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(1) When doing PCA, do you assume the variables to be bell-shaped? Say if I have a bunch of variables, some are bell-shaped but some have characteristic long (right) tails (highly skewed and asymmetric with the median close to the minimum and and a few extremely large values). Should I do log transformation on those long-tailed variables or leave them like that? What are the implications to the results of the PCA with and without transformation?

(2) The second question is, when I do principal component regression (PCR), the response variable have a long tail & one of the PC scores also have long tail. Should I log transform either or both of them?

NB: Tails in this question are all one-sided.

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  • $\begingroup$ That post could be of your interest. $\endgroup$ – ttnphns Apr 14 '13 at 13:04
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Regarding 1). As far as I know, it is not an explicit assumption of PCA that the variables be Normal. However, since PCA is based on correlations/covariances, there may be effects on the results from using variables that are long tailed. In particular, the points at the tails may have large effects on the correlations (how large depends on the exact nature of the distribution). I suggest making a scatterplot matrix of all the variables and seeing if they look reasonable. If you transform the variables, the meaning of the PCA will change somewhat (since the variables change).

Regarding 2). As with any OLS regression, it is not assumed that either the DV or the IV are normal. It is assumed that the errors as estimated by the residuals are normal. You can plot these after running the model

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