On the Importance of Asymmetry and Monotonicity Constraints in Maximal Correlation Analysis

摘要

The maximal correlation coefficient is a wellestablished generalization of the Pearson correlation coefficient for measuring non-linear dependence between random variables. It is appealing from a theoretical standpoint, satisfying Rényi’s axioms for measures of dependence. It is also attractive from a computational point of view due to the celebrated alternating conditional expectation algorithm, allowing to compute its empirical version directly from observed data. Nevertheless, from the outset, it was recognized that the maximal correlation coefficient suffers from some fundamental deficiencies, limiting its usefulness as an indicator of estimation quality. Another well-known measure of dependence is the correlation ratio but it too suffers from some drawbacks. Specifically, the maximal correlation coefficient equals one too easily whereas the correlation ratio equals zero too easily. The present work recounts some attempts that have been made in the past to alter the definition of the maximal correlation coefficient in order to overcome its weaknesses and then proceeds to suggest a natural variant of the maximal correlation coefficient. The proposed dependence measure at the same time resolves the major weakness of the correlation ratio measure and may be viewed as a bridge between the two classical measures.