A Tracking Approach to Parameter Estimation in Linear Ordinary Differential Equations

TL;DR

This paper introduces a novel parameter estimation method for linear ordinary differential equations using optimal control theory, improving robustness and accuracy over classical approaches by avoiding derivative estimation and accounting for model discrepancies.

Contribution

The paper proposes a new estimator based on optimal control that enhances robustness to model misspecification and provides a detailed analysis for linear ODEs, with proven consistency and optimal convergence rates.

Findings

Estimator is more robust to model misspecification.

Achieves parametric root-n convergence rate with regression splines.

Provides a detailed qualitative analysis of model discrepancy.

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-posedness of the 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 estimator and Generalized Smoothing (introduced by Ramsay et al, 2007). We introduce a perturbed model and we use optimal control theory for constructing a criterion that aims at minimizing the discrepancy with data and the 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 cause of inaccuracy in Two-Step estimators. Moreover, we take into account model discrepancy and our estimator is more robust to model misspecification than classical methods. The discrepancy with the parametric ODE model correspond to the minimum perturbation (or control) to apply to the initial model. Its qualitative analysis can be informative for misspecification diagnosis. In the case of well-specified model, we show the consistency of our estimator and that we reach the parametric root-n rate when regression splines are used in the first step.