%PCA Principal Component Analysis%% 	[W,alf] = pca(A,n)% 	[W,n] = pca(A,alf)%% A principal component analysis is performed on the joint % covarianve matrix of the data in A. If A is a labeled dataset the% class contributions are weighted according to their posterior% probabilities.% The routine finds a rotation of the dataset A to a n-dimensional% linear subspace such that at least a fraction alf of the total% variance is preserved.% If n is given (n>=1), alf is maximized. If alf is given (alf<1) % n is minimized. If n < 0 an abs(n)-dimensional subspace is found% that minimizes the preserved variance. If alf<0 (abs(alf<1)) the% maximum n is found for which the preserved variance <= abs(alf).% New objects B can be mapped by B*W, W*B or by A*klm([],n)*B.% Default: the features are decorrelated and ordered, but no% feature reduction is made.%% 	v = pca(A,0)% % Returns the cummulative fraction of the explained variance. v(n) % is the cummulative fraction of the explained variance by using n % eigenvectors.%% Use klm for a pca of the mean class covariance matrix.% Use fisherm for optimizing the linear class separability (LDA).%% See also mappings, datasets, kljlc, klclc, klm, fisherm% Copyright: R.P.W. Duin, duin@ph.tn.tudelft.nl% Faculty of Applied Physics, Delft University of Technology% P.O. Box 5046, 2600 GA Delft, The Netherlandsfunction [W,q] = pca(a,alf)if nargin < 2 | isempty(alf), alf = inf; endif nargin < 1 | isempty(a)   W = mapping('pca',alf); returnend[nlab,lablist,m,k,c,p,fl,imheight] = dataset(a);a = a*scalem(a); % set mean to origina = a/max(abs(a(:))); % occasionally necessary to prevent inf's in covif m <= k	u = reducm(a);	a = a*u;	korg = k;	[m,k] = size(a);else	u = [];end[U,GG] = meancov(a,1); G = zeros(k,k);for i = 1:c	G = G + p(i)*GG(:,:,i);endG = m*(G + cov(+U,1))/(m-1);[F V] = eig(G);[v,I] = sort(-diag(V));if alf == inf	n = k;	q = k;elseif alf >= 1	n = alf;	if n > k		error('Illegal dimensionality requested');	end	q = sum(v(1:n))/sum(v);	I = I(1:n);elseif alf > 0	vv = v'*triu(ones(k,k)) / sum(v) - alf;	J = find(vv > 0);	n = J(1); q = n;	I = I(1:n);elseif alf == 0	W = ones(1,k);	w = v'*triu(ones(k,k)) / sum(v);	W(1:length(w)) = w;	returnelseif alf > -1	alf = abs(alf);	v = flipud(v); I = flipud(I);	vv = v'*triu(ones(k,k)) / sum(v) - alf;	J = find(vv > 0);	n = J(1)-1; q = n;	I = I(1:n);else	n = abs(alf);	v = flipud(v); I = flipud(I);	if n > k		error('Illegal dimensionality requested');	end	q = sum(v(1:n));	sv = sum(v);	if sv ~= 0, q = q/sv; end	I = I(1:n);endif ~isempty(u)	R = double(u)*F(:,I);	k = korg;else	R = [F(:,I); -mean(a*F(:,I))];endW = mapping('affine',R,[],k,n,1,imheight);