[R] Cancor

[Gabor Grothendieck]

Irena Komprej <irena.komprej <at> telemach.net> writes:

> 
> 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.