Parameter estimation strategies for separable grey system models with comparisons and applications

Abstract

Parameter estimation is a vital part of grey system models for dynamical modelling and forecasting time series. In this work, we induce a separable grey system model which covers both linear and nonlinear grey system models with separable structural parameters and propose three least squares-based strategies to estimate structural parameters and initial condition, namely two-step least squares, nonlinear least squares and separable nonlinear least squares. By using matrix computation tricks, the relationship between two-step least squares and separable nonlinear least squares estimates are quantified. Furthermore, all three strategies are comprehensively compared in terms of basic ideas, optimality, and computational efficiency. The numerical results indicate that nonlinear least squares outperforms the other two strategies, especially in the settings with large time intervals and high noise levels. Finally, we present two real applications aimed at forecasting the failure times of products and traffic flows during a given period. The results show that in each individual application, all three strategies generate reasonable estimates and the corresponding separable grey system models outperform the competitive ones.