Generalized Tikhonov regularization in estimation of ordinary differential equations models

Abstract

We consider estimation of parameters in models defined by systems of ordinary differential equations (ODEs). This problem is important because many processes in different fields of science are modelled by systems of ODEs. Various estimation methods based on smoothing have been suggested to bypass numerical integration of the ODE system. In this paper, we do not propose another method based on smoothing but show how some of the existing ones can be brought together under one unifying framework. The framework is based on generalized Tikhonov regularization and extremum estimation. We define an approximation of the ODE solution by viewing the system of ODEs as an operator equation and exploiting the connection with regularization theory. Combining the introduced regularized solution with an extremum criterion function provides a general framework for estimating parameters in ODEs, which can handle partially observed systems. If the extremum criterion function is the negative log‐likelihood, then suitable regularized solutions yield estimators that are consistent and asymptotically efficient. The well‐known generalized profiling procedure fits into the proposed framework. Copyright © 2016 John Wiley & Sons, Ltd.