Correlation of Principal Component Scores after Varimax Rotation

Hi all:

I'm extracting principal components from time series data and use the varimax rotation to interpret the PCs. In addition, I'd like to compute the rotated scores and use them for further analysis. However, the rotated scores are not uncorrelated anymore, although they should (I think) because the rotation matrix is orthornomal.

Here's a simple example in which I pick two PCs:

load hald [C,S] = pca(zscore(ingredients));

[L,T] = rotatefactors(C(:,1:2)); % L = C(:,1:2)*T

cov(S(:,1:2)*T)

Can anybody help?

Peter

Answer 1

[Ayush Aniket]

PCA scores are uncorrelated because they come from the eigen-decomposition of the covariance matrix. When you rotate the loadings (e.g. varimax), the rotation matrix is orthogonal, so the axes remain orthogonal, but the scores no longer stay uncorrelated. This is expected as rotation trades uncorrelated scores for more interpretable loadings.

Hence, if you need uncorrelated variables, use the original PC scores, and if you need interpretability, use the rotated ones (and accept correlation).