Canonical correlation coefficients
Maximum values of correlation coefficients between pairs of linear functions
 |
of two sets of random variables 
and 
for which 
and 
are canonical random variables (see Canonical correlation). The problem of determining the maximum correlation coefficient between 
and 
under the conditions 
and 
can be solved using Lagrange multipliers. The canonical correlation coefficients are the roots 
of the equation
 |
where 
and 
are the covariance matrices of 
and 
, respectively, and 
is the covariance matrix between the variables of the first and second sets. The 
-th root of the equation is called the 
-th canonical correlation coefficient between 
and 
. It is equal to the maximum value of the correlation coefficients between the pair of linear functions 
and 
of canonical random variables, each of which has variance one and is uncorrelated with the first 
pairs of variables 
and 
. The coefficients 
, 
of 
and 
satisfy the equation
 |
when 
.
Comments
See also Correlation; Correlation coefficient.
Canonical correlation coefficients. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Canonical_correlation_coefficients&oldid=46194