function [coeff, score, latent, tsquare] = princomp(x,econFlag)%PRINCOMP Principal Components Analysis (PCA) from raw data.% COEFF = PRINCOMP(X) performs principal components analysis on the N-by-P% data matrix X, and returns the principal component coefficients, also% known as loadings. Rows of X correspond to observations, columns to% variables. COEFF is a P-by-P matrix, each column containing coefficients% for one principal component. The columns are in order of decreasing% component variance.%% PRINCOMP centers X by subtracting off column means, but does not% rescale the columns of X. To perform PCA with standardized variables,% i.e., based on correlations, use PRINCOMP(ZSCORE(X)). To perform PCA% directly on a covariance or correlation matrix, use PCACOV.%% [COEFF, SCORE] = PRINCOMP(X) returns the principal component scores,% i.e., the representation of X in the principal component space. Rows% of SCORE correspond to observations, columns to components.%% [COEFF, SCORE, LATENT] = PRINCOMP(X) returns the principal component% variances, i.e., the eigenvalues of the covariance matrix of X, in% LATENT.%% [COEFF, SCORE, LATENT, TSQUARED] = PRINCOMP(X) returns Hotelling's% T-squared statistic for each observation in X.%% When N <= P, SCORE(:,N:P) and LATENT(N:P) are necessarily zero, and the% columns of COEFF(:,N:P) define directions that are orthogonal to X.%% [...] = PRINCOMP(X,'econ') returns only the elements of LATENT that are% not necessarily zero, i.e., when N <= P, only the first N-1, and the% corresponding columns of COEFF and SCORE. This can be significantly% faster when P >> N.%% See also BARTTEST, BIPLOT, CANONCORR, FACTORAN, PCACOV, PCARES, ROTATEFACTORS.% References:% [1] Jackson, J.E., A User's Guide to Principal Components,% Wiley, 1988.% [2] Jolliffe, I.T. Principal Component Analysis, 2nd ed.,% Springer, 2002.% [3] Krzanowski, W.J., Principles of Multivariate Analysis,% Oxford University Press, 1988.% [4] Seber, G.A.F., Multivariate Observations, Wiley, 1984.% Copyright 1993-2010 The MathWorks, Inc.% $Revision: 1.1.8.4 $ $Date: 2011/05/09 01:26:32 $% When X has more variables than observations, the default behavior is to% return all the pc's, even those that have zero variance. When econFlag% is 'econ', those will not be returned.if nargin < 2, econFlag = 0; end[n,p] = size(x);if isempty(x) pOrZero = ~isequal(econFlag, 'econ') * p; coeff = zeros(p,pOrZero); coeff(1:p+1:end) = 1; score = zeros(n,pOrZero); latent = zeros(pOrZero,1); tsquare = zeros(n,1); returnend% Center X by subtracting off column meansx0 = bsxfun(@minus,x,mean(x,1)); if nargout < 2 if n >= p && (isequal(econFlag,0) || isequal(econFlag,'econ')) % When only coefs are needed, EIG is significantly faster than SVD. [coeff,~] = eig(x0'*x0); coeff = fliplr(coeff); else % The principal component coefficients are the eigenvectors of % S = X0'*X0./(n-1), but computed using SVD. [~,~,coeff] = svd(x0,econFlag); end % When econFlag is 'econ', only (n-1) components should be returned. % See comment below. if (n <= p) && isequal(econFlag, 'econ') coeff(:,n) = []; endelse r = min(n-1,p); % max possible rank of X0 % The principal component coefficients are the eigenvectors of % S = X0'*X0./(n-1), but computed using SVD. [U,sigma,coeff] = svd(x0,econFlag); % put in 1/sqrt(n-1) later % Project X0 onto the principal component axes to get the scores. if n == 1 % sigma might have only 1 row sigma = sigma(1); else sigma = diag(sigma); end score = bsxfun(@times,U,sigma'); % == x0*coeff sigma = sigma ./ sqrt(n-1); % When X has at least as many variables as observations, eigenvalues % n:p of S are exactly zero. if n <= p % When econFlag is 'econ', nothing corresponding to the zero % eigenvalues should be returned. svd(,'econ') won't have % returned anything corresponding to components (n+1):p, so we % just have to cut off the n-th component. if isequal(econFlag, 'econ') sigma(n,:) = []; % make sure this shrinks as a column coeff(:,n) = []; score(:,n) = []; % Otherwise, set those eigenvalues and the corresponding scores to % exactly zero. svd(,0) won't have returned columns of U % corresponding to components (n+1):p, need to fill those out. else sigma(n:p,1) = 0; % make sure this extends as a column score(:,n:p) = 0; end end % The variances of the pc's are the eigenvalues of S = X0'*X0./(n-1). latent = sigma.^2; % Hotelling's T-squared statistic is the sum of squares of the % standardized scores, i.e., Mahalanobis distances. When X appears to % have column rank < r, ignore components that are orthogonal to the % data. if nargout == 4 if n > 1 q = sum(sigma > max(n,p).*eps(sigma(1))); if q < r warning(message('stats:princomp:colRankDefX', q)); end else q = 0; end tsquare = (n-1) .* sum(U(:,1:q).^2,2); % == sum((score*diag(1./sigma)).^2,2) endend% Enforce a sign convention on the coefficients -- the largest element in each% column will have a positive sign.[~,maxind] = max(abs(coeff),[],1);d = size(coeff,2);colsign = sign(coeff(maxind + (0:p:(d-1)*p)));coeff = bsxfun(@times,coeff,colsign);if nargout > 1 score = bsxfun(@times,score,colsign);end
