Singular Value Decomposition versus Principal Component Analysis

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[https://stats.stackexchange.com/questions/134282/relationship-between-svd-and-pca-how-to-use-svd-to-perform-pca Relationship between SVD and PCA].
[https://stats.stackexchange.com/questions/134282/relationship-between-svd-and-pca-how-to-use-svd-to-perform-pca Relationship between SVD and PCA].
<br><i>Standard deviation</i> is square root of variance.
<br><i>Standard deviation</i> is square root of variance.
 +
<br>Sign of principal component vectors are not unique.

Revision as of 19:39, 17 September 2018

from SVD meets PCA slide [17:46] by Cleve Moler.

The Wikipedia pages on SVD and PCA are quite good and contain a number of useful links, although not to each other.
LaTeX: -MATLAB News & Notes, Cleve’s Corner, 2006

%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');

% S  vs. latent
rho   = rank(X);
latent = diag(S(:,1:rho)).^2/(m-1)

% U  vs. score
sense = sign(score).*sign(U*S(:,1:rho));  %account for negated left singular vector
score = U*S(:,1:rho).*sense

% V  vs. coef
sense2 = sign(coef).*sign(V(:,1:rho));    %account for corresponding negated right singular vector
coef = V(:,1:rho).*sense2

coef, score, latent definitions from Matlab pca() command.

Terminology like variance of principal components (PCs) can be found here: Relationship between SVD and PCA.
Standard deviation is square root of variance.
Sign of principal component vectors are not unique.

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