Brownian distance covariance

Abstract

Distance correlation is a new class of multivariate dependence coefficients applicable to random vectors of arbitrary and not necessarily equal dimension. Distance covariance and distance correlation are analogous to product-moment covariance and correlation, but generalize and extend these classical bivariate measures of dependence. Distance correlation characterizes independence: it is zero if and only if the random vectors are independent. The notion of covariance with respect to a stochastic process is introduced, and it is shown that population distance covariance coincides with the covariance with respect to Brownian motion; thus, both can be called Brownian distance covariance. In the bivariate case, Brownian covariance is the natural extension of product-moment covariance, as we obtain Pearson product-moment covariance by replacing the Brownian motion in the definition with identity. The corresponding statistic has an elegantly simple computing formula. Advantages of applying Brownian covariance and correlation vs the classical Pearson covariance and correlation are discussed and illustrated.

This paper discussed in: [arXiv:0912.3295], [arXiv:1010.0822], [arXiv:1010.0825], [arXiv:1010.0828], [arXiv:1010.0836], [arXiv:1010.0838], [arXiv:1010.0839]. Rejoinder at [arXiv:1010.0844]. Published in at http://dx.doi.org/10.1214/09-AOAS312 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Measuring and testing dependence by correlation of distances
• Brownian distance covariance

Cited by in corpus (20)

• Brownian distance covariance
• Estimation and testing for spatially indexed curves with application to ionospheric and magnetic field trends
• On efficient dimension reduction with respect to a statistical functional of interest
• On Quantifying Dependence: A Framework for Developing Interpretable Measures
• Interpreting the Distance Correlation Results for the COMBO-17 Survey
• LP Approach to Statistical Modeling
• Theoretical Foundations of Equitability and the Maximal Information Coefficient
• Measuring Association between Random Vectors
• The Classification Permutation Test: A Nonparametric Test for Equality of Multivariate Distributions
• A measure of association between vectors based on "similarity covariance"
• A nonparametric Bayesian test of dependence
• Fast Computing for Distance Covariance
• Mutual Dependence: A Novel Method for Computing Dependencies Between Random Variables
• Brownian Manifolds, Negative Type and Geo-temporal Covariances
• Learning Markov Network Structure using Brownian Distance Covariance
• Feature Screening via Distance Correlation Learning
• High-dimensional inference on covariance structures via the extended cross-data-matrix methodology
• Distance covariance for stochastic processes
• Capturing collective conflict dynamics with sparse social circuits
• Discovering general multidimensional associations