Cleve's Cubicle

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Singular Value Decomposition versus Principal Component Analysis

SVD meets PCA

MATLAB News & Notes, 2006, Cleve’s Corner

“The Wikipedia pages on SVD and PCA are 
 quite good and contain a number of useful links, 
 although not to each other.”

LaTeX: -

%relationship of pca to svd
m=3;  n=7;
A = randn(m,n);

[coef,score,latent] = pca(A);

X       = A - mean(A);
[U,S,V] = svd(X,'econ');

% U  vs. score
rho   = rank(X);
sense = sign(score).*sign(U*S(:,1:rho));  %account for negated left singular vector
sum(sum(abs(score - U*S(:,1:rho).*sense)))
% S  vs. latent
sum(abs(latent - diag(S(:,1:rho)).^2/(m-1)))
% V  vs. coef
sense2 = sign(coef).*sign(V(:,1:rho));    %account for corresponding negated right singular vector
sum(sum(abs(coef - V(:,1:rho).*sense2)))
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