Generalized fractional grey system models: The memory effects perspective

In recent years, grey models based on fractional-order accumulation and/or derivatives have attracted considerable research interest because they offer better performance in handling limited samples with uncertainty than integer-order grey models; however, there remains room for improvement. This paper considers a more flexible and general structure for the fractional grey model by incorporating a generalized fractional-order derivative (GFOD) that complies by memory effects, resulting in the development of a generalized fractional grey model (denoted as GFGM(1,1)). Specifically, we comprehensively analyse the modelling mechanism of the proposed GFGM(1,1) model, involving model parameter estimation and time response function derivation, and discuss the link between the proposed approach and existing special cases. Then, to further improve the efficacy of the proposed approach, four mainstream metaheuristic algorithms are employed to ascertain the orders of fractional accumulation and derivatives. Finally, we carry out a series of simulation studies and a real-world application case to demonstrate the applicability and advantage of the our approach. The numerical results show that GFGM(1,1) outperforms other benchmarks, and some significant insights are obtained from the numerical experiments.