A multivariate correlation ratio
This paper aims to extend the notion of K. Pearson's correlation ratio \(\eta\) (Y;X) as a measure of predictability of Y from X, to the multivariate case. Let Y and X be jointly distributed random vectors. To account for different importance of the q components of Y, the author introduces an arbitrary, positive definite \(q\times q\) weighting matrix \(\Lambda\) and defines the multivariate correlation ratio of Y upon X, relative to \(\Lambda\), by \[ \eta^ 2_{\Lambda}(Y;X)=tr(\Lambda^{- 1}Cov E(Y| X))/tr(\Lambda^{-1}Cov Y). \] He studies its properties and shows, thereby generalizing known results in the univariate case, that for any \(\Lambda\), among all arbitrary functions g(X), \(g(x)=E(Y| X=x)\) is that one which obtains the minimum of \(E[(Y'- g'(X))\Lambda^{-1}(Y-g(X))]\) with value \([1-\eta^ 2_{\Lambda}(Y;X)]tr(\Lambda^{-1}Cov Y)\), as well as the maximum of \(\rho^ 2_{\Lambda}(Y,g(X))\) with value \(\eta^ 2_{\Lambda}(Y;X)\), where the vector correlation \(\rho_{\Lambda}(S,T)\) between two \(q\times 1\) vectors S,T, relative to \(\Lambda\), is defined by \[ \rho_{\Lambda}(S,T)=tr(\Lambda^{-1}Cov(S,T))[tr(\Lambda^{-1}Cov S))tr(\Lambda^{-1}Cov T)]^{-1/2}. \] The results are applied to the family of elliptically symmetric distributions (enclosing a.o. multivariate normal distributions) and to the multinomial distribution of (X',Y')'. The problem of finding those r linear functions of Y, that are most predictable from X, is solved by determining among all \(r\times q\) matrices B with \(Cov(BY)=\Lambda^{-1}\) positive definite, that one which maximizes \(\eta_{\Lambda}(BY;X)\). Both, the maximizing B and the maximum obtained, imply r eigenvalues and corresponding eigenvectors of the matrix (Cov Y)\({}^{-1/2}Cov E(Y| X)(Cov Y)^{-1/2}\).
scientific article; zbMATH DE number 3844839