function [mappedX, mapping] = pca(X, no_dims)%PCA Perform the PCA algorithm%%   [mappedX, mapping] = pca(X, no_dims)%% The function runs PCA on a set of datapoints X. The variable% no_dims sets the number of dimensions of the feature points in the % embedded feature space (no_dims >= 1, default = 2). % For no_dims, you can also specify a number between 0 and 1, determining % the amount of variance you want to retain in the PCA step.% The function returns the locations of the embedded trainingdata in % mappedX. Furthermore, it returns information on the mapping in mapping.%%% This file is part of the Matlab Toolbox for Dimensionality Reduction v0.3b.% The toolbox can be obtained from http://www.cs.unimaas.nl/l.vandermaaten% You are free to use, change, or redistribute this code in any way you% want for non-commercial purposes. However, it is appreciated if you % maintain the name of the original author.%% (C) Laurens van der Maaten% Maastricht University, 2007    if ~exist('no_dims', 'var')        no_dims = 2;    end		% Make sure data is zero mean    mapping.mean = mean(X, 1);	X = X - repmat(mapping.mean, [size(X, 1) 1]);	% Compute covariance matrix    if size(X, 2) < size(X, 1)        C = cov(X);    else        C = (1 / size(X, 1)) * (X * X');        % if N>D, we better use this matrix for the eigendecomposition    end		% Perform eigendecomposition of C	C(isnan(C)) = 0;	C(isinf(C)) = 0;    [M, lambda] = eig(C);        % Sort eigenvectors in descending order    [lambda, ind] = sort(diag(lambda), 'descend');    if no_dims > size(M, 2)        no_dims = size(M, 2);        warning(['Target dimensionality reduced to ' num2str(no_dims) '.']);    end	M = M(:,ind(1:no_dims));    lambda = lambda(1:no_dims);		% Apply mapping on the data    if ~(size(X, 2) < size(X, 1))        M = (X' * M) .* repmat((1 ./ sqrt(size(X, 1) .* lambda))', [size(X, 2) 1]);     % normalize in order to get eigenvectors of covariance matrix    end    mappedX = X * M;        % Store information for out-of-sample extension    mapping.M = M;	mapping.lambda = lambda;