Parameter estimation of ODE’s via nonparametric estimators
Nicolas J-B. Brunel
Electronic Journal of Statistics
Ordinary differential equations (ODE’s) are widespreadmodels
in physics, chemistry and biology. In particular, this mathematical formal-
ism 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 parame-
ters indexing ODE’s from times series. Our method is able to alleviate the
computational difficulties encountered by the classical parametricmethods.
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
root n-rate for parametric estimators. Some perspectives of refinements of this
new family of parametric estimators are given.