标题： 微分方程模型的松弛法推断 via 动力系统 显示英文标题

标题： Inference for Differential Equation Models using Relaxation via Dynamical Systems

摘要： 以常微分方程（ODE）表示均值函数的统计回归模型可用于描述自然界中动态现象，在生物学、气候学和遗传学等领域中普遍存在。 基于ODE模型的参数估计对于理解其动力学至关重要，但由于ODE缺乏解析解，这使得参数估计变得具有挑战性。 本文旨在通过松弛ODE系统来提出一个通用且快速的基于ODE模型的统计推断框架。 松弛是通过适当选择数值方法（如龙格-库塔法）以及引入具有小方差的加性高斯噪声实现的。 因此，可以应用滤波方法来获得贝叶斯框架下参数的后验分布。 所提出方法的主要优势在于计算速度。 在仿真研究中，该方法比其他方法至少快14倍。 给出了保证近似动力系统后验分布收敛到真实模型后验分布的理论结果。 提供了显式表达式，将龙格-库塔方法的阶数和网格大小与近似后验分布关于样本量的收敛率联系起来。 显示英文摘要

摘要： Statistical regression models whose mean functions are represented by ordinary differential equations (ODEs) can be used to describe phenomenons dynamical in nature, which are abundant in areas such as biology, climatology and genetics. The estimation of parameters of ODE based models is essential for understanding its dynamics, but the lack of an analytical solution of the ODE makes the parameter estimation challenging. The aim of this paper is to propose a general and fast framework of statistical inference for ODE based models by relaxation of the underlying ODE system. Relaxation is achieved by a properly chosen numerical procedure, such as the Runge-Kutta, and by introducing additive Gaussian noises with small variances. Consequently, filtering methods can be applied to obtain the posterior distribution of the parameters in the Bayesian framework. The main advantage of the proposed method is computation speed. In a simulation study, the proposed method was at least 14 times faster than the other methods. Theoretical results which guarantee the convergence of the posterior of the approximated dynamical system to the posterior of true model are presented. Explicit expressions are given that relate the order and the mesh size of the Runge-Kutta procedure to the rate of convergence of the approximated posterior as a function of sample size.