We commonly do a pca on matrices of , say, a hundred variables and, say, 10,000
samples. As an earlier poster said, the information can be obtained
from analysis of the smaller matrix. the run time is not too long on a pc (maybe 2-10
minutes), the problem is really tied to voluminous output. Among our applications,
analysis of remote sensing multispectral data sounds similar to your data structure in
that we may have a pixel matrix that is 5000 x 5000 ("samples") and perhaps 50 spectral
bins (variables). John Imbrie and Ed Klovan showed the way 30 or so years ago and I am
sure that many others have described use of the Q matrix in this way.
Gottfried Helms wrote:
> (Since there were some small errors in the previous text
> I just send a complete correction. Pls excuse for inconvenience)
>
> "Arthur J. Kendall" schrieb:
> >
> > It is unusual.
> > SPSS has had PCA under FACTOR since at least 1972.
> > include the specification
> > /extraction = PC
> > or equivalently
> > /extraction = PA1
> > on the FACTOR procedure.
> >
> > I don't know if it can handle 11,000 variables.
> >
>
> It would need a *lot* of time and memory (at least ~121*16 MB, only for the
> correlation matrix.
>
> I remember to have read articles about "large matrices" or "large spare matrices"
> some years ago... Search via google, also in sci.stat.consult. They talked about
> these matrix-dimensions.
>
> Concerning 150 cases with 11000 variables : you just get at most 150 factors, and
> have linear dependencies after that.
>
> If nothing helps, you could do a PCA on the first 150 variables and save the
> scores.
> Then you can correlate all variables with the factor scores and put the
> correlations together to form a proper factor-loadingsmatrix. This way you can use
> as much variables as your program can handle per run. Putting them all together this
> gives you a factor-loadings-matrix of 11000*150 (~26 MB per matrix), which might be
> *a little* better to handle than 11000*11000. With SPSS you can read a ready factor-
> loadingmatrix directly into the procedure
>
> Only you don't get factor scores then. If you need them, you can use the matrix-
> language facility to build the pseudoinverse of your final factor-loadings matrix,
> (this has one dimension of only 150) and matrix-multiply this with your raw-data.
>
> Say
> V - Array of all 11000 Variables,variables verical, cases horizontal
> --------------------
> V1 - Array of first 150 variables
> V2 - Array of next 150 variables
> ...
> then
> R = corr(V1,V1')
> L0 = cholesky(R) // compute loadingsmatrix, for instance with chlesky method
> I0 = inv(L0) // inverse of loadingsmatrix for scores-calculation
> Fsc1 = I0*V1 // compute raw scores for your 150 factors
>
> now compute loadings for all variables. Their loadings are the
> correlations between factors and variables:
>
> Lad1 = corr(Fsc0,V1) // loadings for the first 150 variables
> Lad2 = corr(Fsc0,V2) // loadings for the next 150 variables
> ...
> Ladx = corr(fsc0,Vx) // loadings for the last 150 variables
>
> put them all together to have a combined loadingsmatrix for rotations
> Lad = {L1,L2,L3...}
>
> After that you can perform the rotations.
>
> -----------
>
> To get scores, you use matrix-algebra:
>
> Since
> [1] Lad * Fsc = V
> [2] Lad'*Lad * Fsc = Lad' * V // here Lad'*Lad is of 150*150
> [3] ILad = inv(Lad'*Lad)
> [4] ILT = ILad*Lad'
> [5] Fsc = ILT * V
>
> you can get factor-scores just by multiplying your variable-values
> by the matrix ILT, which has one dimension of only 150 at most.
>
> HTH
>
> Gottfried Helms
.
.
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