灰色预测建模技术研究
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摘要
灰色系统理论是研究和解决现实世界不确定性问题主要方法之一，灰色预测建模技术是灰色系统理论的重要组成部分，经过近30年的发展，灰色预测建模技术取得了可喜的研究成果，但作为一门新兴横断学科，其理论体系还有待于进一步丰富和完善。本文从建模思想的创新、建模对象的拓展、建模方法的改进、建模序列的优化等方面对灰色预测建模技术进行了深入研究，其主要成果包括以下几个方面。
（1）基于区间灰数及其白化权函数所围成图形的几何重心，研究了区间灰数“核”的计算新方法，有效解决了传统方法仅能在“对称型”白化权函数条件下计算区间灰数“核”的不足，从而能够对基于“非对称型”三角形或梯形白化权函数的区间灰数“核”进行计算；另外，本文还通过面积法对传统区间灰数“灰度”的测度方法进行了拓展，拓展后的测度方法能够对各种白化权函数的区间灰数“灰度”进行计算。
（2）针对既有灰色预测模型的建模对象仅局限于实数序列，尚难以构建面向区间灰数序列预测模型的缺陷，本文在区间灰数白化权函数未知及已知两种情况下，深入研究了基于区间灰数序列的灰色预测模型建模方法，详细探究了模型的建模思路、推导过程及建模步骤，从而将灰色预测模型的建模对象从实数序列延伸至区间灰数序列，丰富并完善了灰色预测模型的理论体系。
（3）离散灰数是灰信息的一种重要表现形式，研究基于离散灰数序列的灰色预测模型具有重要价值。本文提出了标准离散灰数和灰单元格两个概念，按照离散灰数取值分布信息未知及已知两种情况，研究了基于离散灰数序列的灰色预测模型建模方法，详细讨论了模型的推导过程及建模步骤，从而将灰色预测模型的建模对象从实数序列延伸至离散灰数序列，拓展了灰色预测模型的应用范围。
（4）传统灰色预测建模技术通过对满足灰指数规律的累加生成序列建立指数模型来实现对未知世界的预测；然而，当建模序列本身已具有近似非齐次指数特征时，累加生成反而会破坏建模序列的这种特征。本文通过近似非齐次指数增长序列的齐次性转换以及省略累加生成直接建模的方法，构建了基于近似非齐次指数增长序列的两类不同灰色预测模型，从而增强了灰色模型对此类特殊序列的预测能力。
（5）针对振荡序列灰色预测模型的构建，已有的灰色波形预测建模技术不仅需要的数据量大，而且对超出已有数据取值范围的情况无能为力。本文通过构建序列的平滑算法以压缩振荡序列的振幅，提高建模序列的光滑度，从而构建适用于振荡序列的灰色预测模型，以提高灰色预测模型对振荡序列的模拟及预测精度。
Grey system theory is one of the major methods for studying and solving uncertain problems of thepresent world, and modeling technologies of grey prediction is an important component of greysystem theory, undergoing30years’ development, it has obtained many encouraging researchachievements. However, as an emerging cross-discipline, its theory system is still expected to befurther enriched and perfected, based on this, the paper in-depth studied grey prediction modelingtechnologies from modeling thought innovation, modeling object expansion, modeling methodimprovement, modeling sequences optimization, etc, and the main achievements of this paper are asfollowing.
⑴Based on the centre of geometric gravity encircled by interval grey number and whitenizationweight function,the new calculation method for the “kernel” of interval grey number is presented,which effectively figures out the deficiency of the traditional method that only can be used tocalculate the “kernel” of interval grey number on the condition of symmetric form Whitenizationweight function, in this way,the kernel of interval grey number of unsymmetricform triangle andtrapezoid Whitenization weight function can be calculated. Apart from this, the paper also expandsthe traditional measure methods for degree of grey of interval grey by acreage method, which can beused to calculate degree of grey of interval grey of a variety of whitenization weight function
⑵With the limitations of current existing grey prediction models whose modeling objects areonly based on real number sequences and cannot build models for interval grey number sequence, thispaper in-depth researched the grey prediction modeling methods of interval and discussed detailedlythe modeling thought, derivation process and modeling steps. Consequently, the modeling objects ofgrey predction models extend to interval grey number sequence from real number sequence, and itenriches and perfects the theory system of grey prediction models.
⑶Discrete grey number is an important manifestation of grey information, and it has a veryimportant significance to study a grey prediction model based on the sequence of discrete greynumber. This paper puts forward two concepts: standard discrete grey numbers and grey unit cells,and study the modeling methods of grey prediction model based on discrete grey number sequenceaccording to the case of the value distribution information of discrete grey number is known orunknown, introduces detailedly derivation process and modeling steps. Consequently, the modelingobjects of grey prediction models extend to discrete grey number sequence from real numbersequence and it expands the applied scope of grey prediction models.
⑷Traditional grey prediction technologies establish exponent models based on accumulating generation sequence which satisfies the law of quasi-exponent, and accomplish predictions of theunknown world. However, when modeling sequence has alrealy the characteristic of approximatenonhomogeneous index, the procedure of accumulating generation maybe destroy this feature. Thispaper converts an approximate nonhomogeneous exponential growth sequence into a homogeneousone and a direct modeling method that omits the procedure of accumulating generation, through those,two different grey prediction models are built, and it enhances the prediction ability of grey models dowith such kind of specific sequences.
⑸with oscillation sequence grey model’s build, the existing modeling technology of greywaveform prediction not only needs large amount of data but also cannot do such situation of beyondexisting data value range. This paper builds a smooth algorithm of oscillation sequence to compressits amplitude, and enhance the degree of smoothness of modeling sequence, after this, we canestablish a grey prediction model which is suitable for oscillation sequence, and it can improve thesimulation and forecast accuracy of grey prediction model of oscillation sequence.
（1）基于区间灰数及其白化权函数所围成图形的几何重心，研究了区间灰数“核”的计算新方法，有效解决了传统方法仅能在“对称型”白化权函数条件下计算区间灰数“核”的不足，从而能够对基于“非对称型”三角形或梯形白化权函数的区间灰数“核”进行计算；另外，本文还通过面积法对传统区间灰数“灰度”的测度方法进行了拓展，拓展后的测度方法能够对各种白化权函数的区间灰数“灰度”进行计算。
（2）针对既有灰色预测模型的建模对象仅局限于实数序列，尚难以构建面向区间灰数序列预测模型的缺陷，本文在区间灰数白化权函数未知及已知两种情况下，深入研究了基于区间灰数序列的灰色预测模型建模方法，详细探究了模型的建模思路、推导过程及建模步骤，从而将灰色预测模型的建模对象从实数序列延伸至区间灰数序列，丰富并完善了灰色预测模型的理论体系。
（3）离散灰数是灰信息的一种重要表现形式，研究基于离散灰数序列的灰色预测模型具有重要价值。本文提出了标准离散灰数和灰单元格两个概念，按照离散灰数取值分布信息未知及已知两种情况，研究了基于离散灰数序列的灰色预测模型建模方法，详细讨论了模型的推导过程及建模步骤，从而将灰色预测模型的建模对象从实数序列延伸至离散灰数序列，拓展了灰色预测模型的应用范围。
（4）传统灰色预测建模技术通过对满足灰指数规律的累加生成序列建立指数模型来实现对未知世界的预测；然而，当建模序列本身已具有近似非齐次指数特征时，累加生成反而会破坏建模序列的这种特征。本文通过近似非齐次指数增长序列的齐次性转换以及省略累加生成直接建模的方法，构建了基于近似非齐次指数增长序列的两类不同灰色预测模型，从而增强了灰色模型对此类特殊序列的预测能力。
（5）针对振荡序列灰色预测模型的构建，已有的灰色波形预测建模技术不仅需要的数据量大，而且对超出已有数据取值范围的情况无能为力。本文通过构建序列的平滑算法以压缩振荡序列的振幅，提高建模序列的光滑度，从而构建适用于振荡序列的灰色预测模型，以提高灰色预测模型对振荡序列的模拟及预测精度。
Grey system theory is one of the major methods for studying and solving uncertain problems of thepresent world, and modeling technologies of grey prediction is an important component of greysystem theory, undergoing30years’ development, it has obtained many encouraging researchachievements. However, as an emerging cross-discipline, its theory system is still expected to befurther enriched and perfected, based on this, the paper in-depth studied grey prediction modelingtechnologies from modeling thought innovation, modeling object expansion, modeling methodimprovement, modeling sequences optimization, etc, and the main achievements of this paper are asfollowing.
⑴Based on the centre of geometric gravity encircled by interval grey number and whitenizationweight function,the new calculation method for the “kernel” of interval grey number is presented,which effectively figures out the deficiency of the traditional method that only can be used tocalculate the “kernel” of interval grey number on the condition of symmetric form Whitenizationweight function, in this way,the kernel of interval grey number of unsymmetricform triangle andtrapezoid Whitenization weight function can be calculated. Apart from this, the paper also expandsthe traditional measure methods for degree of grey of interval grey by acreage method, which can beused to calculate degree of grey of interval grey of a variety of whitenization weight function
⑵With the limitations of current existing grey prediction models whose modeling objects areonly based on real number sequences and cannot build models for interval grey number sequence, thispaper in-depth researched the grey prediction modeling methods of interval and discussed detailedlythe modeling thought, derivation process and modeling steps. Consequently, the modeling objects ofgrey predction models extend to interval grey number sequence from real number sequence, and itenriches and perfects the theory system of grey prediction models.
⑶Discrete grey number is an important manifestation of grey information, and it has a veryimportant significance to study a grey prediction model based on the sequence of discrete greynumber. This paper puts forward two concepts: standard discrete grey numbers and grey unit cells,and study the modeling methods of grey prediction model based on discrete grey number sequenceaccording to the case of the value distribution information of discrete grey number is known orunknown, introduces detailedly derivation process and modeling steps. Consequently, the modelingobjects of grey prediction models extend to discrete grey number sequence from real numbersequence and it expands the applied scope of grey prediction models.
⑷Traditional grey prediction technologies establish exponent models based on accumulating generation sequence which satisfies the law of quasi-exponent, and accomplish predictions of theunknown world. However, when modeling sequence has alrealy the characteristic of approximatenonhomogeneous index, the procedure of accumulating generation maybe destroy this feature. Thispaper converts an approximate nonhomogeneous exponential growth sequence into a homogeneousone and a direct modeling method that omits the procedure of accumulating generation, through those,two different grey prediction models are built, and it enhances the prediction ability of grey models dowith such kind of specific sequences.
⑸with oscillation sequence grey model’s build, the existing modeling technology of greywaveform prediction not only needs large amount of data but also cannot do such situation of beyondexisting data value range. This paper builds a smooth algorithm of oscillation sequence to compressits amplitude, and enhance the degree of smoothness of modeling sequence, after this, we canestablish a grey prediction model which is suitable for oscillation sequence, and it can improve thesimulation and forecast accuracy of grey prediction model of oscillation sequence.
引文
[1] Zadeh, L. A. Soft computing and fuzzy logic [J]. IEEE Software,1994,11(6):48-56.
[2] Deng J L. The Control problem of grey systems [J]. System&Control Letter.1982,1(5):288-294.
[3] Pawlak Z. Rough Sets: Theoretical aspects of reasoning about data. Dordrecht: Kluwer AcademicPublisher,1991.
[4]王光远.未确知信息及其数学处理[J].哈尔滨建筑大学学报,1990,(4):1-8.
[5]刘思峰,党耀国,方志耕,谢乃明.灰色系统理论及其应用(第五版)[M].北京:科学出版社,2010.
[6] Deng J L. The Primary Methods of Grey Theory[M]．Wuhan: Hua zhong University of Science andTechnology Press,2004:26-34
[7]刘思峰.灰色系统理论的产生与发展[J].南京航空航天大学学报,2004,36(2):267-272.
[8]邓聚龙.灰色控制系统[J],华中工学院学报,1982,10(3):9-18.
[9]肖新平,宋中民,李峰.灰技术基础及其应用[M].北京:科学出版社,2005.
[10]吴惠荣.灰色预测模型的进一步拓广[J].系统工程理论与实践,1994,14(8):31-34.
[11]何斌,蒙清.灰色预测模型拓广方法研究[J].系统工程理论与实践,2002,22(9):137-140.
[12]李翠凤,戴文战.基于函数cotx变换的灰色建模方法[J].系统工程,2005,23(3):110-114.
[13]戴文战,李俊峰.基于函数x-a变换的GM(1,1)模型及在我国农村人均住房面积建模中的应用[J].系统工程理论与实践,2004,24(11):63-67.
[14]戴文战,熊伟,杨爱萍.基于函数cot(xa)变换及背景值优化的灰色建模[J].浙江大学学报(工学版),2010,44(7):1368-1372.
[15]关叶青,刘思峰.基于函数cot(xα)变换的灰色GM(1,1)建模方法[J].系统工程,2008,26(9):89-93.
[16]崔立志,刘思峰.基于数据变换技术的灰色预测模型[J].系统工程,2010,28(5):104-107.
[17] Liu S F. The three axioms of buffer operator and their application[J]. The Journal of Grey System,1991,3(1):39-48.
[18]刘思峰.冲击扰动系统预测陷阱与缓冲算子[J].华中理工大学学报,1997,25(1):28-31.
[19]党耀国,刘思峰.关于弱化缓冲算子的研究[J].中国管理科学,2004,12(2):108-111.
[20]党耀国,刘斌,关叶青.关于强化缓冲算子的研究[J].控制与决策,2005,20(12):1332-1336.
[21]谢乃明,刘思峰.一种新的实用弱化缓冲算子[J].中国管理科学,2003,11:46-48.
[22]关叶青,刘思峰.线性缓冲算子矩阵及其应用研究[J].高校应用数学学报,2008,23(3):357-362.
[23]崔立志,刘思峰,吴正鹏.新的强化缓冲算子的构造及其应用[J].系统工程理论与实践,2010,30(3):484-489.
[24]魏勇,孔新海.几类强弱缓冲算子的构造方法及其内在联系[J].控制与决策,2010,25(2):196-202.
[25]许秀莉,罗键.GM(1,1)模型的改进方法及其应用[J].系统工程与电子技术,2002,24(4):61-63.
[26]尹春华,顾培亮.基于灰色序列生成中缓冲算子的能源预测[J].系统工程学报,2003,18(2):189-192.
[27]邓聚龙.累加生成灰指数律[J].华中工学院学报,1987,15(5):7-12.
[28] Deng J L. Negative power AGO in grey theory[J]. Journal of Grey System,2001,13(3):1-6.
[29] Zhang Q S. Difference information entropy in grey theory[J]. Journal of Grey System.2001,13(2):21-25.
[30] Xiao X P.On parameters in grey models[J].Journal of Grey System,2000,11(4):73-78.
[31] Deng J L. Generating space of grey system[J].Grey System. China Ocean Press,1988:79-90.
[32]宋中民,刘希强,王树泽.生成空间及IM模型[J].系统工程,1992,10(6):37-42.
[33] Deng J L. Moving operator in grey theory[J].The Journal of Grey System,1999,11(1):1-4.
[34] Wen Kunli, John H.Wu. AGO for invariant series[J].The Journal of Grey System.1998,10(1):17-21.
[35] Ai D B, Chen R Q. Frame of AGO Generating space[J].The Journal of Grey System.2001,13(1):13-16.
[36] Song Z M, Wang Z D, Tong X J. Grey Generating space on opposite accurnulation[J]. TheJournal of Grey System,2001,13(4):305-308.
[37] Song Z M, Xiao X P, Deng J L. The character of opposite direction AGO and class ratio[J].TheJournal of Grey System.2002,14(1):9-14.
[38]周慧,王晓光.倒数累加生成灰色GRM(1,1)模型的改进[J].沈阳理工大学学报,2008,27(4):84-86.
[39]宋中民,邓聚龙.反向累加生成及灰色GOM(1,1)模型[J].系统工程,2001,19(1):66-69.
[40]杨知,任鹏,党耀国.反向累加生成与GOM(1,1)模型的优化[J].系统工程理论与实践，2009，29(8):160-164.
[41]黄继,种晓丽.广义累加灰色预测控制模型及其优化算法[J].系统工程理论与实践,2009,29(6):147-156.
[42]陈超英.累加生成的改进和GM(1,1,t)灰色模型[J].数学的实践与认识,2007,37(2):105-109.
[43]戴文战.具有残差校正的n次累加灰色模型[J].系统工程理论与实践,1997,17(12):121-124.
[44]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,(05):38-44.
[45]党耀国,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.
[46]徐涛,冷淑霞.灰色系统模型初始条件的改进及应用[J].山东工程学院学报,1999,13(1):15-19.
[47] Dang Y G, Liu S F. The GM Models That x(n) be Taken as Initial Value [J].The InternationalJournal of Systems&Cyberntics,2004,33(2):247-255.
[48]刘斌,刘思峰,翟振杰等. GM(1,1)模型时间响应函数的最优化[J]中国管理科学,2003,(04):54-57.
[49]熊萍萍,门可佩,吴香华.以x(1)(n)为初始条件的无偏GM(1,1)模型[J].南京信息工程大学学报,2009,1(3):258-263.
[50]钟珞,江琼,张诚等.基于最优初始条件和动态辨识参数的灰色时程数据预测[J]，武汉理工大学学报(交通科学与工程版),2004,28(5):685-691.
[51]周命端,郭际明,文鸿雁等.基于优化初始值的GM(1,1)模型及其在大坝监测中的应用周[J],水电自动化与大坝监测,2008,32(2):52-54.
[52]谭冠军.GM(1,1)模型的背景值构造方法和应用(Ⅰ)[J].系统工程理论与实践,2000,20(4):98-103.
[53]谭冠军.GM(1,1)模型的背景值构造方法和应用(Ⅱ)[J].系统工程理论与实践,2000,20(5):125-128.
[54]谭冠军.GM(1,1)模型的背景值构造方法和应用(Ⅲ)[J].系统工程理论与实践,2000,20(6):70-75.
[55]谢开贵,李春燕,周家启.基于遗传算法的GM(1,1,λ)模型[J].系统工程学报,2000,15(2):168-172.
[56]戴文战,熊伟.基于函数cot(xα)变换及背景值优化的灰色建模[J].浙江大学学报(工学版),2010,44(7):1368-1372.
[57] Wang Z X, Dang Y G, Liu S F. The Optimization of Background Value in GM (1,1) Model[J].The Journal of Grey System,2007,10(2):69-74.
[58]王正新,党耀国,刘思峰.基于离散指数函数优化的GM(1,1)模型王[J].系统工程理论与实践,2008,28(2):61-67.
[59] Hsu C C, Chen CY. A modified grey forecasting model for long-term prediction[J]. Journal of theChinese Institute of Engineers,2003,26(23):301-308.
[60]李俊峰,戴文战.基于插值和Newton-Cores公式的GM(1,1)模型的背景值构造新方法与应用[J].系统工程理论与实践,2004,24(10):122-126.
[61]江南,刘小洋.基于Gauss公式的GM(1,1)模型的背景值构造新方法与应用[J].数学的实践与认识,2008,38(7):90-94.
[62]谢开贵,李春燕等.基于遗传算法的GM(1,1)模型[J].系统工程学报,2000,15(2):168-172.
[63]李翠凤,戴文战.非等间距GM(1,1)模型背景值构造方法及应用[J].清华大学学报(自然科学版),2007,47(S2):1729-1732.
[64]王叶梅,党耀国,王正新.非等间距GM(1,1)模型背景值的优化[J].中国管理科学,2008,16(4):159-162.
[65]董奋义,田军.背景值和初始条件同时优化的GM(1,1)模型[J].系统工程与电子技术,2008,16(4):159-162.
[66] Liu S F, Deng J L. The range suitable for GM(1,1)[J]. The Journal of Grey System,1996,11(1):131-138.
[67] Li X C. On parameters in Grey Model GM(1,1)[J]. The Journal of Grey System,1998,10(2):155-162.
[68] Kuo C Y, Ching T L, Yen T H. Generalized admissible region of class ratio for GM(1,1)[J]. TheJournal of Grey System,2000,12(2):153-156.
[69] Jin F Y, Hung C L. On some of the basic features of GM(1,1) model(I)[J]. The Journal of GreySystem,1996,8(1):19-36.
[70]王文平,邓聚龙.灰色系统中GM(1,1)模型的混沌特性研究[J].系统工程,1997,15(2):13-16.
[71]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.
[72]徐永高.采油工程中灰色预测模型的病态性诊断[J].武汉理工大学学报(交通科学与工程版),2004,28(5):702~705.
[73]崔杰,党耀国,刘思峰.基于矩阵条件数的NGM(1,1,k)模型病态性研究[J].控制与决策,2010,25(7):1050~1054.
[74]王义闹. GM(1,1)的直接建模方法及性质[J].系统工程理论与实践,1988,8(1):27-31.
[75]王义闹,张建生,国春光.SCGM (1,1)a0的一个性质[J].河北建筑科技学院学报(自然科学版),1998,15(4):62-64.
[76]王义闹,刘光珍,刘开第.GM(1,1)的一种逐步优化直接建模方法[J].系统工程理论与实践,2000,20(9):99-104.
[77]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,2001,29(3):54-57.
[78]王义闹,刘开第,李应川.优化灰导数白化值的GM(1,1)建模法[J].系统工程理论与实践,2001,21(5):124-128.
[79]王义闹,李万庆,王本玉等.一种逐步优化灰导数白化值的GM(1,1)建模方法[J].系统工程理论与实践,2002,22(9):128-131.
[80]王义闹. GM(1,1)逐步优化直接建模方法的推广[J].系统工程理论与实践,2003,23(2):120-124.
[81] Shi B Z. Modeling of non-equigap GM(1,1)[J]. The Journal of Grey Systems,1993,5(2):105-114.
[82] Kuo C Y, Ching T L. Fourier modified non-equigap GM(1,1)[J]. The Journal of Grey Systems,2000,12(2):139-142.
[83]王丰效.多变量非等间距GM(1,m)模型及其应用[J].系统工程与电子技术,2007,29(3):388-390.
[84]陈绵云.制订城市总体规划的灰色系统方法[J].华中理工大学学报(自然科学版),1990,18(3).
[85]陈绵云. SCGM(1, h)c模型和灰色预测模态控制[J].华中理工大学学报(自然科学版),1993,(3):47-52.
[86]陈绵云.镗床控制系统的灰色动态[J].华中工学院学报(自然科学版),1982,10(6).
[87]陈绵云,尹平林,熊辉. SCGM(1,h)c残差修正预测模型及其在柳州市总体规划中的应用[J].华中理工大学学报(自然科学版),1993,21(3).
[88]熊鹰飞,陈绵云,熊和金.系统云SCGM(1,h)模型仿真及应用[J].武汉交通科技大学学报,1999,(3):230-233.
[89]李君伟,陈绵云,董鹏宇等. SCGM(1,1)简化模型及应用[J].武汉交通科技大学学报,2000,(6):615-618.
[90]宋中民,同小军,肖新平.中心逼近式灰色GM(1,1)模型[J].系统工程理论与实践,2001,21(5):110-113.
[91] Song Z M, Wang Z D, Tong X J. Grey Generating space on opposite accurnulation[J]. TheJournal of Grey System,2001,13(4):305-308.
[92]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
[93]谢乃明,刘思峰.一类离散灰色模型及其预测效果研究[J].系统工程学报,2006,21(5):520-523.
[94]谢乃明,刘思峰.离散灰色模型的拓展极其最优化求解[J].系统工程理论与实践,2006,26(6):108-112.
[95]姚天祥,刘思峰.改进的离散灰色预测模型[J].系统工程,2007,25(9):103-106.
[96]曾波,刘思峰.近似非齐次指数序列的DGM(1,1)直接建模法[J].系统工程理论与实践,2010,31(2):297-301.
[97]穆勇.无偏灰色GM(1，1)模型的直接建模法[J].系统工程与电子技术,2003,25(9):1094-1095.
[98]穆勇.一种新的灰色无偏GM(1,1)模型建模方法[J].济南大学学报(自然科学版),2002,16(4):367-369.
[99]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.
[100]邹红波,吉培荣.无偏GM(1,1)模型的动态特性分析[J].三峡大学学报(自然科学版),2006,28(4):334-336.
[101] Chen C K, Tien T L.The indirect measurement of tensile strength by the deterministic greydynamic model DGDM(1,1,1)[J]. International Journal of Systems Science,1997,28(7):683–690.
[102] Chen C K, Tien T L. A new forecasting method of discrete dynamic system [J]. AppliedMathematics and Computation,1997,86(1):61-84
[103] Kuang Y H, Chuen J J. A hybrid model for stock market forecasting and portfolio selection basedon ARX, grey system and RS theories [J]. Expert Systems with Applications2009,(36):5387-5392.
[104] Deng J L. Undulating grey model GM(1,1|tan(k-τ)p, sin(k-τ)p), Journal of Grey System,2001,13(3):201-204.
[105] Jiang X,Wang B W, Chen F X. Based GM(1,1|τ,r)-moving object segmentation, Journal of GreySystem,2003,15(2):101-106.
[106] Huang W C, Kuo M S, et al. Application of GM(1,1|τ,r)to analyze the ports for putting inresources, Journal of Grey System,2004,16(6):211-220.
[107] He X J, Sun G Z. A non-equigap grey model NGM(1,1)[J].Journal of Grey System,2001,13(2):217-222.
[108] Yu Q Z, Li F. Digital watermarking via undulating grey model GM(1,1|tan(k-τ)p,sin(k-τ)p),Journal of Grey System,2002,14(3):217-222.
[109] Liu S F, Lin Y. Grey Systems Theory and Applications [M], Berlin Heidelberg: Springer-Verlag,2010:169-190.
[110] Li G D, Daisuke Y, Masatake N. A GM(1,1)-Markov chain combined model with an applicationto predict the number of Chinese international airlined. Technological Forecasting and SocialChange,2007,74(8):1465-1481.
[111] Liu S F,Li B J, Dang Y G. The G-C-D model and technical advance[J]. Kybernetes: TheInternational Journal of Systems&Cybernetics,2004,33(2):303-309
[112] Liu S F, Zhu Y D. Grey-econometrics combined model[J]. Journal of Grey System,1996,8(1):103-110.
[113]汤少梁,李南,巩在武.灰色绝对关联度组合预测模型的性质研究[J].系统工程与电子技术,2008,30(1):89-92.
[114]曾波,刘思峰,方志耕,谢乃明.灰色组合预测模型及其应用[J].中国管理科学,2009,17(5):150-155.
[115]万玉成,何亚群,盛昭瀚.基于灰色系统与神经网络的航材消耗广义加权函数平均组合预测模型研究[J],系统工程理论与实践,2003,23(7):80-87.
[116]巩林明,张振国.基于灰色小波的网络流量组合预测模型[J].计算机工程与设计,2010,31(8):1660-1667.
[117]赵雪花.基于灰色马尔科夫链的径流序列模式挖掘[J].武汉大学学报(工学版),2008,41(1):1-4.
[118]谭华,谢赤,孙柏等.证券市场灰色神经网络组合预测模型应用研究[J].湖南大学学报(自然科学版),2007,34(9):86-89.
[119]方志耕,刘思峰.区间灰数表征与算法改进及GM(1,1)模型应用研究[J].中国工程科学,2005,7(2):57-61.
[120]曾波,刘思峰,谢乃明等.基于灰数带及灰数层的区间灰数预测模型[J].控制与决策,2010,25(10):1585-1588.
[121] Zeng B, Liu S F, Xie N M. Prediction model of interval grey number based on DGM (1,1)[J],Journal of Systems Engineering and Electronics,2010,21(4):598-603.
[122]曾波,刘思峰.一种基于区间灰数几何特征的灰数预测模型[J].系统工程学报,2011,26(2):122-126.
[123]曾波,刘思峰.白化权函数已知的区间灰数预测模型[J].控制与决策,2010,25(10):1815-1820.
[124]曾波.基于核和灰度的区间灰数预测模型[J].系统工程与电子技术,2011,33(4):821-824.
[125]袁潮清,刘思峰,张可.基于发展趋势和认知程度的区间灰数预测[J].控制与决策,2011,26(2):313-315.
[126]谢乃明,刘思峰.近似非齐次指数序列的离散灰色模型特性研究[J].系统工程与电子技术,2008,30(5):863-867.
[127]高明.一种适用于非齐次指数增长序列的直接型离散灰色模型[J].统计与信息论坛,2010,25(4):30-32.
[128]曾波,刘思峰.近似非齐次指数增长序列的间接DGM(1,1)模型分析[J].统计与洗洗论坛,2010,25(8):30-33
[129]陈鹏宇,段新胜.近似非齐次指数序列的离散GM(1,1)模型的建立及其优化[J].西华大学学报(自然科学版),2010,29(1):89-92.
[130]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(3):93-98.
[131]向跃霖.灰色摆动序列建模方法研究[J].贵州环保科技,2004, l0(l):5-8.
[132]向跃霖.灰色摆动序列的GM(1,1)拟合建模法及其应用[J].化工环保,1998,18(5):299-302.
[133]刘以安,陈松灿,张明俊等.缓冲算子及数据融合技术在目标跟踪中的应用[J].应用科学学报,2006,24(2):154-158.
[134] Liu Sifeng. On measure of grey information[J]. The Journal of Grey System,1995,7(2):97-101.
[135]刘思峰,方志耕,谢乃明.基于核和灰度的区间灰数运算法则[J].系统工程与电子技术,2010,32(2):313-316.
[136]邓聚龙.灰理论基础[J].武汉:华中科技大学出版社,2002年2月:63-70.
[137]刘思峰,林益.灰色灰度的一种公理化定义[J].中国工程科学,2004(8):91-93.
[138]袁潮清,刘思峰.一种基于灰色白化权函数的灰数灰度[J].江南大学学报(自然科学版),2007,6(4):494-496.
[139]王伟民,宫俊峰,魏景刚.梯形重心及应用[J].物理教师,2008,29(7):23-24.
[140] Zeng B，Liu S F，Meng W. Development and application of MSGT6.0(Modeling System of GreyTheory6.0) based on Visual C#and XML [J]. The Journal of Grey System,2011,23(2):145-154.
[141]曾波,刘思峰.基于Visual C#的灰色理论建模系统及其应用[M].第19届灰色系统全国会议论文集.2010,268-271.
[2] Deng J L. The Control problem of grey systems [J]. System&Control Letter.1982,1(5):288-294.
[3] Pawlak Z. Rough Sets: Theoretical aspects of reasoning about data. Dordrecht: Kluwer AcademicPublisher,1991.
[4]王光远.未确知信息及其数学处理[J].哈尔滨建筑大学学报,1990,(4):1-8.
[5]刘思峰,党耀国,方志耕,谢乃明.灰色系统理论及其应用(第五版)[M].北京:科学出版社,2010.
[6] Deng J L. The Primary Methods of Grey Theory[M]．Wuhan: Hua zhong University of Science andTechnology Press,2004:26-34
[7]刘思峰.灰色系统理论的产生与发展[J].南京航空航天大学学报,2004,36(2):267-272.
[8]邓聚龙.灰色控制系统[J],华中工学院学报,1982,10(3):9-18.
[9]肖新平,宋中民,李峰.灰技术基础及其应用[M].北京:科学出版社,2005.
[10]吴惠荣.灰色预测模型的进一步拓广[J].系统工程理论与实践,1994,14(8):31-34.
[11]何斌,蒙清.灰色预测模型拓广方法研究[J].系统工程理论与实践,2002,22(9):137-140.
[12]李翠凤,戴文战.基于函数cotx变换的灰色建模方法[J].系统工程,2005,23(3):110-114.
[13]戴文战,李俊峰.基于函数x-a变换的GM(1,1)模型及在我国农村人均住房面积建模中的应用[J].系统工程理论与实践,2004,24(11):63-67.
[14]戴文战,熊伟,杨爱萍.基于函数cot(xa)变换及背景值优化的灰色建模[J].浙江大学学报(工学版),2010,44(7):1368-1372.
[15]关叶青,刘思峰.基于函数cot(xα)变换的灰色GM(1,1)建模方法[J].系统工程,2008,26(9):89-93.
[16]崔立志,刘思峰.基于数据变换技术的灰色预测模型[J].系统工程,2010,28(5):104-107.
[17] Liu S F. The three axioms of buffer operator and their application[J]. The Journal of Grey System,1991,3(1):39-48.
[18]刘思峰.冲击扰动系统预测陷阱与缓冲算子[J].华中理工大学学报,1997,25(1):28-31.
[19]党耀国,刘思峰.关于弱化缓冲算子的研究[J].中国管理科学,2004,12(2):108-111.
[20]党耀国,刘斌,关叶青.关于强化缓冲算子的研究[J].控制与决策,2005,20(12):1332-1336.
[21]谢乃明,刘思峰.一种新的实用弱化缓冲算子[J].中国管理科学,2003,11:46-48.
[22]关叶青,刘思峰.线性缓冲算子矩阵及其应用研究[J].高校应用数学学报,2008,23(3):357-362.
[23]崔立志,刘思峰,吴正鹏.新的强化缓冲算子的构造及其应用[J].系统工程理论与实践,2010,30(3):484-489.
[24]魏勇,孔新海.几类强弱缓冲算子的构造方法及其内在联系[J].控制与决策,2010,25(2):196-202.
[25]许秀莉,罗键.GM(1,1)模型的改进方法及其应用[J].系统工程与电子技术,2002,24(4):61-63.
[26]尹春华,顾培亮.基于灰色序列生成中缓冲算子的能源预测[J].系统工程学报,2003,18(2):189-192.
[27]邓聚龙.累加生成灰指数律[J].华中工学院学报,1987,15(5):7-12.
[28] Deng J L. Negative power AGO in grey theory[J]. Journal of Grey System,2001,13(3):1-6.
[29] Zhang Q S. Difference information entropy in grey theory[J]. Journal of Grey System.2001,13(2):21-25.
[30] Xiao X P.On parameters in grey models[J].Journal of Grey System,2000,11(4):73-78.
[31] Deng J L. Generating space of grey system[J].Grey System. China Ocean Press,1988:79-90.
[32]宋中民,刘希强,王树泽.生成空间及IM模型[J].系统工程,1992,10(6):37-42.
[33] Deng J L. Moving operator in grey theory[J].The Journal of Grey System,1999,11(1):1-4.
[34] Wen Kunli, John H.Wu. AGO for invariant series[J].The Journal of Grey System.1998,10(1):17-21.
[35] Ai D B, Chen R Q. Frame of AGO Generating space[J].The Journal of Grey System.2001,13(1):13-16.
[36] Song Z M, Wang Z D, Tong X J. Grey Generating space on opposite accurnulation[J]. TheJournal of Grey System,2001,13(4):305-308.
[37] Song Z M, Xiao X P, Deng J L. The character of opposite direction AGO and class ratio[J].TheJournal of Grey System.2002,14(1):9-14.
[38]周慧,王晓光.倒数累加生成灰色GRM(1,1)模型的改进[J].沈阳理工大学学报,2008,27(4):84-86.
[39]宋中民,邓聚龙.反向累加生成及灰色GOM(1,1)模型[J].系统工程,2001,19(1):66-69.
[40]杨知,任鹏,党耀国.反向累加生成与GOM(1,1)模型的优化[J].系统工程理论与实践，2009，29(8):160-164.
[41]黄继,种晓丽.广义累加灰色预测控制模型及其优化算法[J].系统工程理论与实践,2009,29(6):147-156.
[42]陈超英.累加生成的改进和GM(1,1,t)灰色模型[J].数学的实践与认识,2007,37(2):105-109.
[43]戴文战.具有残差校正的n次累加灰色模型[J].系统工程理论与实践,1997,17(12):121-124.
[44]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,(05):38-44.
[45]党耀国,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.
[46]徐涛,冷淑霞.灰色系统模型初始条件的改进及应用[J].山东工程学院学报,1999,13(1):15-19.
[47] Dang Y G, Liu S F. The GM Models That x(n) be Taken as Initial Value [J].The InternationalJournal of Systems&Cyberntics,2004,33(2):247-255.
[48]刘斌,刘思峰,翟振杰等. GM(1,1)模型时间响应函数的最优化[J]中国管理科学,2003,(04):54-57.
[49]熊萍萍,门可佩,吴香华.以x(1)(n)为初始条件的无偏GM(1,1)模型[J].南京信息工程大学学报,2009,1(3):258-263.
[50]钟珞,江琼,张诚等.基于最优初始条件和动态辨识参数的灰色时程数据预测[J]，武汉理工大学学报(交通科学与工程版),2004,28(5):685-691.
[51]周命端,郭际明,文鸿雁等.基于优化初始值的GM(1,1)模型及其在大坝监测中的应用周[J],水电自动化与大坝监测,2008,32(2):52-54.
[52]谭冠军.GM(1,1)模型的背景值构造方法和应用(Ⅰ)[J].系统工程理论与实践,2000,20(4):98-103.
[53]谭冠军.GM(1,1)模型的背景值构造方法和应用(Ⅱ)[J].系统工程理论与实践,2000,20(5):125-128.
[54]谭冠军.GM(1,1)模型的背景值构造方法和应用(Ⅲ)[J].系统工程理论与实践,2000,20(6):70-75.
[55]谢开贵,李春燕,周家启.基于遗传算法的GM(1,1,λ)模型[J].系统工程学报,2000,15(2):168-172.
[56]戴文战,熊伟.基于函数cot(xα)变换及背景值优化的灰色建模[J].浙江大学学报(工学版),2010,44(7):1368-1372.
[57] Wang Z X, Dang Y G, Liu S F. The Optimization of Background Value in GM (1,1) Model[J].The Journal of Grey System,2007,10(2):69-74.
[58]王正新,党耀国,刘思峰.基于离散指数函数优化的GM(1,1)模型王[J].系统工程理论与实践,2008,28(2):61-67.
[59] Hsu C C, Chen CY. A modified grey forecasting model for long-term prediction[J]. Journal of theChinese Institute of Engineers,2003,26(23):301-308.
[60]李俊峰,戴文战.基于插值和Newton-Cores公式的GM(1,1)模型的背景值构造新方法与应用[J].系统工程理论与实践,2004,24(10):122-126.
[61]江南,刘小洋.基于Gauss公式的GM(1,1)模型的背景值构造新方法与应用[J].数学的实践与认识,2008,38(7):90-94.
[62]谢开贵,李春燕等.基于遗传算法的GM(1,1)模型[J].系统工程学报,2000,15(2):168-172.
[63]李翠凤,戴文战.非等间距GM(1,1)模型背景值构造方法及应用[J].清华大学学报(自然科学版),2007,47(S2):1729-1732.
[64]王叶梅,党耀国,王正新.非等间距GM(1,1)模型背景值的优化[J].中国管理科学,2008,16(4):159-162.
[65]董奋义,田军.背景值和初始条件同时优化的GM(1,1)模型[J].系统工程与电子技术,2008,16(4):159-162.
[66] Liu S F, Deng J L. The range suitable for GM(1,1)[J]. The Journal of Grey System,1996,11(1):131-138.
[67] Li X C. On parameters in Grey Model GM(1,1)[J]. The Journal of Grey System,1998,10(2):155-162.
[68] Kuo C Y, Ching T L, Yen T H. Generalized admissible region of class ratio for GM(1,1)[J]. TheJournal of Grey System,2000,12(2):153-156.
[69] Jin F Y, Hung C L. On some of the basic features of GM(1,1) model(I)[J]. The Journal of GreySystem,1996,8(1):19-36.
[70]王文平,邓聚龙.灰色系统中GM(1,1)模型的混沌特性研究[J].系统工程,1997,15(2):13-16.
[71]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.
[72]徐永高.采油工程中灰色预测模型的病态性诊断[J].武汉理工大学学报(交通科学与工程版),2004,28(5):702~705.
[73]崔杰,党耀国,刘思峰.基于矩阵条件数的NGM(1,1,k)模型病态性研究[J].控制与决策,2010,25(7):1050~1054.
[74]王义闹. GM(1,1)的直接建模方法及性质[J].系统工程理论与实践,1988,8(1):27-31.
[75]王义闹,张建生,国春光.SCGM (1,1)a0的一个性质[J].河北建筑科技学院学报(自然科学版),1998,15(4):62-64.
[76]王义闹,刘光珍,刘开第.GM(1,1)的一种逐步优化直接建模方法[J].系统工程理论与实践,2000,20(9):99-104.
[77]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,2001,29(3):54-57.
[78]王义闹,刘开第,李应川.优化灰导数白化值的GM(1,1)建模法[J].系统工程理论与实践,2001,21(5):124-128.
[79]王义闹,李万庆,王本玉等.一种逐步优化灰导数白化值的GM(1,1)建模方法[J].系统工程理论与实践,2002,22(9):128-131.
[80]王义闹. GM(1,1)逐步优化直接建模方法的推广[J].系统工程理论与实践,2003,23(2):120-124.
[81] Shi B Z. Modeling of non-equigap GM(1,1)[J]. The Journal of Grey Systems,1993,5(2):105-114.
[82] Kuo C Y, Ching T L. Fourier modified non-equigap GM(1,1)[J]. The Journal of Grey Systems,2000,12(2):139-142.
[83]王丰效.多变量非等间距GM(1,m)模型及其应用[J].系统工程与电子技术,2007,29(3):388-390.
[84]陈绵云.制订城市总体规划的灰色系统方法[J].华中理工大学学报(自然科学版),1990,18(3).
[85]陈绵云. SCGM(1, h)c模型和灰色预测模态控制[J].华中理工大学学报(自然科学版),1993,(3):47-52.
[86]陈绵云.镗床控制系统的灰色动态[J].华中工学院学报(自然科学版),1982,10(6).
[87]陈绵云,尹平林,熊辉. SCGM(1,h)c残差修正预测模型及其在柳州市总体规划中的应用[J].华中理工大学学报(自然科学版),1993,21(3).
[88]熊鹰飞,陈绵云,熊和金.系统云SCGM(1,h)模型仿真及应用[J].武汉交通科技大学学报,1999,(3):230-233.
[89]李君伟,陈绵云,董鹏宇等. SCGM(1,1)简化模型及应用[J].武汉交通科技大学学报,2000,(6):615-618.
[90]宋中民,同小军,肖新平.中心逼近式灰色GM(1,1)模型[J].系统工程理论与实践,2001,21(5):110-113.
[91] Song Z M, Wang Z D, Tong X J. Grey Generating space on opposite accurnulation[J]. TheJournal of Grey System,2001,13(4):305-308.
[92]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
[93]谢乃明,刘思峰.一类离散灰色模型及其预测效果研究[J].系统工程学报,2006,21(5):520-523.
[94]谢乃明,刘思峰.离散灰色模型的拓展极其最优化求解[J].系统工程理论与实践,2006,26(6):108-112.
[95]姚天祥,刘思峰.改进的离散灰色预测模型[J].系统工程,2007,25(9):103-106.
[96]曾波,刘思峰.近似非齐次指数序列的DGM(1,1)直接建模法[J].系统工程理论与实践,2010,31(2):297-301.
[97]穆勇.无偏灰色GM(1，1)模型的直接建模法[J].系统工程与电子技术,2003,25(9):1094-1095.
[98]穆勇.一种新的灰色无偏GM(1,1)模型建模方法[J].济南大学学报(自然科学版),2002,16(4):367-369.
[99]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.
[100]邹红波,吉培荣.无偏GM(1,1)模型的动态特性分析[J].三峡大学学报(自然科学版),2006,28(4):334-336.
[101] Chen C K, Tien T L.The indirect measurement of tensile strength by the deterministic greydynamic model DGDM(1,1,1)[J]. International Journal of Systems Science,1997,28(7):683–690.
[102] Chen C K, Tien T L. A new forecasting method of discrete dynamic system [J]. AppliedMathematics and Computation,1997,86(1):61-84
[103] Kuang Y H, Chuen J J. A hybrid model for stock market forecasting and portfolio selection basedon ARX, grey system and RS theories [J]. Expert Systems with Applications2009,(36):5387-5392.
[104] Deng J L. Undulating grey model GM(1,1|tan(k-τ)p, sin(k-τ)p), Journal of Grey System,2001,13(3):201-204.
[105] Jiang X,Wang B W, Chen F X. Based GM(1,1|τ,r)-moving object segmentation, Journal of GreySystem,2003,15(2):101-106.
[106] Huang W C, Kuo M S, et al. Application of GM(1,1|τ,r)to analyze the ports for putting inresources, Journal of Grey System,2004,16(6):211-220.
[107] He X J, Sun G Z. A non-equigap grey model NGM(1,1)[J].Journal of Grey System,2001,13(2):217-222.
[108] Yu Q Z, Li F. Digital watermarking via undulating grey model GM(1,1|tan(k-τ)p,sin(k-τ)p),Journal of Grey System,2002,14(3):217-222.
[109] Liu S F, Lin Y. Grey Systems Theory and Applications [M], Berlin Heidelberg: Springer-Verlag,2010:169-190.
[110] Li G D, Daisuke Y, Masatake N. A GM(1,1)-Markov chain combined model with an applicationto predict the number of Chinese international airlined. Technological Forecasting and SocialChange,2007,74(8):1465-1481.
[111] Liu S F,Li B J, Dang Y G. The G-C-D model and technical advance[J]. Kybernetes: TheInternational Journal of Systems&Cybernetics,2004,33(2):303-309
[112] Liu S F, Zhu Y D. Grey-econometrics combined model[J]. Journal of Grey System,1996,8(1):103-110.
[113]汤少梁,李南,巩在武.灰色绝对关联度组合预测模型的性质研究[J].系统工程与电子技术,2008,30(1):89-92.
[114]曾波,刘思峰,方志耕,谢乃明.灰色组合预测模型及其应用[J].中国管理科学,2009,17(5):150-155.
[115]万玉成,何亚群,盛昭瀚.基于灰色系统与神经网络的航材消耗广义加权函数平均组合预测模型研究[J],系统工程理论与实践,2003,23(7):80-87.
[116]巩林明,张振国.基于灰色小波的网络流量组合预测模型[J].计算机工程与设计,2010,31(8):1660-1667.
[117]赵雪花.基于灰色马尔科夫链的径流序列模式挖掘[J].武汉大学学报(工学版),2008,41(1):1-4.
[118]谭华,谢赤,孙柏等.证券市场灰色神经网络组合预测模型应用研究[J].湖南大学学报(自然科学版),2007,34(9):86-89.
[119]方志耕,刘思峰.区间灰数表征与算法改进及GM(1,1)模型应用研究[J].中国工程科学,2005,7(2):57-61.
[120]曾波,刘思峰,谢乃明等.基于灰数带及灰数层的区间灰数预测模型[J].控制与决策,2010,25(10):1585-1588.
[121] Zeng B, Liu S F, Xie N M. Prediction model of interval grey number based on DGM (1,1)[J],Journal of Systems Engineering and Electronics,2010,21(4):598-603.
[122]曾波,刘思峰.一种基于区间灰数几何特征的灰数预测模型[J].系统工程学报,2011,26(2):122-126.
[123]曾波,刘思峰.白化权函数已知的区间灰数预测模型[J].控制与决策,2010,25(10):1815-1820.
[124]曾波.基于核和灰度的区间灰数预测模型[J].系统工程与电子技术,2011,33(4):821-824.
[125]袁潮清,刘思峰,张可.基于发展趋势和认知程度的区间灰数预测[J].控制与决策,2011,26(2):313-315.
[126]谢乃明,刘思峰.近似非齐次指数序列的离散灰色模型特性研究[J].系统工程与电子技术,2008,30(5):863-867.
[127]高明.一种适用于非齐次指数增长序列的直接型离散灰色模型[J].统计与信息论坛,2010,25(4):30-32.
[128]曾波,刘思峰.近似非齐次指数增长序列的间接DGM(1,1)模型分析[J].统计与洗洗论坛,2010,25(8):30-33
[129]陈鹏宇,段新胜.近似非齐次指数序列的离散GM(1,1)模型的建立及其优化[J].西华大学学报(自然科学版),2010,29(1):89-92.
[130]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(3):93-98.
[131]向跃霖.灰色摆动序列建模方法研究[J].贵州环保科技,2004, l0(l):5-8.
[132]向跃霖.灰色摆动序列的GM(1,1)拟合建模法及其应用[J].化工环保,1998,18(5):299-302.
[133]刘以安,陈松灿,张明俊等.缓冲算子及数据融合技术在目标跟踪中的应用[J].应用科学学报,2006,24(2):154-158.
[134] Liu Sifeng. On measure of grey information[J]. The Journal of Grey System,1995,7(2):97-101.
[135]刘思峰,方志耕,谢乃明.基于核和灰度的区间灰数运算法则[J].系统工程与电子技术,2010,32(2):313-316.
[136]邓聚龙.灰理论基础[J].武汉:华中科技大学出版社,2002年2月:63-70.
[137]刘思峰,林益.灰色灰度的一种公理化定义[J].中国工程科学,2004(8):91-93.
[138]袁潮清,刘思峰.一种基于灰色白化权函数的灰数灰度[J].江南大学学报(自然科学版),2007,6(4):494-496.
[139]王伟民,宫俊峰,魏景刚.梯形重心及应用[J].物理教师,2008,29(7):23-24.
[140] Zeng B，Liu S F，Meng W. Development and application of MSGT6.0(Modeling System of GreyTheory6.0) based on Visual C#and XML [J]. The Journal of Grey System,2011,23(2):145-154.
[141]曾波,刘思峰.基于Visual C#的灰色理论建模系统及其应用[M].第19届灰色系统全国会议论文集.2010,268-271.