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Langevin diffusions on the torus: estimation and applications

机译：Langevin在圆环上的扩散：估计和应用

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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：我们介绍了角度连续时间演变的随机模型，并发展了它们的估计。我们专注于研究具有恒定分布的Langevin扩散，这些分布等于来自方向统计的众所周知的分布，因为这样的扩散可以被视为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，没有解析表达式，但是可以通过适当的方案通过数值求解Fokker-Planck方程来计算。我们提出了三种在计算上易于处理的近似似然：（i）基于平稳分布的似然； （ii）Euler和Shoji-Ozaki伪似然的环形适应； （iii）基于对包裹的正常​​过程的过渡密度的特定近似的可能性。模拟研究在第一个维度和第二个维度上比较了近似的转换密度与精确的转换密度，并研究了近似可能性的经验性能。最后，在分子动力学模拟过程中，使用两个扩散对蛋白质G（PDB标识符1GB1）的骨架角的演化进行建模。软件包sdetorus实现了本文介绍的估算方法和应用。

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