function model = kpca(X,options)% KPCA Kernel Principal Component Analysis.%  % Synopsis:%  model = kpca(X)%  model = kpca(X,options)%% Description:%  This function is implementation of Kernel Principal Component %  Analysis (KPCA) [Schol98b]. The input data X are non-linearly%  mapped to a new high dimensional space induced by prescribed%  kernel function. The PCA is applied on the non-linearly mapped %  data. The result is a model describing non-linear data projection.%  See 'help kernelproj' for info how to project data.%% Input:%  X [dim x num_data] Training data.%  %  options [struct] Decribes kernel and output dimension:%   .ker [string] Kernel identifier (see 'help kernel'); %     (default 'linear').%   .arg [1 x narg] kernel argument; (default 1).%   .new_dim [1x1] Output dimension (number of used principal %     components); (default dim).%% Output:%  model [struct] Kernel projection:%   .Alpha [num_data x new_dim] Multipliers.%   .b [new_dim x 1] Bias.%   .sv.X [dim x num_data] Training vectors.%  %   .nsv [1x1] Number of training data.%   .eigval [1 x num_data] Eigenvalues of centered kernel matrix.%   .mse [1x1] Mean square representation error of maped data.%   .MsErr [dim x 1] MSE with respect to used basis vectors;%      mse=MsErr(new_dim).%   .kercnt [1x1] Number of used kernel evaluations.%   .options [struct] Copy of used options.%   .cputime [1x1] CPU time used for training.%% Example:%  X = gencircledata([1;1],5,250,1);%  model = kpca( X, struct('ker','rbf','arg',4,'new_dim',2));%  XR = kpcarec( X, model );%  figure; %  ppatterns( X ); ppatterns( XR, '+r' );%  % See also %  KERNELPROJ, PCA, GDA.%   % 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:% 4-may-2004, VF% 10-july-2003, VF, computation of kercnt added% 22-jan-2003, VF% 11-july-2002, VF, mistake "Jt=zeros(N,L)/N" repared %              (reported by SH_Srinivasan@Satyam.com).% 5-July-2001, V.Franc, comments changed% 20-dec-2000, V.Franc, algorithm was implemented% timerstart_time = cputime;% gets dimensions[dim,num_data] = size(X);  % process input arguments%-----------------------------------if nargin < 2, options = []; else options=c2s(options); endif ~isfield(options,'ker'), options.ker = 'linear'; endif ~isfield(options,'arg'), options.arg = 1; endif ~isfield(options,'new_dim'), options.new_dim = dim; end% compute kernel matrixK = kernel(X,options.ker,options.arg);% Centering kernel matrix (non-linearly mapped data).J = ones(num_data,num_data)/num_data;Kc = K - J*K - K*J + J*K*J;% eigen decomposition of the kernel marix[U,D] = eig(Kc);Lambda=real(diag(D));% normalization of eigenvectors to be orthonormal for k = 1:num_data,  if Lambda(k) ~= 0,     U(:,k)=U(:,k)/sqrt(Lambda(k));  endend% Sort the eigenvalues and the eigenvectors in descending order.[Lambda,ordered]=sort(-Lambda);    Lambda=-Lambda;U=U(:,ordered);                    % use first new_dim principal componentsA=U(:,1:options.new_dim);              % compute Alpha and compute bias (implicite centering)% of kernel projectionmodel.Alpha = (eye(num_data,num_data)-J)*A;Jt=ones(num_data,1)/num_data;model.b = A'*(J'*K*Jt-K*Jt);% fill output structuremodel.sv.X = X;model.nsv = num_data;model.options = options;model.eigval = Lambda;model.kercnt = num_data*(num_data+1)/2;model.MsErr = triu(ones(num_data,num_data),1)*model.eigval/num_data;model.mse = model.MsErr(options.new_dim);model.cputime = cputime - start_time;model.fun = 'kernelproj';return;