分数阶累加的离散GM(2,1)模型与应用

A Discrete GM(2,1) Model with Fractional-Order Accumulation and Its Application

【目的】为解决传统二阶单变量灰色预测模型GM(2,1)对某些序列存在预测精度不高的问题。【方法】在GM(2,1)的基础上,引入分数阶累加算子并利用离散灰色模型的建模原理,提出了一种基于分数阶累加的离散GM(2,1)模型(FDGM(2,1))。【结果】给出了FDGM(2,1)模型可用于预测的时间响应式和还原式。为达到最优的预测性能,建立了以平均相对模拟误差最小为目标的优化模型,并给出了确定模型最优阶数的方法。最后用一个数值模拟实验和一个应用实例对新模型进行检验。【结论】结果表明,新模型比其他常用模型具有更高的模拟预测精度,是有效且实用的。

[Purposes]The traditional second-order single-variable grey prediction model(GM(2,1))is studied to solve the problem that the prediction accuracy for some sequences is not high.[Methods]A discrete GM(2,1)model with fractional-order accumulation termed as FDGM(2,1)is proposed on the basis of GM(2,1)by combining the modeling principle of discrete grey model and the idea of fractional-order accumulation.[Findings]The time-response and reduction formulas of FDGM(2,1)for prediction are given.To achieve the optimal prediction performance,an optimization model with the objective of minimizing the average relative simulation error is established,and a method to determine the optimal order of the model is given.Finally,a numerical simulation experiment and an application case are used to verify the new model.[Conclusions]The results show that the new model exhibits the higher simulation and prediction accuracy than the other common models,and it is effective and practical.