Abstract: As one of the main contents of grey system theory, grey prediction is widely used to predict short-time series data because it only needs a small sample and poor information to perform well. Among them, the GM(1,1)model is the core and foundation of grey prediction theory. Scholars have carried out systematic research on the GM(1,1) model to improve its prediction accuracy. They have researched its application range, background value and boundary value correction, response function optimization, and the construction of a generalized discrete GM(1,1) model.
Due to the influence of seasonality or statistical system, the characteristics of seasonal cycle fluctuation generally exist in the time series data of social and economic fields. However, lacking the characterization of seasonal periodic fluctuations in time series, the original GM(1,1) model has poor performance in predicting time series data with seasonal periodic fluctuations. Therefore, some scholars preprocessed original time series data with seasonal fluctuations based on the data grouping method and proposed a model of DGGM(1,1)(data grouping-based grey modelling) to characterize the characteristics of seasonal periodic fluctuations.
However, in practical applications, DGGM(1,1) model may have an obvious overfitting phenomenon on some data sets, which leads to the fact that the model still cannot predict the time series data with seasonal periodic fluctuations. Aiming at the problem of unstable performance of the traditional DGGM(1,1) model in practical application, this study proposed a new PSO-ESM-DGGM(1,1) model to expand the application range of the original DGGM(1,1) model. First, the ESM-DGGM (1,1) model is constructed by introducing smoothing coefficient α and using the exponential smoothing method (ESM) to select appropriate smoothing coefficients for time series with different seasonal fluctuation characteristics. On this basis, to solve the problem of setting the smoothing coefficient α and the background value weight e, the Particle Swarm Optimization (PSO) algorithm is used to optimize the smoothing coefficient α and the background value weight e, aiming at the minimum average absolute percentage error. At the same time, the early stop method is introduced in the process of parameter optimization, and the training samples are divided into three subsets: Training subset, confirmation subset, and test subset. The generalization loss of the confirmation subset is observed in the training process, to reduce the time cost in the process of parameter optimization and avoid the phenomenon of overfitting of the model. Finally, taking the data sets of refrigerator export volume and the LI Keqiang index of China as examples, this paper compares and analyzes the DGGM(1,1) model and its improved form.
The numerical experimental results show that the average absolute percentage error of the PSO-ESM-DGGM(1,1) model compared with the suboptimal model in the test set decreases by 27% and 5%, respectively. It is verified that the model can improve the prediction accuracy on the premise of ensuring that the model can accept the fitting error and shows the feasibility and effectiveness of the model. However, the applicability of different types of DGGM(1,1) models has not been theoretically discussed in this study, which is worthy of further study in the future.

Key words: grey prediction, DGGM(1,1) model, PSO, exponential smoothing method

摘要： 由于受到季节性或统计制度等因素的影响,季节性周期波动特征普遍存在于社会经济领域的时间序列数据。针对传统DGGM(1,1)(Data grouping-based grey modelling)模型在实际运用中性能不稳定的问题,提出了一种新的PSO-ESM-DGGM(1,1)模型。首先,引入平滑系数α,使用指数平滑法(Exponential smoothing method,ESM),对具有不同季节性波动特征的时间序列进行处理,构建ESM-DGGM(1,1)模型。在此基础上,为解决平滑系数α和背景值权重e的设定问题,以平均绝对百分比误差最小为目标,使用粒子群优化算法(Particle Swarm Optimization,PSO)对二者进行调优。同时,在参数寻优过程中引入提前停止法,以降低参数寻优过程中的时间成本、避免模型过拟合问题。最后,以我国冰箱出口量和克强指数两个数据集为例,对DGGM(1,1)模型及其改进形式进行了对比分析,发现PSO-ESM-DGGM(1,1)模型较次优模型在测试集的平均绝对百分比误差分别减少了27%和5%,验证了其在保证模型可接受拟合误差的前提下,能够提升预测精度。

关键词: 灰色预测, DGGM(1,1), PSO, 指数平滑