Asymptotic efficiency and finite-sample properties of the generalized profiling estimation of parameters in ordinary differential equations

By Xin Qi, Hongyu Zhao. 2009

72 citations recorded by the source corpus.

Ordinary differential equations (ODEs) are commonly used to model dynamic behavior of a system. Because many parameters are unknown and have to be estimated from the observed data, there is growing interest in statistics to develop efficient estimation procedures for these parameters. Among the proposed methods in the literature, the generalized profiling estimation method developed by Ramsay and colleagues is particularly promising for its computational efficiency and good performance. In this approach, the ODE solution is approximated with a linear combination of basis functions. The coefficients of the basis functions are estimated by a penalized smoothing procedure with an ODE-defined penalty. However, the statistical properties of this procedure are not known. In this paper, we first give an upper bound on the uniform norm of the difference between the true solutions and their approximations. Then we use this bound to prove the consistency and asymptotic normality of this estimation procedure. We show that the asymptotic covariance matrix is the same as that of the maximum likelihood estimation. Therefore, this procedure is asymptotically efficient. For a fixed sample and fixed basis functions, we study the limiting behavior of the approximation when the smoothing parameter tends to infinity. We propose an algorithm to choose the smoothing parameters and a method to compute the deviation of the spline approximation from solution without solving the ODES.

Subfields: Artificial Intelligence, Control and Systems Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8c6147400f93051bd75db4b8fd0d4d41db5aa859

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In Artificial Intelligence

In Control and Systems Engineering

In Statistics and Probability

Citations in the corpus

Statistical inference for dynamical systems: A review (2012, background): “For statements regarding asymptotic normality of some estimation schemes in some ODE settings, see [138, 140, 174].”
√n-consistent parameter estimation for systems of ordinary differential equations : bypassing numerical integration via smoothing (2010, unknown): “For information on its asymptotic properties we refer to Qi and Zhao (2010).”
Sparse Additive Ordinary Differential Equations for Dynamic Gene Regulatory Network Modeling (2014, methodology): “There exist several classes of parameter estimation methods for ODE models, which include the nonlinear least-square (NLS) 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 2008”
Network Reconstruction From High-Dimensional Ordinary Differential Equations (2016, methodology): “√ n-consistency of the limiting estimator was later established by Qi and Zhao (2010). Zhang et al.”
Optimal rate of direct estimators in systems of ordinary differential equations linear in functions of the parameters (2013, methodology): “, [38, 37] and [20]), especially if one deals with a large system and if the underlying system has “fast” and “slow” components.”