Statistics and Computing

, Volume 29, Issue 1, pp 1–22 | Cite as

Langevin diffusions on the torus: estimation and applications

  • Eduardo García-PortuguésEmail author
  • Michael Sørensen
  • Kanti V. Mardia
  • Thomas Hamelryck


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.


Circular data Directional statistics Likelihood Protein structure Stochastic Differential Equation Wrapped normal 

Mathematics Subject Classification

60J60 62M05 62H11 



We acknowledge the insightful discussions with John Kent, Jotun Hein, and Michael Golden that led to the key motivation for the manuscript. We are grateful to Sandro Bottaro for the providing the molecular dynamics data used in the illustration. We acknowledge the valuable comments and remarks provided by two anonymous referees and an Associate Editor, which significantly improved the manuscript.


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Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark
  2. 2.Department of StatisticsCarlos III University of MadridMadridSpain
  3. 3.UC3M-BS Institute of Financial Big DataCarlos III University of MadridMadridSpain
  4. 4.Department of StatisticsUniversity of LeedsLeedsUK
  5. 5.Department of StatisticsUniversity of OxfordOxfordUK
  6. 6.Bioinformatics Centre, Section for Computational and RNA Biology, Department of BiologyUniversity of CopenhagenCopenhagenDenmark
  7. 7.Image Section, Department of Computer ScienceUniversity of CopenhagenCopenhagenDenmark

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