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

TL;DR

This paper unifies continuous-time grey models into a common framework using integral matching, simplifying their structure and enabling simultaneous parameter and initial value estimation, thus enhancing understanding and application.

Contribution

It introduces a unified form of grey models, reconstructs them via integral matching, and compares their performance through simulations and real-world data.

Findings

Unified grey models into a common framework

Integral matching effectively estimates parameters and initial values

Reconstructed models perform well in simulations and real-world applications

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 models is proposed and simplified into a reduced-order ordinary differential equation. Then, the integral matching that consists of integral transformation 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 transformation formula. 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 simulation studies, and a real-world example shows how to apply and further verify the reconstructed model.