一类离散灰色预测模型的统一处理方法及应用
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摘要
在加权最小二乘框架下构建了离散灰色预测模型DGMP(1,1,N),论证了最小均方误差准则、最小均方相对误差准则和最小平均绝对百分误差准则下的DGM(1,1)模型、NDGM(1,1)模型和NGM(1,1,k~α)模型均是DGMP(1,1,N)模型的特殊形式,给出了模型阶数N取值的判定准则,证明了模型的仿射变换不变性和无偏性.将DGMP(1,1,N)模型运用到人均能源生活消费量预测中,结果表明该模型具有高拟合和预测精度,验证了模型的可行性与有效性.
A weighted least square grey forecasting model termed as DGMP(1, 1,N) is proposed, which proves to be the unified form of DGM(1,1) model, NDGM(1,1) model and DGM(1,1, ka) model under least mean square error criterion, least mean square relative error criterion and least mean absolute percentage error criterion. Then, a criterion to determine the value of N in DGMP(1,1, N) model is also put forward.Furthermore, the modeling accuracy proves to be independent on the affine transformation of 1-AGO sequence and it is unbiased for N order homogenous exponential sequence. Finally, DGMP(1,1, N) model is applied to predicting per capita living energy consumption. The result shows that this model is with high fitting and forecasting accuracy, which verified the feasibility and effectiveness.
A weighted least square grey forecasting model termed as DGMP(1, 1,N) is proposed, which proves to be the unified form of DGM(1,1) model, NDGM(1,1) model and DGM(1,1, ka) model under least mean square error criterion, least mean square relative error criterion and least mean absolute percentage error criterion. Then, a criterion to determine the value of N in DGMP(1,1, N) model is also put forward.Furthermore, the modeling accuracy proves to be independent on the affine transformation of 1-AGO sequence and it is unbiased for N order homogenous exponential sequence. Finally, DGMP(1,1, N) model is applied to predicting per capita living energy consumption. The result shows that this model is with high fitting and forecasting accuracy, which verified the feasibility and effectiveness.
引文
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[3] Li D C, Chang C J, Chen C C, et al. Forecasting short-term electricity consumption using the adaptive grey-based approach—An Asian case[J]. Omega, 2012, 40(6):767-773.
[4] Samvedi A, Jain V. A grey approach for forecasting in a supply chain during intermittent disruptions[J]. Engi?neering Applications of Artificial Intelligence, 2013, 26(3):1044-1051.
[5] Luo D. Risk evaluation of ice-jam disasters using gray systems theory:The case of Ningxia-Inner Mongolia reaches of the Yellow River[J]. Natural Hazards, 2014, 71(3):1419-1431.
[6]Xia M, Wong W K. A seasonal discrete grey forecasting model for fashion retailing[J]. Knowledge-Based Systems,2014, 57(2):119-126.
[7] Zeng B, Li C. Forecasting the natural gas demand in China using a self-adapting intelligent grey model[J]. Energy,2016, 112(1):810-825.
[8] Chen Y, He K, Zhang C. A novel grey wave forecasting method for predicting metal prices[J]. Resources Policy,2016, 49:323-331.
[9] Bezuglov A, Comert G. Short-term freeway traffic parameter prediction:Application of grey system theory models[J]. Expert Systems with Applications, 2016, 62(15):284-292.
[10]吉培荣,黄巍松,胡翔勇.无偏灰色预测模型[J]·系统工程与电子技术,2000,22(6):6-7.Ji P R, Huang W S, Hu X Y. An unbiased grey forecasting model[J]. Systems Engineering and Electronics, 2000,22(6):6-7.
[11]王正新,党耀国,刘思峰.无偏GM(1,1)模型的混沌特性分析[J]·系统工程理论与实践,2007, 27(11):153-158.Wang Z X, Dang Y G, Liu S F. Analysis of chaotic charateristics of unbiased GM(1,1)[J]. Systems Engineering-Theory h Practice, 2007, 27(11):153-158.
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[14]罗党,刘思峰,党耀国.灰色模型GM(1,1)优化[J].中国工程科学,2003, 5(8):50-53.Luo D, Liu S F, Dang Y G. The optimization of grey model GM(1,1)[J]. Engineering Science, 2003, 5(8):50-53.
[15]徐宁,党耀国,丁松.基于误差最小化的GM(1,1)模型背景值优化方法[J].控制与决策,2015, 30(2):283-288.Xu N, Dang Y G, Ding S. Optimization method of background value in GM(1,1)model based on least error[J].Control&Decision, 2015, 30(2):283-288.
[16] Jin X, Tao T, Mao T, et al. Improvement of grey models by least squares[J]. Expert Systems with Applications,2011, 38(11):13961-13966.
[17]郭金海,杨锦伟.GM(1,1)模型初始条件和初始点的优化[J].系统工程理论与实践,2015, 35(9):2333-2338.Guo J H, Yang J W. Optimizing the initial condition and the initial point of GM(1,1)[J]. Systems Engineering-Theory&Practice, 2015, 35(9):2333-2338.
[18]王义闹,李应川,陈智洁.逐步优化灰导数白化值的GM(1,1)直接建模法[J].华中科技大学学报(自然科学版),2001,29(3):54-57.Wang Y N, Li Y C, Chen Z J. A direct modeling method of GM(1,1)with a step by step optimum grey derivative's whitening values[J]. Journal of Huangzhong University of Science&Technology(Natural Science Edition), 2001,29(3):54-57.
[19] Liu J, Xiao X P, Guo J H, et al. Error and its upper bound estimation between the solutions of GM(1,1)grey forecasting models[J]. Applied Mathematics&Computation, 2014, 246(C):648-660.
[20] Xiao X P, Guo H, Mao S H. The modeling mechanism, extension and optimization of grey GM(1,1)model[J].Applied Mathematical Modelling, 2014, 38(s5-6):1896-1910.
[21] Liu S F, Zeng B, Liu J F, et al. Four basic models of GM(1,1)and their suitable sequences[J]. Grey Systems Theory&Application, 2015, 5(2):141-156.
[22]崔杰,党耀国,刘思峰.一种新的灰色预测模型及其建模机理[J]·控制与决策,2009, 24(11):1702-1706.Cui J, Dang Y G, Liu S F. Novel grey forecasting model and its modeling mechanism[J]. Control&Decision,2009, 24(11):1702-1706.
[23]钱吴永,党耀国,刘思峰.含时间幕次项的灰色GM(1,1,t~α)模型及其应用[J].系统工程理论与实践,2012, 32(10):2247-2252.Qian W Y, Dang Y G, Liu S F. Grey GM(1,1,t~α)model with time power and its application[J]. Systems Engineering—Theory&Practice, 2012, 32(10):2247-2252.
[24]谢乃明,刘思峰.离散GM(1,1)模型与灰色预测模型建模机理[J].系统工程理论与实践,2005, 25(1):93-99.Xie N M, Liu S F. Discrete GM(1,1)and mechanism of grey forecasting model[J]. Systems Engineering—Theory&Practice, 2005, 25(1):93-99.
[25] Xie N M, Liu S F. Discrete grey forecasting model and its optimization[J]. Applied Mathematical Modelling,2009, 33(2):1173-1186.
[26] Xie N M, Liu S F, Yang Y J, et al. On novel grey forecasting model based on non-homogeneous index sequence[J].Applied Mathematical Modelling, 2013, 37(7):5059-5068.
[27]崔杰,刘思峰,马红燕.含有时间幕次项的灰色预测模型病态特性[J]·控制与决策,2016, 31(5):953-956.Cui J, Liu S F, Ma H Y. Morbid property of grey prediction model with time-power[J]. Control&Decision, 2016,31(5):953-956.
[28] Xie N M, Zhu C Y, Liu S F, et al. On discrete grey system forecasting model corresponding with polynomial time-vary sequence[J]. Journal of Grey System, 2013, 25(4):1-18.
[29] Tan P N, Steinbach M, Kumar V. Introduction to data mining[M]. Pearson Addison-Wesley, 2006:106-114.
[30]张贤达.矩阵分析与应用[M]. 2版·北京:清华大学出版社,2014:384-395.Zhang X D. Matrix analysis and applications[M]. 2nd ed. Beijing:Tsinghua University Press, 2014:384-395.
1.数据来源:中华人民共和国国家统计局;网址:http://data.stats.gov cn/easyquery.htm;访问时间:2017-04-16.
[2]刘思峰,杨英杰,吴利丰,等.灰色系统理论及其应用[M]. 7版.北京:科学出版社,2014:1-15, 42-62.Liu S F, Yang Y J, Wu L F, et al. Grey system theory and applications[M]. 7th ed. Beijing:Science Press, 2014:1-15, 42-62.
[3] Li D C, Chang C J, Chen C C, et al. Forecasting short-term electricity consumption using the adaptive grey-based approach—An Asian case[J]. Omega, 2012, 40(6):767-773.
[4] Samvedi A, Jain V. A grey approach for forecasting in a supply chain during intermittent disruptions[J]. Engi?neering Applications of Artificial Intelligence, 2013, 26(3):1044-1051.
[5] Luo D. Risk evaluation of ice-jam disasters using gray systems theory:The case of Ningxia-Inner Mongolia reaches of the Yellow River[J]. Natural Hazards, 2014, 71(3):1419-1431.
[6]Xia M, Wong W K. A seasonal discrete grey forecasting model for fashion retailing[J]. Knowledge-Based Systems,2014, 57(2):119-126.
[7] Zeng B, Li C. Forecasting the natural gas demand in China using a self-adapting intelligent grey model[J]. Energy,2016, 112(1):810-825.
[8] Chen Y, He K, Zhang C. A novel grey wave forecasting method for predicting metal prices[J]. Resources Policy,2016, 49:323-331.
[9] Bezuglov A, Comert G. Short-term freeway traffic parameter prediction:Application of grey system theory models[J]. Expert Systems with Applications, 2016, 62(15):284-292.
[10]吉培荣,黄巍松,胡翔勇.无偏灰色预测模型[J]·系统工程与电子技术,2000,22(6):6-7.Ji P R, Huang W S, Hu X Y. An unbiased grey forecasting model[J]. Systems Engineering and Electronics, 2000,22(6):6-7.
[11]王正新,党耀国,刘思峰.无偏GM(1,1)模型的混沌特性分析[J]·系统工程理论与实践,2007, 27(11):153-158.Wang Z X, Dang Y G, Liu S F. Analysis of chaotic charateristics of unbiased GM(1,1)[J]. Systems Engineering-Theory h Practice, 2007, 27(11):153-158.
[12]郑照宁,武玉英,包涵龄.GM模型的病态性问题[J].中国管理科学,2001, 9(5):38-44.Zheng Z N, Wu Y Y,Bao H L. Morbidity problem in grey model[J]. Chinese Journal of Management Science,2001, 9(5):38-44.
[13]i*.,王正新,刘思峰.灰色模型的病态问题研究[J].系统工程理论与实践,2008, 28(1):156-160.Dang Y G, Wang Z X, Liu S F. Study on morbidity problem in grey model[J]. Systems Engineering—Theory&Practice, 2008, 28(1):156-160.
[14]罗党,刘思峰,党耀国.灰色模型GM(1,1)优化[J].中国工程科学,2003, 5(8):50-53.Luo D, Liu S F, Dang Y G. The optimization of grey model GM(1,1)[J]. Engineering Science, 2003, 5(8):50-53.
[15]徐宁,党耀国,丁松.基于误差最小化的GM(1,1)模型背景值优化方法[J].控制与决策,2015, 30(2):283-288.Xu N, Dang Y G, Ding S. Optimization method of background value in GM(1,1)model based on least error[J].Control&Decision, 2015, 30(2):283-288.
[16] Jin X, Tao T, Mao T, et al. Improvement of grey models by least squares[J]. Expert Systems with Applications,2011, 38(11):13961-13966.
[17]郭金海,杨锦伟.GM(1,1)模型初始条件和初始点的优化[J].系统工程理论与实践,2015, 35(9):2333-2338.Guo J H, Yang J W. Optimizing the initial condition and the initial point of GM(1,1)[J]. Systems Engineering-Theory&Practice, 2015, 35(9):2333-2338.
[18]王义闹,李应川,陈智洁.逐步优化灰导数白化值的GM(1,1)直接建模法[J].华中科技大学学报(自然科学版),2001,29(3):54-57.Wang Y N, Li Y C, Chen Z J. A direct modeling method of GM(1,1)with a step by step optimum grey derivative's whitening values[J]. Journal of Huangzhong University of Science&Technology(Natural Science Edition), 2001,29(3):54-57.
[19] Liu J, Xiao X P, Guo J H, et al. Error and its upper bound estimation between the solutions of GM(1,1)grey forecasting models[J]. Applied Mathematics&Computation, 2014, 246(C):648-660.
[20] Xiao X P, Guo H, Mao S H. The modeling mechanism, extension and optimization of grey GM(1,1)model[J].Applied Mathematical Modelling, 2014, 38(s5-6):1896-1910.
[21] Liu S F, Zeng B, Liu J F, et al. Four basic models of GM(1,1)and their suitable sequences[J]. Grey Systems Theory&Application, 2015, 5(2):141-156.
[22]崔杰,党耀国,刘思峰.一种新的灰色预测模型及其建模机理[J]·控制与决策,2009, 24(11):1702-1706.Cui J, Dang Y G, Liu S F. Novel grey forecasting model and its modeling mechanism[J]. Control&Decision,2009, 24(11):1702-1706.
[23]钱吴永,党耀国,刘思峰.含时间幕次项的灰色GM(1,1,t~α)模型及其应用[J].系统工程理论与实践,2012, 32(10):2247-2252.Qian W Y, Dang Y G, Liu S F. Grey GM(1,1,t~α)model with time power and its application[J]. Systems Engineering—Theory&Practice, 2012, 32(10):2247-2252.
[24]谢乃明,刘思峰.离散GM(1,1)模型与灰色预测模型建模机理[J].系统工程理论与实践,2005, 25(1):93-99.Xie N M, Liu S F. Discrete GM(1,1)and mechanism of grey forecasting model[J]. Systems Engineering—Theory&Practice, 2005, 25(1):93-99.
[25] Xie N M, Liu S F. Discrete grey forecasting model and its optimization[J]. Applied Mathematical Modelling,2009, 33(2):1173-1186.
[26] Xie N M, Liu S F, Yang Y J, et al. On novel grey forecasting model based on non-homogeneous index sequence[J].Applied Mathematical Modelling, 2013, 37(7):5059-5068.
[27]崔杰,刘思峰,马红燕.含有时间幕次项的灰色预测模型病态特性[J]·控制与决策,2016, 31(5):953-956.Cui J, Liu S F, Ma H Y. Morbid property of grey prediction model with time-power[J]. Control&Decision, 2016,31(5):953-956.
[28] Xie N M, Zhu C Y, Liu S F, et al. On discrete grey system forecasting model corresponding with polynomial time-vary sequence[J]. Journal of Grey System, 2013, 25(4):1-18.
[29] Tan P N, Steinbach M, Kumar V. Introduction to data mining[M]. Pearson Addison-Wesley, 2006:106-114.
[30]张贤达.矩阵分析与应用[M]. 2版·北京:清华大学出版社,2014:384-395.Zhang X D. Matrix analysis and applications[M]. 2nd ed. Beijing:Tsinghua University Press, 2014:384-395.
1.数据来源:中华人民共和国国家统计局;网址:http://data.stats.gov cn/easyquery.htm;访问时间:2017-04-16.