Copula Correlation: An Equitable and Consistently Estimable Measure for Association

• Activity Number: 509
• Type: Contributed
• Date/Time: Wednesday, August 6, 2014 : 10:30 AM to 12:20 PM
• Sponsor: Section on Statistical Learning and Data Mining
• Author(s): Aidong Ding*+ and Yi Li
• Companies: Northeastern University and Northeastern University
• Keywords: Equitability ; maximal information coefficient ; Copula ; rate of convergence ; mutual information ; distance correlation

Abstract

Reshef et al. (Science, 2011) proposed an important new concept of equitability that measures of association should satisfy: treating all types of functional relationships equally. Traditional measures such as Pearson's correlation prefer linear relationship and are not adequate for the needs in data mining. To this end, Reshef et al. (2011) proposed a novel measure, the maximal information coefficient (MIC). However, Kinney and Atwal (2013) showed that MIC is in fact not equitable under a strict mathematical definition. In addition, the mutual information (MI), a measure criticized in Reshef et al. (2011), is in fact equitable. We explain the paradoxical results by the fact that MI is very hard to estimate. We provide a first mathematical result qualifying this fact: MI's minimax risk is infinite and thus not consistently estimable. We propose a new correlation measure, the copula correlation (Ccor). This measure is shown to be both equitable and consistently estimable. It is the only known measure in literature that has both theoretical properties. Numerical studies were conducted to confirm its good performance in ranking strengths of nonlinear relationships.