Canonical Correlation Analysis

Abstract

The association between two sets of variables may be quantified by canonical correlation analysis (CCA). Given a set of variables $X \in \mathbb{R}^{q}$ and another set $Y \in \mathbb{R}^{p}$, one asks for the linear combination $a^{\top }X$ that “best matches” a linear combination $b^{\top }Y$. The best match in CCA is defined through maximal correlation. The task of CCA is therefore to find $a \in \mathbb{R}^{q}$ and $b \in \mathbb{R}^{p}$ so that the correlation $\rho (a,b) =\rho _{a^{\top }X,b^{\top }Y }$ is maximized. These best-matching linear combinations a ^⊤ X and b ^⊤ Y are then called canonical correlation variables; their correlation is the canonical correlation coefficient. The coefficients a and b of the canonical correlation variables are the canonical vectors.