Regularized Semiparametric Estimation for Ordinary Differential Equations

Abstract

Ordinary differential equations (ODEs) are widely used in modeling dynamic systems and have ample applications in the fields of physics, engineering, economics, and biological sciences. The ODE parameters often possess physiological meanings and can help scientists gain better understanding of the system. One key interest is thus to well estimate these parameters. Ideally, constant parameters are preferred due to their easy interpretation. In reality, however, constant parameters can be too restrictive such that even after incorporating error terms, there could still be unknown sources of disturbance that lead to poor agreement between observed data and the estimated ODE system. In this article, we address this issue and accommodate short-term interferences by allowing parameters to vary with time. We propose a new regularized estimation procedure on the time-varying parameters of an ODE system so that these parameters could change with time during transitions but remain constants within stable stages. We found, through simulation studies, that the proposed method performs well and tends to have less variation in comparison to the nonregularized approach. On the theoretical front, we derive finite-sample estimation error bounds for the proposed method. Applications of the proposed method to modeling the hare–lynx relationship and the measles incidence dynamic in Ontario, Canada lead to satisfactory and meaningful results. Supplementary materials for this article are available online.

• B-spline
• Nonparametric
• Penalized estimation