[R] Cancor
Gabor Grothendieck
ggrothendieck at myway.com
Mon Sep 13 14:00:11 CEST 2004
Irena Komprej <irena.komprej <at> telemach.net> writes:
: thank you for your answer, but I am still a little confused because the
: values of the xcoef and ycoef are so small. It is true, that I receive
: very similar results to the cancor, if I use the proposed formula with
:
: z <- svd(cov(x,y) %*% solve(var(y), cov(y,x)) %*% solve(var(x)))
: sqrt(z$d) # canonical correlations
: isqrt((nrow(x)-1)*var(x)) %*% z$u # xcoef
:
: But, why do I need the nrow(x)-1)* in isqrt()?
I don't think the scaling really matters. If you want to define
the scaling so that the canonical variables have identity variance
matrix or other scaling you can rescale. I chose the above
scaling since the objective was to give the same result as
cancor.
If the values we calculate above are slightly different than cancor's
I would go with cancor since it uses the QR decomposition which is
presumably more stable numerically than what we have above.
Don't know about the rest of your questions.
:
: >
: > I am strugling with cancor procedure in R. I cannot figure out the
: > meaning of xcoef and of yxcoef.
: > Are these:
: > 1. standardized coefficients
: > 2. structural coefficients
: > 3. something else?
: >
:
: Look at the examples at the bottom of ?cancor from which its evident
: xcoef is such that x %*% cxy$xcoef are the canonical variables. (More
: at the end of this post.)
:
: > I have tried to simulate canonical correlation analysis by checking
: the
: > eigenstructure of the expression:
: >
: > Sigma_xx %*% Sigma_xy %*% Sigma_yy %*% t(Sigma_xy).
: >
: > The resulting eigenvalues were the same as the squared values of
: > cancor$cor. I have normalized the resulting eigenvectors, the a's with
: >
: > sqrt(a'%*%Sigma_xx%*%t(a)), and similarly the b's with
: > sqrt(b'%*%Sigma_yy%*%t(b)).
: >
: > The results differed considerably from xcoef and ycoef of the cancor.
:
: Run the example in the help page to get some data and some
: output:
:
: set.seed(1)
: example(cancor)
:
: # Also, define isqrt as the inverse square root of a postive def matrix
:
: isqrt <- function(x) {
: e <- eigen(x)
: stopifnot( all(e$values > 0) )
: e$vectors %*% diag(1/sqrt(e$values)) %*% solve(e$vectors)
: }
:
: # we can reconstruct the canonical correlations and xcoef
: # in the way you presumably intended like this:
:
: z <- svd(cov(x,y) %*% solve(var(y), cov(y,x)) %*% solve(var(x)))
: sqrt(z$d) # canonical correlations
: isqrt((nrow(x)-1)*var(x)) %*% z$u # xcoef
:
: Another thing you can do is to type
:
: cancor
:
: at the R prompt to view its source and see how it works using
: the QR decomposition.
:
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