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Langevin diffusions on the torus: estimation and applications

机译：兰辛在托伦上的扩散：估算和应用

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摘要

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.
机译：我们介绍了角度的角度延时的随机模型，并发展了他们的估计。我们专注于研究Langevin扩散与静止分布等于来自方向统计的众所周知的分布，因为这些扩散可以被视为ornstein-Uhlenbeck过程的环形类似物。它们的似然函数是过渡密度的乘积，没有分析表达，但这可以通过通过足够方案来求解Fokker-Planck方程来计算。我们提出了三种近似可能性，这些似乎是计算贸易的：（i）基于静止分布的可能性; （ii）欧拉和绍兴奥扎希伪可能性的环形调整; （iii）基于对包裹正常过程的过渡密度的特定近似的可能性。仿真研究比较了尺寸的一个和两个，对确切的近似过渡密度，并研究了近似似然的经验性能。最后，两个扩散用于在分子动力学模拟期间模拟蛋白G（PDB标识符1GB1）的骨干角的演变。软件包Sdetorus实现了纸张中呈现的估计方法和应用程序。

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