On Mon, 15 Aug 2005, Dennis Shea wrote: > [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]. Which are variables and which are samples here? In standard statistical parlance you have 195 variables at 10368 samples. In some fields there are the concepts of R-mode and Q-mode PCA, and you seem to be in Q-mode, which is why you have a transpose. > 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. Eigenvalues of what? Eckart-Young is about the SVD, see e.g. http://voteview.com/ideal_point_Eckart_Young_Theorem.htm as Googling easily shows. (It is used to prove some of the approximation properties of PCA, e.g. in http://www.stats.ox.ac.uk/~ripley/MultAnal_MT2004/PCA.pdf) > 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. You speculate incorrectly, even in your Q-mode view of the world. The real point is that is solves a different problem, which is what my answer to the original post was about. > PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html It really would be a good idea to do the homework it suggests. -- Brian D. Ripley, [EMAIL PROTECTED] Professor of Applied Statistics, http://www.stats.ox.ac.uk/~ripley/ University of Oxford, Tel: +44 1865 272861 (self) 1 South Parks Road, +44 1865 272866 (PA) Oxford OX1 3TG, UK Fax: +44 1865 272595 ______________________________________________ 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
