version 1.2, 2006-05-10 ------------------------------ added the "erfint" function to "binaryLaplaceGP.m". The erfint function was missing by mistake, preventing the use of the "logistic" likelihood. NEW in version 1.1, 2006-04-12 ------------------------------ added files: "covProd.m" and "covPeriodic.m" changes: "covSEiso.m" was changed slightly to avoid the use of persistent variables NEW in version 1.0, 2006-03-29 ------------------------------ initial version shipped with the book gpml-matlab-v3.2-2013-01-15/inf/000755 000765 000024 00000000000 12077253603 015632 5ustar00carlstaff000000 000000 gpml-matlab-v3.2-2013-01-15/infMethods.m000644 000765 000024 00000004673 12077253575 017356 0ustar00carlstaff000000 000000 % 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 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 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 Leave-One-Out predictive probability and Least-Squares Approxim. % % The interface to the approximation methods is the following: % % function [post nlZ dnlZ] = inf..(hyp, cov, lik, x, y) % % where: % % hyp is a struct of hyperparameters % cov is the name of the covariance function (see covFunctions.m) % lik is the name of the likelihood function (see likFunctions.m) % x is a n by D matrix of training inputs % y is a (column) vector (of size n) of targets % % nlZ is the returned value of the negative log marginal likelihood % dnlZ is a (column) vector of partial derivatives of the negative % log marginal likelihood w.r.t. each hyperparameter % post struct representation of the (approximate) posterior containing % alpha is a (sparse or full column vector) containing inv(K)*m, where K % is the prior covariance matrix and m the approx posterior mean % sW is a (sparse or full column) vector containing diagonal of sqrt(W) % the approximate posterior covariance matrix is inv(inv(K)+W) % L is a (sparse or full) matrix, L = chol(sW*K*sW+eye(n)) % % 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 separate inf??.m files. See also gp.m % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2013-01-21 gpml-matlab-v3.2-2013-01-15/meanFunctions.m000644 000765 000024 00000003112 12077253575 020052 0ustar00carlstaff000000 000000 % mean functions to be use by Gaussian process functions. There are two % different kinds of mean functions: simple and composite: % % simple mean functions: % % meanZero - zero mean function % meanOne - one mean function % meanConst - constant mean function % meanLinear - linear mean function % % composite covariance functions (see explanation at the bottom): % % meanScale - scaled version of a mean function % meanPow - power of a mean function % meanProd - products of mean functions % meanSum - sums of mean functions % meanMask - mask some dimensions of the data % % Naming convention: all mean functions are named "mean/mean*.m". % % % 1) With no or only a single input argument: % % s = meanNAME or s = meanNAME(hyp) % % The mean function returns a string s telling how many hyperparameters hyp it % expects, using the convention that "D" is the dimension of the input space. % For example, calling "meanLinear" returns the string 'D'. % % 2) With two input arguments: % % m = meanNAME(hyp, x) % % The function computes and returns the mean vector where hyp are the % hyperparameters and x is an n by D matrix of cases, where D is the dimension % of the input space. The returned mean vector is of size n by 1. % % 3) With three input arguments: % % dm = meanNAME(hyp, x, i) % % The function computes and returns the n by 1 vector of partial derivatives % of the mean vector w.r.t. hyp(i) i.e. hyperparameter number i. % % See also doc/usageMean.m. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2013-01-21 gpml-matlab-v3.2-2013-01-15/lik/000755 000765 000024 00000000000 12077253603 015635 5ustar00carlstaff000000 000000 gpml-matlab-v3.2-2013-01-15/doc/000755 000765 000024 00000000000 12077253603 015623 5ustar00carlstaff000000 000000 gpml-matlab-v3.2-2013-01-15/Copyright000644 000765 000024 00000003456 12077213602 016754 0ustar00carlstaff000000 000000 GAUSSIAN PROCESS REGRESSION AND CLASSIFICATION Toolbox version 3.2 for GNU Octave 3.2.x and Matlab 7.x The code is released under the FreeBSD License. Copyright (c) 2005-2013 Carl Edward Rasmussen & Hannes Nickisch. All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: 1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. 2. 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 CARL EDWARD RASMUSSEN & HANNES NICKISCH ``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 CARL EDWARD RASMUSSEN & HANNES NICKISCH 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. The views and conclusions contained in the software and documentation are those of the authors and should not be interpreted as representing official policies, either expressed or implied, of Carl Edward Rasmussen & Hannes Nickisch. The code and associated documentation is available from http://gaussianprocess.org/gpml/code. gpml-matlab-v3.2-2013-01-15/cov/000755 000765 000024 00000000000 12077253603 015645 5ustar00carlstaff000000 000000 gpml-matlab-v3.2-2013-01-15/startup.m000644 000765 000024 00000001410 11511036276 016727 0ustar00carlstaff000000 000000 % startup script to make Octave/Matlab aware of the GPML package % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch 2010-07-05. disp(['executing gpml startup script...']); OCT = exist('OCTAVE_VERSION') ~= 0; % check if we run Matlab or Octave me = mfilename; % what is my filename mydir = which(me); mydir = mydir(1:end-2-numel(me)); % where am I located if OCT && numel(mydir)==2 if strcmp(mydir,'./'), mydir = [pwd,mydir(2:end)]; end end % OCTAVE 3.0.x relative, MATLAB and newer have absolute path addpath(mydir(1:end-1)) addpath([mydir,'cov']) addpath([mydir,'doc']) addpath([mydir,'inf']) addpath([mydir,'lik']) addpath([mydir,'mean']) addpath([mydir,'util']) clear me mydir gpml-matlab-v3.2-2013-01-15/cov/covLIN.m000644 000765 000024 00000002036 11511036246 017150 0ustar00carlstaff000000 000000 function K = covLIN(hyp, x, z, i) % Linear covariance function. The covariance function is parameterized as: % % k(x^p,x^q) = x^p'*x^q % % The are no hyperparameters: % % hyp = [ ] % % Note that there is no bias or scale term; use covConst to add these. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if nargin<2, K = '0'; return; end % report number of parameters if nargin<3, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode % compute inner products if dg % vector kxx K = sum(x.*x,2); else if xeqz % symmetric matrix Kxx K = x*x'; else % cross covariances Kxz K = x*z'; end end if nargin>3 % derivatives error('Unknown hyperparameter') endgpml-matlab-v3.2-2013-01-15/cov/covLINard.m000644 000765 000024 00000003011 11511036246 017631 0ustar00carlstaff000000 000000 function K = covLINard(hyp, x, z, i) % Linear covariance function with Automatic Relevance Determination (ARD). The % covariance function is parameterized as: % % k(x^p,x^q) = x^p'*inv(P)*x^q % % where the P matrix is diagonal with ARD parameters ell_1^2,...,ell_D^2, where % D is the dimension of the input space. The hyperparameters are: % % hyp = [ log(ell_1) % log(ell_2) % .. % log(ell_D) ] % % Note that there is no bias term; use covConst to add a bias. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if nargin<2, K = 'D'; return; end % report number of parameters if nargin<3, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode ell = exp(hyp); [n,D] = size(x); x = x*diag(1./ell); % precompute inner products if dg % vector kxx K = sum(x.*x,2); else if xeqz % symmetric matrix Kxx K = x*x'; else % cross covariances Kxz z = z*diag(1./ell); K = x*z'; end end if nargin>3 % derivatives if i<=D if dg K = -2*x(:,i).*x(:,i); else if xeqz K = -2*x(:,i)*x(:,i)'; else K = -2*x(:,i)*z(:,i)'; end end else error('Unknown hyperparameter') end endgpml-matlab-v3.2-2013-01-15/cov/covPoly.m000644 000765 000024 00000002717 11511036246 017457 0ustar00carlstaff000000 000000 function K = covPoly(d, hyp, x, z, i) % Polynomial covariance function. The covariance function is parameterized as: % % k(x^p,x^q) = sf^2 * ( c + (x^p)'*(x^q) )^d % % The hyperparameters are: % % hyp = [ log(c) % log(sf) ] % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if nargin<3, K = '2'; return; end % report number of parameters if nargin<4, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode c = exp(hyp(1)); % inhomogeneous offset sf2 = exp(2*hyp(2)); % signal variance if d~=max(1,fix(d)), error('only nonzero integers allowed for d'), end % degree % precompute inner products if dg % vector kxx K = sum(x.*x,2); else if xeqz % symmetric matrix Kxx K = x*x'; else % cross covariances Kxz K = x*z'; end end if nargin<5 % covariances K = sf2*( c + K ).^d; else % derivatives if i==1 K = c*d*sf2*( c + K ).^(d-1); elseif i==2 K = 2*sf2*( c + K ).^d; else error('Unknown hyperparameter') end endgpml-matlab-v3.2-2013-01-15/cov/covLINone.m000644 000765 000024 00000002516 11511036246 017655 0ustar00carlstaff000000 000000 function K = covLINone(hyp, x, z, i) % Linear covariance function with a single hyperparameter. The covariance % function is parameterized as: % % k(x^p,x^q) = (x^p'*x^q + 1)/t2; % % where the P matrix is t2 times the unit matrix. The second term plays the % role of the bias. The hyperparameter is: % % hyp = [ log(sqrt(t2)) ] % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if nargin<2, K = '1'; return; end % report number of parameters if nargin<3, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode it2 = exp(-2*hyp); % t2 inverse % precompute inner products if dg % vector kxx K = sum(x.*x,2); else if xeqz % symmetric matrix Kxx K = x*x'; else % cross covariances Kxz K = x*z'; end end if nargin<4 % covariances K = it2*(1+K); else % derivatives if i==1 K = -2*it2*(1+K); else error('Unknown hyperparameter') end endgpml-matlab-v3.2-2013-01-15/cov/covSEisoU.m000644 000765 000024 00000002760 11511036246 017701 0ustar00carlstaff000000 000000 function K = covSEisoU(hyp, x, z, i) % Squared Exponential covariance function with isotropic distance measure with % unit magnitude. The covariance function is parameterized as: % % k(x^p,x^q) = exp(-(x^p - x^q)'*inv(P)*(x^p - x^q)/2) % % where the P matrix is ell^2 times the unit matrix and sf2 is the signal % variance. The hyperparameters are: % % hyp = [ log(ell) ] % % For more help on design of covariance functions, try "help covFunctions". % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if nargin<2, K = '1'; return; end % report number of parameters if nargin<3, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode ell = exp(hyp(1)); % characteristic length scale % precompute squared distances if dg % vector kxx K = zeros(size(x,1),1); else if xeqz % symmetric matrix Kxx K = sq_dist(x'/ell); else % cross covariances Kxz K = sq_dist(x'/ell,z'/ell); end end if nargin<4 % covariances K = exp(-K/2); else % derivatives if i==1 K = exp(-K/2).*K; else error('Unknown hyperparameter') end endgpml-matlab-v3.2-2013-01-15/cov/covConst.m000644 000765 000024 00000002172 11511036246 017615 0ustar00carlstaff000000 000000 function K = covConst(hyp, x, z, i) % covariance function for a constant function. The covariance function is % parameterized as: % % k(x^p,x^q) = s2; % % The scalar hyperparameter is: % % hyp = [ log(sqrt(s2)) ] % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if nargin<2, K = '1'; return; end % report number of parameters if nargin<3, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode s2 = exp(2*hyp); % s2 n = size(x,1); if dg % vector kxx K = s2*ones(n,1); else if xeqz % symmetric matrix Kxx K = s2*ones(n); else % cross covariances Kxz K = s2*ones(n,size(z,1)); end end if nargin>3 % derivatives if i==1 K = 2*K; else error('Unknown hyperparameter') end endgpml-matlab-v3.2-2013-01-15/cov/covADD.m000644 000765 000024 00000007120 12077070412 017116 0ustar00carlstaff000000 000000 function K = covADD(cov, hyp, x, z, i) % Additive covariance function using a 1d base covariance function % cov(x^p,x^q;hyp) with individual hyperparameters hyp. % % k(x^p,x^q) = \sum_{r \in R} sf_r \sum_{|I|=r} % \prod_{i \in I} cov(x^p_i,x^q_i;hyp_i) % % hyp = [ hyp_1 % hyp_2 % ... % hyp_D % log(sf_R(1)) % ... % log(sf_R(end)) ] % % where hyp_d are the parameters of the 1d covariance function which are shared % over the different values of R(1) to R(end). % % Please see the paper Additive Gaussian Processes by Duvenaud, Nickisch and % Rasmussen, NIPS, 2011 for details. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. R = cov{1}; nh = eval(feval(cov{2})); % number of hypers per individual covariance nr = numel(R); % number of different degrees of interaction if nargin<3 % report number of hyper parameters K = ['D*', int2str(nh), '+', int2str(nr)]; return end if nargin<4, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode [n,D] = size(x); % dimensionality sf2 = exp( 2*hyp(D*nh+(1:nr)) ); % signal variances of individual degrees Kd = Kdim(cov{2},hyp,x,z); % evaluate dimensionwise covariances K if nargin<5 % covariances EE = elsympol(Kd,max(R)); % Rth elementary symmetric polynomials K = 0; for ii=1:nr, K = K + sf2(ii)*EE(:,:,R(ii)+1); end % sf2 weighted sum else % derivatives if i <= D*nh % individual covariance function parameters j = fix(1+(i-1)/nh); % j is the dimension of the hyperparameter if dg, zj='diag'; else if xeqz, zj=[]; else zj=z(:,j); end, end dKj = feval(cov{2},hyp(nh*(j-1)+(1:nh)),x(:,j),zj,i-(j-1)*nh); % other dK=0 % the final derivative is a sum of multilinear terms, so if only one term % depends on the hyperparameter under consideration, we can factorise it % out and compute the sum with one degree less E = elsympol(Kd(:,:,[1:j-1,j+1:D]),max(R)-1); % R-1th elementary sym polyn K = 0; for ii=1:nr, K = K + sf2(ii)*E(:,:,R(ii)); end % sf2 weighted sum K = dKj.*K; elseif i <= D*nh+nr EE = elsympol(Kd,max(R)); % Rth elementary symmetric polynomials j = i-D*nh; K = 2*sf2(j)*EE(:,:,R(j)+1); % rest of the sf2 weighted sum else error('Unknown hyperparameter') end end % evaluate dimensionwise covariances K function K = Kdim(cov,hyp,x,z) [n,D] = size(x); % dimensionality nh = eval(feval(cov)); % number of hypers per individual covariance if nargin<4, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode if dg % allocate memory K = zeros(n,1,D); else if xeqz, K = zeros(n,n,D); else K = zeros(n,size(z,1),D); end end for d=1:D hyp_d = hyp(nh*(d-1)+(1:nh)); % hyperparamter of dimension d if dg K(:,:,d) = feval(cov,hyp_d,x(:,d),'diag'); else if xeqz K(:,:,d) = feval(cov,hyp_d,x(:,d)); else K(:,:,d) = feval(cov,hyp_d,x(:,d),z(:,d)); end end end gpml-matlab-v3.2-2013-01-15/cov/covFITC.m000644 000765 000024 00000003343 12040247015 017251 0ustar00carlstaff000000 000000 function [K,Kuu,Ku] = covFITC(cov, xu, hyp, x, z, i) % 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. In fact, it wraps a proper % covariance function such that it can be used together with infFITC.m. % Instead of outputing the full covariance, it returns cross-covariances between % the inputs x, z and the inducing inputs xu as needed by infFITC.m % % Copyright (c) by Ed Snelson, Carl Edward Rasmussen % and Hannes Nickisch, 2011-11-02. % % See also COVFUNCTIONS.M, INFFITC.M. if nargin<4, K = feval(cov{:}); return, end if nargin<5, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode if size(xu,2) ~= size(x,2) error('Dimensionality of inducing inputs must match training inputs'); end if nargin<6 % covariances if dg K = feval(cov{:},hyp,x,'diag'); else if xeqz K = feval(cov{:},hyp,x,'diag'); if nargout>1, Kuu = feval(cov{:},hyp,xu); end if nargout>2, Ku = feval(cov{:},hyp,xu,x); end else K = feval(cov{:},hyp,xu,z); end end else % derivatives if dg K = feval(cov{:},hyp,x,'diag',i); else if xeqz K = feval(cov{:},hyp,x,'diag',i); if nargout>1, Kuu = feval(cov{:},hyp,xu,[],i); end if nargout>2, Ku = feval(cov{:},hyp,xu,x,i); end else K = feval(cov{:},hyp,xu,z,i); end end end gpml-matlab-v3.2-2013-01-15/cov/covPeriodic.m000644 000765 000024 00000003015 11541741254 020267 0ustar00carlstaff000000 000000 function K = covPeriodic(hyp, x, z, i) % Stationary covariance function for a smooth periodic function, with period p: % % k(x,y) = sf2 * exp( -2*sin^2( pi*||x-y||/p )/ell^2 ) % % where the hyperparameters are: % % hyp = [ log(ell) % log(p) % log(sqrt(sf2)) ] % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-01-05. % % See also COVFUNCTIONS.M. if nargin<2, K = '3'; return; end % report number of parameters if nargin<3, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode n = size(x,1); ell = exp(hyp(1)); p = exp(hyp(2)); sf2 = exp(2*hyp(3)); % precompute distances if dg % vector kxx K = zeros(size(x,1),1); else if xeqz % symmetric matrix Kxx K = sqrt(sq_dist(x')); else % cross covariances Kxz K = sqrt(sq_dist(x',z')); end end K = pi*K/p; if nargin<4 % covariances K = sin(K)/ell; K = K.*K; K = sf2*exp(-2*K); else % derivatives if i==1 K = sin(K)/ell; K = K.*K; K = 4*sf2*exp(-2*K).*K; elseif i==2 R = sin(K)/ell; K = 4*sf2/ell*exp(-2*R.*R).*R.*cos(K).*K; elseif i==3 K = sin(K)/ell; K = K.*K; K = 2*sf2*exp(-2*K); else error('Unknown hyperparameter') end endgpml-matlab-v3.2-2013-01-15/cov/covNNone.m000644 000765 000024 00000004016 11511036246 017543 0ustar00carlstaff000000 000000 function K = covNNone(hyp, x, z, i) % Neural network covariance function with a single parameter for the distance % measure. The covariance function is parameterized as: % % k(x^p,x^q) = sf2 * asin(x^p'*P*x^q / sqrt[(1+x^p'*P*x^p)*(1+x^q'*P*x^q)]) % % where the x^p and x^q vectors on the right hand side have an added extra bias % entry with unit value. P is ell^-2 times the unit matrix and sf2 controls the % signal variance. The hyperparameters are: % % hyp = [ log(ell) % log(sqrt(sf2) ] % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if nargin<2, K = '2'; return; end % report number of parameters if nargin<3, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode n = size(x,1); ell2 = exp(2*hyp(1)); sf2 = exp(2*hyp(2)); sx = 1 + sum(x.*x,2); if dg % vector kxx K = sx./(sx+ell2); else if xeqz % symmetric matrix Kxx S = 1 + x*x'; K = S./(sqrt(ell2+sx)*sqrt(ell2+sx)'); else % cross covariances Kxz S = 1 + x*z'; sz = 1 + sum(z.*z,2); K = S./(sqrt(ell2+sx)*sqrt(ell2+sz)'); end end if nargin<4 % covariances K = sf2*asin(K); else % derivatives if i==1 % lengthscale if dg V = K; else vx = sx./(ell2+sx); if xeqz V = repmat(vx/2,1,n) + repmat(vx'/2,n,1); else vz = sz./(ell2+sz); nz = size(z,1); V = repmat(vx/2,1,nz) + repmat(vz'/2,n,1); end end K = -2*sf2*(K-K.*V)./sqrt(1-K.*K); elseif i==2 % magnitude K = 2*sf2*asin(K); else error('Unknown hyperparameter') end endgpml-matlab-v3.2-2013-01-15/cov/covRQard.m000644 000765 000024 00000004320 11511036246 017535 0ustar00carlstaff000000 000000 function K = covRQard(hyp, x, z, i) % Rational Quadratic covariance function with Automatic Relevance Determination % (ARD) distance measure. The covariance function is parameterized as: % % k(x^p,x^q) = sf2 * [1 + (x^p - x^q)'*inv(P)*(x^p - x^q)/(2*alpha)]^(-alpha) % % where the P matrix is diagonal with ARD parameters ell_1^2,...,ell_D^2, where % D is the dimension of the input space, sf2 is the signal variance and alpha % is the shape parameter for the RQ covariance. The hyperparameters are: % % hyp = [ log(ell_1) % log(ell_2) % .. % log(ell_D) % log(sqrt(sf2)) % log(alpha) ] % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-08-04. % % See also COVFUNCTIONS.M. if nargin<2, K = '(D+2)'; return; end % report number of parameters if nargin<3, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode [n,D] = size(x); ell = exp(hyp(1:D)); sf2 = exp(2*hyp(D+1)); alpha = exp(hyp(D+2)); % precompute squared distances if dg % vector kxx D2 = zeros(size(x,1),1); else if xeqz % symmetric matrix Kxx D2 = sq_dist(diag(1./ell)*x'); else % cross covariances Kxz D2 = sq_dist(diag(1./ell)*x',diag(1./ell)*z'); end end if nargin<4 % covariances K = sf2*((1+0.5*D2/alpha).^(-alpha)); else % derivatives if i<=D % length scale parameter if dg K = D2*0; else if xeqz K = sf2*(1+0.5*D2/alpha).^(-alpha-1).*sq_dist(x(:,i)'/ell(i)); else K = sf2*(1+0.5*D2/alpha).^(-alpha-1).*sq_dist(x(:,i)'/ell(i),z(:,i)'/ell(i)); end end elseif i==D+1 % magnitude parameter K = 2*sf2*((1+0.5*D2/alpha).^(-alpha)); elseif i==D+2 K = (1+0.5*D2/alpha); K = sf2*K.^(-alpha).*(0.5*D2./K - alpha*log(K)); else error('Unknown hyperparameter') end endgpml-matlab-v3.2-2013-01-15/cov/covSum.m000644 000765 000024 00000003742 11511036246 017277 0ustar00carlstaff000000 000000 function K = covSum(cov, hyp, x, z, i) % covSum - compose a covariance function as the sum of other covariance % functions. This function doesn't actually compute very much on its own, it % merely does some bookkeeping, and calls other covariance functions to do the % actual work. % % Copyright (c) by Carl Edward Rasmussen & Hannes Nickisch 2010-09-10. % % See also COVFUNCTIONS.M. if numel(cov)==0, error('We require at least one summand.'), end for ii = 1:numel(cov) % iterate over covariance functions f = cov(ii); if iscell(f{:}), f = f{:}; end % expand cell array if necessary j(ii) = cellstr(feval(f{:})); % collect number hypers end if nargin<3 % report number of parameters K = char(j(1)); for ii=2:length(cov), K = [K, '+', char(j(ii))]; end, return end if nargin<4, z = []; end % make sure, z exists [n,D] = size(x); v = []; % v vector indicates to which covariance parameters belong for ii = 1:length(cov), v = [v repmat(ii, 1, eval(char(j(ii))))]; end if nargin<5 % covariances K = 0; if nargin==3, z = []; end % set default for ii = 1:length(cov) % iteration over summand functions f = cov(ii); if iscell(f{:}), f = f{:}; end % expand cell array if necessary K = K + feval(f{:}, hyp(v==ii), x, z); % accumulate covariances end else % derivatives if i<=length(v) vi = v(i); % which covariance function j = sum(v(1:i)==vi); % which parameter in that covariance f = cov(vi); if iscell(f{:}), f = f{:}; end % dereference cell array if necessary K = feval(f{:}, hyp(v==vi), x, z, j); % compute derivative else error('Unknown hyperparameter') end endgpml-matlab-v3.2-2013-01-15/cov/covPPiso.m000644 000765 000024 00000004601 11511036246 017560 0ustar00carlstaff000000 000000 function K = covPPiso(v, hyp, x, z, i) % Piecewise polynomial covariance function with compact support, v = 0,1,2,3. % The covariance functions are 2v times contin. diff'ble and the corresponding % processes are hence v times mean-square diffble. The covariance function is: % % k(x^p,x^q) = s2f * (1-r)_+.^j * f(r,j) % % where r is the distance sqrt((x^p-x^q)'*inv(P)*(x^p-x^q)), P is ell^2 times % the unit matrix and sf2 is the signal variance. The hyperparameters are: % % hyp = [ log(ell) % log(sqrt(sf2)) ] % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if nargin<3, K = '2'; return; end % report number of parameters if nargin<4, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode ell = exp(hyp(1)); sf2 = exp(2*hyp(2)); if all(v~=[0,1,2,3]), error('only 0,1,2 and 3 allowed for v'), end % degree j = floor(size(x,2)/2)+v+1; % exponent switch v case 0, f = @(r,j) 1; df = @(r,j) 0; case 1, f = @(r,j) 1 + (j+1)*r; df = @(r,j) (j+1); case 2, f = @(r,j) 1 + (j+2)*r + ( j^2+ 4*j+ 3)/ 3*r.^2; df = @(r,j) (j+2) + 2*( j^2+ 4*j+ 3)/ 3*r; case 3, f = @(r,j) 1 + (j+3)*r + (6*j^2+36*j+45)/15*r.^2 ... + (j^3+9*j^2+23*j+15)/15*r.^3; df = @(r,j) (j+3) + 2*(6*j^2+36*j+45)/15*r ... + (j^3+9*j^2+23*j+15)/ 5*r.^2; end pp = @(r,j,v,f) max(1-r,0).^(j+v).*f(r,j); dpp = @(r,j,v,f) max(1-r,0).^(j+v-1).*r.*( (j+v)*f(r,j) - max(1-r,0).*df(r,j) ); % precompute squared distances if dg % vector kxx K = zeros(size(x,1),1); else if xeqz % symmetric matrix Kxx K = sqrt( sq_dist(x'/ell) ); else % cross covariances Kxz K = sqrt( sq_dist(x'/ell,z'/ell) ); end end if nargin<5 % covariances K = sf2*pp( K, j, v, f ); else % derivatives if i==1 K = sf2*dpp( K, j, v, f ); elseif i==2 K = 2*sf2*pp( K, j, v, f ); else error('Unknown hyperparameter') end endgpml-matlab-v3.2-2013-01-15/cov/covMask.m000644 000765 000024 00000004133 12077070412 017422 0ustar00carlstaff000000 000000 function K = covMask(cov, hyp, x, z, i) % Apply a covariance function to a subset of the dimensions only. The subset can % either be specified by a 0/1 mask by a boolean mask or by an index set. % % This function doesn't actually compute very much on its own, it merely does % some bookkeeping, and calls another covariance function to do the actual work. % % The function was suggested by Iain Murray, 2010-02-18 and is based on an % earlier implementation of his dating back to 2009-06-16. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2012-11-17. % % See also COVFUNCTIONS.M. mask = fix(cov{1}(:)); % either a binary mask or an index set cov = cov(2); % covariance function to be masked if iscell(cov{:}), cov = cov{:}; end % properly unwrap nested cell arrays nh_string = feval(cov{:}); % number of hyperparameters of the full covariance if max(mask)<2 && length(mask)>1, mask = find(mask); end % convert 1/0->index D = length(mask); % masked dimension if nargin<3, K = num2str(eval(nh_string)); return, end % number of parameters if nargin<4, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode if eval(nh_string)~=length(hyp) % check hyperparameters error('number of hyperparameters does not match size of masked data') end if nargin<5 % covariances if dg K = feval(cov{:}, hyp, x(:,mask), 'diag'); else if xeqz K = feval(cov{:}, hyp, x(:,mask)); else K = feval(cov{:}, hyp, x(:,mask), z(:,mask)); end end else % derivatives if i <= eval(nh_string) if dg K = feval(cov{:}, hyp, x(:,mask), 'diag', i); else if xeqz K = feval(cov{:}, hyp, x(:,mask), [], i); else K = feval(cov{:}, hyp, x(:,mask), z(:,mask), i); end end else error('Unknown hyperparameter') end endgpml-matlab-v3.2-2013-01-15/cov/covScale.m000644 000765 000024 00000002067 11511036246 017561 0ustar00carlstaff000000 000000 function K = covScale(cov, hyp, x, z, i) % meanScale - compose a mean function as a scaled version of another one. % % k(x^p,x^q) = sf^2 * k_0(x^p,x^q) % % The hyperparameter is: % % hyp = [ log(sf) ] % % This function doesn't actually compute very much on its own, it merely does % some bookkeeping, and calls other mean function to do the actual work. % % Copyright (c) by Carl Edward Rasmussen & Hannes Nickisch 2010-09-10. % % See also MEANFUNCTIONS.M. if nargin<3 % report number of parameters K = [feval(cov{:}),'+1']; return end if nargin<4, z = []; end % make sure, z exists [n,D] = size(x); sf2 = exp(2*hyp(1)); % signal variance if nargin<5 % covariances K = sf2*feval(cov{:},hyp(2:end),x,z); else % derivatives if i==1 K = 2*sf2*feval(cov{:},hyp(2:end),x,z); else K = sf2*feval(cov{:},hyp(2:end),x,z,i-1); end endgpml-matlab-v3.2-2013-01-15/cov/covRQiso.m000644 000765 000024 00000003505 11511036246 017565 0ustar00carlstaff000000 000000 function K = covRQiso(hyp, x, z, i) % Rational Quadratic covariance function with isotropic distance measure. The % covariance function is parameterized as: % % k(x^p,x^q) = sf2 * [1 + (x^p - x^q)'*inv(P)*(x^p - x^q)/(2*alpha)]^(-alpha) % % where the P matrix is ell^2 times the unit matrix, sf2 is the signal % variance and alpha is the shape parameter for the RQ covariance. The % hyperparameters are: % % hyp = [ log(ell) % log(sqrt(sf2)) % log(alpha) ] % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if nargin<2, K = '3'; return; end % report number of parameters if nargin<3, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode ell = exp(hyp(1)); sf2 = exp(2*hyp(2)); alpha = exp(hyp(3)); % precompute squared distances if dg % vector kxx D2 = zeros(size(x,1),1); else if xeqz % symmetric matrix Kxx D2 = sq_dist(x'/ell); else % cross covariances Kxz D2 = sq_dist(x'/ell,z'/ell); end end if nargin<4 % covariances K = sf2*((1+0.5*D2/alpha).^(-alpha)); else % derivatives if i==1 % length scale parameter K = sf2*(1+0.5*D2/alpha).^(-alpha-1).*D2; elseif i==2 % magnitude parameter K = 2*sf2*((1+0.5*D2/alpha).^(-alpha)); elseif i==3 K = (1+0.5*D2/alpha); K = sf2*K.^(-alpha).*(0.5*D2./K - alpha*log(K)); else error('Unknown hyperparameter') end endgpml-matlab-v3.2-2013-01-15/cov/covNoise.m000644 000765 000024 00000003303 12077070412 017602 0ustar00carlstaff000000 000000 function K = covNoise(hyp, x, z, i) % Independent covariance function, ie "white noise", with specified variance. % The covariance function is specified as: % % k(x^p,x^q) = s2 * \delta(p,q) % % where s2 is the noise variance and \delta(p,q) is a Kronecker delta function % which is 1 iff p=q and zero otherwise. Two data points p and q are considered % equal if their norm is less than 1e-9. The hyperparameter is % % hyp = [ log(sqrt(s2)) ] % % For more help on design of covariance functions, try "help covFunctions". % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2012-12-19. % % See also COVFUNCTIONS.M. tol = 1e-9; % threshold on the norm when two vectors are considered to be equal if nargin<2, K = '1'; return; end % report number of parameters if nargin<3, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode if ndims(x)==ndims(z) && all(size(x)==size(z)), xeqz = norm(x-z,'inf')0; % determine mode ell = exp(hyp(1)); % characteristic length scale sf2 = exp(2*hyp(2)); % signal variance % precompute squared distances if dg % vector kxx K = zeros(size(x,1),1); else if xeqz % symmetric matrix Kxx K = sq_dist(x'/ell); else % cross covariances Kxz K = sq_dist(x'/ell,z'/ell); end end if nargin<4 % covariances K = sf2*exp(-K/2); else % derivatives if i==1 K = sf2*exp(-K/2).*K; elseif i==2 K = 2*sf2*exp(-K/2); else error('Unknown hyperparameter') end endgpml-matlab-v3.2-2013-01-15/cov/covProd.m000644 000765 000024 00000004247 11511036246 017440 0ustar00carlstaff000000 000000 function K = covProd(cov, hyp, x, z, i) % covProd - compose a covariance function as the product of other covariance % functions. This function doesn't actually compute very much on its own, it % merely does some bookkeeping, and calls other covariance functions to do the % actual work. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if numel(cov)==0, error('We require at least one factor.'), end for ii = 1:numel(cov) % iterate over covariance functions f = cov(ii); if iscell(f{:}), f = f{:}; end % expand cell array if necessary j(ii) = cellstr(feval(f{:})); % collect number hypers end if nargin<3 % report number of parameters K = char(j(1)); for ii=2:length(cov), K = [K, '+', char(j(ii))]; end, return end if nargin<4, z = []; end % make sure, z exists [n,D] = size(x); v = []; % v vector indicates to which covariance parameters belong for ii = 1:length(cov), v = [v repmat(ii, 1, eval(char(j(ii))))]; end if nargin<5 % covariances K = 1; if nargin==3, z = []; end % set default for ii = 1:length(cov) % iteration over factor functions f = cov(ii); if iscell(f{:}), f = f{:}; end % expand cell array if necessary K = K .* feval(f{:}, hyp(v==ii), x, z); % accumulate covariances end else % derivatives if i<=length(v) K = 1; vi = v(i); % which covariance function j = sum(v(1:i)==vi); % which parameter in that covariance for ii = 1:length(cov) % iteration over factor functions f = cov(ii); if iscell(f{:}), f = f{:}; end % expand cell if necessary if ii==vi K = K .* feval(f{:}, hyp(v==ii), x, z, j); % accumulate covariances else K = K .* feval(f{:}, hyp(v==ii), x, z); % accumulate covariances end end else error('Unknown hyperparameter') end endgpml-matlab-v3.2-2013-01-15/cov/covSEard.m000644 000765 000024 00000003754 11511036246 017534 0ustar00carlstaff000000 000000 function K = covSEard(hyp, x, z, i) % Squared Exponential covariance function with Automatic Relevance Detemination % (ARD) distance measure. The covariance function is parameterized as: % % k(x^p,x^q) = sf2 * exp(-(x^p - x^q)'*inv(P)*(x^p - x^q)/2) % % where the P matrix is diagonal with ARD parameters ell_1^2,...,ell_D^2, where % D is the dimension of the input space and sf2 is the signal variance. The % hyperparameters are: % % hyp = [ log(ell_1) % log(ell_2) % . % log(ell_D) % log(sqrt(sf2)) ] % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if nargin<2, K = '(D+1)'; return; end % report number of parameters if nargin<3, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode [n,D] = size(x); ell = exp(hyp(1:D)); % characteristic length scale sf2 = exp(2*hyp(D+1)); % signal variance % precompute squared distances if dg % vector kxx K = zeros(size(x,1),1); else if xeqz % symmetric matrix Kxx K = sq_dist(diag(1./ell)*x'); else % cross covariances Kxz K = sq_dist(diag(1./ell)*x',diag(1./ell)*z'); end end K = sf2*exp(-K/2); % covariance if nargin>3 % derivatives if i<=D % length scale parameters if dg K = K*0; else if xeqz K = K.*sq_dist(x(:,i)'/ell(i)); else K = K.*sq_dist(x(:,i)'/ell(i),z(:,i)'/ell(i)); end end elseif i==D+1 % magnitude parameter K = 2*K; else error('Unknown hyperparameter') end endgpml-matlab-v3.2-2013-01-15/cov/covMaterniso.m000644 000765 000024 00000003731 11511036246 020472 0ustar00carlstaff000000 000000 function K = covMaterniso(d, hyp, x, z, i) % 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. The covariance function is: % % k(x^p,x^q) = s2f * f( sqrt(d)*r ) * exp(-sqrt(d)*r) % % with f(t)=1 for d=1, f(t)=1+t for d=3 and f(t)=1+t+t²/3 for d=5. % Here r is the distance sqrt((x^p-x^q)'*inv(P)*(x^p-x^q)), P is ell times % the unit matrix and sf2 is the signal variance. The hyperparameters are: % % hyp = [ log(ell) % log(sqrt(sf2)) ] % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if nargin<3, K = '2'; return; end % report number of parameters if nargin<4, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode ell = exp(hyp(1)); sf2 = exp(2*hyp(2)); if all(d~=[1,3,5]), error('only 1, 3 and 5 allowed for d'), end % degree switch d case 1, f = @(t) 1; df = @(t) 1; case 3, f = @(t) 1 + t; df = @(t) t; case 5, f = @(t) 1 + t.*(1+t/3); df = @(t) t.*(1+t)/3; end m = @(t,f) f(t).*exp(-t); dm = @(t,f) df(t).*t.*exp(-t); % precompute distances if dg % vector kxx K = zeros(size(x,1),1); else if xeqz % symmetric matrix Kxx K = sqrt( sq_dist(sqrt(d)*x'/ell) ); else % cross covariances Kxz K = sqrt( sq_dist(sqrt(d)*x'/ell,sqrt(d)*z'/ell) ); end end if nargin<5 % covariances K = sf2*m(K,f); else % derivatives if i==1 K = sf2*dm(K,f); elseif i==2 K = 2*sf2*m(K,f); else error('Unknown hyperparameter') end endgpml-matlab-v3.2-2013-01-15/doc/f8.gif000644 000765 000024 00000034552 12077070414 016635 0ustar00carlstaff000000 000000 GIF89a€ÆAŸ¯¿ÏßïÿÿŸ¯ÿ¿Ïßï/ÿÿ?ÿÿOÿÿ_ÿÿ/oÿÿ?ÿÿOÿÿ_Ÿÿÿo¯ÿÿ¿ÿÿÏÿÿŸßÿÿ¯ïÿÿ¿ÿÿÿÏÿïÿß/ÿÏ?ÿ¿Oÿ¯ÿß_ÿŸoÿÿÿoŸÿ_ÿï¯ÿO¿ÿ?Ïÿ/ßÿïÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ!þCreated with GIMP,€þ€A‚ƒ„…†‡ˆ‰Š‹ŒŽ‘’“”•–—˜™š›œžŸ ¡¢£¤¥¦§¨©ª«¬­®¯°±²³´µ¶·¸¹º»¼½¾¿ÀÁÂÃÄÅÆÇÈÉÊËÌÍÎÏÐÑÒÓÔÕÖ×ØÙÚÛÜÝÞßàáâãäåæçèéêëìíîïðñòóôõö÷øùúûüýþÿ H° Áƒ*\È°¡Ã‡#JœH±¢EI\ÜÈñVÆŽ C²ÊøQ¤É“¢HjDɲ%&•+]ÊœÙfLš8s*YR§Oš®2÷÷ Ä[ï5 úF÷gJýfíð5×a;ùSž%ŽÂŸ‚íeÅÀƒF(á„Ê·škÐX˜yœ( Š¦€è›ƒ–X¢†Ü(ŠØp艇ë±"bpÙadâ²ø†õ¸ ŒÌ"8•áh$:fþ£bIFã£(@F‰§P9‰GšØäó«Ö[W‚;í¤®ŸÔWoýõØWÿUQpïz‹þÊ /ˆò‚,mû%Íç.ûºYeïþû7ý¶¢ð­Eç¦é‡/‹ø³—ϼי8ôÖ7ʼï€×‹è7¿í]þöwXðÌ _Á»¥O\Þ²V$½E(”€¨ŸF°/¤Ä)8Âñ ‚g!áoHÀ¢%…Œ‰üè—ÖÍï{Ô˜¡ñ`¥åé,‡…ЧHE„n‡þCD߀Q’Pà‹` 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ÇÁ…rðiâ(OyÅOÂ)ŒK[㑨n¼­è’»ÐæmþìOžòžC|ÛãsŒøÌ«T˜Íã¼ xp‰ü’ûL¯6G|NuËØŒÎh2ÏÒ(ÓXÖxؘڸÜHd¸k;Gtó#ŽT„ŒÆhŽ'Ž›GŽÕ¦ŽåHŒ½·uÝ7t߈^á(> (Ñ莕ӊèÈWÿõ,%`ùQúèŒìèh6x[Q ´¤xËhãf÷¨l&g‘ïÈÏh‡Äׄ¶ñÇdÉbɲɄV‘ 9’²Hjïx’ÙX .i!Y’9w¨•h†Â’Ø'}=9“þå“?‰“:)”E‰þ˜5ù’¹‘Ð×”ÊH<ƒ0ýH‘W9•H¹ŽY=T™ŒVédX9æ• ¹’*ù”¹ÈnÅ8–:†‘`É[_Ùf]™‘ÞG‹ðXº¨’–§—\9¿Xƒé•¨—k…yØ2É’ˆ˜÷h„l‹“i3è[u™ŽVX™¯È™²á™©‰G¹™{kI§™™šÙ—ɘû’šp™AZ‰—¬IŠ®y›¸™›º¹›¼Ù›¾;gpml-matlab-v3.2-2013-01-15/doc/index.html000644 000765 000024 00000114765 12077253464 017643 0ustar00carlstaff000000 000000 Documentation for GPML Matlab Code

Documentation for GPML Matlab Code version 3.2

What?

The code provided here originally demonstrated the main algorithms from Rasmussen and Williams: Gaussian Processes for Machine Learning. It has since grown to allow more likelihood functions, further inference methods and a flexible framework for specifying GPs. Other GP packages can be found here.

The code is written by Carl Edward Rasmussen and Hannes Nickisch; it runs on both Octave 3.2.x and Matlab® 7.x and later. The code is based on previous versions written by Carl Edward Rasmussen and Chris Williams.

Download, Install and Documentation

All the code including demonstrations and html documentation can be downloaded in a tar or zip archive file.

Minor changes and incremental bugfixes to the current version are documented in the changelog, changes from previous versions are documented in README.

Please read the copyright notice.

After unpacking the tar or zip file you will find 6 subdirectories: cov, doc, inf, lik, mean and util. It is not necessary to install anything to get started, just run the startup script to set your path.

Details about the directory contents and on how to compile mex files can be found in the README. The getting started guide is the remainder of the html file you are currently reading (also available at http://gaussianprocess.org/gpml/code/matlab/doc). A Developer's Guide containing technical documentation is found in manual.pdf, but for the casual user, the guide is below.

Theory

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 complicated covariance functions 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. This is what the GPML software package does.

Before going straight to the examples, just a brief note about the organization of the package. There are four types of objects which you need to know about:

Gaussian Process
A Gaussian Process is fully specified by a mean function and a covariance function. These functions are specified separately, and consist of a specification of a functional form as well as a set of parameters called hyperparameters, see below.
Mean functions
Several mean functions are available, all start with the four letters mean and reside in the mean directory. An overview is provided by the meanFunctions help function (type help meanFunctions to get help), and an example is the meanLinear function.
Covariance functions
There are many covariance functions available, all start with the three letters cov and reside in the cov directory. An overview is provided by the covFunctions help function (type help covFunctions to get help), and an example is the covSEard "Squared Exponential with Automatic Relevance Determination" covariance function.
For both mean functions and covariance functions, two types exist: simple and composite. Whereas simple types are specified by the function name (or function pointer), composite functions join together several components using cell arrays. Composite functions can be composed of other composite functions, allowing for very flexible and interesting structures.
Examples are given below and in the usageMean and usageCov functions.
Hyperparameters
GPs are typically specified using mean and covariance functions which have free parameters called hyperparameters. Also likelihood functions may have such parameters. These are encoded in a struct with the fields mean, cov and lik (some of which may be empty). When specifying hyperparameters, it is important that the number of elements in each of these struct fields, precisely match the number of parameters expected by the mean function, the covariance function and the likelihood functions respectively (otherwise an error will result). Hyperparameters whose natural domain is positive are represented by their logarithms.
Likelihood Functions
The likelihood function specifies the probability of the observations given the latent function, i.e. the GP (and the hyperparameters). All likelihood functions begin with the three letters lik and reside in the lik directory. An overview is provided by the likFunctions help function (type help likFunctions to get help). Some examples are likGauss the Gaussian likelihood for regression, likPoisson the Poisson likelihood for count data, and likLogistic the logistic function used in classification (a.k.a. logistic regression). As for the mean and covariance functions there is also a composite likelihood likMix that can be used to generate mixtures of other likelihood functions.
Examples are given below and in the usageLik function.
Inference Methods
The inference methods specify how to compute with the model, i.e. how to infer the (approximate) posterior process, how to find hyperparameters, evaluate the log marginal likelihood and how to make predictions. Inference methods all begin with the three letters inf and reside in the inf directory. An overview is provided by the infMethods help file (type help infMethods to get help). Some examples are infExact for exact inference (regression with Gaussian likelihood), infFITC for large scale Gaussian regression, infEP for the Expectation Propagation algorithm, or infFITC_Laplace for large scale Laplace's approximation. Further usage examples are provided for both regression and classification. However, not all combinations of likelihood function and inference method are possible (e.g. you cannot do exact inference with a Laplace likelihood). An exhaustive compatibility matrix between likelihoods (rows) and inference methods (columns) is given in the table below:
Inference GPML name Exact
infExact		 EP
infEP		 Laplace
infLaplace		 Variational Bayes
infVB		 Sampling
infMCMC		 LOO
infLOO		 type, output domain
 		 alternate names
 
Likelihood FITC
infFITC		 FITC-EP
infFITC_EP		 FITC-Laplace
infFITC_Laplace
Gaussian likGauss $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ regression, IR  
Sech-squared likSech2   $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ regression, IR  
Laplacian likLaplace   $\checkmark$   $\checkmark$ $\checkmark$ $\checkmark$ regression, IR double exponential
Student's t likT     $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ regression, IR  
Mixture likMix   $\checkmark$ $\checkmark$   $\checkmark$ $\checkmark$ classification, ±1 and regression, IR mixing meta likelihood
Uniform likUni   $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ classification, ±1 label noise
Error function likErf   $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ classification, ±1 probit regression
Logistic function likLogistic   $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ classification, ±1 logistic regression
logit regression
Poisson likPoisson   $\checkmark$ $\checkmark$   $\checkmark$ $\checkmark$ count data, IN Poisson regression

All of the objects described above are written in a modular way, so you can add functionality if you feel constrained despite the considerable flexibility provided. Details about how to do this are provided in the developer documentation.

Inference by MCMC sampling is the only inference method that cannot be used as a black box. Also gradient-based marginal likelihood optimisation is not possible with MCMC. Please see usageSampling for a toy example illustrating the usage of the implemented samplers.

Practice

Using the GPML package is simple. There is only one single function to call: gp, it does posterior inference, learns hyperparameters, computes the marginal likelihood and makes predictions. Generally, the gp function takes the following arguments: a hyperparameter struct, an inference method, a mean function, a covariance function, a likelihood function, training inputs, training targets, and possibly test cases. The exact computations done by the function is controlled by the number of input and output arguments in the call. Here is part of the help message for the gp function (follow the link to see the whole thing):

  function [varargout] = gp(hyp, inf, mean, cov, lik, x, y, xs, ys)

  [ ... snip ...]

  Two modes are possible: training or prediction: if no test cases are
  supplied, then the negative log marginal likelihood and its partial
  derivatives wrt the hyperparameters is computed; this mode is used to fit the
  hyperparameters. If test cases are given, then the test set predictive
  probabilities are returned. Usage:

    training: [nlZ dnlZ          ] = gp(hyp, inf, mean, cov, lik, x, y);
  prediction: [ymu ys2 fmu fs2   ] = gp(hyp, inf, mean, cov, lik, x, y, xs);
          or: [ymu ys2 fmu fs2 lp] = gp(hyp, inf, mean, cov, lik, x, y, xs, ys);

  [ .. snip ...]

Here x and y are training inputs and outputs, and xs and ys are test set inputs and outputs, nlZ is the negative log marginal likelihood and dnlZ its partial derivatives wrt the hyperparameters (which are used for training the hyperparameters). The prediction outputs are ymu and ys2 for test output mean and covariance, and fmu and fs2 are the equivalent quenteties for the corresponding latent variables. Finally, lp are the test output log probabilities.

Instead of exhaustively explaining all the possibilities, we will give two illustrative examples to give you the idea; one for regression and one for classification. You can either follow the example here on this page, or using the two scripts demoRegression and demoClassification (using the scripts, you still need to follow the explanation on this page).

Regression

You can either follow the example here on this page, or use the script demoRegression.

This is a simple example, where we first generate n=20 data points from a GP, where the inputs are scalar (so that it is easy to plot what is going on). We then use various other GPs to make inferences about the underlying function.

First, generate some data from a Gaussian process (it is not essential to understand the details of this):

  clear all, close all
 
  meanfunc = {@meanSum, {@meanLinear, @meanConst}}; hyp.mean = [0.5; 1];
  covfunc = {@covMaterniso, 3}; ell = 1/4; sf = 1; hyp.cov = log([ell; sf]);
  likfunc = @likGauss; sn = 0.1; hyp.lik = log(sn);
 
  n = 20;
  x = gpml_randn(0.3, n, 1);
  K = feval(covfunc{:}, hyp.cov, x);
  mu = feval(meanfunc{:}, hyp.mean, x);
  y = chol(K)'*gpml_randn(0.15, n, 1) + mu + exp(hyp.lik)*gpml_randn(0.2, n, 1);

  plot(x, y, '+')

Above, we first specify the mean function meanfunc, covariance function covfunc of a GP and a likelihood function, likfunc. The corresponding hyperparameters are specified in the hyp structure:

The mean function is composite, adding (using meanSum function) a linear (meanLinear) and a constant (meanConst) to get an affine function. Note, how the different components are composed using cell arrays. The hyperparameters for the mean are given in hyp.mean and consists of a single (because the input will one dimensional, i.e. D=1) slope (set to 0.5) and an off-set (set to 1). The number and the order of these hyperparameters conform to the mean function specification. You can find out how many hyperparameters a mean (or covariance or likelihood function) expects by calling it without arguments, such as feval(@meanfunc{:}). For more information on mean functions see meanFunctions and the directory mean/.

The covariance function is of the Matérn form with isotropic distance measure covMaterniso. This covariance function is also composite, as it takes a constant (related to the smoothness of the GP), which in this case is set to 3. The covariance function takes two hyperparameters, a characteristic length-scale ell and the standard deviation of the signal sf. Note, that these positive parameters are represented in hyp.cov using their logarithms. For more information on covariance functions see covFunctions and cov/.

Finally, the likelihood function is specified to be Gaussian. The standard deviation of the noise sn is set to 0.1. Again, the representation in the hyp.lik is given in terms of its logarithm. For more information about likelihood functions, see likFunctions and lik/.

Then, we generate a dataset with n=20 examples. The inputs x are drawn from a unit Gaussian (using the gpml_randn utility, which generates unit Gaussian pseudo random numbers with a specified seed). We then evaluate the covariance matrix K and the mean vector m by calling the corresponding functions with the hyperparameters and the input locations x. Finally, the targets y are computed by drawing randomly from a Gaussian with the desired covariance and mean and adding Gaussian noise with standard deviation exp(hyp.lik). The above code is a bit special because we explicitly call the mean and covariance functions (in order to generate samples from a GP); ordinarily, we would only directly call the gp function.

f1.gif

Let's ask the model to compute the (joint) negative log probability (density) nlml (also called marginal likelihood or evidence) and to generalize from the training data to other (test) inputs z:

  nlml = gp(hyp, @infExact, meanfunc, covfunc, likfunc, x, y)

  z = linspace(-1.9, 1.9, 101)';
  [m s2] = gp(hyp, @infExact, meanfunc, covfunc, likfunc, x, y, z);

  f = [m+2*sqrt(s2); flipdim(m-2*sqrt(s2),1)]; 
  fill([z; flipdim(z,1)], f, [7 7 7]/8)
  hold on; plot(z, m); plot(x, y, '+')

The gp function is called with a struct of hyperparameters hyp, and inference method, in this case @infExact for exact inference and the mean, covariance and likelihood functions, as well as the inputs and outputs of the training data. With no test inputs, gp returns the negative log probability of the training data, in this example nlml=11.97.

To compute the predictions at test locations we add the test inputs z as a final argument, and gp returns the mean m variance s2 at the test location. The program is using algorithm 2.1 from the GPML book. Plotting the mean function plus/minus two standard deviations (corresponding to a 95% confidence interval):

f2.gif

Typically, we would not a priori know the values of the hyperparameters hyp, let alone the form of the mean, covariance or likelihood functions. So, let's pretend we didn't know any of this. We assume a particular structure and learn suitable hyperparameters:

  covfunc = @covSEiso; hyp2.cov = [0; 0]; hyp2.lik = log(0.1);

  hyp2 = minimize(hyp2, @gp, -100, @infExact, [], covfunc, likfunc, x, y);
  exp(hyp2.lik)
  nlml2 = gp(hyp2, @infExact, [], covfunc, likfunc, x, y)

  [m s2] = gp(hyp2, @infExact, [], covfunc, likfunc, x, y, z);
  f = [m+2*sqrt(s2); flipdim(m-2*sqrt(s2),1)];
  fill([z; flipdim(z,1)], f, [7 7 7]/8)
  hold on; plot(z, m); plot(x, y, '+')

First, we guess that a squared exponential covariance function covSEiso may be suitable. This covariance function takes two hyperparameters: a characteristic length-scale and a signal standard deviation (magnitude). These hyperparameters are non-negative and represented by their logarithms; thus, initializing hyp2.cov to zero, correspond to unit characteristic length-scale and unit signal standard deviation. The likelihood hyperparameter in hyp2.lik is also initialized. We assume that the mean function is zero, so we simply ignore it (and when in the following we call gp, we give an empty argument for the mean function).

In the following line, we optimize over the hyperparameters, by minimizing the negative log marginal likelihood w.r.t. the hyperparameters. The third parameter in the call to minimize limits the number of function evaluations to a maximum of 100. The inferred noise standard deviation is exp(hyp2.lik)=0.15, somewhat larger than the one used to generate the data (0.1). The final negative log marginal likelihood is nlml2=14.13, showing that the joint probability (density) of the training data is about exp(14.13-11.97)=8.7 times smaller than for the setup actually generating the data. Finally, we plot the predictive distribution.

f3.gif

This plot shows clearly, that the model is indeed quite different from the generating process. This is due to the different specifications of both the mean and covariance functions. Below we'll try to do a better job, by allowing more flexibility in the specification.

Note that the confidence interval in this plot is the confidence for the distribution of the (noisy) data. If instead you want the confidence region for the underlying function, you should use the 3rd and 4th output arguments from gp as these refer to the latent process, rather than the data points.

  hyp.cov = [0; 0]; hyp.mean = [0; 0]; hyp.lik = log(0.1);
  hyp = minimize(hyp, @gp, -100, @infExact, meanfunc, covfunc, likfunc, x, y);
  [m s2] = gp(hyp, @infExact, meanfunc, covfunc, likfunc, x, y, z);
 
  f = [m+2*sqrt(s2); flipdim(m-2*sqrt(s2),1)];
  fill([z; flipdim(z,1)], f, [7 7 7]/8)
  hold on; plot(z, m); plot(x, y, '+');

Here, we have changed the specification by adding the affine mean function. All the hyperparameters are learnt by optimizing the marginal likelihood.

f4.gif

This shows that a much better fit is achieved when allowing a mean function (although the covariance function is still different from that of the generating process).

Large scale regression

In case the number of training inputs x exceeds a few thousands, exact inference using infExact.m 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.

Using the FITC approximation is very simple, we just have to wrap the covariance function covfunc into covFITC.m and call gp.m with the inference method infFITC.m as demonstrated by the following lines of code. 

  nu = fix(n/2); u = linspace(-1.3,1.3,nu)';
  covfuncF = {@covFITC, {covfunc}, u};
  [mF s2F] = gp(hyp, @infFITC, meanfunc, covfuncF, likfunc, x, y, z);

We define equispaced inducing points u that are shown in the figure as black circles. 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 simply use a random subset of the training points. 

  nu = fix(n/2); iu = randperm(n); iu = iu(1:nu); u = x(iu,:);
f5.gif

Exercises for the reader

Inference Methods
Try using Expectation Propagation instead of exact inference in the above, by exchanging @infExact with @infEP. You get exactly identical results, why?
Mean or Covariance
Try training a GP where the affine part of the function is captured by the covariance function instead of the mean function. That is, use a GP with no explicit mean function, but further additive contributions to the covariance. How would you expect the marginal likelihood to compare to the previous case?

Classification

You can either follow the example here on this page, or use the script demoClassification.

The difference between regression and classification isn't of fundamental nature. We can use a Gaussian process latent function in essentially the same way, it is just that the Gaussian likelihood function often used for regression is inappropriate for classification. And since exact inference is only possible for Gaussian likelihood, we also need an alternative, approximate, inference method.

Here, we will demonstrate binary classification, using two partially overlapping Gaussian sources of data in two dimensions. First we generate the data:

  clear all, close all
 
  n1 = 80; n2 = 40;                   % number of data points from each class
  S1 = eye(2); S2 = [1 0.95; 0.95 1];           % the two covariance matrices
  m1 = [0.75; 0]; m2 = [-0.75; 0];                            % the two means
 
  x1 = bsxfun(@plus, chol(S1)'*gpml_randn(0.2, 2, n1), m1);
  x2 = bsxfun(@plus, chol(S2)'*gpml_randn(0.3, 2, n2), m2);
 
  x = [x1 x2]'; y = [-ones(1,n1) ones(1,n2)]';
  plot(x1(1,:), x1(2,:), 'b+'); hold on;
  plot(x2(1,:), x2(2,:), 'r+');

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 (and not 0/1).

In the plot, 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] = meshgrid(-4:0.1:4,-4:0.1:4);
  t = [t1(:) t2(:)]; n = length(t);                 % these are the test inputs
  tmm = bsxfun(@minus, t, m1');
  p1 = n1*exp(-sum(tmm*inv(S1).*tmm/2,2))/sqrt(det(S1));
  tmm = bsxfun(@minus, t, m2');
  p2 = n2*exp(-sum(tmm*inv(S2).*tmm/2,2))/sqrt(det(S2));
  contour(t1, t2, reshape(p2./(p1+p2), size(t1)), [0.1:0.1:0.9]);
f6.gif

We specify a Gaussian process model as follows: a constant mean function, with initial parameter set to 0, a squared exponential with automatic relevance determination (ARD) covariance function covSEard. This covariance function has one characteristic length-scale parameter for each dimension of the input space, and a signal magnitude parameter, for a total of 3 parameters (as the input dimension is D=2). ARD with separate length-scales for each input dimension is a very powerful tool to learn which inputs are important for predictions: if length-scales are short, inputs are very important, and when they grow very long (compared to the spread of the data), the corresponding inputs will be largely ignored. Both length-scales and the signal magnitude are initialized to 1 (and represented in the log space). Finally, the likelihood function likErf has the shape of the error-function (or cumulative Gaussian), which doesn't take any hyperparameters (so hyp.lik does not exist).

  meanfunc = @meanConst; hyp.mean = 0;
  covfunc = @covSEard; ell = 1.0; sf = 1.0; hyp.cov = log([ell ell sf]);
  likfunc = @likErf;

  hyp = minimize(hyp, @gp, -40, @infEP, meanfunc, covfunc, likfunc, x, y);
  [a b c d lp] = gp(hyp, @infEP, meanfunc, covfunc, likfunc, x, y, t, ones(n, 1));

  plot(x1(1,:), x1(2,:), 'b+'); hold on; plot(x2(1,:), x2(2,:), 'r+')
  contour(t1, t2, reshape(exp(lp), size(t1)), [0.1:0.1:0.9]);

We train the hyperparameters using minimize, to minimize the negative log marginal likelihood. We allow for 40 function evaluations, and specify that inference should be done with the Expectation Propagation (EP) inference method @infEP, and pass the usual parameters. Training is done using algorithm 3.5 and 5.2 from the gpml book. When computing test probabilities, we call gp with additional test inputs, and as the last argument a vector of targets for which the log probabilities lp should be computed. The fist four output arguments of the function are mean and variance for the targets and corresponding latent variables respectively. The test set predictions are computed using algorithm 3.6 from the GPML book. 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.

f7.gif

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 inputs important.

Large scale classification

In case the number of training inputs x exceeds a few hundreds, approximate inference using infLaplace.m, infEP.m and infVB.m 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.

Using the FITC approximation is very simple, we just have to wrap the covariance function covfunc into covFITC.m and call gp.m with the inference methods infFITC_Laplace.m and infFITC_EP.m as demonstrated by the following lines of code. 

  [u1,u2] = meshgrid(linspace(-2,2,5)); u = [u1(:),u2(:)]; clear u1; clear u2
  nu = size(u,1);
  covfuncF = {@covFITC, {covfunc}, u};
  inffunc = @infFITC_EP;                       % also @infFITC_Laplace is possible
  hyp = minimize(hyp, @gp, -40, inffunc, meanfunc, covfuncF, likfunc, x, y);
  [a b c d lp] = gp(hyp, inffunc, meanfunc, covfuncF, likfunc, x, y, t, ones(n,1));

We define equispaced inducing points u that are shown in the figure as black circles. Alternatively, a random subset of the training points can be used as inducing points.

f8.gif

Exercise for the reader

Inference Methods
Use the Laplace Approximation for inference @infLaplace (or @infFITC_Laplace in the large scale example), and compare the approximate marginal likelihood for the two methods. Which approximation is best?
Covariance Function
Try using the squared exponential with isotropic distance measure covSEiso instead of ARD distance measure covSEard. Which is best?

Acknowledgements

Innumerable colleagues have helped to improve this software. Some of these are: John Cunningham, Máté Lengyel, Joris Mooij, Iain Murray, David Duvenaud, James Lloyd, Andrew McHutchon, Rowan McAllister, Daniel Marthaler and Chris Williams. Especially Ed Snelson helped to improve the code and to include sparse approximations.

Last modified: January 15th 2013 gpml-matlab-v3.2-2013-01-15/doc/changelog000644 000765 000024 00000000374 12077070414 017476 0ustar00carlstaff000000 000000 GAUSSIAN PROCESS REGRESSION AND CLASSIFICATION Toolbox version 3.2 for GNU Octave 3.2.x and Matlab 7.x Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2013-01-15. Changes and small incremental bug fixes applied since last release. gpml-matlab-v3.2-2013-01-15/doc/usageRegression.m000644 000765 000024 00000005177 12077070414 021155 0ustar00carlstaff000000 000000 % demonstrate usage of regression % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch 2010-07-21. clear all, close all %% SAY WHICH CODE WE WISH TO EXERCISE id = [1,1]; % use Exact % id = [1,2; 3,2; 4,2]; % compare Laplace id = [1,3; 2,3; 3,3]; % compare EP id = [1,4; 2,4; 3,4; 4,4]; % compare VB seed = 197; randn('seed',seed), rand('seed',seed) ntr = 50; nte = 1e4; % number of training and test points xtr = 10*sort(rand(ntr,1)); % sample dataset f = @(x) sin(x)+sqrt(x); % "true" underlying function sn = 0.2; ytr = f(xtr) + randn(ntr,1)*sn; % add Gaussian noise i = randperm(ntr); nout = 3; % add outliers ytr(i(1:nout)) = 5; xte = linspace(0,10,1e4)'; % support, we test our function on cov = {@covSEiso}; sf = 1; ell = 0.4; % setup the GP hyp0.cov = log([ell;sf]); mean = {@meanSum,{@meanLinear,@meanConst}}; a = 1/5; b = 1; % m(x) = a*x+b hyp0.mean = [a;b]; lik_list = {'likGauss','likLaplace','likSech2','likT'}; % possible likelihoods inf_list = {'infExact','infLaplace','infEP','infVB'}; % allowable inference algs Ncg = 50; % number of conjugate gradient steps sdscale = 0.5; % how many sd wide should the error bars become? col = {'k',[.8,0,0],[0,.5,0],'b',[0,.75,.75],[.7,0,.5]}; % colors ymu{1} = f(xte); ys2{1} = sn^2; nlZ(1) = -Inf; for i=1:size(id,1) lik = lik_list{id(i,1)}; % setup the likelihood if strcmp(lik,'likT') nu = 4; hyp0.lik = log([nu-1;sqrt((nu-2)/nu)*sn]); else hyp0.lik = log(sn); end inf = inf_list{id(i,2)}; fprintf('OPT: %s/%s\n',lik_list{id(i,1)},inf_list{id(i,2)}) if Ncg==0 hyp = hyp0; else hyp = minimize(hyp0,'gp', -Ncg, inf, mean, cov, lik, xtr, ytr); % opt hypers end [ymu{i+1}, ys2{i+1}] = gp(hyp, inf, mean, cov, lik, xtr, ytr, xte); % predict [nlZ(i+1)] = gp(hyp, inf, mean, cov, lik, xtr, ytr); end figure, hold on for i=1:size(id,1)+1 plot(xte,ymu{i},'Color',col{i},'LineWidth',2) if i==1 leg = {'function'}; else leg{end+1} = sprintf('%s/%s -lZ=%1.2f',... lik_list{id(i-1,1)},inf_list{id(i-1,2)},nlZ(i)); end end for i=1:size(id,1)+1 ysd = sdscale*sqrt(ys2{i}); fill([xte;flipud(xte)],[ymu{i}+ysd;flipud(ymu{i}-ysd)],... col{i},'EdgeColor',col{i},'FaceAlpha',0.1,'EdgeAlpha',0.3); end for i=1:size(id,1)+1, plot(xte,ymu{i},'Color',col{i},'LineWidth',2), end plot(xtr,ytr,'k+'), plot(xtr,ytr,'ko'), legend(leg) gpml-matlab-v3.2-2013-01-15/doc/gpml_randn.m000644 000765 000024 00000002125 11511036275 020116 0ustar00carlstaff000000 000000 function x = gpml_randn(seed,varargin) % Generate pseudo-random numbers in a quick and dirty way. % The function makes sure, we obtain the same random numbers using Octave and % Matlab for the demo scripts. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch 2010-07-07. if nargin==2 sz = varargin{1}; else sz = zeros(1,nargin-1); for i=1:nargin-1 sz(i) = varargin{i}; end end n = prod(sz); N = ceil(n/2)*2; % minimal uniform random number generator for uniform deviates from [0,1] % by Park and Miller a = 7^5; m = 2^31-1; % using Schrage's algorithm q = fix(m/a); r = mod(m,a); % m = a*q+r u = zeros(N+1,1); u(1) = fix(seed*2^31); for i=2:N+1 % Schrage's algorithm for mod(a*u(i),m) u(i) = a*mod(u(i-1),q) - r*fix(u(i-1)/q); if u(i)<0, u(i) = u(i)+m; end end u = u(2:N+1)/2^31; % Box-Muller transform: Numerical Recipies, 2nd Edition, $7.2.8 % http://en.wikipedia.org/wiki/Box-Muller_transform w = sqrt(- 2*log(u(1:N/2))); % split into two groups x = [w.*cos(2*pi*u(N/2+1:N)); w.*sin(2*pi*u(N/2+1:N))]; x = reshape(x(1:n),sz); gpml-matlab-v3.2-2013-01-15/doc/usageCov.m000644 000765 000024 00000004122 12077253575 017564 0ustar00carlstaff000000 000000 % demonstrate usage of covariance functions % % See also covFunctions.m. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2013-01-21 clear all, close all n = 5; D = 3; x = randn(n,D); xs = randn(3,D); % create a data set % set up simple covariance functions cn = {'covNoise'}; sn = .1; hypn = log(sn); % one hyperparameter cc = {@covConst}; sf = 2; hypc = log(sf); % function handles OK cl = {@covLIN}; hypl = []; % linear is parameter-free cla = {'covLINard'}; L = rand(D,1); hypla = log(L); % linear (ARD) clo = {@covLINone}; ell = .9; hyplo = log(ell); % linear with bias cp = {@covPoly,3}; c = 2; hypp = log([c;sf]); % third order poly cga = {@covSEard}; hypga = log([L;sf]); % Gaussian with ARD cgi = {'covSEiso'}; hypgi = log([ell;sf]); % isotropic Gaussian cgu = {'covSEisoU'}; hypgu = log(ell); % isotropic Gauss no scale cra = {'covRQard'}; al = 2; hypra = log([L;sf;al]); % ration. quad. cri = {@covRQiso}; hypri = log([ell;sf;al]); % isotropic cm = {'covMaterniso',3}; hypm = log([ell;sf]); % Matern class q=3 cnn = {'covNNone'}; hypnn = log([L;sf]); % neural network cpe = {'covPeriodic'}; om = 2; hyppe = log([ell;om;sf]); % periodic ccc = {'covPPiso',2}; hypcc = hypm; % compact support poly degree 2 % set up composite covariance functions csc = {'covScale',{cgu}}; hypsc = [log(3); hypgu]; % scale by 9 csu = {'covSum',{cn,cc,cl}}; hypsu = [hypn; hypc; hypl]; % sum cpr = {@covProd,{cc,ccc}}; hyppr = [hypc; hypcc]; % product mask = [0,1,0]; % binary mask excluding all but the 2nd component cma = {'covMask',{mask,cgi{:}}}; hypma = hypgi; % additive based on SEiso using unary and pairwise interactions cad = {'covADD',{[1,2],'covSEiso'}}; % 0) specify covariance function cov = cma; hyp = hypma; % 1) query the number of parameters feval(cov{:}) % 2) evaluate the function on x feval(cov{:},hyp,x) % 3) evaluate the function on x and xs to get cross-terms kss = feval(cov{:},hyp,xs,'diag') Ks = feval(cov{:},hyp,x,xs) % 4) compute the derivatives w.r.t. to hyperparameter i i = 1; feval(cov{:},hyp,x,[],i) gpml-matlab-v3.2-2013-01-15/doc/README000644 000765 000024 00000026333 12077252501 016506 0ustar00carlstaff000000 000000 GAUSSIAN PROCESS REGRESSION AND CLASSIFICATION Toolbox version 3.2 for GNU Octave 3.2.x and Matlab 7.x Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2013-01-15. 0) HOW TO READ ============== If you want to get started without further delay, then read section 1) below and jump right to the examples in doc/index.html. 1) ABOUT THESE PROGRAMS ======================= This collection of matlab programs implements and demonstrates some of the algorithms described in a) the book by Rasmussen and Williams: "Gaussian Processes for Machine Learning", the MIT Press 2006, in b) the article by Nickisch and Rasmussen: "Approximations for Binary Gaussian Process Classification", JMLR 2008, in c) the article by Candela and Rasmussen: "A Unifying View of Sparse Approximate Gaussian Process Regression", JMLR 2005, in d) the paper by Murray, Adams and Mackay: "Elliptical slice sampling", AISTATS 2010, in e) the report by Neal: "Annealed Importance Sampling", UToronto 1998, in f) the paper by Naish-Guzman and Holden: "The Generalized FITC Approximation", NIPS, 2007, and in g) the paper by Duvenaud, Nickisch and Rasmussen: "Additive Gaussian Processes", NIPS, 2011. There are 6 code subdirectories: cov, doc, inf, lik, mean and util. cov/ contains covariance functions cov*.m => see covFunctions.m doc/ contains an index.html file providing documentation. This information is also available from http://www.GaussianProcess.org/gpml/code. Usage of mean, cov, classification and regression is demonstrated in usage*.m. Further details can be found in the developer documentation manual.pdf. inf/ contains the inference algorithms inf*.m => see infMethods.m lik/ contains the likelihood functions lik*.m => see likFunctions.m mean/ contains the mean functions mean*.m => see meanFunctions.m util/ contains optimisation routines, backward compatibility programs and small auxiliary scripts Before running the demos or any other part of the code, you should execute startup.m to add all necessary directories to your path. In Octave, you can also use the .octaverc file. 2) ABOUT MEX FILES ================== Some of the programs make use of the MEX facility in matlab for more efficient implementation. However, if you don't know about how to compile MEX files, you do not need to worry about this -- the code runs anyway. If you do compile the MEX files, this is automatically detected, and the program will run more efficiently. If you do not have a compiler installed on your system, you might want to consult [2]. Two components of the toolbox can be accelerated by MEX. First, there is an auxiliary functions solve_chol.m having a MEX equivalent. This can be compiled by either executing util/make.m from the Matlab/Octave command line. In addition to that, we provide a Makefile for Matlab in util/ that you can run by simply typing make in your shell after having supplied your Matlab path and your operating system. We recommend the make.m script because it works both under Matlab and Octave. Second, in order to use the L-BFGS minimiser for hyperparameter optimisation as an equivalent to minimize.m, you have to compile Peter Carbonetto's "Matlab interface for L-BFGS-B" [1]. The challenge here is the Fortran 77 code. We provide a Makefile suitable for Linux 32/64 bit and Mac whenever you have 1) g77 or 2) gfortran properly set up. Under Ubuntu, you can achieve this by installing the packages fort77 and gfortran, respectively. Compilation is done by editing util/lbfgsb/Makefile. In any case, you need to provide $MATLAB_HOME which can be found by the commands 'locate matlab' or 'find / -name "matlab"'. You can choose between two compilation modes: a) using the mex utility by Matlab [default] provide $MEX, then type 'make mex' b) without mex utility by Matlab provide $MEX_SUFFIX and $MATLAB_LIB, then type 'make nomex' In Ubuntu 10.04 LTS, the libg2c library needed for both 1)+a) and 1)+b) is not included per default. If 'ls /usr/lib/libg2c.*' does not list anything this is the case on your machine. You then whant to install the packages gcc-3.4-base and libg2c0 e.g. from http://packages.ubuntu.com/hardy/gcc-3.4-base and http://packages.ubuntu.com/hardy/libg2c0. After installation, you have to create a symbolic link by 'cd /usr/lib' and 'ln -s libg2c.so.0 libg2c.so'. [1] http://www.cs.ubc.ca/~pcarbo/lbfgsb-for-matlab.html [2] http://www.mathworks.com/support/compilers/R2010a 3) CURRENT VERSION ================== The current version of the programs is 3.2. Previous versions of the code are available at http://www.gaussianprocess.org/gpml/code/oldcode.html. 4) DIFFERENCES TO PREVIOUS VERSIONS =================================== NEW in version 3.2, 2013-01-15 ------------------------------ - added inf/infFITC_Laplace.m and inf/infFITC_EP.m for large scale approximate inference - added inf/infMCMC.m to allow for Markov Chain Monte Carlo sampling and Annealed Importance Sampling; we currently support two samplers: Elliptical Slice Sampling (ESS) and Hybrid Monte Carlo (HMC) - improved accuracy of the digamma function lik/likT.m/dloggamma from three to ten digits precision (thanks to Edward Challis) - added gfortran support to util/lbfgsb/Makefile (thanks to Ernst Kloppenburg) - changed parametrisation in cov/covPeriodic.m (thanks to Philipp Hennig) - fixed a stupid bug in cov/covFITC.m (thanks to Krzysztof Chalupka) - fixed a severe stability bug in the derivatives of inf/infFITC.m (thanks to Andrew McHutchon) - changed scalars in util/solve_chol.c to long to provide 64bit compatibility (thanks to Ernst Kloppenburg) - added infLOO to allow pseudo-likelihood optimisation instead of the marginal likelihood - fixed a performance issue in util/sq_dist.m (thanks to Krzysztof Chalupka) - fixed a bug causing numerical problems in infFITC and covFITC whenever Kuu is close to singular (thanks to Joris M. Mooij) - fixed an issue in covNoise (thanks to Daniel Marthaler) - fixed implicit derivative w.r.t. likelihood in infLaplace* - added two more likelihood functions likMix.m for mixtures and likUni for classification noise - slightly changed the interface for the likelihood functions - added likPoisson for Poisson regression of count data NEW in version 3.1, 2010-10-10 ------------------------------ - following a suggestion by Ed Snelson we now support FITC regression - cov/covFITC.m and inf/infFITC.m have been added with Ed Snelson's help - the covariance interface was slightly changed to make that possible NEW in version 3.0, 2010-07-23 ------------------------------ A major code reorganisation effort did take place in the current release. First, classification and regression are now done by a single file gp.m which is completely generic in the likelihood. The previous regression program gpr.m corresponds to gp.m with Gaussian likelihood. Several other likelihoods for robust regresssion were added. Further, the code now supports mean functions with a similar specification mechanism as already used by the covariance functions. Previous implementations correspond to using meanZero.m. We merged the covariance functions covMatern3iso.m and covMatern5.iso into a single covMaterniso.m and added a covariance function for additive functions as well as the possibility to use only certain components of the data. Finally, we included covPPiso.m a piecewise polynomial covariance function with compact support. New likelihood functions in lik/: likGauss.m, likLaplace.m, likLogistic.m and likT.m New mean functions in mean/: meanConst.m, meanLinear.m, meanPow.m, meanProd.m, meanSum.m and meanZero.m New covariance functions in cov/: covADD.m, covMask.m, covPPiso.m The gprSRPP.m function which previously provided "Subset of Regressors" and the "Projected Process" approximation has now been removed. NEW in version 2.1, 2007-07-25 ------------------------------ covConst.m: fixed a bug which caused an error in the derivative of the log marginal likelihood for certain combinations of covariance functions and approximation methods. (Thanks to Antonio Eleuteri for reporting the problem) gauher.m: added the function "gauher.m" which was mistakenly missing from the previous release. This caused an error for certain combinations of approximation method and likelihood function. logistic.m: modified the approximation of moments calculation to use a mixture of cumulative Gaussian, rather than Gauss-Hermite quadrature, as the former turns out to be more accurate. NEW in version 2.0, 2007-06-25 ------------------------------ Some code restructuring has taken place for the classification code to make it more modular, to facilitate addition of new likelihood functions and approximations methods. Now, all classification is done using the binaryGP function, which (among other things) takes an approximation method and a likelihood function as an arguments. Thus, binaryGP replaces both binaryEPGP and binaryLapaceGP, although wrapper functions are still provided for backward compatibility. This gives added flexibility: now EP can also be used wth the logistic likelihood function (implemented using Gauss-Hermite quadrature). approxEP.m: New file, containing the Expectation Propagation approximation method, which was previously contained in binaryEPGP.m approxLA.m: New file, containing Laplaces approximation method, which was previously contained in binaryLaplace.m approximations.m: New file, help for the approximation methods. binaryEPGP.m: This file has been replaced by a wrapper (for backward compatibility) which calls the more general binaryGP function. binaryGP.m: New general function to do binary classification. binaryLaplaceGP.m: This file has been replaced by a wrapper (for backward compatibility) which calls the more general binaryGP function. covMatern3iso.m, covMatern5iso.m, covNNone.m, covRQard.m, covRQiso.m, cosSEard, covSEiso: now check more carefully, that persistent variables have the correct sizes, and some variable names have been modified. cumGauss.m: New file, containing code for the cumulative Gaussian likelihood function likelihoods.m: New file, help for likelihood functions logistic.m: New file, logistic likelihood NEW in version 1.3, 2006-09-08 ------------------------------ covRQard.m: bugfix: replaced x with x' and z with z' in line 36 covRQiso.m: bugfix: replaced x with x' and z with z' in line 28 minimize.m: correction: replaced "error()" with "error('')", and made a few cosmetic changes binaryEPGP.m: added the line "lml = -n*log(2);" in line 77. This change should be largely inconsequential, but occationally may save things when the covariance matrix is exceptionally badly conditioned. NEW in version 1.2, 2006-05-10 ------------------------------ added the "erfint" function to "binaryLaplaceGP.m". The erfint function was missing by mistake, preventing the use of the "logistic" likelihood. NEW in version 1.1, 2006-04-12 ------------------------------ added files: "covProd.m" and "covPeriodic.m" changes: "covSEiso.m" was changed slightly to avoid the use of persistent variables NEW in version 1.0, 2006-03-29 ------------------------------ initial version shipped with the book gpml-matlab-v3.2-2013-01-15/doc/checkmark.png000644 000765 000024 00000000306 11511036275 020254 0ustar00carlstaff000000 000000 ‰PNG  IHDR§óa*PLTE³³³¨¨¨œœœ„„„xxxlll```TTTHHH<<<$$$ ÍÍtRNS@æØfJIDAT™c````d€6 N`€8,Åà~€Àe€ˆ‹ RS!K”Áe0O€Ðâ åPÎë`ʳÁ&Õ]¹ ‡NIEND®B`‚gpml-matlab-v3.2-2013-01-15/doc/demoRegression.m000644 000765 000024 00000011772 12077070414 020773 0ustar00carlstaff000000 000000 disp('See http://www.gaussianprocess.org/gpml/code/matlab/doc/ for details.') disp('Hit any key to continue...'); pause disp(' '); disp('clear all, close all') clear all, close all write_fig = 0; disp(' ') disp('meanfunc = {@meanSum, {@meanLinear, @meanConst}}; hyp.mean = [0.5; 1];') meanfunc = {@meanSum, {@meanLinear, @meanConst}}; hyp.mean = [0.5; 1]; disp('covfunc = {@covMaterniso, 3}; ell = 1/4; sf = 1; hyp.cov = log([ell; sf]);') covfunc = {@covMaterniso, 3}; ell = 1/4; sf = 1; hyp.cov = log([ell; sf]); disp('likfunc = @likGauss; sn = 0.1; hyp.lik = log(sn);') likfunc = @likGauss; sn = 0.1; hyp.lik = log(sn); disp(' ') disp('n = 20;') n = 20; disp('x = gpml_randn(0.3, n, 1);') x = gpml_randn(0.3, n, 1); disp('K = feval(covfunc{:}, hyp.cov, x);') K = feval(covfunc{:}, hyp.cov, x); disp('mu = feval(meanfunc{:}, hyp.mean, x);') mu = feval(meanfunc{:}, hyp.mean, x); disp('y = chol(K)''*gpml_randn(0.15, n, 1) + mu + exp(hyp.lik)*gpml_randn(0.2, n, 1);') y = chol(K)'*gpml_randn(0.15, n, 1) + mu + exp(hyp.lik)*gpml_randn(0.2, n, 1); figure(1) set(gca, 'FontSize', 24) disp(' '); disp('plot(x, y, ''+'')') plot(x, y, '+', 'MarkerSize', 12) axis([-1.9 1.9 -0.9 3.9]) grid on xlabel('input, x') ylabel('output, y') if write_fig, print -depsc f1.eps; end disp(' '); disp('Hit any key to continue...'); pause disp(' ') disp('nlml = gp(hyp, @infExact, meanfunc, covfunc, likfunc, x, y)') nlml = gp(hyp, @infExact, meanfunc, covfunc, likfunc, x, y) disp(' ') disp('z = linspace(-1.9, 1.9, 101)'';') z = linspace(-1.9, 1.9, 101)'; disp('[m s2] = gp(hyp, @infExact, meanfunc, covfunc, likfunc, x, y, z);') [m s2] = gp(hyp, @infExact, meanfunc, covfunc, likfunc, x, y, z); figure(2) set(gca, 'FontSize', 24) disp(' ') disp('f = [m+2*sqrt(s2); flipdim(m-2*sqrt(s2),1)];') f = [m+2*sqrt(s2); flipdim(m-2*sqrt(s2),1)]; disp('fill([z; flipdim(z,1)], f, [7 7 7]/8);') fill([z; flipdim(z,1)], f, [7 7 7]/8); disp('hold on; plot(z, m); plot(x, y, ''+'')') hold on; plot(z, m, 'LineWidth', 2); plot(x, y, '+', 'MarkerSize', 12) axis([-1.9 1.9 -0.9 3.9]) grid on xlabel('input, x') ylabel('output, y') if write_fig, print -depsc f2.eps; end disp(' '); disp('Hit any key to continue...'); pause disp(' ') disp('covfunc = @covSEiso; hyp2.cov = [0; 0]; hyp2.lik = log(0.1);') covfunc = @covSEiso; hyp2.cov = [0; 0]; hyp2.lik = log(0.1); disp('hyp2 = minimize(hyp2, @gp, -100, @infExact, [], covfunc, likfunc, x, y)') hyp2 = minimize(hyp2, @gp, -100, @infExact, [], covfunc, likfunc, x, y); disp(' ') disp('exp(hyp2.lik)') exp(hyp2.lik) disp('nlml2 = gp(hyp2, @infExact, [], covfunc, likfunc, x, y)') nlml2 = gp(hyp2, @infExact, [], covfunc, likfunc, x, y) disp('[m s2] = gp(hyp2, @infExact, [], covfunc, likfunc, x, y, z);') [m s2] = gp(hyp2, @infExact, [], covfunc, likfunc, x, y, z); disp(' ') figure(3) set(gca, 'FontSize', 24) f = [m+2*sqrt(s2); flipdim(m-2*sqrt(s2),1)]; disp('fill([z; flipdim(z,1)], f, [7 7 7]/8)'); fill([z; flipdim(z,1)], f, [7 7 7]/8) disp('hold on; plot(z, m); plot(x, y, ''+'')'); hold on; plot(z, m, 'LineWidth', 2); plot(x, y, '+', 'MarkerSize', 12) grid on xlabel('input, x') ylabel('output, y') axis([-1.9 1.9 -0.9 3.9]) if write_fig, print -depsc f3.eps; end disp(' '); disp('Hit any key to continue...'); pause disp(' ') disp('hyp.cov = [0; 0]; hyp.mean = [0; 0]; hyp.lik = log(0.1);') hyp.cov = [0; 0]; hyp.mean = [0; 0]; hyp.lik = log(0.1); disp('hyp = minimize(hyp, @gp, -100, @infExact, meanfunc, covfunc, likfunc, x, y);') hyp = minimize(hyp, @gp, -100, @infExact, meanfunc, covfunc, likfunc, x, y); disp('[m s2] = gp(hyp, @infExact, meanfunc, covfunc, likfunc, x, y, z);') [m s2] = gp(hyp, @infExact, meanfunc, covfunc, likfunc, x, y, z); figure(4) set(gca, 'FontSize', 24) disp(' ') disp('f = [m+2*sqrt(s2); flipdim(m-2*sqrt(s2),1)];') f = [m+2*sqrt(s2); flipdim(m-2*sqrt(s2),1)]; disp('fill([z; flipdim(z,1)], f, [7 7 7]/8)') fill([z; flipdim(z,1)], f, [7 7 7]/8) disp('hold on; plot(z, m); plot(x, y, ''+'');') hold on; plot(z, m, 'LineWidth', 2); plot(x, y, '+', 'MarkerSize', 12) grid on xlabel('input, x') ylabel('output, y') axis([-1.9 1.9 -0.9 3.9]) if write_fig, print -depsc f4.eps; end disp(' '); disp('Hit any key to continue...'); pause disp('large scale regression using the FITC approximation') disp('nu = fix(n/2); u = linspace(-1.3,1.3,nu)'';') nu = fix(n/2); u = linspace(-1.3,1.3,nu)'; disp('covfuncF = {@covFITC, {covfunc}, u};') covfuncF = {@covFITC, {covfunc}, u}; disp('[mF s2F] = gp(hyp, @infFITC, meanfunc, covfuncF, likfunc, x, y, z);') [mF s2F] = gp(hyp, @infFITC, meanfunc, covfuncF, likfunc, x, y, z); figure(5) set(gca, 'FontSize', 24) disp(' ') disp('f = [mF+2*sqrt(s2F); flipdim(mF-2*sqrt(s2F),1)];') f = [mF+2*sqrt(s2F); flipdim(mF-2*sqrt(s2F),1)]; disp('fill([z; flipdim(z,1)], f, [7 7 7]/8)') fill([z; flipdim(z,1)], f, [7 7 7]/8) disp('hold on; plot(z, mF); plot(x, y, ''+'');') hold on; plot(z, mF, 'LineWidth', 2); plot(x, y, '+', 'MarkerSize', 12) disp('plot(u,1,''o'')') plot(u,1,'ko', 'MarkerSize', 12) grid on xlabel('input, x') ylabel('output, y') axis([-1.9 1.9 -0.9 3.9]) if write_fig, print -depsc f5.eps; end disp(' ')gpml-matlab-v3.2-2013-01-15/doc/usageClassification.m000644 000765 000024 00000005057 12046775407 021777 0ustar00carlstaff000000 000000 % demonstrate usage of classification % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch 2012-11-07. clear all, close all %% SAY WHICH CODE WE WISH TO EXERCISE % id = [1,1]; % use Gauss/Exact id = [1,2; 2,2; 3,2]; % compare Laplace % id = [1,3; 2,3; 3,3]; % compare EP % id = [1,4; 3,4]; % compare VB seed = 943; randn('seed',seed), rand('seed',seed) ntr = 50; nte = 1e4; % number of training and test points xtr = 10*sort(rand(ntr,1)); % sample dataset p = @(x) 1./(1+exp(-5*sin(x))); % "true" underlying probability ytr = 2*(p(xtr)>rand(ntr,1))-1; % draw labels +1/-1 i = randperm(ntr); nout = 3; % add outliers ytr(i(1:nout)) = -ytr(i(1:nout)); xte = linspace(0,10,1e4)'; % support, we test our function on cov = {@covSEiso}; sf = 1; ell = 0.7; % setup the GP hyp0.cov = log([ell;sf]); mean = {@meanZero}; % m(x) = 0 hyp0.mean = []; lik_list = {'likGauss','likErf','likLogistic'}; % allowable likelihoods inf_list = {'infExact','infLaplace','infEP','infVB'}; % possible inference algs Ncg = 50; % number of conjugate gradient steps sdscale = 0.5; % how many sd wide should the error bars become? col = {'k',[.8,0,0],[0,.5,0],'b',[0,.75,.75],[.7,0,.5]}; % colors ymu{1} = 2*p(xte)-1; ys2{1} = 0; for i=1:size(id,1) lik = lik_list(id(i,1)); % setup the likelihood if strcmp(lik,'likGauss') sn = .2; hyp0.lik = log(sn); else hyp0.lik = []; end inf = inf_list{id(i,2)}; fprintf('OPT: %s/%s\n',lik_list{id(i,1)},inf_list{id(i,2)}) hyp = minimize(hyp0,'gp', -Ncg, inf, mean, cov, lik, xtr, ytr); % opt hypers [ymu{i+1}, ys2{i+1}] = gp(hyp, inf, mean, cov, lik, xtr, ytr, xte); % predict [nlZ(i+1)] = gp(hyp, inf, mean, cov, lik, xtr, ytr); end figure, hold on for i=1:size(id,1)+1 plot(xte,ymu{i},'Color',col{i},'LineWidth',2) if i==1 leg = {'function'}; else leg{end+1} = sprintf('%s/%s -lZ=%1.2f',... lik_list{id(i-1,1)},inf_list{id(i-1,2)},nlZ(i)); end end for i=1:size(id,1)+1 ysd = sdscale*sqrt(ys2{i}); fill([xte;flipud(xte)],[ymu{i}+ysd;flipud(ymu{i}-ysd)],... col{i},'EdgeColor',col{i},'FaceAlpha',0.1,'EdgeAlpha',0.3); end for i=1:size(id,1)+1, plot(xte,ymu{i},'Color',col{i},'LineWidth',2), end plot(xtr,ytr,'k+'), plot(xtr,ytr,'ko'), legend(leg) gpml-matlab-v3.2-2013-01-15/doc/style.css000644 000765 000024 00000000115 11511036275 017466 0ustar00carlstaff000000 000000 body {font-family: sans-serif; font-size: 16px} table {font-size: inherit;} gpml-matlab-v3.2-2013-01-15/doc/usageMean.m000644 000765 000024 00000002406 12077253575 017720 0ustar00carlstaff000000 000000 % demonstrate usage of mean functions % % See also meanFunctions.m. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2013-01-21 clear all, close all n = 5; D = 2; x = randn(n,D); % create a random data set % set up simple mean functions m0 = {'meanZero'}; hyp0 = []; % no hyperparameters are needed m1 = {'meanOne'}; hyp1 = []; % no hyperparameters are needed mc = {@meanConst}; hypc = 2; % also function handles are possible ml = {@meanLinear}; hypl = [2;3]; % m(x) = 2*x1 + 3*x2 % set up composite mean functions msc = {'meanScale',{m1}}; hypsc = [3; hyp1]; % scale by 3 msu = {'meanSum',{m0,mc,ml}}; hypsu = [hyp0; hypc; hypl]; % sum mpr = {@meanProd,{mc,ml}}; hyppr = [hypc; hypl]; % product mpo = {'meanPow',{3,msu}}; hyppo = hypsu; % third power mask = [0,1,0]; % binary mask excluding all but the 2nd component mma = {'meanMask',{mask,mpo{:}}}; hypma = hyppo; % 0) specify mean function % mean = m0; hyp = hyp0; % mean = msu; hyp = hypsu; % mean = mpr; hyp = hyppr; mean = mpo; hyp = hyppo; % 1) query the number of parameters feval(mean{:}) % 2) evaluate the function on x feval(mean{:},hyp,x) % 3) compute the derivatives w.r.t. to hyperparameter i i = 2; feval(mean{:},hyp,x,i) gpml-matlab-v3.2-2013-01-15/doc/usageLik.m000644 000765 000024 00000003314 12077253575 017556 0ustar00carlstaff000000 000000 % demonstrate usage of likelihood functions % % See also likFunctions.m. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2013-01-21 clear all, close all n = 5; f = randn(n,1); % create random latent function values % set up simple classification likelihood functions yc = sign(f); lc0 = {'likErf'}; hypc0 = []; % no hyperparameters are needed lc1 = {@likLogistic}; hypc1 = []; % also function handles are OK lc2 = {'likUni'}; hypc2 = []; lc3 = {'likMix',{'likUni',@likErf}}; hypc3 = log([1,2]); %mixture % set up simple regression likelihood functions yr = f + randn(n,1)/20; sn = 0.1; % noise standard deviation lr0 = {'likGauss'}; hypr0 = log(sn); lr1 = {'likLaplace'}; hypr1 = log(sn); lr2 = {'likSech2'}; hypr2 = log(sn); nu = 4; % number of degrees of freedom lr3 = {'likT'}; hypr3 = [log(nu-1); log(sn)]; lr4 = {'likMix',{lr0,lr1}}; hypr4 = [log([1,2]),hypr0,hypr1]; % set up Poisson regression yp = fix(abs(f)) + 1; lp0 = {@likPoisson,'logistic'}; hypp0 = []; lp1 = {@likPoisson,'exp'}; hypp1 = []; % 0) specify the likelihood function lik = lc0; hyp = hypc0; y = yc; % lik = lr4; hyp = hypr4; y = yr; % lik = lp1; hyp = hypp1; y = yp; % 1) query the number of parameters feval(lik{:}) % 2) evaluate the likelihood 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n2=5; % number of data points from each class n1/n2 sc=1; % change the scale of the initial problem % 1) draw data from two Gaussians S1 = eye(2); S2 = [1 0.95; 0.95 1]; m1 = [0.75; 0]; m2 = [-0.75; 0]; randn('seed',17); rand('seed',17) % coordinates x1 = chol(S1)'*randn(2,n1)+repmat(m1,1,n1); x2 = chol(S2)'*randn(2,n2)+repmat(m2,1,n2); x1(:,5) = x1(:,2); x1(:,1) = x1(:,4); x2(:,1) = x2(:,2); % rank deficiency in K xtr = [x1 x2]'; % class labels ytr = [repmat(-1,[1,n1]) repmat(1,[1,n2])]'; tt = (-5:.3:5)*sc; [t1 t2] = meshgrid(tt,tt); x1 = x1*sc; x2=x2*sc; xtr=xtr*sc; m1=m1*sc; m2=m2*sc; t = [t1(:) t2(:)]; z1 = exp(-sum((t-repmat(m1',length(t),1))*inv(S1).*(t-repmat(m1',length(t),1)),2)/2); z2 = exp(-sum((t-repmat(m2',length(t),1))*inv(S2).*(t-repmat(m2',length(t),1)),2)/2); xte = t; clear t xtr(end,:)=[2.9,2.3]; ytr(end)=-1; % modif % 2) set GP parameters cov = {@covSum,{@covSEiso,@covNoise}}; hyp.cov = [0; 2; -Inf]; % ell,sf,sn lik = {'likLogistic'}; hyp.lik = []; % likLogistic or likErf mn = {'meanZero'}; hyp.mean = []; inf = 'infEP'; % 3) predict using approximated inference [nlZ,dnlZ,post] = gp(hyp, inf, mn, cov, lik, xtr, ytr); [ymu,ys2,fmu,fs2,junk,post] = gp(hyp,inf,mn,cov,lik,xtr,post,xte); % 4) prediction using MCMC sampling % set MCMC parameters, see some more details in inf/infMCMC.m % We have two samplers implemented, namely % hmc - Hybrid Monte Carlo, and % ess - Elliptical Slice Sampling. par.sampler = 'hmc'; par.Nsample = 20; % par.sampler = 'ess'; % par.Nais = 5; par.Nsample = 200; par.Nskip = 100; tic [posts,nlZs,dnlZs] = infMCMC(hyp,mn,cov,lik,xtr,ytr,par); toc [ymus,ys2s,fmus,fs2s,junk,posts] = gp(hyp,@infMCMC,mn,cov,lik,xtr,posts,xte); % 5a) echo results fprintf('nlZ-EP=%f, nlZ-AIS=%f\n', nlZ, nlZs) fprintf('acceptance rate (MCMC) = %1.2f%%\n',100*posts.acceptance_rate_MCMC) for r=1:length(posts.acceptance_rate_AIS) fprintf('acceptance rate (AIS) = %1.2f%%\n',100*posts.acceptance_rate_AIS(r)) end % 5b) print results figure subplot(221) plot([-12,12],[-12,12],'r'), hold on, plot(fmus,fmu,'k.'), title('\mu_f') xlabel('MCMC'), ylabel(inf) subplot(222) plot([0,10],[0,10],'r'), hold on, plot(sqrt(fs2s),sqrt(fs2),'k.') title('\sigma_f') xlabel('MCMC'), ylabel(inf) subplot(223) plot([-1,1],[-1,1],'r'), hold on, plot(ymus,ymu,'k.'), title('\mu_y') xlabel('MCMC'), ylabel(inf) subplot(224) plot([0,1],[0,1],'r'), hold on, plot(sqrt(ys2s),sqrt(ys2),'k.') title('\sigma_y') xlabel('MCMC'), ylabel(inf)gpml-matlab-v3.2-2013-01-15/doc/demoClassification.m000644 000765 000024 00000011117 12077070414 021577 0ustar00carlstaff000000 000000 disp('See http://www.gaussianprocess.org/gpml/code/matlab/doc/ for details.') disp('Hit any key to continue...'); pause disp(' '); disp('clear all, close all') clear all, close all write_fig = 0; disp(' ') disp('n1 = 80; n2 = 40; % number of data points from each class') n1 = 80; n2 = 40; disp('S1 = eye(2); S2 = [1 0.95; 0.95 1]; % the two covariance matrices') S1 = eye(2); S2 = [1 0.95; 0.95 1]; disp('m1 = [0.75; 0]; m2 = [-0.75; 0]; % the two means') m1 = [0.75; 0]; m2 = [-0.75; 0]; disp(' ') disp('x1 = bsxfun(@plus, chol(S1)''*gpml_randn(0.2, 2, n1), m1);') x1 = bsxfun(@plus, chol(S1)'*gpml_randn(0.2, 2, n1), m1); disp('x2 = bsxfun(@plus, chol(S2)''*gpml_randn(0.3, 2, n2), m2);') x2 = bsxfun(@plus, chol(S2)'*gpml_randn(0.3, 2, n2), m2); disp(' ') disp('x = [x1 x2]''; y = [-ones(1,n1) ones(1,n2)]'';') x = [x1 x2]'; y = [-ones(1,n1) ones(1,n2)]'; figure(6) disp('plot(x1(1,:), x1(2,:), ''b+''); hold on;'); plot(x1(1,:), x1(2,:), 'b+', 'MarkerSize', 12); hold on disp('plot(x2(1,:), x2(2,:), ''r+'');'); plot(x2(1,:), x2(2,:), 'r+', 'MarkerSize', 12); disp(' ') disp('[t1 t2] = meshgrid(-4:0.1:4,-4:0.1:4);') [t1 t2] = meshgrid(-4:0.1:4,-4:0.1:4); disp('t = [t1(:) t2(:)]; n = length(t); % these are the test inputs') t = [t1(:) t2(:)]; n = length(t); % these are the test inputs disp('tmm = bsxfun(@minus, t, m1'');') tmm = bsxfun(@minus, t, m1'); disp('p1 = n1*exp(-sum(tmm*inv(S1).*tmm/2,2))/sqrt(det(S1));') p1 = n1*exp(-sum(tmm*inv(S1).*tmm/2,2))/sqrt(det(S1)); disp('tmm = bsxfun(@minus, t, m2'');') tmm = bsxfun(@minus, t, m2'); disp('p2 = n2*exp(-sum(tmm*inv(S2).*tmm/2,2))/sqrt(det(S2));') p2 = n2*exp(-sum(tmm*inv(S2).*tmm/2,2))/sqrt(det(S2)); set(gca, 'FontSize', 24) disp('contour(t1, t2, reshape(p2./(p1+p2), size(t1)), [0.1:0.1:0.9])') contour(t1, t2, reshape(p2./(p1+p2), size(t1)), [0.1:0.1:0.9]) [c h] = contour(t1, t2, reshape(p2./(p1+p2), size(t1)), [0.5 0.5]); set(h, 'LineWidth', 2) colorbar grid axis([-4 4 -4 4]) if write_fig, print -depsc f6.eps; end disp(' '); disp('Hit any key to continue...'); pause disp(' ') disp('meanfunc = @meanConst; hyp.mean = 0;') meanfunc = @meanConst; hyp.mean = 0; disp('covfunc = @covSEard; hyp.cov = log([1 1 1]);') covfunc = @covSEard; hyp.cov = log([1 1 1]); disp('likfunc = @likErf;') likfunc = @likErf; disp(' ') disp('hyp = minimize(hyp, @gp, -40, @infEP, meanfunc, covfunc, likfunc, x, y);') hyp = minimize(hyp, @gp, -40, @infEP, meanfunc, covfunc, likfunc, x, y); disp('[a b c d lp] = gp(hyp, @infEP, meanfunc, covfunc, likfunc, x, y, t, ones(n, 1));') [a b c d lp] = gp(hyp, @infEP, meanfunc, covfunc, likfunc, x, y, t, ones(n,1)); disp(' ') figure(7) set(gca, 'FontSize', 24) disp('plot(x1(1,:), x1(2,:), ''b+''); hold on') plot(x1(1,:), x1(2,:), 'b+', 'MarkerSize', 12); hold on disp('plot(x2(1,:), x2(2,:), ''r+'')') plot(x2(1,:), x2(2,:), 'r+', 'MarkerSize', 12) disp('contour(t1, t2, reshape(exp(lp), size(t1)), 0.1:0.1:0.9)') contour(t1, t2, reshape(exp(lp), size(t1)), 0.1:0.1:0.9) [c h] = contour(t1, t2, reshape(exp(lp), size(t1)), [0.5 0.5]); set(h, 'LineWidth', 2) colorbar grid axis([-4 4 -4 4]) if write_fig, print -depsc f7.eps; end disp(' '); disp('Hit any key to continue...'); pause disp('large scale classification using the FITC approximation') disp('[u1,u2] = meshgrid(linspace(-2,2,5)); u = [u1(:),u2(:)];') [u1,u2] = meshgrid(linspace(-2,2,5)); u = [u1(:),u2(:)]; clear u1; clear u2 disp('nu = size(u,1);') nu = size(u,1); disp('covfuncF = {@covFITC, {covfunc}, u};') covfuncF = {@covFITC, {covfunc}, u}; disp('inffunc = @infFITC_Laplace;') inffunc = @infFITC_EP; % one could also use @infFITC_Laplace disp('hyp = minimize(hyp, @gp, -40, inffunc, meanfunc, covfuncF, likfunc, x, y);') hyp = minimize(hyp, @gp, -40, inffunc, meanfunc, covfuncF, likfunc, x, y); disp('[a b c d lp] = gp(hyp, inffunc, meanfunc, covfuncF, likfunc, x, y, t, ones(n,1));') [a b c d lp] = gp(hyp, inffunc, meanfunc, covfuncF, likfunc, x, y, t, ones(n,1)); disp(' ') figure(8) set(gca, 'FontSize', 24) disp('plot(x1(1,:), x1(2,:), ''b+''); hold on') plot(x1(1,:), x1(2,:), 'b+', 'MarkerSize', 12); hold on disp('plot(x2(1,:), x2(2,:), ''r+'')') plot(x2(1,:), x2(2,:), 'r+', 'MarkerSize', 12) disp('contour(t1, t2, reshape(exp(lp), size(t1)), 0.1:0.1:0.9)') contour(t1, t2, reshape(exp(lp), size(t1)), 0.1:0.1:0.9) [c h] = contour(t1, t2, reshape(exp(lp), size(t1)), [0.5 0.5]); set(h, 'LineWidth', 2) disp('plot(u(:,1),u(:,2),''ko'', ''MarkerSize'', 12)') plot(u(:,1),u(:,2),'ko', 'MarkerSize', 12) colorbar grid axis([-4 4 -4 4]) if write_fig, print -depsc f8.eps; end disp(' ')gpml-matlab-v3.2-2013-01-15/doc/f3.gif000644 000765 000024 00000033052 12077070414 016622 0ustar00carlstaff000000 000000 GIF89aõš÷µ !!!"""&&&(((***---...///000111222333555888:::<<<>>>???@@@AAADDDKKKMMMNNNOOOPPPQQQRRRSSSUUUYYY[[[\\\]]]^^^___```aaabbbcccdddfffggnhhhiiijjjkkklllmmmnnnooopppqqqrrrssstttuuuwwwxxxzzz}}}~~~^^€WWº99Öÿÿ þýþþþýý>>ç ó!!û%%û''û ü//ú22ù11ú44ù77ù66ú==øBB÷FF÷HH÷MMöOOöRRõUUõVVõ\\ôïaaóccó``ôjjòkkòppñppòttñvvñ€€€ƒƒƒ„„„………†††‡‡‡ˆˆˆ‰‰‰ŒŒŒ•••–––———˜˜˜™™™šššœœœ’’·   £££¥¥¥¦¦¦§§§¨¨¨©©©ªªª«««®®®°°°±±±³³³´´´¶¶¶¹¹¹ººº»»»¿¿¿··Ó파ìë¡¡ë««é¹¹çÀÀÀÃÃÃÆÆÆÉÉÉÊÊÊËËËÌÌÌÏÏÏÒÒÒ×××ØØØÜÜÜÝÝÝÆÆåÈÈåÔÔãàààéééîîî÷÷÷ûûûýýýÿÿÿ!ù,õšþk H° Áƒ*\È°¡Ã‡#JœH±¢Å‹3jÜȱ£Ç CŠI²¤É“(Sª\ɲ¥Ë—0cÊœI³¦Í›8sêÜɳ§ÏŸ@ƒ J´¨Ñ£H“*]Ê´©Ó§P£JJµªÕ«X³jÝʵ«W¯'PK¶¬Ù³hÓª]˶­[·&¼K·®Ý»t(‹·¯ß¿€ËŠ-+áÔªWˆ+^̸±ãÇ#Kf< «Ç«&Ô|ÁaÒW„>>S¤DIôÄÐ/L^ͺµkÈž qYBÍ€@˜.(dwDJž}?D-‘×È“+oÜÉÈ«Ü6D#¸oâÂN²ž}!ö –þ‹ßz ’3‘mÓla蔆ì4º7¤DH>ÃøAßÏ¿ql$—a&Ýz†¼rB)ÂÅbC .˜Pƒµr\î×ÉH`!™z3±÷ (-8(âˆõI…(Š×†H´Ù*Êä¡«hãipXŠ<¾ÖHöÀtˆˆã‘6–BCL¶KL6$‰yág"YK–Gfib“`J¶…sR‰Xuh©¦|ú…é¦cKœW&;¾¢Ûšxš¶J o¾ÉJ€¶‘†lhêbc†LÙa‰ÑÀežnH}VŠØ%HL!Ù)hFebFŠV£–£Y#§XZé-JF"þlÆh•WvUë·’¨ ªVç&‘E¢[¬Ÿ&VG‘&‹•’uöÚd'²Iv‚Ä.ÊØ{Êf;•':fsšB&DšµT+Ó ³&fãڶ»”Þº‰a@Êåš©X¨^±{£¿$RrB¼`º"›']Pª¹1͸˜îF¬TwÜ$&™BÖv Ãä°bw "ñÈEµªÅ=¢‚AA¿ôqb§„HòÍ@©†ræaòXu0ëûðW¥šŠ¤’;óømˆb'PÌ.Íœ˜"Ûªf®"Ö‘nÒýa¨¡cÓuKRi3ÎhçÄ+×(²z(c¯46Ke#vAÚxÛËþÚlSøkc €ÐÜ+Õý Õy'"uôMáÒ-÷tÍî{¶V“˜ù‚=„ç8^7v ă Í銧¾R›Ÿïçvcw’á*~øæªçîQ,·¾ßß×*D{J¶ƒâ´îÈ{ĸïäA‹c«rèEbzRÁàˆÔ˜T®‹™×%Nç*’ÄŠvÊŠû´GßôŽ¢³Š ήŒ! ¹D@Næ‰QZŒ!ÈUCÊM¦ÈZdMôFG.æ‹pkáÓÁúiRƒ¬óäcæ…žÅòFt êª"JùÔò3TåbɘWRp&èbnyÊÄMbIºt $£HÉJƲ1ù+¦µ–LÇ4§ ¡|ˆG‚G»I•ÕtŒé¨MYµ}T ´NÓä2œ¼t%D¶)’n¾"dVþI§|ôÙ/„“1Otc ÑLgZ‹5 Ðß7ù‡É~Ò9Lk1é{æq¡û3Y'“9NyΓ”Ž9?1:²T:ˆ"¤/ayPÖà“*ÄNLµÂ;‡*cáRÌ#BQ°g£Ž1Iu·<›"Fe,“d&ËùÌǬt¨8óçŒÚ3¸qgŠ éH¡š-“þ“²IibN ‘žJ$>@p€Ô*˜®yiÕÔί¼³šOÌ©bÈÊÓ“¬"kÝÀ€¢YôChdÊV}³X¬4Ò¨¯eðúj’À­‰…e_@Ã^”«xëA!ÛQÅœB¡5 `‹ € puþHA‹7xA6ž„œdYOâƒÞd­ÝÚjäJÛ›ÙÖ¨Ë áU±ÚËzJ žMèTfºêZi•ìbl°X³RD°k@Õ +ÜÕ<µ¸Ê¢&iå4Éxw"á½lAJPÞÉ”‰E©¸¤ßðw¿ùpf\‹þ ä¿þ p‚¼àØÀ[Rð%aOØÂvpƒ!Œà kØæp‡3b øÄ"N±‰Uüa?øÅþ5Ùm½ 8³Æëí b”’U"€gï9^ôFÌ@UercJæ¶y“üéF¥®òÁ2M{gTí*FŠ»} pÉ«œó¥±¦A3þUQ±õ¶r1ucjI"là c®Å_1K#k‹[T$n£¾0“d®­"À¡= a¹~Ž”Wù䡹÷$ˆ2„&Øê¹¾“s¤³¥³.“©1Mv2IJX;o ÓòÈž±ëŠÆÖRÁ®QY‰@Š¼÷!«ðÖ\5¶Èéó¨ó¤^›:1@-e[ŠœB/R-4 +ýå¥N[F uA¯­¦»&ÓË„†´ªîM¥)ZÎN¼­² º9¤ùõFf]H¨¨ù3ÿ† ›gDǼ۠à æMîìH²Ü&$¿Ínm?¦Enø‘&]MtÓLÚ÷X»þ%ce#Éܺ,­$ÇÍTj'ÇÌCÁõWdî”c|‹yt¾³:e›|DÍvhÄÅðè[#³FŒÎþ‘Iô· ÇÏ}êÆÐÐ×<—Ì#ËtŠL4E<‚:D]&(W%¯³Šƒ#œÝä™xRŠ®'ˆ¬B4HUú ³¿$ö¶)Xô˜«_š&ÈVŽÏÏ|5ˆh‡ñ!#Ñ÷£Õ¦8… ÈC.3CX2ÖîzE|@)ÄÌ1”÷;K¾dT¤>Æ›Ÿó¹F~Io‹^"“àrdÍXy¢¤Ý“UuŒ!2Îù¨‚<5»=Eb¡“™Þ1€9NJíÐÁ‹5Œê®cÖ'þ#ýŸÐ¼+¢œÙé3®˜#ó{ w-rôŒ$}¬Ê—"Ο˜ñ+ÉBihëW–°‹´#‰§B gñÇ° b([`(€²/Ý×!ÎçPÁÇvn÷v"·°W€&Ìå¿HÖrW|dC{æÅt'* q\Ÿ$øç5eT¡Ó2\·n"÷y‘±xD¡z4¥¤@Ša‹q/?ñX6õ:¯wƒø¨ § D„aD|8ážgTÀã}}Ú·Ç·Ä%…á?Ö4îáS§Jó3`ˆƒÇ%g†q,q}g{8‚pTƒÁWì·}«ÁƒBñ}þ\±9.¸½—éY2ÔþgNüaˆØ@äQLø“YØ„Qƒ‚«A¤`‡*•èƒñ=6A‘ñ€¿”pü¡[¨(« {ã‘j“]H0>ö²‰dD'ü‘|s§&ªpt0Z•¸‚$á‡]‰¯Çr&H7¤˜‚IÁŠX‘+m˜‘Š+asÎFcQq(‡£H!³…Šª ‹û66¡5:èACF4÷ñ~9wŽLW{P!Ñ{0ÑŽäqä4‰M…;…ŠÀ#8ÐØDk–†.7w—ˆ¸ì2ƒ‘‘€’ÃŇ/QoFõl!‹³w¢’Ëö¥þ#498Wx5¡•y’ŽÃL×XŠÎˆ^Ýè *“)^àq%7 ;žÖ0N® ‰9˜d†±ÐÁ ó²2å æ±’×2. 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sech-square likelihood function for regression. Often, the sech- % square distribution is also referred to as the logistic distribution not to be % confused with the logistic function for classification. The expression for the % likelihood is % likSech2(t) = Z / cosh(tau*(y-t))^2 where % tau = pi/(2*sqrt(3)*sn) and Z = tau/2 % and y is the mean and sn^2 is the variance. % % hyp = [ log(sn) ] % % Several modes are provided, for computing likelihoods, derivatives and moments % respectively, see likFunctions.m for the details. In general, care is taken % to avoid numerical issues when the arguments are extreme. The moments % \int f^k likSech2(y,f) N(f|mu,var) df are calculated via a Gaussian % scale mixture approximation. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2012-11-04. % % See also LIKFUNCTIONS.M, LIKLOGISTIC.M. if nargin<3, varargout = {'1'}; return; end % report number of hyperparameters sn = exp(hyp); tau = pi/(2*sqrt(3)*sn); lZ = log(pi) - log(sn) - log(4*sqrt(3)); if nargin<5 % prediction mode if inf is not present if numel(y)==0, y = zeros(size(mu)); end s2zero = 1; if nargin>3, if norm(s2)>0, s2zero = 0; end, end % s2==0 ? if s2zero % log probability evaluation lp = lZ - 2*logcosh(tau*(y-mu)); s2 = 0; else % prediction lp = likSech2(hyp, y, mu, s2, 'infEP'); end ymu = {}; ys2 = {}; if nargout>1 ymu = mu; % first y moment if nargout>2 ys2 = s2 + sn.^2; % second y moment end end varargout = {lp,ymu,ys2}; else % inference mode switch inf case 'infLaplace' r = y-mu; [g,dg,d2g,d3g] = logcosh(tau.*r); % precompute derivatives if nargin<6 % no derivative mode dlp = {}; d2lp = {}; d3lp = {}; lp = lZ - 2*g; if nargout>1 % first derivatives dlp = 2*tau.*dg; if nargout>2 % 2nd derivative of log likelihood d2lp = -2*tau.^2.*d2g; if nargout>3 % 3rd derivative of log likelihood d3lp = 2*tau.^3.*d3g; end end end varargout = {lp,dlp,d2lp,d3lp}; else % derivative mode lp_dhyp = 2*tau.*r.*dg - 1; % derivative w.r.t. sn dlp_dhyp = -2*tau.*(dg+tau.*r.*d2g); d2lp_dhyp = 2*tau.^2.*(2*d2g + tau.*r.*d3g); varargout = {lp_dhyp,dlp_dhyp,d2lp_dhyp}; end case 'infEP' n = max([length(y),length(mu),length(s2),length(sn)]); on = ones(n,1); y = y.*on; mu = mu.*on; s2 = s2.*on; sn = sn.*on; % vectors only fac = 1e3; % factor between the widths of the two distributions ... % ... from when one considered a delta peak, we use 3 orders of magnitude idlik = fac*sn1/(2*tau^2) & ~idup; % interesting zone is in between [g,dg,d2g] = inv_xcothx(2*tau^2*ga(id)); thg = tanh(g); h(id) = 4*logcosh(g) -2*g.*thg; h = h + y.^2./ga - 2*lZ; g1mt2 = g.*(1-thg.^2); % first derivatives dh(idup) = 4*tau^2; dh(id) = 4*tau^2*dg.*( thg - g1mt2 ); dh = dh - (y./ga).^2; d2h(id) = 8*tau^4*( g1mt2.*(2*dg.^2.*thg-d2g) +d2g.*thg ); % second derivs d2h = d2h + 2*y.^2./ga.^3; id = ga<0; h(id) = Inf; dh(id) = 0; d2h(id) = 0; % neg. var. treatment varargout = {h,b,dh,db,d2h,d2b}; else ga = s2; n = numel(ga); dhhyp = zeros(n,1); idup = 50<=ga(:); % asymptotic behavior for large gammas dhhyp(idup) = -2*pi^2/3/sn^2*ga(idup); % tau = pi/(2*sqrt(3)*sn) id = ga(:)>1/(2*tau^2) & ~idup; % interesting zone is in between [g,dg] = inv_xcothx(2*tau^2*ga(id)); thg = tanh(g); [f,df] = logcosh(g); % h = 4*f -2*g.*tanh(g) if any(id) dhhyp(id) = 8*tau^2*ga(id).*dg.*( thg - 2*df + g.*(1-thg.^2) ); end dhhyp = dhhyp + 2; % from lZ = log(pi) - log(sn) - log(4*sqrt(3)) dhhyp(ga<0) = 0; % negative variances get a special treatment varargout = {dhhyp}; % deriv. wrt hyp.lik end end end % Invert f(x) = x.*coth(x) = y, return the positive value % uses Newton's method to minimize (y-f(x))² w.r.t. x function [x,dx,d2x] = inv_xcothx(y) % f = @(x) x.*coth(x); % df = @(x) x + (1-f(x)).*coth(x); % d2f = @(x) 2*(1-f(x)).*(1-coth(x).^2); if numel(y)==0, x=[]; dx=[]; d2x=[]; return, end x = sqrt(y.^2-1); i = 1; % init ep = eps./(1+abs(x)); th=tanh(x); fx=(x+ep)./(th+ep); % function r = fx-y; % init while i==1 || (i<10 && max(abs(r))>1e-12) dfx = x + (1-fx)./(th+ep); % first derivative d2fx = 2*( 1-fx + (x-th+ep/3)./(th.*th.*th+ep) ); % second derivative x = x - r.*dfx./(dfx.^2+r.*d2fx+ep); % Newton step ep = eps./(1+abs(x)); th=tanh(x); fx=(x+ep)./(th+ep); r = fx-y; i = i+1; end % first derivative dx/dy; derivatives of inverse functions are reciprocals dx = 1./dfx; % second derivative d2x/dy2; quotient rule and chaine rule d2x = -d2fx.*dx./dfx./dfx; % numerically safe version of log(cosh(x)) = log(exp(x)+exp(-x))-log(2) function [f,df,d2f,d3f] = logcosh(x) a = exp(-2*abs(x)); % always between 0 and 1 and therefore safe to evaluate f = abs(x) + log(1+a) - log(2); df = sign(x).*( 1 - 2*a./(1+a) ); d2f = 4*a./(1+a).^2; d3f = -8*sign(x).*a.*(1-a)./(1+a).^3; % computes y = log( exp(A)*x ) in a numerically safe way by subtracting the % maximal value in each row to avoid cancelation after taking the exp function y = log_expA_x(A,x) N = size(A,2); maxA = max(A,[],2); % number of columns, max over columns y = log(exp(A-maxA*ones(1,N))*x) + maxA; % exp(A) = exp(A-max(A))*exp(max(A)) % 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. function y = expABz_expAx(A,x,B,z) N = size(A,2); maxA = max(A,[],2); % number of columns, max over columns A = A-maxA*ones(1,N); % subtract maximum value y = ( (exp(A).*B)*z ) ./ ( exp(A)*x ); gpml-matlab-v3.2-2013-01-15/lik/likT.m000644 000765 000024 00000014021 12077070411 016706 0ustar00carlstaff000000 000000 function [varargout] = likT(hyp, y, mu, s2, inf, i) % likT - Student's t likelihood function for regression. % The expression for the likelihood is % likT(t) = Z * ( 1 + (t-y)^2/(nu*sn^2) ).^(-(nu+1)/2), % where Z = gamma((nu+1)/2) / (gamma(nu/2)*sqrt(nu*pi)*sn) % and y is the mean (for nu>1) and nu*sn^2/(nu-2) is the variance (for nu>2). % % The hyperparameters are: % % hyp = [ log(nu-1) % log(sn) ] % % Note that the parametrisation guarantees nu>1, thus the mean always exists. % % Several modes are provided, for computing likelihoods, derivatives and moments % respectively, see likFunctions.m for the details. In general, care is taken % to avoid numerical issues when the arguments are extreme. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2012-10-27. % % See also LIKFUNCTIONS.M. if nargin<3, varargout = {'2'}; return; end % report number of hyperparameters numin = 1; % minimum value of nu nu = exp(hyp(1))+numin; sn2 = exp(2*hyp(2)); % extract hyperparameters lZ = loggamma(nu/2+1/2) - loggamma(nu/2) - log(nu*pi*sn2)/2; if nargin<5 % prediction mode if inf is not present if numel(y)==0, y = zeros(size(mu)); end s2zero = 1; if nargin>3, if norm(s2)>0, s2zero = 0; end, end % s2==0 ? if s2zero % log probability evaluation lp = lZ - (nu+1)*log( 1+(y-mu).^2./(nu.*sn2) )/2; s2 = 0; else % prediction lp = likT(hyp, y, mu, s2, 'infEP'); end ymu = {}; ys2 = {}; if nargout>1 ymu = mu; % first y moment; for nu<=1 we this is the mode if nargout>2 if nu<=2 ys2 = Inf(size(mu)); else ys2 = (s2 + nu*sn2/(nu-2)).*ones(size(mu)); % second y moment end end end varargout = {lp,ymu,ys2}; else switch inf case 'infLaplace' r = y-mu; r2 = r.*r; if nargin<6 % no derivative mode dlp = {}; d2lp = {}; d3lp = {}; lp = lZ - (nu+1)*log( 1+r2./(nu.*sn2) )/2; if nargout>1 a = r2+nu*sn2; dlp = (nu+1)*r./a; % dlp, derivative of log likelihood if nargout>2 % d2lp, 2nd derivative of log likelihood d2lp = (nu+1)*(r2-nu*sn2)./a.^2; if nargout>3 % d3lp, 3rd derivative of log likelihood d3lp = (nu+1)*2*r.*(r2-3*nu*sn2)./a.^3; end end end varargout = {lp,dlp,d2lp,d3lp}; else % derivative mode a = r2+nu*sn2; a2 = a.*a; a3 = a2.*a; if i==1 % derivative w.r.t. nu lp_dhyp = nu*( dloggamma(nu/2+1/2)-dloggamma(nu/2) )/2 - 1/2 ... -nu*log(1+r2/(nu*sn2))/2 +(nu/2+1/2)*r2./(nu*sn2+r2); lp_dhyp = (1-numin/nu)*lp_dhyp; % correct for lower bound on nu dlp_dhyp = nu*r.*( a - sn2*(nu+1) )./a2; dlp_dhyp = (1-numin/nu)*dlp_dhyp; % correct for lower bound on nu d2lp_dhyp = nu*( r2.*(r2-3*sn2*(1+nu)) + nu*sn2^2 )./a3; d2lp_dhyp = (1-numin/nu)*d2lp_dhyp; % correct for lower bound on nu else % derivative w.r.t. sn lp_dhyp = (nu+1)*r2./a - 1; dlp_dhyp = -(nu+1)*2*nu*sn2*r./a2; d2lp_dhyp = (nu+1)*2*nu*sn2*(a-4*r2)./a3; end varargout = {lp_dhyp,dlp_dhyp,d2lp_dhyp}; end case 'infEP' if nargout>1 error('infEP not supported since likT is not log-concave') end n = max([length(y),length(mu),length(s2)]); on = ones(n,1); y = y(:).*on; mu = mu(:).*on; sig = sqrt(s2(:)).*on; % vectors only % since we are not aware of an analytical expression of the integral, % we use Gaussian-Hermite quadrature N = 20; [t,w] = gauher(N); oN = ones(1,N); lZ = likT(hyp, y*oN, sig*t'+mu*oN, []); lZ = log_expA_x(lZ,w); % log( exp(lZ)*w ) varargout = {lZ}; case 'infVB' if nargin<6 % variational lower site bound % t(s) \propto (1+(s-y)^2/(nu*s2))^(-nu/2+1/2) % the bound has the form: b*s - s.^2/(2*ga) - h(ga)/2 with b=y/ga!! ga = s2; n = numel(ga); b = y./ga; y = y.*ones(n,1); db = -y./ga.^2; d2b = 2*y./ga.^3; id = ga<=sn2*nu/(nu+1); h = (nu+1)*( log(ga*(1+1/nu)/sn2) - 1 ) + (nu*sn2+y.^2)./ga; h(id) = y(id).^2./ga(id); h = h - 2*lZ; dh = (nu+1)./ga - (nu*sn2+y.^2)./ga.^2; dh(id) = -y(id).^2./ga(id).^2; d2h = -(nu+1)./ga.^2 + 2*(nu*sn2+y.^2)./ga.^3; d2h(id) = 2*y(id).^2./ga(id).^3; id = ga<0; h(id) = Inf; dh(id) = 0; d2h(id) = 0; % neg. var. treatment varargout = {h,b,dh,db,d2h,d2b}; else ga = s2; n = numel(ga); dhhyp = zeros(n,1); id = ga>sn2*nu/(nu+1); % dhhyp(~id) = 0 if i==1 % log(nu) % h = (nu+1)*log(1+1/nu) - nu + nu*sn2./ga; dhhyp(id) = nu*log(ga(id)*(1+1/nu)/sn2) - 1 - nu + nu*sn2./ga(id); % lZ = loggamma(nu/2+1/2) - loggamma(nu/2) - log(nu)/2 dhhyp = dhhyp - nu*dloggamma(nu/2+1/2) + nu*dloggamma(nu/2) + 1; % -2*lZ dhhyp = (1-numin/nu)*dhhyp; % correct for lower bound on nu else % log(sn) % h = (nu+1)*log(1/sn2) + nu*sn2./ga; dhhyp(id) = -2*(nu+1) + 2*nu*sn2./ga(id); % lZ = - log(sn2)/2 dhhyp = dhhyp + 2; % -2*lZ end dhhyp(ga<0) = 0; % negative variances get a special treatment varargout = {dhhyp}; % deriv. wrt hyp.lik end end end function f = loggamma(x) f = gammaln(x); function df = dloggamma(x) df = psi(x); % computes y = log( exp(A)*x ) in a numerically safe way by subtracting the % maximal value in each row to avoid cancelation after taking the exp function y = log_expA_x(A,x) N = size(A,2); maxA = max(A,[],2); % number of columns, max over columns y = log(exp(A-maxA*ones(1,N))*x) + maxA; % exp(A) = exp(A-max(A))*exp(max(A)) gpml-matlab-v3.2-2013-01-15/lik/likMix.m000644 000765 000024 00000020616 12077070411 017247 0ustar00carlstaff000000 000000 function [varargout] = likMix(lik, hyp, varargin) % likMix - Mixture of likelihoods for regression/classification. % The expression for the likelihood is % log( likMix(t) ) = sum_i=1..m w_i * log( lik_i(t) ), % where lik_i are the m individual likelihood functions combined by a weighted % sum in the log domain with weights wi. % % The hyperparameters are: % % hyp = [ log(w_1) % log(w_2) % ... % log(w_m) % hyp_1 % hyp_2 % ... % hyp_m ] % % Note that the mixture weights are normalised using the softmax function, so % that 1 = sum w_i. Here, hyp_i are the hyperparameters for the individual % likelihoods. % % Several modes are provided, for computing likelihoods, derivatives and moments % respectively, see likFunctions.m for the details. In general, care is taken % to avoid numerical issues when the arguments are extreme. % % Copyright (c) by Hannes Nickisch and Rowan McAllister, 2013-01-17 % % See also LIKFUNCTIONS.M. m = numel(lik); % number of mixture components if m==0, error('We require at least one mixture component.'), end for i = 1:m % iterate over mixture components f = lik(i); if iscell(f{:}), f = f{:}; end % expand cell array if necessary j(i) = cellstr(feval(f{:})); % collect number hypers end if nargin<4 % report number of parameters L = char(j(1)); for i=2:length(lik), L = [L, '+', char(j(i))]; end varargout = {[num2str(m),'+',L]}; return end mu = varargin{2}; n = length(mu); % number of query points if numel(varargin)>3 inf = varargin{4}; % varargin = {y, mu, s2, inf, i} else inf = ''; end v = []; % v vector indicates to which likelihood parameters belong nhyp = m; % number of hyperparameters; the first m are the mixture weights for i = 1:m ni = eval(char(j(i))); nhyp = nhyp + ni; v = [v, repmat(i,1,ni)]; end if nhyp>length(hyp), error('not enough hyperparameters'), end if strcmp(inf,'infVB'), error('infVB not supported'); end na = nargout; if nargout>1 && strcmp(inf,'infLaplace') && nargin>6, na = max(na,3); end argout = zeros(n,m,na); % collect output arguments from individual lik fun for i=1:m % iteration over mixture components f = lik(i); if iscell(f{:}), f = f{:}; end % expand cell array if necessary id = find(v==i)+m; % hyperparameter indices out = cell(size(argout,3),1); nn = min(nargin,6)-2; % do not ask for derivs [out{:}] = feval(f{:}, hyp(id), varargin{1:nn}); % ask individual likelihoods for j=1:size(argout,3), argout(:,i,j) = out{j}; end end lw = reshape(hyp(1:m),1,[]); lwn = lw-s(lw); % w = exp(lwn), sum(w)=1, weights one_lwn = ones(n,1)*lwn; % the same as above only that the weights are repeated varargout = cell(nargout,1); % allocate memory for output arguments if nargin<6 % prediction mode if inf is not present % lp and ymu outputs are of the weighted sum of integral structure if nargout>0, varargout{1} = s(one_lwn + argout(:,:,1)); end if nargout>1, varargout{2} = sum(exp(one_lwn).*argout(:,:,2), 2); end if nargout>2 % ys2 output computed using the parallel axis theorem argout3corr = argout(:,:,3) + (argout(:,:,2)-varargout{2}*ones(1,m)).^2; varargout{3} = sum(exp(one_lwn).*argout3corr, 2); end else if nargin>6, i = varargin{5}; end % make arguments available switch inf case {'infEP','infLaplace'} % the two cases are structurally similar if nargin<7 % no derivative mode if nargout>0 % lf={lZ,lp} for inf={'infEP','infLaplace'} lfi = argout(:,:,1); % partition functions of individual likelihoods lf = s(lfi + one_lwn); % weighted sum in the exp domain varargout{1} = lf; if nargout>1 % dlf dlfi = argout(:,:,2); % derivatives of individual terms % Using f*dlf=df, f=exp(lf), f=sum_i wi*fi, df=sum_i wi*dfi we obtain % dlf = sum_i exp(lfi-lf+lwi)*dlfi = sum_i ai*dlfi. a = exp(lfi - lf*ones(1,m) + one_lwn); dlf = sum(a.*dlfi,2); varargout{2} = dlf; if nargout>2 % d2lf d2lfi = argout(:,:,3); % 2nd derivatives of individual terms % Using d2lf=d(df/f)=(d2f*f-df^2)/f^2 <=> d2f=f*(d2lf+dlf^2) and % d2f = sum_i wi*d2fi, we get d2lf = sum_i ai*(d2lfi+dlfi^2)-dlf^2. d2lf = sum(a.*(d2lfi+dlfi.*dlfi),2) - dlf.*dlf; varargout{3} = d2lf; if nargout>3 && strcmp(inf,'infLaplace') % d3lf % Using d3f = sum_i wi*d3fi, d3lf=d((d2f*f-df^2)/f^2), we obtain % d3f = f*(d3lf+dlf*(3*d2lf+dlf^2)) <=> % d3lf = d3f/f - dlf*(3*d2lf+dlf^2). d3lfi = argout(:,:,4); % 3rd derivatives of individual terms d3lf = sum(a.*(d3lfi+dlfi.*(3*d2lfi+dlfi.*dlfi)),2); d3lf = d3lf - dlf.*(3*d2lf+dlf.*dlf); varargout{4} = d3lf; end end end end else % d(lf)/dhyp and d(d2lf)/dhyp if nargout>0 lfi = argout(:,:,1); % partition functions of individual likelihoods lf = s(lfi + one_lwn); % weighted sum in the exp domain a = exp(lfi - lf*ones(1,m) + one_lwn); if i<=m % the first m hyperparameters are mixture weights lf_dhyp = a(:,i) - exp(lwn(i)); varargout{1} = lf_dhyp; if nargout>1 && strcmp(inf,'infLaplace') % d(dlf)/dhyp dlfi = argout(:,:,2); dlwn = zeros(size(lw)); dlwn(i) = 1; dlwn = dlwn - exp(lw(i)-s(lw)); dla = ones(n,1)*dlwn - lf_dhyp*ones(1,m); % d log(v) / dhyp da = a.*dla; dlf_dhyp = sum(da.*dlfi,2); varargout{2} = dlf_dhyp; if nargout>2 % d(d2lf)/dhyp d2lfi = argout(:,:,3); dlf = sum(a.*dlfi,2); d2lf_dhyp = sum(da.*(d2lfi+dlfi.*dlfi),2) - 2*dlf.*dlf_dhyp; varargout{3} = d2lf_dhyp; end end else j = v(i-m); % index of mixture component under consideration id = find(v==j)+m; % hyper parameter indices i = i-min(id)+1; % index of likelihood parameter of mixture j f = lik(j); if iscell(f{:}), f = f{:}; end % expand cell if necessary if nargout==1 lfi_dhyp = feval(f{:}, hyp(id), varargin{1:end-1}, i); elseif nargout==2 [lfi_dhyp,dlfi_dhyp] = feval(f{:}, hyp(id), varargin{1:end-1}, i); else [lfi_dhyp,dlfi_dhyp,d2lfi_dhyp] = ... feval(f{:}, hyp(id), varargin{1:end-1}, i); end lf_dhyp = exp(lwn(j)+lfi(:,j)-lf) .* lfi_dhyp(:); % d(lf)/dhyp varargout{1} = lf_dhyp; if nargout>1 % d(dlf)/dhyp dlfi = argout(:,:,2); d2lfi = argout(:,:,3); dlf = sum(a.*dlfi,2); dla = -lf_dhyp*ones(1,m); dla(:,j) = dla(:,j)+lfi_dhyp(:); da = a.*dla; dlf_dhyp = sum(da.*dlfi,2) + a(:,j).*dlfi_dhyp(:); varargout{2} = dlf_dhyp; if nargout>2 % d(d2lf)/dhyp p = d2lfi+dlfi.*dlfi; dpj = 2*dlfi(:,j).*dlfi_dhyp(:) + d2lfi_dhyp(:); d2lf_dhyp = a(:,j).*dpj + sum(da.*p,2) - 2*dlf.*dlf_dhyp; varargout{3} = d2lf_dhyp; end end end end end case 'infVB' error('infVB not supported') end end % computes y = log( sum(exp(x),2) ), the softmax in a numerically safe way by % subtracting the row maximum to avoid cancelation after taking the exp % the sum is done along the rows function [y,x] = s(logx) N = size(logx,2); max_logx = max(logx,[],2); % we have all values in the log domain, and want to calculate a sum x = exp(logx-max_logx*ones(1,N)); y = log(sum(x,2)) + max_logx; gpml-matlab-v3.2-2013-01-15/lik/likUni.m000644 000765 000024 00000005570 12077070411 017247 0ustar00carlstaff000000 000000 function [varargout] = likUni(hyp, y, mu, s2, inf, i) % likUni - Uniform likelihood function for classification. The expression for % the likelihood is % likUni(t) = 1/2. % % There are no hyperparameters: % % hyp = [ ] % % Several modes are provided, for computing likelihoods, derivatives and moments % respectively, see likFunctions.m for the details. In general, care is taken % to avoid numerical issues when the arguments are extreme. % % Copyright (c) by Hannes Nickisch, 2012-11-07 % % See also LIKFUNCTIONS.M. if nargin<3, varargout = {'0'}; return; end % report number of hyperparameters if nargin<5 % prediction mode if inf is not present lp = -log(2)*ones(size(mu)); ymu = {}; ys2 = {}; if nargout>1 p = exp(lp); ymu = 2*p-1; % first y moment if nargout>2 ys2 = 4*p.*(1-p); % second y moment end end varargout = {lp,ymu,ys2}; else switch inf case 'infLaplace' if nargin<6 % no derivative mode lp = -log(2)*ones(size(mu)); dlp = {}; d2lp = {}; d3lp = {}; if nargout>1 dlp = zeros(size(mu)); % dlp, derivative of log likelihood if nargout>2 % d2lp, 2nd derivative of log likelihood d2lp = zeros(size(mu)); if nargout>3 % d3lp, 3rd derivative of log likelihood d3lp = zeros(size(mu)); end end end varargout = {lp,dlp,d2lp,d3lp}; else % derivative mode varargout = {[],[],[]}; % derivative w.r.t. hypers end case 'infEP' if nargin<6 % no derivative mode lZ = -log(2)*ones(size(mu)); % log part function dlZ = {}; d2lZ = {}; if nargout>1 dlZ = zeros(size(mu)); % 1st derivative w.r.t. mean if nargout>2 d2lZ = zeros(size(mu)); % 2nd derivative w.r.t. mean end end varargout = {lZ,dlZ,d2lZ}; else % derivative mode varargout = {[]}; % deriv. wrt hyp.lik end case 'infVB' if nargin<6 % no derivative mode % variational lower site bound % t(s) = 1/2 % the bound has the form: b*s - s.^2/(2*ga) - h(ga)/2 with b=y/ga ga = s2; n = numel(ga); b = y./ga; y = y.*ones(n,1); db = -y./ga.^2; d2b = 2*y./ga.^3; h = y.^2./ga + 2*log(2); dh = -y.^2./ga.^2; d2h = 2*y.^2./ga.^3; varargout = {h,b,dh,db,d2h,d2b}; else varargout = {[]}; % deriv. wrt hyp.lik end end end gpml-matlab-v3.2-2013-01-15/lik/likPoisson.m000644 000765 000024 00000014253 12077070411 020144 0ustar00carlstaff000000 000000 function [varargout] = likPoisson(kind, hyp, y, mu, s2, inf, i) % likPoisson - Poisson likelihood function for count data y. The expression for % the likelihood is % likPoisson(f) = mu^y * exp(-mu) / y! with mean=variance=mu % where mu = g(f) is the Poisson intensity, f is a % Gaussian process, y is the non-negative integer count data and % y! = gamma(y+1) its factorial. Hence, we have -- with Zy = gamma(y+1) = y! -- % llik(f) = log(likPoisson(f)) = log(g(f))*y - g(f) - log(Zy). % % We provide two kinds of intensities 'exp' and 'logistic': % For g(f) = exp(f), we have lik(f) = exp(f*y-exp(f)) / Zy. % For g(f) = log(1+exp(f))), we have lik(f) = log^y(1+exp(f)))(1+exp(f)) / Zy. % % Note that for both intensities g(f) the likelihood lik(f) is log concave. % % Several modes are provided, for computing likelihoods, derivatives and moments % respectively, see likFunctions.m for the details. In general, care is taken % to avoid numerical issues when the arguments are extreme. % % See also LIKFUNCTIONS.M. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2013-01-17 if nargin<4, varargout = {'0'}; return; end % report number of hyperparameters if nargin<6 % prediction mode if inf is not present if numel(y)==0, y = zeros(size(mu)); end s2zero = 1; if nargin>4, if norm(s2)>0, s2zero = 0; end, end % s2==0 ? if s2zero % log probability lg = g(mu,kind); lp = lg.*y - exp(lg) - gammaln(y+1); else lp = likPoisson(kind, hyp, y, mu, s2, 'infEP'); end ymu = {}; ys2 = {}; if nargout>1 % compute y moments by quadrature n = max([length(y),length(mu),length(s2)]); on = ones(n,1); N = 20; [t,w] = gauher(N); oN = ones(1,N); lw = ones(n,1)*log(w'); mu = mu(:).*on; sig = sqrt(s2(:)).*on; % vectors only ymu = exp(s(g(sig*t'+mu*oN,kind)+lw)); % exp(mu+s2/2) for kind='exp' if nargout>2 ys2 = ymu; % second y moment equals first end end varargout = {lp,ymu,ys2}; else switch inf case 'infLaplace' if nargin<7 % no derivative mode [lg,dlg,d2lg,d3lg] = g(mu,kind); elg = exp(lg); lp = lg.*y - elg - gammaln(y+1); dlp = {}; d2lp = {}; d3lp = {}; % return arguments if nargout>1 dlp = dlg.*(y-elg); % dlp, derivative of log likelihood if nargout>2 % d2lp, 2nd derivative of log likelihood d2lp = d2lg.*(y-elg) - dlg.*dlg.*elg; if nargout>3 % d3lp, 3rd derivative of log likelihood d3lp = d3lg.*(y-elg) - dlg.*(dlg.*dlg+3*d2lg).*elg; end end end varargout = {lp,dlp,d2lp,d3lp}; else % derivative mode varargout = {[],[],[]}; % derivative w.r.t. hypers end case 'infEP' if nargin<7 % no derivative mode % Since we are not aware of an analytical expression of the integral, % we use Gaussian-Hermite quadrature. % The following code is generic in the intensity g and relies on the % implementation of the infLaplace part. n = max([length(y),length(mu),length(s2)]); on = ones(n,1); N = 20; [t,w] = gauher(N); oN = ones(1,N); lw = ones(n,1)*log(w'); y = y(:).*on; mu = mu(:).*on; sig = sqrt(s2(:)).*on; % vectors only [lpi,dlpi,d2lpi] = likPoisson(kind,hyp,y*oN,sig*t'+mu*oN,[],'infLaplace'); lZ = s(lpi+lw); dlZ = {}; d2lZ = {}; if nargout>1 % 1st derivative wrt mean % Using p*dlp=dp, p=exp(lp), Z=sum_i wi*pi, dZ = sum_i wi*dpi we obtain % dlZ = sum_i exp(lpi-lZ+lwi)*dlpi = sum_i ai*dlpi. a = exp(lpi - lZ*oN + lw); dlZ = sum(a.*dlpi,2); if nargout>2 % 2nd derivative wrt mean % Using d2lZ=(d2Z*Z-dZ^2)/Z^2 <=> d2Z=Z*(d2lZ+dlZ^2) and % d2Z = sum_i wi*d2Zi, we get d2lZ = sum_i ai*(d2lpi+dlpi^2)-dlZ^2. d2lZ = sum(a.*(d2lpi+dlpi.*dlpi),2) - dlZ.*dlZ; end end varargout = {lZ,dlZ,d2lZ}; else % derivative mode varargout = {[]}; % deriv. wrt hyp.lik end case 'infVB' error('infVB not supported') end end % debug code % h = 1e-4; ff = [-1e6,-100,-15.01,-14.99,-10,-1,2,100,1e6]; % [lg,dlg,d2lg,d3lg] = g(ff,kind); [lg_h,dlg_h,d2lg_h] = g(ff+h,kind); % dlgn = (lg_h-lg)/h; d2lgn = (dlg_h-dlg)/h; d3lgn = (d2lg_h-d2lg)/h; % % [dlg(:),dlgn(:), d2lg(:),d2lgn(:), d3lg(:),d3lgn(:)] % keyboard % compute the log Poisson intensity function varargout = g(f,kind) varargout = cell(nargout, 1); % allocate the right number of output arguments if strcmp(kind,'exp') [varargout{:}] = g_exp(f); else [varargout{:}] = g_logistic(f); end % compute the log Poisson intensity for g(f) = exp(f) function [lg,dlg,d2lg,d3lg] = g_exp(f) lg = f; if nargout>1 dlg = ones(size(f)); if nargout>2 d2lg = zeros(size(f)); if nargout>2 d3lg = zeros(size(f)); end end end % compute the log Poisson intensity for g(f) = log(1+exp(f))) function [lg,dlg,d2lg,d3lg] = g_logistic(f) l1pef = max(0,f) + log(1+exp(-abs(f))); % safely compute log(1+exp(f)) lg = log(l1pef); id = f<-15; lg(id) = f(id); % fix log(log(1+exp(f))) limits if nargout>1 sm = 1./(1+exp(-f)); dlg = sm./l1pef; dlg(f<-15) = 1; if nargout>2 sp = 1./(1+exp(f)); d2lg = dlg.*(sp-dlg); if nargout>2 d3lg = d2lg.*(sp-2*dlg) - dlg.*sp.*sm; end end end % computes y = log( sum(exp(x),2) ), the softmax in a numerically safe way by % subtracting the row maximum to avoid cancelation after taking the exp % the sum is done along the rows function [y,x] = s(logx) N = size(logx,2); max_logx = max(logx,[],2); % we have all values in the log domain, and want to calculate a sum x = exp(logx-max_logx*ones(1,N)); y = log(sum(x,2)) + max_logx; gpml-matlab-v3.2-2013-01-15/lik/likLaplace.m000644 000765 000024 00000025620 12077070411 020053 0ustar00carlstaff000000 000000 function [varargout] = likLaplace(hyp, y, mu, s2, inf, i) % likLaplace - Laplacian likelihood function for regression. % The expression for the likelihood is % likLaplace(t) = exp(-|t-y|/b)/(2*b) with b = sn/sqrt(2), % where y is the mean and sn^2 is the variance. % % The hyperparameters are: % % hyp = [ log(sn) ] % % Several modes are provided, for computing likelihoods, derivatives and moments % respectively, see likFunctions.m for the details. In general, care is taken % to avoid numerical issues when the arguments are extreme. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2012-11-04. % % See also LIKFUNCTIONS.M. if nargin<3, varargout = {'1'}; return; end % report number of hyperparameters sn = exp(hyp); b = sn/sqrt(2); if nargin<5 % prediction mode if inf is not present if numel(y)==0, y = zeros(size(mu)); end s2zero = 1; if nargin>3, if norm(s2)>0, s2zero = 0; end, end % s2==0 ? if s2zero % log probability evaluation lp = -abs(y-mu)./b -log(2*b); s2 = 0; else % prediction lp = likLaplace(hyp, y, mu, s2, 'infEP'); end ymu = {}; ys2 = {}; if nargout>1 ymu = mu; % first y moment if nargout>2 ys2 = s2 + sn.^2; % second y moment end end varargout = {lp,ymu,ys2}; else % inference mode switch inf case 'infLaplace' if nargin<6 % no derivative mode if numel(y)==0, y=0; end ymmu = y-mu; dlp = {}; d2lp = {}; d3lp = {}; lp = -abs(ymmu)/b -log(2*b); if nargout>1 dlp = sign(ymmu)/b; % dlp, derivative of log likelihood if nargout>2 % d2lp, 2nd derivative of log likelihood d2lp = zeros(size(ymmu)); if nargout>3 % d3lp, 3rd derivative of log likelihood d3lp = zeros(size(ymmu)); end end end varargout = {lp,dlp,d2lp,d3lp}; else % derivative mode lp_dhyp = abs(y-mu)/b - 1; % derivative of log likelihood w.r.t. hypers dlp_dhyp = sign(mu-y)/b; % first derivative, d2lp_dhyp = zeros(size(mu)); % and also of the second mu derivative varargout = {lp_dhyp,dlp_dhyp,d2lp_dhyp}; end case 'infEP' n = max([numel(y),numel(mu),numel(s2),numel(sn)]); on = ones(n,1); y = y(:).*on; mu = mu(:).*on; s2 = s2(:).*on; sn = sn(:).*on; % vectors only fac = 1e3; % factor between the widths of the two distributions ... % ... from when one considered a delta peak, we use 3 orders of magnitude idlik = fac*sn1 % 1st moment % M1p = sqrt(tvar).*( zp.*exp(lM0p) - exp(-zp.*zp)/sqrt(pi) ); % M1m = sqrt(tvar).*( zm.*exp(lM0m) - exp(-zm.*zm)/sqrt(pi) ); % lZmax = max([vp-lZ(id),vm-lZ(id)],[],2); % m1m0 = exp(lZmax).*( exp(vp-lZ(id)-lZmax).*M1p ... % -exp(vm-lZ(id)-lZmax).*M1m )/2 + y(id); % dlZ(id) = (m1m0-mu(id))./s2(id); % if nargout>2 % 2nd moment % M2p = 2*tvar.*exp(vp-lZ(id)+lM0p)/2+(2*tvar+sqrt(2)*tmu) ... % .*M1p.*exp(vp-lZ(id)); % M2m = 2*tvar.*exp(vm-lZ(id)+lM0m)/2+(2*tvar-sqrt(2)*tmu) ... % .*M1m.*exp(vm-lZ(id)); % m2m0 = (M2p+M2m).*sn(id)/sqrt(8) + 2*y(id).*m1m0 - y(id).^2; % d2lZ(id) = (m2m0 - m1m0.^2)./s2(id).^2 - 1./s2(id); % end % end % an implementation based on logphi(t) = log(normcdf(t)) zp = (tmu+sqrt(2)*tvar)./sqrt(tvar); zm = (tmu-sqrt(2)*tvar)./sqrt(tvar); ap = logphi(-zp)+sqrt(2)*tmu; am = logphi( zm)-sqrt(2)*tmu; lZ(id) = logsum2exp([ap,am]) + tvar - log(sn(id)*sqrt(2)); if nargout>1 lqp = -zp.^2/2 - log(2*pi)/2 - logphi(-zp); % log( N(z)/Phi(z) ) lqm = -zm.^2/2 - log(2*pi)/2 - logphi( zm); dap = -exp(lqp-log(s2(id))/2) + sqrt(2)./sn(id); dam = exp(lqm-log(s2(id))/2) - sqrt(2)./sn(id); % ( exp(ap).*dap + exp(am).*dam )./( exp(ap) + exp(am) ) dlZ(id) = expABz_expAx([ap,am],[1;1],[dap,dam],[1;1]); if nargout>2 a = sqrt(8)./sn(id)./sqrt(s2(id)); bp = 2./sn(id).^2 - (a - zp./s2(id)).*exp(lqp); bm = 2./sn(id).^2 - (a + zm./s2(id)).*exp(lqm); % d2lZ(id) = ( exp(ap).*bp + exp(am).*bm )./( exp(ap) + exp(am) ) ... % - dlZ(id).^2; d2lZ(id) = expABz_expAx([ap,am],[1;1],[bp,bm],[1;1]) - dlZ(id).^2; end end end varargout = {lZ,dlZ,d2lZ}; else % derivative mode dlZhyp = zeros(n,1); if any(idlik) dlZhyp(idlik) = 0; end if any(idgau) dlZhyp(idgau) = ... likLaplace(log(sn(idgau)), mu(idgau), y(idgau), 'infLaplace', 1); end if any(id) % substitution to obtain unit variance, zero mean Laplacian tmu = (mu(id)-y(id))./sn(id); tvar = s2(id)./sn(id).^2; zp = (tvar+tmu/sqrt(2))./sqrt(tvar); vp = tvar+sqrt(2)*tmu; zm = (tvar-tmu/sqrt(2))./sqrt(tvar); vm = tvar-sqrt(2)*tmu; dzp = (-s2(id)./sn(id)+tmu.*sn(id)/sqrt(2)) ./ sqrt(s2(id)); dvp = -2*tvar - sqrt(2)*tmu; dzm = (-s2(id)./sn(id)-tmu.*sn(id)/sqrt(2)) ./ sqrt(s2(id)); dvm = -2*tvar + sqrt(2)*tmu; lezp = lerfc(zp); % ap = exp(vp).*ezp lezm = lerfc(zm); % am = exp(vm).*ezm vmax = max([vp+lezp,vm+lezm],[],2); % subtract max to avoid numerical pb ep = exp(vp+lezp-vmax); em = exp(vm+lezm-vmax); dap = ep.*(dvp - 2/sqrt(pi)*exp(-zp.^2-lezp).*dzp); dam = em.*(dvm - 2/sqrt(pi)*exp(-zm.^2-lezm).*dzm); dlZhyp(id) = (dap+dam)./(ep+em) - 1; end varargout = {dlZhyp}; % deriv. wrt hypers end case 'infVB' if nargin<6 % variational lower site bound % t(s) = exp(-sqrt(2)|y-s|/sn) / sqrt(2*sn²) % the bound has the form: b*s - s.^2/(2*ga) - h(ga)/2 with b=y/ga!! ga = s2; n = numel(ga); b = y./ga; y = y.*ones(n,1); db = -y./ga.^2; d2b = 2*y./ga.^3; h = 2*ga/sn^2 + log(2*sn^2) + y.^2./ga; dh = 2/sn^2 - y.^2./ga.^2; d2h = 2*y.^2./ga.^3; id = ga<0; h(id) = Inf; dh(id) = 0; d2h(id) = 0; % neg. var. treatment varargout = {h,b,dh,db,d2h,d2b}; else ga = s2; dhhyp = -4*ga/sn^2 + 2; dhhyp(ga<0) = 0; % negative variances get a special treatment varargout = {dhhyp}; % deriv. wrt hyp.lik end end end % 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 function [y,x] = logsum2exp(logx) N = size(logx,2); max_logx = max(logx,[],2); % we have all values in the log domain, and want to calculate a sum x = exp(logx-max_logx*ones(1,N)); y = log(sum(x,2)) + max_logx; % numerically safe implementation of f(t) = log(1-erf(t)) = log(erfc(t)) function f = lerfc(t) f = zeros(size(t)); tmin = 20; tmax = 25; ok = ttmax; % evaluate tight bound interp = ~ok & ~bd; % linearly interpolate between both of them f(~ok) = log(2/sqrt(pi)) -t(~ok).^2 -log(t(~ok)+sqrt( t(~ok).^2+4/pi )); lam = 1./(1+exp( 12*(1/2-(t(interp)-tmin)/(tmax-tmin)) )); % interp. weights f(interp) = lam.*f(interp) + (1-lam).*log(erfc( t(interp) )); f(ok) = f(ok) + log(erfc( t(ok) )); % safe eval % 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. function y = expABz_expAx(A,x,B,z) N = size(A,2); maxA = max(A,[],2); % number of columns, max over columns A = A-maxA*ones(1,N); % subtract maximum value y = ( (exp(A).*B)*z ) ./ ( exp(A)*x ); % safe implementation of the log of phi(x) = \int_{-\infty}^x N(f|0,1) df % logphi(z) = log(normcdf(z)) function lp = logphi(z) lp = zeros(size(z)); % allocate memory zmin = -6.2; zmax = -5.5; ok = z>zmax; % safe evaluation for large values bd = z3, if norm(s2)>0, s2zero = 0; end, end % s2==0 ? if s2zero % log probability lp = -(y-mu).^2./sn2/2-log(2*pi*sn2)/2; s2 = 0; else lp = likGauss(hyp, y, mu, s2, 'infEP'); % prediction end ymu = {}; ys2 = {}; if nargout>1 ymu = mu; % first y moment if nargout>2 ys2 = s2 + sn2; % second y moment end end varargout = {lp,ymu,ys2}; else switch inf case 'infLaplace' if nargin<6 % no derivative mode if numel(y)==0, y=0; end ymmu = y-mu; dlp = {}; d2lp = {}; d3lp = {}; lp = -ymmu.^2/(2*sn2) - log(2*pi*sn2)/2; if nargout>1 dlp = ymmu/sn2; % dlp, derivative of log likelihood if nargout>2 % d2lp, 2nd derivative of log likelihood d2lp = -ones(size(ymmu))/sn2; if nargout>3 % d3lp, 3rd derivative of log likelihood d3lp = zeros(size(ymmu)); end end end varargout = {lp,dlp,d2lp,d3lp}; 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*ones(size(mu))/sn2; % and also of the second mu derivative varargout = {lp_dhyp,dlp_dhyp,d2lp_dhyp}; end case 'infEP' if nargin<6 % no derivative mode lZ = -(y-mu).^2./(sn2+s2)/2 - log(2*pi*(sn2+s2))/2; % log part function dlZ = {}; d2lZ = {}; 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 end end varargout = {lZ,dlZ,d2lZ}; else % derivative mode dlZhyp = ((y-mu).^2./(sn2+s2)-1) ./ (1+s2./sn2); % deriv. w.r.t. hyp.lik varargout = {dlZhyp}; end case 'infVB' if nargin<6 % 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 = numel(ga); b = y./ga; y = y.*ones(n,1); db = -y./ga.^2; d2b = 2*y./ga.^3; h = 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) + log(2*pi*sn2); h(~id) = Inf; dh(id) = -y(id).^2./ga(id).^2; d2h(id) = 2*y(id).^2./ga(id).^3; id = ga<0; h(id) = Inf; dh(id) = 0; d2h(id) = 0; % neg. var. treatment varargout = {h,b,dh,db,d2h,d2b}; else ga = s2; n = numel(ga); dhhyp = 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 end end end gpml-matlab-v3.2-2013-01-15/lik/likErf.m000644 000765 000024 00000013505 12077070411 017225 0ustar00carlstaff000000 000000 function [varargout] = likErf(hyp, y, mu, s2, inf, i) % likErf - Error function or cumulative Gaussian likelihood function for binary % classification or probit regression. The expression for the likelihood is % likErf(t) = (1+erf(t/sqrt(2)))/2 = normcdf(t). % % Several modes are provided, for computing likelihoods, derivatives and moments % respectively, see likFunctions.m for the details. In general, care is taken % to avoid numerical issues when the arguments are extreme. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-07-22. % % See also LIKFUNCTIONS.M. if nargin<3, varargout = {'0'}; return; end % report number of hyperparameters if nargin>1, y = sign(y); y(y==0) = 1; else y = 1; end % allow only +/- 1 values if numel(y)==0, y = 1; end if nargin<5 % prediction mode if inf is not present y = y.*ones(size(mu)); % make y a vector s2zero = 1; if nargin>3, if norm(s2)>0, s2zero = 0; end, end % s2==0 ? if s2zero % log probability evaluation [p,lp] = cumGauss(y,mu); else % prediction lp = likErf(hyp, y, mu, s2, 'infEP'); p = exp(lp); end ymu = {}; ys2 = {}; if nargout>1 ymu = 2*p-1; % first y moment if nargout>2 ys2 = 4*p.*(1-p); % second y moment end end varargout = {lp,ymu,ys2}; else % inference mode switch inf case 'infLaplace' if nargin<6 % no derivative mode f = mu; yf = y.*f; % product latents and labels dlp = {}; d2lp = {}; d3lp = {}; % return arguments [p,lp] = cumGauss(y,f); if nargout>1 % derivative of log likelihood n_p = 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; end end end varargout = {lp,dlp,d2lp,d3lp}; else % derivative mode varargout = {[],[],[]}; % derivative w.r.t. hypers end case 'infEP' if nargin<6 % no derivative mode z = mu./sqrt(1+s2); dlZ = {}; d2lZ = {}; [junk,lZ] = cumGauss(y,z); % log part function if numel(y)>0, z=z.*y; end if nargout>1 if numel(y)==0, y=1; end n_p = gauOverCumGauss(z,exp(lZ)); dlZ = y.*n_p./sqrt(1+s2); % 1st derivative wrt mean if nargout>2 d2lZ = -n_p.*(z+n_p)./(1+s2); % 2nd derivative wrt mean end end varargout = {lZ,dlZ,d2lZ}; else % derivative mode varargout = {[]}; % deriv. wrt hyp.lik end case 'infVB' if nargin<6 % no derivative mode % naive variational lower bound based on asymptotical properties of lik % normcdf(t) -> -(t²-2dt+c)/2 for t->-oo (tight lower bound) d = 0.158482605320942; c = -1.785873318175113; ga = s2; n = numel(ga); b = d*y.*ones(n,1); db = zeros(n,1); d2b = db; h = -2*c*ones(n,1); h(ga>1) = Inf; dh = zeros(n,1); d2h = dh; varargout = {h,b,dh,db,d2h,d2b}; else % derivative mode varargout = {[]}; % deriv. wrt hyp.lik end end end function [p,lp] = cumGauss(y,f) if numel(y)>0, yf = y.*f; else yf = f; end % product of latents and labels p = (1+erf(yf/sqrt(2)))/2; % likelihood if nargout>1, lp = logphi(yf,p); end % log likelihood % safe implementation of the log of phi(x) = \int_{-\infty}^x N(f|0,1) df % logphi(z) = log(normcdf(z)) function lp = logphi(z,p) lp = zeros(size(z)); % allocate memory zmin = -6.2; zmax = -5.5; ok = z>zmax; % safe evaluation for large values bd = z-5; % naive evaluation for large values of f n_p(ok) = (exp(-f(ok).^2/2)/sqrt(2*pi)) ./ p(ok); bd = f<-6; % tight upper bound evaluation n_p(bd) = sqrt(f(bd).^2/4+1)-f(bd)/2; interp = ~ok & ~bd; % linearly interpolate between both of them tmp = f(interp); lam = -5-f(interp); n_p(interp) = (1-lam).*(exp(-tmp.^2/2)/sqrt(2*pi))./p(interp) + ... lam .*(sqrt(tmp.^2/4+1)-tmp/2); gpml-matlab-v3.2-2013-01-15/lik/likLogistic.m000644 000765 000024 00000017765 12077070411 020302 0ustar00carlstaff000000 000000 function [varargout] = likLogistic(hyp, y, mu, s2, inf, i) % likLogistic - logistic function for binary classification or logit regression. % The expression for the likelihood is % likLogistic(t) = 1./(1+exp(-t)). % % Several modes are provided, for computing likelihoods, derivatives and moments % respectively, see likFunctions.m for the details. In general, care is taken % to avoid numerical issues when the arguments are extreme. The moments % \int f^k likLogistic(y,f) N(f|mu,var) df are calculated via a cumulative % Gaussian scale mixture approximation. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-07-22. % % See also LIKFUNCTIONS.M. if nargin<3, varargout = {'0'}; return; end % report number of hyperparameters if nargin>1, y = sign(y); y(y==0) = 1; else y = 1; end % allow only +/- 1 values if numel(y)==0, y = 1; end if nargin<5 % prediction mode if inf is not present y = y.*ones(size(mu)); % make y a vector s2zero = 1; if nargin>3, if norm(s2)>0, s2zero = 0; end, end % s2==0 ? if s2zero % log probability evaluation yf = y.*mu; % product latents and labels lp = yf; ok = -351 p = exp(lp); ymu = 2*p-1; % first y moment if nargout>2 ys2 = 4*p.*(1-p); % second y moment end end varargout = {lp,ymu,ys2}; else % inference mode switch inf case 'infLaplace' if nargin<6 % no derivative mode f = mu; yf = y.*f; s = -yf; % product latents and labels dlp = {}; d2lp = {}; d3lp = {}; % return arguments ps = max(0,s); lp = -(ps+log(exp(-ps)+exp(s-ps))); % lp = -(log(1+exp(s))) if nargout>1 % first derivatives s = min(0,f); p = exp(s)./(exp(s)+exp(s-f)); % p = 1./(1+exp(-f)) dlp = (y+1)/2-p; % derivative of log likelihood if nargout>2 % 2nd derivative of log likelihood d2lp = -exp(2*s-f)./(exp(s)+exp(s-f)).^2; if nargout>3 % 3rd derivative of log likelihood d3lp = 2*d2lp.*(0.5-p); end end end varargout = {lp,dlp,d2lp,d3lp}; else % derivative mode varargout = {[],[],[]}; % derivative w.r.t. hypers end case 'infEP' if nargin<6 % no derivative mode y = y.*ones(size(mu)); % make y a vector % likLogistic(t) \approx 1/2 + \sum_{i=1}^5 (c_i/2) erf(lam_i/sqrt(2)t) lam = sqrt(2)*[0.44 0.41 0.40 0.39 0.36]; % approx coeffs lam_i and c_i c = [1.146480988574439e+02; -1.508871030070582e+03; 2.676085036831241e+03; -1.356294962039222e+03; 7.543285642111850e+01 ]; [lZc,dlZc,d2lZc] = likErf([], y*ones(1,5), mu*lam, s2*(lam.^2), inf); lZ = log_expA_x(lZc,c); % A=lZc, B=dlZc, d=c.*lam', lZ=log(exp(A)*c) dlZ = expABz_expAx(lZc, c, dlZc, c.*lam'); % ((exp(A).*B)*d)./(exp(A)*c) % d2lZ = ((exp(A).*Z)*e)./(exp(A)*c) - dlZ.^2 where e = c.*(lam.^2)' d2lZ = expABz_expAx(lZc, c, dlZc.^2+d2lZc, c.*(lam.^2)') - dlZ.^2; % The scale mixture approximation does not capture the correct asymptotic % behavior; we have linear decay instead of quadratic decay as suggested % by the scale mixture approximation. By observing that for large values % of -f*y ln(p(y|f)) of likLogistic is linear in f with slope y, we are % able to analytically integrate the tail region; there is no contribution % to the second derivative val = abs(mu)-196/200*s2-4; % empirically determined bound at val==0 lam = 1./(1+exp(-10*val)); % interpolation weights lZtail = min(s2/2-abs(mu),-.1); % apply the same to p(y|f) = 1 - p(-y|f) dlZtail = -sign(mu); id = y.*mu>0; lZtail(id) = log(1-exp(lZtail(id))); % label and mean agree dlZtail(id) = 0; lZ = (1-lam).* lZ + lam.* lZtail; % interpolate between scale mixture .. dlZ = (1-lam).*dlZ + lam.*dlZtail; % .. and tail approximation varargout = {lZ,dlZ,d2lZ}; else % derivative mode varargout = {[]}; % deriv. wrt hyp.lik end case 'infVB' if nargin<6 % no derivative mode % variational lower site bound % using -log(1+exp(-s)) = s/2 -log( 2*cosh(s/2) ); % the bound has the form: b*s - s.^2/(2*ga) - h(ga)/2 with b=1/2 ga = s2; n = numel(ga); b = (y/2).*ones(n,1); db = zeros(n,1); d2b = db; h = zeros(n,1); dh = h; d2h = h; % allocate memory for return argument idlo = ga(:)<=4; h(idlo) = 2*log(2); % constant value below 4 idup = 50<=ga(:); zn = zeros(n,1); % linear behavior for large gammas h(idup) = ga(idup)/4; dh(idup) = zn(idup)+1/4; d2h(idup) = zn(idup); id = ~idlo & ~idup; % interesting zone is in between [g,dg,d2g] = inv_xcothx(1/4*ga(id)); thg = tanh(g); h(id) = 2*logcosh(g) +2*log(2) -g.*thg; g1mt2 = g.*(1-thg.^2); dh(id) = dg.*( thg - g1mt2 )/4; d2h(id) = ( g1mt2.*(2*dg.^2.*thg-d2g) + d2g.*thg )/16; varargout = {h,b,dh,db,d2h,d2b}; else % derivative mode varargout = {[]}; % deriv. wrt hyp.lik end end end % numerically safe version of log(cosh(x)) = log(exp(x)+exp(-x))-log(2) function f = logcosh(x) f = abs(x) + log(1+exp(-2*abs(x))) - log(2); % Invert f(x) = x.*coth(x) = y, return the positive value % uses Newton's method to minimize (y-f(x))² w.r.t. x function [x,dx,d2x] = inv_xcothx(y) % f = @(x) x.*coth(x); % df = @(x) x + (1-f(x)).*coth(x); % d2f = @(x) 2*(1-f(x)).*(1-coth(x).^2); if numel(y)==0, x=[]; dx=[]; d2x=[]; return, end x = sqrt(y.^2-1); i = 1; % init ep = eps./(1+abs(x)); th=tanh(x); fx=(x+ep)./(th+ep); % function r = fx-y; % init while i==1 || (i<10 && max(abs(r))>1e-12) dfx = x + (1-fx)./(th+ep); % first derivative d2fx = 2*( 1-fx + (x-th+ep/3)./(th.*th.*th+ep) ); % second derivative x = x - r.*dfx./(dfx.^2+r.*d2fx+ep); % Newton step ep = eps./(1+abs(x)); th=tanh(x); fx=(x+ep)./(th+ep); r = fx-y; i = i+1; end % first derivative dx/dy % derivatives of inverse functions are reciprocals dx = 1./dfx; % second derivative d2x/dy2 % quotient rule and chaine rule d2x = -d2fx.*dx./dfx./dfx; % computes y = log( exp(A)*x ) in a numerically safe way by subtracting the % maximal value in each row to avoid cancelation after taking the exp function y = log_expA_x(A,x) N = size(A,2); maxA = max(A,[],2); % number of columns, max over columns y = log(exp(A-maxA*ones(1,N))*x) + maxA; % exp(A) = exp(A-max(A))*exp(max(A)) % 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. function y = expABz_expAx(A,x,B,z) N = size(A,2); maxA = max(A,[],2); % number of columns, max over columns A = A-maxA*ones(1,N); % subtract maximum value y = ( (exp(A).*B)*z ) ./ ( exp(A)*x ); gpml-matlab-v3.2-2013-01-15/inf/infEP.m000644 000765 000024 00000013514 12046775407 017024 0ustar00carlstaff000000 000000 function [post nlZ dnlZ] = infEP(hyp, mean, cov, lik, x, y) % Expectation Propagation approximation to the posterior Gaussian Process. % The function takes a specified covariance function (see covFunction.m) and % likelihood function (see likFunction.m), and is designed to be used with % gp.m. See also infFunctions.m. In the EP algorithm, the sites are % updated in random order, for better performance when cases are ordered % according to the targets. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch 2012-11-07. % % See also INFMETHODS.M. persistent last_ttau last_tnu % keep tilde parameters between calls tol = 1e-4; max_sweep = 10; min_sweep = 2; % tolerance to stop EP iterations inf = 'infEP'; n = size(x,1); K = feval(cov{:}, hyp.cov, x); % evaluate the covariance matrix m = feval(mean{:}, hyp.mean, x); % evaluate the mean vector % 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 = -sum(feval(lik{:}, hyp.lik, y, m, diag(K), inf)); if any(size(last_ttau) ~= [n 1]) % find starting point for tilde parameters ttau = zeros(n,1); % initialize to zero if we have no better guess tnu = zeros(n,1); Sigma = K; % initialize Sigma and mu, the parameters of .. mu = zeros(n,1); % .. the Gaussian posterior approximation nlZ = nlZ0; else ttau = last_ttau; % try the tilde values from previous call tnu = last_tnu; [Sigma, mu, nlZ, L] = epComputeParams(K, y, ttau, tnu, lik, hyp, m, inf); if nlZ > nlZ0 % if zero is better .. ttau = zeros(n,1); % .. then initialize with zero instead tnu = zeros(n,1); Sigma = K; % initialize Sigma and mu, the parameters of .. mu = zeros(n,1); % .. the Gaussian posterior approximation nlZ = nlZ0; end end nlZ_old = Inf; sweep = 0; % converged, max. sweeps or min. sweeps? while (abs(nlZ-nlZ_old) > tol && sweep < max_sweep) || sweep2 % do we want derivatives? dnlZ = hyp; % allocate space for derivatives V = L'\(repmat(sW,1,n).*K); Sigma = K - V'*V; mu = Sigma*tnu; tau_n = 1./diag(Sigma)-ttau; % compute the log marginal likelihood nu_n = mu./diag(Sigma)-tnu+m.*tau_n; % vectors of cavity parameters F = alpha*alpha'-repmat(sW,1,n).*solve_chol(L,diag(sW)); % covariance hypers for i=1:length(hyp.cov) dK = feval(cov{:}, hyp.cov, x, [], i); dnlZ.cov(i) = -sum(sum(F.*dK))/2; end for i = 1:numel(hyp.lik) % likelihood hypers dlik = feval(lik{:}, hyp.lik, y, nu_n./tau_n, 1./tau_n, inf, i); dnlZ.lik(i) = -sum(dlik); end [junk,dlZ] = feval(lik{:}, hyp.lik, y, nu_n./tau_n, 1./tau_n, inf);% mean hyps for i = 1:numel(hyp.mean) dm = feval(mean{:}, hyp.mean, x, i); dnlZ.mean(i) = -dlZ'*dm; end end % function to compute the parameters of the Gaussian approximation, Sigma and % mu, and the negative log marginal likelihood, nlZ, from the current site % parameters, ttau and tnu. Also returns L (useful for predictions). function [Sigma,mu,nlZ,L] = epComputeParams(K, y, ttau, tnu, lik, hyp, m, inf) n = length(y); % number of training cases sW = sqrt(ttau); % compute Sigma and mu L = chol(eye(n)+sW*sW'.*K); % L'*L=B=eye(n)+sW*K*sW V = L'\(repmat(sW,1,n).*K); Sigma = K - V'*V; mu = Sigma*tnu; 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 = feval(lik{:}, hyp.lik, y, nu_n./tau_n, 1./tau_n, inf); nlZ = sum(log(diag(L))) -sum(lZ) -tnu'*Sigma*tnu/2 ... -(nu_n-m.*tau_n)'*((ttau./tau_n.*(nu_n-m.*tau_n)-2*tnu)./(ttau+tau_n))/2 ... +sum(tnu.^2./(tau_n+ttau))/2-sum(log(1+ttau./tau_n))/2; gpml-matlab-v3.2-2013-01-15/inf/infFITC.m000644 000765 000024 00000010015 12077070414 017224 0ustar00carlstaff000000 000000 function [post nlZ dnlZ] = infFITC(hyp, mean, cov, lik, x, y) % 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). % We fixed the standard deviation of the inducing inputs snu to be a one per mil % of the measurement noise's standard deviation sn. % The implementation exploits the Woodbury matrix identity % inv(Kt) = inv(G) - inv(G)*V'*inv(eye(nu)+V*inv(G)*V')*V*inv(G) % in order to be applicable to large datasets. The computational complexity % is O(n nu^2) where n is the number of data points x and nu the number of % inducing inputs in xu. % The function takes a specified covariance function (see covFunction.m) and % likelihood function (see likFunction.m), and is designed to be used with % gp.m and in conjunction with covFITC and likGauss. % % Copyright (c) by Ed Snelson, Carl Edward Rasmussen % and Hannes Nickisch, 2012-11-20. % % See also INFMETHODS.M, COVFITC.M. if iscell(lik), likstr = lik{1}; else likstr = lik; end if ~ischar(likstr), likstr = func2str(likstr); end if ~strcmp(likstr,'likGauss') % NOTE: no explicit call to likGauss error('Inference with inFITC only possible with Gaussian likelihood.'); end cov1 = cov{1}; if isa(cov1, 'function_handle'), cov1 = func2str(cov1); end if ~strcmp(cov1,'covFITC'); error('Only covFITC supported.'), end % check cov [diagK,Kuu,Ku] = feval(cov{:}, hyp.cov, x); % evaluate covariance matrix m = feval(mean{:}, hyp.mean, x); % evaluate mean vector [n, D] = size(x); nu = size(Kuu,1); sn2 = exp(2*hyp.lik); % noise variance of likGauss snu2 = 1e-6*sn2; % hard coded inducing inputs noise Luu = chol(Kuu+snu2*eye(nu)); % Kuu + snu2*I = Luu'*Luu V = Luu'\Ku; % V = inv(Luu')*Ku => V'*V = Q g_sn2 = diagK + sn2 - sum(V.*V,1)'; % g + sn2 = diag(K) + sn2 - diag(Q) Lu = chol(eye(nu) + (V./repmat(g_sn2',nu,1))*V'); % Lu'*Lu=I+V*diag(1/g_sn2)*V' r = (y-m)./sqrt(g_sn2); be = Lu'\(V*(r./sqrt(g_sn2))); iKuu = solve_chol(Luu,eye(nu)); % inv(Kuu + snu2*I) = iKuu post.alpha = Luu\(Lu\be); % return the posterior parameters post.L = solve_chol(Lu*Luu,eye(nu)) - iKuu; % Sigma-inv(Kuu) post.sW = ones(n,1)/sqrt(sn2); % unused for FITC prediction with gp.m if nargout>1 % do we want the marginal likelihood nlZ = sum(log(diag(Lu))) + (sum(log(g_sn2)) + n*log(2*pi) + r'*r - be'*be)/2; if nargout>2 % do we want derivatives? dnlZ = hyp; % allocate space for derivatives al = r./sqrt(g_sn2) - (V'*(Lu\be))./g_sn2; % al = (Kt+sn2*eye(n))\y B = iKuu*Ku; w = B*al; W = Lu'\(V./repmat(g_sn2',nu,1)); for i = 1:numel(hyp.cov) [ddiagKi,dKuui,dKui] = feval(cov{:}, hyp.cov, x, [], i); % eval cov deriv R = 2*dKui-dKuui*B; v = ddiagKi - sum(R.*B,1)'; % diag part of cov deriv dnlZ.cov(i) = (ddiagKi'*(1./g_sn2)+w'*(dKuui*w-2*(dKui*al))-al'*(v.*al)... - sum(W.*W,1)*v - sum(sum((R*W').*(B*W'))) )/2; end dnlZ.lik = sn2*(sum(1./g_sn2) -sum(sum(W.*W,1)) -al'*al); % since snu2 is a fixed fraction of sn2, there is a covariance-like term in % the derivative as well dKuui = 2*snu2; R = -dKuui*B; v = -sum(R.*B,1)'; % diag part of cov deriv dnlZ.lik = dnlZ.lik + (w'*dKuui*w -al'*(v.*al)... - sum(W.*W,1)*v - sum(sum((R*W').*(B*W'))) )/2; for i = 1:numel(hyp.mean) dnlZ.mean(i) = -feval(mean{:}, hyp.mean, x, i)'*al; end end end gpml-matlab-v3.2-2013-01-15/inf/infVB.m000644 000765 000024 00000013646 12046775407 017035 0ustar00carlstaff000000 000000 function [post, nlZ, dnlZ] = infVB(hyp, mean, cov, lik, x, y) % Variational approximation to the posterior Gaussian process with MKL % covariance function hyperparameter optimisation. % The function takes a likelihood function (see likFunction.m), and is designed % to be used with gp.m. See also infFunctions.m. % % Minimisation of an upper bound on the negative marginal likelihood % \int N(f|0,K) p(y|f) df \ge Psi(ga,theta) with hyperparameters % theta=cov.hyp and likelihood lik(y,f) = p(y|f). % % Psi(ga,theta) = nlZ = % = (ln|K+ga|-ln|ga| + h(ga) - b'*inv(inv(K)+inv(ga)))*b)/2 % = (ln|A| + ln|K| + h(ga) - b'*inv(A)*b)/2 with A = inv(K)+inv(ga) % % The map (ga,K) |-> Psi(ga,theta) - ln|K| is jointly convex and ln|K| is % concave in its linear parameters theta. % % We optimise the convex Psi(ga,theta)-ln|K|+z'*theta/2 + B(t,theta) criterion % in the outer loop using the barrier function B(t,theta) = -sum(log(theta))/t % to enforce positivity w.r.t. theta. % The linear term z'*theta/2 is an upper bound on the concave ln|K| term. % We use an interleaved optimisation with inner loops doing Newton steps in ga % and an outer loop doing joint Newton updates in (ga,theta). Line searches are % done using derivative-free 'Brent's minimum' search. % % Copyright (c) by Hannes Nickisch 2012-11-07. % % See also INFMETHODS.M. maxitinner = 20; % number of inner Newton steps in ga % some less important parameters ep = 1e-8; % small constant for the ridge added to the stab. Newton direction smax = 5; Nline = 15; thr = 1e-4; % line search parameters tol = 1e-7; % tolerance for when to stop the Newton iterations inf = 'infVB'; [n,D] = size(x); K = feval(cov{:}, hyp.cov, x); % evaluate the covariance matrix m = feval(mean{:}, hyp.mean, x); % evaluate the mean vector if iscell(lik), likstr = lik{1}; else likstr = lik; end if ~ischar(likstr), likstr = func2str(likstr); end if (norm(m)>1e-10 || numel(hyp.mean)>0) ... && (strcmp(likstr,'likErf')||strcmp(likstr,'likLogistic')) error('only meanZero implemented for classification') end y = y-m; % no we have either zero mean or non-classification likelihoods if strcmp(likstr,'likGauss') ga = exp(2*hyp.lik)*ones(n,1); % best ga is known for Gaussian likelihood elseif strcmp(likstr,'likErf') ga = ones(n,1); % use a fixed ga for errf likelihood due to asymptotics else % INNER compute the Newton direction of ga ga = ones(n,1); % initial values itinner = 0; nlZ_new = 1e100; nlZ_old = Inf; % make sure while loop starts while nlZ_old-nlZ_new>tol && itinner0 Psi_s = @(s,ddga,ga,K,inf,hyp,lik,y) Psi(ga+s*ddga,K,inf,hyp,lik,y); [s,nlZ_new] = brentmin(0,s,Nline,thr, Psi_s,0,ddga,ga,K,inf,hyp,lik,y); ga = abs(ga + s*ddga); % update variational parameters end end [nlZ,dnlZ,d2nlZ,b] = Psi(ga,K,inf,hyp,lik,y); % upp bd on neg log marg lik W = 1./ga; sW = sqrt(W); % return the posterior parameters L = chol(eye(n)+sW*sW'.*K); % L'*L = B =eye(n)+sW*K*sW iKtil = repmat(sW,1,n).*solve_chol(L,diag(sW)); % sW*B^-1*sW=inv(K+inv(W)) alpha = b - iKtil*(K*b); post.alpha = alpha; post.sW = sW; post.L = L; if nargout>2 % do we want derivatives? dnlZ = hyp; % allocate space for derivatives for j=1:length(hyp.cov) % covariance hypers dK = feval(cov{:}, hyp.cov, x, [], j); if j==1, v = iKtil*(b./W); end dnlZ.cov(j) = sum(sum(iKtil.*dK))/2 - (v'*dK*v)/2; % implicit derivative = 0 end if ~strcmp(likstr,'likGauss') % likelihood hypers for j=1:length(hyp.lik) dhhyp = feval(lik{:},hyp.lik,y,[],ga,inf,j); dnlZ.lik(j) = sum(dhhyp)/2; % implicit derivative = 0 end else % special treatment for the Gaussian case dnlZ.lik = sum(sum( (L'\eye(n)).^2 )) - exp(2*hyp.lik)*(alpha'*alpha); end for j=1:length(hyp.mean) % mean hypers dm = feval(mean{:}, hyp.mean, x, j); dnlZ.mean(j) = -alpha'*dm; % implicit derivative = 0 end end % variational lower bound along with derivatives w.r.t. ga and theta=hyp.cov % psi = (ln|K+ga|-ln|ga| + h(ga) - b'*inv(inv(K)+inv(ga)))*b)/2 % = (ln|A| + ln|K| + h(ga) - b'*inv(A)*b)/2 with A = inv(K)+inv(ga) % the code is numerically stable function [nlZ,dga,d2ga,b] = Psi(ga,K,inf,hyp,lik,y) n = size(K,1); [h,b,dh,db,d2h,d2b] = feval(lik{:},hyp.lik,y,[],ga,inf); W = 1./ga; sW = sqrt(W); L = chol(eye(n)+sW*sW'.*K); % sum(log(diag(L))) = log|B|/2 C = L'\(repmat(sW,1,n).*K); t = C*b; % t'*t-b'*K*b = -b'*inv(inv(K)+diag(1./ga))*b nlZ = sum(log(diag(L))) + ( sum(h) + t'*t-b'*K*b )/2; if nargout>1 % Hessian w.r.t. variational parameters gamma iKtil = repmat(sW,1,n).*solve_chol(L,diag(sW)); % sW*inv(B)*sW=inv(K+inv(W)) Khat = K-C'*C; v = Khat*b; % K-K*sW*inv(B)*sW*K=inv(inv(K)+W) dga = ( diag(iKtil)-1./ga + dh - (v./ga).^2 - 2*v.*db )/2; if nargout>2 % gradient w.r.t. variational parameters gamma w = v./ga.^2; d2ga = ( -iKtil.^2+diag(1./ga.^2) + diag(d2h) )/2 ... -Khat.*(w*(w+2*db)') -Khat.*(db*db') +diag(w.^2.*ga)-diag(v.*d2b); d2ga = (d2ga+d2ga')/2; % symmetrise end end gpml-matlab-v3.2-2013-01-15/inf/infLaplace.m000644 000765 000024 00000015722 12047277603 020060 0ustar00carlstaff000000 000000 function [post nlZ dnlZ] = infLaplace(hyp, mean, cov, lik, x, y) % Laplace approximation to the posterior Gaussian process. % The function takes a specified covariance function (see covFunction.m) and % likelihood function (see likFunction.m), and is designed to be used with % gp.m. See also infFunctions.m. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch 2012-11-09. % % See also INFMETHODS.M. persistent last_alpha % copy of the last alpha if any(isnan(last_alpha)), last_alpha = zeros(size(last_alpha)); end % prevent tol = 1e-6; % tolerance for when to stop the Newton iterations smax = 2; Nline = 10; thr = 1e-4; % line search parameters maxit = 20; % max number of Newton steps in f inf = 'infLaplace'; n = size(x,1); K = feval(cov{:}, hyp.cov, x); % evaluate the covariance matrix m = feval(mean{:}, hyp.mean, x); % evaluate the mean vector Psi_old = Inf; % make sure while loop starts by the largest old objective val if any(size(last_alpha)~=[n,1]) % find a good starting point for alpha and f alpha = zeros(n,1); f = K*alpha+m; % start at mean if sizes do not match [lp,dlp,d2lp] = feval(lik{:},hyp.lik,y,f,[],inf); W = -d2lp; Psi_new=-sum(lp); else alpha = last_alpha; f = K*alpha+m; % try last one [lp,dlp,d2lp] = feval(lik{:},hyp.lik,y,f,[],inf); W = -d2lp; Psi_new = alpha'*(f-m)/2 - sum(lp); % objective for last alpha Psi_def = -sum(feval(lik{:},hyp.lik,y,m,[],inf));% objective for def init f==m if Psi_def < Psi_new % if default is better, we use it alpha = zeros(n,1); f = K*alpha+m; [lp,dlp,d2lp] = feval(lik{:},hyp.lik,y,f,[],inf); W=-d2lp; Psi_new=-sum(lp); end end isWneg = 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 && it2 % do we want derivatives? dnlZ = hyp; % allocate space for derivatives if isWneg % switch between Cholesky and LU decomposition mode Z = -post.L; % inv(K+inv(W)) g = sum(iA.*K,2)/2; % deriv. of ln|B| wrt W; g = diag(inv(inv(K)+diag(W)))/2 else Z = repmat(sW,1,n).*solve_chol(post.L,diag(sW)); %sW*inv(B)*sW=inv(K+inv(W)) C = post.L'\(repmat(sW,1,n).*K); % deriv. of ln|B| wrt W g = (diag(K)-sum(C.^2,1)')/2; % g = diag(inv(inv(K)+W))/2 end dfhat = g.*d3lp; % deriv. of nlZ wrt. fhat: dfhat=diag(inv(inv(K)+W)).*d3lp/2 for i=1:length(hyp.cov) % covariance hypers dK = feval(cov{:}, hyp.cov, x, [], i); dnlZ.cov(i) = sum(sum(Z.*dK))/2 - alpha'*dK*alpha/2; % explicit part b = dK*dlp; % b-K*(Z*b) = inv(eye(n)+K*diag(W))*b dnlZ.cov(i) = dnlZ.cov(i) - dfhat'*( b-K*(Z*b) ); % implicit part end for i=1:length(hyp.lik) % likelihood hypers [lp_dhyp,dlp_dhyp,d2lp_dhyp] = feval(lik{:},hyp.lik,y,f,[],inf,i); dnlZ.lik(i) = -g'*d2lp_dhyp - sum(lp_dhyp); % explicit part b = K*dlp_dhyp; % b-K*(Z*b) = inv(eye(n)+K*diag(W))*b dnlZ.lik(i) = dnlZ.lik(i) - dfhat'*( b-K*(Z*b) ); % implicit part end for i=1:length(hyp.mean) % mean hypers dm = feval(mean{:}, hyp.mean, x, i); dnlZ.mean(i) = -alpha'*dm; % explicit part dnlZ.mean(i) = dnlZ.mean(i) - dfhat'*(dm-K*(Z*dm)); % implicit part end end % criterion Psi at alpha + s*dalpha for line search function [Psi,alpha,f,dlp,W] = Psi_line(s,dalpha,alpha,hyp,K,m,lik,y,inf) alpha = alpha + s*dalpha; f = K*alpha+m; [lp,dlp,d2lp] = feval(lik{:},hyp.lik,y,f,[],inf); W = -d2lp; Psi = alpha'*(f-m)/2 - sum(lp); % 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). function [ldA,iA,mwiA] = logdetA(K,w) [m,n] = size(K); if m~=n, error('K has to be nxn'), end A = eye(n)+K.*repmat(w',n,1); [L,U,P] = lu(A); u = diag(U); % compute LU decomposition, A = P'*L*U signU = prod(sign(u)); % sign of U detP = 1; % compute sign (and det) of the permutation matrix P p = P*(1:n)'; for i=1:n % swap entries if i~=p(i), detP = -detP; j = find(p==i); p([i,j]) = p([j,i]); end end if signU~=detP % log becomes complex for negative values, encoded by infinity ldA = Inf; else % det(L) = 1 and U triangular => det(A) = det(P)*prod(diag(U)) ldA = sum(log(abs(u))); end if nargout>1, iA = inv(U)*inv(L)*P; end % return the inverse, as well if nargout>2, mwiA = -repmat(w,1,n).*iA; end gpml-matlab-v3.2-2013-01-15/inf/infFITC_Laplace.m000644 000765 000024 00000022734 12047277603 020666 0ustar00carlstaff000000 000000 function [post nlZ dnlZ] = infFITC_Laplace(hyp, mean, cov, lik, x, y) % 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. % % The implementation exploits the Woodbury matrix identity % inv(Kt) = inv(G) - inv(G)*Ku'*inv(Kuu+Ku*inv(G)*Ku')*Ku*inv(G) % in order to be applicable to large datasets. The computational complexity % is O(n nu^2) where n is the number of data points x and nu the number of % inducing inputs in xu. % The posterior N(f|h,Sigma) is given by h = m+mu with mu = nn + P'*gg and % Sigma = inv(inv(K)+diag(W)) = diag(d) + P'*R0'*R'*R*R0*P. % % The function takes a specified covariance function (see covFunction.m) and % likelihood function (see likFunction.m), and is designed to be used with % gp.m and in conjunction with covFITC. % % Copyright (c) by Hannes Nickisch, 2012-11-07. % % See also INFMETHODS.M, COVFITC.M. persistent last_alpha % copy of the last alpha if any(isnan(last_alpha)), last_alpha = zeros(size(last_alpha)); end % prevent tol = 1e-6; % tolerance for when to stop the Newton iterations smax = 2; Nline = 10; thr = 1e-4; % line search parameters maxit = 20; % max number of Newton steps in f inf = 'infLaplace'; n = size(x,1); cov1 = cov{1}; if isa(cov1, 'function_handle'), cov1 = func2str(cov1); end if ~strcmp(cov1,'covFITC'); error('Only covFITC supported.'), end % check cov [diagK,Kuu,Ku] = feval(cov{:}, hyp.cov, x); % evaluate covariance matrix if ~isempty(hyp.lik) % hard coded inducing inputs noise sn2 = exp(2*hyp.lik(end)); snu2 = 1e-6*sn2; % similar to infFITC else snu2 = 1e-6; end nu = size(Kuu,1); m = feval(mean{:}, hyp.mean, x); % evaluate the mean vector rot180 = @(A) rot90(rot90(A)); % little helper functions chol_inv = @(A) rot180(chol(rot180(A))')\eye(nu); % chol(inv(A)) R0 = chol_inv(Kuu+snu2*eye(nu)); % initial R, used for refresh O(nu^3) V = R0*Ku; d0 = diagK-sum(V.*V,1)'; % initial d, needed for refresh O(n*nu^2) Psi_old = Inf; % make sure while loop starts by the largest old objective val if any(size(last_alpha)~=[n,1]) % find a good starting point for alpha and f alpha = zeros(n,1); f = mvmK(alpha,V,d0)+m; % start at mean if sizes not match [lp,dlp,d2lp] = feval(lik{:},hyp.lik,y,f,[],inf); W=-d2lp; Psi_new = -sum(lp); else alpha = last_alpha; f = mvmK(alpha,V,d0)+m; % try last one [lp,dlp,d2lp] = feval(lik{:},hyp.lik,y,f,[],inf); W=-d2lp; Psi_new = alpha'*(f-m)/2 - sum(lp); % objective for last alpha Psi_def = -feval(lik{:},hyp.lik,y,m,[],inf); % objective for default init f==m if Psi_def < Psi_new % if default is better, we use it alpha = zeros(n,1); f = mvmK(alpha,V,d0)+m; [lp,dlp,d2lp] = feval(lik{:},hyp.lik,y,f,[],inf); W=-d2lp; Psi_new=-sum(lp); end end isWneg = 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 && it2 % do we want derivatives? dnlZ = hyp; % allocate space for derivatives [d,P,R] = fitcRefresh(d0,Ku,R0,V,W); % g = diag(inv(inv(K)+W))/2 g = d/2 + sum(((R*R0)*P).^2,1)'/2; t = W./(1+W.*d0); dfhat = g.*d3lp; % deriv. of nlZ wrt. fhat: dfhat=diag(inv(inv(K)+W)).*d3lp/2 for i=1:length(hyp.cov) % covariance hypers [ddiagK,dKuu,dKu] = feval(cov{:}, hyp.cov, x, [], i); % eval cov derivatives dA = 2*dKu'-R0tV'*dKuu; % dQ = dA*R0tV w = sum(dA.*R0tV',2); v = ddiagK-w; % w = diag(dQ); v = diag(dK)-diag(dQ); dnlZ.cov(i) = ddiagK'*t - sum(RVdd.*RVdd,1)*v; % explicit part dnlZ.cov(i) = dnlZ.cov(i)-sum(sum((RVdd*dA).*(RVdd*R0tV'))); % explicit part dnlZ.cov(i) = dnlZ.cov(i)/2-alpha'*(dA*(R0tV*alpha)+v.*alpha)/2; % explicit b = dA*(R0tV*dlp) + v.*dlp; % b-K*(Z*b) = inv(eye(n)+K*diag(W))*b KZb = mvmK(mvmZ(b,RVdd,t),V,d0); dnlZ.cov(i) = dnlZ.cov(i) - dfhat'*( b-KZb ); % implicit part end for i=1:length(hyp.lik) % likelihood hypers [lp_dhyp,dlp_dhyp,d2lp_dhyp] = feval(lik{:},hyp.lik,y,f,[],inf,i); dnlZ.lik(i) = -g'*d2lp_dhyp - sum(lp_dhyp); % explicit part b = mvmK(dlp_dhyp,V,d0); % implicit part dnlZ.lik(i) = dnlZ.lik(i) - dfhat'*(b-mvmK(mvmZ(b,RVdd,t),V,d0)); if i==numel(hyp.lik) % since snu2 is a fixed fraction of sn2, there is a covariance-like term % in the derivative as well snu = sqrt(snu2); T = chol_inv(Kuu + snu2*eye(nu)); T = T'*(T*(snu*Ku)); t = sum(T.*T,1)'; z = alpha'*(T'*(T*alpha)-t.*alpha) - sum(RVdd.*RVdd,1)*t; z = z + sum(sum( (RVdd*T').^2 )); b = (t.*dlp-T'*(T*dlp))/2; KZb = mvmK(mvmZ(b,RVdd,t),V,d0); z = z - dfhat'*( b-KZb ); dnlZ.lik(i) = dnlZ.lik(i) + z; end end for i=1:length(hyp.mean) % mean hypers dm = feval(mean{:}, hyp.mean, x, i); dnlZ.mean(i) = -alpha'*dm; % explicit part Zdm = mvmZ(dm,RVdd,t); dnlZ.mean(i) = dnlZ.mean(i) - dfhat'*(dm-mvmK(Zdm,V,d0)); % implicit part end end % matrix vector multiplication with Z=inv(K+inv(W)) function Zx = mvmZ(x,RVdd,t) Zx = t.*x - RVdd'*(RVdd*x); % matrix vector multiplication with approximate covariance matrix function Kal = mvmK(al,V,d0) Kal = V'*(V*al) + d0.*al; % criterion Psi at alpha + s*dalpha for line search function [Psi,alpha,f,dlp,W] = Psi_line(s,dalpha,alpha,hyp,V,d0,m,lik,y,inf) alpha = alpha + s*dalpha; f = mvmK(alpha,V,d0)+m; [lp,dlp,d2lp] = feval(lik{:},hyp.lik,y,f,[],inf); W = -d2lp; Psi = alpha'*(f-m)/2 - sum(lp); % 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 = (y-m)'*alpha/2 + sum(log(diag(L))) + n*log(2*pi*sn2)/2; % -log marg lik if nargout>2 % do we want derivatives? dnlZ = hyp; % allocate space for derivatives Q = solve_chol(L,eye(n))/sn2 - alpha*alpha'; % precompute for convenience for i = 1:numel(hyp.cov) dnlZ.cov(i) = sum(sum(Q.*feval(cov{:}, hyp.cov, x, [], i)))/2; end dnlZ.lik = sn2*trace(Q); for i = 1:numel(hyp.mean), dnlZ.mean(i) = -feval(mean{:}, hyp.mean, x, i)'*alpha; end end end gpml-matlab-v3.2-2013-01-15/inf/infFITC_EP.m000644 000765 000024 00000024420 12047277603 017623 0ustar00carlstaff000000 000000 function [post nlZ dnlZ] = infFITC_EP(hyp, mean, cov, lik, x, y) % 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. % % The implementation exploits the Woodbury matrix identity % inv(Kt) = inv(G) - inv(G)*Ku'*inv(Kuu+Ku*inv(G)*Ku')*Ku*inv(G) % in order to be applicable to large datasets. The computational complexity % is O(n nu^2) where n is the number of data points x and nu the number of % inducing inputs in xu. % The posterior N(f|h,Sigma) is given by h = m+mu with mu = nn + P'*gg and % Sigma = inv(inv(K)+diag(W)) = diag(d) + P'*R0'*R'*R*R0*P. Here, we use the % site parameters: b,w=$b,\pi$=tnu,ttau, P=$P'$, nn=$\nu$, gg=$\gamma$ % % The function takes a specified covariance function (see covFunction.m) and % likelihood function (see likFunction.m), and is designed to be used with % gp.m and in conjunction with covFITC. % % Copyright (c) by Hannes Nickisch, 2012-11-09. % % See also INFMETHODS.M, COVFITC.M. cov1 = cov{1}; if isa(cov1, 'function_handle'), cov1 = func2str(cov1); end if ~strcmp(cov1,'covFITC'); error('Only covFITC supported.'), end % check cov persistent last_ttau last_tnu % keep tilde parameters between calls tol = 1e-4; max_sweep = 20; min_sweep = 2; % tolerance to stop EP iterations inf = 'infEP'; n = size(x,1); [diagK,Kuu,Ku] = feval(cov{:}, hyp.cov, x); % evaluate covariance matrix if ~isempty(hyp.lik) % hard coded inducing inputs noise sn2 = exp(2*hyp.lik(end)); snu2 = 1e-6*sn2; % similar to infFITC else snu2 = 1e-6; end nu = size(Kuu,1); m = feval(mean{:}, hyp.mean, x); % evaluate the mean vector rot180 = @(A) rot90(rot90(A)); % little helper functions chol_inv = @(A) rot180(chol(rot180(A))')\eye(nu); % chol(inv(A)) R0 = chol_inv(Kuu+snu2*eye(nu)); % initial R, used for refresh O(nu^3) V = R0*Ku; d0 = diagK-sum(V.*V,1)'; % 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 = -sum(feval(lik{:}, hyp.lik, y, m, diagK, inf)); if any(size(last_ttau) ~= [n 1]) % find starting point for tilde parameters ttau = zeros(n,1); % initialize to zero if we have no better guess tnu = zeros(n,1); [d,P,R,nn,gg] = epfitcRefresh(d0,Ku,R0,V, ttau,tnu); % compute initial repres. nlZ = nlZ0; else ttau = last_ttau; % try the tilde values from previous call tnu = last_tnu; [d,P,R,nn,gg] = epfitcRefresh(d0,Ku,R0,V, ttau,tnu); % compute initial repres. nlZ = epfitcZ(d,P,R,nn,gg,ttau,tnu,d0,R0,Ku,y,lik,hyp,m,inf); if nlZ > nlZ0 % if zero is better .. ttau = zeros(n,1); % .. then initialize with zero instead tnu = zeros(n,1); [d,P,R,nn,gg] = epfitcRefresh(d0,Ku,R0,V, ttau,tnu); % initial repres. nlZ = nlZ0; end end nlZ_old = Inf; sweep = 0; % converged, max. sweeps or min. sweeps? while (abs(nlZ-nlZ_old) > tol && sweep < max_sweep) || sweep2 % do we want derivatives? dnlZ = hyp; % allocate space for derivatives RVdd = RV.*repmat(dd',nu,1); for i=1:length(hyp.cov) [ddiagK,dKuu,dKu] = feval(cov{:}, hyp.cov, x, [], i); % eval cov derivatives dA = 2*dKu'-R0tV'*dKuu; % dQ = dA*R0tV w = sum(dA.*R0tV',2); v = ddiagK-w; % w = diag(dQ); v = diag(dK)-diag(dQ); z = dd'*(v+w) - sum(RVdd.*RVdd,1)*v - sum(sum( (RVdd*dA)'.*(R0tV*RVdd') )); dnlZ.cov(i) = (z - alpha'*(alpha.*v) - (alpha'*dA)*(R0tV*alpha))/2; end for i = 1:numel(hyp.lik) % likelihood hypers dlik = feval(lik{:}, hyp.lik, y, nu_n./tau_n, 1./tau_n, inf, i); dnlZ.lik(i) = -sum(dlik); if i==numel(hyp.lik) % since snu2 is a fixed fraction of sn2, there is a covariance-like term % in the derivative as well v = sum(R0tV.*R0tV,1)'; z = sum(sum( (RVdd*R0tV').^2 )) - sum(RVdd.*RVdd,1)*v; z = z + post.alpha'*post.alpha - alpha'*(v.*alpha); dnlZ.lik(i) = dnlZ.lik(i) + snu2*z; end end [junk,dlZ] = feval(lik{:}, hyp.lik, y, nu_n./tau_n, 1./tau_n, inf);% mean hyps for i = 1:numel(hyp.mean) dm = feval(mean{:}, hyp.mean, x, i); dnlZ.mean(i) = -dlZ'*dm; end end % 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 nu0 R = cholupdate(R,sqrt( r)*R'*v,'-'); else R = cholupdate(R,sqrt(-r)*R'*v,'+'); end gg = gg + ((dbi-dwi*(hi-m(i)))/t)*(R0'*(R'*(R*(R0*Pi)))); % O(nu^2) w(i) = wi; b(i) = bi; % update site parameters O(1) Pi = Pi/t; % O(nu) gpml-matlab-v3.2-2013-01-15/inf/infMCMC.m000644 000765 000024 00000024661 12077070414 017232 0ustar00carlstaff000000 000000 function [post nlZ dnlZ] = infMCMC(hyp, mean, cov, lik, x, y, par) % Markov Chain Monte Carlo (MCMC) sampling from posterior and % Annealed Importance Sampling (AIS) for marginal likelihood estimation. % % The algorithms are not to be used as a black box, since the acceptance rate % of the samplers need to be carefully monitored. Also, there are no derivatives % of the marginal likelihood available. % % There are additional parameters: % - par.sampler switch between the samplers 'hmc' or 'ess' % - par.Nsample num of samples % - par.Nskip num of steps out of which one sample kept % - par.Nburnin num of burn in samples (corresponds to Nskip*Nburning steps) % - par.Nais num of AIS runs to remove finite temperature bias % Default values are 'sampler=hmc', Nsample=200, Nskip=40, Nburnin=10, Nais=3. % % The Hybrid Monte Carlo Sampler (HMC) is implemented as described in the % technical report: Probabilistic Inference using MCMC Methods by Radford Neal, % CRG-TR-93-1, 1993. % % Instead of sampling from f ~ 1/Zf * N(f|m,K) P(y|f), we use a % parametrisation in terms of alpha = inv(K)*(f-m) and sample from % alpha ~ P(a) = 1/Za * N(a|0,inv(K)) P(y|K*a+m) to increase numerical % stability since log P(a) = -(a'*K*a)/2 + log P(y|K*a+m) + C and its % gradient can be computed safely. % % The leapfrog stepsize as a time discretisation stepsize comes with a % tradeoff: % - too small: frequently accept, slow exploration, accurate dynamics % - too large: seldomly reject, fast exploration, inaccurate dynamics % In order to balance between the two extremes, we adaptively adjust the % stepsize in order to keep the acceptance rate close to a target acceptance % rate. Taken from http://deeplearning.net/tutorial/hmc.html. % % The code issues a warning in case the overall acceptance rate did deviate % strongly from the target. This can indicate problems with the sampler. % % The Elliptical Slice Sampler (ESS) is a straight implementation that is % inspired by the code shipped with the paper: Elliptical slice sampling by % Iain Murray, Ryan Prescott Adams and David J.C. MacKay, AISTATS 2010. % % Annealed Importance Sampling (AIS) to determine the marginal likelihood is % described in the technical report: AIS, Radford Neal, 1998. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch 2012-11-07. % % See also INFMETHODS.M. if nargin<7, par = []; end % analyse parameter structure if isfield(par,'sampler'), alg=par.sampler; else alg = 'hmc'; end if isfield(par,'Nsample'), N =par.Nsample; else N = 200; end if isfield(par,'Nskip'), Ns =par.Nskip; else Ns = 40; end if isfield(par,'Nburnin'), Nb =par.Nburnin; else Nb = 10; end if isfield(par,'Nais'), R =par.Nais; else R = 3; end K = feval(cov{:}, hyp.cov, x); % evaluate the covariance matrix m = feval(mean{:}, hyp.mean, x); % evaluate the mean vector n = size(K,1); [cK,fail] = chol(K); % try an ordinary Cholesky decomposition if fail, sr2 = 1e-8*sum(diag(K))/n; cK = chol(K+sr2*eye(n)); end % regularise T = (N+Nb)*Ns; % overall number of steps [alpha,Na] = sample(K,cK,m,y,lik,hyp.lik, N,Nb,Ns, alg); % sample w/o annealing post.alpha = alpha; al = sum(alpha,2)/N; post.L = -(cK\(cK'\eye(n)) + al*al' - alpha*alpha'/N); % inv(K) - cov(alpha) post.sW = []; post.acceptance_rate_MCMC = Na/T; % additional output parameter if nargout>1 % annealed importance sampling % discrete time t from 1 to T and temperature tau from tau(1)=0 to tau(T)=1 taus = [zeros(1,Nb),linspace(0,1,N)].^4; % annealing schedule, effort at start % the fourth power idea is taken from the Kuss&Rasmussen paper, 2005 JMLR % Z(t) := \int N(f|m,K) lik(f)^taus(t) df hence Z(1) = 1 and Z(T) = Z % Z = Z(T)/Z(1) = prod_t Z(t)/Z(t-1); % ln Z(t)/Z(t-1) = ( tau(t)-tau(t-1) ) * loglik(f_t) lZ = zeros(R,1); dtaus = diff(taus(Nb+(1:N))); % we have: sum(dtaus)==1 for r=1:R [A,Na] = sample(K,cK,m,y,lik,hyp.lik, N,Nb,Ns, alg, taus); % AIS for t=2:N % evaluate the likelihood sample-wise lp = feval(lik{:},hyp.lik,y,K*A(:,t)+m,[],'infLaplace'); lZ(r) = lZ(r)+dtaus(t-1)*sum(lp); end post.acceptance_rate_AIS(r) = Na/T; % additional output parameters end nlZ = log(R)-logsumexp(lZ); % remove finite temperature bias, softmax average if nargout>2 % marginal likelihood derivatives are not computed dnlZ = struct('cov',0*hyp.cov, 'mean',0*hyp.mean, 'lik',0*hyp.lik); end end %% choose between HMC and ESS depending on the alg string function [alpha,Na] = sample(K,cK,m,y,lik,hyp, N,Nb,Ns, alg, varargin) if strcmpi(alg,'hmc') [alpha,Na] = sample_hmc(K, m,y,lik,hyp, N,Nb,Ns, varargin{:}); else [alpha,Na] = sample_ess(cK,m,y,lik,hyp, N,Nb,Ns, varargin{:}); end %% sample using elliptical slices function [alpha,T] = sample_ess(cK,m,y,lik,hyp, N,Nb,Ns, taus) if nargin>=9, tau = taus(1); else tau = 1; end % default is no annealing T = (N+Nb)*Ns; % overall number of steps F = zeros(size(m,1),N); for t=1:T if nargin>=9, tau = taus(1+floor((t-1)/Ns)); end % parameter from schedule if t==1, f=0*m; l=sum(feval(lik{:},hyp,y,f+m)); end % init sample f & lik l r = cK'*randn(size(m)); % random sample from N(0,K) [f,l] = sample_ess_step(f,l,r,m,y,lik,hyp,tau); if mod(t,Ns)==0 % keep one state out of Ns if t/Ns>Nb, F(:,t/Ns-Nb) = f; end % wait for Nb burn-in steps end end alpha = cK\(cK'\F); %% elliptical slice sampling: one step function [f,l] = sample_ess_step(f,l,r,m,y,lik,hyp,tau) if nargin<7, tau=1; end, if tau>1, tau=1; end, if tau<0, tau=0; end h = log(rand) + tau*l; % acceptance threshold a = rand*2*pi; amin = a-2*pi; amax = a; % bracket whole ellipse k = 0; % emergency break while true % slice sampling loop; f for proposed angle diff; check if on slice fp = f*cos(a) + r*sin(a); % move on ellipsis defined by r l = sum(feval(lik{:},hyp,y,fp+m)); if tau*l>h || k>20, break, end % exit if new point is on slice or exhausted if a>0, amax=a; elseif a<0, amin=a; end % shrink slice to rejected point a = rand*(amax-amin) + amin; k = k+1; % propose new angle difference; break end f = fp; % accept %% sample using Hamiltonian dynamics as proposal algorithm function [alpha,Na] = sample_hmc(K,m,y,lik,hyp, N,Nb,Ns, taus) % use adaptive stepsize rule to enforce a specific acceptance rate epmin = 1e-6; % minimum leapfrog stepsize epmax = 9e-1; % maximum leapfrog stepsize ep = 1e-2; % initial leapfrog stepsize acc_t = 0.9; % target acceptance rate acc = 0; % current acceptance rate epinc = 1.02; % increase factor of stepsize if acceptance rate is below target epdec = 0.98; % decrease factor of stepsize if acceptance rate is above target lam = 0.01; % exponential moving average computation of the acceptance rate % 2/(3*lam) steps half height; lam/nstep = 0.02/33, 0.01/66, 0.005/133 l = 20; % number of leapfrog steps to perform for one step n = size(K,1); T = (N+Nb)*Ns; % overall number of steps alpha = zeros(n,N); % sample points al = zeros(n,1); % current position if nargin>=9, tau = taus(1); else tau = 1; end % default is no annealing [gold,eold] = E(al,K,m,y,lik,hyp,tau); % initial energy, gradient Na = 0; % number of accepted points for t=1:T if nargin>=9, tau = taus(1+floor((t-1)/Ns)); end % parameter from schedule p = randn(n,1); % random initial momentum q = al; g = gold; Hold = (p'*p)/2 + eold; % H = Ekin + Epot => Hamiltonian for ll=1:l % leapfrog discretization steps, Euler like p = p - (ep/2)*g; % half step in momentum p q = q + ep*p; % full step in position q g = E(q,K,m,y,lik,hyp,tau); % compute new gradient g p = p - (ep/2)*g; % half step in momentum p end [g,e] = E(q,K,m,y,lik,hyp,tau); H = (p'*p)/2 + e; % recompute Hamiltonian acc = (1-lam)*acc; % decay current acceptance rate if log(rand) < Hold-H % accept with p = min(1,exp(Hold-H)) al = q; % keep new state, gold = g; % gradient eold = e; % and potential energy acc = acc + lam; % increase rate due to acceptance Na = Na+1; % increase number accepted steps end if acc>acc_t % too large acceptance rate => increase leapfrog stepsize ep = epinc*ep; else % too small acceptance rate => decrease leapfrog stepsize ep = epdec*ep; end if epepmax, ep = epmax; end if mod(t,Ns)==0 % keep one state out of Ns if t/Ns>Nb, alpha(:,t/Ns-Nb) = al; end % wait for Nb burn-in steps end end if Na/T dE/dal = K*( al-dloglik(f) ) if nargin<7, tau=1; end, if tau>1, tau=1; end, if tau<0, tau=0; end Kal = K*al; [lp,dlp] = feval(lik{:},hyp,y,Kal+m,[],'infLaplace'); g = Kal-K*(tau*dlp); if nargout>1, e = (al'*Kal)/2 - tau*sum(lp); end %% y = logsumexp(x) = log(sum(exp(x(:))) avoiding overflow function y=logsumexp(x) mx = max(x(:)); y = log(sum(exp(x(:)-mx)))+mx; gpml-matlab-v3.2-2013-01-15/inf/infLOO.m000644 000765 000024 00000010234 12077070414 017133 0ustar00carlstaff000000 000000 function [post nloo dnloo loo] = infLOO(hyp, mean, cov, lik, x, y) % Leave-One-Out - Perform Least-Squares GP predictions in the style of the % Pseudo Likelihood in §5.4.2 p. 117 and the Probabilistic Least-squares % Classifier of §6.5 p. 146 of the GPML book. % For Gaussian likelihood, the prediction is the same as with infExact, for % lik{Erf,Logistic} the result corresponds to probabilistic least-squares % classification. In any case, the objective returned is not the marginal % likelihood but the predictive log probability. % The standard deviation of the noise either comes from the sn parameter of the % likelihood function lik{Gauss,Laplace,Sech2} or from the covariance function % by means of using covSum and covNoise i.e. for lik{Uni,Erf,Logistic}. % We then compute the negative leave-one-out predictive probability and its % derivatives w.r.t. the hyperparameters. See also "help infMethods". % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2012-11-07 % % See also INFMETHODS.M. [n, D] = size(x); K = feval(cov{:}, hyp.cov, x); % evaluate covariance matrix m = feval(mean{:}, hyp.mean, x); % evaluate mean vector if numel(hyp.lik)==0 cov1 = cov{1}; if isa(cov1, 'function_handle'), cov1 = func2str(cov1); end if ~strcmp(cov1,'covSum'); error('Only covSum supported.'), end % check cov cov2 = cov{2}; npar = 0; for i=1:length(cov2) cov2i = cov2{i}; if numel(cov2i)==1 if isa(cov2i, 'function_handle'), cov2i = func2str(cov2i); end if strcmp(cov2i,'covNoise'), isn = npar+1; sn2 = exp(2*hyp.cov(isn)); end end npar = npar + eval(feval(cov2i)); end if ~exist('sn2','var'), error('We need covNoise in the covSum.'), end else % Gauss, Laplace, Sech2 sn2 = exp(2*hyp.lik(end)); isn = 0; K = K + sn2*eye(n); end L = chol(K)/sqrt(sn2); % Cholesky factor of covariance with noise alpha = solve_chol(L,y-m)/sn2; post.alpha = alpha; % return the posterior parameters post.sW = ones(n,1)/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? sdeff = 1./post.sW; % effective Gaussian width mneff = K*alpha + m; % alpha = inv(K)*(mneff-m), effective Gaussian center % all loo marginal means and variances at once V = post.L'\diag(1./sdeff); r = sum(V.*V,1)'; % r=diag( inv(K) ) loo.fmu = mneff - alpha./r; % GPML book, eq. 5.12 loo.fs2 = 1./r - sn2; % GPML book, eq. 5.12 [loo.lp, loo.ymu, loo.ys2] = feval(lik{:},hyp.lik,y,loo.fmu,loo.fs2); nloo = -sum(loo.lp); % negative leave-one-out predictive probability if nargout>2 % do we want derivatives? [lZ,dlZmn,d2lZmn] = feval(lik{:},hyp.lik,y,loo.fmu,loo.fs2,'infEP'); dlZva = (d2lZmn+dlZmn.*dlZmn)/2; % derivative w.r.t. variance dnloo = hyp; iK = solve_chol(L,eye(n))/sn2; for j=1:numel(hyp.cov) dKj = feval(cov{:}, hyp.cov, x, [], j); Zj = solve_chol(L,dKj)/sn2; % GPML book, eq. 5.13 dva = sum(Zj.*iK,2)./(r.*r); % GPML book, eq. 5.13 dmn = (Zj*alpha)./r - alpha.*dva; % GPML book, eq. 5.13 if j==isn, dva = dva - 2*sn2; end % additional part for covNoise dnloo.cov(j) = -dlZmn'*dmn -dlZva'*dva; end for j=1:numel(hyp.lik) dnloo.lik(j) = -sum( feval(lik{:},hyp.lik,y,loo.fmu,loo.fs2,'infEP',j) ); if j==numel(hyp.lik) dva = 2*sn2*sum(iK.*iK,2)./(r.*r); % GPML book, eq. 5.13 dmn = 2*solve_chol(L,alpha)./r - alpha.*dva; % GPML book, eq. 5.13 dva = dva - 2*sn2; dnloo.lik(j) = dnloo.lik(j) -dlZmn'*dmn -dlZva'*dva; end end for j=1:numel(hyp.mean) dmj = feval(mean{:}, hyp.mean, x, j); dmn = solve_chol(L,dmj)./(sn2*r); dnloo.mean(j) = -dlZmn'*dmn; end end end gpml-matlab-v3.2-2013-01-15/util/binaryEPGP.m000644 000765 000024 00000005753 11511036276 020160 0ustar00carlstaff000000 000000 function varargout = binaryEPGP(hyper, covfunc, x, y, xstar) % binaryEPGP - The Expectation Propagation approximation for binary Gaussian % process classification. Two modes are possible: training or testing: if no % test cases are supplied, then the approximate negative log marginal % likelihood and its partial derivatives wrt the hyperparameters is computed; % this mode is used to fit the hyperparameters. If test cases are given, then % the test set predictive probabilities are returned. The program is flexible % in allowing a multitude of covariance functions. % % usage: [nlZ, dnlZ ] = binaryEPGP(hyper, covfunc, x, y); % or: [p, mu, s2, nlZ] = binaryEPGP(hyper, covfunc, x, y, xstar); % % where: % % hyper is a (column) vector of hyperparameters % covfunc is the name of the covariance function (see below) % lik is the name of the likelihood function (see below) % x is a n by D matrix of training inputs % y is a (column) vector (of size n) of binary +1/-1 targets % xstar is a nn by D matrix of test inputs % nlZ is the returned value of the negative log marginal likelihood % dnlZ is a (column) vector of partial derivatives of the negative % log marginal likelihood wrt each log hyperparameter % p is a (column) vector (of length nn) of predictive probabilities % mu is a (column) vector (of length nn) of predictive latent means % s2 is a (column) vector (of length nn) of predictive latent variances % % The length of the vector of hyperparameters depends on the covariance % function, as specified by the "covfunc" input to the function, specifying the % name of a covariance function. A number of different covariance function are % implemented, and it is not difficult to add new ones. See "help covFunctions" % for the details % % The function can conveniently be used with the "minimize" function to train % a Gaussian process, eg: % % [hyper, fX, i] = minimize(hyper, 'binaryEPGP', length, 'covSEiso', % 'logistic', x, y); % % Copyright (c) 2004, 2005, 2006, 2007 Carl Edward Rasmussen, 2010-07-21. if nargin<4 || nargin>5 disp('Usage: [nlZ, dnlZ ] = binaryEPGP(hyper, covfunc, x, y);') disp(' or: [p, mu, s2, nlZ] = binaryEPGP(hyper, covfunc, x, y, xstar);') return end % Note, this function is just a wrapper provided for backward compatibility, % the functionality is now provided by the more general gp function. lik = @likErf; inf = @infEP; mean = @meanZero; varargout = cell(nargout, 1); % allocate the right number of output arguments hyp.cov = hyper; if nargin==4 [varargout{:}] = gp(hyp,inf,mean,covfunc,lik,x,y); if nargout>1, varargout{2} = varargout{2}.cov; end else [ymu,ys2,fmu,fs2] = gp(hyp,inf,mean,covfunc,lik,x,y,xstar); if nargout>0, varargout{1} = (1+ymu)/2; end if nargout>1, varargout{2} = fmu; end if nargout>2, varargout{3} = fs2; end if nargout>3, varargout{4} = gp(hyp,inf,mean,covfunc,lik,x,y); end % nlZ end gpml-matlab-v3.2-2013-01-15/util/binaryLaplaceGP.m000644 000765 000024 00000007030 11511036276 021203 0ustar00carlstaff000000 000000 function varargout = binaryLaplaceGP(hyper, covfunc, lik, x, y, xstar) % binaryLaplaceGP - Laplace's approximation for binary Gaussian process % classification. Two modes are possible: training or testing: if no test % cases are supplied, then the approximate negative log marginal likelihood % and its partial derivatives wrt the hyperparameters is computed; this mode is % used to fit the hyperparameters. If test cases are given, then the test set % predictive probabilities are returned. The program is flexible in allowing % several different likelihood functions and a multitude of covariance % functions. % % usage: [nlZ, dnlZ ] = binaryLaplaceGP(hyper, covfunc, lik, x, y); % or: [p, mu, s2, nlZ] = binaryLaplaceGP(hyper, covfunc, lik, x, y, xstar); % % where: % % hyper is a (column) vector of hyperparameters % covfunc is the name of the covariance function (see below) % lik is the name of the likelihood function (see below) % x is a n by D matrix of training inputs % y is a (column) vector (of size n) of binary +1/-1 targets % xstar is a nn by D matrix of test inputs % nlZ is the returned value of the negative log marginal likelihood % dnlZ is a (column) vector of partial derivatives of the negative % log marginal likelihood wrt each log hyperparameter % p is a (column) vector (of length nn) of predictive probabilities % mu is a (column) vector (of length nn) of predictive latent means % s2 is a (column) vector (of length nn) of predictive latent variances % % The length of the vector of log hyperparameters depends on the covariance % function, as specified by the "covfunc" input to the function, specifying the % name of a covariance function. A number of different covariance function are % implemented, and it is not difficult to add new ones. See "help covFunctions" % for the details. % % The shape of the likelihood function is given by the "lik" input to the % function, specifying the name of the likelihood function. The two implemented % likelihood functions are: % % logistic the logistic function: 1/(1+exp(-x)) % cumGauss the cumulative Gaussian (error function) % % The function can conveniently be used with the "minimize" function to train % a Gaussian process, eg: % % [hyper, fX, i] = minimize(hyper, 'binaryLaplaceGP', length, 'covSEiso', % 'logistic', x, y); % % Copyright (c) 2004, 2005, 2006, 2007 Carl Edward Rasmussen, 2010-07-21. if nargin<5 || nargin>6 disp('Usage: [nlZ, dnlZ ] = binaryLaplaceGP(hyper, covfunc, lik, x, y);') disp(' or: [p, mu, s2, nlZ] = binaryLaplaceGP(hyper, covfunc, lik, x, y, xstar);') return end % Note, this function is just a wrapper provided for backward compatibility, % the functionality is now provided by the more general gp function. if strcmp(lik,'logistic') lik = @likLogistic; elseif strcmpi(lik,'cumgauss') lik = @likErf; else error('Allowable likelihoods: logistic and cumGauss.'); end inf = @infLaplace; mean = @meanZero; varargout = cell(nargout, 1); % allocate the right number of output arguments hyp.cov = hyper; if nargin==5 [varargout{:}] = gp(hyp,inf,mean,covfunc,lik,x,y); if nargout>1, varargout{2} = varargout{2}.cov; end else [ymu,ys2,fmu,fs2] = gp(hyp,inf,mean,covfunc,lik,x,y,xstar); if nargout>0, varargout{1} = (1+ymu)/2; end if nargout>1, varargout{2} = fmu; end if nargout>2, varargout{3} = fs2; end if nargout>3, varargout{4} = gp(hyp,inf,mean,covfunc,lik,x,y); end % nlZ end gpml-matlab-v3.2-2013-01-15/util/solve_chol.c000644 000765 000024 00000002404 11511036276 020331 0ustar00carlstaff000000 000000 /* solve_chol - solve a linear system A*X = B using the cholesky factorization of A (where A is square, symmetric and positive definite. Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch 2010-12-16. */ #include "mex.h" #include #include extern int dpotrs_(char *, long *, long *, double *, long *, double *, long *, long *); void mexFunction(int nlhs, mxArray *plhs[], int nrhs, const mxArray *prhs[]) { double *C; long n, m, q; if (nrhs != 2 || nlhs > 1) /* check the input */ mexErrMsgTxt("Usage: X = solve_chol(R, B)"); n = mxGetN(prhs[0]); if (n != mxGetM(prhs[0])) mexErrMsgTxt("Error: First argument matrix must be square"); if (n != mxGetM(prhs[1])) mexErrMsgTxt("Error: First and second argument matrices must have same number of rows"); m = mxGetN(prhs[1]); plhs[0] = mxCreateDoubleMatrix(n, m, mxREAL); /* allocate space for output */ C = mxGetPr(plhs[0]); if (n==0) return; /* if argument was empty matrix, do no more */ memcpy( C, mxGetPr(prhs[1]), n*m*sizeof(double) );/* copy argument matrix */ dpotrs_("U", &n, &m, mxGetPr(prhs[0]), &n, C, &n, &q); /* solve system */ if (q > 0) mexErrMsgTxt("Error: illegal input to solve_chol"); } gpml-matlab-v3.2-2013-01-15/util/solve_chol.m000644 000765 000024 00000001741 11511036276 020346 0ustar00carlstaff000000 000000 % solve_chol - 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); % % NOTE: The program code is written in the C language for efficiency and is % contained in the file solve_chol.c, and should be compiled using matlabs mex % facility. However, this file also contains a (less efficient) matlab % implementation, supplied only as a help to people unfamiliar with mex. If % the C code has been properly compiled and is avaiable, it automatically % takes precendence over the matlab code in this file. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch 2010-09-18. function X = solve_chol(L, B) if nargin ~= 2 || nargout > 1 error('Wrong number of arguments.'); end if size(L,1) ~= size(L,2) || size(L,1) ~= size(B,1) error('Wrong sizes of matrix arguments.'); end X = L\(L'\B);gpml-matlab-v3.2-2013-01-15/util/lbfgsb.m000644 000765 000024 00000011601 11511036276 017444 0ustar00carlstaff000000 000000 % LBFGSB Call the nonlinear bound-constrained solver that uses % limited-memory BFGS quasi-Newton updates. % % The basic function call is % % LBFGSB(x0,lb,ub,objfunc,gradfunc) % % The first input argument x0 is either a matrix or a cell array of % matrices. It declares the starting point for the solver. % % The second and third input arguments lb and ub must be of the exact same % structure as x0. They declare the lower and upper bounds on the % variables, respectively. Set an entry to Inf to declare no bound. % % The next two input arguments must be the names of MATLAB routines % (M-files): % % objfunc Calculates the objective function at the current % point. The routine must accept as many inputs as cell % entry in x0 (or one input if x0 is a matrix). The % output must be a scalar representing the objective % evaluated at the current point. % % gradfunc Computes the gradient of the objective at the current % point. The input is the same as objfunc, but it must % return as many outputs as there are inputs, and each % of the outputs have the same matrix structure as its % corresponding input. % % Optionally, one may choose to pass additional auxiliary data to the % MATLAB callback routines listed above through the function call % LBFGSB(...,auxdata). If auxdata is the empty matrix, no extra % information is passed. It is important to observe that the auxiliary % data MAY NOT CHANGE through the course of the L-BFGS optimization! The % auxiliary data keep the same values as they possessed in the initial % call. If you need variables that change over time, you may want to % consider global variables (type HELP GLOBAL). % % LBFGSB(...,auxdata,iterfunc) specifies an additional callback routine % which is called once per algorithm iteration. The callback routine must % take the form ITERFUNC(T,F,X1,X2,...,AUXDATA). T is the current % iteration of the algorithm. F is the current value of the objective. The % inputs X1, X2,... are exactly the same as the ones passed to objfunc and % gradfunc. Finally, extra information may be passed through the input % AUXDATA. No outputs are expected from iterfunc. If iterfunc is the empty % string, no routine is called. % % These inputs may be followed by parameter/value pairs to modify the % default algorithm options. The algorithm option 'maxiter' sets the % maximum number of iterations. The rest of the options 'm', 'factr' and % 'pgtol' are described in the L-BFGS-B documentation. Note that in this % implementation we divide factr by the machine precision, so that the % convergence condition is satisfied when it is less than factr, not less % than factr times the machine precision. % % Interface has been written by Peter Carbonetto. % Compilation instructions by Hannes Nickisch, 2010-11-24 function x = lbfgsb(x,varargin) fprintf('\n\n\n') fprintf('##########################################################################\n') fprintf('In order to use the L-BFGS minimiser you have to compile Peter Carbonetto''s\n') fprintf('"Matlab interface for L-BFGS-B" [1]. The challenge here is the Fortran 77 code.\n') fprintf('We provide a Makefile suitable for Linux 32/64 bit and Mac whenever you have g77\n') fprintf('properly installed.Under Ubuntu, you can achieve this by installing the package\n') fprintf('fort77, for example.\n\n') fprintf('Compilation is done by editing lbfgsb/Makefile. In any case, you need to\n') fprintf('provide MATLAB_HOME which can be found by the commands ''locate matlab'' or \n') fprintf('''find / -name "matlab"''. You can choose between two compilation modes:\n') fprintf('a) using the mex utility by Matlab [default]\n') fprintf(' provide MEX, then type ''make mex''\n\n') fprintf('b) without mex utility by Matlab\n') fprintf(' provide MEX_SUFFIX and MATLAB_LIB, then type ''make nomex''\n\n') fprintf('In Ubuntu 10.04 LTS, the libg2c library needed for both a) and b) is not\n') fprintf('included per default. If ''ls /usr/lib/libg2c.*'' does not list anything\n') fprintf('this is the case on your machine. You then whant to install the packages\n') fprintf('gcc-3.4-base and libg2c0 e.g. from\n') fprintf('http://packages.ubuntu.com/hardy/gcc-3.4-base and\n') fprintf('http://packages.ubuntu.com/hardy/libg2c0.\n') fprintf('After installation, you have to create a symbolic link by ''cd /usr/lib'' and\n') fprintf('''ln -s libg2c.so.0 libg2c.so''.\n\n') fprintf('[1] http://www.cs.ubc.ca/~pcarbo/lbfgsb-for-matlab.html\n') fprintf('[2] http://www.mathworks.com/support/compilers/R2010a\n') fprintf('##########################################################################\n') fprintf('\n\n\n') error gpml-matlab-v3.2-2013-01-15/util/minimize_new.m000644 000765 000024 00000027260 12044426143 020705 0ustar00carlstaff000000 000000 % minimize.m - minimize a smooth differentiable multivariate function using % LBFGS (Limited memory LBFGS) or CG (Conjugate Gradients) % Usage: [X, fX, i] = minimize(X, F, p, other, ... ) % where % X is an initial guess (any type: vector, matrix, cell array, struct) % F is the objective function (function pointer or name) % p parameters - if p is a number, it corresponds to p.length below % p.length allowed 1) # linesearches or 2) if -ve minus # func evals % p.method minimization method, 'BFGS', 'LBFGS' or 'CG' % p.verbosity 0 quiet, 1 line, 2 line + warnings (default), 3 graphical % p.mem number of directions used in LBFGS (default 100) % other, ... other parameters, passed to the function F % X returned minimizer % fX vector of function values showing minimization progress % i final number of linesearches or function evaluations % The function F must take the following syntax [f, df] = F(X, other, ...) % where f is the function value and df its partial derivatives. The types of X % and df must be identical (vector, matrix, cell array, struct, etc). % % Copyright (C) 1996 - 2011 by Carl Edward Rasmussen, 2011-10-13. % Permission is hereby granted, free of charge, to any person OBTAINING A COPY % OF THIS SOFTWARE AND ASSOCIATED DOCUMENTATION FILES (THE "SOFTWARE"), TO DEAL % IN THE SOFTWARE WITHOUT RESTRICTION, INCLUDING WITHOUT LIMITATION THE RIGHTS % to use, copy, modify, merge, publish, distribute, sublicense, and/or sell % copies of the Software, and to permit persons to whom the Software is % furnished to do so, subject to the following conditions: % % The above copyright notice and this permission notice shall be included in % all copies or substantial portions of the Software. % % THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR % IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, % FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE % AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER % LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, % OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE % SOFTWARE. function [X, fX, i] = minimize(X, F, p, varargin) if isnumeric(p), p = struct('length', p); end % convert p to struct if p.length > 0, p.S = 'linesearch #'; else p.S = 'function evaluation #'; end; x = unwrap(X); % convert initial guess to vector if ~isfield(p,'method'), if length(x) > 1000, p.method = @LBFGS; else p.method = @BFGS; end; end % set default method if ~isfield(p,'verbosity'), p.verbosity = 2; end % default 1 line text output if ~isfield(p,'MFEPLS'), p.MFEPLS = 10; end % Max Func Evals Per Line Search if ~isfield(p,'MSR'), p.MSR = 100; end % Max Slope Ratio default f(F, X, varargin{:}); % set up the function f [fx, dfx] = f(x); % initial function value and derivatives if p.verbosity, printf('Initial Function Value %4.6e\r', fx); end if p.verbosity > 2, clf; subplot(211); hold on; xlabel(p.S); ylabel('function value'); plot(p.length < 0, fx, '+'); drawnow; end [x, fX, i] = feval(p.method, x, fx, dfx, p); % minimize using direction method X = rewrap(X, x); % convert answer to original representation if p.verbosity, printf('\n'); end function [x, fx, i] = CG(x0, fx0, dfx0, p) if ~isfield(p, 'SIG'), p.SIG = 0.1; end % default for line search quality i = p.length < 0; ok = 0; % initialize resource counter r = -dfx0; s = -r'*r; b = -1/(s-1); bs = -1; fx = fx0; % steepest descent while i < abs(p.length) b = b*bs/min(b*s,bs/p.MSR); % suitable initial step size using slope ratio [x, b, fx0, dfx, i] = lineSearch(x0, fx0, dfx0, r, s, b, i, p); if i < 0 % if line search failed i = -i; if ok, ok = 0; r = -dfx; else break; end % try steepest or stop else ok = 1; bs = b*s; % record step times slope (for slope ratio method) r = (dfx'*(dfx-dfx0))/(dfx0'*dfx0)*r - dfx; % Polack-Ribiere CG end s = r'*dfx; if s >= 0, r = -dfx; s = r'*dfx; ok = 0; end % slope must be -ve x0 = x; dfx0 = dfx; fx = [fx; fx0]; % replace old values with new ones end function [x, fx, i] = BFGS(x0, fx0, dfx0, p) if ~isfield(p, 'SIG'), p.SIG = 0.5; end % default for line search quality i = p.length < 0; ok = 0; % initialize resource counter x = x0; fx = fx0; r = -dfx0; s = -r'*r; b = -1/(s-1); H = eye(length(x0)); while i < abs(p.length) [x, b, fx0, dfx, i] = lineSearch(x0, fx0, dfx0, r, s, b, i, p); if i < 0 i = -i; if ok, ok = 0; else break; end; % try steepest or stop else ok = 1; t = x - x0; y = dfx - dfx0; ty = t'*y; Hy = H*y; H = H + (ty+y'*Hy)/ty^2*t*t' - 1/ty*Hy*t' - 1/ty*t*Hy'; % BFGS update end r = -H*dfx; s = r'*dfx; x0 = x; dfx0 = dfx; fx = [fx; fx0]; end function [x, fx, i] = LBFGS(x0, fx0, dfx0, p) if ~isfield(p, 'SIG'), p.SIG = 0.5; end % default for line search quality n = length(x0); k = 0; ok = 0; x = x0; fx = fx0; bs = -1/p.MSR; if isfield(p, 'mem'), m = p.mem; else m = min(100, n); end % set memory size a = zeros(1, m); t = zeros(n, m); y = zeros(n, m); % allocate memory i = p.length < 0; % initialize resource counter while i < abs(p.length) q = dfx0; for j = rem(k-1:-1:max(0,k-m),m)+1 a(j) = t(:,j)'*q/rho(j); q = q-a(j)*y(:,j); end if k == 0, r = -q/(q'*q); else r = -t(:,j)'*y(:,j)/(y(:,j)'*y(:,j))*q; end for j = rem(max(0,k-m):k-1,m)+1 r = r-t(:,j)*(a(j)+y(:,j)'*r/rho(j)); end s = r'*dfx0; if s >= 0, r = -dfx0; s = r'*dfx0; k = 0; ok = 0; end b = bs/min(bs,s/p.MSR); % suitable initial step size (usually 1) if isnan(r) | isinf(r) % if nonsense direction i = -i; % try steepest or stop else [x, b, fx0, dfx, i] = lineSearch(x0, fx0, dfx0, r, s, b, i, p); end if i < 0 % if line search failed i = -i; if ok, ok = 0; k = 0; else break; end % try steepest or stop else j = rem(k,m)+1; t(:,j) = x-x0; y(:,j) = dfx-dfx0; rho(j) = t(:,j)'*y(:,j); ok = 1; k = k+1; bs = b*s; end x0 = x; dfx0 = dfx; fx = [fx; fx0]; % replace and add values end function [x, a, fx, df, i] = lineSearch(x0, f0, df0, d, s, a, i, p) if p.length < 0, LIMIT = min(p.MFEPLS, -i-p.length); else LIMIT = p.MFEPLS; end p0.x = 0.0; p0.f = f0; p0.df = df0; p0.s = s; p1 = p0; % init p0 and p1 j = 0; p3.x = a; wp(p0, p.SIG, 0); % set step & Wolfe-Powell conditions if p.verbosity > 2 A = [-a a]/5; nd = norm(d); subplot(212); hold off; plot(0, f0, 'k+'); hold on; plot(nd*A, f0+s*A, 'k-'); xlabel('distance in line search direction'); ylabel('function value'); end while 1 % keep extrapolating as long as necessary ok = 0; while ~ok & j < LIMIT try % try, catch and bisect to safeguard extrapolation evaluation j = j+1; [p3.f p3.df] = f(x0+p3.x*d); p3.s = p3.df'*d; ok = 1; if isnan(p3.f+p3.s) | isinf(p3.f+p3.s) error('Objective function returned Inf or NaN',''); end; catch if p.verbosity > 1, printf('\n'); warning(lasterr); end % warn or silence p3.x = (p1.x+p3.x)/2; ok = 0; p3.f = NaN; % bisect, and retry end end if p.verbosity > 2 plot(nd*p3.x, p3.f, 'b+'); plot(nd*(p3.x+A), p3.f+p3.s*A, 'b-'); drawnow end if wp(p3) | j >= LIMIT, break; end % done? p0 = p1; p1 = p3; % move points back one unit p3.x = p0.x + minCubic(p1.x-p0.x, p1.f-p0.f, p0.s, p1.s, 1); % cubic extrap end while 1 % keep interpolating as long as necessary if p1.f > p3.f, p2 = p3; else p2 = p1; end % make p2 the best so far if wp(p2) > 1 | j >= LIMIT, break; end % done? p2.x = p1.x + minCubic(p3.x-p1.x, p3.f-p1.f, p1.s, p3.s, 0); % cubic interp j = j+1; [p2.f p2.df] = f(x0+p2.x*d); p2.s = p2.df'*d; if p.verbosity > 2 plot(nd*p2.x, p2.f, 'r+'); plot(nd*(p2.x+A), p2.f+p2.s*A, 'r'); drawnow end if wp(p2) > -1 & p2.s > 0 | wp(p2) < -1, p3 = p2; else p1 = p2; end % bracket end x = x0+p2.x*d; fx = p2.f; df = p2.df; a = p2.x; % return the value found if p.length < 0, i = i+j; else i = i+1; end % count func evals or line searches if p.verbosity, printf('%s %6i; value %4.6e\r', p.S, i, fx); end if wp(p2) < 2, i = -i; end % indicate faliure if p.verbosity > 2 if i>0, plot(norm(d)*p2.x, fx, 'go'); end subplot(211); plot(abs(i), fx, '+'); drawnow; end function z = minCubic(x, df, s0, s1, extr) % minimizer of approximating cubic INT = 0.1; EXT = 5.0; % interpolate and extrapolation limits A = -6*df+3*(s0+s1)*x; B = 3*df-(2*s0+s1)*x; if B<0, z = s0*x/(s0-s1); else z = -s0*x*x/(B+sqrt(B*B-A*s0*x)); end if extr % are we extrapolating? if ~isreal(z) | ~isfinite(z) | z < x | z > x*EXT, z = EXT*x; end % fix bad z z = max(z, (1+INT)*x); % extrapolate by at least INT else % else, we are interpolating if ~isreal(z) | ~isfinite(z) | z < 0 | z > x, z = x/2; end; % fix bad z z = min(max(z, INT*x), (1-INT)*x); % at least INT away from the boundaries end function y = wp(p, SIG, RHO) persistent a b c sig rho; if nargin == 3 % if three arguments, then set up the Wolfe-Powell conditions a = RHO*p.s; b = p.f; c = -SIG*p.s; sig = SIG; rho = RHO; y= 0; else if p.f > a*p.x+b % function value too large? if a > 0, y = -1; else y = -2; end else if p.s < -c, y = 0; elseif p.s > c, y = 1; else y = 2; end % if sig*abs(p.s) > c, a = rho*p.s; b = p.f-a*p.x; c = sig*abs(p.s); end end end function [fx, dfx] = f(varargin) persistent F p; if nargout == 0 p = varargin; if ischar(p{1}), F = str2func(p{1}); else F = p{1}; end else [fx, dfx] = F(rewrap(p{2}, varargin{1}), p{3:end}); dfx = unwrap(dfx); end function v = unwrap(s) % extract num elements of s (any type) into v (vector) v = []; if isnumeric(s) v = s(:); % numeric values are recast to column vector elseif isstruct(s) v = unwrap(struct2cell(orderfields(s))); % alphabetize, conv to cell, recurse elseif iscell(s) % cell array elements are for i = 1:numel(s), v = [v; unwrap(s{i})]; end % handled sequentially end % other types are ignored function [s v] = rewrap(s, v) % map elements of v (vector) onto s (any type) if isnumeric(s) if numel(v) < numel(s) error('The vector for conversion contains too few elements') end s = reshape(v(1:numel(s)), size(s)); % numeric values are reshaped v = v(numel(s)+1:end); % remaining arguments passed on elseif isstruct(s) [s p] = orderfields(s); p(p) = 1:numel(p); % alphabetize, store ordering [t v] = rewrap(struct2cell(s), v); % convert to cell, recurse s = orderfields(cell2struct(t,fieldnames(s),1),p); % conv to struct, reorder elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially [s{i} v] = rewrap(s{i}, v); end end % other types are not processed function printf(varargin) fprintf(varargin{:}); if exist('fflush','builtin'), fflush(stdout); end gpml-matlab-v3.2-2013-01-15/util/lbfgsb/000755 000765 000024 00000000000 12077253603 017272 5ustar00carlstaff000000 000000 gpml-matlab-v3.2-2013-01-15/util/minimize_lbfgsb_gradfun.m000644 000765 000024 00000004526 11511036276 023063 0ustar00carlstaff000000 000000 function G = minimize_lbfgsb_gradfun(X,varargin) % extract input arguments varargin = varargin{1}; strctX = varargin{2}; f = varargin{1}; % global variables serve as communication interface between calls global minimize_lbfgsb_iteration_number global minimize_lbfgsb_objective global minimize_lbfgsb_gradient global minimize_lbfgsb_X if norm(unwrap(X)-unwrap(minimize_lbfgsb_X))>1e-10 [y,G] = feval(f,rewrap(strctX,X),varargin{3:end}); else y = minimize_lbfgsb_objective(minimize_lbfgsb_iteration_number); G = minimize_lbfgsb_gradient; end % memorise gradient and position minimize_lbfgsb_gradient = G; minimize_lbfgsb_X = X; G = unwrap(G); % Extract the numerical values from "s" into the column vector "v". The % variable "s" can be of any type, including struct and cell array. % Non-numerical elements are ignored. See also the reverse rewrap.m. function v = unwrap(s) v = []; if isnumeric(s) v = s(:); % numeric values are recast to column vector elseif isstruct(s) v = unwrap(struct2cell(orderfields(s)));% alphabetize, conv to cell, recurse elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially v = [v; unwrap(s{i})]; end end % other types are ignored % Map the numerical elements in the vector "v" onto the variables "s" which can % be of any type. The number of numerical elements must match; on exit "v" % should be empty. Non-numerical entries are just copied. See also unwrap.m. function [s v] = rewrap(s, v) if isnumeric(s) if numel(v) < numel(s) error('The vector for conversion contains too few elements') end s = reshape(v(1:numel(s)), size(s)); % numeric values are reshaped v = v(numel(s)+1:end); % remaining arguments passed on elseif isstruct(s) [s p] = orderfields(s); p(p) = 1:numel(p); % alphabetize, store ordering [t v] = rewrap(struct2cell(s), v); % convert to cell, recurse s = orderfields(cell2struct(t,fieldnames(s),1),p); % conv to struct, reorder elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially [s{i} v] = rewrap(s{i}, v); end end % other types are not processed gpml-matlab-v3.2-2013-01-15/util/minimize_lbfgsb.m000644 000765 000024 00000010574 11511036276 021355 0ustar00carlstaff000000 000000 function [X, fX, i] = minimize_lbfgsb(X, f, length, varargin) % Minimize a differentiable multivariate function using quasi Newton. % % Usage: [X, fX, i] = minimize_lbfgsb(X, f, length, P1, P2, P3, ... ) % % X initial guess; may be of any type, including struct and cell array % f the name or pointer to the function to be minimized. The function % f must return two arguments, the value of the function, and it's % partial derivatives wrt the elements of X. The partial derivative % must have the same type as X. % length length of the run; if it is positive, it gives the maximum number of % line searches, if negative its absolute gives the maximum allowed % number of function evaluations. Optionally, length can have a second % component, which will indicate the reduction in function value to be % expected in the first line-search (defaults to 1.0). % P1, P2 ... parameters are passed to the function f. % % X the returned solution % fX vector of function values indicating progress made % i number of iterations (line searches or function evaluations, % depending on the sign of "length") used at termination. % % The function returns when either its length is up, or if no further progress % can be made (ie, we are at a (local) minimum, or so close that due to % numerical problems, we cannot get any closer). NOTE: If the function % terminates within a few iterations, it could be an indication that the % function values and derivatives are not consistent (ie, there may be a bug in % the implementation of your "f" function). % % Copyright (C) 2010 by Hannes Nickisch, 2010-02-05 % global variables serve as communication interface between calls global minimize_lbfgsb_iteration_number global minimize_lbfgsb_objective global minimize_lbfgsb_gradient global minimize_lbfgsb_X % init global variables minimize_lbfgsb_iteration_number = 0; minimize_lbfgsb_objective = Inf; minimize_lbfgsb_gradient = 0*unwrap(X); minimize_lbfgsb_X = X; X0 = X; lb = -Inf*ones(size(unwrap(X0))); ub = Inf*ones(size(unwrap(X0))); maxiter = abs(length); % max number of iterations % no callback routine used so far % m is the number of saved vectors used to estimate the Hessian % factr is the precision 1e-12 X = lbfgsb( unwrap(X0), lb, ub, 'minimize_lbfgsb_objfun', ... 'minimize_lbfgsb_gradfun', ... {f,X0,varargin{:}}, [], ... 'maxiter',maxiter, 'm',4, 'factr',1e-12, 'pgtol',1e-5); i = minimize_lbfgsb_iteration_number; fX = minimize_lbfgsb_objective; X = rewrap(X0,X); % clear global variables clear minimize_lbfgsb_iteration_number clear minimize_lbfgsb_objective clear minimize_lbfgsb_gradient clear minimize_lbfgsb_X % Extract the numerical values from "s" into the column vector "v". The % variable "s" can be of any type, including struct and cell array. % Non-numerical elements are ignored. See also the reverse rewrap.m. function v = unwrap(s) v = []; if isnumeric(s) v = s(:); % numeric values are recast to column vector elseif isstruct(s) v = unwrap(struct2cell(orderfields(s)));% alphabetize, conv to cell, recurse elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially v = [v; unwrap(s{i})]; end end % other types are ignored % Map the numerical elements in the vector "v" onto the variables "s" which can % be of any type. The number of numerical elements must match; on exit "v" % should be empty. Non-numerical entries are just copied. See also unwrap.m. function [s v] = rewrap(s, v) if isnumeric(s) if numel(v) < numel(s) error('The vector for conversion contains too few elements') end s = reshape(v(1:numel(s)), size(s)); % numeric values are reshaped v = v(numel(s)+1:end); % remaining arguments passed on elseif isstruct(s) [s p] = orderfields(s); p(p) = 1:numel(p); % alphabetize, store ordering [t v] = rewrap(struct2cell(s), v); % convert to cell, recurse s = orderfields(cell2struct(t,fieldnames(s),1),p); % conv to struct, reorder elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially [s{i} v] = rewrap(s{i}, v); end end % other types are not processed gpml-matlab-v3.2-2013-01-15/util/gpr.m000644 000765 000024 00000004667 11511036276 017013 0ustar00carlstaff000000 000000 function varargout = gpr(hyper, cov, x, y, xs) % gpr - Gaussian process regression, with a named covariance function. Two % modes are possible: training and prediction: if no test data are given, the % function returns minus the log likelihood and its partial derivatives with % respect to the hyperparameters; this mode is used to fit the hyperparameters. % If test data are given, then (marginal) Gaussian predictions are computed, % whose mean and variance are returned. Note that in cases where the covariance % function has noise contributions, the variance returned in s2 is for noisy % test targets; if you want the variance of the noise-free latent function, you % must substract the noise variance. % % usage: [nlZ dnlZ] = gpr(hyp, cov, x, y) % or: [mu s2] = gpr(hyp, cov, x, y, xs) % % where: % % hyp is a (column) vector of log hyperparameters % cov is the covariance function % x is a n by D matrix of training inputs % y is a (column) vector (of size n) of targets % xs is a ns by D matrix of test inputs % nlZ is the returned value of the negative log marginal likelihood % dnlZ is a (column) vector of partial derivatives of the negative % log marginal likelihood wrt each log hyperparameter % mu is a (column) vector (of size nn) of prediced means % s2 is a (column) vector (of size nn) of predicted variances % % For more help on covariance functions, see "help covFunctions". % % Copyright (c) 2010 Carl Edward Rasmussen and Hannes Nickisch 2010-06-18. err = 'we need cov = {''covSum'', {cov1, ..,''covNoise'', .., covP} }; to map to gp.m'; if ~strcmp(cov{1},'covSum'), error(err), end id = 0; nhyp = zeros(length(cov{2}),1); for i=1:length(cov{2}) if strcmp(cov{2}{i},'covNoise'), id = i; break, end nhyp(i) = eval(feval(cov{2}{i})); end if id==0, error(err), else nhyp = sum(nhyp(1:id-1)); end cov{2} = cov{2}([1:id-1,id+1:end]); hyp.lik = hyper(nhyp+1); hyp.cov = hyper([1:nhyp,nhyp+2:end]); % Note, this function is just a wrapper provided for backward compatibility, % the functionality is now provided by the more general gp function. lik = @likGauss; inf = @infExact; mean = @meanZero; varargout = cell(nargout, 1); % allocate the right number of output arguments if nargin==4 [varargout{:}] = gp(hyp,inf,mean,cov,lik,x,y); if nargout>1, varargout{2} = [varargout{2}.lik; varargout{2}.cov]; end else [varargout{:}] = gp(hyp,inf,mean,cov,lik,x,y,xs); end gpml-matlab-v3.2-2013-01-15/util/sq_dist.m000644 000765 000024 00000003657 11511036276 017667 0ustar00carlstaff000000 000000 % sq_dist - a function to compute a matrix of all pairwise squared distances % between two sets of vectors, stored in the columns of the two matrices, a % (of size D by n) and b (of size D by m). If only a single argument is given % or the second matrix is empty, the missing matrix is taken to be identical % to the first. % % Usage: C = sq_dist(a, b) % or: C = sq_dist(a) or equiv.: C = sq_dist(a, []) % % Where a is of size Dxn, b is of size Dxm (or empty), C is of size nxm. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-12-13. function C = sq_dist(a, b) if nargin<1 || nargin>3 || nargout>1, error('Wrong number of arguments.'); end bsx = exist('bsxfun','builtin'); % since Matlab R2007a 7.4.0 and Octave 3.0 if ~bsx, bsx = exist('bsxfun'); end % bsxfun is not yes "builtin" in Octave [D, n] = size(a); % Computation of a^2 - 2*a*b + b^2 is less stable than (a-b)^2 because numerical % precision can be lost when both a and b have very large absolute value and the % same sign. For that reason, we subtract the mean from the data beforehand to % stabilise the computations. This is OK because the squared error is % independent of the mean. if nargin==1 % subtract mean mu = mean(a,2); if bsx a = bsxfun(@minus,a,mu); else a = a - repmat(mu,1,size(a,2)); end b = a; m = n; else [d, m] = size(b); if d ~= D, error('Error: column lengths must agree.'); end mu = (m/(n+m))*mean(b,2) + (n/(n+m))*mean(a,2); if bsx a = bsxfun(@minus,a,mu); b = bsxfun(@minus,b,mu); else a = a - repmat(mu,1,n); b = b - repmat(mu,1,m); end end if bsx % compute squared distances C = bsxfun(@plus,sum(a.*a,1)',bsxfun(@minus,sum(b.*b,1),2*a'*b)); else C = repmat(sum(a.*a,1)',1,m) + repmat(sum(b.*b,1),n,1) - 2*a'*b; end C = max(C,0); % numerical noise can cause C to negative i.e. C > -1e-14 gpml-matlab-v3.2-2013-01-15/util/make.m000644 000765 000024 00000004064 11511036276 017127 0ustar00carlstaff000000 000000 % For compilation under Linux/Octave you need to install the package % octavex.x-headers. % Under Windows/Octave, compilation works out of the box. % % There is a big variety of platforms running Matlab around i.e. % 'SUN4','SOL2','HP700','SGI','SGI64','IBM_RS','ALPHA', % 'LNX86','GLNX86','GLNXA64', % 'MAC', 'MACI' % the following script is only tested for the most common ones (Mac,Linux,Win) % but should work with minor modifications on your architecture. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch 2010-12-16. OCTAVE = exist('OCTAVE_VERSION') ~= 0; % check if we run Matlab or Octave %% 0) change directory me = mfilename; % what is my filename mydir = which(me); mydir = mydir(1:end-2-numel(me)); % where am I located cd(mydir) if OCTAVE %% compile solve_chol mkoctfile --mex solve_chol.c delete solve_chol.o else % MATLAB %% compile solve_chol if ispc % Windows libext = 'lib'; % remove the trailing underscore after the Lapack function "dpotrs" % we do that in a clumsy way since there is no sed/awk under Windows fid = fopen('solve_chol.c','r'); fid_win = fopen('solve_chol2XXXX.c','w'); while 1 tline = fgetl(fid); if ~ischar(tline), break, end id = strfind(tline,'dpotrs'); if numel(id) tline = [tline(1:id-1),'dpotrs',tline(id+7:end)]; end fprintf(fid_win,[tline,'\n']); end fclose(fid_win); fclose(fid); if numel(strfind(computer,'64')) % 64 bit ospath = 'extern/lib/win64/microsoft'; else % 32 bit ospath = 'extern/lib/win32/lcc'; end % compile comp_cmd = 'mex -O solve_chol2XXXX.c -output solve_chol'; eval([comp_cmd,' ''',matlabroot,'/',ospath,'/libmwlapack.',libext,'''']) delete solve_chol2XXXX* else % Linux, MAC, Solaris mex -O -lmwlapack solve_chol.c end endgpml-matlab-v3.2-2013-01-15/util/unwrap.m000644 000765 000024 00000001213 11511036276 017517 0ustar00carlstaff000000 000000 % Extract the numerical values from "s" into the column vector "v". The % variable "s" can be of any type, including struct and cell array. % Non-numerical elements are ignored. See also the reverse rewrap.m. function v = unwrap(s) v = []; if isnumeric(s) v = s(:); % numeric values are recast to column vector elseif isstruct(s) v = unwrap(struct2cell(orderfields(s))); % alphabetize, conv to cell, recurse elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially v = [v; unwrap(s{i})]; end end % other types are ignored gpml-matlab-v3.2-2013-01-15/util/brentmin.m000644 000765 000024 00000007574 11511036276 020041 0ustar00carlstaff000000 000000 %% BRENTMIN: Brent's minimization method in one dimension function [xmin,fmin,funccount,varargout] = ... brentmin(xlow,xupp,Nitmax,tol,f,nout,varargin) % code taken from % § 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 % % Copyright (c) by Hannes Nickisch 2010-01-10. if nargin<6, nout = 0; end varargout = cell(nout,1); % tolerance is no smaller than machine's floating point precision tol = max(tol,eps); % Evaluate endpoints fa = f(xlow,varargin{:}); fb = f(xupp,varargin{:}); 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 = 0.0; e = 0.0; x = xf; [fx,varargout{:}] = f(x,varargin{:}); funccount = 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 = 1; % Is a parabolic fit possible if abs(e) > tol1 % Yes, so fit parabola gs = 0; 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; end q = abs(q); r = e; e = d; % Is the parabola acceptable if ( (abs(p)q*(a-xf)) && (p= xm, e = a-xf; else e = b-xf; end d = c*e; end % The function must not be evaluated too close to xf si = sign(d) + (d == 0); x = xf + si * max( abs(d), tol1 ); [fu,varargout{:}] = f(x,varargin{:}); funccount = funccount + 1; % Update a, b, v, w, x, xm, tol1, tol2 if fu <= fx if x >= xf, a = xf; else b = xf; end v = w; fv = fw; w = xf; fw = fx; xf = x; fx = fu; else % fu > fx if x < xf, a = x; else b = x; end if ( (fu <= fw) || (w == xf) ) v = w; fv = fw; w = x; fw = fu; elseif ( (fu <= fv) || (v == xf) || (v == w) ) v = x; fv = fu; end end 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 % warning(sprintf(['Maximum number of iterations (%d) exceeded:', ... % 'precision is not guaranteed'],Nitmax)) % fprintf('[%1.3f,%1.3f,%1.3f]\n',xlow,xf,xupp) break end end % while % check that endpoints are less than the minimum found if ( (fa < fx) && (fa <= fb) ) xf = xlow; fx = fa; elseif fb < fx xf = xupp; fx = fb; end fmin = fx; xmin = xf; gpml-matlab-v3.2-2013-01-15/util/minimize.m000644 000765 000024 00000025667 12047277603 020055 0ustar00carlstaff000000 000000 function [X, fX, i] = minimize(X, f, length, varargin) % Minimize a differentiable multivariate function using conjugate gradients. % % Usage: [X, fX, i] = minimize(X, f, length, P1, P2, P3, ... ) % % X initial guess; may be of any type, including struct and cell array % f the name or pointer to the function to be minimized. The function % f must return two arguments, the value of the function, and it's % partial derivatives wrt the elements of X. The partial derivative % must have the same type as X. % length length of the run; if it is positive, it gives the maximum number of % line searches, if negative its absolute gives the maximum allowed % number of function evaluations. Optionally, length can have a second % component, which will indicate the reduction in function value to be % expected in the first line-search (defaults to 1.0). % P1, P2, ... parameters are passed to the function f. % % X the returned solution % fX vector of function values indicating progress made % i number of iterations (line searches or function evaluations, % depending on the sign of "length") used at termination. % % The function returns when either its length is up, or if no further progress % can be made (ie, we are at a (local) minimum, or so close that due to % numerical problems, we cannot get any closer). NOTE: If the function % terminates within a few iterations, it could be an indication that the % function values and derivatives are not consistent (ie, there may be a bug in % the implementation of your "f" function). % % The Polack-Ribiere flavour of conjugate gradients is used to compute search % directions, and a line search using quadratic and cubic polynomial % approximations and the Wolfe-Powell stopping criteria is used together with % the slope ratio method for guessing initial step sizes. Additionally a bunch % of checks are made to make sure that exploration is taking place and that % extrapolation will not be unboundedly large. % % See also: checkgrad % % Copyright (C) 2001 - 2010 by Carl Edward Rasmussen, 2010-01-03 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. % The code falls naturally into 3 parts, after the initial line search is % started in the direction of steepest descent. 1) we first enter a while loop % which uses point 1 (p1) and (p2) to compute an extrapolation (p3), until we % have extrapolated far enough (Wolfe-Powell conditions). 2) if necessary, we % enter the second loop which takes p2, p3 and p4 chooses the subinterval % containing a (local) minimum, and interpolates it, unil an acceptable point % is found (Wolfe-Powell conditions). Note, that points are always maintained % in order p0 <= p1 <= p2 < p3 < p4. 3) compute a new search direction using % conjugate gradients (Polack-Ribiere flavour), or revert to steepest if there % was a problem in the previous line-search. Return the best value so far, if % two consecutive line-searches fail, or whenever we run out of function % evaluations or line-searches. During extrapolation, the "f" function may fail % either with an error or returning Nan or Inf, and minimize should handle this % gracefully. if max(size(length)) == 2, red=length(2); length=length(1); else red=1; end if length>0, S='Linesearch'; else S='Function evaluation'; end i = 0; % zero the run length counter ls_failed = 0; % no previous line search has failed [f0 df0] = feval(f, X, varargin{:}); % get function value and gradient Z = X; X = unwrap(X); df0 = unwrap(df0); fprintf('%s %6i; Value %4.6e\r', S, i, f0); if exist('fflush','builtin') fflush(stdout); end fX = f0; i = i + (length<0); % count epochs?! s = -df0; d0 = -s'*s; % initial search direction (steepest) and slope x3 = red/(1-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); end while 1 % keep extrapolating as long as necessary x2 = 0; f2 = f0; d2 = d0; f3 = f0; df3 = df0; success = 0; while ~success && M > 0 try M = M - 1; i = i + (length<0); % count epochs?! [f3 df3] = feval(f, rewrap(Z,X+x3*s), varargin{:}); df3 = unwrap(df3); if isnan(f3) || isinf(f3) || any(isnan(df3)+isinf(df3)), error(''), end success = 1; catch % catch any error which occured in f x3 = (x2+x3)/2; % bisect and try again end end if f3 < F0, X0 = X+x3*s; F0 = f3; dF0 = df3; end % keep best values d3 = df3'*s; % new slope if d3 > SIG*d0 || f3 > f0+x3*RHO*d0 || M == 0 % are we done extrapolating? break end 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); x3 = x1-d1*(x2-x1)^2/(B+sqrt(B*B-A*d1*(x2-x1))); % num. error possible, ok! if ~isreal(x3) || isnan(x3) || isinf(x3) || x3 < 0 % num prob | wrong sign? x3 = x2*EXT; % extrapolate maximum amount elseif x3 > x2*EXT % new point beyond extrapolation limit? x3 = x2*EXT; % extrapolate maximum amount elseif x3 < x2+INT*(x2-x1) % new point too close to previous point? x3 = x2+INT*(x2-x1); end end % end extrapolation while (abs(d3) > -SIG*d0 || f3 > f0+x3*RHO*d0) && M > 0 % keep interpolating if d3 > 0 || 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 end 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); x3 = x2+(sqrt(B*B-A*d2*(x4-x2)^2)-B)/A; % num. error possible, ok! end if isnan(x3) || isinf(x3) x3 = (x2+x4)/2; % if we had a numerical problem then bisect end x3 = max(min(x3, x4-INT*(x4-x2)),x2+INT*(x4-x2)); % don't accept too close [f3 df3] = feval(f, rewrap(Z,X+x3*s), varargin{:}); df3 = unwrap(df3); if f3 < F0, X0 = X+x3*s; F0 = f3; dF0 = df3; end % keep best values M = M - 1; i = i + (length<0); % count epochs?! d3 = df3'*s; % new slope end % end interpolation if abs(d3) < -SIG*d0 && f3 < f0+x3*RHO*d0 % if line search succeeded X = X+x3*s; f0 = f3; fX = [fX' f0]'; % update variables fprintf('%s %6i; Value %4.6e\r', S, i, f0); if exist('fflush','builtin') fflush(stdout); end s = (df3'*df3-df0'*df3)/(df0'*df0)*s - df3; % Polack-Ribiere CG direction df0 = df3; % swap derivatives d3 = d0; d0 = df0'*s; if d0 > 0 % new slope must be negative s = -df0; d0 = -s'*s; % otherwise use steepest direction end x3 = x3 * min(RATIO, d3/(d0-realmin)); % 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 || i > abs(length) % line search failed twice in a row break; % or we ran out of time, so we give up end s = -df0; d0 = -s'*s; % try steepest x3 = 1/(1-d0); ls_failed = 1; % this line search failed end end X = rewrap(Z,X); fprintf('\n'); if exist('fflush','builtin') fflush(stdout); end function v = unwrap(s) % Extract the numerical values from "s" into the column vector "v". The % variable "s" can be of any type, including struct and cell array. % Non-numerical elements are ignored. See also the reverse rewrap.m. v = []; if isnumeric(s) v = s(:); % numeric values are recast to column vector elseif isstruct(s) v = unwrap(struct2cell(orderfields(s))); % alphabetize, conv to cell, recurse elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially v = [v; unwrap(s{i})]; end end % other types are ignored function [s v] = rewrap(s, v) % Map the numerical elements in the vector "v" onto the variables "s" which can % be of any type. The number of numerical elements must match; on exit "v" % should be empty. Non-numerical entries are just copied. See also unwrap.m. if isnumeric(s) if numel(v) < numel(s) error('The vector for conversion contains too few elements') end s = reshape(v(1:numel(s)), size(s)); % numeric values are reshaped v = v(numel(s)+1:end); % remaining arguments passed on elseif isstruct(s) [s p] = orderfields(s); p(p) = 1:numel(p); % alphabetize, store ordering [t v] = rewrap(struct2cell(s), v); % convert to cell, recurse s = orderfields(cell2struct(t,fieldnames(s),1),p); % conv to struct, reorder elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially [s{i} v] = rewrap(s{i}, v); end end % other types are not processed gpml-matlab-v3.2-2013-01-15/util/minimize_lbfgsb_objfun.m000644 000765 000024 00000005207 11511036276 022715 0ustar00carlstaff000000 000000 function y = minimize_lbfgsb_objfun(X,varargin) % extract input arguments varargin = varargin{1}; strctX = varargin{2}; f = varargin{1}; % global variables serve as communication interface between calls global minimize_lbfgsb_iteration_number global minimize_lbfgsb_objective global minimize_lbfgsb_gradient global minimize_lbfgsb_X recompute = 0; if minimize_lbfgsb_iteration_number==0 recompute = 1; else if norm(unwrap(X)-unwrap(minimize_lbfgsb_X))>1e-10 recompute = 1; end end if recompute [y,G] = feval(f,rewrap(strctX,X),varargin{3:end}); else y = minimize_lbfgsb_objective(minimize_lbfgsb_iteration_number); G = minimize_lbfgsb_gradient; end % increase global counter, memorise objective function, gradient and position minimize_lbfgsb_iteration_number = minimize_lbfgsb_iteration_number+1; minimize_lbfgsb_objective(minimize_lbfgsb_iteration_number,1) = y; minimize_lbfgsb_gradient = G; minimize_lbfgsb_X = X; % Extract the numerical values from "s" into the column vector "v". The % variable "s" can be of any type, including struct and cell array. % Non-numerical elements are ignored. See also the reverse rewrap.m. function v = unwrap(s) v = []; if isnumeric(s) v = s(:); % numeric values are recast to column vector elseif isstruct(s) v = unwrap(struct2cell(orderfields(s)));% alphabetize, conv to cell, recurse elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially v = [v; unwrap(s{i})]; end end % other types are ignored % Map the numerical elements in the vector "v" onto the variables "s" which can % be of any type. The number of numerical elements must match; on exit "v" % should be empty. Non-numerical entries are just copied. See also unwrap.m. function [s v] = rewrap(s, v) if isnumeric(s) if numel(v) < numel(s) error('The vector for conversion contains too few elements') end s = reshape(v(1:numel(s)), size(s)); % numeric values are reshaped v = v(numel(s)+1:end); % remaining arguments passed on elseif isstruct(s) [s p] = orderfields(s); p(p) = 1:numel(p); % alphabetize, store ordering [t v] = rewrap(struct2cell(s), v); % convert to cell, recurse s = orderfields(cell2struct(t,fieldnames(s),1),p); % conv to struct, reorder elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially [s{i} v] = rewrap(s{i}, v); end end % other types are not processed gpml-matlab-v3.2-2013-01-15/util/gauher.m000644 000765 000024 00000004305 12077070414 017463 0ustar00carlstaff000000 000000 % compute abscissas and weight factors for Gaussian-Hermite quadrature % % CALL: [x,w] = gauher(N) % % x = base points (abscissas) % w = weight factors % N = number of base points (abscissas) (integrates an up to (2N-1)th order % polynomial exactly) % % p(x)=exp(-x^2/2)/sqrt(2*pi), a =-Inf, b = Inf % % The Gaussian Quadrature integrates a (2n-1)th order % polynomial exactly and the integral is of the form % b N % Int ( p(x)* F(x) ) dx = Sum ( w_j* F( x_j ) ) % a j=1 % % this procedure uses the coefficients a(j), b(j) of the % recurrence relation % % b p (x) = (x - a ) p (x) - b p (x) % j j j j-1 j-1 j-2 % % for the various classical (normalized) orthogonal polynomials, % and the zero-th moment % % 1 = integral w(x) dx % % of the given polynomial's weight function w(x). Since the % polynomials are orthonormalized, the tridiagonal matrix is % guaranteed to be symmetric. function [x,w]=gauher(N) if N==20 % return precalculated values x=[ -7.619048541679757;-6.510590157013656;-5.578738805893203; -4.734581334046057;-3.943967350657318;-3.18901481655339 ; -2.458663611172367;-1.745247320814127;-1.042945348802751; -0.346964157081356; 0.346964157081356; 1.042945348802751; 1.745247320814127; 2.458663611172367; 3.18901481655339 ; 3.943967350657316; 4.734581334046057; 5.578738805893202; 6.510590157013653; 7.619048541679757]; w=[ 0.000000000000126; 0.000000000248206; 0.000000061274903; 0.00000440212109 ; 0.000128826279962; 0.00183010313108 ; 0.013997837447101; 0.061506372063977; 0.161739333984 ; 0.260793063449555; 0.260793063449555; 0.161739333984 ; 0.061506372063977; 0.013997837447101; 0.00183010313108 ; 0.000128826279962; 0.00000440212109 ; 0.000000061274903; 0.000000000248206; 0.000000000000126 ]; else b = sqrt( (1:N-1)/2 )'; [V,D] = eig( diag(b,1) + diag(b,-1) ); w = V(1,:)'.^2; x = sqrt(2)*diag(D); end gpml-matlab-v3.2-2013-01-15/util/rewrap.m000644 000765 000024 00000001766 11511036276 017520 0ustar00carlstaff000000 000000 % Map the numerical elements in the vector "v" onto the variables "s" which can % be of any type. The number of numerical elements must match; on exit "v" % should be empty. Non-numerical entries are just copied. See also unwrap.m. function [s v] = rewrap(s, v) if isnumeric(s) if numel(v) < numel(s) error('The vector for conversion contains too few elements') end s = reshape(v(1:numel(s)), size(s)); % numeric values are reshaped v = v(numel(s)+1:end); % remaining arguments passed on elseif isstruct(s) [s p] = orderfields(s); p(p) = 1:numel(p); % alphabetize, store ordering [t v] = rewrap(struct2cell(s), v); % convert to cell, recurse s = orderfields(cell2struct(t,fieldnames(s),1),p); % conv to struct, reorder elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially [s{i} v] = rewrap(s{i}, v); end end % other types are not processed gpml-matlab-v3.2-2013-01-15/util/binaryGP.m000644 000765 000024 00000011547 11511036276 017731 0ustar00carlstaff000000 000000 function varargout = binaryGP(hyper, approx, covfunc, lik, x, y, xstar) % Approximate binary Gaussian Process classification. Two modes are possible: % training or testing: if no test cases are supplied, then the approximate % negative log marginal likelihood and its partial derivatives wrt the % hyperparameters is computed; this mode is used to fit the hyperparameters. If % test cases are given, then the test set predictive probabilities are % returned. Exact inference is intractible, the function uses a specified % approximation method (see approximations.m), flexible covariance functions % (see covFunctions.m) and likelihood functions (see likelihoods.m). % % usage: [nlZ, dnlZ ] = binaryGP(hyper, approx, covfunc, lik, x, y); % or: [p,mu,s2,nlZ] = binaryGP(hyper, approx, covfunc, lik, x, y, xstar); % % where: % % hyper is a column vector of hyperparameters % approx is a function specifying an approximation method for inference % covfunc is the name of the covariance function (see below) % lik is the name of the likelihood function % x is a n by D matrix of training inputs % y is a (column) vector (of size n) of binary +1/-1 targets % xstar is a nn by D matrix of test inputs % nlZ is the returned value of the negative log marginal likelihood % dnlZ is a (column) vector of partial derivatives of the negative % log marginal likelihood wrt each hyperparameter % p is a (column) vector (of length nn) of predictive probabilities % mu is a (column) vector (of length nn) of predictive latent means % s2 is a (column) vector (of length nn) of predictive latent variances % % The length of the vector of hyperparameters depends on the covariance % function, as specified by the "covfunc" input to the function, specifying the % name of a covariance function. A number of different covariance function are % implemented, and it is not difficult to add new ones. See covFunctions.m for % the details. % % The "lik" input argument specifies the name of the likelihood function (see % likelihoods.m). % % The "approx" input argument to the function specifies an approximation method % (see approximations.m). An approximation method returns a representation of % the approximate Gaussian posterior. Usually, the approximate posterior admits % the form N(m=K*alpha, V=inv(inv(K)+W)), where alpha is a vector and W is % diagonal. The approximation method returns: % % alpha is a (sparse or full column vector) containing inv(K)*m, where K % is the prior covariance matrix and m the approx posterior mean % sW is a (sparse or full column) vector containing diagonal of sqrt(W) % the approximate posterior covariance matrix is inv(inv(K)+W) % L is a (sparse or full) matrix, L = chol(sW*K*sW+eye(n)) % % In cases where the approximate posterior variance does not admit the form % V=inv(inv(K)+W) with diagonal W, L contains instead -inv(K+inv(W)), and sW % is unused. % % The alpha parameter may be sparse. In that case sW and L can either be sparse % or full (retaining only the non-zero rows and columns, as indicated by the % sparsity structure of alpha). The L paramter is allowed to be empty, in % which case it will be computed. % % The function can conveniently be used with the "minimize" function to train % a Gaussian Process, eg: % % [hyper, fX, i] = minimize(hyper, 'binaryGP', length, 'approxEP', 'covSEiso', 'logistic', x, y); % % where "length" gives the length of the run: if it is positive, it gives the % maximum number of line searches, if negative its absolute gives the maximum % allowed number of function evaluations. % % Copyright (c) 2007 Carl Edward Rasmussen and Hannes Nickisch, 2010-07-21. if nargin<6 || nargin>7 disp('Usage: [nlZ, dnlZ ] = binaryGP(hyper,approx,covfunc,lik,x,y);') disp(' or: [p,mu,s2,nlZ] = binaryGP(hyper,approx,covfunc,lik,x,y,xstar);') return end % Note, this function is just a wrapper provided for backward compatibility, % the functionality is now provided by the more general gp function. if strcmp(lik,'logistic') lik = @likLogistic; elseif strcmpi(lik,'cumgauss') lik = @likErf; else error('Allowable likelihoods: logistic and cumGauss.'); end if strcmp(approx,'approxLA') inf = @infLaplace; elseif strcmp(approx,'approxEP') inf = @infEP; else error('Allowable approximations: approxLA and approxEP.'); end mean = @meanZero; varargout = cell(nargout, 1); % allocate the right number of output arguments hyp.cov = hyper; if nargin==6 [varargout{:}] = gp(hyp,inf,mean,covfunc,lik,x,y); if nargout>1, varargout{2} = varargout{2}.cov; end else [ymu,ys2,fmu,fs2] = gp(hyp,inf,mean,covfunc,lik,x,y,xstar); if nargout>0, varargout{1} = (1+ymu)/2; end if nargout>1, varargout{2} = fmu; end if nargout>2, varargout{3} = fs2; end if nargout>3, varargout{4} = gp(hyp,inf,mean,covfunc,lik,x,y); end % nlZ endgpml-matlab-v3.2-2013-01-15/util/elsympol.m000644 000765 000024 00000001273 11511036276 020055 0ustar00carlstaff000000 000000 % Evaluate the order R elementary symmetric polynomial Newton's identity aka % the Newton–Girard formulae: http://en.wikipedia.org/wiki/Newton's_identities % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-01-10. function E = elsympol(Z,R) % evaluate 'power sums' of the individual terms in Z sz = size(Z); P = zeros([sz(1:2),R]); for r=1:R, P(:,:,r) = sum(Z.^r,3); end E = zeros([sz(1:2),R+1]); % E(:,:,r+1) yields polynomial r E(:,:,1) = ones(sz(1:2)); if R==0, return, end % init recursion E(:,:,2) = P(:,:,1); if R==1, return, end % init recursion for r=2:R for i=1:r E(:,:,r+1) = E(:,:,r+1) + P(:,:,i).*E(:,:,r+1-i)*(-1)^(i-1)/r; end end gpml-matlab-v3.2-2013-01-15/util/lbfgsb/matlabscalar.cpp000644 000765 000024 00000001730 11511036275 022421 0ustar00carlstaff000000 000000 #include "matlabscalar.h" #include "matlabexception.h" // Function definitions. // ----------------------------------------------------------------- double& getMatlabScalar (const mxArray* ptr) { if (!mxIsDouble(ptr)) throw MatlabException("Matlab array must be of type double"); if (mxGetNumberOfElements(ptr) != 1) throw MatlabException("The Matlab array must be a scalar"); return *mxGetPr(ptr); } double& createMatlabScalar (mxArray*& ptr) { ptr = mxCreateDoubleScalar(0); return *mxGetPr(ptr); } // Function definitions for class MatlabScalar. // --------------------------------------------------------------- MatlabScalar::MatlabScalar (const mxArray* ptr) : x(getMatlabScalar(ptr)) { } MatlabScalar::MatlabScalar (mxArray*& ptr, double value) : x(createMatlabScalar(ptr)) { x = value; } MatlabScalar::MatlabScalar (MatlabScalar& source) : x(source.x) { } MatlabScalar& MatlabScalar::operator= (double value) { x = value; return *this; } gpml-matlab-v3.2-2013-01-15/util/lbfgsb/program.cpp000644 000765 000024 00000007675 11511036275 021460 0ustar00carlstaff000000 000000 #include "program.h" #include // Function definitions. // ----------------------------------------------------------------- // Copy a C-style string (a null-terminated character array) to a // non-C-style string (a simple character array). The length of the // destination character array is given by "ndest". If the source is // shorter than the destination, the destination is padded with blank // characters. void copyCStrToCharArray (const char* source, char* dest, int ndest) { // Get the length of the source C string. int nsource = strlen(source); // Only perform the copy if the source can fit into the destination. if (nsource < ndest) { // Copy the string. strcpy(dest,source); // Fill in the rest of the string with blanks. for (int i = nsource; i < ndest; i++) dest[i] = ' '; } } // Return true if the two strings are the same. The second input // argument must be a C-style string (a null-terminated character // array), but the first input argument need not be one. We only // compare the two strings up to the length of "cstr". bool strIsEqualToCStr (const char* str, const char* cstr) { // Get the length of the C string. int n = strlen(cstr); return !strncmp(str,cstr,n); } // Function definitions for class Program. // ----------------------------------------------------------------- Program::Program (int n, double* x, double* lb, double* ub, int* btype, int m, int maxiter, double factr, double pgtol) { this->n = n; this->x = x; this->lb = lb; this->ub = ub; this->btype = btype; this->m = m; this->maxiter = maxiter; this->factr = factr; this->pgtol = pgtol; this->iprint = defaultprintlevel; this->owner = false; initStructures(); } Program::Program (int n, int m, int maxiter, double factr, double pgtol) { this->n = n; this->x = new double[n]; this->lb = new double[n]; this->ub = new double[n]; this->btype = new int[n]; this->m = m; this->maxiter = maxiter; this->factr = factr; this->pgtol = pgtol; this->iprint = defaultprintlevel; this->owner = true; initStructures(); } Program::~Program() { if (owner) { delete[] x; delete[] lb; delete[] ub; delete[] btype; } delete[] g; delete[] wa; delete[] iwa; } SolverExitStatus Program::runSolver() { SolverExitStatus status = success; // The return value. // Initialize the objective function and gradient to zero. f = 0; for (int i = 0; i < n; i++) g[i] = 0; // This initial call sets up the structures for L-BFGS. callLBFGS("START"); // Repeat until we've reached the maximum number of iterations. int t = 0; while (true) { // Do something according to the "task" from the previous call to // L-BFGS. if (strIsEqualToCStr(task,"FG")) { // Evaluate the objective function and the gradient of the // objective at the current point. f = computeObjective(n,x); computeGradient(n,x,g); } else if (strIsEqualToCStr(task,"NEW_X")) { // Go to the next iteration and call the iterative callback // routine. t++; iterCallback(t,x,f); // If we've reached the maximum number of iterations, terminate // the optimization. if (t == maxiter) { callLBFGS("STOP"); break; } } else if (strIsEqualToCStr(task,"CONV")) break; else if (strIsEqualToCStr(task,"ABNO")) { status = abnormalTermination; break; } else if (strIsEqualToCStr(task,"ERROR")) { status = errorOnInput; break; } // Call L-BFGS again. callLBFGS(); } return status; } void Program::callLBFGS (const char* cmd) { if (cmd) copyCStrToCharArray(cmd,task,60); setulb_(&n,&m,x,lb,ub,btype,&f,g,&factr,&pgtol,wa,iwa,task,&iprint, csave,lsave,isave,dsave); } void Program::initStructures() { f = 0; g = new double[n]; wa = new double[(2*m + 4)*n + 12*m*(m + 1)]; iwa = new int[3*n]; } gpml-matlab-v3.2-2013-01-15/util/lbfgsb/matlabstring.cpp000644 000765 000024 00000002007 11511036275 022460 0ustar00carlstaff000000 000000 #include "matlabstring.h" #include "matlabexception.h" // Function definitions. // ----------------------------------------------------------------- char* copystring (const char* source) { int n = strlen(source); // The length of the string. char* dest = new char[n+1]; // The return value. strcpy(dest,source); return dest; } // Function definitions for class MatlabString. // ----------------------------------------------------------------- MatlabString::MatlabString (const mxArray* ptr) { s = 0; // Check to make sure the Matlab array is a string. if (!mxIsChar(ptr)) throw MatlabException("Matlab array must be a string (of type CHAR)"); // Get the string passed as a Matlab array. s = mxArrayToString(ptr); if (s == 0) throw MatlabException("Unable to obtain string from Matlab array"); } MatlabString::MatlabString (const MatlabString& source) { s = 0; // Copy the source string. s = copystring(source.s); } MatlabString::~MatlabString() { if (s) mxFree(s); } gpml-matlab-v3.2-2013-01-15/util/lbfgsb/matlabexception.h000644 000765 000024 00000001545 11511036275 022623 0ustar00carlstaff000000 000000 #ifndef INCLUDE_MATLABEXCEPTION #define INCLUDE_MATLABEXCEPTION #include // Class MatlabException // ----------------------------------------------------------------- // This class just makes it easier for me to throw exceptions. Its // functionality really has nothing to do with MATLAB. class MatlabException : public std::exception { public: MatlabException (const char* message) throw(); ~MatlabException() throw() { }; // The copy constructor makes a shallow copy. MatlabException (const MatlabException& source) throw(); // The copy assignment operator makes a shallow copy as well. MatlabException& operator= (const MatlabException& source); // Return the message string. virtual const char* what () const throw() { return message; }; private: const char* message; // The error message. }; #endif gpml-matlab-v3.2-2013-01-15/util/lbfgsb/Makefile000644 000765 000024 00000004265 11541741254 020740 0ustar00carlstaff000000 000000 # path to your Matlab installation # you can find it by the commands 'locate matlab' or 'find / -name "matlab"' MATLAB_HOME=/agbs/share/sw/matlab #MATLAB_HOME=/usr/local/share/matlabR2008b # for 'make mex' you need to provide the variable MEX and have mex set up # for 'make nomex' you need to provide the variables MEX_SUFFIX and MATLAB_LIB # Choose Operating System ###################################################### # For 32 bit Linux, uncomment the following three lines.------------------------ MEX = $(MATLAB_HOME)/bin/mex MEX_SUFFIX = mexglx MATLAB_LIB = -L$(MATLAB_HOME)/bin/glnx86 -lmex # For 64 bit Linux, uncomment the following three lines.------------------------ #MEX = $(MATLAB_HOME)/bin/mex #MEX_SUFFIX = mexa64 #MATLAB_LIB = -L$(MATLAB_HOME)/bin/glnxa64 -lmex # For Mac OS X, uncomment the following three lines.---------------------------- #MEX = /Applications/MATLAB7/bin/mex #MEX_SUFFIX = mexmac #MATLAB_LIB = -L$(MATLAB_HOME)/bin/maci -lmex # Choose FORTRAN compiler ###################################################### # To work with f77, uncomment the following two lines.-------------------------- F77 = f77 FORTRAN_LIB = g2c # To use gfortran, uncomment the following two lines.--------------------------- #F77 = gfortran #FORTRAN_LIB = gfortran # Do not edit below ############################################################ CXX = g++ CFLAGS = -O3 -ffast-math -fomit-frame-pointer -fPIC -Werror -pthread -Wall -ansi FFLAGS = -O3 -fPIC -fexceptions MATLAB_INCLUDE=-I$(MATLAB_HOME)/extern/include TARGET = lbfgsb OBJS = solver.o matlabexception.o matlabscalar.o matlabstring.o \ matlabmatrix.o arrayofmatrices.o program.o matlabprogram.o \ lbfgsb.o %.o: %.cpp $(CXX) $(CFLAGS) $(MATLAB_INCLUDE) -o $@ -c $^ %.o: %.f $(F77) $(FFLAGS) -o $@ -c $^ mex: $(TARGET) clean nomex: $(TARGET)_nomex clean $(TARGET): $(OBJS) $(MEX) -cxx CXX=$(CXX) CC=$(CXX) FC=$(FCC) LD=$(CXX) -l$(FORTRAN_LIB) -lm \ -O -output $@ $^ $(TARGET)_nomex: $(OBJS) $(CXX) $^ -shared -o $(TARGET).$(MEX_SUFFIX) $(MATLAB_LIB) -l$(FORTRAN_LIB) -lm clean: rm -f *.o cp $(TARGET).$(MEX_SUFFIX) ../$(TARGET).mex mv $(TARGET).$(MEX_SUFFIX) .. gpml-matlab-v3.2-2013-01-15/util/lbfgsb/solver.f000644 000765 000024 00000423410 11511036275 020753 0ustar00carlstaff000000 000000 c================ L-BFGS-B (version 2.4) ========================== subroutine setulb(n, m, x, l, u, nbd, f, g, factr, pgtol, wa, iwa, + task, iprint, csave, lsave, isave, dsave) character*60 task, csave logical lsave(4) integer n, m, iprint, + nbd(n), iwa(3*n), isave(44) double precision f, factr, pgtol, x(n), l(n), u(n), g(n), + wa(2*m*n+4*n+11*m*m+8*m), dsave(29) c ************ c c Subroutine setulb c c This subroutine partitions the working arrays wa and iwa, and c then uses the limited memory BFGS method to solve the bound c constrained optimization problem by calling mainlb. c (The direct method will be used in the subspace minimization.) c c n is an integer variable. c On entry n is the dimension of the problem. c On exit n is unchanged. c c m is an integer variable. c On entry m is the maximum number of variable metric corrections c used to define the limited memory matrix. c On exit m is unchanged. c c x is a double precision array of dimension n. c On entry x is an approximation to the solution. c On exit x is the current approximation. c c l is a double precision array of dimension n. c On entry l is the lower bound on x. c On exit l is unchanged. c c u is a double precision array of dimension n. c On entry u is the upper bound on x. c On exit u is unchanged. c c nbd is an integer array of dimension n. c On entry nbd represents the type of bounds imposed on the c variables, and must be specified as follows: c nbd(i)=0 if x(i) is unbounded, c 1 if x(i) has only a lower bound, c 2 if x(i) has both lower and upper bounds, and c 3 if x(i) has only an upper bound. c On exit nbd is unchanged. c c f is a double precision variable. c On first entry f is unspecified. c On final exit f is the value of the function at x. c c g is a double precision array of dimension n. c On first entry g is unspecified. c On final exit g is the value of the gradient at x. c c factr is a double precision variable. c On entry factr >= 0 is specified by the user. The iteration c will stop when c c (f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr*epsmch c c where epsmch is the machine precision, which is automatically c generated by the code. Typical values for factr: 1.d+12 for c low accuracy; 1.d+7 for moderate accuracy; 1.d+1 for extremely c high accuracy. c On exit factr is unchanged. c c pgtol is a double precision variable. c On entry pgtol >= 0 is specified by the user. The iteration c will stop when c c max{|proj g_i | i = 1, ..., n} <= pgtol c c where pg_i is the ith component of the projected gradient. c On exit pgtol is unchanged. c c wa is a double precision working array of length c (2mmax + 4)nmax + 11mmax^2 + 8mmax. c c iwa is an integer working array of length 3nmax. c c task is a working string of characters of length 60 indicating c the current job when entering and quitting this subroutine. c c iprint is an integer variable that must be set by the user. c It controls the frequency and type of output generated: c iprint<0 no output is generated; c iprint=0 print only one line at the last iteration; c 0100 print details of every iteration including x and g; c When iprint > 0, the file iterate.dat will be created to c summarize the iteration. c c csave is a working string of characters of length 60. c c lsave is a logical working array of dimension 4. c On exit with 'task' = NEW_X, the following information is c available: c If lsave(1) = .true. then the initial X has been replaced by c its projection in the feasible set; c If lsave(2) = .true. then the problem is constrained; c If lsave(3) = .true. then each variable has upper and lower c bounds; c c isave is an integer working array of dimension 44. c On exit with 'task' = NEW_X, the following information is c available: c isave(22) = the total number of intervals explored in the c search of Cauchy points; c isave(26) = the total number of skipped BFGS updates before c the current iteration; c isave(30) = the number of current iteration; c isave(31) = the total number of BFGS updates prior the current c iteration; c isave(33) = the number of intervals explored in the search of c Cauchy point in the current iteration; c isave(34) = the total number of function and gradient c evaluations; c isave(36) = the number of function value or gradient c evaluations in the current iteration; c if isave(37) = 0 then the subspace argmin is within the box; c if isave(37) = 1 then the subspace argmin is beyond the box; c isave(38) = the number of free variables in the current c iteration; c isave(39) = the number of active constraints in the current c iteration; c n + 1 - isave(40) = the number of variables leaving the set of c active constraints in the current iteration; c isave(41) = the number of variables entering the set of active c constraints in the current iteration. c c dsave is a double precision working array of dimension 29. c On exit with 'task' = NEW_X, the following information is c available: c dsave(1) = current 'theta' in the BFGS matrix; c dsave(2) = f(x) in the previous iteration; c dsave(3) = factr*epsmch; c dsave(4) = 2-norm of the line search direction vector; c dsave(5) = the machine precision epsmch generated by the code; c dsave(7) = the accumulated time spent on searching for c Cauchy points; c dsave(8) = the accumulated time spent on c subspace minimization; c dsave(9) = the accumulated time spent on line search; c dsave(11) = the slope of the line search function at c the current point of line search; c dsave(12) = the maximum relative step length imposed in c line search; c dsave(13) = the infinity norm of the projected gradient; c dsave(14) = the relative step length in the line search; c dsave(15) = the slope of the line search function at c the starting point of the line search; c dsave(16) = the square of the 2-norm of the line search c direction vector. c c Subprograms called: c c L-BFGS-B Library ... mainlb. c c c References: c c [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited c memory algorithm for bound constrained optimization'', c SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208. c c [2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, ``L-BFGS-B: a c limited memory FORTRAN code for solving bound constrained c optimization problems'', Tech. Report, NAM-11, EECS Department, c Northwestern University, 1994. c c (Postscript files of these papers are available via anonymous c ftp to ece.nwu.edu in the directory pub/lbfgs/lbfgs_bcm.) c c * * * c c NEOS, November 1994. (Latest revision April 1997.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer l1,l2,l3,lws,lr,lz,lt,ld,lwa,lwy,lsy,lss,lwt,lwn,lsnd if (task .eq. 'START') then isave(1) = m*n isave(2) = m**2 isave(3) = 4*m**2 isave(4) = 1 isave(5) = isave(4) + isave(1) isave(6) = isave(5) + isave(1) isave(7) = isave(6) + isave(2) isave(8) = isave(7) + isave(2) isave(9) = isave(8) isave(10) = isave(9) + isave(2) isave(11) = isave(10) + isave(3) isave(12) = isave(11) + isave(3) isave(13) = isave(12) + n isave(14) = isave(13) + n isave(15) = isave(14) + n isave(16) = isave(15) + n endif l1 = isave(1) l2 = isave(2) l3 = isave(3) lws = isave(4) lwy = isave(5) lsy = isave(6) lss = isave(7) lwt = isave(9) lwn = isave(10) lsnd = isave(11) lz = isave(12) lr = isave(13) ld = isave(14) lt = isave(15) lwa = isave(16) call mainlb(n,m,x,l,u,nbd,f,g,factr,pgtol, + wa(lws),wa(lwy),wa(lsy),wa(lss),wa(lwt), + wa(lwn),wa(lsnd),wa(lz),wa(lr),wa(ld),wa(lt), + wa(lwa),iwa(1),iwa(n+1),iwa(2*n+1),task,iprint, + csave,lsave,isave(22),dsave) return end c======================= The end of setulb ============================= subroutine mainlb(n, m, x, l, u, nbd, f, g, factr, pgtol, ws, wy, + sy, ss, wt, wn, snd, z, r, d, t, wa, + index, iwhere, indx2, task, iprint, + csave, lsave, isave, dsave) character*60 task, csave logical lsave(4) integer n, m, iprint, nbd(n), index(n), + iwhere(n), indx2(n), isave(23) double precision f, factr, pgtol, + x(n), l(n), u(n), g(n), z(n), r(n), d(n), t(n), + wa(8*m), ws(n, m), wy(n, m), sy(m, m), ss(m, m), + wt(m, m), wn(2*m, 2*m), snd(2*m, 2*m), dsave(29) c ************ c c Subroutine mainlb c c This subroutine solves bound constrained optimization problems by c using the compact formula of the limited memory BFGS updates. c c n is an integer variable. c On entry n is the number of variables. c On exit n is unchanged. c c m is an integer variable. c On entry m is the maximum number of variable metric c corrections allowed in the limited memory matrix. c On exit m is unchanged. c c x is a double precision array of dimension n. c On entry x is an approximation to the solution. c On exit x is the current approximation. c c l is a double precision array of dimension n. c On entry l is the lower bound of x. c On exit l is unchanged. c c u is a double precision array of dimension n. c On entry u is the upper bound of x. c On exit u is unchanged. c c nbd is an integer array of dimension n. c On entry nbd represents the type of bounds imposed on the c variables, and must be specified as follows: c nbd(i)=0 if x(i) is unbounded, c 1 if x(i) has only a lower bound, c 2 if x(i) has both lower and upper bounds, c 3 if x(i) has only an upper bound. c On exit nbd is unchanged. c c f is a double precision variable. c On first entry f is unspecified. c On final exit f is the value of the function at x. c c g is a double precision array of dimension n. c On first entry g is unspecified. c On final exit g is the value of the gradient at x. c c factr is a double precision variable. c On entry factr >= 0 is specified by the user. The iteration c will stop when c c (f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr*epsmch c c where epsmch is the machine precision, which is automatically c generated by the code. c On exit factr is unchanged. c c pgtol is a double precision variable. c On entry pgtol >= 0 is specified by the user. The iteration c will stop when c c max{|proj g_i | i = 1, ..., n} <= pgtol c c where pg_i is the ith component of the projected gradient. c On exit pgtol is unchanged. c c ws, wy, sy, and wt are double precision working arrays used to c store the following information defining the limited memory c BFGS matrix: c ws, of dimension n x m, stores S, the matrix of s-vectors; c wy, of dimension n x m, stores Y, the matrix of y-vectors; c sy, of dimension m x m, stores S'Y; c ss, of dimension m x m, stores S'S; c wt, of dimension m x m, stores the Cholesky factorization c of (theta*S'S+LD^(-1)L'); see eq. c (2.26) in [3]. c c wn is a double precision working array of dimension 2m x 2m c used to store the LEL^T factorization of the indefinite matrix c K = [-D -Y'ZZ'Y/theta L_a'-R_z' ] c [L_a -R_z theta*S'AA'S ] c c where E = [-I 0] c [ 0 I] c c snd is a double precision working array of dimension 2m x 2m c used to store the lower triangular part of c N = [Y' ZZ'Y L_a'+R_z'] c [L_a +R_z S'AA'S ] c c z(n),r(n),d(n),t(n),wa(8*m) are double precision working arrays. c z is used at different times to store the Cauchy point and c the Newton point. c c c index is an integer working array of dimension n. c In subroutine freev, index is used to store the free and fixed c variables at the Generalized Cauchy Point (GCP). c c iwhere is an integer working array of dimension n used to record c the status of the vector x for GCP computation. c iwhere(i)=0 or -3 if x(i) is free and has bounds, c 1 if x(i) is fixed at l(i), and l(i) .ne. u(i) c 2 if x(i) is fixed at u(i), and u(i) .ne. l(i) c 3 if x(i) is always fixed, i.e., u(i)=x(i)=l(i) c -1 if x(i) is always free, i.e., no bounds on it. c c indx2 is an integer working array of dimension n. c Within subroutine cauchy, indx2 corresponds to the array iorder. c In subroutine freev, a list of variables entering and leaving c the free set is stored in indx2, and it is passed on to c subroutine formk with this information. c c task is a working string of characters of length 60 indicating c the current job when entering and leaving this subroutine. c c iprint is an INTEGER variable that must be set by the user. c It controls the frequency and type of output generated: c iprint<0 no output is generated; c iprint=0 print only one line at the last iteration; c 0100 print details of every iteration including x and g; c When iprint > 0, the file iterate.dat will be created to c summarize the iteration. c c csave is a working string of characters of length 60. c c lsave is a logical working array of dimension 4. c c isave is an integer working array of dimension 23. c c dsave is a double precision working array of dimension 29. c c c Subprograms called c c L-BFGS-B Library ... cauchy, subsm, lnsrlb, formk, c c errclb, prn1lb, prn2lb, prn3lb, active, projgr, c c freev, cmprlb, matupd, formt. c c Minpack2 Library ... timer, dpmeps. c c Linpack Library ... dcopy, ddot. c c c References: c c [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited c memory algorithm for bound constrained optimization'', c SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208. c c [2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, ``L-BFGS-B: FORTRAN c Subroutines for Large Scale Bound Constrained Optimization'' c Tech. Report, NAM-11, EECS Department, Northwestern University, c 1994. c c [3] R. Byrd, J. Nocedal and R. Schnabel "Representations of c Quasi-Newton Matrices and their use in Limited Memory Methods'', c Mathematical Programming 63 (1994), no. 4, pp. 129-156. c c (Postscript files of these papers are available via anonymous c ftp to ece.nwu.edu in the directory pub/lbfgs/lbfgs_bcm.) c c * * * c c NEOS, November 1994. (Latest revision April 1997.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ logical prjctd,cnstnd,boxed,updatd,wrk character*3 word integer i,k,nintol,itfile,iback,nskip, + head,col,iter,itail,iupdat, + nint,nfgv,info,ifun, + iword,nfree,nact,ileave,nenter double precision theta,fold,ddot,dr,rr,tol,dpmeps, + xstep,sbgnrm,ddum,dnorm,dtd,epsmch, + cpu1,cpu2,cachyt,sbtime,lnscht,time1,time2, + gd,gdold,stp,stpmx,time double precision one,zero parameter (one=1.0d0,zero=0.0d0) if (task .eq. 'START') then call timer(time1) c Generate the current machine precision. epsmch = dpmeps() fold = 0.0d0 dnorm = 0.0d0 cpu1 = 0.0d0 gd = 0.0d0 sbgnrm = 0.0d0 stp = 0.0d0 stpmx = 0.0d0 gdold = 0.0d0 dtd = 0.0d0 c Initialize counters and scalars when task='START'. c for the limited memory BFGS matrices: col = 0 head = 1 theta = one iupdat = 0 updatd = .false. iback = 0 itail = 0 ifun = 0 iword = 0 nact = 0 ileave = 0 nenter = 0 c for operation counts: iter = 0 nfgv = 0 nint = 0 nintol = 0 nskip = 0 nfree = n c for stopping tolerance: tol = factr*epsmch c for measuring running time: cachyt = 0 sbtime = 0 lnscht = 0 c 'word' records the status of subspace solutions. word = '---' c 'info' records the termination information. info = 0 itfile = 0 if (iprint .ge. 1) then c open a summary file 'iterate.dat' open (8, file = 'iterate.dat', status = 'unknown') itfile = 8 endif c Check the input arguments for errors. call errclb(n,m,factr,l,u,nbd,task,info,k) if (task(1:5) .eq. 'ERROR') then call prn3lb(n,x,f,task,iprint,info,itfile, + iter,nfgv,nintol,nskip,nact,sbgnrm, + zero,nint,word,iback,stp,xstep,k, + cachyt,sbtime,lnscht) return endif call prn1lb(n,m,l,u,x,iprint,itfile,epsmch) c Initialize iwhere & project x onto the feasible set. call active(n,l,u,nbd,x,iwhere,iprint,prjctd,cnstnd,boxed) c The end of the initialization. else c restore local variables. prjctd = lsave(1) cnstnd = lsave(2) boxed = lsave(3) updatd = lsave(4) nintol = isave(1) itfile = isave(3) iback = isave(4) nskip = isave(5) head = isave(6) col = isave(7) itail = isave(8) iter = isave(9) iupdat = isave(10) nint = isave(12) nfgv = isave(13) info = isave(14) ifun = isave(15) iword = isave(16) nfree = isave(17) nact = isave(18) ileave = isave(19) nenter = isave(20) theta = dsave(1) fold = dsave(2) tol = dsave(3) dnorm = dsave(4) epsmch = dsave(5) cpu1 = dsave(6) cachyt = dsave(7) sbtime = dsave(8) lnscht = dsave(9) time1 = dsave(10) gd = dsave(11) stpmx = dsave(12) sbgnrm = dsave(13) stp = dsave(14) gdold = dsave(15) dtd = dsave(16) c After returning from the driver go to the point where execution c is to resume. if (task(1:5) .eq. 'FG_LN') goto 666 if (task(1:5) .eq. 'NEW_X') goto 777 if (task(1:5) .eq. 'FG_ST') goto 111 if (task(1:4) .eq. 'STOP') then if (task(7:9) .eq. 'CPU') then c restore the previous iterate. call dcopy(n,t,1,x,1) call dcopy(n,r,1,g,1) f = fold endif goto 999 endif endif c Compute f0 and g0. task = 'FG_START' c return to the driver to calculate f and g; reenter at 111. goto 1000 111 continue nfgv = 1 c Compute the infinity norm of the (-) projected gradient. call projgr(n,l,u,nbd,x,g,sbgnrm) if (iprint .ge. 1) then write (6,1002) iter,f,sbgnrm write (itfile,1003) iter,nfgv,sbgnrm,f endif if (sbgnrm .le. pgtol) then c terminate the algorithm. task = 'CONVERGENCE: NORM OF PROJECTED GRADIENT <= PGTOL' goto 999 endif c ----------------- the beginning of the loop -------------------------- 222 continue if (iprint .ge. 99) write (6,1001) iter + 1 iword = -1 c if (.not. cnstnd .and. col .gt. 0) then c skip the search for GCP. call dcopy(n,x,1,z,1) wrk = updatd nint = 0 goto 333 endif cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c c Compute the Generalized Cauchy Point (GCP). c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc call timer(cpu1) call cauchy(n,x,l,u,nbd,g,indx2,iwhere,t,d,z, + m,wy,ws,sy,wt,theta,col,head, + wa(1),wa(2*m+1),wa(4*m+1),wa(6*m+1),nint, + iprint,sbgnrm,info,epsmch) if (info .ne. 0) then c singular triangular system detected; refresh the lbfgs memory. if(iprint .ge. 1) write (6, 1005) info = 0 col = 0 head = 1 theta = one iupdat = 0 updatd = .false. call timer(cpu2) cachyt = cachyt + cpu2 - cpu1 goto 222 endif call timer(cpu2) cachyt = cachyt + cpu2 - cpu1 nintol = nintol + nint c Count the entering and leaving variables for iter > 0; c find the index set of free and active variables at the GCP. call freev(n,nfree,index,nenter,ileave,indx2, + iwhere,wrk,updatd,cnstnd,iprint,iter) nact = n - nfree 333 continue c If there are no free variables or B=theta*I, then c skip the subspace minimization. if (nfree .eq. 0 .or. col .eq. 0) goto 555 cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c c Subspace minimization. c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc call timer(cpu1) c Form the LEL^T factorization of the indefinite c matrix K = [-D -Y'ZZ'Y/theta L_a'-R_z' ] c [L_a -R_z theta*S'AA'S ] c where E = [-I 0] c [ 0 I] if (wrk) call formk(n,nfree,index,nenter,ileave,indx2,iupdat, + updatd,wn,snd,m,ws,wy,sy,theta,col,head,info) if (info .ne. 0) then c nonpositive definiteness in Cholesky factorization; c refresh the lbfgs memory and restart the iteration. if(iprint .ge. 1) write (6, 1006) info = 0 col = 0 head = 1 theta = one iupdat = 0 updatd = .false. call timer(cpu2) sbtime = sbtime + cpu2 - cpu1 goto 222 endif c compute r=-Z'B(xcp-xk)-Z'g (using wa(2m+1)=W'(xcp-x) c from 'cauchy'). call cmprlb(n,m,x,g,ws,wy,sy,wt,z,r,wa,index, + theta,col,head,nfree,cnstnd,info) if (info .ne. 0) goto 444 c call the direct method. call subsm(n,m,nfree,index,l,u,nbd,z,r,ws,wy,theta, + col,head,iword,wa,wn,iprint,info) 444 continue if (info .ne. 0) then c singular triangular system detected; c refresh the lbfgs memory and restart the iteration. if(iprint .ge. 1) write (6, 1005) info = 0 col = 0 head = 1 theta = one iupdat = 0 updatd = .false. call timer(cpu2) sbtime = sbtime + cpu2 - cpu1 goto 222 endif call timer(cpu2) sbtime = sbtime + cpu2 - cpu1 555 continue cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c c Line search and optimality tests. c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c Generate the search direction d:=z-x. do 40 i = 1, n d(i) = z(i) - x(i) 40 continue call timer(cpu1) 666 continue call lnsrlb(n,l,u,nbd,x,f,fold,gd,gdold,g,d,r,t,z,stp,dnorm, + dtd,xstep,stpmx,iter,ifun,iback,nfgv,info,task, + boxed,cnstnd,csave,isave(22),dsave(17)) if (info .ne. 0 .or. iback .ge. 20) then c restore the previous iterate. call dcopy(n,t,1,x,1) call dcopy(n,r,1,g,1) f = fold if (col .eq. 0) then c abnormal termination. if (info .eq. 0) then info = -9 c restore the actual number of f and g evaluations etc. nfgv = nfgv - 1 ifun = ifun - 1 iback = iback - 1 endif task = 'ABNORMAL_TERMINATION_IN_LNSRCH' iter = iter + 1 goto 999 else c refresh the lbfgs memory and restart the iteration. if(iprint .ge. 1) write (6, 1008) if (info .eq. 0) nfgv = nfgv - 1 info = 0 col = 0 head = 1 theta = one iupdat = 0 updatd = .false. task = 'RESTART_FROM_LNSRCH' call timer(cpu2) lnscht = lnscht + cpu2 - cpu1 goto 222 endif else if (task(1:5) .eq. 'FG_LN') then c return to the driver for calculating f and g; reenter at 666. goto 1000 else c calculate and print out the quantities related to the new X. call timer(cpu2) lnscht = lnscht + cpu2 - cpu1 iter = iter + 1 c Compute the infinity norm of the projected (-)gradient. call projgr(n,l,u,nbd,x,g,sbgnrm) c Print iteration information. call prn2lb(n,x,f,g,iprint,itfile,iter,nfgv,nact, + sbgnrm,nint,word,iword,iback,stp,xstep) goto 1000 endif 777 continue c Test for termination. if (sbgnrm .le. pgtol) then c terminate the algorithm. task = 'CONVERGENCE: NORM OF PROJECTED GRADIENT <= PGTOL' goto 999 endif ddum = max(abs(fold), abs(f), one) if ((fold - f) .le. tol*ddum) then c terminate the algorithm. task = 'CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH' if (iback .ge. 10) info = -5 c i.e., to issue a warning if iback>10 in the line search. goto 999 endif c Compute d=newx-oldx, r=newg-oldg, rr=y'y and dr=y's. do 42 i = 1, n r(i) = g(i) - r(i) 42 continue rr = ddot(n,r,1,r,1) if (stp .eq. one) then dr = gd - gdold ddum = -gdold else dr = (gd - gdold)*stp call dscal(n,stp,d,1) ddum = -gdold*stp endif if (dr .le. epsmch*ddum) then c skip the L-BFGS update. nskip = nskip + 1 updatd = .false. if (iprint .ge. 1) write (6,1004) dr, ddum goto 888 endif cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c c Update the L-BFGS matrix. c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc updatd = .true. iupdat = iupdat + 1 c Update matrices WS and WY and form the middle matrix in B. call matupd(n,m,ws,wy,sy,ss,d,r,itail, + iupdat,col,head,theta,rr,dr,stp,dtd) c Form the upper half of the pds T = theta*SS + L*D^(-1)*L'; c Store T in the upper triangular of the array wt; c Cholesky factorize T to J*J' with c J' stored in the upper triangular of wt. call formt(m,wt,sy,ss,col,theta,info) if (info .ne. 0) then c nonpositive definiteness in Cholesky factorization; c refresh the lbfgs memory and restart the iteration. if(iprint .ge. 1) write (6, 1007) info = 0 col = 0 head = 1 theta = one iupdat = 0 updatd = .false. goto 222 endif c Now the inverse of the middle matrix in B is c [ D^(1/2) O ] [ -D^(1/2) D^(-1/2)*L' ] c [ -L*D^(-1/2) J ] [ 0 J' ] 888 continue c -------------------- the end of the loop ----------------------------- goto 222 999 continue call timer(time2) time = time2 - time1 call prn3lb(n,x,f,task,iprint,info,itfile, + iter,nfgv,nintol,nskip,nact,sbgnrm, + time,nint,word,iback,stp,xstep,k, + cachyt,sbtime,lnscht) 1000 continue c Save local variables. lsave(1) = prjctd lsave(2) = cnstnd lsave(3) = boxed lsave(4) = updatd isave(1) = nintol isave(3) = itfile isave(4) = iback isave(5) = nskip isave(6) = head isave(7) = col isave(8) = itail isave(9) = iter isave(10) = iupdat isave(12) = nint isave(13) = nfgv isave(14) = info isave(15) = ifun isave(16) = iword isave(17) = nfree isave(18) = nact isave(19) = ileave isave(20) = nenter dsave(1) = theta dsave(2) = fold dsave(3) = tol dsave(4) = dnorm dsave(5) = epsmch dsave(6) = cpu1 dsave(7) = cachyt dsave(8) = sbtime dsave(9) = lnscht dsave(10) = time1 dsave(11) = gd dsave(12) = stpmx dsave(13) = sbgnrm dsave(14) = stp dsave(15) = gdold dsave(16) = dtd 1001 format (//,'ITERATION ',i5) 1002 format + (/,'At iterate',i5,4x,'f= ',1p,d12.5,4x,'|proj g|= ',1p,d12.5) 1003 format (2(1x,i4),5x,'-',5x,'-',3x,'-',5x,'-',5x,'-',8x,'-',3x, + 1p,2(1x,d10.3)) 1004 format (' ys=',1p,e10.3,' -gs=',1p,e10.3,' BFGS update SKIPPED') 1005 format (/, +' Singular triangular system detected;',/, +' refresh the lbfgs memory and restart the iteration.') 1006 format (/, +' Nonpositive definiteness in Cholesky factorization in formk;',/, +' refresh the lbfgs memory and restart the iteration.') 1007 format (/, +' Nonpositive definiteness in Cholesky factorization in formt;',/, +' refresh the lbfgs memory and restart the iteration.') 1008 format (/, +' Bad direction in the line search;',/, +' refresh the lbfgs memory and restart the iteration.') return end c======================= The end of mainlb ============================= subroutine active(n, l, u, nbd, x, iwhere, iprint, + prjctd, cnstnd, boxed) logical prjctd, cnstnd, boxed integer n, iprint, nbd(n), iwhere(n) double precision x(n), l(n), u(n) c ************ c c Subroutine active c c This subroutine initializes iwhere and projects the initial x to c the feasible set if necessary. c c iwhere is an integer array of dimension n. c On entry iwhere is unspecified. c On exit iwhere(i)=-1 if x(i) has no bounds c 3 if l(i)=u(i) c 0 otherwise. c In cauchy, iwhere is given finer gradations. c c c * * * c c NEOS, November 1994. (Latest revision June 1996.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer nbdd,i double precision zero parameter (zero=0.0d0) c Initialize nbdd, prjctd, cnstnd and boxed. nbdd = 0 prjctd = .false. cnstnd = .false. boxed = .true. c Project the initial x to the easible set if necessary. do 10 i = 1, n if (nbd(i) .gt. 0) then if (nbd(i) .le. 2 .and. x(i) .le. l(i)) then if (x(i) .lt. l(i)) then prjctd = .true. x(i) = l(i) endif nbdd = nbdd + 1 else if (nbd(i) .ge. 2 .and. x(i) .ge. u(i)) then if (x(i) .gt. u(i)) then prjctd = .true. x(i) = u(i) endif nbdd = nbdd + 1 endif endif 10 continue c Initialize iwhere and assign values to cnstnd and boxed. do 20 i = 1, n if (nbd(i) .ne. 2) boxed = .false. if (nbd(i) .eq. 0) then c this variable is always free iwhere(i) = -1 c otherwise set x(i)=mid(x(i), u(i), l(i)). else cnstnd = .true. if (nbd(i) .eq. 2 .and. u(i) - l(i) .le. zero) then c this variable is always fixed iwhere(i) = 3 else iwhere(i) = 0 endif endif 20 continue if (iprint .ge. 0) then if (prjctd) write (6,*) + 'The initial X is infeasible. Restart with its projection.' if (.not. cnstnd) + write (6,*) 'This problem is unconstrained.' endif if (iprint .gt. 0) write (6,1001) nbdd 1001 format (/,'At X0 ',i9,' variables are exactly at the bounds') return end c======================= The end of active ============================= subroutine bmv(m, sy, wt, col, v, p, info) integer m, col, info double precision sy(m, m), wt(m, m), v(2*col), p(2*col) c ************ c c Subroutine bmv c c This subroutine computes the product of the 2m x 2m middle matrix c in the compact L-BFGS formula of B and a 2m vector v; c it returns the product in p. c c m is an integer variable. c On entry m is the maximum number of variable metric corrections c used to define the limited memory matrix. c On exit m is unchanged. c c sy is a double precision array of dimension m x m. c On entry sy specifies the matrix S'Y. c On exit sy is unchanged. c c wt is a double precision array of dimension m x m. c On entry wt specifies the upper triangular matrix J' which is c the Cholesky factor of (thetaS'S+LD^(-1)L'). c On exit wt is unchanged. c c col is an integer variable. c On entry col specifies the number of s-vectors (or y-vectors) c stored in the compact L-BFGS formula. c On exit col is unchanged. c c v is a double precision array of dimension 2col. c On entry v specifies vector v. c On exit v is unchanged. c c p is a double precision array of dimension 2col. c On entry p is unspecified. c On exit p is the product Mv. c c info is an integer variable. c On entry info is unspecified. c On exit info = 0 for normal return, c = nonzero for abnormal return when the system c to be solved by dtrsl is singular. c c Subprograms called: c c Linpack ... dtrsl. c c c * * * c c NEOS, November 1994. (Latest revision June 1996.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer i,k,i2 double precision sum if (col .eq. 0) return c PART I: solve [ D^(1/2) O ] [ p1 ] = [ v1 ] c [ -L*D^(-1/2) J ] [ p2 ] [ v2 ]. c solve Jp2=v2+LD^(-1)v1. p(col + 1) = v(col + 1) do 20 i = 2, col i2 = col + i sum = 0.0d0 do 10 k = 1, i - 1 sum = sum + sy(i,k)*v(k)/sy(k,k) 10 continue p(i2) = v(i2) + sum 20 continue c Solve the triangular system call dtrsl(wt,m,col,p(col+1),11,info) if (info .ne. 0) return c solve D^(1/2)p1=v1. cc do 30 i = 1, col cc p(i) = v(i)/sqrt(sy(i,i)) cc30 continue c PART II: solve [ -D^(1/2) D^(-1/2)*L' ] [ p1 ] = [ p1 ] c [ 0 J' ] [ p2 ] [ p2 ]. c solve J^Tp2=p2. call dtrsl(wt,m,col,p(col+1),01,info) if (info .ne. 0) return c compute p1=-D^(-1/2)(p1-D^(-1/2)L'p2) c =-D^(-1/2)p1+D^(-1)L'p2. do 40 i = 1, col cc p(i) = -p(i)/sqrt(sy(i,i)) combined with do 30 loop cc into the next line p(i) = -v(i)/sy(i,i) 40 continue do 60 i = 1, col sum = 0.d0 do 50 k = i + 1, col sum = sum + sy(k,i)*p(col+k)/sy(i,i) 50 continue p(i) = p(i) + sum 60 continue return end c======================== The end of bmv =============================== subroutine cauchy(n, x, l, u, nbd, g, iorder, iwhere, t, d, xcp, + m, wy, ws, sy, wt, theta, col, head, p, c, wbp, + v, nint, iprint, sbgnrm, info, epsmch) integer n, m, head, col, nint, iprint, info, + nbd(n), iorder(n), iwhere(n) double precision theta, epsmch, + x(n), l(n), u(n), g(n), t(n), d(n), xcp(n), + wy(n, col), ws(n, col), sy(m, m), + wt(m, m), p(2*m), c(2*m), wbp(2*m), v(2*m) c ************ c c Subroutine cauchy c c For given x, l, u, g (with sbgnrm > 0), and a limited memory c BFGS matrix B defined in terms of matrices WY, WS, WT, and c scalars head, col, and theta, this subroutine computes the c generalized Cauchy point (GCP), defined as the first local c minimizer of the quadratic c c Q(x + s) = g's + 1/2 s'Bs c c along the projected gradient direction P(x-tg,l,u). c The routine returns the GCP in xcp. c c n is an integer variable. c On entry n is the dimension of the problem. c On exit n is unchanged. c c x is a double precision array of dimension n. c On entry x is the starting point for the GCP computation. c On exit x is unchanged. c c l is a double precision array of dimension n. c On entry l is the lower bound of x. c On exit l is unchanged. c c u is a double precision array of dimension n. c On entry u is the upper bound of x. c On exit u is unchanged. c c nbd is an integer array of dimension n. c On entry nbd represents the type of bounds imposed on the c variables, and must be specified as follows: c nbd(i)=0 if x(i) is unbounded, c 1 if x(i) has only a lower bound, c 2 if x(i) has both lower and upper bounds, and c 3 if x(i) has only an upper bound. c On exit nbd is unchanged. c c g is a double precision array of dimension n. c On entry g is the gradient of f(x). g must be a nonzero vector. c On exit g is unchanged. c c iorder is an integer working array of dimension n. c iorder will be used to store the breakpoints in the piecewise c linear path and free variables encountered. On exit, c iorder(1),...,iorder(nleft) are indices of breakpoints c which have not been encountered; c iorder(nleft+1),...,iorder(nbreak) are indices of c encountered breakpoints; and c iorder(nfree),...,iorder(n) are indices of variables which c have no bound constraits along the search direction. c c iwhere is an integer array of dimension n. c On entry iwhere indicates only the permanently fixed (iwhere=3) c or free (iwhere= -1) components of x. c On exit iwhere records the status of the current x variables. c iwhere(i)=-3 if x(i) is free and has bounds, but is not moved c 0 if x(i) is free and has bounds, and is moved c 1 if x(i) is fixed at l(i), and l(i) .ne. u(i) c 2 if x(i) is fixed at u(i), and u(i) .ne. l(i) c 3 if x(i) is always fixed, i.e., u(i)=x(i)=l(i) c -1 if x(i) is always free, i.e., it has no bounds. c c t is a double precision working array of dimension n. c t will be used to store the break points. c c d is a double precision array of dimension n used to store c the Cauchy direction P(x-tg)-x. c c xcp is a double precision array of dimension n used to return the c GCP on exit. c c m is an integer variable. c On entry m is the maximum number of variable metric corrections c used to define the limited memory matrix. c On exit m is unchanged. c c ws, wy, sy, and wt are double precision arrays. c On entry they store information that defines the c limited memory BFGS matrix: c ws(n,m) stores S, a set of s-vectors; c wy(n,m) stores Y, a set of y-vectors; c sy(m,m) stores S'Y; c wt(m,m) stores the c Cholesky factorization of (theta*S'S+LD^(-1)L'). c On exit these arrays are unchanged. c c theta is a double precision variable. c On entry theta is the scaling factor specifying B_0 = theta I. c On exit theta is unchanged. c c col is an integer variable. c On entry col is the actual number of variable metric c corrections stored so far. c On exit col is unchanged. c c head is an integer variable. c On entry head is the location of the first s-vector c (or y-vector) in S (or Y). c On exit col is unchanged. c c p is a double precision working array of dimension 2m. c p will be used to store the vector p = W^(T)d. c c c is a double precision working array of dimension 2m. c c will be used to store the vector c = W^(T)(xcp-x). c c wbp is a double precision working array of dimension 2m. c wbp will be used to store the row of W corresponding c to a breakpoint. c c v is a double precision working array of dimension 2m. c c nint is an integer variable. c On exit nint records the number of quadratic segments explored c in searching for the GCP. c c iprint is an INTEGER variable that must be set by the user. c It controls the frequency and type of output generated: c iprint<0 no output is generated; c iprint=0 print only one line at the last iteration; c 0100 print details of every iteration including x and g; c When iprint > 0, the file iterate.dat will be created to c summarize the iteration. c c sbgnrm is a double precision variable. c On entry sbgnrm is the norm of the projected gradient at x. c On exit sbgnrm is unchanged. c c info is an integer variable. c On entry info is 0. c On exit info = 0 for normal return, c = nonzero for abnormal return when the the system c used in routine bmv is singular. c c Subprograms called: c c L-BFGS-B Library ... hpsolb, bmv. c c Linpack ... dscal dcopy, daxpy. c c c References: c c [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited c memory algorithm for bound constrained optimization'', c SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208. c c [2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, ``L-BFGS-B: FORTRAN c Subroutines for Large Scale Bound Constrained Optimization'' c Tech. Report, NAM-11, EECS Department, Northwestern University, c 1994. c c (Postscript files of these papers are available via anonymous c ftp to ece.nwu.edu in the directory pub/lbfgs/lbfgs_bcm.) c c * * * c c NEOS, November 1994. (Latest revision April 1997.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ logical xlower,xupper,bnded integer i,j,col2,nfree,nbreak,pointr, + ibp,nleft,ibkmin,iter double precision f1,f2,dt,dtm,tsum,dibp,zibp,dibp2,bkmin, + tu,tl,wmc,wmp,wmw,ddot,tj,tj0,neggi,sbgnrm, + f2_org double precision one,zero parameter (one=1.0d0,zero=0.0d0) c Check the status of the variables, reset iwhere(i) if necessary; c compute the Cauchy direction d and the breakpoints t; initialize c the derivative f1 and the vector p = W'd (for theta = 1). if (sbgnrm .le. zero) then if (iprint .ge. 0) write (6,*) 'Subgnorm = 0. GCP = X.' call dcopy(n,x,1,xcp,1) return endif bnded = .true. nfree = n + 1 nbreak = 0 ibkmin = 0 bkmin = zero col2 = 2*col f1 = zero if (iprint .ge. 99) write (6,3010) c We set p to zero and build it up as we determine d. do 20 i = 1, col2 p(i) = zero 20 continue c In the following loop we determine for each variable its bound c status and its breakpoint, and update p accordingly. c Smallest breakpoint is identified. do 50 i = 1, n neggi = -g(i) if (iwhere(i) .ne. 3 .and. iwhere(i) .ne. -1) then c if x(i) is not a constant and has bounds, c compute the difference between x(i) and its bounds. if (nbd(i) .le. 2) tl = x(i) - l(i) if (nbd(i) .ge. 2) tu = u(i) - x(i) c If a variable is close enough to a bound c we treat it as at bound. xlower = nbd(i) .le. 2 .and. tl .le. zero xupper = nbd(i) .ge. 2 .and. tu .le. zero c reset iwhere(i). iwhere(i) = 0 if (xlower) then if (neggi .le. zero) iwhere(i) = 1 else if (xupper) then if (neggi .ge. zero) iwhere(i) = 2 else if (abs(neggi) .le. zero) iwhere(i) = -3 endif endif pointr = head if (iwhere(i) .ne. 0 .and. iwhere(i) .ne. -1) then d(i) = zero else d(i) = neggi f1 = f1 - neggi*neggi c calculate p := p - W'e_i* (g_i). do 40 j = 1, col p(j) = p(j) + wy(i,pointr)* neggi p(col + j) = p(col + j) + ws(i,pointr)*neggi pointr = mod(pointr,m) + 1 40 continue if (nbd(i) .le. 2 .and. nbd(i) .ne. 0 + .and. neggi .lt. zero) then c x(i) + d(i) is bounded; compute t(i). nbreak = nbreak + 1 iorder(nbreak) = i t(nbreak) = tl/(-neggi) if (nbreak .eq. 1 .or. t(nbreak) .lt. bkmin) then bkmin = t(nbreak) ibkmin = nbreak endif else if (nbd(i) .ge. 2 .and. neggi .gt. zero) then c x(i) + d(i) is bounded; compute t(i). nbreak = nbreak + 1 iorder(nbreak) = i t(nbreak) = tu/neggi if (nbreak .eq. 1 .or. t(nbreak) .lt. bkmin) then bkmin = t(nbreak) ibkmin = nbreak endif else c x(i) + d(i) is not bounded. nfree = nfree - 1 iorder(nfree) = i if (abs(neggi) .gt. zero) bnded = .false. endif endif 50 continue c The indices of the nonzero components of d are now stored c in iorder(1),...,iorder(nbreak) and iorder(nfree),...,iorder(n). c The smallest of the nbreak breakpoints is in t(ibkmin)=bkmin. if (theta .ne. one) then c complete the initialization of p for theta not= one. call dscal(col,theta,p(col+1),1) endif c Initialize GCP xcp = x. call dcopy(n,x,1,xcp,1) if (nbreak .eq. 0 .and. nfree .eq. n + 1) then c is a zero vector, return with the initial xcp as GCP. if (iprint .gt. 100) write (6,1010) (xcp(i), i = 1, n) return endif c Initialize c = W'(xcp - x) = 0. do 60 j = 1, col2 c(j) = zero 60 continue c Initialize derivative f2. f2 = -theta*f1 f2_org = f2 if (col .gt. 0) then call bmv(m,sy,wt,col,p,v,info) if (info .ne. 0) return f2 = f2 - ddot(col2,v,1,p,1) endif dtm = -f1/f2 tsum = zero nint = 1 if (iprint .ge. 99) + write (6,*) 'There are ',nbreak,' breakpoints ' c If there are no breakpoints, locate the GCP and return. if (nbreak .eq. 0) goto 888 nleft = nbreak iter = 1 tj = zero c------------------- the beginning of the loop ------------------------- 777 continue c Find the next smallest breakpoint; c compute dt = t(nleft) - t(nleft + 1). tj0 = tj if (iter .eq. 1) then c Since we already have the smallest breakpoint we need not do c heapsort yet. Often only one breakpoint is used and the c cost of heapsort is avoided. tj = bkmin ibp = iorder(ibkmin) else if (iter .eq. 2) then c Replace the already used smallest breakpoint with the c breakpoint numbered nbreak > nlast, before heapsort call. if (ibkmin .ne. nbreak) then t(ibkmin) = t(nbreak) iorder(ibkmin) = iorder(nbreak) endif c Update heap structure of breakpoints c (if iter=2, initialize heap). endif call hpsolb(nleft,t,iorder,iter-2) tj = t(nleft) ibp = iorder(nleft) endif dt = tj - tj0 if (dt .ne. zero .and. iprint .ge. 100) then write (6,4011) nint,f1,f2 write (6,5010) dt write (6,6010) dtm endif c If a minimizer is within this interval, c locate the GCP and return. if (dtm .lt. dt) goto 888 c Otherwise fix one variable and c reset the corresponding component of d to zero. tsum = tsum + dt nleft = nleft - 1 iter = iter + 1 dibp = d(ibp) d(ibp) = zero if (dibp .gt. zero) then zibp = u(ibp) - x(ibp) xcp(ibp) = u(ibp) iwhere(ibp) = 2 else zibp = l(ibp) - x(ibp) xcp(ibp) = l(ibp) iwhere(ibp) = 1 endif if (iprint .ge. 100) write (6,*) 'Variable ',ibp,' is fixed.' if (nleft .eq. 0 .and. nbreak .eq. n) then c all n variables are fixed, c return with xcp as GCP. dtm = dt goto 999 endif c Update the derivative information. nint = nint + 1 dibp2 = dibp**2 c Update f1 and f2. c temporarily set f1 and f2 for col=0. f1 = f1 + dt*f2 + dibp2 - theta*dibp*zibp f2 = f2 - theta*dibp2 if (col .gt. 0) then c update c = c + dt*p. call daxpy(col2,dt,p,1,c,1) c choose wbp, c the row of W corresponding to the breakpoint encountered. pointr = head do 70 j = 1,col wbp(j) = wy(ibp,pointr) wbp(col + j) = theta*ws(ibp,pointr) pointr = mod(pointr,m) + 1 70 continue c compute (wbp)Mc, (wbp)Mp, and (wbp)M(wbp)'. call bmv(m,sy,wt,col,wbp,v,info) if (info .ne. 0) return wmc = ddot(col2,c,1,v,1) wmp = ddot(col2,p,1,v,1) wmw = ddot(col2,wbp,1,v,1) c update p = p - dibp*wbp. call daxpy(col2,-dibp,wbp,1,p,1) c complete updating f1 and f2 while col > 0. f1 = f1 + dibp*wmc f2 = f2 + 2.0d0*dibp*wmp - dibp2*wmw endif f2 = max(epsmch*f2_org,f2) if (nleft .gt. 0) then dtm = -f1/f2 goto 777 c to repeat the loop for unsearched intervals. else if(bnded) then f1 = zero f2 = zero dtm = zero else dtm = -f1/f2 endif c------------------- the end of the loop ------------------------------- 888 continue if (iprint .ge. 99) then write (6,*) write (6,*) 'GCP found in this segment' write (6,4010) nint,f1,f2 write (6,6010) dtm endif if (dtm .le. zero) dtm = zero tsum = tsum + dtm c Move free variables (i.e., the ones w/o breakpoints) and c the variables whose breakpoints haven't been reached. call daxpy(n,tsum,d,1,xcp,1) 999 continue c Update c = c + dtm*p = W'(x^c - x) c which will be used in computing r = Z'(B(x^c - x) + g). if (col .gt. 0) call daxpy(col2,dtm,p,1,c,1) if (iprint .gt. 100) write (6,1010) (xcp(i),i = 1,n) if (iprint .ge. 99) write (6,2010) 1010 format ('Cauchy X = ',/,(4x,1p,6(1x,d11.4))) 2010 format (/,'---------------- exit CAUCHY----------------------',/) 3010 format (/,'---------------- CAUCHY entered-------------------') 4010 format ('Piece ',i3,' --f1, f2 at start point ',1p,2(1x,d11.4)) 4011 format (/,'Piece ',i3,' --f1, f2 at start point ', + 1p,2(1x,d11.4)) 5010 format ('Distance to the next break point = ',1p,d11.4) 6010 format ('Distance to the stationary point = ',1p,d11.4) return end c====================== The end of cauchy ============================== subroutine cmprlb(n, m, x, g, ws, wy, sy, wt, z, r, wa, index, + theta, col, head, nfree, cnstnd, info) logical cnstnd integer n, m, col, head, nfree, info, index(n) double precision theta, + x(n), g(n), z(n), r(n), wa(4*m), + ws(n, m), wy(n, m), sy(m, m), wt(m, m) c ************ c c Subroutine cmprlb c c This subroutine computes r=-Z'B(xcp-xk)-Z'g by using c wa(2m+1)=W'(xcp-x) from subroutine cauchy. c c Subprograms called: c c L-BFGS-B Library ... bmv. c c c * * * c c NEOS, November 1994. (Latest revision June 1996.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer i,j,k,pointr double precision a1,a2 if (.not. cnstnd .and. col .gt. 0) then do 26 i = 1, n r(i) = -g(i) 26 continue else do 30 i = 1, nfree k = index(i) r(i) = -theta*(z(k) - x(k)) - g(k) 30 continue call bmv(m,sy,wt,col,wa(2*m+1),wa(1),info) if (info .ne. 0) then info = -8 return endif pointr = head do 34 j = 1, col a1 = wa(j) a2 = theta*wa(col + j) do 32 i = 1, nfree k = index(i) r(i) = r(i) + wy(k,pointr)*a1 + ws(k,pointr)*a2 32 continue pointr = mod(pointr,m) + 1 34 continue endif return end c======================= The end of cmprlb ============================= subroutine errclb(n, m, factr, l, u, nbd, task, info, k) character*60 task integer n, m, info, k, nbd(n) double precision factr, l(n), u(n) c ************ c c Subroutine errclb c c This subroutine checks the validity of the input data. c c c * * * c c NEOS, November 1994. (Latest revision April 1997.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer i double precision zero parameter (zero=0.0d0) c Check the input arguments for errors. if (n .le. 0) task = 'ERROR: N .LE. 0' if (m .le. 0) task = 'ERROR: M .LE. 0' if (factr .lt. zero) task = 'ERROR: FACTR .LT. 0' c Check the validity of the arrays nbd(i), u(i), and l(i). do 10 i = 1, n if (nbd(i) .lt. 0 .or. nbd(i) .gt. 3) then c return task = 'ERROR: INVALID NBD' info = -6 k = i endif if (nbd(i) .eq. 2) then if (l(i) .gt. u(i)) then c return task = 'ERROR: NO FEASIBLE SOLUTION' info = -7 k = i endif endif 10 continue return end c======================= The end of errclb ============================= subroutine formk(n, nsub, ind, nenter, ileave, indx2, iupdat, + updatd, wn, wn1, m, ws, wy, sy, theta, col, + head, info) integer n, nsub, m, col, head, nenter, ileave, iupdat, + info, ind(n), indx2(n) double precision theta, wn(2*m, 2*m), wn1(2*m, 2*m), + ws(n, m), wy(n, m), sy(m, m) logical updatd c ************ c c Subroutine formk c c This subroutine forms the LEL^T factorization of the indefinite c c matrix K = [-D -Y'ZZ'Y/theta L_a'-R_z' ] c [L_a -R_z theta*S'AA'S ] c where E = [-I 0] c [ 0 I] c The matrix K can be shown to be equal to the matrix M^[-1]N c occurring in section 5.1 of [1], as well as to the matrix c Mbar^[-1] Nbar in section 5.3. c c n is an integer variable. c On entry n is the dimension of the problem. c On exit n is unchanged. c c nsub is an integer variable c On entry nsub is the number of subspace variables in free set. c On exit nsub is not changed. c c ind is an integer array of dimension nsub. c On entry ind specifies the indices of subspace variables. c On exit ind is unchanged. c c nenter is an integer variable. c On entry nenter is the number of variables entering the c free set. c On exit nenter is unchanged. c c ileave is an integer variable. c On entry indx2(ileave),...,indx2(n) are the variables leaving c the free set. c On exit ileave is unchanged. c c indx2 is an integer array of dimension n. c On entry indx2(1),...,indx2(nenter) are the variables entering c the free set, while indx2(ileave),...,indx2(n) are the c variables leaving the free set. c On exit indx2 is unchanged. c c iupdat is an integer variable. c On entry iupdat is the total number of BFGS updates made so far. c On exit iupdat is unchanged. c c updatd is a logical variable. c On entry 'updatd' is true if the L-BFGS matrix is updatd. c On exit 'updatd' is unchanged. c c wn is a double precision array of dimension 2m x 2m. c On entry wn is unspecified. c On exit the upper triangle of wn stores the LEL^T factorization c of the 2*col x 2*col indefinite matrix c [-D -Y'ZZ'Y/theta L_a'-R_z' ] c [L_a -R_z theta*S'AA'S ] c c wn1 is a double precision array of dimension 2m x 2m. c On entry wn1 stores the lower triangular part of c [Y' ZZ'Y L_a'+R_z'] c [L_a+R_z S'AA'S ] c in the previous iteration. c On exit wn1 stores the corresponding updated matrices. c The purpose of wn1 is just to store these inner products c so they can be easily updated and inserted into wn. c c m is an integer variable. c On entry m is the maximum number of variable metric corrections c used to define the limited memory matrix. c On exit m is unchanged. c c ws, wy, sy, and wtyy are double precision arrays; c theta is a double precision variable; c col is an integer variable; c head is an integer variable. c On entry they store the information defining the c limited memory BFGS matrix: c ws(n,m) stores S, a set of s-vectors; c wy(n,m) stores Y, a set of y-vectors; c sy(m,m) stores S'Y; c wtyy(m,m) stores the Cholesky factorization c of (theta*S'S+LD^(-1)L') c theta is the scaling factor specifying B_0 = theta I; c col is the number of variable metric corrections stored; c head is the location of the 1st s- (or y-) vector in S (or Y). c On exit they are unchanged. c c info is an integer variable. c On entry info is unspecified. c On exit info = 0 for normal return; c = -1 when the 1st Cholesky factorization failed; c = -2 when the 2st Cholesky factorization failed. c c Subprograms called: c c Linpack ... dcopy, dpofa, dtrsl. c c c References: c [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited c memory algorithm for bound constrained optimization'', c SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208. c c [2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, ``L-BFGS-B: a c limited memory FORTRAN code for solving bound constrained c optimization problems'', Tech. Report, NAM-11, EECS Department, c Northwestern University, 1994. c c (Postscript files of these papers are available via anonymous c ftp to ece.nwu.edu in the directory pub/lbfgs/lbfgs_bcm.) c c * * * c c NEOS, November 1994. (Latest revision April 1997.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer m2,ipntr,jpntr,iy,is,jy,js,is1,js1,k1,i,k, + col2,pbegin,pend,dbegin,dend,upcl double precision ddot,temp1,temp2,temp3,temp4 double precision zero parameter (zero=0.0d0) c Form the lower triangular part of c WN1 = [Y' ZZ'Y L_a'+R_z'] c [L_a+R_z S'AA'S ] c where L_a is the strictly lower triangular part of S'AA'Y c R_z is the upper triangular part of S'ZZ'Y. if (updatd) then if (iupdat .gt. m) then c shift old part of WN1. do 10 jy = 1, m - 1 js = m + jy call dcopy(m-jy,wn1(jy+1,jy+1),1,wn1(jy,jy),1) call dcopy(m-jy,wn1(js+1,js+1),1,wn1(js,js),1) call dcopy(m-1,wn1(m+2,jy+1),1,wn1(m+1,jy),1) 10 continue endif c put new rows in blocks (1,1), (2,1) and (2,2). pbegin = 1 pend = nsub dbegin = nsub + 1 dend = n iy = col is = m + col ipntr = head + col - 1 if (ipntr .gt. m) ipntr = ipntr - m jpntr = head do 20 jy = 1, col js = m + jy temp1 = zero temp2 = zero temp3 = zero c compute element jy of row 'col' of Y'ZZ'Y do 15 k = pbegin, pend k1 = ind(k) temp1 = temp1 + wy(k1,ipntr)*wy(k1,jpntr) 15 continue c compute elements jy of row 'col' of L_a and S'AA'S do 16 k = dbegin, dend k1 = ind(k) temp2 = temp2 + ws(k1,ipntr)*ws(k1,jpntr) temp3 = temp3 + ws(k1,ipntr)*wy(k1,jpntr) 16 continue wn1(iy,jy) = temp1 wn1(is,js) = temp2 wn1(is,jy) = temp3 jpntr = mod(jpntr,m) + 1 20 continue c put new column in block (2,1). jy = col jpntr = head + col - 1 if (jpntr .gt. m) jpntr = jpntr - m ipntr = head do 30 i = 1, col is = m + i temp3 = zero c compute element i of column 'col' of R_z do 25 k = pbegin, pend k1 = ind(k) temp3 = temp3 + ws(k1,ipntr)*wy(k1,jpntr) 25 continue ipntr = mod(ipntr,m) + 1 wn1(is,jy) = temp3 30 continue upcl = col - 1 else upcl = col endif c modify the old parts in blocks (1,1) and (2,2) due to changes c in the set of free variables. ipntr = head do 45 iy = 1, upcl is = m + iy jpntr = head do 40 jy = 1, iy js = m + jy temp1 = zero temp2 = zero temp3 = zero temp4 = zero do 35 k = 1, nenter k1 = indx2(k) temp1 = temp1 + wy(k1,ipntr)*wy(k1,jpntr) temp2 = temp2 + ws(k1,ipntr)*ws(k1,jpntr) 35 continue do 36 k = ileave, n k1 = indx2(k) temp3 = temp3 + wy(k1,ipntr)*wy(k1,jpntr) temp4 = temp4 + ws(k1,ipntr)*ws(k1,jpntr) 36 continue wn1(iy,jy) = wn1(iy,jy) + temp1 - temp3 wn1(is,js) = wn1(is,js) - temp2 + temp4 jpntr = mod(jpntr,m) + 1 40 continue ipntr = mod(ipntr,m) + 1 45 continue c modify the old parts in block (2,1). ipntr = head do 60 is = m + 1, m + upcl jpntr = head do 55 jy = 1, upcl temp1 = zero temp3 = zero do 50 k = 1, nenter k1 = indx2(k) temp1 = temp1 + ws(k1,ipntr)*wy(k1,jpntr) 50 continue do 51 k = ileave, n k1 = indx2(k) temp3 = temp3 + ws(k1,ipntr)*wy(k1,jpntr) 51 continue if (is .le. jy + m) then wn1(is,jy) = wn1(is,jy) + temp1 - temp3 else wn1(is,jy) = wn1(is,jy) - temp1 + temp3 endif jpntr = mod(jpntr,m) + 1 55 continue ipntr = mod(ipntr,m) + 1 60 continue c Form the upper triangle of WN = [D+Y' ZZ'Y/theta -L_a'+R_z' ] c [-L_a +R_z S'AA'S*theta] m2 = 2*m do 70 iy = 1, col is = col + iy is1 = m + iy do 65 jy = 1, iy js = col + jy js1 = m + jy wn(jy,iy) = wn1(iy,jy)/theta wn(js,is) = wn1(is1,js1)*theta 65 continue do 66 jy = 1, iy - 1 wn(jy,is) = -wn1(is1,jy) 66 continue do 67 jy = iy, col wn(jy,is) = wn1(is1,jy) 67 continue wn(iy,iy) = wn(iy,iy) + sy(iy,iy) 70 continue c Form the upper triangle of c WN= [ LL' L^-1(-L_a'+R_z')] c [(-L_a +R_z)L'^-1 S'AA'S*theta ] c first Cholesky factor (1,1) block of wn to get LL' c with L' stored in the upper triangle of wn. call dpofa(wn,m2,col,info) if (info .ne. 0) then info = -1 return endif c then form L^-1(-L_a'+R_z') in the (1,2) block. col2 = 2*col do 71 js = col+1 ,col2 call dtrsl(wn,m2,col,wn(1,js),11,info) 71 continue c Form S'AA'S*theta + (L^-1(-L_a'+R_z'))'L^-1(-L_a'+R_z') in the c upper triangle of (2,2) block of wn. do 72 is = col+1, col2 do 74 js = is, col2 wn(is,js) = wn(is,js) + ddot(col,wn(1,is),1,wn(1,js),1) 74 continue 72 continue c Cholesky factorization of (2,2) block of wn. call dpofa(wn(col+1,col+1),m2,col,info) if (info .ne. 0) then info = -2 return endif return end c======================= The end of formk ============================== subroutine formt(m, wt, sy, ss, col, theta, info) integer m, col, info double precision theta, wt(m, m), sy(m, m), ss(m, m) c ************ c c Subroutine formt c c This subroutine forms the upper half of the pos. def. and symm. c T = theta*SS + L*D^(-1)*L', stores T in the upper triangle c of the array wt, and performs the Cholesky factorization of T c to produce J*J', with J' stored in the upper triangle of wt. c c Subprograms called: c c Linpack ... dpofa. c c c * * * c c NEOS, November 1994. (Latest revision June 1996.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer i,j,k,k1 double precision ddum double precision zero parameter (zero=0.0d0) c Form the upper half of T = theta*SS + L*D^(-1)*L', c store T in the upper triangle of the array wt. do 52 j = 1, col wt(1,j) = theta*ss(1,j) 52 continue do 55 i = 2, col do 54 j = i, col k1 = min(i,j) - 1 ddum = zero do 53 k = 1, k1 ddum = ddum + sy(i,k)*sy(j,k)/sy(k,k) 53 continue wt(i,j) = ddum + theta*ss(i,j) 54 continue 55 continue c Cholesky factorize T to J*J' with c J' stored in the upper triangle of wt. call dpofa(wt,m,col,info) if (info .ne. 0) then info = -3 endif return end c======================= The end of formt ============================== subroutine freev(n, nfree, index, nenter, ileave, indx2, + iwhere, wrk, updatd, cnstnd, iprint, iter) integer n, nfree, nenter, ileave, iprint, iter, + index(n), indx2(n), iwhere(n) logical wrk, updatd, cnstnd c ************ c c Subroutine freev c c This subroutine counts the entering and leaving variables when c iter > 0, and finds the index set of free and active variables c at the GCP. c c cnstnd is a logical variable indicating whether bounds are present c c index is an integer array of dimension n c for i=1,...,nfree, index(i) are the indices of free variables c for i=nfree+1,...,n, index(i) are the indices of bound variables c On entry after the first iteration, index gives c the free variables at the previous iteration. c On exit it gives the free variables based on the determination c in cauchy using the array iwhere. c c indx2 is an integer array of dimension n c On entry indx2 is unspecified. c On exit with iter>0, indx2 indicates which variables c have changed status since the previous iteration. c For i= 1,...,nenter, indx2(i) have changed from bound to free. c For i= ileave+1,...,n, indx2(i) have changed from free to bound. c c c * * * c c NEOS, November 1994. (Latest revision June 1996.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer iact,i,k nenter = 0 ileave = n + 1 if (iter .gt. 0 .and. cnstnd) then c count the entering and leaving variables. do 20 i = 1, nfree k = index(i) if (iwhere(k) .gt. 0) then ileave = ileave - 1 indx2(ileave) = k if (iprint .ge. 100) write (6,*) + 'Variable ',k,' leaves the set of free variables' endif 20 continue do 22 i = 1 + nfree, n k = index(i) if (iwhere(k) .le. 0) then nenter = nenter + 1 indx2(nenter) = k if (iprint .ge. 100) write (6,*) + 'Variable ',k,' enters the set of free variables' endif 22 continue if (iprint .ge. 99) write (6,*) + n+1-ileave,' variables leave; ',nenter,' variables enter' endif wrk = (ileave .lt. n+1) .or. (nenter .gt. 0) .or. updatd c Find the index set of free and active variables at the GCP. nfree = 0 iact = n + 1 do 24 i = 1, n if (iwhere(i) .le. 0) then nfree = nfree + 1 index(nfree) = i else iact = iact - 1 index(iact) = i endif 24 continue if (iprint .ge. 99) write (6,*) + nfree,' variables are free at GCP ',iter + 1 return end c======================= The end of freev ============================== subroutine hpsolb(n, t, iorder, iheap) integer iheap, n, iorder(n) double precision t(n) c ************ c c Subroutine hpsolb c c This subroutine sorts out the least element of t, and puts the c remaining elements of t in a heap. c c n is an integer variable. c On entry n is the dimension of the arrays t and iorder. c On exit n is unchanged. c c t is a double precision array of dimension n. c On entry t stores the elements to be sorted, c On exit t(n) stores the least elements of t, and t(1) to t(n-1) c stores the remaining elements in the form of a heap. c c iorder is an integer array of dimension n. c On entry iorder(i) is the index of t(i). c On exit iorder(i) is still the index of t(i), but iorder may be c permuted in accordance with t. c c iheap is an integer variable specifying the task. c On entry iheap should be set as follows: c iheap .eq. 0 if t(1) to t(n) is not in the form of a heap, c iheap .ne. 0 if otherwise. c On exit iheap is unchanged. c c c References: c Algorithm 232 of CACM (J. W. J. Williams): HEAPSORT. c c * * * c c NEOS, November 1994. (Latest revision June 1996.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c ************ integer i,j,k,indxin,indxou double precision ddum,out if (iheap .eq. 0) then c Rearrange the elements t(1) to t(n) to form a heap. do 20 k = 2, n ddum = t(k) indxin = iorder(k) c Add ddum to the heap. i = k 10 continue if (i.gt.1) then j = i/2 if (ddum .lt. t(j)) then t(i) = t(j) iorder(i) = iorder(j) i = j goto 10 endif endif t(i) = ddum iorder(i) = indxin 20 continue endif c Assign to 'out' the value of t(1), the least member of the heap, c and rearrange the remaining members to form a heap as c elements 1 to n-1 of t. if (n .gt. 1) then i = 1 out = t(1) indxou = iorder(1) ddum = t(n) indxin = iorder(n) c Restore the heap 30 continue j = i+i if (j .le. n-1) then if (t(j+1) .lt. t(j)) j = j+1 if (t(j) .lt. ddum ) then t(i) = t(j) iorder(i) = iorder(j) i = j goto 30 endif endif t(i) = ddum iorder(i) = indxin c Put the least member in t(n). t(n) = out iorder(n) = indxou endif return end c====================== The end of hpsolb ============================== subroutine lnsrlb(n, l, u, nbd, x, f, fold, gd, gdold, g, d, r, t, + z, stp, dnorm, dtd, xstep, stpmx, iter, ifun, + iback, nfgv, info, task, boxed, cnstnd, csave, + isave, dsave) character*60 task, csave logical boxed, cnstnd integer n, iter, ifun, iback, nfgv, info, + nbd(n), isave(2) double precision f, fold, gd, gdold, stp, dnorm, dtd, xstep, + stpmx, x(n), l(n), u(n), g(n), d(n), r(n), t(n), + z(n), dsave(13) c ********** c c Subroutine lnsrlb c c This subroutine calls subroutine dcsrch from the Minpack2 library c to perform the line search. Subroutine dscrch is safeguarded so c that all trial points lie within the feasible region. c c Subprograms called: c c Minpack2 Library ... dcsrch. c c Linpack ... dtrsl, ddot. c c c * * * c c NEOS, November 1994. (Latest revision June 1996.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ********** integer i double precision ddot,a1,a2 double precision one,zero,big parameter (one=1.0d0,zero=0.0d0,big=1.0d+10) double precision ftol,gtol,xtol parameter (ftol=1.0d-3,gtol=0.9d0,xtol=0.1d0) if (task(1:5) .eq. 'FG_LN') goto 556 dtd = ddot(n,d,1,d,1) dnorm = sqrt(dtd) c Determine the maximum step length. stpmx = big if (cnstnd) then if (iter .eq. 0) then stpmx = one else do 43 i = 1, n a1 = d(i) if (nbd(i) .ne. 0) then if (a1 .lt. zero .and. nbd(i) .le. 2) then a2 = l(i) - x(i) if (a2 .ge. zero) then stpmx = zero else if (a1*stpmx .lt. a2) then stpmx = a2/a1 endif else if (a1 .gt. zero .and. nbd(i) .ge. 2) then a2 = u(i) - x(i) if (a2 .le. zero) then stpmx = zero else if (a1*stpmx .gt. a2) then stpmx = a2/a1 endif endif endif 43 continue endif endif if (iter .eq. 0 .and. .not. boxed) then stp = min(one/dnorm, stpmx) else stp = one endif call dcopy(n,x,1,t,1) call dcopy(n,g,1,r,1) fold = f ifun = 0 iback = 0 csave = 'START' 556 continue gd = ddot(n,g,1,d,1) if (ifun .eq. 0) then gdold=gd if (gd .ge. zero) then c the directional derivative >=0. c Line search is impossible. info = -4 return endif endif call dcsrch(f,gd,stp,ftol,gtol,xtol,zero,stpmx,csave,isave,dsave) xstep = stp*dnorm if (csave(1:4) .ne. 'CONV' .and. csave(1:4) .ne. 'WARN') then task = 'FG_LNSRCH' ifun = ifun + 1 nfgv = nfgv + 1 iback = ifun - 1 if (stp .eq. one) then call dcopy(n,z,1,x,1) else do 41 i = 1, n x(i) = stp*d(i) + t(i) 41 continue endif else task = 'NEW_X' endif return end c======================= The end of lnsrlb ============================= subroutine matupd(n, m, ws, wy, sy, ss, d, r, itail, + iupdat, col, head, theta, rr, dr, stp, dtd) integer n, m, itail, iupdat, col, head double precision theta, rr, dr, stp, dtd, d(n), r(n), + ws(n, m), wy(n, m), sy(m, m), ss(m, m) c ************ c c Subroutine matupd c c This subroutine updates matrices WS and WY, and forms the c middle matrix in B. c c Subprograms called: c c Linpack ... dcopy, ddot. c c c * * * c c NEOS, November 1994. (Latest revision June 1996.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer j,pointr double precision ddot double precision one parameter (one=1.0d0) c Set pointers for matrices WS and WY. if (iupdat .le. m) then col = iupdat itail = mod(head+iupdat-2,m) + 1 else itail = mod(itail,m) + 1 head = mod(head,m) + 1 endif c Update matrices WS and WY. call dcopy(n,d,1,ws(1,itail),1) call dcopy(n,r,1,wy(1,itail),1) c Set theta=yy/ys. theta = rr/dr c Form the middle matrix in B. c update the upper triangle of SS, c and the lower triangle of SY: if (iupdat .gt. m) then c move old information do 50 j = 1, col - 1 call dcopy(j,ss(2,j+1),1,ss(1,j),1) call dcopy(col-j,sy(j+1,j+1),1,sy(j,j),1) 50 continue endif c add new information: the last row of SY c and the last column of SS: pointr = head do 51 j = 1, col - 1 sy(col,j) = ddot(n,d,1,wy(1,pointr),1) ss(j,col) = ddot(n,ws(1,pointr),1,d,1) pointr = mod(pointr,m) + 1 51 continue if (stp .eq. one) then ss(col,col) = dtd else ss(col,col) = stp*stp*dtd endif sy(col,col) = dr return end c======================= The end of matupd ============================= subroutine prn1lb(n, m, l, u, x, iprint, itfile, epsmch) integer n, m, iprint, itfile double precision epsmch, x(n), l(n), u(n) c ************ c c Subroutine prn1lb c c This subroutine prints the input data, initial point, upper and c lower bounds of each variable, machine precision, as well as c the headings of the output. c c c * * * c c NEOS, November 1994. (Latest revision June 1996.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer i if (iprint .ge. 0) then write (6,7001) epsmch write (6,*) 'N = ',n,' M = ',m if (iprint .ge. 1) then write (itfile,2001) epsmch write (itfile,*)'N = ',n,' M = ',m write (itfile,9001) if (iprint .gt. 100) then write (6,1004) 'L =',(l(i),i = 1,n) write (6,1004) 'X0 =',(x(i),i = 1,n) write (6,1004) 'U =',(u(i),i = 1,n) endif endif endif 1004 format (/,a4, 1p, 6(1x,d11.4),/,(4x,1p,6(1x,d11.4))) 2001 format ('RUNNING THE L-BFGS-B CODE',/,/, + 'it = iteration number',/, + 'nf = number of function evaluations',/, + 'nint = number of segments explored during the Cauchy search',/, + 'nact = number of active bounds at the generalized Cauchy point' + ,/, + 'sub = manner in which the subspace minimization terminated:' + ,/,' con = converged, bnd = a bound was reached',/, + 'itls = number of iterations performed in the line search',/, + 'stepl = step length used',/, + 'tstep = norm of the displacement (total step)',/, + 'projg = norm of the projected gradient',/, + 'f = function value',/,/, + ' * * *',/,/, + 'Machine precision =',1p,d10.3) 7001 format ('RUNNING THE L-BFGS-B CODE',/,/, + ' * * *',/,/, + 'Machine precision =',1p,d10.3) 9001 format (/,3x,'it',3x,'nf',2x,'nint',2x,'nact',2x,'sub',2x,'itls', + 2x,'stepl',4x,'tstep',5x,'projg',8x,'f') return end c======================= The end of prn1lb ============================= subroutine prn2lb(n, x, f, g, iprint, itfile, iter, nfgv, nact, + sbgnrm, nint, word, iword, iback, stp, xstep) character*3 word integer n, iprint, itfile, iter, nfgv, nact, nint, + iword, iback double precision f, sbgnrm, stp, xstep, x(n), g(n) c ************ c c Subroutine prn2lb c c This subroutine prints out new information after a successful c line search. c c c * * * c c NEOS, November 1994. (Latest revision June 1996.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer i,imod c 'word' records the status of subspace solutions. if (iword .eq. 0) then c the subspace minimization converged. word = 'con' else if (iword .eq. 1) then c the subspace minimization stopped at a bound. word = 'bnd' else if (iword .eq. 5) then c the truncated Newton step has been used. word = 'TNT' else word = '---' endif if (iprint .ge. 99) then write (6,*) 'LINE SEARCH',iback,' times; norm of step = ',xstep write (6,2001) iter,f,sbgnrm if (iprint .gt. 100) then write (6,1004) 'X =',(x(i), i = 1, n) write (6,1004) 'G =',(g(i), i = 1, n) endif else if (iprint .gt. 0) then imod = mod(iter,iprint) if (imod .eq. 0) write (6,2001) iter,f,sbgnrm endif if (iprint .ge. 1) write (itfile,3001) + iter,nfgv,nint,nact,word,iback,stp,xstep,sbgnrm,f 1004 format (/,a4, 1p, 6(1x,d11.4),/,(4x,1p,6(1x,d11.4))) 2001 format + (/,'At iterate',i5,4x,'f= ',1p,d12.5,4x,'|proj g|= ',1p,d12.5) 3001 format(2(1x,i4),2(1x,i5),2x,a3,1x,i4,1p,2(2x,d7.1),1p,2(1x,d10.3)) return end c======================= The end of prn2lb ============================= subroutine prn3lb(n, x, f, task, iprint, info, itfile, + iter, nfgv, nintol, nskip, nact, sbgnrm, + time, nint, word, iback, stp, xstep, k, + cachyt, sbtime, lnscht) character*60 task character*3 word integer n, iprint, info, itfile, iter, nfgv, nintol, + nskip, nact, nint, iback, k double precision f, sbgnrm, time, stp, xstep, cachyt, sbtime, + lnscht, x(n) c ************ c c Subroutine prn3lb c c This subroutine prints out information when either a built-in c convergence test is satisfied or when an error message is c generated. c c c * * * c c NEOS, November 1994. (Latest revision April 1997.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer i if (task(1:5) .eq. 'ERROR') goto 999 if (iprint .ge. 0) then write (6,3003) write (6,3004) write(6,3005) n,iter,nfgv,nintol,nskip,nact,sbgnrm,f if (iprint .ge. 100) then write (6,1004) 'X =',(x(i),i = 1,n) endif if (iprint .ge. 1) write (6,*) ' F =',f endif 999 continue if (iprint .ge. 0) then write (6,3009) task if (info .ne. 0) then if (info .eq. -1) write (6,9011) if (info .eq. -2) write (6,9012) if (info .eq. -3) write (6,9013) if (info .eq. -4) write (6,9014) if (info .eq. -5) write (6,9015) if (info .eq. -6) write (6,*)' Input nbd(',k,') is invalid.' if (info .eq. -7) + write (6,*)' l(',k,') > u(',k,'). No feasible solution.' if (info .eq. -8) write (6,9018) if (info .eq. -9) write (6,9019) endif if (iprint .ge. 1) write (6,3007) cachyt,sbtime,lnscht write (6,3008) time if (iprint .ge. 1) then if (info .eq. -4 .or. info .eq. -9) then write (itfile,3002) + iter,nfgv,nint,nact,word,iback,stp,xstep endif write (itfile,3009) task if (info .ne. 0) then if (info .eq. -1) write (itfile,9011) if (info .eq. -2) write (itfile,9012) if (info .eq. -3) write (itfile,9013) if (info .eq. -4) write (itfile,9014) if (info .eq. -5) write (itfile,9015) if (info .eq. -8) write (itfile,9018) if (info .eq. -9) write (itfile,9019) endif write (itfile,3008) time endif endif 1004 format (/,a4, 1p, 6(1x,d11.4),/,(4x,1p,6(1x,d11.4))) 3002 format(2(1x,i4),2(1x,i5),2x,a3,1x,i4,1p,2(2x,d7.1),6x,'-',10x,'-') 3003 format (/, + ' * * *',/,/, + 'Tit = total number of iterations',/, + 'Tnf = total number of function evaluations',/, + 'Tnint = total number of segments explored during', + ' Cauchy searches',/, + 'Skip = number of BFGS updates skipped',/, + 'Nact = number of active bounds at final generalized', + ' Cauchy point',/, + 'Projg = norm of the final projected gradient',/, + 'F = final function value',/,/, + ' * * *') 3004 format (/,3x,'N',3x,'Tit',2x,'Tnf',2x,'Tnint',2x, + 'Skip',2x,'Nact',5x,'Projg',8x,'F') 3005 format (i5,2(1x,i4),(1x,i6),(2x,i4),(1x,i5),1p,2(2x,d10.3)) 3007 format (/,' Cauchy time',1p,e10.3,' seconds.',/ + ' Subspace minimization time',1p,e10.3,' seconds.',/ + ' Line search time',1p,e10.3,' seconds.') 3008 format (/,' Total User time',1p,e10.3,' seconds.',/) 3009 format (/,a60) 9011 format (/, +' Matrix in 1st Cholesky factorization in formk is not Pos. Def.') 9012 format (/, +' Matrix in 2st Cholesky factorization in formk is not Pos. Def.') 9013 format (/, +' Matrix in the Cholesky factorization in formt is not Pos. Def.') 9014 format (/, +' Derivative >= 0, backtracking line search impossible.',/, +' Previous x, f and g restored.',/, +' Possible causes: 1 error in function or gradient evaluation;',/, +' 2 rounding errors dominate computation.') 9015 format (/, +' Warning: more than 10 function and gradient',/, +' evaluations in the last line search. Termination',/, +' may possibly be caused by a bad search direction.') 9018 format (/,' The triangular system is singular.') 9019 format (/, +' Line search cannot locate an adequate point after 20 function',/ +,' and gradient evaluations. Previous x, f and g restored.',/, +' Possible causes: 1 error in function or gradient evaluation;',/, +' 2 rounding error dominate computation.') return end c======================= The end of prn3lb ============================= subroutine projgr(n, l, u, nbd, x, g, sbgnrm) integer n, nbd(n) double precision sbgnrm, x(n), l(n), u(n), g(n) c ************ c c Subroutine projgr c c This subroutine computes the infinity norm of the projected c gradient. c c c * * * c c NEOS, November 1994. (Latest revision April 1997.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer i double precision gi double precision zero parameter (zero=0.0d0) sbgnrm = zero do 15 i = 1, n gi = g(i) if (nbd(i) .ne. 0) then if (gi .lt. zero) then if (nbd(i) .ge. 2) gi = max((x(i)-u(i)),gi) else if (nbd(i) .le. 2) gi = min((x(i)-l(i)),gi) endif endif sbgnrm = max(sbgnrm,abs(gi)) 15 continue return end c======================= The end of projgr ============================= subroutine subsm(n, m, nsub, ind, l, u, nbd, x, d, ws, wy, theta, + col, head, iword, wv, wn, iprint, info) integer n, m, nsub, col, head, iword, iprint, info, + ind(nsub), nbd(n) double precision theta, + l(n), u(n), x(n), d(n), + ws(n, m), wy(n, m), + wv(2*m), wn(2*m, 2*m) c ************ c c Subroutine subsm c c Given xcp, l, u, r, an index set that specifies c the active set at xcp, and an l-BFGS matrix B c (in terms of WY, WS, SY, WT, head, col, and theta), c this subroutine computes an approximate solution c of the subspace problem c c (P) min Q(x) = r'(x-xcp) + 1/2 (x-xcp)' B (x-xcp) c c subject to l<=x<=u c x_i=xcp_i for all i in A(xcp) c c along the subspace unconstrained Newton direction c c d = -(Z'BZ)^(-1) r. c c The formula for the Newton direction, given the L-BFGS matrix c and the Sherman-Morrison formula, is c c d = (1/theta)r + (1/theta*2) Z'WK^(-1)W'Z r. c c where c K = [-D -Y'ZZ'Y/theta L_a'-R_z' ] c [L_a -R_z theta*S'AA'S ] c c Note that this procedure for computing d differs c from that described in [1]. One can show that the matrix K is c equal to the matrix M^[-1]N in that paper. c c n is an integer variable. c On entry n is the dimension of the problem. c On exit n is unchanged. c c m is an integer variable. c On entry m is the maximum number of variable metric corrections c used to define the limited memory matrix. c On exit m is unchanged. c c nsub is an integer variable. c On entry nsub is the number of free variables. c On exit nsub is unchanged. c c ind is an integer array of dimension nsub. c On entry ind specifies the coordinate indices of free variables. c On exit ind is unchanged. c c l is a double precision array of dimension n. c On entry l is the lower bound of x. c On exit l is unchanged. c c u is a double precision array of dimension n. c On entry u is the upper bound of x. c On exit u is unchanged. c c nbd is a integer array of dimension n. c On entry nbd represents the type of bounds imposed on the c variables, and must be specified as follows: c nbd(i)=0 if x(i) is unbounded, c 1 if x(i) has only a lower bound, c 2 if x(i) has both lower and upper bounds, and c 3 if x(i) has only an upper bound. c On exit nbd is unchanged. c c x is a double precision array of dimension n. c On entry x specifies the Cauchy point xcp. c On exit x(i) is the minimizer of Q over the subspace of c free variables. c c d is a double precision array of dimension n. c On entry d is the reduced gradient of Q at xcp. c On exit d is the Newton direction of Q. c c ws and wy are double precision arrays; c theta is a double precision variable; c col is an integer variable; c head is an integer variable. c On entry they store the information defining the c limited memory BFGS matrix: c ws(n,m) stores S, a set of s-vectors; c wy(n,m) stores Y, a set of y-vectors; c theta is the scaling factor specifying B_0 = theta I; c col is the number of variable metric corrections stored; c head is the location of the 1st s- (or y-) vector in S (or Y). c On exit they are unchanged. c c iword is an integer variable. c On entry iword is unspecified. c On exit iword specifies the status of the subspace solution. c iword = 0 if the solution is in the box, c 1 if some bound is encountered. c c wv is a double precision working array of dimension 2m. c c wn is a double precision array of dimension 2m x 2m. c On entry the upper triangle of wn stores the LEL^T factorization c of the indefinite matrix c c K = [-D -Y'ZZ'Y/theta L_a'-R_z' ] c [L_a -R_z theta*S'AA'S ] c where E = [-I 0] c [ 0 I] c On exit wn is unchanged. c c iprint is an INTEGER variable that must be set by the user. c It controls the frequency and type of output generated: c iprint<0 no output is generated; c iprint=0 print only one line at the last iteration; c 0100 print details of every iteration including x and g; c When iprint > 0, the file iterate.dat will be created to c summarize the iteration. c c info is an integer variable. c On entry info is unspecified. c On exit info = 0 for normal return, c = nonzero for abnormal return c when the matrix K is ill-conditioned. c c Subprograms called: c c Linpack dtrsl. c c c References: c c [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited c memory algorithm for bound constrained optimization'', c SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208. c c c c * * * c c NEOS, November 1994. (Latest revision June 1996.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ logical temp1_updated integer pointr,m2,col2,ibd,jy,js,i,j,k double precision alpha,dk,temp1,temp2 double precision one,zero parameter (one=1.0d0,zero=0.0d0) if (nsub .le. 0) return if (iprint .ge. 99) write (6,1001) c Compute wv = W'Zd. pointr = head do 20 i = 1, col temp1 = zero temp2 = zero do 10 j = 1, nsub k = ind(j) temp1 = temp1 + wy(k,pointr)*d(j) temp2 = temp2 + ws(k,pointr)*d(j) 10 continue wv(i) = temp1 wv(col + i) = theta*temp2 pointr = mod(pointr,m) + 1 20 continue c Compute wv:=K^(-1)wv. m2 = 2*m col2 = 2*col call dtrsl(wn,m2,col2,wv,11,info) if (info .ne. 0) return do 25 i = 1, col wv(i) = -wv(i) 25 continue call dtrsl(wn,m2,col2,wv,01,info) if (info .ne. 0) return c Compute d = (1/theta)d + (1/theta**2)Z'W wv. pointr = head do 40 jy = 1, col js = col + jy do 30 i = 1, nsub k = ind(i) d(i) = d(i) + wy(k,pointr)*wv(jy)/theta + + ws(k,pointr)*wv(js) 30 continue pointr = mod(pointr,m) + 1 40 continue do 50 i = 1, nsub d(i) = d(i)/theta 50 continue c Backtrack to the feasible region. alpha = one temp1 = alpha do 60 i = 1, nsub k = ind(i) dk = d(i) if (nbd(k) .ne. 0) then temp1_updated = .false. if (dk .lt. zero .and. nbd(k) .le. 2) then temp2 = l(k) - x(k) if (temp2 .ge. zero) then temp1 = zero temp1_updated = .true. else if (dk*alpha .lt. temp2) then temp1 = temp2/dk temp1_updated = .true. endif else if (dk .gt. zero .and. nbd(k) .ge. 2) then temp2 = u(k) - x(k) if (temp2 .le. zero) then temp1 = zero temp1_updated = .true. else if (dk*alpha .gt. temp2) then temp1 = temp2/dk temp1_updated = .true. endif endif cc logical variable temp1_updated added to eliminate unexpected cc trigger of the if statement due to possible difference between cc hardware precision and double precision. if (temp1_updated .and. temp1 .lt. alpha) then alpha = temp1 ibd = i endif endif 60 continue if (alpha .lt. one) then dk = d(ibd) k = ind(ibd) if (dk .gt. zero) then x(k) = u(k) d(ibd) = zero else if (dk .lt. zero) then x(k) = l(k) d(ibd) = zero endif endif do 70 i = 1, nsub k = ind(i) x(k) = x(k) + alpha*d(i) 70 continue if (iprint .ge. 99) then if (alpha .lt. one) then write (6,1002) alpha else write (6,*) 'SM solution inside the box' end if if (iprint .gt.100) write (6,1003) (x(i),i=1,n) endif if (alpha .lt. one) then iword = 1 else iword = 0 endif if (iprint .ge. 99) write (6,1004) 1001 format (/,'----------------SUBSM entered-----------------',/) 1002 format ( 'ALPHA = ',f7.5,' backtrack to the BOX') 1003 format ('Subspace solution X = ',/,(4x,1p,6(1x,d11.4))) 1004 format (/,'----------------exit SUBSM --------------------',/) return end c====================== The end of subsm =============================== subroutine dcsrch(f,g,stp,ftol,gtol,xtol,stpmin,stpmax, + task,isave,dsave) character*(*) task integer isave(2) double precision f,g,stp,ftol,gtol,xtol,stpmin,stpmax double precision dsave(13) c ********** c c Subroutine dcsrch c c This subroutine finds a step that satisfies a sufficient c decrease condition and a curvature condition. c c Each call of the subroutine updates an interval with c endpoints stx and sty. The interval is initially chosen c so that it contains a minimizer of the modified function c c psi(stp) = f(stp) - f(0) - ftol*stp*f'(0). c c If psi(stp) <= 0 and f'(stp) >= 0 for some step, then the c interval is chosen so that it contains a minimizer of f. c c The algorithm is designed to find a step that satisfies c the sufficient decrease condition c c f(stp) <= f(0) + ftol*stp*f'(0), c c and the curvature condition c c abs(f'(stp)) <= gtol*abs(f'(0)). c c If ftol is less than gtol and if, for example, the function c is bounded below, then there is always a step which satisfies c both conditions. c c If no step can be found that satisfies both conditions, then c the algorithm stops with a warning. In this case stp only c satisfies the sufficient decrease condition. c c A typical invocation of dcsrch has the following outline: c c task = 'START' c 10 continue c call dcsrch( ... ) c if (task .eq. 'FG') then c Evaluate the function and the gradient at stp c goto 10 c end if c c NOTE: The user must no alter work arrays between calls. c c The subroutine statement is c c subroutine dcsrch(f,g,stp,ftol,gtol,xtol,stpmin,stpmax, c task,isave,dsave) c where c c f is a double precision variable. c On initial entry f is the value of the function at 0. c On subsequent entries f is the value of the c function at stp. c On exit f is the value of the function at stp. c c g is a double precision variable. c On initial entry g is the derivative of the function at 0. c On subsequent entries g is the derivative of the c function at stp. c On exit g is the derivative of the function at stp. c c stp is a double precision variable. c On entry stp is the current estimate of a satisfactory c step. On initial entry, a positive initial estimate c must be provided. c On exit stp is the current estimate of a satisfactory step c if task = 'FG'. If task = 'CONV' then stp satisfies c the sufficient decrease and curvature condition. c c ftol is a double precision variable. c On entry ftol specifies a nonnegative tolerance for the c sufficient decrease condition. c On exit ftol is unchanged. c c gtol is a double precision variable. c On entry gtol specifies a nonnegative tolerance for the c curvature condition. c On exit gtol is unchanged. c c xtol is a double precision variable. c On entry xtol specifies a nonnegative relative tolerance c for an acceptable step. The subroutine exits with a c warning if the relative difference between sty and stx c is less than xtol. c On exit xtol is unchanged. c c stpmin is a double precision variable. c On entry stpmin is a nonnegative lower bound for the step. c On exit stpmin is unchanged. c c stpmax is a double precision variable. c On entry stpmax is a nonnegative upper bound for the step. c On exit stpmax is unchanged. c c task is a character variable of length at least 60. c On initial entry task must be set to 'START'. c On exit task indicates the required action: c c If task(1:2) = 'FG' then evaluate the function and c derivative at stp and call dcsrch again. c c If task(1:4) = 'CONV' then the search is successful. c c If task(1:4) = 'WARN' then the subroutine is not able c to satisfy the convergence conditions. The exit value of c stp contains the best point found during the search. c c If task(1:5) = 'ERROR' then there is an error in the c input arguments. c c On exit with convergence, a warning or an error, the c variable task contains additional information. c c isave is an integer work array of dimension 2. c c dsave is a double precision work array of dimension 13. c c Subprograms called c c MINPACK-2 ... dcstep c c MINPACK-1 Project. June 1983. c Argonne National Laboratory. c Jorge J. More' and David J. Thuente. c c MINPACK-2 Project. October 1993. c Argonne National Laboratory and University of Minnesota. c Brett M. Averick, Richard G. Carter, and Jorge J. More'. c c ********** double precision zero,p5,p66 parameter(zero=0.0d0,p5=0.5d0,p66=0.66d0) double precision xtrapl,xtrapu parameter(xtrapl=1.1d0,xtrapu=4.0d0) logical brackt integer stage double precision finit,ftest,fm,fx,fxm,fy,fym,ginit,gtest, + gm,gx,gxm,gy,gym,stx,sty,stmin,stmax,width,width1 c Initialization block. if (task(1:5) .eq. 'START') then c Check the input arguments for errors. if (stp .lt. stpmin) task = 'ERROR: STP .LT. STPMIN' if (stp .gt. stpmax) task = 'ERROR: STP .GT. STPMAX' if (g .ge. zero) task = 'ERROR: INITIAL G .GE. ZERO' if (ftol .lt. zero) task = 'ERROR: FTOL .LT. ZERO' if (gtol .lt. zero) task = 'ERROR: GTOL .LT. ZERO' if (xtol .lt. zero) task = 'ERROR: XTOL .LT. ZERO' if (stpmin .lt. zero) task = 'ERROR: STPMIN .LT. ZERO' if (stpmax .lt. stpmin) task = 'ERROR: STPMAX .LT. STPMIN' c Exit if there are errors on input. if (task(1:5) .eq. 'ERROR') return c Initialize local variables. brackt = .false. stage = 1 finit = f ginit = g gtest = ftol*ginit width = stpmax - stpmin width1 = width/p5 c The variables stx, fx, gx contain the values of the step, c function, and derivative at the best step. c The variables sty, fy, gy contain the value of the step, c function, and derivative at sty. c The variables stp, f, g contain the values of the step, c function, and derivative at stp. stx = zero fx = finit gx = ginit sty = zero fy = finit gy = ginit stmin = zero stmax = stp + xtrapu*stp task = 'FG' goto 1000 else c Restore local variables. if (isave(1) .eq. 1) then brackt = .true. else brackt = .false. endif stage = isave(2) ginit = dsave(1) gtest = dsave(2) gx = dsave(3) gy = dsave(4) finit = dsave(5) fx = dsave(6) fy = dsave(7) stx = dsave(8) sty = dsave(9) stmin = dsave(10) stmax = dsave(11) width = dsave(12) width1 = dsave(13) endif c If psi(stp) <= 0 and f'(stp) >= 0 for some step, then the c algorithm enters the second stage. ftest = finit + stp*gtest if (stage .eq. 1 .and. f .le. ftest .and. g .ge. zero) + stage = 2 c Test for warnings. if (brackt .and. (stp .le. stmin .or. stp .ge. stmax)) + task = 'WARNING: ROUNDING ERRORS PREVENT PROGRESS' if (brackt .and. stmax - stmin .le. xtol*stmax) + task = 'WARNING: XTOL TEST SATISFIED' if (stp .eq. stpmax .and. f .le. ftest .and. g .le. gtest) + task = 'WARNING: STP = STPMAX' if (stp .eq. stpmin .and. (f .gt. ftest .or. g .ge. gtest)) + task = 'WARNING: STP = STPMIN' cc New warning statement added to eliminate the unexpected case cc of stp=stx due to possible difference between hardware precision cc and double precision. if (stp .eq. stx) + task = 'WARNING: STP = STX' c Test for convergence. if (f .le. ftest .and. abs(g) .le. gtol*(-ginit)) + task = 'CONVERGENCE' c Test for termination. if (task(1:4) .eq. 'WARN' .or. task(1:4) .eq. 'CONV') goto 1000 c A modified function is used to predict the step during the c first stage if a lower function value has been obtained but c the decrease is not sufficient. if (stage .eq. 1 .and. f .le. fx .and. f .gt. ftest) then c Define the modified function and derivative values. fm = f - stp*gtest fxm = fx - stx*gtest fym = fy - sty*gtest gm = g - gtest gxm = gx - gtest gym = gy - gtest c Call dcstep to update stx, sty, and to compute the new step. call dcstep(stx,fxm,gxm,sty,fym,gym,stp,fm,gm, + brackt,stmin,stmax) c Reset the function and derivative values for f. fx = fxm + stx*gtest fy = fym + sty*gtest gx = gxm + gtest gy = gym + gtest else c Call dcstep to update stx, sty, and to compute the new step. call dcstep(stx,fx,gx,sty,fy,gy,stp,f,g, + brackt,stmin,stmax) endif c Decide if a bisection step is needed. if (brackt) then if (abs(sty-stx) .ge. p66*width1) stp = stx + p5*(sty - stx) width1 = width width = abs(sty-stx) endif c Set the minimum and maximum steps allowed for stp. if (brackt) then stmin = min(stx,sty) stmax = max(stx,sty) else stmin = stp + xtrapl*(stp - stx) stmax = stp + xtrapu*(stp - stx) endif c Force the step to be within the bounds stpmax and stpmin. stp = max(stp,stpmin) stp = min(stp,stpmax) c If further progress is not possible, let stp be the best c point obtained during the search. if (brackt .and. (stp .le. stmin .or. stp .ge. stmax) + .or. (brackt .and. stmax-stmin .le. xtol*stmax)) stp = stx c Obtain another function and derivative. task = 'FG' 1000 continue c Save local variables. if (brackt) then isave(1) = 1 else isave(1) = 0 endif isave(2) = stage dsave(1) = ginit dsave(2) = gtest dsave(3) = gx dsave(4) = gy dsave(5) = finit dsave(6) = fx dsave(7) = fy dsave(8) = stx dsave(9) = sty dsave(10) = stmin dsave(11) = stmax dsave(12) = width dsave(13) = width1 end c====================== The end of dcsrch ============================== subroutine dcstep(stx,fx,dx,sty,fy,dy,stp,fp,dp,brackt, + stpmin,stpmax) logical brackt double precision stx,fx,dx,sty,fy,dy,stp,fp,dp,stpmin,stpmax c ********** c c Subroutine dcstep c c This subroutine computes a safeguarded step for a search c procedure and updates an interval that contains a step that c satisfies a sufficient decrease and a curvature condition. c c The parameter stx contains the step with the least function c value. If brackt is set to .true. then a minimizer has c been bracketed in an interval with endpoints stx and sty. c The parameter stp contains the current step. c The subroutine assumes that if brackt is set to .true. then c c min(stx,sty) < stp < max(stx,sty), c c and that the derivative at stx is negative in the direction c of the step. c c The subroutine statement is c c subroutine dcstep(stx,fx,dx,sty,fy,dy,stp,fp,dp,brackt, c stpmin,stpmax) c c where c c stx is a double precision variable. c On entry stx is the best step obtained so far and is an c endpoint of the interval that contains the minimizer. c On exit stx is the updated best step. c c fx is a double precision variable. c On entry fx is the function at stx. c On exit fx is the function at stx. c c dx is a double precision variable. c On entry dx is the derivative of the function at c stx. The derivative must be negative in the direction of c the step, that is, dx and stp - stx must have opposite c signs. c On exit dx is the derivative of the function at stx. c c sty is a double precision variable. c On entry sty is the second endpoint of the interval that c contains the minimizer. c On exit sty is the updated endpoint of the interval that c contains the minimizer. c c fy is a double precision variable. c On entry fy is the function at sty. c On exit fy is the function at sty. c c dy is a double precision variable. c On entry dy is the derivative of the function at sty. c On exit dy is the derivative of the function at the exit sty. c c stp is a double precision variable. c On entry stp is the current step. If brackt is set to .true. c then on input stp must be between stx and sty. c On exit stp is a new trial step. c c fp is a double precision variable. c On entry fp is the function at stp c On exit fp is unchanged. c c dp is a double precision variable. c On entry dp is the the derivative of the function at stp. c On exit dp is unchanged. c c brackt is an logical variable. c On entry brackt specifies if a minimizer has been bracketed. c Initially brackt must be set to .false. c On exit brackt specifies if a minimizer has been bracketed. c When a minimizer is bracketed brackt is set to .true. c c stpmin is a double precision variable. c On entry stpmin is a lower bound for the step. c On exit stpmin is unchanged. c c stpmax is a double precision variable. c On entry stpmax is an upper bound for the step. c On exit stpmax is unchanged. c c MINPACK-1 Project. June 1983 c Argonne National Laboratory. c Jorge J. More' and David J. Thuente. c c MINPACK-2 Project. October 1993. c Argonne National Laboratory and University of Minnesota. c Brett M. Averick and Jorge J. More'. c c ********** double precision zero,p66,two,three parameter(zero=0.0d0,p66=0.66d0,two=2.0d0,three=3.0d0) double precision gamma,p,q,r,s,sgnd,stpc,stpf,stpq,theta sgnd = dp*(dx/abs(dx)) c First case: A higher function value. The minimum is bracketed. c If the cubic step is closer to stx than the quadratic step, the c cubic step is taken, otherwise the average of the cubic and c quadratic steps is taken. if (fp .gt. fx) then theta = three*(fx - fp)/(stp - stx) + dx + dp s = max(abs(theta),abs(dx),abs(dp)) gamma = s*sqrt((theta/s)**2 - (dx/s)*(dp/s)) if (stp .lt. stx) gamma = -gamma p = (gamma - dx) + theta q = ((gamma - dx) + gamma) + dp r = p/q stpc = stx + r*(stp - stx) stpq = stx + ((dx/((fx - fp)/(stp - stx) + dx))/two)* + (stp - stx) if (abs(stpc-stx) .lt. abs(stpq-stx)) then stpf = stpc else stpf = stpc + (stpq - stpc)/two endif brackt = .true. c Second case: A lower function value and derivatives of opposite c sign. The minimum is bracketed. If the cubic step is farther from c stp than the secant step, the cubic step is taken, otherwise the c secant step is taken. else if (sgnd .lt. zero) then theta = three*(fx - fp)/(stp - stx) + dx + dp s = max(abs(theta),abs(dx),abs(dp)) gamma = s*sqrt((theta/s)**2 - (dx/s)*(dp/s)) if (stp .gt. stx) gamma = -gamma p = (gamma - dp) + theta q = ((gamma - dp) + gamma) + dx r = p/q stpc = stp + r*(stx - stp) stpq = stp + (dp/(dp - dx))*(stx - stp) if (abs(stpc-stp) .gt. abs(stpq-stp)) then stpf = stpc else stpf = stpq endif brackt = .true. c Third case: A lower function value, derivatives of the same sign, c and the magnitude of the derivative decreases. else if (abs(dp) .lt. abs(dx)) then c The cubic step is computed only if the cubic tends to infinity c in the direction of the step or if the minimum of the cubic c is beyond stp. Otherwise the cubic step is defined to be the c secant step. theta = three*(fx - fp)/(stp - stx) + dx + dp s = max(abs(theta),abs(dx),abs(dp)) c The case gamma = 0 only arises if the cubic does not tend c to infinity in the direction of the step. gamma = s*sqrt(max(zero,(theta/s)**2-(dx/s)*(dp/s))) if (stp .gt. stx) gamma = -gamma p = (gamma - dp) + theta q = (gamma + (dx - dp)) + gamma r = p/q if (r .lt. zero .and. gamma .ne. zero) then stpc = stp + r*(stx - stp) else if (stp .gt. stx) then stpc = stpmax else stpc = stpmin endif stpq = stp + (dp/(dp - dx))*(stx - stp) if (brackt) then c A minimizer has been bracketed. If the cubic step is c closer to stp than the secant step, the cubic step is c taken, otherwise the secant step is taken. if (abs(stpc-stp) .lt. abs(stpq-stp)) then stpf = stpc else stpf = stpq endif if (stp .gt. stx) then stpf = min(stp+p66*(sty-stp),stpf) else stpf = max(stp+p66*(sty-stp),stpf) endif else c A minimizer has not been bracketed. If the cubic step is c farther from stp than the secant step, the cubic step is c taken, otherwise the secant step is taken. if (abs(stpc-stp) .gt. abs(stpq-stp)) then stpf = stpc else stpf = stpq endif stpf = min(stpmax,stpf) stpf = max(stpmin,stpf) endif c Fourth case: A lower function value, derivatives of the c same sign, and the magnitude of the derivative does not c decrease. If the minimum is not bracketed, the step is either c stpmin or stpmax, otherwise the cubic step is taken. else if (brackt) then theta = three*(fp - fy)/(sty - stp) + dy + dp s = max(abs(theta),abs(dy),abs(dp)) gamma = s*sqrt((theta/s)**2 - (dy/s)*(dp/s)) if (stp .gt. sty) gamma = -gamma p = (gamma - dp) + theta q = ((gamma - dp) + gamma) + dy r = p/q stpc = stp + r*(sty - stp) stpf = stpc else if (stp .gt. stx) then stpf = stpmax else stpf = stpmin endif endif c Update the interval which contains a minimizer. if (fp .gt. fx) then sty = stp fy = fp dy = dp else if (sgnd .lt. zero) then sty = stx fy = fx dy = dx endif stx = stp fx = fp dx = dp endif c Compute the new step. stp = stpf end c====================== The end of dcstep ============================== subroutine timer(ttime) double precision ttime c ********* c c Subroutine timer c c This subroutine is used to determine user time. In a typical c application, the user time for a code segment requires calls c to subroutine timer to determine the initial and final time. c c The subroutine statement is c c subroutine timer(ttime) c c where c c ttime is an output variable which specifies the user time. c c Argonne National Laboratory and University of Minnesota. c MINPACK-2 Project. c c Modified October 1990 by Brett M. Averick. c c ********** real temp real tarray(2) real etime c The first element of the array tarray specifies user time temp = etime(tarray) ttime = dble(tarray(1)) return end c====================== The end of timer =============================== double precision function dnrm2(n,x,incx) integer n,incx double precision x(n) c ********** c c Function dnrm2 c c Given a vector x of length n, this function calculates the c Euclidean norm of x with stride incx. c c The function statement is c c double precision function dnrm2(n,x,incx) c c where c c n is a positive integer input variable. c c x is an input array of length n. c c incx is a positive integer variable that specifies the c stride of the vector. c c Subprograms called c c FORTRAN-supplied ... abs, max, sqrt c c MINPACK-2 Project. February 1991. c Argonne National Laboratory. c Brett M. Averick. c c ********** integer i double precision scale dnrm2 = 0.0d0 scale = 0.0d0 do 10 i = 1, n, incx scale = max(scale, abs(x(i))) 10 continue if (scale .eq. 0.0d0) return do 20 i = 1, n, incx dnrm2 = dnrm2 + (x(i)/scale)**2 20 continue dnrm2 = scale*sqrt(dnrm2) return end c====================== The end of dnrm2 =============================== double precision function dpmeps() c ********** c c Subroutine dpeps c c This subroutine computes the machine precision parameter c dpmeps as the smallest floating point number such that c 1 + dpmeps differs from 1. c c This subroutine is based on the subroutine machar described in c c W. J. Cody, c MACHAR: A subroutine to dynamically determine machine parameters, c ACM Trans. Math. Soft., 14, 1988, pages 303-311. c c The subroutine statement is: c c subroutine dpeps(dpmeps) c c where c c dpmeps is a double precision variable. c On entry dpmeps need not be specified. c On exit dpmeps is the machine precision. c c MINPACK-2 Project. February 1991. c Argonne National Laboratory and University of Minnesota. c Brett M. Averick. c c ******* integer i,ibeta,irnd,it,itemp,negep double precision a,b,beta,betain,betah,temp,tempa,temp1, + zero,one,two data zero,one,two /0.0d0,1.0d0,2.0d0/ c determine ibeta, beta ala malcolm. a = one b = one 10 continue a = a + a temp = a + one temp1 = temp - a if (temp1 - one .eq. zero) go to 10 20 continue b = b + b temp = a + b itemp = int(temp - a) if (itemp .eq. 0) go to 20 ibeta = itemp beta = dble(ibeta) c determine it, irnd. it = 0 b = one 30 continue it = it + 1 b = b * beta temp = b + one temp1 = temp - b if (temp1 - one .eq. zero) go to 30 irnd = 0 betah = beta/two temp = a + betah if (temp - a .ne. zero) irnd = 1 tempa = a + beta temp = tempa + betah if ((irnd .eq. 0) .and. (temp - tempa .ne. zero)) irnd = 2 c determine dpmeps. negep = it + 3 betain = one/beta a = one do 40 i = 1, negep a = a*betain 40 continue 50 continue temp = one + a if (temp - one .ne. zero) go to 60 a = a*beta go to 50 60 continue dpmeps = a if ((ibeta .eq. 2) .or. (irnd .eq. 0)) go to 70 a = (a*(one + a))/two temp = one + a if (temp - one .ne. zero) dpmeps = a 70 return end c====================== The end of dpmeps ============================== subroutine daxpy(n,da,dx,incx,dy,incy) c c constant times a vector plus a vector. c uses unrolled loops for increments equal to one. c jack dongarra, linpack, 3/11/78. c double precision dx(*),dy(*),da integer i,incx,incy,ix,iy,m,mp1,n c if(n.le.0)return if (da .eq. 0.0d0) return if(incx.eq.1.and.incy.eq.1)go to 20 c c code for unequal increments or equal increments c not equal to 1 c ix = 1 iy = 1 if(incx.lt.0)ix = (-n+1)*incx + 1 if(incy.lt.0)iy = (-n+1)*incy + 1 do 10 i = 1,n dy(iy) = dy(iy) + da*dx(ix) ix = ix + incx iy = iy + incy 10 continue return c c code for both increments equal to 1 c c c clean-up loop c 20 m = mod(n,4) if( m .eq. 0 ) go to 40 do 30 i = 1,m dy(i) = dy(i) + da*dx(i) 30 continue if( n .lt. 4 ) return 40 mp1 = m + 1 do 50 i = mp1,n,4 dy(i) = dy(i) + da*dx(i) dy(i + 1) = dy(i + 1) + da*dx(i + 1) dy(i + 2) = dy(i + 2) + da*dx(i + 2) dy(i + 3) = dy(i + 3) + da*dx(i + 3) 50 continue return end c====================== The end of daxpy =============================== subroutine dcopy(n,dx,incx,dy,incy) c c copies a vector, x, to a vector, y. c uses unrolled loops for increments equal to one. c jack dongarra, linpack, 3/11/78. c double precision dx(*),dy(*) integer i,incx,incy,ix,iy,m,mp1,n c if(n.le.0)return if(incx.eq.1.and.incy.eq.1)go to 20 c c code for unequal increments or equal increments c not equal to 1 c ix = 1 iy = 1 if(incx.lt.0)ix = (-n+1)*incx + 1 if(incy.lt.0)iy = (-n+1)*incy + 1 do 10 i = 1,n dy(iy) = dx(ix) ix = ix + incx iy = iy + incy 10 continue return c c code for both increments equal to 1 c c c clean-up loop c 20 m = mod(n,7) if( m .eq. 0 ) go to 40 do 30 i = 1,m dy(i) = dx(i) 30 continue if( n .lt. 7 ) return 40 mp1 = m + 1 do 50 i = mp1,n,7 dy(i) = dx(i) dy(i + 1) = dx(i + 1) dy(i + 2) = dx(i + 2) dy(i + 3) = dx(i + 3) dy(i + 4) = dx(i + 4) dy(i + 5) = dx(i + 5) dy(i + 6) = dx(i + 6) 50 continue return end c====================== The end of dcopy =============================== double precision function ddot(n,dx,incx,dy,incy) c c forms the dot product of two vectors. c uses unrolled loops for increments equal to one. c jack dongarra, linpack, 3/11/78. c double precision dx(*),dy(*),dtemp integer i,incx,incy,ix,iy,m,mp1,n c ddot = 0.0d0 dtemp = 0.0d0 if(n.le.0)return if(incx.eq.1.and.incy.eq.1)go to 20 c c code for unequal increments or equal increments c not equal to 1 c ix = 1 iy = 1 if(incx.lt.0)ix = (-n+1)*incx + 1 if(incy.lt.0)iy = (-n+1)*incy + 1 do 10 i = 1,n dtemp = dtemp + dx(ix)*dy(iy) ix = ix + incx iy = iy + incy 10 continue ddot = dtemp return c c code for both increments equal to 1 c c c clean-up loop c 20 m = mod(n,5) if( m .eq. 0 ) go to 40 do 30 i = 1,m dtemp = dtemp + dx(i)*dy(i) 30 continue if( n .lt. 5 ) go to 60 40 mp1 = m + 1 do 50 i = mp1,n,5 dtemp = dtemp + dx(i)*dy(i) + dx(i + 1)*dy(i + 1) + * dx(i + 2)*dy(i + 2) + dx(i + 3)*dy(i + 3) + dx(i + 4)*dy(i + 4) 50 continue 60 ddot = dtemp return end c====================== The end of ddot ================================ subroutine dpofa(a,lda,n,info) integer lda,n,info double precision a(lda,*) c c dpofa factors a double precision symmetric positive definite c matrix. c c dpofa is usually called by dpoco, but it can be called c directly with a saving in time if rcond is not needed. c (time for dpoco) = (1 + 18/n)*(time for dpofa) . c c on entry c c a double precision(lda, n) c the symmetric matrix to be factored. only the c diagonal and upper triangle are used. c c lda integer c the leading dimension of the array a . c c n integer c the order of the matrix a . c c on return c c a an upper triangular matrix r so that a = trans(r)*r c where trans(r) is the transpose. c the strict lower triangle is unaltered. c if info .ne. 0 , the factorization is not complete. c c info integer c = 0 for normal return. c = k signals an error condition. the leading minor c of order k is not positive definite. c c linpack. this version dated 08/14/78 . c cleve moler, university of new mexico, argonne national lab. c c subroutines and functions c c blas ddot c fortran sqrt c c internal variables c double precision ddot,t double precision s integer j,jm1,k c begin block with ...exits to 40 c c do 30 j = 1, n info = j s = 0.0d0 jm1 = j - 1 if (jm1 .lt. 1) go to 20 do 10 k = 1, jm1 t = a(k,j) - ddot(k-1,a(1,k),1,a(1,j),1) t = t/a(k,k) a(k,j) = t s = s + t*t 10 continue 20 continue s = a(j,j) - s c ......exit if (s .le. 0.0d0) go to 40 a(j,j) = sqrt(s) 30 continue info = 0 40 continue return end c====================== The end of dpofa =============================== subroutine dscal(n,da,dx,incx) c c scales a vector by a constant. c uses unrolled loops for increment equal to one. c jack dongarra, linpack, 3/11/78. c modified 3/93 to return if incx .le. 0. c double precision da,dx(*) integer i,incx,m,mp1,n,nincx c if( n.le.0 .or. incx.le.0 )return if(incx.eq.1)go to 20 c c code for increment not equal to 1 c nincx = n*incx do 10 i = 1,nincx,incx dx(i) = da*dx(i) 10 continue return c c code for increment equal to 1 c c c clean-up loop c 20 m = mod(n,5) if( m .eq. 0 ) go to 40 do 30 i = 1,m dx(i) = da*dx(i) 30 continue if( n .lt. 5 ) return 40 mp1 = m + 1 do 50 i = mp1,n,5 dx(i) = da*dx(i) dx(i + 1) = da*dx(i + 1) dx(i + 2) = da*dx(i + 2) dx(i + 3) = da*dx(i + 3) dx(i + 4) = da*dx(i + 4) 50 continue return end c====================== The end of dscal =============================== subroutine dtrsl(t,ldt,n,b,job,info) integer ldt,n,job,info double precision t(ldt,*),b(*) c c c dtrsl solves systems of the form c c t * x = b c or c trans(t) * x = b c c where t is a triangular matrix of order n. here trans(t) c denotes the transpose of the matrix t. c c on entry c c t double precision(ldt,n) c t contains the matrix of the system. the zero c elements of the matrix are not referenced, and c the corresponding elements of the array can be c used to store other information. c c ldt integer c ldt is the leading dimension of the array t. c c n integer c n is the order of the system. c c b double precision(n). c b contains the right hand side of the system. c c job integer c job specifies what kind of system is to be solved. c if job is c c 00 solve t*x=b, t lower triangular, c 01 solve t*x=b, t upper triangular, c 10 solve trans(t)*x=b, t lower triangular, c 11 solve trans(t)*x=b, t upper triangular. c c on return c c b b contains the solution, if info .eq. 0. c otherwise b is unaltered. c c info integer c info contains zero if the system is nonsingular. c otherwise info contains the index of c the first zero diagonal element of t. c c linpack. this version dated 08/14/78 . c g. w. stewart, university of maryland, argonne national lab. c c subroutines and functions c c blas daxpy,ddot c fortran mod c c internal variables c double precision ddot,temp integer case,j,jj c c begin block permitting ...exits to 150 c c check for zero diagonal elements. c do 10 info = 1, n c ......exit if (t(info,info) .eq. 0.0d0) go to 150 10 continue info = 0 c c determine the task and go to it. c case = 1 if (mod(job,10) .ne. 0) case = 2 if (mod(job,100)/10 .ne. 0) case = case + 2 go to (20,50,80,110), case c c solve t*x=b for t lower triangular c 20 continue b(1) = b(1)/t(1,1) if (n .lt. 2) go to 40 do 30 j = 2, n temp = -b(j-1) call daxpy(n-j+1,temp,t(j,j-1),1,b(j),1) b(j) = b(j)/t(j,j) 30 continue 40 continue go to 140 c c solve t*x=b for t upper triangular. c 50 continue b(n) = b(n)/t(n,n) if (n .lt. 2) go to 70 do 60 jj = 2, n j = n - jj + 1 temp = -b(j+1) call daxpy(j,temp,t(1,j+1),1,b(1),1) b(j) = b(j)/t(j,j) 60 continue 70 continue go to 140 c c solve trans(t)*x=b for t lower triangular. c 80 continue b(n) = b(n)/t(n,n) if (n .lt. 2) go to 100 do 90 jj = 2, n j = n - jj + 1 b(j) = b(j) - ddot(jj-1,t(j+1,j),1,b(j+1),1) b(j) = b(j)/t(j,j) 90 continue 100 continue go to 140 c c solve trans(t)*x=b for t upper triangular. c 110 continue b(1) = b(1)/t(1,1) if (n .lt. 2) go to 130 do 120 j = 2, n b(j) = b(j) - ddot(j-1,t(1,j),1,b(1),1) b(j) = b(j)/t(j,j) 120 continue 130 continue 140 continue 150 continue return end c====================== The end of dtrsl =============================== gpml-matlab-v3.2-2013-01-15/util/lbfgsb/lbfgsb.cpp000644 000765 000024 00000007734 11511036275 021244 0ustar00carlstaff000000 000000 #include "mex.h" #include "matrix.h" #include "matlabexception.h" #include "matlabscalar.h" #include "matlabstring.h" #include "matlabmatrix.h" #include "arrayofmatrices.h" #include "program.h" #include "matlabprogram.h" #include #include extern void _main(); // Constants. // ----------------------------------------------------------------- const int minNumInputArgs = 5; // Function definitions. // ----------------------------------------------------------------- void mexFunction (int nlhs, mxArray *plhs[], int nrhs, const mxArray *prhs[]) try { // Check to see if we have the correct number of input and output // arguments. if (nrhs < minNumInputArgs) throw MatlabException("Incorrect number of input arguments"); // Get the starting point for the variables. This is specified in // the first input argument. The variables must be either a single // matrix or a cell array of matrices. int k = 0; // The index of the current input argument. ArrayOfMatrices x0(prhs[k++]); // Create the output, which stores the solution obtained from // running IPOPT. There should be as many output arguments as cell // entries in X. if (nlhs != x0.length()) throw MatlabException("Incorrect number of output arguments"); ArrayOfMatrices x(plhs,x0); // Load the lower and upper bounds on the variables as // ArrayOfMatrices objects. They should have the same structure as // the ArrayOfMatrices object "x". ArrayOfMatrices lb(prhs[k++]); ArrayOfMatrices ub(prhs[k++]); // Check to make sure the bounds make sense. if (lb != x || ub != x) throw MatlabException("Input arguments LB and UB must have the same \ structure as X"); // Get the Matlab callback functions. MatlabString objFunc(prhs[k++]); MatlabString gradFunc(prhs[k++]); // Get the auxiliary data. const mxArray* auxData; const mxArray* ptr = prhs[k++]; if (nrhs > 5) { if (mxIsEmpty(ptr)) auxData = 0; else auxData = ptr; } else auxData = 0; // Get the intermediate callback function. MatlabString* iterFunc; ptr = prhs[k++]; if (nrhs > 6) { if (mxIsEmpty(ptr)) iterFunc = 0; else iterFunc = new MatlabString(ptr); } else iterFunc = 0; // Set the options for the L-BFGS algorithm to their defaults. int maxiter = defaultmaxiter; int m = defaultm; double factr = defaultfactr; double pgtol = defaultpgtol; // Process the remaining input arguments, which set options for // the IPOPT algorithm. while (k < nrhs) { // Get the option label from the Matlab input argument. MatlabString optionLabel(prhs[k++]); if (k < nrhs) { // Get the option value from the Matlab input argument. MatlabScalar optionValue(prhs[k++]); double value = optionValue; if (!strcmp(optionLabel,"maxiter")) maxiter = (int) value; else if (!strcmp(optionLabel,"m")) m = (int) value; else if (!strcmp(optionLabel,"factr")) factr = value / mxGetEps(); else if (!strcmp(optionLabel,"pgtol")) pgtol = value; else { if (iterFunc) delete iterFunc; throw MatlabException("Nonexistent option"); } } } // Create a new instance of the optimization problem. x = x0; MatlabProgram program(x,lb,ub,&objFunc,&gradFunc,iterFunc, (mxArray*) auxData,m,maxiter,factr,pgtol); // Run the L-BFGS-B solver. SolverExitStatus exitStatus = program.runSolver(); if (exitStatus == abnormalTermination) { if (iterFunc) delete iterFunc; throw MatlabException("Solver unable to satisfy convergence \ criteria due to abnormal termination"); } else if (exitStatus == errorOnInput) { if (iterFunc) delete iterFunc; throw MatlabException("Invalid inputs to L-BFGS routine"); } // Free the dynamically allocated memory. if (iterFunc) delete iterFunc; } catch (std::exception& error) { mexErrMsgTxt(error.what()); } gpml-matlab-v3.2-2013-01-15/util/lbfgsb/matlabstring.h000644 000765 000024 00000002202 11511036275 022122 0ustar00carlstaff000000 000000 #ifndef INCLUDE_MATLABSTRING #define INCLUDE_MATLABSTRING #include "mex.h" #include // Function declarations. // ----------------------------------------------------------------- // Copy a C-style string (i.e. a null-terminated character array). char* copystring (const char* source); // Class MatlabString. // ----------------------------------------------------------------- // This class encapsulates read-only access to a MATLAB character // array. class MatlabString { public: // The constructor accepts as input a pointer to a Matlab array, // which must be a valid string (array of type CHAR). explicit MatlabString (const mxArray* ptr); // The copy constructor makes a full copy of the source string. MatlabString (const MatlabString& source); // The destructor. ~MatlabString(); // Conversion operator for null-terminated string. operator const char* () const { return s; }; protected: char* s; // The null-terminated string. // The copy assignment operator is not proper, thus remains // protected. MatlabString& operator= (const MatlabString& source) { return *this; }; }; #endif gpml-matlab-v3.2-2013-01-15/util/lbfgsb/README.html000644 000765 000024 00000043521 11511036275 021116 0ustar00carlstaff000000 000000 MATLAB interface for L-BFGS-B

A MATLAB interface for L-BFGS-B

by Peter Carbonetto
Dept. of Computer Science
University of British Columbia

L-BFGS-B is a collection of Fortran 77 routines for solving large-scale nonlinear optimization problems with bound constraints on the variables. One of the key features of the nonlinear solver is that knowledge of the Hessian is not required; the solver computes search directions by keeping track of a quadratic model of the objective function with a limited-memory BFGS (Broyden-Fletcher-Goldfarb-Shanno) approximation to the Hessian.1 The algorithm was developed by Ciyou Zhu, Richard Byrd and Jorge Nocedal. For more information, go to the original distribution site for the L-BFGS-B software package.

I've designed an interface to the L-BFGS-B solver so that it can be called like any other function in MATLAB.2 See the text below for more information on installing and calling it in MATLAB. Along the way, I've also developed a C++ class that encapsulates all the "messy" details in executing the L-BFGS-B code. See below for instructions on how to use this class for your own C++ code.

If you have any questions, praise, or comments, or would like to report a bug, do not hesitate to contact the author. I've tested this software in MATLAB version 7.2 on both the Mac OS X and Linux operating systems.

Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Canada License. Please cite the source as Peter Carbonetto, Department of Computer Science, University of British Columbia.

Installation

These installation instructions assume you have a UNIX-based operating system, such as Mac OS X or Linux. It is surely possible to install this software on Windows, I'm just not sure how. These instructions also assume you have GNU make installed.

Download the source. First, download and unpack the tar archive. It contains some MATLAB files (ending in .m), some C++ source and header files (ending in .h and .cpp), a single Fortran 77 source file solver.f containing the L-BFGS-B routines, and a Makefile. I've included a version of the Fortran routines that is more recent than what is available for download at the distribution site at Northwestern University.

What we will do is create a MEX file, which is basically a file that contains a routine that can be called from MATLAB as if it were a built-it function. To learn about MEX files, I refer you to this document at the MathWorks website.

Install the C++ and Fortran 77 compilers. In order to build the MEX file, you will need both a C++ compiler and a Fortran 77 compiler. Unfortunately, you can't just use any compiler. You have to use the precise one supported by MATLAB. For instance, if you are running Mac OS X 7.2 on a Linux operating system, you will need to install the GNU compiler collection (GCC) version 3.4.4. Even if you already have a compiler installed on Linux, it may be the wrong version, and if you use the wrong version things could go horribly wrong! It is important that you use the correct version of the compiler, otherwise you will encounter linking errors. Refer to this document to find out which compiler is supported by your version of MATLAB.

Configure MATLAB. Next, you need to set up and configure MATLAB to build MEX Files. This is explained quite nicely in this MathWorks support document.

Modify the Makefile. You are almost ready to build the MEX file. But before you do so, you need to edit the Makefile to coincide with your system setup. At the top of the file are several variables you may need to modify. The variable MEX is the executable called to build the MEX file. The variables CXX and F77 must be the names of your C++ and Fortran 77 compilers. MEXSUFFIX is the MEX file extension specific to your platform. (For instance, the extension for Linux is mexglx.) The variable MATLAB_HOME must be the base directory of your MATLAB installation. Finally, CFLAGS and FFLAGS are options passed to the C++ and Fortran compilers, respectively. Often these flags coincide with your MEX options file (see here for more information on the options file). But often the settings in the MEX option file are incorrect, and so the options must be set manually. MATLAB requires some special compilation flags for various reasons, one being that it requires position-independent code. These instructions are vague (I apologize), and this step may require a bit of trial and error until before you get it right.

On my laptop running Mac OS X 10.3.9 with MATLAB 7.2, my settings for the Makefile are

  MEX         = mex
  MEXSUFFIX   = mexmac
  MATLAB_HOME = /Applications/MATLAB72
  CXX         = g++
  F77         = g77 
  CFLAGS      = -O3 -fPIC -fno-common -fexceptions \
                -no-cpp-precomp 
  FFLAGS      = -O3 -x f77-cpp-input -fPIC -fno-common

On my Linux machine, I set the variables in the Makefile like so:

  MEX         = mex
  MEXSUFFIX   = mexglx
  MATLAB_HOME = /cs/local/generic/lib/pkg/matlab-7.2
  CXX         = g++-3.4.5
  F77         = g77-3.4.5
  CFLAGS      = -O3 -fPIC -pthread 
  FFLAGS      = -O3 -fPIC -fexceptions

It may be helpful to look at the GCC documentation in order to understand what these various compiler flags mean.

Build the MEX file. If you are in the directory containing all the source files, typing make in the command prompt will first compile the Fortran and C++ source files into object code (.o files). After that, the make program calls the MEX script, which in turn links all the object files together into a single MEX file. If you didn't get any errors, then you are ready to try out the bound-constrained solver in MATLAB. Note that even if you didn't get any errors, there's still a possibility that you didn't link the MEX file properly, in which case executing the MEX file will cause MATLAB to crash. But there's only one way to find out: the hard way.

A brief tutorial

I've written a short script examplehs038.m which demonstrates how to call lbfgsb for solving a small optimization problem, the Hock & Schittkowski test problem no. 38.3 The optimization problem has four variables. They are bounded from below by -10 and bounded from above by +10. We've set the starting point to (-3, -1, -3, -1). We've written two MATLAB functions for computing the objective function and the gradient of the objective function at a given point. These functions can be found in the files computeObjectiveHS038.m and computeGradientHS038.m. If you run the script, the solver should very quickly progress toward the point (1, 1, 1, 1). The objective is 0 at this point.

The basic MATLAB function call is

  lbfgsb(x0,lb,ub,objfunc,gradfunc)

The first input argument x0 declares the starting point for the solver. The second and third input arguments define the lower and upper bounds on the variables, respectively. If an entry of lb is set to -Inf, then this is equivalent to no lower bound for that entry (the same applies to upper bounds, except that it must be +Inf instead). The next two input arguments must be the names of MATLAB routines (M-files). The first routine must calculate the objective function at the current point. The output must be a scalar representing the objective evaluated at the current point. The second function must compute the gradient of the objective at the current point. The input is the same as objfunc, but it must return as many outputs as there are inputs. For a complete guide on using the MATLAB interface, type help lbfgsb in the MATLAB prompt.

The script exampleldaimages.m is a more complicated example. It uses the L-BFGS-B solver to compute some statistics of a posterior distribution for a text analysis model called latent Dirichlet allication (LDA). The model is used to estimate the topics for a collection of documents. The optimization problem is to compute an approximation to the posterior, since it is not possible to compute the posterior exactly. In this case, the optimization problem corresponds to minimizing the distance between the true posterior distribution and an approximate distribution. For more information, see the paper of Blei, Ng and Jordan (2003).4 In order to run this example, you will need to install the lightspeed toolbox for MATLAB developed by Tom Minka at Microsoft Research.

The script starts by generating a bunch of canonical images that represent topics; if a pixel is white then a word at that position is more likely to appear in that topic. Note that the order of the pixels in the image is unimportant, since LDA is a "bag of words" model. In another figure we show a small sample of the data generated from the topics. Each image represents a document, and the pixels in the image depict the word proportions in that document.5

Next, the script runs the L-BFGS-B solver to find a local minimum to the variational objective. Buried in the M-file mflda.m is the call to the solver: 

  [xi gamma Phi] = lbfgsb({xi gamma Phi},...
      {repmat(lb,W,K)  repmat(lb,K,D)  repmat(lb,K,M)},...
      {repmat(inf,W,K) repmat(inf,K,D) repmat(inf,K,M)},...
      'computeObjectiveMFLDA','computeGradientMFLDA',...
      {nu eta L w lnZconst},'callbackMFLDA',...
      'm',m_lbfgs,'factr',1e-12,'pgtol',...
      tolerance,'maxiter',maxiter);

This example will give us the opportunity to demonstrate some of the more complicated aspects of the our MATLAB interface. The first input to lbfgsb sets the initial iterate of the optimization algorithm. Notice that we've passed a cell array with entries that are matrices. Indeed, it is possible to pass either a matrix or a cell array. Whatever structure is passed, the other input arguments (and outputs) must also abide by this structure. The lower and upper bounds on variables are also cell arrays. All the upper bounds are set to infinity, which means that the variables are only bounded from below.

It is instructive to examine the callback routines. They look like: 

  f = computeObjectiveMFLDA(xi,gamma,Phi,auxdata)

  [dxi, dgamma, dPhi] = ...
      computeGradientMFLDA(xi,gamma,Phi,auxdata)

Notice that each entry to the cell array is its own input argument to the callback routines. The same applies for the output arguments of the gradient callback function.

The sixth input argument is a cell array that contains auxiliary data. This data will be passed to the MATLAB functions that evaluate the objective and gradient. The seventh input argument specifies a callback routine that is called exactly once for every iteration. This callback routine is most useful for examining the progress of the solver. The remaining input arguments are label/value pairs that override some of the default settings of L-BFGS-B.

After converging to a solution, we display topic samples drawn from the variational approximation. A good solution should come close to recovering the true topics from which the data was generated (although they might not be in the same order).

Don't be surprised if it takes many iterations before the optimization algorithm converges on a local minimum. (In repeated trials, I found it as many as two thousand iterations to converge to a stationary point of the objective.) This particular problem demonstrates the limitations of limited-memory quasi-Newton approximations to the Hessian; the low storage requirements can come at the cost of slow convergence to the solution.

The C++ interface

One of the nice byproducts of writing a MATLAB interface for L-BFGS-B is that I've ended up with a neat little C++ class that encapsulates the nuts and bolts of executing the solver. A brief description of the Program class can be found in the header file program.h.

Program is an abstract class, which means that its impossible to instantiate a Program object. This is because the class has member functions that aren't yet implemented. These are called pure virtual functions. In order to use the Program class, one needs to define another class that inherits the Program class and that implements the pure virtual functions. In fact, the class MatlabProgram (in matlabprogram.h) is an example of such a class.

The new child class must declare and implement these two functions:

  virtual double computeObjective (int n, double* x);
  virtual void computeGradient (int n, double* x, double* g);

See the header file for a detailed description of these functions. The only remaining detail is calling the Program constructor. After that, it is just a matter of declaring a new object of type MyProgram then calling the function runSolver.

Contents of the tar archive

solver.f
Fortran 77 source for the L-BFGS-B solver routines.

program.h
program.cpp
Header and source file for the C++ interface.

array.h
arrayofmatrices.h
matlabexception.h
matlabmatrix.h
matlabprogram.h
matlabscalar.h
matlabstring.h
arrayofmatrices.cpp
matlabexception.cpp
matlabmatrix.cpp
matlabprogram.cpp
matlabscalar.cpp
matlabstring.cpp
lbfgsb.cpp
Header and source files for the MATLAB interface.

lbfgsb.m
Usage instructions for the MEX file.

examplehs038.m
MATLAB script for solving the H&S test example no. 38.

computeObjectiveHS038.m
computeGradientHS038.m
genericcallback.m
MATLAB functions used by the script examplehs038.m.

exampleldaimages.m
MATLAB script that generates synthetic documents and topics, then computes a mean field variational approximation to the posterior distribution of the latent Dirichlet allocation model, then displays the result.

mflda.m
This function computes a mean field variational approximation to the posterior distribution of the latent Dirichlet allocation model by minimizing the distance between the variational distribution and the true distribution. It uses the L-BFGS-B to minimize the objective function subject to the bound constraints. The objective function also acts as a lower bound on the logarithm of the denominator that appears from the application of Bayes' rule.4

callbackMFLDA.m
computeObjectiveMFLDA.m
computeGradientMFLDA.m
computelnZ.m
computelnZconst.m
computem.m
dirichletrnd.m
gammarnd.m
createsyntheticdata.m
mfldainit.m
Some functions used by exampleldaimages.m and mflda.m.

Footnotes

1 Ciyou Zhu, Richard H. Byrd, Peihuang Lu and Jorge Nocedal (1997). Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization. ACM Transactions on Mathematical Software, Vol. 23, No. 4., pp. 550-560.

1 Another MATLAB interface for the L-BFGS-B routines has been developed by Liam Stewart at the University of Toronto.

3 Willi Hock and Klaus Schittkowski (1981). Test Examples for Nonlinear Programming Codes. Lecture Notes in Economics and Mathematical Systems, Vol. 187, Springer-Verlag.

4 David M. Blei, Andrew Y. Ng, Michael I. Jordan (2003). Latent Dirichlet allocation. Journal of Machine Learning Research, Vol. 3, pp. 993-1022.

5 Tom L. Griffiths and Mark Steyvers (2004). Finding scientific topics. Proceedings of the National Academy of Sciences, Vol. 101, pp. 5228-5235.


March 3, 2007
gpml-matlab-v3.2-2013-01-15/util/lbfgsb/matlabprogram.h000644 000765 000024 00000005501 11511036275 022270 0ustar00carlstaff000000 000000 #ifndef INCLUDE_MATLABPROGRAM #define INCLUDE_MATLABPROGRAM #include "mex.h" #include "program.h" #include "matlabscalar.h" #include "matlabstring.h" #include "arrayofmatrices.h" // Class MatlabProgram. // ----------------------------------------------------------------- // This is an implementation of the abstract class Program. class MatlabProgram: public Program { public: // On input, "variables" should contain the initial point for the // optimization routine. If no iterative callback function is // specified, "iterFunc" may be set to 0. Also, if no auxiliary data // is needed, it may also be set to 0. MatlabProgram (ArrayOfMatrices& variables, const ArrayOfMatrices& lowerbounds, const ArrayOfMatrices& upperbounds, const MatlabString* objFunc, const MatlabString* gradFunc, const MatlabString* iterFunc, mxArray* auxData, int m = defaultm, int maxiter = defaultmaxiter, double factr = defaultfactr, double pgtol = defaultpgtol); // The destructor. virtual ~MatlabProgram(); // These provide definitions for the pure virtual functions of the // abstract parent class. virtual double computeObjective (int n, double* x); virtual void computeGradient (int n, double* x, double* g); virtual void iterCallback (int t, double* x, double f); // Run the solver. Upon completion, the solution is stored in // "variables". SolverExitStatus runSolver(); protected: ArrayOfMatrices& variables; // Storage for the initial value and // solution. const MatlabString* objFunc; // The objective callback function. const MatlabString* gradFunc; // The gradient callback function. const MatlabString* iterFunc; // The iterative callback function. ArrayOfMatrices* varMatlab; // Inputs to the Matlab callback mxArray** varInputs; // functions representing the // current values of the variables. MatlabScalar* fMatlab; // Input to the Matlab callback // functions representing the // current value of the objective. MatlabScalar* tMatlab; // Input to the Matlab callback // functions representing the // current iteration. int numInputsObjFunc; // The number of inputs passed to the // objective callback function. int numInputsGradFunc; // The number of inputs passed to the // gradient callback function. int numInputsIterFunc; // The number of inputs passed to the // iterative callback function. mxArray** inputsObjFunc; // The inputs passed to the objective // callback function. mxArray** inputsGradFunc; // The inputs passed to the gradient // callback function. mxArray** inputsIterFunc; // The inputs passed to the iterative // callback function. }; #endif gpml-matlab-v3.2-2013-01-15/util/lbfgsb/arrayofmatrices.h000644 000765 000024 00000004144 11511036275 022635 0ustar00carlstaff000000 000000 #ifndef INCLUDE_ARRAYOFMATRICES #define INCLUDE_ARRAYOFMATRICES #include "array.h" #include "matlabmatrix.h" #include "mex.h" // Class ArrayOfMatrices. // ----------------------------------------------------------------- class ArrayOfMatrices : public Array { public: // This version of the constructor behaves just like its parent. explicit ArrayOfMatrices (int length) : Array(length) { }; // This constructor creates an array of matrices from a Matlab // array. It accepts either a matrix in double precision, or a cell // array with entries that are matrices. explicit ArrayOfMatrices (const mxArray* ptr); // This constructor creates an array of matrices from a collection // of Matlab arrays. The Matlab arrays must be matrices. ArrayOfMatrices (const mxArray* ptrs[], int numptrs); // This constructor creates an array of matrices and the // associated Matlab structures. The Matlab structures are // matrices. The second input argument acts as a template for the // creation of the matrices, but the data from "model" is not // actually copied into the new ArrayOfMatrices object. It is up // to the user to make sure that the array of mxArray pointers has // enough room for the pointers to the Matlab arrays. ArrayOfMatrices (mxArray* ptrs[], const ArrayOfMatrices& model); // This constructor creates an array of matrices using the second // input argument as a model. The input argument "data" contains // the element data. Note that the information is NOT copied from // the model! ArrayOfMatrices (double* data, const ArrayOfMatrices& model); // The copy constructor makes a shallow copy of the source object. ArrayOfMatrices (const ArrayOfMatrices& source); // The destructor. ~ArrayOfMatrices(); // Copy assignment operator that observes the same behaviour as // the Array copy assignment operator. ArrayOfMatrices& operator= (const ArrayOfMatrices& source); // Return the total number of elements in all the matrices. int numelems() const; protected: static int getnummatlabmatrices (const mxArray* ptr); }; #endif gpml-matlab-v3.2-2013-01-15/util/lbfgsb/array.h000644 000765 000024 00000010714 11511036275 020560 0ustar00carlstaff000000 000000 #ifndef INCLUDE_ARRAY #define INCLUDE_ARRAY #include "matlabexception.h" #include // Function definitions. // ----------------------------------------------------------------- template void copymemory (const Type* source, Type* dest, int length) { memcpy(dest,source,sizeof(Type)*length); } // Class Array // ----------------------------------------------------------------- // An Array object stores a collection of ordered elements. The key // aspect of Array objects is that they do not necessarily achieve // encapsulation; they do not necessarily retain independent // ownership of their data. That is, the data could be modified // externally by another agent. This behaviour is determined by the // data member "owner". If "owner" is false, the array does not take // care of allocation and deallocation of the data in memory. // // Copy assignment behaves different than usual---it copies the // data, but requires that the destination already have the proper // resources allocated in memory. // // The copy constructor performs a shallow copy. template class Array { public: // This constructor allocates memory for an array of elements of // the specificed length. explicit Array (int length); // This constructor is set to point to an already existing // array. As such, it does not gain ownership of the elements. Array (Type* data, int length); // The copy constructor performs a shallow copy, so that both // arrays share the same copy of the data. Array (const Array& source); // The destructor. ~Array(); // Set all the elements of the array to the specified value. void setvalue (const Type& value); // Copy the elements to from the source array. Both the source and // destination array must have the same length. void inject (const Type* source); void inject (const Array& source); Array& operator= (const Type* source); Array& operator= (const Array& source); // Get the number of elements in the array. int length() const { return n; }; // Elements of an array A can be accessed the subscript operator; // e.g. A[i]. For the sake of efficiency, no checks are made to // ensure that the subscripting is legal. It is up to the user to // ensure that i is non-negative and less than A.length(). Note // that subscripts, as per C++ convention, start at 0. Type& operator[] (int i); const Type& operator[] (int i) const; // Returns true if the two arrays have the same dimensions // (i.e. the same lengths). bool operator== (const Array& a) const { return n == a.n; }; bool operator!= (const Array& a) const { return !(*this == a); }; protected: Type* elems; int n; // The number of elements in the array. bool owner; // Whether the object has ownership of the data. }; // Function definitions for class Array. // ----------------------------------------------------------------- template Array::Array (int length) { n = length; owner = true; elems = 0; elems = new Type[length]; } template Array::Array (Type* data, int length) { n = length; owner = false; elems = data; } template Array::Array (const Array& source) { n = source.n; owner = false; elems = source.elems; } template Array::~Array () { if (owner && elems) delete[] elems; } template void Array::setvalue (const Type& value) { for (int i = 0; i < n; i++) elems[i] = value; } template void Array::inject (const Type* source) { // Check for self-assignment. If there is self-assignment, there // is no need to copy because the objects share the same data! if (elems != source) copymemory(source,elems,n); } template void Array::inject (const Array& source) { if (n != source.n) throw MatlabException("Unable to perform copy; arrays have \ different lengths"); inject(source.elems); } template Array& Array::operator= (const Type* source) { inject(source); return *this; } template Array& Array::operator= (const Array& source) { inject(source); return *this; } template Type& Array::operator[] (int i) { return elems[i]; } template const Type& Array::operator[] (int i) const { return elems[i]; } #endif gpml-matlab-v3.2-2013-01-15/util/lbfgsb/matlabexception.cpp000644 000765 000024 00000001000 11511036275 023140 0ustar00carlstaff000000 000000 #include "matlabexception.h" // Function definitions for class MatlabException. // ----------------------------------------------------------------- MatlabException::MatlabException (const char* message) throw() : exception() { this->message = message; } MatlabException::MatlabException (const MatlabException& source) throw() : exception() { message = source.message; } MatlabException& MatlabException::operator= (const MatlabException& source) { message = source.message; return *this; } gpml-matlab-v3.2-2013-01-15/util/lbfgsb/arrayofmatrices.cpp000644 000765 000024 00000004655 11511036275 023177 0ustar00carlstaff000000 000000 #include "arrayofmatrices.h" #include "matrix.h" // Function definitions for class ArrayOfMatrices. // ----------------------------------------------------------------- ArrayOfMatrices::ArrayOfMatrices (const mxArray* ptr) : Array(getnummatlabmatrices(ptr)) { setvalue(0); if (mxIsCell(ptr)) // Fill in the entries of the array with the entries of the Matlab // cell array. for (int i = 0; i < n; i++) elems[i] = new Matrix(mxGetCell(ptr,i)); else // Fill in the single entry of the array with the Matlab matrix. elems[0] = new Matrix(ptr); } ArrayOfMatrices::ArrayOfMatrices (const mxArray* ptrs[], int numptrs) : Array(numptrs) { setvalue(0); for (int i = 0; i < numptrs; i++) elems[i] = new Matrix(ptrs[i]); } ArrayOfMatrices::ArrayOfMatrices (mxArray* ptrs[], const ArrayOfMatrices& source) : Array(source.length()) { setvalue(0); // Create the new Matlab matrices and copy the data from the // source. for (int i = 0; i < n; i++) { int h = source[i]->height(); int w = source[i]->width(); elems[i] = new Matrix(ptrs[i],h,w); *elems[i] = *source[i]; } } ArrayOfMatrices::ArrayOfMatrices (const ArrayOfMatrices& source) : Array(source) { } ArrayOfMatrices::ArrayOfMatrices (double* data, const ArrayOfMatrices& model) : Array(model.length()) { setvalue(0); // Repeat for each matrix. for (int i = 0; i < n; i++) { const Matrix* A = model[i]; elems[i] = new Matrix(data,A->height(),A->width()); data += A->length(); } } ArrayOfMatrices::~ArrayOfMatrices() { for (int i = 0; i < n; i++) if (elems[i]) delete elems[i]; } int ArrayOfMatrices::numelems() const { int m = 0; // The return value. // Repeat for each matrix. for (int i = 0; i < n; i++) m += elems[i]->length(); return m; } ArrayOfMatrices& ArrayOfMatrices::operator= (const ArrayOfMatrices& source) { // Copy each matrix. for (int i = 0; i < n; i++) *elems[i] = *source.elems[i]; return *this; } int ArrayOfMatrices::getnummatlabmatrices (const mxArray* ptr) { int n; // The return value. if (mxIsCell(ptr)) n = mxGetNumberOfElements(ptr); else if (mxGetNumberOfDimensions(ptr) == 2 && mxIsDouble(ptr)) n = 1; else throw MatlabException("The Matlab array must either be a cell \ array or a double precision matrix"); return n; } gpml-matlab-v3.2-2013-01-15/util/lbfgsb/program.h000644 000765 000024 00000011647 11511036275 021117 0ustar00carlstaff000000 000000 #ifndef INCLUDE_PROGRAM #define INCLUDE_PROGRAM // Type definitions. // ----------------------------------------------------------------- // This defines the possible results of running the L-BFGS-B solver. enum SolverExitStatus { success, // The algorithm has converged to a stationary // point or has reached the maximum number of // iterations. abnormalTermination, // The algorithm has terminated abnormally and // was unable to satisfy the convergence // criteria. errorOnInput // The routine has detected an error in the // input parameters. }; // Constants. // ----------------------------------------------------------------- const int defaultm = 5; const int defaultmaxiter = 100; const double defaultfactr = 1e7; const double defaultpgtol = 1e-5; const int defaultprintlevel = -1; // Function declarations. // ----------------------------------------------------------------- // This is the L-BFGS-B routine implemented in Fortran 77. extern "C" void setulb_ (int* n, int* m, double x[], double l[], double u[], int nbd[], double* f, double g[], double* factr, double* pgtol, double wa[], int iwa[], char task[], int* iprint, char csave[], bool lsave[], int isave[], double dsave[]); // Class Program. // ----------------------------------------------------------------- // This class encapsulates execution of the L-BFGS-B routine for // solving a nonlinear optimization problem with bound constraints // using limited-memory approximations to the Hessian. // // This is an abstract class since some of the class methods have not // been implemented (they are "pure virtual" functions). In order to // use this class, one needs to define a child class and provide // definitions for the pure virtual methods. class Program { public: // The first input argument "n" is the number of variables, and "x" // must be set to the suggested starting point for the optimization // algorithm. See the L-BFGS-B documentation for more information on // the inputs to the constructor. Program (int n, double* x, double* lb, double* ub, int* btype, int m = defaultm, int maxiter = defaultmaxiter, double factr = defaultfactr, double pgtol = defaultpgtol); // This is the same constructor as above, except that the // appropriate amount of memory is dynamically allocated for the // variables, the bounds, and the bound types. The destructor also // makes sure that the memory is properly deallocated, but not so // for the other constructor. Program (int n, int m = defaultm, int maxiter = defaultmaxiter, double factr = defaultfactr, double pgtol = defaultpgtol); // The destructor. virtual ~Program(); // An implementation of this method should return the value of the // objective at the current value of the variables "x", where "n" is // the number of variables. virtual double computeObjective (int n, double* x) = 0; // An implementation of this method should fill in the values of the // gradient "g" and the current point "x". virtual void computeGradient (int n, double* x, double* g) = 0; // The child class may optionally override this method which is // called once per iteration of the L-BFGS-B routine. The input // arguments are the current iteration "t", the current value of the // variables "x" and the current value of the objective "f". virtual void iterCallback (int t, double* x, double f) { }; // Run the solver. Upon completion, the solution is stored in "x". SolverExitStatus runSolver(); protected: // The copy constructor and copy assignment operator are kept // private so that they are not used. Program (const Program& source) { }; Program& operator= (const Program& source) { return *this; }; int n; // The number of variables. double* x; // The current point. double* lb; // The lower bounds. double* ub; // The upper bounds. int* btype; // The bound types. private: // Execute a single step the L-BFGS-B solver routine. void callLBFGS (const char* cmd = 0); // Initialize some of the structures used by the L-BFGS-B routine. void initStructures(); bool owner; // If true, then the blocks of memory pointed to by // x, lb, ub and btype must be deallocated in the // destructor. double f; // The value of the objective. double* g; // The value of the gradient. int iprint; // The print level. int maxiter; // The maximum number of iterations. double pgtol; // Convergence parameter passed to L-BFGS-B. double factr; // Convergence parameter passed to L-BFGS-B. int m; // The number of variable corrections to the // limited-memory approximation to the Hessian. // These are structures used by the L-BFGS-B routine. double* wa; int* iwa; char task[60]; char csave[60]; bool lsave[4]; int isave[44]; double dsave[29]; }; #endif gpml-matlab-v3.2-2013-01-15/util/lbfgsb/license.txt000644 000765 000024 00000003027 11511036275 021453 0ustar00carlstaff000000 000000 Copyright (c) 2009, Peter Carbonetto 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 * Neither the name of the University of British Columbia nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission. 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 OWNER 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. gpml-matlab-v3.2-2013-01-15/util/lbfgsb/matlabscalar.h000644 000765 000024 00000002264 11511036275 022071 0ustar00carlstaff000000 000000 #ifndef INCLUDE_MATLABSCALAR #define INCLUDE_MATLABSCALAR #include "mex.h" // Class MatlabScalar // ----------------------------------------------------------------- // The main appeal of this class is that one can create a scalar // object that accesses a MATLAB array. // // Note that the copy assignment operator is not valid for this class // because we cannot reassign a reference. class MatlabScalar { public: // This constructor accepts as input a pointer to a Matlab array // which must be a scalar in double precision. explicit MatlabScalar (const mxArray* ptr); // This constructor creates a new Matlab array which is a scalar // in double precision. MatlabScalar (mxArray*& ptr, double value); // The copy constructor. MatlabScalar (MatlabScalar& source); // The destructor. ~MatlabScalar() { }; // Access the value of the scalar. operator const double () const { return x; }; // Assign the value of the scalar. MatlabScalar& operator= (double value); protected: double& x; // The copy assignment operator is kept protected because it is // invalid. MatlabScalar& operator= (const MatlabScalar& source) { return *this; }; }; #endif gpml-matlab-v3.2-2013-01-15/util/lbfgsb/matlabmatrix.cpp000644 000765 000024 00000003156 11511036275 022464 0ustar00carlstaff000000 000000 #include "matlabmatrix.h" #include "matlabexception.h" // Function definitions. // ----------------------------------------------------------------- double* getMatlabMatrixDouble (const mxArray* ptr) { if (mxGetNumberOfDimensions(ptr) != 2) throw MatlabException("Matlab array must be a matrix"); if (!mxIsDouble(ptr)) throw MatlabException("Matlab array must be of type double"); return mxGetPr(ptr); } double* createMatlabMatrixDouble (mxArray*& ptr, int height, int width) { ptr = mxCreateDoubleMatrix(height,width,mxREAL); return mxGetPr(ptr); } // Function definitions for class Matrix. // ----------------------------------------------------------------- Matrix::Matrix (int height, int width) : Array(height*width) { h = height; w = width; } Matrix::Matrix (double* data, int height, int width) : Array(data,height*width) { h = height; w = width; } Matrix::Matrix (const mxArray* ptr) : Array(getMatlabMatrixDouble(ptr),mxGetNumberOfElements(ptr)) { h = mxGetM(ptr); w = mxGetN(ptr); } Matrix::Matrix (mxArray*& ptr, int height, int width) : Array(createMatlabMatrixDouble(ptr,height,width), height*width) { h = height; w = width; } Matrix::Matrix (const Matrix& source) : Array(source) { h = source.h; w = source.w; } Matrix& Matrix::operator= (const Matrix& source) { inject(source); return *this; } bool Matrix::operator== (const Matrix& X) const { return (h == X.h) && (w == X.w); } double& Matrix::entry (int r, int c) { return elems[h*c + r]; } double Matrix::entry (int r, int c) const { return elems[h*c + r]; } gpml-matlab-v3.2-2013-01-15/util/lbfgsb/matlabmatrix.h000644 000765 000024 00000004562 11511036275 022133 0ustar00carlstaff000000 000000 #ifndef INCLUDE_MATLABMATRIX #define INCLUDE_MATLABMATRIX #include "array.h" #include "mex.h" // Class Matrix // --------------------------------------------------------------- // A matrix object stores its elements in column-major format, as in // Fortran and Matlab. This means that columns are stored one after // another. For example, the matrix // // 1 2 3 // 4 5 6 // // is stored in memory as // // 1 4 2 5 3 6 // // Like Array objects, Matrix objects are not necessarily // encapsulated, and they exhibit analogous behaviour. class Matrix : public Array { public: // This constructor allocates memory for matrix of the specified // heigth and width. Matrix (int height, int width); // This constructor basically follows the lead of the analagous // constructor for the Array class. Matrix (double* data, int height, int width); // This constructor retrieves a matrix from a Matlab array. This // particular constructor is only defined for Matrix. // Since Matlab handles storage, the object created by this // constructor is not encapsulated. explicit Matrix (const mxArray* ptr); // This constructor creates a new Matlab array as a side effect. // Since Matlab handles storage, the object created by this // constructor is not encapsulated. Matrix (mxArray*& ptr, int height, int width); // The copy constructor makes a shallow copy of the data. Matrix (const Matrix& source); // The destructor. ~Matrix() { }; // Copy assignment operator that observes the same behaviour as // the Array copy assignment operator. Matrix& operator= (const Matrix& source); // Get the height and width of the matrix. int height() const { return h; }; int width () const { return w; }; // Returns true if the two matrices have the same dimensions // (i.e. the same height and width). bool operator== (const Matrix& X) const; bool operator!= (const Matrix& X) const { return !(*this == X); }; // If X is an object of type Matrix, X.entry(r,c) accesses the // entry of the rth row and cth column. double entry (int r, int c) const; double& entry (int r, int c); double operator() (int r, int c) const { return entry(r,c); }; double& operator() (int r, int c) { return entry(r,c); }; protected: int h; // The height of the matrix. int w; // The width of the matrix. }; #endif gpml-matlab-v3.2-2013-01-15/util/lbfgsb/matlabprogram.cpp000644 000765 000024 00000016452 11511036275 022632 0ustar00carlstaff000000 000000 #include "matlabprogram.h" #include "array.h" // Function definitions. // ----------------------------------------------------------------- // Copy the matrix elements to the destination array. It is assumed // that sufficient memory has been allocated for the destination // array. The copying preserves the column-major ordering of the // matrix elements. void copyElemsFromMatrix (const Matrix& source, double* dest) { Matrix destmatrix(dest,source.height(),source.width()); destmatrix = source; } // Copy the elements from the source array into the matrix. void copyElemsToMatrix (const double* source, Matrix& dest) { dest.inject(source); } // Copy the elements contained in all the matrices to the destination // array. It is assumed that sufficient memory has been allocated for // the destination array. The copying preserves the column-major // ordering of the matrix elements. void copyElemsFromArrayOfMatrices (const ArrayOfMatrices& source, double* dest) { // Repeat for each matrix. for (int i = 0; i < source.length(); i++) { copyElemsFromMatrix(*source[i],dest); dest += source[i]->length(); } } // Copy the elements from the source array into the array of matrices. void copyElemsToArrayOfMatrices (const double* source, ArrayOfMatrices& dest) { // Repeat for each matrix. for (int i = 0; i < dest.length(); i++) { copyElemsToMatrix(source,*dest[i]); source += dest[i]->length(); } } int getBoundType (double& lb, double& ub) { bool haslb = !mxIsInf(lb); bool hasub = !mxIsInf(ub); int btype = haslb + 3*hasub - 2*(haslb && hasub); lb = haslb ? lb : 0; ub = hasub ? ub : 0; return btype; } // Function definitions for class MatlabProgram. // ----------------------------------------------------------------- MatlabProgram::MatlabProgram (ArrayOfMatrices& variables, const ArrayOfMatrices& lowerbounds, const ArrayOfMatrices& upperbounds, const MatlabString* objFunc, const MatlabString* gradFunc, const MatlabString* iterFunc, mxArray* auxData, int m, int maxiter, double factr, double pgtol) : Program(variables.numelems(),0,0,0,0,m,maxiter,factr,pgtol), variables(variables) { x = new double[n]; lb = new double[n]; ub = new double[n]; btype = new int[n]; this->objFunc = objFunc; this->gradFunc = gradFunc; this->iterFunc = iterFunc; // Copy some pieces of information from the input arguments to the // respective program structures. copyElemsFromArrayOfMatrices(variables,x); copyElemsFromArrayOfMatrices(lowerbounds,lb); copyElemsFromArrayOfMatrices(upperbounds,ub); // Set the bound types. for (int i = 0; i < n; i++) btype[i] = getBoundType(lb[i],ub[i]); // Create a Matlab array for the variables. int nv = variables.length(); varInputs = new mxArray*[nv]; varMatlab = new ArrayOfMatrices(varInputs,variables); // Set up the inputs to the objective callback function. numInputsObjFunc = nv + (bool) auxData; inputsObjFunc = new mxArray*[numInputsObjFunc]; copymemory(varInputs,inputsObjFunc,nv); if (auxData) inputsObjFunc[nv] = auxData; // Set up the inputs to the gradient callback function. numInputsGradFunc = numInputsObjFunc; inputsGradFunc = inputsObjFunc; // If necessary, set up the inputs to the iterative callback // function. The first two inputs are the iteration and the function // objective. The next of inputs are the values of the variables // then, optionally, the auxiliary data. if (iterFunc) { numInputsIterFunc = 2 + nv + (bool) auxData; inputsIterFunc = new mxArray*[numInputsIterFunc]; mxArray** ptr = inputsIterFunc; tMatlab = new MatlabScalar(*ptr++,0); fMatlab = new MatlabScalar(*ptr++,0); copymemory(varInputs,ptr,nv); ptr += nv; if (auxData) *ptr = auxData; } } MatlabProgram::~MatlabProgram() { delete[] x; delete[] lb; delete[] ub; delete[] btype; // Deallocate the Matlab arrays. int nv = variables.length(); for (int i = 0; i < nv; i++) mxDestroyArray(varInputs[i]); delete varMatlab; delete[] varInputs; // If necessary, deallocate the additional Matlab arrays that act as // inputs to the iterative callback function. if (iterFunc) { delete fMatlab; delete tMatlab; // delete[] inputsIterFunc; } } double MatlabProgram::computeObjective (int n, double* x) { int nlhs = 1; // The number of outputs from Matlab. mxArray* plhs[nlhs]; // The outputs from the Matlab routine. // Copy the current value of the optimization variables. copyElemsToArrayOfMatrices(x,*varMatlab); // Call the designated Matlab routine for evaluating the objective // function. It takes as input the values of the variables, and // returns a single output, the value of the objective function. if (mexCallMATLAB(nlhs,plhs,numInputsObjFunc,inputsObjFunc,*objFunc)) throw MatlabException("Evaluation of objective function failed in \ call to MATLAB routine"); // Get the result passed back from the Matlab routine. MatlabScalar matlabOutput(plhs[0]); double objective = matlabOutput; // Free the dynamically allocated memory. mxDestroyArray(plhs[0]); return objective; } void MatlabProgram::computeGradient (int n, double* x, double* g) { int nlhs = variables.length(); // The number of outputs from Matlab. mxArray* plhs[nlhs]; // The outputs from the Matlab routine. // Copy the current value of the optimization variables. copyElemsToArrayOfMatrices(x,*varMatlab); // Call the designated Matlab routine for computing the gradient // of the objective function. It takes as input the values of the // variables, and returns as many outputs corresponding to the // partial derivatives of the objective function with respect to // the variables at their curent values. if (mexCallMATLAB(nlhs,plhs,numInputsGradFunc,inputsGradFunc,*gradFunc)) throw MatlabException("Evaluation of objective gradient failed in \ call to MATLAB routine"); // Get the result passed back from the Matlab routine. ArrayOfMatrices matlabOutput((const mxArray**) plhs,nlhs); for (int i = 0; i < variables.length(); i++) if (matlabOutput[i]->length() != variables[i]->length()) throw MatlabException("Invalid gradient passed back from MATLAB \ routine"); copyElemsFromArrayOfMatrices(matlabOutput,g); // Free the dynamically allocated Matlab outputs. for (int i = 0; i < variables.length(); i++) mxDestroyArray(plhs[i]); } void MatlabProgram::iterCallback (int t, double* x, double f) { if (iterFunc) { // Copy the current iteration, the current value of the objective // function, and the current value of the optimization variables. *tMatlab = t; *fMatlab = f; copyElemsToArrayOfMatrices(x,*varMatlab); // Call the Matlab iterative callback routine. Since there are no // outputs, there is no memory to deallocate. if (mexCallMATLAB(0,0,numInputsIterFunc,inputsIterFunc,*iterFunc)) throw MatlabException("Call to MATLAB iterative callback function \ failed"); } } SolverExitStatus MatlabProgram::runSolver() { SolverExitStatus status; // The return value. status = Program::runSolver(); // Copy the solution to the class member "variables". copyElemsToArrayOfMatrices(x,variables); return status; } gpml-matlab-v3.2-2013-01-15/mean/meanConst.m000644 000765 000024 00000001217 11511036244 020074 0ustar00carlstaff000000 000000 function A = meanConst(hyp, x, i) % Constant mean function. The mean function is parameterized as: % % m(x) = c % % The hyperparameter is: % % hyp = [ c ] % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-08-04. % % See also MEANFUNCTIONS.M. if nargin<2, A = '1'; return; end % report number of hyperparameters if numel(hyp)~=1, error('Exactly one hyperparameter needed.'), end c = hyp; if nargin==2 A = c*ones(size(x,1),1); % evaluate mean else if i==1 A = ones(size(x,1),1); % derivative else A = zeros(size(x,1),1); end endgpml-matlab-v3.2-2013-01-15/mean/meanSum.m000644 000765 000024 00000003511 11511036244 017551 0ustar00carlstaff000000 000000 function A = meanSum(mean, hyp, x, i) % meanSum - compose a mean function as the sum of other mean functions. % This function doesn't actually compute very much on its own, it merely does % some bookkeeping, and calls other mean functions to do the actual work. % % m(x) = \sum_i m_i(x) % % Copyright (c) by Carl Edward Rasmussen & Hannes Nickisch 2010-08-04. % % See also MEANFUNCTIONS.M. for ii = 1:numel(mean) % iterate over mean functions f = mean(ii); if iscell(f{:}), f = f{:}; end % expand cell array if necessary j(ii) = cellstr(feval(f{:})); % collect number hypers end if nargin<3 % report number of parameters A = char(j(1)); for ii=2:length(mean), A = [A, '+', char(j(ii))]; end; return end [n,D] = size(x); v = []; % v vector indicates to which mean parameters belong for ii = 1:length(mean), v = [v repmat(ii, 1, eval(char(j(ii))))]; end if nargin==3 % compute mean vector A = zeros(n,1); % allocate space for ii = 1:length(mean) % iteration over summand functions f = mean(ii); if iscell(f{:}), f = f{:}; end % expand cell array if needed A = A + feval(f{:}, hyp(v==ii), x); % accumulate means end else % compute derivative vector if i<=length(v) ii = v(i); % which mean function j = sum(v(1:i)==ii); % which parameter in that mean f = mean(ii); if iscell(f{:}), f = f{:}; end % dereference cell array if necessary A = feval(f{:}, hyp(v==ii), x, j); % compute derivative else A = zeros(n,1); end endgpml-matlab-v3.2-2013-01-15/mean/meanZero.m000644 000765 000024 00000000603 11511036244 017723 0ustar00carlstaff000000 000000 function A = meanZero(hyp, x, i) % Zero mean function. The mean function does not have any parameters. % % m(x) = 0 % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-01-10. % % See also MEANFUNCTIONS.M. if nargin<2, A = '0'; return; end % report number of hyperparameters A = zeros(size(x,1),1); % derivative and mean gpml-matlab-v3.2-2013-01-15/mean/meanPow.m000644 000765 000024 00000001721 11511036244 017553 0ustar00carlstaff000000 000000 function A = meanPow(mean, hyp, x, i) % meanPow - compose a mean function as the power of another mean function. % % The degree d has to be a strictly positive integer. % % m(x) = m_0(x) ^ d % % This function doesn't actually compute very much on its own, it merely does % some bookkeeping, and calls other mean function to do the actual work. % % Copyright (c) by Carl Edward Rasmussen & Hannes Nickisch 2010-06-18. % % See also MEANFUNCTIONS.M. d = mean{1}; d = abs(floor(d)); d = max(d,1); % positive integer degree mean = mean{2}; if ~iscell(mean), mean = {mean}; end if nargin<3 % report number of parameters A = feval(mean{:}); return end [n,D] = size(x); if nargin==3 % compute mean vector A = feval(mean{:},hyp,x).^d; else % compute derivative vector A = ( d*feval(mean{:},hyp,x).^(d-1) ).* feval(mean{:},hyp,x,i); endgpml-matlab-v3.2-2013-01-15/mean/meanMask.m000644 000765 000024 00000002512 12077070412 017703 0ustar00carlstaff000000 000000 function A = meanMask(mean, hyp, x, i) % Apply a mean function to a subset of the dimensions only. The subset can % either be specified by a 0/1 mask by a boolean mask or by an index set. % % This function doesn't actually compute very much on its own, it merely does % some bookkeeping, and calls another mean function to do the actual work. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2013-01-16. % % See also MEANFUNCTIONS.M. mask = fix(mean{1}(:)); % either a binary mask or an index set mean = mean(2); % mean function to be masked if iscell(mean{:}), mean = mean{:}; end % properly unwrap nested cell arrays nh_string = feval(mean{:}); % number of hyperparameters of the full mean if max(mask)<2 && length(mask)>1, mask = find(mask); end % convert 1/0->index D = length(mask); % masked dimension if nargin<3, A = num2str(eval(nh_string)); return; end % number of parameters if eval(nh_string)~=length(hyp) % check hyperparameters error('number of hyperparameters does not match size of masked data') end if nargin==3 A = feval(mean{:}, hyp, x(:,mask)); else % compute derivative A = feval(mean{:}, hyp, x(:,mask), i); endgpml-matlab-v3.2-2013-01-15/mean/meanLinear.m000644 000765 000024 00000001330 11511036244 020214 0ustar00carlstaff000000 000000 function A = meanLinear(hyp, x, i) % Linear mean function. The mean function is parameterized as: % % m(x) = sum_i a_i * x_i; % % The hyperparameter is: % % hyp = [ a_1 % a_2 % .. % a_D ] % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-01-10. % % See also MEANFUNCTIONS.M. if nargin<2, A = 'D'; return; end % report number of hyperparameters [n,D] = size(x); if any(size(hyp)~=[D,1]), error('Exactly D hyperparameters needed.'), end a = hyp; if nargin==2 A = x*a; % evaluate mean else if i<=D A = x(:,i); % derivative else A = zeros(n,1); end endgpml-matlab-v3.2-2013-01-15/mean/meanOne.m000644 000765 000024 00000000746 11511036244 017535 0ustar00carlstaff000000 000000 function A = meanOne(hyp, x, i) % One mean function. The mean function does not have any parameters. % % m(x) = 1 % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-08-04. % % See also MEANFUNCTIONS.M. if nargin<2, A = '0'; return; end % report number of hyperparameters if nargin==2 A = ones(size(x,1),1); % evaluate mean else A = zeros(size(x,1),1); % derivative endgpml-matlab-v3.2-2013-01-15/mean/meanProd.m000644 000765 000024 00000003754 11511036244 017722 0ustar00carlstaff000000 000000 function A = meanProd(mean, hyp, x, i) % meanProd - compose a mean function as the product of other mean functions. % This function doesn't actually compute very much on its own, it merely does % some bookkeeping, and calls other mean functions to do the actual work. % % m(x) = \prod_i m_i(x) % % Copyright (c) by Carl Edward Rasmussen & Hannes Nickisch 2010-08-04. % % See also MEANFUNCTIONS.M. for ii = 1:numel(mean) % iterate over mean functions f = mean(ii); if iscell(f{:}), f = f{:}; end % expand cell array if necessary j(ii) = cellstr(feval(f{:})); % collect number hypers end if nargin<3 % report number of parameters A = char(j(1)); for ii=2:length(mean), A = [A, '+', char(j(ii))]; end; return end [n,D] = size(x); v = []; % v vector indicates to which mean parameters belong for ii = 1:length(mean), v = [v repmat(ii, 1, eval(char(j(ii))))]; end A = ones(n,1); % allocate space if nargin==3 % compute mean vector for ii = 1:length(mean) % iteration over summand functions f = mean(ii); if iscell(f{:}), f = f{:}; end % expand cell array if needed A = A.*feval(f{:}, hyp(v==ii), x); % accumulate means end else % compute derivative vector if i<=length(v) ii = v(i); % which mean function j = sum(v(1:i)==ii); % which parameter in that mean for jj = 1:length(mean) f = mean(jj); if iscell(f{:}), f = f{:}; end % dereference cell array if necessary if jj==ii A = A .* feval(f{:}, hyp(v==jj), x, j); % multiply derivative else A = A .* feval(f{:}, hyp(v==jj), x); % multiply mean end end else A = zeros(n,1); end endgpml-matlab-v3.2-2013-01-15/mean/meanScale.m000644 000765 000024 00000001575 11511036244 020044 0ustar00carlstaff000000 000000 function A = meanScale(mean, hyp, x, i) % meanScale - compose a mean function as a scaled version of another one. % % m(x) = a * m_0(x) % % The hyperparameter is: % % hyp = [ a ] % % This function doesn't actually compute very much on its own, it merely does % some bookkeeping, and calls other mean function to do the actual work. % % Copyright (c) by Carl Edward Rasmussen & Hannes Nickisch 2010-07-15. % % See also MEANFUNCTIONS.M. if nargin<3 % report number of parameters A = [feval(mean{:}),'+1']; return end [n,D] = size(x); a = hyp(1); if nargin==3 % compute mean vector A = a*feval(mean{:},hyp(2:end),x); else % compute derivative vector if i==1 A = feval(mean{:},hyp(2:end),x); else A = a*feval(mean{:},hyp(2:end),x,i-1); end end