% DEMO_SVMPOUT Fitting a posteriory probability to SVM output.%% A posteriory probability p(y==1|f(x)) of the first class% given SVM output f(x) is assumed to be sigmoid function. % Parameters A(1) and A(2) of the sigmoid function %   p(y==1|f(x)) = 1/(1+exp(A(1)*f(x)+A(2)) % are estimated using Maximum-Likelihood [Platt99a]. %% The Gaussian mixture model (GMM) is fitted to the SVM output% and the a posteriory probability are computed for % comparison to the ML estimate.% % The ML estimation of the sigmoid function is imlemented % in 'mlsigmoid' (see 'help mlsigmoid' for more info).%% About: Statistical Pattern Recognition Toolbox% (C) 1999-2003, Written by Vojtech Franc and Vaclav Hlavac% <a href="http://www.cvut.cz">Czech Technical University Prague</a>% <a href="http://www.feld.cvut.cz">Faculty of Electrical Engineering</a>% <a href="http://cmp.felk.cvut.cz">Center for Machine Perception</a>% Modifications:% 03-jun-2004, VF% 6-May-2003, VFhelp demo_svmpout;echo on;% load training datadata = load('riply_trn');% train SVM modelsvm_model = smo(data,struct('ker','rbf','arg',1,'C',10));% plot SVM decision boundary figure; ppatterns(data); psvm(svm_model);% compute SVM output[dummy,svm_output.X] = svmclass(data.X,svm_model);svm_output.y = data.y;% ML fitting of sigmod to SVM ouput%-------------------------------------------------------sigmoid_model = mlsigmoid(svm_output);% plot fitted probabilityfx = linspace(min(svm_output.X), max(svm_output.X), 200);sigmoid_apost = sigmoid(fx,sigmoid_model);figure; hold on; xlabel('svm output f(x)'); ylabel('p(y=1|f(x))');hsigmoid = plot(fx,sigmoid_apost,'k');ppatterns(svm_output);% ML estimation of GMM model of SVM output%-------------------------------------------------------gmm_model = mlcgmm( svm_output );% compute a posteriory probabilitypcond = pdfgauss( fx, gmm_model);gmm_apost = (pcond(1,:)*gmm_model.Prior(1))./...    (pcond(1,:)*gmm_model.Prior(1)+(pcond(2,:)*gmm_model.Prior(2)));hgmm = plot(fx,gmm_apost,'g');hcomp = pgauss(gmm_model); legend([hsigmoid,hgmm,hcomp],'P(y=1|f(x)) ML-Sigmoid',...    'P(y=1|f(x)) ML-GMM','P(f(x)|y=1) ML-GMM','P(f(x)|y=2) ML-GMM');echo off;% EOF