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Multi-variable GMU(1,N) Grey Prediction Model Considering Unknown Factors
Ye Li, Yuanping Ding, Jianping Wang
The Journal of Grey System
2022, 34 (1):
17-33.
The multi-variable grey prediction model represented by the GM(1,N) is an important casual relationship forecasting model. However, the traditional GM(1,N) model shows some defects which affect the modeling accuracy and applicability. In this paper, the modeling process of the traditional GM(1,N) model is studied, and three defects are observed in terms of “modeling mechanism,” “modeling structure,” and “parameter estimation.” To address these defects, a novel multi-variable GMU(1,N) grey prediction model considering unknown factors is proposed by introducing an exponential function \beta e^(\alpha(k-1)) in this paper, the modeling assumption in the traditional GM(1,N) model that \sum b_i X_i (k) can be treated as a grey constant with a small variation range of X_i (1) (i=2,3,……,N) is dropped, and the derivation form of the GMU(1,N) model is defined, which solve these three defects in the traditional GM(1,N) model effectively. Meanwhile, the genetic algorithm toolbox and recursive method are used to solve the parameter \alpha and time response function, respectively. Additionally, it is theoretically proved that the GMU(1,N) model can be completely compatible with the GM(1,1) model, GM(1,1,e^at) model, GM(1,N) model, and GMC(1,n) model by adjusting the parameters’ values. The GMU(1,N) model is used to simulate and predict grain production in Henan province to verify the effectiveness of these improvements. The mean average simulated and predicted relative errors of the GMU(1,N) model are 0.000% and 0.811%, in comparison with the traditional GM(1,1) model and the GM(1,N) model, which are 1.234%, 1.487% and 8.105%, 8.874% respectively. Results show that the GMU(1,N) model has superior performance, which confirms the effectiveness of the model improvement.
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