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