残差非等间距GM(1,1)模型及应用

Residual unequal GM(1,1) Model and its Application

为了提高灰色系统的预测精度,人们从理论与实践中不断探索新的建模方法,从模型自身特性出发改进灰色模型。通过对传统的GM(1,1)模型进行差分运算,推导出非等间距GM(1,1)模型,扩展了模型的应用范围;从背景值的几何意义出发,指出背景值的积分构造形式比均值生成的形式更加合理,并给出了积分背景值的数学表达式;在一次拟合结果的基础上,为了进一步提高模型的精度,对一次拟合的结果进行了残差分析,并建立了灰色残差模型,通过把灰色残差模型得到的数据依次补偿到一次拟合结果上,得到了残差修正的灰色GM(1,1)模型。通过对热处理实验中渗碳浓度的数据处理结果显示,残差GM(1,1)模型较好的描叙了渗碳浓度和深度的关系。该模型对工程数据的处理具有一定的参考价值和指导意义,为灰色模型的应用提供了一个有效的方法。

In order to improve the accuracy of the grey system people research the establishment method of the grey model and improve the grey model from its model character. The paper adopts the difference method on the traditional GM（1,1） model and gets the unequal GM（1,1） model which expands the application of the GM（1,1） model; From the geometry point the integral construction form of the background values is more reasonable than the average form and it gives the mathematic form of the background values; In order to improve the accuracy on the basis of the first fitting results it puts the residual analysis on the fitting data and establish the grey residual model. Through compensating the residual sequence to the first fitting result it can get the residual GM（1,1） model. By processing the data of carbon concentration in the heat process experiment it shows the residual GM（1,1） model can describe the relation of the carton concentration and depth. The model is of great value for reference and guidance to the data processing in the engineering experiment and it provides an effective way to the application of the grey model.