A tracking approach to parameter estimation in linear ordinary differential equations

Abstract

Ordinary Differential Equations are widespread tools to model chemical, physical, biological process but they usually rely on parameters which are of critical importance in terms of dynamic and need to be estimated directly from the data. Classical statistical approaches (nonlinear least squares, maximum likelihood estimator) can give unsatisfactory results because of computational difficulties and ill-posed statistical problem. New estimation methods that use some nonparametric devices have been proposed to circumvent these issues. We present a new estimator that shares properties with Two-Step estimators and Generalized Smoothing (introduced by Ramsay et al. [37]). Our estimation method relies on a relaxation and penalization scheme to regularize the inverse problem. We introduce a perturbed model and we use optimal control theory for constructing a criterion that aims at minimizing the discrepancy between data and the original model. Here, we focus on the case of linear Ordinary Differential Equations as our criterion has a closed-form expression that permits a detailed analysis. Our approach avoids the use of a nonparametric estimator of the derivative, which is one of the main causes of inaccuracy in Two-Step estimators. Regarding the theoretical asymptotic behavior of our estimator, we show its consistency and that we reach the parametric $\sqrt{n}$-rate when regression splines are used in the first step. We consider the estimation of two models possessing sloppy parameters, which usually makes the estimation of ODE models an ill-posed problem in applications [20, 41] and shows the efficiency of the Tracking estimator. Quite interestingly, our relaxation scheme makes the estimator robust to some kind of model misspecification, as shown in simulations.