# Re: [R] PCA problem in R

From: Dennis Shea <shea_at_cgd.ucar.edu>
Date: Tue 16 Aug 2005 - 02:56:28 EST

[SNIP]>>
>>>On Sat, 13 Aug 2005, Alan Zhao wrote:
>>>
>>>>When I have more variables than units, say a 195*10896 matrix which has
>>>>10896 variables and 195 samples. prcomp will give only 195 principal
>>>>components. I checked in the help, but there is no explanation that why
>>>>this happen.

[SNIP]
>Sincerely,
>Zheng Zhao
>Aug-14-2005
>______________________________________________

Just yesterday I subscribed to r-help because I am planning on learning the basics of R ... today. :-) Thus, I am not sure about the history of this question.

The above situation, more variables than samples, is commonly encounterd in the climate studies. Consider annual mean temperatures for 195 years on a coarse 72 [lat] x 144 [lon] grid [72*144=10368 spatial variables].

Let S be the number of grid points and T be the number of years. I think there is a theorem (?Eckart-Young?) which states that the maximum number of unique eigenvalues is min(S,T). In your case 195 eigenvalues is correct. I speculate that the underlying function transposes the input data matrix and computes the the TxT [rather than SxS] covariance matrix and solves for the eigenvalues/vectors. It then uses a linear transformation to get the results for the original input data matrix.

Computationally, the above is much faster and uses less memory.

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