> From: Dennis Shea > > [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.
It is usually a good idea to consult the help page before speculating. ?prcomp has, in its `Detail' section: The calculation is done by a singular value decomposition of the (centered and possibly scaled) data matrix, not by using eigen on the covariance matrix. This is generally the preferred method for numerical accuracy. Andy > ______________________________________________ > R-help@stat.math.ethz.ch mailing list > https://stat.ethz.ch/mailman/listinfo/r-help > PLEASE do read the posting guide! > http://www.R-project.org/posting-guide.html > > > ______________________________________________ R-help@stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
