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 (Dcorr)—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 using 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 Dcorr and other popular methods. Supplementary materials for this article are available online.

Keywords:

• Generalized distance correlation
• Nearest neighbor graph
• Testing independence

Acknowledgments

The authors thank the anonymous reviewers for a very diligent and constructive review leading to significant improvement of the article; and thank Dr. Minh Tang and Dr. Shangsi Wang for discussions and suggestions.

Additional information

Funding

This work was partially supported by the National Science Foundation award DMS-1712947, and the Defense Advanced Research Projects Agency’s (DARPA) SIMPLEX program through SPAWAR contract N66001-15-C-4041.