From Distance Correlation to Multiscale Graph Correlation

Abstract

Understanding and developing a correlation measure that can detect general dependencies is not only imperative to statistics and machine learning, but also crucial to general scientific discovery in the big data age. In this paper, we establish a new framework that generalizes distance correlation --- a correlation measure that was recently proposed and shown to be universally consistent for dependence testing against all joint distributions of finite moments --- to the Multiscale Graph Correlation (MGC). By utilizing the characteristic functions and incorporating the nearest neighbor machinery, we formalize the population version of local distance correlations, define the optimal scale in a given dependency, and name the optimal local correlation as MGC. The new theoretical framework motivates a theoretically sound Sample MGC and allows a number of desirable properties to be proved, including the universal consistency, convergence and almost unbiasedness of the sample version. The advantages of MGC are illustrated via a comprehensive set of simulations with linear, nonlinear, univariate, multivariate, and noisy dependencies, where it loses almost no power in monotone dependencies while achieving better performance in general dependencies, compared to distance correlation and other popular methods.

39 pages + Appendix 22 pages, 6 figures

References in corpus (3)

• Measuring and testing dependence by correlation of distances
• Asymptotic local efficiency of Cramér--von Mises tests for multivariate independence
• Rejoinder: Brownian distance covariance

Cited by in corpus (6)

• The Chi-Square Test of Distance Correlation
• The Exact Equivalence of Distance and Kernel Methods for Hypothesis Testing
• Universally Consistent K-Sample Tests via Dependence Measures
• Correcting a Nonparametric Two-sample Graph Hypothesis Test for Graphs with Different Numbers of Vertices with Applications to Connectomics
• Discovering the Signal Subgraph: An Iterative Screening Approach on Graphs
• Network Dependence Testing via Diffusion Maps and Distance-Based Correlations