[R] Cancor

Gabor Grothendieck ggrothendieck at myway.com
Sun Sep 12 20:04:22 CEST 2004

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


# 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


at the R prompt to view its source and see how it works using
the QR decomposition.

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