Definition
Given two column vectors and of random variables with finite second moments, one may define the cross-covariance to be the matrix whose entry is the covariance . In practice, we would estimate the covariance matrix based on sampled data from and (i.e. from a pair of data matrices).
Canonical correlation analysis seeks vectors and such that the random variables and maximize the correlation . The random variables and are the first pair of canonical variables. Then one seeks vectors maximizing the same correlation subject to the constraint that they are to be uncorrelated with the first pair of canonical variables; this gives the second pair of canonical variables. This procedure may be continued up to times.
Read more about this topic: Canonical Correlation
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