On unified framework for nonlinear grey system models: an integro-differential equation perspective

TL;DR

This paper introduces a unified framework for nonlinear grey system models using an integro-differential equation approach, enabling better understanding, estimation, and improved forecasting accuracy in time series analysis.

Contribution

It reconstructs diverse grey models into a unified integro-differential form, facilitating simultaneous parameter estimation and demonstrating superior performance through simulations and real-world applications.

Findings

Unified model encompasses various nonlinear grey models.

Integral matching improves parameter estimation accuracy.

Model shows robustness and high accuracy in simulations and applications.

Abstract

Nonlinear grey system models, serving to time series forecasting, are extensively used in diverse areas of science and engineering. However, most research concerns improving classical models and developing novel models, relatively limited attention has been paid to the relationship among diverse models and the modelling mechanism. The current paper proposes a unified framework and reconstructs the unified model from an integro-differential equation perspective. First, we propose a methodological framework that subsumes various nonlinear grey system models as special cases, providing a cumulative sum series-orientated modelling paradigm. Then, by introducing an integral operator, the unified model is reduced to an equivalent integro-differential equation; on this basis, the structural parameters and initial value are estimated simultaneously via the integral matching approach. The modelling procedure comparison further indicates that the integral matching-based integro-differential equation provides a direct modelling paradigm. Next, large-scale Monte Carlo simulations are conducted to compare the finite sample performance, and the results show that the reduced model has higher accuracy and robustness to noise. Applications of forecasting the municipal sewage discharge and water consumption in the Yangtze River Delta of China further illustrate the effectiveness of the reconstructed nonlinear grey models.