For the classical GM(1,1) model, the prediction accuracy is not high, and the optimization of the initial and background values is one-sided. In this paper, the Lagrange mean value theorem is used to construct the background value as a variable related to k. At the same time, the initial value is set as a variable, and the corresponding optimal parameter and the time response formula are determined according to the minimum value of mean relative error (MRE). Combined with the domestic natural gas annual consumption data, the classical model and the improved GM(1,1) model are applied to the calculation and error comparison respectively. It proves that the improved model is better than any other models.

中文翻译：

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基于多变量动态优化的GM(1,1)模型的改进与应用

对于经典的GM(1,1)模型，预测精度不高，对初始值和背景值的优化是片面的。本文利用拉格朗日均值定理将背景值构造为与k相关的变量。同时，将初始值设为变量，根据平均相对误差（MRE）的最小值确定相应的最优参数和时间响应公式。结合国内天然气年消费数据，分别采用经典模型和改进的GM(1,1)模型进行计算和误差比较。证明改进后的模型优于任何其他模型。