On the copula correlation ratio and its generalization

The correlation ratio has been used to measure how much the behavior of one variable can be predicted by the other variable. In this paper, we derive a new expression of the correlation ratio based on copulas. We represent the copula correlation ratio in terms of Spearman's rho of the ∗-product of two copulas. Our expression provides a new way to obtain the copula correlation ratio, which is especially useful when a copula is closed under the ∗-product operation. Moreover, we propose a Kendall's tau copula correlation ratio that has not been considered in the literature. We apply the new expressions to investigate the theoretical properties of the copula correlation ratios, including difference and discontinuity. For multivariate copulas, we propose to define the copula correlation ratio matrices, and show their invariance property.