A Generalized Smoother for Linear Ordinary Differential Equations

Ordinary differential equations (ODEs) are equalities involving a function and its derivatives that define the evolution of the function over a prespecified domain. The applications of ODEs range from simulation and prediction to control and diagnosis in diverse fields such as engineering, physics, medicine, and finance. Parameter estimation is often required to calibrate these theoretical models to data. While there are many methods for estimating ODE parameters from partially observed data, they are invariably subject to several problems including high computational cost, complex estimation procedures, biased estimates, and large sampling variance. We propose a method that overcomes these issues and produces estimates of the ODE parameters that have less bias, a smaller sampling variance, and a 10-fold improvement in computational efficiency. The package <i>GenPen</i> containing the Matlab code to perform the methods described in this article is available online.

常微分方程（Ordinary Differential Equations，ODEs）是一类包含某一函数及其各阶导数的等式，用于刻画该函数在预先指定的定义域内的演化规律。ODEs的应用场景广泛，覆盖工程学、物理学、医学、金融学等诸多领域，可用于模拟、预测、控制与诊断等任务。为将这类理论模型适配实测数据，通常需要开展参数估计工作。尽管目前已有诸多基于部分观测数据估计ODE参数的方法，但这类方法普遍存在若干局限：计算成本高昂、估计流程复杂、估计结果存在偏倚，且抽样方差较大。本文提出一种可解决上述问题的参数估计方法，其得到的ODE参数估计结果偏倚更低、抽样方差更小，且计算效率提升可达10倍。可在线获取包含本文所述方法对应Matlab代码的GenPen工具包。