Copula Correlation: An Equitable Dependence Measure and Extension of Pearson's Correlation

TL;DR

This paper introduces copula correlation (Ccor), a new dependence measure based on copula density, which is shown to be more equitable and easier to estimate than mutual information, with theoretical and numerical validation.

Contribution

The paper proposes Ccor, a novel dependence measure based on copula density, demonstrating its equitability and estimation advantages over mutual information.

Findings

Ccor is equitable under new definitions of equitability.

Ccor is easier to estimate than mutual information.

Numerical studies confirm the theoretical properties of Ccor.

Abstract

In Science, Reshef et al. (2011) proposed the concept of equitability for measures of dependence between two random variables. To this end, they proposed a novel measure, the maximal information coefficient (MIC). Recently a PNAS paper (Kinney and Atwal, 2014) gave a mathematical definition for equitability. They proved that MIC in fact is not equitable, while a fundamental information theoretic measure, the mutual information (MI), is self-equitable. In this paper, we show that MI also does not correctly reflect the proportion of deterministic signals hidden in noisy data. We propose a new equitability definition based on this scenario. The copula correlation (Ccor), based on the L1-distance of copula density, is shown to be equitable under both definitions. We also prove theoretically that Ccor is much easier to estimate than MI. Numerical studies illustrate the properties of the measures.