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.
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31 October 2023
A Correction to this paper has been published: https://doi.org/10.1007/s11222-023-10316-1
Notes
Note that \(-(x_2-x_1\text {e}^{-(t_2-t_1)}+2k\pi \text {e}^{-t_2})^2\) should be in the exponential’s denominator of Liu (2013)’s (15) and (16).
Note the similar argument given in Roberts and Stramer (2002), albeit in their equation (24) the covariance matrix is not symmetric, probably because of a typo in (25), which should have been \((J(x)a_{x,h})'=J(x)a_{x,h}\).
In Shoji and Ozaki (1998) the drift approximation is done by Itô’s formula. To obtain a simpler pseudo-likelihood, we use a local linear approximation of b as in Ozaki (1985) (for the case \(p=1\)). Without this extra simplification, the expectation becomes \(\tilde{E}_\varDelta ({\varvec{\varphi }})=E_\varDelta ({\varvec{\varphi }})+J({\varvec{\varphi }})^{-2}(\exp \{J({\varvec{\varphi }})\varDelta \}-{\mathbf {I}}-J({\varvec{\varphi }})\varDelta )M({\varvec{\varphi }})\) with \(M({\varvec{\varphi }})=\frac{1}{2}\left( \mathrm {tr}\left[ {\mathbf {V}}({\varvec{\varphi }}){\mathbf {H}}_1({\varvec{\varphi }})\right] ,\ldots ,\mathrm {tr}\left[ {\mathbf {V}}({\varvec{\varphi }}){\mathbf {H}}_n({\varvec{\varphi }})\right] \right) '\) and \({\mathbf {H}}_i({\varvec{\varphi }})=\left( \tfrac{\partial ^2b_i({\varvec{\varphi }})}{\partial \phi _k\partial \phi _l}\right) _{1\le k,l\le p}\), \(i=1,\ldots ,p\).
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Acknowledgements
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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This work is part of the Dynamical Systems Interdisciplinary Network, University of Copenhagen. It was funded by the University of Copenhagen 2016 Excellence Programme for Interdisciplinary Research (UCPH2016-DSIN) and by Project MTM2016-76969-P from the Spanish Ministry of Economy, Industry and Competitiveness, and European Regional Development Fund (ERDF).
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García-Portugués, E., Sørensen, M., Mardia, K.V. et al. Langevin diffusions on the torus: estimation and applications. Stat Comput 29, 1–22 (2019). https://doi.org/10.1007/s11222-017-9790-2
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DOI: https://doi.org/10.1007/s11222-017-9790-2
Keywords
- Circular data
- Directional statistics
- Likelihood
- Protein structure
- Stochastic Differential Equation
- Wrapped normal