A multivariate correlation ratio

A multivariate correlation ratio of a random vector $\text{Y}$ upon a random vector $\text{X}$ is defined by

$\text{η}{}_{\mathrm{\delta }}\text{(Y;X)={}\text{tr}\text{(δ}{}^{\mathrm{<mtext>?1\; </mtext><mtext>Cov</mtext><mtext>E(Y|X))\}</mtext>}}\text{1}\text{2}\text{{}\text{tr}\text{(δ}{}^{\mathrm{?1}}\text{∑}{}_{\mathrm{Y}}\text{)}}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$

where Λ, a fixed positive definite matrix, is related to the relative importance of predictability for the entries of $\text{Y}$. The properties of η_Λ are discussed, with particular attention paid to a ‘correlation-maximizing’ property. Given are applications of η_Λ to the elliptically symmetric family of distributions and the multinomial distribution. Also discussed is the problem of finding those r linear functions of $\text{Y}$ that are most predictable (in a correlation ratio sense) from $\text{X}$.