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

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    $\begingroup$ 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? $\endgroup$
    – whuber
    Commented Dec 7, 2018 at 20:24
  • $\begingroup$ 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. $\endgroup$
    – Nickpick
    Commented Dec 7, 2018 at 22:13

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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.

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    $\begingroup$ +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!) $\endgroup$
    – Dave
    Commented Aug 25, 2023 at 18:38

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