Distance covariance for random fields

TL;DR

This paper develops an independence test for random fields using distance correlation, providing asymptotic theory and bootstrap methods for different observation schemes, with applications to meteorological data.

Contribution

It introduces a novel independence testing approach for random fields based on distance correlation, with theoretical guarantees and practical implementation for various observation settings.

Findings

Bootstrap test is consistent for independence detection.

Method performs well in simulations with fractional Brownian and stable fields.

Applied successfully to Japanese meteorological data.

Abstract

We study an independence test based on distance correlation for random fields $(X,Y)$. We consider the situations when $(X,Y)$ is observed on a lattice with equidistant grid sizes and when $(X,Y)$ is observed at random locations. We provide asymptotic theory for the sample distance correlation in both situations and show bootstrap consistency. The latter fact allows one to build a test for independence of $X$ and $Y$ based on the considered discretizations of these fields. We illustrate the performance of the bootstrap test in a simulation study involving fractional Brownian and infinite variance stable fields. The independence test is applied to Japanese meteorological data, which are observed over the entire area of Japan.