A family of toroidal diffusions with exact likelihood inference

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 prespecified 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 forgroups 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.