Interaction Measures, Partition Lattices and Kernel Tests for High-Order Interactions
This work addresses the problem of capturing non-trivial dependencies beyond pairwise relationships in fields like socio-economic, ecological, or biomedical systems, representing an incremental advancement in statistical methodology.
The paper tackles the challenge of extracting high-order interactions from complex multivariate data by introducing a hierarchy of interaction measures and non-parametric kernel-based tests to assess their statistical significance, demonstrating results through validation on synthetic data and an application to neuroimaging data.
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.