princomp.zip_matlab princomp_princomp_princomp工具箱

function [coeff, score, latent, tsquare] = princomp(x,econFlag) %PRINCOMP Principal Components Analysis. % 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 FACTORAN, PCACOV, PCARES. % 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 1993-2005 The MathWorks, Inc. % $Revision: 2.9.2.8 $ $Date: 2005/03/23 20:25:42 $ % 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); return end % Center X by subtracting off column means x0 = x - repmat(mean(x,1),n,1); 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 if nargout < 2 % When econFlag is 'econ', only (n-1) components should be returned. % See comment below. if (n <= p) && isequal(econFlag, 'econ') coeff(:,n) = []; end else % 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 = U .* repmat(sigma',n,1); % == 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('stats:princomp:colRankDefX', ... ['Columns of X are linearly dependent to within machine precision.\n' ... 'Using only the first %d components to compute TSQUARED.'],q); end else q = 0; end tsquare = (n-1) .* sum(U(:,1:q).^2,2); % == sum(score*diag(1./sigma),2) end end