Abstract: Accumulation generation is one of the key steps in GM(1,1) modeling, which has an important impact on model accuracy. The damping accumulated generation is a new type of accumulated generation method based on the principle of “new information priority”, and the established damping accumulated GM(1,1) model (DAGM(1,1) model) can adjust the exponential growth rate of the predicted values freely. The damping accumulated generation is equivalent to the function transformation method with parametric variables in the way the data are processed, but the range of values of parameters is different. Then which parameter value range is more reasonable? This paper will carry out related research. In addition, no study has focused on whether damping accumulation or the function transformation method with parametric variables can realize the unbiased prediction of white exponential series, which is very important for the effective improvement of the fitting and prediction accuracy of the GM(1,1) model.
In this paper, the DAGM(1,1) model is taken as the research object. The damping accumulation is regarded as a function transformation method. The effect of the damping accumulation on the simulation accuracy is analyzed in terms of background value error and restoring error, and it is proved that the DAGM(1,1) model has the asymptotic unbiasedness, and can achieve unbiased prediction of white exponential sequence within a negligible range of error. For pure exponential series, this paper expands the range of values of damping coefficient. For the existence of a large number of approximate exponential series in the social and economic data, they are not necessarily strictly increasing series, especially for low-growth approximate exponential series, and affected by data fluctuations, the value of the damping coefficient cannot be taken with reference to the pure exponential series. At this time, the role of the damping coefficient is to regulate the growth rate of the simulated series in order to obtain the best fitting accuracy. Pattern search method, genetic algorithm or particle swarm optimization algorithm and other optimization algorithms should be used to obtain the optimal damping coefficient. In this paper, Matlab programming combined with its optimization toolbox is used to solve the optimal damping coefficients. The results of example applications show that after widening the range of the damping coefficient, it can not only effectively reduce the influence of the background value error on the simulation accuracy, but also adjust the growth rate of the simulated sequences in order to obtain the best simulation accuracy, and the simulation and prediction accuracy of the DAGM(1,1) model is higher than that of the GM(1,1) model and the discrete GM(1,1) model, and is comparable to that of the damping accumulated discrete GM(1,1) model.
Similar to the GM(1,1) model, the DAGM(1,1) model is only applicable to approximate homogeneous exponential series, and the DAGM(1,1) model may fail to predict approximate non-homogeneous exponential series that are widely available in socio and economic data. In order to retain the advantages of damped accumulation while expanding the suitable modeling series to non-homogeneous exponential series, this paper combines damped accumulation with translational transformation to construct the DANGM(1,1) model applicable to approximate non-homogeneous exponential series. The example application results show that the simulation and prediction accuracy of the DANGM(1,1) model is higher than that of the NDGM model and the ONGM(1,1,k,c) model, which verifies the validity of the proposed model.
There exists a class of fluctuation series with seasonal characteristics in social and economic data, such as quarterly GDP, quarterly electricity consumption and so on. For this kind of seasonal fluctuation series, the DAGM(1,1) model is more likely to fail in prediction, and then the seasonal GM(1,1) model (SGM(1,1) model) applicable to seasonal fluctuation series can be adopted. However, the SGM(1,1) model retains the background value construction of the GM(1,1) model, which still affects the fitting accuracy, and for this reason, this paper combines the damped accumulation with the SGM(1,1) model and constructs the damped accumulated SGM(1,1) model (DASGM(1,1) model). The results of the example application show that the simulation and prediction accuracy of the DASGM(1,1) model is higher than that of the SGM(1,1) model and the GM(1,1,T) model, which verifies the validity of the proposed model.

Key words: GM(1,1) model, damping accumulation, asymptotic unbiasedness, function transformation, suitable series for modeling

摘要： 以阻尼累加GM(1,1)模型(DAGM(1,1)模型)为研究对象,将阻尼累加视作一种函数变换方法,从背景值误差和还原误差的角度分析了阻尼累加对模拟精度的影响,证明了DAGM(1,1)模型具有逼近白指数序列预测无偏性,拓展了阻尼系数的取值范围,确定了阻尼系数的取值方法。实例应用结果表明DAGM(1,1)模型的模拟和预测精度高于GM(1,1)模型和离散GM(1,1)模型,与阻尼累加离散GM(1,1)模型相当。为将适宜建模序列拓展至近似非齐次指数序列和季节波动序列,同时保留阻尼系数可以降低背景值误差和调节模拟序列增长率的优势,将阻尼累加与平移变换相结合构建了DANGM(1,1)模型,将阻尼累加与季节性GM(1,1)模型(SGM(1,1)模型)相结合构建了DASGM(1,1)模型。实例应用结果表明DANGM(1,1)模型的模拟和预测精度高于NDGM模型和ONGM(1,1,k,c)模型,DASGM(1,1)模型的模拟和预测精度高于SGM(1,1)模型和GM(1,1,T)模型,验证了两种模型的有效性。

关键词: GM(1,1)模型, 阻尼累加, 逼近无偏性, 函数变换, 适宜建模序列