广义灰色关联分析模型的统一表述及性质
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摘要
系统地研究了初值化、始点零化和初值-始点零化三种关联序列算子的幂等、线性和仿射变换性质,给出了灰色关联接近性公理的数学定义,阐述了灰色关联度正定性、仿射性和仿射保序性的物理意义.揭示广义灰色关联度及其改进均是以序列插值曲线之间面积为基础进行构造的内涵,通过引入分辨系数,分别给出了广义灰色关联度和改进广义灰色关联度的统一表述形式,证明了分辨系数与关联序的独立性,研究了其正定性、平行性、一致性、仿射性、平行保序性、一致保序性、仿射保序性和适用性.
The idempotent, linear and affine properties of initial sequence operator, zero-starting sequence operators and initial zero-starting sequence operator were systematically studied. The closeness axiom of the grey relational degree was defined mathematically. Mathematical definition and physical significance of positive definiteness, affinity and affine transformation isotonicity were also expounded. Generalized grey relational analysis models and their corresponding improvements were essentially constructed based on the areas between sequence interpolation curves. By introducing distinguishing coefficient, the unified representations of generalized relational analysis models and improved models were given respectively. Furthermore, the parallelism, uniformity, affinity, parallel transformation isotonicity, multiple transformation isotonicity and affine transformation isotonicity properties, as well as the applicability were subsequently researched.
The idempotent, linear and affine properties of initial sequence operator, zero-starting sequence operators and initial zero-starting sequence operator were systematically studied. The closeness axiom of the grey relational degree was defined mathematically. Mathematical definition and physical significance of positive definiteness, affinity and affine transformation isotonicity were also expounded. Generalized grey relational analysis models and their corresponding improvements were essentially constructed based on the areas between sequence interpolation curves. By introducing distinguishing coefficient, the unified representations of generalized relational analysis models and improved models were given respectively. Furthermore, the parallelism, uniformity, affinity, parallel transformation isotonicity, multiple transformation isotonicity and affine transformation isotonicity properties, as well as the applicability were subsequently researched.
引文
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[10]刘震,党耀国,周伟杰,等.新型灰色接近关联模型及其拓展[J].控制与决策,2014, 29(6):1071-1075.Liu Z, Dang Y G, Zhou W J, et al. New grey nearness incidence model and its extension[J]. Control&Decision,2014, 29(6):1071-1075.
[11] Luo D, Wei B L, Lin P Y. The optimization of several grey incidence analysis models[J]. Journal of Grey System,2015, 27(4):1-11.
[12]张可,刘思峰.灰色关联聚类在面板数据中的扩展及应用[J].系统工程理论与实践,2010, 30(7):1253-1259.Zhang K, Liu S F. Extended clusters of grey incidences for panel data and its application[J]. Systems Engineering—Theory&Practice, 2010, 30(7):1253-1259.
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[14]吴鸿华,穆勇,屈忠锋,等.基于面板数据的接近性和相似性关联度模型[J].控制与决策,2016, 31(3):555-558.Wu H H, Mu Y, Qu Z F, et al. Similarity and nearness relational degree based on panel data[J]. Control&Decision, 2016, 31(3):555-558.
[15]谢乃明,刘思峰.几类关联度模型的平行性和一致性[J].系统工程,2007, 25(8):98-103.Xie N M, Liu S F. The parallel and uniform properties of several relational models[J]. Systems Engineering, 2007,25(8):98-103.
[16]崔杰,党耀国,刘思峰.几类关联分析模型的新性质[J].系统工程,2009, 27(4):65-70.Cui J, Dang Y G, Liu S F. Novel properties of some grey relational analysis models[J]. Systems Engineering,2009, 27(4):65-70.
[17] Liu S F, Xie N M, Yang Y J, et al. Explanation of terms of sequence operators and grey data mining[J]. Emerald,2016, 6(3):442-447.
[18]蒋诗泉,刘思峰,刘中侠,等.基于面积的灰色关联决策模型[J].控制与决策,2015, 30(4):685-690.Jiang S Q, Liu S F, Liu Z X, et al. Grey incidence decision making model based on area[J]. Control&Decision,2015, 30(4):685-690.
[19]蒋诗泉,刘思峰,刘中侠,等.基于“功能驱动”和“差异驱动”原理的灰关联贴近度决策方法[J].控制与决策,2016,31(1):84-90.Jiang S Q, Liu S F, Liu Z X, et al. Grey incidence decision-making method of close degree based on the principle of “function driver” and “difference driver”[J]. Control&Decision, 2016, 31(1):84-90.
[20]吴利丰,刘思峰.基于灰色凸关联度的面板数据聚类方法及应用[J].控制与决策,2013,28(7):1033-1036.Wu L F, Liu S F. Panel data clustering method based on grey convex relation and its application[J]. Control&Decision, 2013, 28(7):1033-1036.
[21]刘震,党耀国,钱吴永,等.基于面板数据的灰色网格关联度模型[J].系统工程理论与实践,2014, 34(4):991-996.Liu Z,Dang Y G, Qian W Y, et al. Grey grid incidence model based on panel data[J].Systems Engineering—Theory&Practice, 2014, 34(4):991-996.
[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,2014:63-112.Liu S F, Yang Y J, Wu L F, et al. Grey system theory and applications[M]. 7th ed. Beijing:Science Press, 2014:63-112.
[3]刘思峰,谢乃明,FORREST Jeffery.基于相似性和接近性视角的新型灰色关联分析模型[J].系统工程理论与实践,2010,30(5):881-887.Liu S F, Xie N M, Forrest J. On new models of grey incidence analysis based On visual angle of similarity and nearness[J]. Systems Engineering—Theory&Practice, 2010, 30(5):881-887.
[4]刘思峰,蔡华,杨英杰,等.灰色关联分析模型研究进展[J].系统工程理论与实践,2013, 33(8):2041-2046.Liu S F, Cai H, Yang Y J, et al. Advance in grey incidence analysis modelling[J]. Systems Engineering—Theory&Practice, 2013, 33(8):2041-2046.
[5] Liu S F, Yang Y J, Cao Y, et al. A summary on the research of GRA models[J]. Grey Systems:Theory&Applications, 2013, 3(1):7-15.
[6]魏勇,曾柯方.关联度公理的简化与特殊关联度的公理化定义[J].系统工程理论与实践,2015, 35(6):1528-1534.Wei Y, Zeng K F. The simplified relational axioms and the axiomatic definition of special incidence degrees[J].Systems Engineering—Theory&Practice, 2015, 35(6):1528-1534.
[7]陈勇明,张明.灰色样条绝对关联度模型[J].系统工程理论与实践,2015, 35(5):1304-1310.Chen Y M, Zhang M. Cubic spline based grey absolute relational grade model[J]. Systems Engineering—Theory&Practice, 2015, 35(5):1304-1310.
[8]张明,陈勇明.基于相似性和接近性视角的B样条灰色关联分析模型[J].统计与决策,2016(4):12-15.
[9]王靖程,诸文智,张彦斌.基于面积的改进灰关联度算法[J].系统工程与电子技术,2010, 32(4):777-779.Wang J C, Zhu W Z, Zhang Y B. Improved algorithm of grey incidence degree based on area[J]. SystemsEngineering&Electronics, 2010, 32(4):777-779.
[10]刘震,党耀国,周伟杰,等.新型灰色接近关联模型及其拓展[J].控制与决策,2014, 29(6):1071-1075.Liu Z, Dang Y G, Zhou W J, et al. New grey nearness incidence model and its extension[J]. Control&Decision,2014, 29(6):1071-1075.
[11] Luo D, Wei B L, Lin P Y. The optimization of several grey incidence analysis models[J]. Journal of Grey System,2015, 27(4):1-11.
[12]张可,刘思峰.灰色关联聚类在面板数据中的扩展及应用[J].系统工程理论与实践,2010, 30(7):1253-1259.Zhang K, Liu S F. Extended clusters of grey incidences for panel data and its application[J]. Systems Engineering—Theory&Practice, 2010, 30(7):1253-1259.
[13]崔立志,刘思峰.面板数据的灰色矩阵相似关联模型及其应用[J].中国管理科学,2015,23(11):171-176.Cui L Z, Liu S F. Grey matrix similar incidence model for panel data and its application[J]. Chinese Journal of Management Science, 2015, 23(11):171-176.
[14]吴鸿华,穆勇,屈忠锋,等.基于面板数据的接近性和相似性关联度模型[J].控制与决策,2016, 31(3):555-558.Wu H H, Mu Y, Qu Z F, et al. Similarity and nearness relational degree based on panel data[J]. Control&Decision, 2016, 31(3):555-558.
[15]谢乃明,刘思峰.几类关联度模型的平行性和一致性[J].系统工程,2007, 25(8):98-103.Xie N M, Liu S F. The parallel and uniform properties of several relational models[J]. Systems Engineering, 2007,25(8):98-103.
[16]崔杰,党耀国,刘思峰.几类关联分析模型的新性质[J].系统工程,2009, 27(4):65-70.Cui J, Dang Y G, Liu S F. Novel properties of some grey relational analysis models[J]. Systems Engineering,2009, 27(4):65-70.
[17] Liu S F, Xie N M, Yang Y J, et al. Explanation of terms of sequence operators and grey data mining[J]. Emerald,2016, 6(3):442-447.
[18]蒋诗泉,刘思峰,刘中侠,等.基于面积的灰色关联决策模型[J].控制与决策,2015, 30(4):685-690.Jiang S Q, Liu S F, Liu Z X, et al. Grey incidence decision making model based on area[J]. Control&Decision,2015, 30(4):685-690.
[19]蒋诗泉,刘思峰,刘中侠,等.基于“功能驱动”和“差异驱动”原理的灰关联贴近度决策方法[J].控制与决策,2016,31(1):84-90.Jiang S Q, Liu S F, Liu Z X, et al. Grey incidence decision-making method of close degree based on the principle of “function driver” and “difference driver”[J]. Control&Decision, 2016, 31(1):84-90.
[20]吴利丰,刘思峰.基于灰色凸关联度的面板数据聚类方法及应用[J].控制与决策,2013,28(7):1033-1036.Wu L F, Liu S F. Panel data clustering method based on grey convex relation and its application[J]. Control&Decision, 2013, 28(7):1033-1036.
[21]刘震,党耀国,钱吴永,等.基于面板数据的灰色网格关联度模型[J].系统工程理论与实践,2014, 34(4):991-996.Liu Z,Dang Y G, Qian W Y, et al. Grey grid incidence model based on panel data[J].Systems Engineering—Theory&Practice, 2014, 34(4):991-996.