function [sig_e, bay,model] = bay_errorbar(model,Xt, type, nb, bay)% Compute the error bars for a one dimensional regression problem% % >> sig_e = bay_errorbar({X,Y,'function',gam,sig2}, Xt)% >> sig_e = bay_errorbar(model, Xt)% % The computation takes into account the estimated noise variance% and the uncertainty of the model parameters, estimated by% Bayesian inference. sig_e is the estimated standard deviation of% the error bars of the points Xt. A plot is obtained by replacing% Xt by the string 'figure'.% %% Full syntax% %     1. Using the functional interface:% % >> sig_e = bay_errorbar({X,Y,'function',gam,sig2,kernel,preprocess}, Xt)% >> sig_e = bay_errorbar({X,Y,'function',gam,sig2,kernel,preprocess}, Xt, type)% >> sig_e = bay_errorbar({X,Y,'function',gam,sig2,kernel,preprocess}, Xt, type, nb)% >> sig_e = bay_errorbar({X,Y,'function',gam,sig2,kernel,preprocess}, 'figure')% >> sig_e = bay_errorbar({X,Y,'function',gam,sig2,kernel,preprocess}, 'figure', type)% >> sig_e = bay_errorbar({X,Y,'function',gam,sig2,kernel,preprocess}, 'figure', type, nb)% %       Outputs    %         sig_e         : Nt x 1 vector with the [$ \sigma^2$] errorbands of the test data%       Inputs    %         X             : N x d matrix with the inputs of the training data%         Y             : N x 1 vector with the inputs of the training data%         type          : 'function estimation' ('f')%         gam           : Regularization parameter%         sig2          : Kernel parameter%         kernel(*)     : Kernel type (by default 'RBF_kernel')%         preprocess(*) : 'preprocess'(*) or 'original'%         Xt            : Nt x d matrix with the inputs of the test data%         type(*)       : 'svd'(*), 'eig', 'eigs' or 'eign'%         nb(*)         : Number of eigenvalues/eigenvectors used in the eigenvalue decomposition approximation%%     2. Using the object oriented interface:% % >> [sig_e, bay, model] = bay_errorbar(model, Xt)% >> [sig_e, bay, model] = bay_errorbar(model, Xt,       type)% >> [sig_e, bay, model] = bay_errorbar(model, Xt,       type, nb)% >> [sig_e, bay, model] = bay_errorbar(model, 'figure')% >> [sig_e, bay, model] = bay_errorbar(model, 'figure', type)% >> [sig_e, bay, model] = bay_errorbar(model, 'figure', type, nb)% %       Outputs    %         sig_e     : Nt x 1 vector with the [$ \sigma^2$] errorbands of the test data%         model(*)  : Object oriented representation of the LS-SVM model%         bay(*)    : Object oriented representation of the results of the Bayesian inference%       Inputs    %         model     : Object oriented representation of the LS-SVM model%         Xt        : Nt x d matrix with the inputs of the test data%         type(*)   : 'svd'(*), 'eig', 'eigs' or 'eign'%         nb(*)     : Number of eigenvalues/eigenvectors used in the eigenvalue decomposition approximation% % See also:%   bay_lssvm, bay_optimize, bay_modoutClass, plotlssvm% Copyright (c) 2002,  KULeuven-ESAT-SCD, License & help @ http://www.esat.kuleuven.ac.be/sista/lssvmlabif iscell(model), model = initlssvm(model{:}); endif model.type(1)~='f',  error(['confidence bounds only for function estimation. For' ...	 ' classification, use ''bay_modoutClass(...)'' instead;']);endeval('type;','type=''svd'';');eval('nb;','nb=model.nb_data;');if ~(strcmpi(type,'svd') | strcmpi(type,'eig') | strcmpi(type,'eigs') | strcmpi(type,'eign')),  error('Eigenvalue decomposition via ''svd'', ''eig'', ''eigs'' or ''eign''...');endif strcmpi(type,'eign')  warning('The resulting errorbars are most probably not very usefull...');  endif ~isstr(Xt),  eval('[sig_e, bay] = bay_confb(model,Xt,type,nb,bay);',...       '[sig_e, bay] = bay_confb(model,Xt,type,nb);');  else  grid = 50;  [X,Y] = postlssvm(model,model.xtrain,model.ytrain);  eval('[sig_e, bay] = bay_confb(model,X,type,nb,bay);',...       '[sig_e, bay] = bay_confb(model,X,type,nb);');  % plot the curve including confidence bound  sige = sqrt(sig_e);  Yt = simlssvm(model,X);    figure;  hold on;  title(['LS-SVM_{\gamma=' num2str(model.gam(1)) ', \sigma^2=' num2str(model.kernel_pars(1)) ...	 '}^{' model.kernel_type(1:3) '} and its 95% (2\sigma) error bands']);  if model.x_dim==1,    xlabel('X');    ylabel('Y');    [s,si] = sort(X);    plot(X(si),Yt(si),'k'); hold on;     plot(X(si),Yt(si)+2.*sige(si),'-.r');        plot(X(si),Yt(si)-2.*sige(si),':r');    plot(X(si),Y(si),'k*'); hold off;  else    xlabel('time');    ylabel('Y');    plot(Yt,'k'); hold on;     plot(Yt+2.*sige,'-.r');     plot(Yt-2.*sige,':r');    plot(Y,'k*'); hold off;  end endfunction [sig_e, bay] = bay_confb(model,X,type,nb,bay)% see formula's thesis TvG blz 126nD = size(X,1);%tol = .0001;%% calculate the eigenvalues%eval('bay;','[c1,c2,c3,bay] = bay_lssvm(model,1,type,nb);');omega = kernel_matrix(model.xtrain(model.selector,1:model.x_dim), ...		      model.kernel_type, model.kernel_pars);oo = ones(1,model.nb_data)*omega;% kernel values of  Xtheta = kernel_matrix(model.xtrain(model.selector, 1:model.x_dim), ...		      model.kernel_type, model.kernel_pars, X);for i=1:nD,  kxx(i,1) = feval(model.kernel_type, X(i,:),X(i,:), model.kernel_pars);endZc = eye(model.nb_data) - ones(model.nb_data)./model.nb_data;Hd = (Zc*bay.Rscores);Hd = Hd*diag(1./bay.mu - (bay.mu+ bay.zeta*bay.eigvals).^-1)*Hd';% forall xfor i=1:nD,  term1(i,1) = bay.zeta^-1 + kxx(i)/bay.mu - theta(:,i)'*Hd*theta(:,i);  term2(i,1) = 2/model.nb_data*sum(theta(:,i)'*Hd*omega) - 2/bay.mu/model.nb_data* sum(theta(:,i));end% onceterm3 = 1/(bay.zeta*model.nb_data) ...	+ 1/(bay.mu*model.nb_data^2)* sum(oo)  ...	-1/(model.nb_data^2)* oo*Hd*oo';sig_e = term1+term2+term3;