A family of toroidal diffusions with exact likelihood inference

TL;DR

This paper introduces a new class of toroidal diffusion processes with explicit transition densities, enabling exact likelihood inference and applications to biological movement data and protein structure modeling.

Contribution

It provides a novel family of reversible diffusions on the torus with exact likelihood inference and simulation methods, applicable to multimodal stationary distributions.

Findings

Exact likelihood inference demonstrated on multivariate angular data.

Successful simulation of diffusion bridges in complex toroidal spaces.

Applications include testing homogeneity in animal movement and protein structure modeling.

Abstract

We provide a class of diffusion processes for continuous time-varying multivariate angular data with explicit transition probability densities, enabling exact likelihood inference. The presented diffusions are time-reversible and can be constructed for any pre-specified stationary distribution on the torus, including highly-multimodal mixtures. We give results on asymptotic likelihood theory allowing one-sample inference and tests of linear hypotheses for $k$ groups of diffusions, including homogeneity. We show that exact and direct diffusion bridge simulation is possible too. A class of circular jump processes with similar properties is also proposed. Several numerical experiments illustrate the methodology for the circular and two-dimensional torus cases. The new family of diffusions is applied (i) to test several homogeneity hypotheses on the movement of ants and (ii) to simulate bridges between the three-dimensional backbones of two related proteins.