Parameter estimation for a chaotic dynamical system with partial observations

• Published in: Journal of inverse and ill-posed problems Vol. 29; no. 4; pp. 515 - 524
• Main Authors: Zhang, Jiantang, Huang, Sixun, Cheng, Jin
• Format: Journal Article
• Language: English
• Published: Berlin Walter de Gruyter GmbH 01.08.2021
• Subjects: 34K29; 65D10; 90C31; Approximation; Chaos theory; chaotic system; Dependent variables; derivative reconstruction; Dynamical systems; Initial conditions; Least squares method; Lorenz equations; Lorenz model; Numerical differentiation; Parameter estimation; Preprocessing; random noise
• ISSN: 0928-0219; 1569-3945
• DOI: 10.1515/jiip-2021-0030

Parameter estimation in chaotic dynamical systems is an important and practical issue. Nevertheless, the high-dimensionality and the sensitive dependence on initial conditions typically makes the problem difficult to solve. In this paper, we propose an innovative parameter estimation approach, utilizing numerical differentiation for observation data preprocessing. Given plenty of noisy observations on a portion of dependent variables, numerical differentiation allows them and their derivatives to be accurately approximated. Substituting those approximations into the original system can effectively simplify the parameter estimation problem. As an example, we consider parameter estimation in the well-known Lorenz model given partial noisy observations. According to the Lorenz equations, the estimated parameters can be simply given by least squares regression using the approximated functions provided by data preprocessing. Numerical examples show the effectiveness and accuracy of our method. We also prove the uniqueness and stability of the solution.