Parametric Estimation of Ordinary Differential Equations with Orthogonality Conditions

TL;DR

This paper introduces an orthogonality condition-based estimator for noisy parametric ODEs, demonstrating its consistency, normality, and practical advantages over classical methods through simulations and real data applications.

Contribution

It proposes a novel gradient matching estimator using orthogonality conditions for parametric ODEs, including delay differential equations, with proven statistical properties.

Findings

Estimator is root-n consistent and asymptotically normal.

Provides confidence sets with a closed-form asymptotic variance.

Outperforms classical estimators in simulations and real data, including influenza modeling.

Abstract

Differential equations are commonly used to model dynamical deterministic systems in applications. When statistical parameter estimation is required to calibrate theoretical models to data, classical statistical estimators are often confronted to complex and potentially ill-posed optimization problem. As a consequence, alternative estimators to classical parametric estimators are needed for obtaining reliable estimates. We propose a gradient matching approach for the estimation of parametric Ordinary Differential Equations observed with noise. Starting from a nonparametric proxy of a true solution of the ODE, we build a parametric estimator based on a variational characterization of the solution. As a Generalized Moment Estimator, our estimator must satisfy a set of orthogonal conditions that are solved in the least squares sense. Despite the use of a nonparametric estimator, we prove the root-$n$ consistency and asymptotic normality of the Orthogonal Conditions estimator. We can derive confidence sets thanks to a closed-form expression for the asymptotic variance. Finally, the OC estimator is compared to classical estimators in several (simulated and real) experiments and ODE models in order to show its versatility and relevance with respect to classical Gradient Matching and Nonlinear Least Squares estimators. In particular, we show on a real dataset of influenza infection that the approach gives reliable estimates. Moreover, we show that our approach can deal directly with more elaborated models such as Delay Differential Equation (DDE).