# Langevin diffusions on the torus: estimation and applications

## TL;DR

This paper develops methods for estimating Langevin diffusions on the torus with applications to directional data and molecular dynamics, introducing computationally feasible likelihood approximations and demonstrating their effectiveness through simulations and real data.

## Contribution

It introduces novel likelihood approximation techniques for Langevin diffusions on the torus, enabling practical estimation and application to complex directional data.

## Key findings

- Approximate likelihoods perform well in simulations.
- Methods successfully model protein backbone angles.
- Software package sdetorus implements the proposed methods.

## Abstract

We introduce stochastic models for continuous-time evolution of angles and develop their estimation. We focus on studying Langevin diffusions with stationary distributions equal to well-known distributions from directional statistics, since such diffusions can be regarded as toroidal analogues of the Ornstein-Uhlenbeck process. Their likelihood function is a product of transition densities with no analytical expression, but that can be calculated by solving the Fokker-Planck equation numerically through adequate schemes. We propose three approximate likelihoods that are computationally tractable: (i) a likelihood based on the stationary distribution; (ii) toroidal adaptations of the Euler and Shoji-Ozaki pseudo-likelihoods; (iii) a likelihood based on a specific approximation to the transition density of the wrapped normal process. A simulation study compares, in dimensions one and two, the approximate transition densities to the exact ones, and investigates the empirical performance of the approximate likelihoods. Finally, two diffusions are used to model the evolution of the backbone angles of the protein G (PDB identifier 1GB1) during a molecular dynamics simulation. The software package sdetorus implements the estimation methods and applications presented in the paper.