1

Are principal factors in principal component analysis always uncorrelated, or can they end up being correlated? If so, how and why?

kjetil b halvorsen
  • 77,844
Nickpick
  • 321
  • 3
    It might depend on what you mean by "principal factors," because that language appears to combine the language of principal components analysis and factor analysis. Could you clarify? And could you explain what you mean by "correlated"? Would that mean their correlation coefficient as a dataset is zero, or would it refer to the random variables one might use to model the data? – whuber Dec 07 '18 at 20:24
  • I want to know if principal factors in a principal component linear factor model can be correlated with each other. It’s not related to factor analysis. – Nickpick Dec 07 '18 at 22:13

1 Answers1

2

You are using some nonstandard terminology, but you seem to mean simply the principal components in pca. For terminology see What exactly is called "principal component" in PCA?

Then the answer is NO. The principal components are constructed as to be uncorrelated, in the sense of the usual product-moment correlation. If you should mean some other kind of correlation, for example rank correlation (you did not specify), then the principal components could well be correlated.

kjetil b halvorsen
  • 77,844
  • +1 Also possible is to have out-of-sample correlation when you apply the in-sample eigenvectors to the out-of-sample data. (In think a lack of correlation in such a setting would be quite the coincidence!) – Dave Aug 25 '23 at 18:38