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I have a list of 25 air pollutants many of which are strongly correlated. I was hoping to reduce down to a short list of eigenvectors which would each be composed of a small number of the pollutants.

I mostly followed this tutorial video so far.

When I tried varimax rotation on all of the PCs (as seems to be done at the end of this video) I get each of the 25 PCa having a score of 1 or -1 for only one variable, with every variable accounted for, i.e. all PCs and variables accounted for by unique pairings.

Have I done something wrong or if not then why did the example in the video not also do the same? Could it be something to do with my data having many more dimensions?

Depending on the selection criteria (Kaiser criterion; levelling-off of scree plot) the top 7 or top 13 are the important ones. Putting just these 7 or 13 components in gives more the sort of output I had expected.

Does anyone have any suggestions which of these options (7 or 13 components) I should choose or is it a fairly arbitrary decision based on how much I want to compress the data?

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    $\begingroup$ top 7 or top 13 are the important ones Strictly speaking, this not so. "top 7 through top 13 might the important ones" is better said. $\endgroup$ – ttnphns May 15 '13 at 6:09
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    $\begingroup$ Rotation is being done for the purpose of interpretation of PCs. If you have such a purpose that means that you treat PCA as factor analysis, a latent-variable analysis, and not just as data compression technique. You should rotate every solution from 7 thru 13 components and interpet; leaving in the end the most interpretable one. $\endgroup$ – ttnphns May 15 '13 at 6:17
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I would look at the two analyses and see which was more interpretable, more useful etc.

If they are equally so, I'd go with the 7 - parsimony isn't everything, but it is something.

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These really aren't your only choices. There should be a lot of science that tells you which pollutants hang together and direct analysis of the scatter plot matrix and the correlation matrix is often as or more informative. In particular, it's important to know what underlies even the high correlations, ranging from simple approximate linear relationships to any outlier effects.

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