Statistical Parameter Estimation for Ordinary Differential Equations
Title: Statistical Parameter Estimation for Ordinary Differential Equations
Chair: Professor Naisyin Wang, Associate Professor Ji Zhu
Cognate Member: Professor Peter X.K. Song
Member: Professor Kerby Shedden
Abstract: Ordinary differential equations (ODEs) are widely used in modeling a dynamic system and have ample applications in the fields of physics, engineering, economics and biological sciences. One fundamental statistical question is then how to estimate the ODE parameters. Ramsay et al. (2007) proposed a parameter cascading method that tries to strike a balance between the data and the ODE structure via a "loss + penalty" framework. It takes advantages of basis-function approximation and profile-based estimation, and it has become a very popular method for ODE parameter estimation. In the first part of the presentation, we investigate the parameter cascading method in detail and take an alternative view through variance evaluation on this method. We found, through both theoretical evaluation and finite sample numerical experiments, that the penalty term used in Ramsay et al. (2007) could unnecessarily increase estimation variation. Consequently, we propose a simpler alternative structure for parameter cascading that achieves the minimum variation. Further, contrary to the traditional belief established from the penalized spline literature, the "loss + penalty" setup in parameter cascading also makes the procedure more sensitive to the number of basis functions. We observe this phenomenon in numerical studies regardless whether the ODE structure is correctly specified. We provide theoretical explanations behind the observed phenomenon and report numerical findings on both simulated data sets and a lynx-hare predator-prey dynamic data set. In the second part of the presentation, we consider the estimation problem in which ODE parameters are allowed to vary with time. This is often necessary when there are unknown sources of disturbances that lead to deviations from the standard constant-parameter ODE system. On the other hand, to gain sufficient understanding of the system, one key interest is also to keep the structure of the parameters simple. Specifically, we propose a novel regularization method for estimating time-varying ODE parameters, in which the estimated parameters are allowed to change with time when there are disturbances to the system and are encouraged to remain constants within stable stages. Our numerical studies suggest that the proposed approach works better than competing methods. On the theoretical front, we provide finite-sample estimation error bounds under certain regularity conditions. The application of the proposed method to model the hare-lynx relationship and the measles incidence dynamic in Ontario, Canada lead to satisfactory and meaningful results.