  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.
