Canonical correlations

Abstract

Harold Hotelling proposed the method of canonical correlations in 1936 to describe linear relationships between two subsets of variates as simply and as efficiently as possible. The vector of the complete set of all variables, X= [X ₁,... X _q], is subdivided into two subvectors, X₁ and X₂,

$$X = \left[ {{{X}_{1}}|{{X}_{2}}} \right] = \left[ {{{X}_{{11}}}, \ldots {{X}_{{1{{q}_{1}}}}}|{{X}_{{21}}}, \ldots {{X}_{{2{{q}_{2}}}}}} \right]$$

of which the first has q ₁ and the second has q ₂ elements. Conventionally, the variates are ordered so that q ₁ ≤ q ₂ The mean vector µ and the covariance matrix Σ of the q = q ₁ + q ₂ variates are subdivided in conformable fashion:

$$\mu = \left[ {{{\mu }_{1}}|{{\mu }_{2}}} \right]{\text{ and }}\Sigma {\text{ = }}\left[ {\begin{array}{*{20}{c}} {{{\Sigma }_{{11}}}} \hfill & {{{\Sigma }_{{12}}}} \hfill \\ {{{\Sigma }_{{21}}}} \hfill & {{{\Sigma }_{{22}}}} \hfill \\ \end{array} } \right]$$

Canonical correlations possess algebraic properties which are unaffected by the nature of the probability distribution of variates X = [X₁ | X₂], but current methods for testing hypotheses assume that the multivariate normal distribution applies to at least one of the two subvectors X₁ and X₂, the variates having possibly been submitted to prior transformations if necessary.