Interaction Measures, Partition Lattices and Kernel Tests for High-Order Interactions

TL;DR

This paper introduces a hierarchy of high-order interaction measures and kernel-based tests to detect complex multivariate dependencies, supported by mathematical links to lattice theory and validated on synthetic and neuroimaging data.

Contribution

It presents a novel framework for systematically measuring and testing high-order interactions in multivariate data using kernel methods and lattice theory insights.

Findings

Effective detection of high-order interactions in synthetic data.

Application to neuroimaging data reveals complex dependencies.

Enhanced computational efficiency in interaction testing.

Abstract

Models that rely solely on pairwise relationships often fail to capture the complete statistical structure of the complex multivariate data found in diverse domains, such as socio-economic, ecological, or biomedical systems. Non-trivial dependencies between groups of more than two variables can play a significant role in the analysis and modelling of such systems, yet extracting such high-order interactions from data remains challenging. Here, we introduce a hierarchy of $d$-order ($d \geq 2$) interaction measures, increasingly inclusive of possible factorisations of the joint probability distribution, and define non-parametric, kernel-based tests to establish systematically the statistical significance of $d$-order interactions. We also establish mathematical links with lattice theory, which elucidate the derivation of the interaction measures and their composite permutation tests; clarify the connection of simplicial complexes with kernel matrix centring; and provide a means to enhance computational efficiency. We illustrate our results numerically with validations on synthetic data, and through an application to neuroimaging data.