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

Abstract

In this paper, we study application of Le Cam's one‐step method to parameter estimation in ordinary differential equation models. This computationally simple technique can serve as an alternative to numerical evaluation of the popular non‐linear least squares estimator, which typically requires the use of a multistep iterative algorithm and repetitive numerical integration of the ordinary differential equation system. The one‐step method starts from a preliminary n‐consistent estimator of the parameter of interest and next turns it into an asymptotic (as the sample size n→∞) equivalent of the least squares estimator through a numerically straightforward procedure. We demonstrate performance of the one‐step estimator via extensive simulations and real data examples. The method enables the researcher to obtain both point and interval estimates. The preliminary n‐consistent estimator that we use depends on non‐parametric smoothing, and we provide a data‐driven methodology for choosing its tuning parameter and support it by theory. An easy implementation scheme of the one‐step method for practical use is pointed out.

simode: R Package for Statistical Inference of Ordinary Differential Equations using Separable Integral-Matching

“…The statistical methodologies applied in the package are based on recent publications that study theoretical and applied aspects of smoothing methods in the context of ordinary differential equations (Dattner, 2015;Dattner & Gugushvili, 2018;Dattner & Klaassen, 2015;Dattner et al, 2017;. In that sense simode is close in spirit to the CollocInfer R package of (Hooker, Ramsay, & Xiao, 2015) and the episode R package of (Mikkelsen & Hansen, 2017).…”

Differential equations in data analysis

“…The authors established a Bernstein–von Mises theorem for the asymptotic behavior of the posterior distribution of the parameters, so that credible intervals have an asymptotically correct frequentist coverage. Dattner and Gugushvili (2018) applied Le Cam's one‐step method (Van der Vaart, 2000) to parameter estimation in ODE models. The one‐step method starts with a preliminary ‐consistent estimator of the parameter vector θ , and then modifies it to yield an efficient estimator.…”

Incremental Parameter Estimation under Rank-Deficient Measurement Conditions

“…To explore the applicability of the novel incremental parameter estimation method with realistic laboratory data, we execute parameter estimation for the α-pinene batch experiment first reported in [11] and later studied in [12][13][14][15][16]. The reaction network consists of 5 species.…”