Adaptive Geo-Topological Independence Criterion

TL;DR

This paper introduces adaptive independence tests based on nonlinear transformations of distances, improving robustness and power in detecting dependence between multivariate variables with finite samples.

Contribution

It develops a new family of adaptive dependence criteria building on distance correlation, enhancing statistical sensitivity and robustness across diverse scenarios.

Findings

Empirically outperforms existing tests in sensitivity.

Provides reliable dependence indicators.

Offers insights into data relationships.

Abstract

Testing two potentially multivariate variables for statistical dependence on the basis finite samples is a fundamental statistical challenge. Here we explore a family of tests that adapt to the complexity of the relationship between the variables, promising robust power across scenarios. Building on the distance correlation, we introduce a family of adaptive independence criteria based on nonlinear monotonic transformations of distances. We show that these criteria, like the distance correlation and RKHS-based criteria, provide dependence indicators. We propose a class of adaptive (multi-threshold) test statistics, which form the basis for permutation tests. These tests empirically outperform some of the established tests in average and worst-case statistical sensitivity across a range of univariate and multivariate relationships, offer useful insights to the data and may deserve further exploration.