Canonical Correlation Analysis

Abstract

This chapter covers classical and robust canonical correlation analysis (CCA). Let $\varvec{x}$ be the $p \times 1$ vector of predictors, and partition ${\varvec{x}} = ({\varvec{w}}^T, {\varvec{y}}^T)^T$ where ${\varvec{w}}$ is $m \times 1$ and ${\varvec{y}}$ is $q \times 1$ with $m = p-q \le q$ and $m, q \ge 1$. If $m = 1$ and $q = 1$, then the canonical correlation is the usual correlation. Hence usually $q > 1$ and $m > 1$. The population canonical correlation analysis seeks m pairs of linear combinations $({\varvec{a}}_1^T {\varvec{w}}, {\varvec{b}}_1^T {\varvec{y}}), ..., ({\varvec{a}}_m^T {\varvec{w}}, {\varvec{b}}_m^T {\varvec{y}})$ such that corr(${\varvec{a}}_i^T {\varvec{w}}, {\varvec{b}}_i^T {\varvec{y}})$ is large under some constraints on the ${\varvec{a}}_i$ and ${\varvec{b}}_i$ where $i = 1, ..., m$.