Parameter estimation of ODE’s via nonparametric estimators

By Nicolas J.-B. Brunel. 2007

167 citations recorded by the source corpus.

Ordinary differential equations (ODE's) are widespread models in physics, chemistry and biology. In particular, this mathematical formalism is used for describing the evolution of complex systems and it might consist of high-dimensional sets of coupled nonlinear differential equations. In this setting, we propose a general method for estimating the parameters indexing ODE's from times series. Our method is able to alleviate the computational difficulties encountered by the classical parametric methods. These difficulties are due to the implicit definition of the model. We propose the use of a nonparametric estimator of regression functions as a first-step in the construction of an M-estimator, and we show the consistency of the derived estimator under general conditions. In the case of spline estimators, we prove asymptotic normality, and that the rate of convergence is the usual $\sqrt{n}$-rate for parametric estimators. Some perspectives of refinements of this new family of parametric estimators are given.

Subfields: Genetics, Statistics and Probability, Signal Processing

View Paper: https://www.semanticscholar.org/paper/7fb2e6311eac82e655ea7cf15ceafa1fd14ef655

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Citations in the corpus

√n-consistent parameter estimation for systems of ordinary differential equations : bypassing numerical integration via smoothing (2010, unknown): “Our exposition in the present paper is similar to that in Brunel (2008) to some degree, one of the diﬀerences being that instead of spline estimators we use kernel-type estimators for estimation of x θ and x ′ θ .”
Optimal rate of direct estimators in systems of ordinary differential equations linear in functions of the parameters (2013, background): “By estimating the “true” trajectories of the system and their derivatives, the two-step approach bypasses the need to integrate the system numerically and as a result, the parameter estimates can be computed extremely fast ([7, 30]).”
Learning Large Scale Ordinary Differential Equation Systems (2017, methodology): “Calderhead et al. (2009) demonstrated this in a Bayesian setting and proposed a collocation method based on Gaussian processes, Babtie et al.”
Bayesian Solution Uncertainty Quantification for Differential Equations (2013, methodology): “…model inputs, θ , given measurements of the model states (Bock, 1983; Ionides et al., 2006; Dowd, 2007; Ramsay et al., 2007; Xue et al., 2010; Brunel, 2008; Liang and Wu, 2008; Calderhead and Girolami, 2011; Campbell and Steele, 2011; Gugushvili and Klaassen, 2012; Campbell and Chkrebtii,…”
Parameter Estimation in Stochastic Differential Equations (2011, methodology): “, by proposing a totally new criterion for simultaneous estimation of both parameters [36], partially estimate of the parameters [9] or, lastly, Varziri’s estimation [45] with some modifications.”
Sparse Additive Ordinary Differential Equations for Dynamic Gene Regulatory Network Modeling (2014, methodology): “…method (Hemker 1972; Bard 1974; Li, Osborne, and Pravan 2005; Xue, Miao, and Wu 2010), the two-stage smoothing-based estimation method (Varah 1982; Brunel 2008; Chen and Wu 2008a,b; Liang and Wu 2008; Wu, Xue, and Kumar 2012), the principal differential analysis (PDA) and its extensions (Ramsay…”