On unified framework for continuous-time grey models: An integral matching perspective

Abstract

Since most of the research about grey forecasting models is focused on developing novel models and improving accuracy, relatively limited attention has been paid to the modelling mechanism and relationships among diverse kinds of models. This paper aims to unify and reconstruct continuous-time grey models, highlighting the differences and similarities among different models. First, the unified form of grey forecasting models is proposed and simplified into a reduced-order ordinary differential equation. Then, the integral matching that consists of integral operator and least squares, is proposed to estimate the structural parameter and initial value simultaneously. The cumulative sum operator, an essential element in grey modelling, proves to be the discrete approximation of the integral operator. Next, grey models are reconstructed by the integral matching-based ordinary differential equations. Finally, the existing grey models are compared with the reconstructed models through extensive simulations, and a real-world example shows how to apply and further verify the reconstructed model.