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When doing the PCA analysis, you end up with eigenvalues which are ordered by how much variance they explained for each eigenvector. Say, the eigenvectors since they are orthogonal, do not represent the reality - I want to stick with the original factors only, but determine which are the most important and which add up to providing, say, 90% of the variance?

Also, is Jacobian rotation more conducive to finding dominant original factors?

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What do you mean by original factors? – John May 20 '14 at 19:39
@John, By original i mean the factors that are used for PCA, not the eigenvectors which are orthogonal and do not represent a factor but some combination of them. eg, in HJM framework, you may have years on the forward curve as factors. And, say, you are only interested in which year(factor) contributed the most to the eigenvalues. – user12348 May 20 '14 at 20:20
It is usually possible to say a how much a particular entry (year in your example) contributed to a particular eigenvector (which I can provide an answer for), but I'm not sure you can do the same across all of them, if that's what you're asking. – John May 20 '14 at 20:54
Not sure if what you are saying can be translated into what I am asking. If year 1 happens to coincide ( or nearly coincide or contribute the max to PC1) with PC1 then that is what I want. – user12348 May 20 '14 at 23:30
up vote 2 down vote accepted

When I use PCA, I follow a few typical steps. First, I would apply PCA to the covariance matrix, I would then designate certain eigenvalues as dominant or significant (such as by those that contribute up to $x\%$ of variance or by RMT), and then I would identify the eigenvectors that match up with those significant eigenvalues.

I think you're with me at this point. It appears you want to know how to determine which of the inputs to the covariance matrix match up with the eigenvectors (i.e. how much does year $Y$ contribute to eigenvalue $N$, in your example). One way to make that determination is to square the eigenvector $N$. This squared eigenvector should sum up to $1$. Thus, you can consider each of these squared values like a percent of contribution to the eigenvector (and thus the eigenvalues). For your example, you could plot these squared values against the years for each to get a sense of how it changes as you change eigenvectors.

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Do you have a good reference on RMT? – user12348 May 21 '14 at 22:40

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