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This question is an extension of one I asked a few weeks back: Minimum sample size for PCA or FA when the main goal is to estimate only few components?

I will restate that I am interested in the use of PCA in situations where $n \le p$; and generally am only interested in using the first few PC axes for descriptive purposes or as "synthetic" variables that reduce several dimensions into one.

My question today revolves around a text "Numerical Ecology", 3rd edition by Legendre & Legendre. On page 450, they state:

A full-rank dispersion matrix $\mathbf S$ [variance-covariance] cannot be estimated using a number of observations $n$ smaller than or equal to the number of descriptors $p$. When $n \le p$, since there are $n-1$ DF in total, the rank of the resulting $\mathbf S$ matrix of order $p$ is $(n-1)$. In such a case, the eigen-decomposition of $\mathbf S$ produces $(n-1)$ real and $p-(n-1)$ null eigenvalues. Positioning $n$ objects while respecting their distances requires $(n-1)$ dimensions only. A PCA where $n \le p$ produces $(n-1)$ eigenvalues larger than $0$ and the $(n-1)$ corresponding eigenvectors and principle components."

In other words, I believe they are implying that it is OK to use PCA on a dataset where $n \le p$ as long as you are only interested in using $(n-1)$ or fewer of the PCs (as I am).

I am interested in your opinion regarding this (their claim and my interpretation) if you have one; and would appreciate any additional literature that might corroborate this claim.

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    $\begingroup$ You can even apply PCA when $p$ is infinite. For example, your observations might be periodic signals and you could choose to treat them as functions (rather than in terms of some discretized version of those functions). Nevertheless, the set of all real linear combinations of those functions is a finite dimensional vector space endowed with a Euclidean metric (the $L^2$ distance): it makes perfect sense to seek principal directions within this "point cloud." $\endgroup$ – whuber Dec 24 '12 at 15:58
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Yes, you surely can do that. I don’t know applications in ecology, but you may be interested to know that this is widely used in genetics (epidemiology and population genetics), with $n \ll p$, typically $n = 1000$ or $5000$ individuals and $p = 500\,000$ genotypes.

To adjust analyses for population mixture, the first 10 or 50 PC are used. The first two PC give already lots of informations, as shown in Novembre J (2008). Pay special attention to figure 1 where you see that the two first PC obtained from genomic data retrieve roughly the spacial arrangement off populations within Europe.

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