Abstract: Over the past 40 years, through the unremitting efforts of several generations, the grey system theory has made vigorous development on the basis of original ideas and theoretical frameworks. The grey prediction model, as one of the core contents of grey system theory, has been widely applied to various fields of life. In order to avoid the jump error caused by the conversion from difference equation to differential equation in the traditional grey prediction model during the modeling process, and to better explore the inherent laws of the development and change of the data sequence, this paper combines the discrete grey modeling method and the direct modeling idea, and at the same time introduces a quadratic time term to construct an approximate non-homogeneous direct discrete grey prediction model with nonlinear time-varying parameters, called NDDGM(1,1,k,k²).
First, some relevant knowledge and theories required in this article are given, the reasons for the low accuracy of the traditional model are analyzed, and the basic form of the novel model is given. Then, the parameter estimates of the novel model are obtained based on the least squares method, and the predictive expression of the model is derived. Next, the applicable scope of the novel model is discussed, and the applicable scope of the traditional grey prediction model is expanded from approximate homogeneous exponential sequence to approximate non-homogeneous exponential sequence, exponential linear combination sequence, exponential parabolic combination sequence, and cubic curve sequence. In addition, the properties of the novel model are also studied, and it is theoretically proven that the novel model has whitening coincidence for exponential parabolic combination sequence and cubic curve sequence. Finally, through the numerical simulation of approximate non-homogeneous exponential sequence, exponential linear combination sequence, exponential parabolic combination sequence, and cubic curve sequence and prediction analysis of soft soil foundation settlement, the results show that the errors of the novel model are all minimal, thus the effectiveness and practicability of the novel model are verified.
Natural gas, as a clean energy, plays an important role in China's path to achieving the goals of “carbon peaking” and “carbon neutrality”, and there is also a lot of research on grey prediction models. To this end, the model in this article is applied to the prediction of total natural gas consumption in China's automobile manufacturing industry, and four representative models are selected for comparative analysis. The results show that the model in this paper minimizes the mean absolute percentage error, the absolute percentage error and the root mean square percentage error, and its simulation and prediction accuracies are significantly higher than those of the traditional model, which further reflects its superiority. It is an effective supplement to the existing grey prediction model and has certain theoretical and application value.

Key words: approximate non-homogeneous, time power terms, quadratic time-varying parameters, discrete grey model, direct modeling

摘要： 为避免传统灰色预测模型从差分到微分之间转换产生的跳跃误差,更好地挖掘数据序列发展变化的内在规律,结合离散灰色建模方法和直接建模思想,引入时间二次项,构建了一种含非线性时变参数的近似非齐次直接离散灰色模型NDDGM(1,1,k,k²)。将传统灰色模型的适用范围拓展到近似非齐次指数序列、指数线性组合型序列、指数抛物组合型序列、三次曲线型序列。从理论上证明了新模型对指数抛物组合型、三次曲线型序列具有白化重合性。最后通过五类不同特征序列的数值模拟和对软土地基沉降及我国汽车制造业天然气消耗总量的预测分析,结果表明:新模型显著提高了灰色模型的模拟和预测精度,验证了其有效性和实用性。

关键词: 近似非齐次, 时间幂次项, 二次时变参数, 离散灰色模型, 直接建模