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-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. [34]). 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 √ n− rate when regression splines are used in the first step.

Optimal control for estimation in partially observed elliptic and hypoelliptic linear stochastic differential equations

“…We can cite the pioneer work of Martin et al (2001) for nonparametric estimation of B-splines. Parametric approaches have been proposed more recently by Brunel and Clairon (2015), Clairon and Brunel (2018a, b) and Iolov et al (2017). One way of using the optimal control theory is to write an estimation criterion, typically a likelihood, as a tracking problem.…”

“…This idea has already been successfully developed by Brunel and Clairon (2015) to estimate the parameters of ordinary differential equations. It proves to be numerically efficient and stable, especially when the problem is ill-conditioned.…”

Application of one‐step method to parameter estimation in ODE models

“…For each state of the system, the data include only eight observations in time. This is a challenging problem to deal with, a point raised also in Tjoa and Biegler (), Rodriguez‐Fernandez et al (2006 Nov, ) and Brunel and Clairon (). In Table , we see the resulting point and interval estimates based on the real data, using the one‐step method.…”

“…The parameter estimates that we obtained are similar to those in Tjoa and Biegler (1991), except for parameters θ 4 , θ 5 : The estimates computed in Tjoa and Biegler (1991) are not contained in our confidence intervals. As explained in detail in Brunel and Clairon (2015), these two parameters are the most difficult to estimate, and those authors also raise a question whether the values obtained in Tjoa and Biegler (1991) are reliable and speculate the estimates in their own work could be in fact more accurate. Without offering a resolution of this difficult question, here we simply remark that alternative estimates computed in Brunel and Clairon (2015) are contained in our confidence intervals.…”

“…As explained in detail in Brunel and Clairon (2015), these two parameters are the most difficult to estimate, and those authors also raise a question whether the values obtained in Tjoa and Biegler (1991) are reliable and speculate the estimates in their own work could be in fact more accurate. Without offering a resolution of this difficult question, here we simply remark that alternative estimates computed in Brunel and Clairon (2015) are contained in our confidence intervals. Next, we conducted two simulation studies, corresponding to two different measurement error variances.…”