Robust estimation of constant and time-varying parameters in nonlinear ordinary differential equation models

Abstract

Ordinary differential equation (ODE) models are quite popular for modelling complex dynamic processes in many scientific fields, and the parameters in these models are usually unknown, and we need to estimate them using statistical methods. When some observations are contaminated, regular estimation methods, such as nonlinear least-square estimation, will bring large bias. In this paper, robust estimations of both constant and time-varying parameters in ODE models using M-estimators are proposed, and their asymptotic properties are obtained under some mild conditions. We focus on Huber M-estimator, and also provide a method to adjust the Huber parameter automatically to the observations. The proposed method is compared to existing methods in numerical simulations and CD8+ T cell data analysis. It is demonstrated that our method gain substantial efficiency as well as robust properties.

Keywords:

• robust estimation
• ordinary differential equation
• time-varying parameters
• Huber parameter
• asymptotic properties

Acknowledgements

The authors wish to thank the Editor, Dr Iréne Gijbels, the Associate Editor and two reviewers for their many insightful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

Tao Hu's research was supported by the Natural Science Foundation of China [No.11201317], Doctoral Fund of Ministry of Education of China [20111108120002].