Moment estimations for parameters in uncertain delay differential equations

Article type: Research Article

Authors: Liu, Zhe^a | Jia, Lifen^{b; *}

Affiliations: [a] Department of Mathematical Sciences, Tsinghua University, Beijing, China | [b] School of Management and Engineering, Capital University of Economics and Business, Beijing, China

Correspondence: [*] Corresponding author. Lifen Jia, School of Management and Engineering, Capital University of Economics and Business, Beijing, China. E-mail: jialifen@cueb.edu.cn.

Abstract: As a type of differential equations driven by Liu process, uncertain delay differential equations (UDDEs) model dynamic systems with after-effects or memories in uncertain environment by incorporating time delay terms. Because it is natural for UDDEs to incorporate some unknown parameters, how to estimate them is a crucial problem in practice. This paper undertakes this issue by applying the method of moments based on discrete observations of solutions. With the Euler difference form of UDDEs, a function with respect to unknown parameters is proved to follow a standard normal uncertainty distribution. The moment estimations for unknown parameters are obtained by solving a system of equations which uses sample moments to approximate population moments. Analytic solutions for some types of UDDEs are derived. Numerical examples show that estimations give small biases and standard deviations as long as time steps are not too large. Applications to population growth models further illustrate the practicability of our method.

Keywords: Uncertain differential equation, parameter estimation, moments method, uncertainty theory

DOI: 10.3233/JIFS-191751

Journal: Journal of Intelligent & Fuzzy Systems, vol. 39, no. 1, pp. 841-849, 2020