PC rotation question

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PC rotation question

Sasha Pustota
On p. 48 of "Statistics Complements" to the 3rd MASS edition,
http://www.stats.ox.ac.uk/pub/MASS3/VR3stat.pdf

I read that the orthogonal rotations of Z Lambda^-1 remain
uncorrelated, where Z is the PC and Lambda is the diag matrix of
singular values.

However, the example below that text is

   > A <- loadings(ir.pca) %*% diag(ir.pca$sdev)

If ir.pca$sdev are the singular values, should that be diag(1 /
ir.pca$sdev), or is it some discrepancy between S+ and R that I'm
missing?

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Re: PC rotation question

Sasha Pustota
I wrote:
> On p. 48 of "Statistics Complements" to the 3rd MASS edition,
> http://www.stats.ox.ac.uk/pub/MASS3/VR3stat.pdf
> I read that the orthogonal rotations of Z Lambda^-1 remain
> uncorrelated, where Z is the PC and Lambda is the diag matrix of
> singular values. However, the example below that text is
>    > A <- loadings(ir.pca) %*% diag(ir.pca$sdev)
> If ir.pca$sdev are the singular values, should that be diag(1 /
> ir.pca$sdev), or is it some discrepancy between S+ and R that I'm
> missing?


To dwell some more on this and in hopes to get replies I did a small
experimantation (below) that makes me to suspect that the correct
syntax
is A <- loadings(ir.pca) %*% diag(1/ir.pca$sdev). Any comments?

library(MASS)
a <- 1; b <- 0.2; c <- -.3; d <- .7
x <- scale(mvrnorm(n=1000,c(0,0,0),matrix(c(a,b,c, b,a,d, c,d,a),3,3)))
e <- eigen(cov(x))

cat("expect identity corr for orthogonal rotations:\n")
zs <- e$vectors %*% diag(1/sqrt(e$values))
pdx <- x %*% varimax(zs, normalize = FALSE)$loadings
print.table(cor(pdx), digits=2)

cat("expect zero: ", prcomp(x)$sdev - sqrt(e$values), "\n")

cat("Now zero corr between projections is not preserved:\n")
zs <- e$vectors %*% diag(sqrt(e$values))
pdx <- x %*% varimax(zs, normalize = FALSE)$loadings
print.table(cor(pdx), digits=2)

Output:
expect identity corr for orthogonal rotations:
     [,1]     [,2]     [,3]
[1,]  1.0e+00 -3.0e-16 -4.5e-16
[2,] -3.0e-16  1.0e+00  2.3e-16
[3,] -4.5e-16  2.3e-16  1.0e+00
expect zero:  4.440892e-16 2.220446e-16 2.775558e-16
Now zero corr between projections is not preserved:
     [,1]  [,2]  [,3]
[1,]  1.00 -0.35 -0.88
[2,] -0.35  1.00 -0.11
[3,] -0.88 -0.11  1.00

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