Estimation of parameters of nonlinear recurrent relations

Abstract

In this paper, estimates of the parameters of nonlinear recurrent sequences are constructed from inaccurate observations. We are talking about the logistic growth model, the Rikker model and the discretized Lorentz model. In the Lorentz model, the differential problem was reduced to a finite-difference scheme. Additive and multiplicative models of introducing errors into observations are considered, while the error distributions do not always have a normal distribution law. The main idea of the work is to represent the parameters of models through the time averages of certain functions and to estimate these averages from observations. The questions of the existence of time averages of some functions of model variables are the subject of the theory of dynamical systems. They are determined by the presence of limit cycles or limit distributions of a dynamical system. Since dynamic systems are observed against the background of random errors, the estimated parameters are expressed in terms of the trajectory averages and in terms of the variance of the observation errors. An important step in this work was the proof of the convergence in probability of the estimated parameters of a deterministic system to exact values. This procedure is based on classical probabilistic inequalities of the Chebyshev inequality type. The obtained results are verified in the course of computational experiments, in which polygons of the frequencies of the estimated parameters are constructed and compared with their exact values. The models of dynamical systems considered in this paper are nonlinear, which makes it difficult to use the least squares method to estimate their parameters. The results of a numerical experiment known to the authors for such an estimation of parameters contain rather large errors. Whereas the method proposed in this paper for estimating the parameters of nonlinear dynamical systems is analytically rigorous.