# PCA components changing over time

If I do the PCA on the whole dataset I get 7 components that can explain 90% of the variance, if I split the dataset into 2 (sorted by time), the number of significant components in the first half goes to 5 (with 15 variables present in one or more components) and in the second half goes to 8 (with 21 variables present in one or more components), can we infer that some of these variables become more significant in the latter half compared to first half?

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What is "significant" component or variable for you? –  ttnphns Jul 5 '12 at 19:28
@ttnphns I surmize that he considers the number of components that it takes to get 90% variance explained represents the significant ones (assuming you can get that much of the variance explained with a subset of the components). –  Michael Chernick Jul 5 '12 at 20:33
I just think that when you split the data you change the variability. So the number of components needed to explain 90% of the variance should be expected to change. You get fewer in the first half and more in the second just because the second half happened to be more variable. This could just be a chance occurrence rather than something meaningful. –  Michael Chernick Jul 5 '12 at 20:37
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## 3 Answers

I would say that it depends on your data. If it is some sample (which I am sure it is), then you need to be careful about such inferences, since the first half your observations might not have been representative of your underlying population, and of course the same can be said about the latter half. So much in general about variable importance.

As far as 'becoming more important over time' is concerned, I have not heard about analyzing the change of variable importance in PCA (though I no PCA expert!). In either case, I would imagine that you would probably have to account for the importance of the variable within the component, as well as for the component loading itself. Assuming that your sample data does represent the reality reasonably well, you could probably make statements about it, but unless you find some well documented method that mathematically proves that it means anything, it is vague at best.

In your case, I would rather cross-validate the PCA in both datasets and check whether the differences are not purely random.

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Crossvalidation seems dicey, perhaps arbitrary and not necessarily very helpful, when doing PCA. It's not like regression where we might say, "in the training sample these 3 predictors are all effective based on ___ ; are they similarly effective in the test sample?" I don't see an analogous, clear, very testable hypothesis for crossvalidating PCA results. –  rolando2 Aug 5 '12 at 16:00
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Never test variable significance using PCA! This way you will only get overfitted junk.

Try fitting a model and test its parameters or use some machine learning feature selection method.

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With PCA you just calculated the eigenvectors of your data, and reduced the amount of information available. It does not prove "variable importance".

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