"Roland Neumann" <softwarepca@o2online.de> wrote in message <gdq7qd$p1g$1@fred.mathworks.com>...
> I came across a strange behaviour while computing principal components with the Statistics Toolbox (either princomp or pcacov, doesn't matter).
>
> Background is calculating independent software metrics using PCA. This works quite well, when I'm doing like:
>
> EV Lambda =pcacov(corr(M))
> ...(compute FL out of EV and Lambda)
> C= zscore(M)*EV
>
> leads to nicely uncorrelated C (my independent software metrics). Now the strange thing... I want the principal components better suiting the metrics and use orthogonal rotation like varimax or quartimax with:
>
> RotFL = rotatefactors(FL, 'method', 'varimax', 'normalization', 'off')
>
> This is supposed to be (according to help) an orthogonal operation.
> But after another:
> CRot= zscore(M)*EVRot
> I get correlated C.
>
> It's already visible in RotFL: Computing the angles between each 2 columnvektors (A*B/(A*B) shows no 90? anymore (as for EV and FL).
>
> I'm lost with this supposingly orthogonal rotation and will appreciate any hints. Thank you.

Roland, just because a pair of random variables, x and y, are uncorrelated doesn't mean that a rotation of them will yield uncorrelated variables. Let E{x} = E{y} = E(x*y) = 0 so that x and y are uncorrelated. Then rotate them to u = a*x+b*y and v = b*x+a*y where a^2+b^2 = 1. Then their (cross) covariance will be:
E{u*v} =
E{(a*x+b*y)*(b*x+a*y)} =
(a^2b^2)*E{x*y) + a*b*(E{x^2}E{y^2}) =
a*b*(E{x^2}E{y^2})
which will not be zero unless x and y have equal variances. Thus the two quantities u and v are not necessarily uncorrelated.
Roger Stafford
