Canonical correlation coefficients
Maximum values of correlation coefficients between pairs of linear functions
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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
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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
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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