I think PCA decomposes matrix A according to A'A, not to COV (A).
But if A is centered then A'A = (n + 1) COV (A).
So for non-centered A, you want to look at A'A instead:
> crossprod(A) %*% evec[,1] / (nrow (A) - 1) - eval [1] * evec [,1]
[,1]
[1,] 0.000e+00
[2,] 0.000e+00
[3,] 1.066e-14
If I'm telling crap, someone please correct me!
Hope that helps,
Claudia
On 11/10/2010 02:41 PM, kicker wrote:
Hello,
I have a short question about the prcomp function. First I cite the
associated help page (help(prcomp)):
"Value:
...
SDEV the standard deviations of the principal components (i.e., the square
roots of the eigenvalues of the covariance/correlation matrix, though the
calculation is actually done with the singular values of the data matrix).
ROTATION the matrix of variable loadings (i.e., a matrix whose columns
contain the eigenvectors). The function princomp returns this in the element
loadings.
..."
Now please take a look at the following easy example:
first I define a matrix A
A<-matrix(c(0,1,4,1,0,3,4,3,0),3,3)
then I apply PCA on A
trans<-prcomp(A,retx=T,center=F,scale.=F,tol=NULL)
eval<-trans$sdev*trans$sdev #eval is the vector of the eigenvalues of
cov(A) (according to the cited help text above)
evec<-trans$rotation #evec is the matrix with the eigenvectors of cov(A) as
columns (according to the cited help text above)
now the eigenvalue equation should be valid, i.e. it should hold
cov(A)%*%ev[,1]=ew[1]*ev[,1]. But it doesn´t, my result:
cov(A)%*%ev[,1]= t(-0.8244927, -0.8325664,0.8244927)
ew[1]*ev[,1]=t(-8.695427,-7.129314,-10.194816)
So my question is : why does the eigenvalue equation not hold ?
The eigenvalue equation holds when I set center=T in the options of the
prcomp function. But as far as I know and as I understand the help text it
should have no influence on the eigenvalue equation whether the data are
centered or not. I know about the advantages of centered date but I want to
understand how the prcomp function works in the case of uncentered data.
Thank you very much for your efforts.
--
Claudia Beleites
Dipartimento dei Materiali e delle Risorse Naturali
Università degli Studi di Trieste
Via Alfonso Valerio 6/a
I-34127 Trieste
phone: +39 0 40 5 58-37 68
email: cbelei...@units.it
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