Xie Wanli, Wu Wen-Ze, Liu Chong, Goh Mark
Institute of EduInfo Science and Engineering, Nanjing Normal University, Nanjing 210097, China.
School of Economics and Business Administration, Central China Normal University, Wuhan 430079, China.
ISA Trans. 2022 Jul;126:36-46. doi: 10.1016/j.isatra.2021.07.037. Epub 2021 Jul 29.
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.
近年来，基于分数阶累加和/或导数的灰色模型引起了相当大的研究兴趣，因为它们在处理具有不确定性的有限样本方面比整数阶灰色模型具有更好的性能；然而，仍有改进的空间。本文通过纳入符合记忆效应的广义分数阶导数（GFOD），考虑了分数阶灰色模型更灵活、更通用的结构，从而开发了广义分数阶灰色模型（记为GFGM(1,1)）。具体而言，我们全面分析了所提出的GFGM(1,1)模型的建模机制，包括模型参数估计和时间响应函数推导，并讨论了所提出方法与现有特殊情况之间的联系。然后，为了进一步提高所提出方法的有效性，采用了四种主流的元启发式算法来确定分数阶累加和导数的阶数。最后，我们进行了一系列的仿真研究和一个实际应用案例，以证明我们方法的适用性和优势。数值结果表明，GFGM(1,1)优于其他基准模型，并且从数值实验中获得了一些重要的见解。
