Interaction Measures, Partition Lattices and Kernel Tests for High-Order
Interactions
- URL: http://arxiv.org/abs/2306.00904v3
- Date: Tue, 7 Nov 2023 17:26:35 GMT
- Title: Interaction Measures, Partition Lattices and Kernel Tests for High-Order
Interactions
- Authors: Zhaolu Liu, Robert L. Peach, Pedro A.M. Mediano, and Mauricio Barahona
- Abstract summary: Non-trivial dependencies between groups of more than two variables can play a significant role in the analysis and modelling of such systems.
We introduce a hierarchy of $d$-order ($d geq 2$) interaction measures, increasingly inclusive of possible factorisations of the joint probability distribution.
We also establish mathematical links with lattice theory, which elucidate the derivation of the interaction measures and their composite permutation tests.
- Score: 1.9457612782595313
- License: http://creativecommons.org/licenses/by/4.0/
- 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.
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