Re: eigenvalues of the covarience matrix (princomp)

yakir gagnon wrote:
and princomp(zscore( X )) is a CORRECT PCA...
There is absolutely no point in doing this

why? doing princomp(X) or princomp(zscore(X)) yields two different answers. and zscore(X) = zscore(zscore(X))

Yes, princomp(X) and princomp(zscore(X)) do give different results. All I meant was that princomp(X./repmat(std(X,1),size(X,1),1)) and princomp(zscore(X)) will give the same results, because princomp already centers the data to have zero mean, and so the centering step in zscore is redundant. On the other hand, since it's easier to type zscore(X) than X./repmat(std(X,1),size(X,1),1), choosing the former does no harm.


(as opposed to what you've called "correlation PCA"), since PRINCOMP already
centers the data.

here you say 'centre the data' which makes me confused since I thought you were talking about the zscoring (in which case I thought it was called standardizing), but I might be wrong.

ZSCORE centers each column to have zero mean, and normalizes each column to have unit variance. "Standardized" is kind of an ambiguous term; the best description of what ZSCORE does is "type zscore".

PRINCOMP always centers the data to have zero mean before doing anything. There's limited use in doing PCA on non-centered data, because the first component will typically describe the mean of the data, and that's not what most people want out of PCA (some would argue with that).


so why would I choose to do a so called "correlation PCA"? what is it good for?

There are a lot of differing opinions on this. My own opinion is that doing PCA on unstandardized variables implies that you think that the scales on which the different variables are measured are somehow "natural" and "comparable", in the sense that variation of some absolute magnitude in one variable is no more or less important than the same amount of absolute variation in another variable. Doing PCA on standardized variables (scaling each column by the inverse of its sample std dev) implies that you think that the scales of the different variables are an artifact of the units in which you measured them, and that you need to rescale in order to make the variation in the different variables "comparable". The classic example is doing PCA on things like body measurements. Should your PCA results differ if you choose to measure weight in grams vs. stones? Probably it shouldn't.

Whether or not you center the data before doing PCA affects these arguments too.

I would not describe either as "correct", but would apply a method as appropriate to circumstances. Again some would argue with that.

Hope this helps.

- Peter Perkins
The MathWorks, Inc.
.

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