时间序列预测是当下最重要的研究课题之一。现实中，因为数据收集成本高、难度大、周期长、环境影响因素复杂，导致样本“量小”且“不确定”。灰色系统理论以其结构简单、建模样本少、适用范围广泛等优点，在众多领域得到了广泛应用。灰色预测模型研究的时间序列根据其特征可分为单调序列、S形序列、波动序列和随机振荡序列四类。随着预测问题和场景的复杂化，数据特征日益多变。其中，在随机因素的影响下，S形序列会具有一定的振荡和非饱和性，表现出非线性特征；在固定因素直接或间接的影响下，波动序列也会呈现复杂的趋势性，表现为非线性趋势特征。此时，传统灰色预测模型难以获得理想的预测精度，需要根据数据特征对预测模型进行扩展，以契合相应的应用场景。本文在现有研究基础上，分析了灰色广义Verhulst模型和灰色波形模型对应不同特征序列的局限性，并根据数据特征对这两类灰色预测模型进行改进。主要研究内容如下：

（1）针对非线性S形序列的灰色广义Verhulst模型改进。灰色广义Verhulst模型在S形序列预测中具有天然优势。然而，传统Verhulst模型过于依赖S形序列，且实际中是更为复杂的非线性S形序列。因此，本文引入初始值加权控制函数和背景值权重系数，构建了不同参数间的非线性协同优化模型。加权控制函数在符合新旧信息变化规律和新信息优先原理的基础上，充分利用了小样本序列中蕴含的“信息”；背景值权重系数可以动态修正序列波动造成的固有误差。最后，通过实例预测证明了新模型具有更好的灵活性和拟合预测性能，与其他经典灰色预测模型相比具有显著优势。

（2）针对非线性趋势波动序列的灰色波形模型改进。灰色波形模型可以很好的利用时间序列波动图形进行预测。然而，传统灰色波形模型对具有非线性趋势的波动序列预测效果较差。因此，本文引入一元二次泛等高线，借用MATLAB动态拟合等高线系数，从而获取数据的非线性趋势性。最后，通过两个不同增长趋势的实际案例验证了改进模型的有效性和优势。

（3）将新模型应用于我国环境问题相关要素预测。科学的预测结果能够为环境问题的进一步决策提供有力支撑，对于打赢污染防治攻坚战具有重要意义。由于我国环境问题缺乏相关的统计数据，且过于陈旧的数据丧失了时效性，使得整个系统呈现出“小样本”的特点，本文改进的模型对这种特点具有很好的适应性。因此，利用这两个模型来预测我国环境问题中的危险固体废弃物和空气污染物浓度的发展趋势，从而为环境问题以及符合数据特征的其他领域预测问题提供新的解决方法。

Time series forecasting is one of the most important research topics today. In reality, due to the high cost, difficulty, long cycle and complex environmental impact factors of data collection, the sample size is "small" and "uncertain". Grey system theory has been widely used in many fields due to its advantages of simple structure, few modeling samples and wide application range. The time series studied by grey forecasting model can be divided into four categories according to their characteristics: monotonic sequence, S-shaped sequence, fluctuation sequence and random oscillation sequence. As prediction problems and scenarios become more complex, data characteristics are increasingly variable. Among them, under the influence of random factors, the S-shaped sequence will have certain oscillation and non-saturation, showing nonlinear characteristics; under the direct or indirect influence of fixed factors, the fluctuation sequence will also show a complex trend, which is expressed as nonlinear trend feature. At this time, it is difficult for the traditional grey prediction model to obtain the ideal prediction accuracy, and the prediction model needs to be expanded according to the data characteristics to fit the corresponding application scenarios. Based on the existing research, this paper analyzes the limitations of the grey generalized Verhulst model and the grey wave model corresponding to different feature sequences, and improves the two types of grey prediction models according to the data characteristics. The main research contents are as follows:

 

(1) Improvement of grey generalized Verhulst model for nonlinear sigmoid sequences. The grey generalized Verhulst model has natural advantages for sigmoid sequence prediction. However, the traditional Verhulst model relies too much on the sigmoid sequence and is actually a more complex nonlinear sigmoid sequence. Therefore, this paper introduces the initial value weighted control function and the background value weight coefficient, and constructs a nonlinear collaborative optimization model between different parameters. The weighted control function makes full use of the "information" contained in the small sample sequence on the basis of conforming to the changing law of old and new information and the principle of prioritizing new information; the background value weight coefficient can dynamically correct the inherent error caused by the fluctuation of the sequence. Finally, the new model is proved to have better flexibility and fitting prediction performance through instance prediction, which has significant advantages compared with other classical grey prediction models.

 

(2) Improvement of the grey wave model for nonlinear trend fluctuation series. The grey wave model can make good use of time series fluctuation patterns for forecasting. However, the traditional grey wave model is less effective for predicting fluctuation series with nonlinear trends. Therefore, this paper introduces a unary quadratic contour line and uses MATLAB to dynamically fit the contour line coefficients to obtain the nonlinear trend of the data. Finally, the effectiveness and advantages of the improved model are verified by two real cases with different growth trends.

 

(3) The new model is applied to the prediction of relevant elements of environmental problems in my country. Scientific prediction results can provide strong support for further decision-making on environmental issues, and are of great significance to winning the tough battle of pollution prevention and control. Due to the lack of relevant statistical data on environmental problems in our country, and the data that is too old loses its timeliness, the entire system presents a "small sample" feature. The improved model in this paper has a good adaptability to this feature. Therefore, these two models are used to predict the development trend of the concentration of hazardous solid waste and air pollutants in my country's environmental problems, thereby providing new solutions for environmental problems and other field prediction problems that conform to the characteristics of the data.

[1]MA X. A Brief Introduction to the Grey Machine Learning [J]. Journal of Grey System, 2019, 31(1): 1-12.
[2]ZENG B, TONG M Y, MA X. A new-structure grey Verhulst model: Development and performance comparison [J]. Applied Mathematical Modelling, 2020, 81: 522-37.
[3]ZENG B, ZHOU M, LIU X Z, et al. Application of a new grey prediction model and grey average weakening buffer operator to forecast China's shale gas output [J]. Energy Reports, 2020, 6: 1608-18.
[4]ZENG B, MA X, SHI J J. Modeling Method of the Grey GM(1,1) Model with Interval Grey Action Quantity and Its Application [J]. Complexity, 2020, 2020.
[5]LI S L, ZENG B, MAI X, et al. A Novel Grey Model with A Three-parameter Background Value and Its Application in Forecasting Average Annual Water Consumption Per Capita in Urban Areas Along The Yangtze River Basin [J]. Journal of Grey System, 2020, 32(1): 118-32.
[6]ZHOU M, ZENG B, ZHOU W H. A Hybrid Grey Prediction Model for Small Oscillation Sequence Based on Information Decomposition [J]. Complexity, 2020, 2020.
[7]MA X, XIE M, WU W Q, et al. The novel fractional discrete multivariate grey system model and its applications [J]. Applied Mathematical Modelling, 2019, 70: 402-24.
[8]DUAN H M, XIAO X P. A Multimode Dynamic Short-Term Traffic Flow Grey Prediction Model of High-Dimension Tensors [J]. Complexity, 2019.
[9]刘思峰, 郭天榜, 党耀国. 灰色系统理论及其应用 [M]. 科学出版社, 2010.
[10]周伟杰, 党耀国. 灰色广义Verhulst模型的构建及其应用 [J]. 系统工程理论与实践, 2020, 40(01): 230-9.
[11]DANG Y, LIU S F, CHEN K. The GM models that x ( n ) be taken as initial value [J]. Kybernetes, 2004, 33(2).
[12]王海城, 徐进军. Verhulst模型优化及其在建筑物沉降监测中的应用 [J]. 测绘通报, 2016, (S2): 130-3.
[13]付洁, 黄洪. 改进Verhulst模型的铁路客运量预测研究 [J]. 信息技术, 2019, 43(07): 159-61+66.
[14]吕敏蓉, 吕闽晖. 改进灰色Verhulst模型的优化设计 [J]. 统计与决策, 2017, (01): 35-8.
[15]沈琴琴, 王玥, 黄悦, 等. 改进初值的灰色Verhulst-Markov模型及其应用 [J]. 统计与决策, 2020, 36(07): 30-3.
[16]熊萍萍, 党耀国, 姚天祥, 等. 灰色Verhulst模型背景值优化的建模方法研究 [J]. 中国管理科学, 2012, 20(06): 154-9.
[17]张怡. 灰色Verhulst模型的背景值的改进及应用 [J]. 系统工程理论与实践, 2013, 33(12): 3168-71.
[18]杨孝良, 周猛, 曾波. 灰色预测模型背景值构造的新方法 [J]. 统计与决策, 2018, 34(19): 14-8.
[19]高媛媛, 魏勇. 灰色模型背景值优化的一种新方法 [J]. 统计与决策, 2020, 36(07): 21-6.
[20]XIONG P P, DANG Y G, QIAN W Y. The Optimization of Time Response Function in Non-Equidistant Verhulst Model [J]. Journal of Grey System, 2010, 22(3): 249-56.
[21]龙钊, 龙霞. 基于三参数的新灰色Verhulst离散直接建模模型 [J]. 数学的实践与认识, 2019, 49(04): 287-90.
[22]TANG L W, LU Y Y. Study of the grey Verhulst model based on the weighted least square method [J]. Physica A: Statistical Mechanics and its Applications, 2020, 545: 123615.
[23]王正新, 党耀国, 刘思峰. 无偏灰色Verhulst模型及其应用 [J]. 系统工程理论与实践, 2009, 29(10): 138-44.
[24]蒋红梅, 魏勇. 灰色离散Verhulst直接建模模型的构建 [J]. 统计与决策, 2015, (21): 63-5.
[25]李晔, 丁圆苹. 白化权函数已知的区间灰数Verhulst预测模型 [J]. 数学的实践与认识, 2020, 50(20): 308-14.
[26]MA X, WU W Q, ZENG B, et al. The conformable fractional grey system model [J]. ISA Transactions, 2020, 96: 255-71.
[27]ZHOU W, PEI L. The grey generalized Verhulst model and its application for forecasting Chinese pig price index [J]. Soft Computing, 2020, 24(7): 4977-90.
[28]钱吴永, 党耀国. 基于振荡序列的GM(1,1)模型 [J]. 系统工程理论与实践, 2009, 29(03): 149-54.
[29]ZENG B, MENG W, LIU S, et al. Research on prediction model of oscillatory sequence based on GM (1,1) and its application in electricity demand prediction [J]. Journal of Grey System, 2013, 25(4): 31-40.
[30]MAO S, CHEN Y, XIAO X. City Traffic Flow Prediction Based on Improved GM(1,1) Model [J]. Journal of Grey System, 2012, 24(4): 337-46.
[31]曾亮. 基于振荡序列的灰色GM(1,1|sin）幂模型及其应用 [J]. 浙江大学学报(理学版), 2019, 46(06): 697-704.
[32]王正新, 党耀国, 刘思峰, 等. GM(1,1)幂模型求解方法及其解的性质 [J]. 系统工程与电子技术, 2009, 31(10): 2380-3.
[33]丁松, 李若瑾, 党耀国. 基于初始条件优化的GM(1,1）幂模型及其应用 [J]. 中国管理科学, 2020, 28(01): 153-61.
[34]王正新. 振荡型GM(1,1)幂模型及其应用 [J]. 控制与决策, 2013, 28(10): 1459-64+72.
[35]陈彦晖, 甘爱平. 灰色波形预测模型的改进及应用 [J]. 统计与决策, 2016, (06): 75-8.
[36]陈彦晖, 刘斌. 基于广义等高线的灰色波形预测模型及其应用 [J]. 中国管理科学, 2017, 25(08): 134-9.
[37]郝慧慧, 朱涵钰. 改进灰色波形预测方法在降水量预测中的应用 [J]. 节水灌溉, 2021, (09): 41-4+50.
[38]任旺, 徐国宾. 基于GA-灰色波形预测模型的白洋淀天然入淀水量 [J]. 南水北调与水利科技, 2017, 15(05): 9-14+49.
[39]张琼楠, 陈元芳, 顾圣华, 等. 优化的灰色拓扑模型在年径流预测中的应用 [J]. 水电能源科学, 2014, 32(04): 22-5.
[40]石永杰, 林燕凌, 邹燕歆, 等. 基于灰色拓扑的中国交通事故死亡人数预测模型 [J]. 数学的实践与认识, 2013, 43(20): 110-6.
[41]林添枝, 吴卢荣, 陈绩馨. 基于时间序列与灰色拓扑的春节火灾损失预测 [J]. 数学的实践与认识, 2014, 44(17): 176-83.
[42]陈彦晖. 波罗的海干散货指数预测的非等间隔灰色波形预测方法 [J]. 大连海事大学学报, 2015, 41(04): 96-101.
[43]曹飞飞, 付晓, 李嘉珣, 等. 基于灰色拓扑理论的草地生态系统损害基线动态预测 [J]. 生态学报, 2020, 40(02): 540-8.
[44]马永梅, 葛国菊. GM(1,1)幂模型的研究现状及展望 [J]. 统计与决策, 2020, 36(17): 42-5.
[45]曾波, 李惠, 余乐安, 等. 季节波动数据特征提取与分数阶灰色预测建模 [J]. 系统工程理论与实践: 1-18.
[46]ZHANG P, MA X, SHE K. A Novel Power-Driven Grey Model with Whale Optimization Algorithm and Its Application in Forecasting the Residential Energy Consumption in China [J]. Complexity, 2019, 2019.
[47]CEYLAN Z, BULKAN S, ELEVLI S. Prediction of medical waste generation using SVR, GM (1,1) and ARIMA models: a case study for megacity Istanbul [J]. Journal of Environmental Health Science and Engineering, 2020, 18(2): 687-97.
[48]梁小珍, 邬志坤, 杨明歌, 等. 基于二次分解策略和模糊时间序列模型的航空客运需求预测研究 [J]. 中国管理科学, 2020, 28(12): 108-17.
[49]LUO D, AMBREEN M, LATIF A, et al. Forecasting Pakistan's electricity based on improved discrete grey polynomial model [J]. Grey Systems-Theory and Application, 2020, 10(2): 215-30.
[50]ZENG B, TONG M, MA X. A new-structure grey Verhulst model: Development and performance comparison [J]. Applied Mathematical Modelling, 2020, 81: 522-37.
[51]LIANG X, YAN F, YANG X. Research on Industrial Hazardous Waste Generation in China Based on Combination Forecasting Model; proceedings of the 6th International Conference on Environment and Renewable Energy (ICERE), Hanoi, VIETNAM, F 2020.Sep 28-30, 2020 [C]. 2020.
[52]ZHOU W, JIANG R, DING S, et al. A novel grey prediction model for seasonal time series [J]. Knowledge-Based Systems, 2021, 229: 107363.
[53]WANG Z-X, LI Q, PEI L-L. Grey forecasting method of quarterly hydropower production in China based on a data grouping approach [J]. Applied Mathematical Modelling, 2017, 51: 302-16.
[54]徐冰涵, 孙云莲, 易仕敏, 等. 基于模糊聚类分类与Elman神经网络算法的居民用户短期用电量预测及修正方法 [J]. 电测与仪表, 2020, 57(05): 1-7+13.
[55]SANDBERG M, KLOCKARS K, WILéN K. Green growth or degrowth? Assessing the normative justifications for environmental sustainability and economic growth through critical social theory [J]. Journal of Cleaner Production, 2019, 206: 133-41.
[56]朱文杰. 我国固体废弃物处理现状及对策分析 [J]. 资源节约与环保, 2021, (01): 106-7.
[57]TANG J X, WANG Q W, CHOI G. Efficiency assessment of industrial solid waste generation and treatment processes with carry-over in China [J]. Science of The Total Environment, 2020, 726: 138274.
[58]FAGHIH-ROOHI S, ONG Y S, ASIAN S, et al. Dynamic conditional value-at-risk model for routing and scheduling of hazardous material transportation networks [J]. Ann Oper Res, 2016, 247(2): 715-34.
[59]FAZZO L, MINICHILLI F, SANTORO M, et al. Hazardous waste and health impact: a systematic review of the scientific literature [J]. Environmental Health, 2017, 16.
[60]AL-KHATIB I A, ELEYAN D, GARFIELD J. A system dynamics approach for hospital waste management in a city in a developing country: the case of Nablus, Palestine [J]. Environmental Monitoring and Assessment, 2016, 188(9).
[61]PETRIDIS N E, STIAKAKIS E, PETRIDIS K, et al. Estimation of computer waste quantities using forecasting techniques [J]. Journal of Cleaner Production, 2016, 112: 3072-85.
[62]AZARMI S L, OLADIPO A A, VAZIRI R, et al. Comparative Modelling and Artificial Neural Network Inspired Prediction of Waste Generation Rates of Hospitality Industry: The Case of North Cyprus [J]. Sustainability, 2018, 10(9).
[63]ABBASI M, ABDULI M A, OMIDVAR B, et al. Forecasting Municipal Solid waste Generation by Hybrid Support Vector Machine and Partial Least Square Model [J]. International Journal of Environmental Research, 2013, 7(1): 27-38.
[64]ZENG B, MA X, ZHOU M. A new-structure grey Verhulst model for China’s tight gas production forecasting [J]. Applied Soft Computing, 2020, 96: 106600.