基于级差格式的GM(2,1)模型参数估计优化研究

参数估计的优化是提高灰色模型精度的一个重要途径,级差格式的提出避免了背景值的复杂构造.现有的GM(2,1)模型计算较为复杂,且参数估计基于目标函数是原始序列一次差分序列的拟合误差平方和最小化来确定,同时,参数估计中微分到差分的转换以及背景值构造存在较大误差.针对这些问题,本文基于GM(2,1)模型微分方程的时间响应函数推导了级差格式,给出了最小二乘法的参数估计方法,然后基于原始序列误差平方和最小的目标函数,优化了模型的两个初始条件,同时,推导出GM(1,1)回归模型和GM(1,1,exp)模型是该模型的特殊情况,最后通过实例比较本文优化方法与现有方法估计的GM(2,1)模型拟合精度与预测精度.实例结果显示,本文的优化方法估计的GM(2,1)模型具有较好的效果.

The optimization of GM(2,1) model based on parameter estimation of grade difference format

The optimization of parameter estimation is an important way to improve the accuracy of grey model, the grade difference format proposed avoids to constructing complex background value. The calculation of existing GM(2,1) models are complicated, and the parameter estimation is determined by the objective function which is the minimization of squares fitting error on a difference of the original sequence. At the same time, there is more error in parameter estimation when differential form converts to differential form and constructs background value. To solve these problems, we derive the grade difference format of the GM(2,1) model based on the time response function of differential equation, and give the parameter estimation method based on least squares method. Then based on the minimization of objective function on summing of the squared errors of original series we optimize the two initial conditions of the model. At the same time, we find that GM(1,1) regression model and the GM(1,1,exp) model are two forms of GM(2,1) model in special circumstances. Finally, through example analysis, we compare to the fitted and prediction accuracy of our model and the existing GM(2,1) models. The result shows that the optimization method we proposed is better.