粗糙集模型与粗糙代数的研究
|
摘要
粗糙集(RS)理论不但是一种新型的处理模糊和不确定知识的数学工具,而且是一个新颖、有效的软计算方法,目前已经在机器学习、知识发现、决策分析、金融数据分析、人工智能、数据挖掘、模式识别等方面得到了广泛的应用。同时,纯粹的数学理论与粗糙集理论结合起来进行研究已经有文章出现,并不断有新的数学概念出现,如:“半群中的粗理想”、“粗糙不变子群”、“粗糙群与粗糙子群”、“粗糙群的同态与同构”。当然,随着粗糙结构与代数结构、拓扑结构、序结构等各种结构的不断整合,必将不断涌现新的富有生机的数学分支。此外,Pawlak粗糙集模型的推广也一直是RS理论研究的主流方向,目前主要有构造性方法和代数性(公理化)方法。
本文在现有的Pawlak粗糙集模型基础上,对模型进行推广;根据粗糙结构和代数结构及已有的研究成果的基础上,更进一步地研究了粗糙集理论在代数系统一群上的应用,以此建立较为完善的粗糙代数系统。主要结果如下:
1、把Pawlak粗糙集模型中的论域U向其幂集上提升,提出粗糙幂集的概念;把Pawlak粗糙集模型中单一论域的模型推广为双论域的情形;研究了S-粗集模型的代数性质。
2、在粗糙集代数系统中,引入粗糙模糊群的概念,讨论了模糊群与环上的粗糙集特性;在粗糙集类中引入超代数运算,提出群上的粗糙幂群的概念,并由此讨论它的一些性质;介绍粗糙集理论在决策分析中的应用,利用粗糙集间的相似度量,讨论决策条件与结论匹配的一类决策问题。
论文分为六章:
第一章对粗糙集理论的研究现状和发展前景进行介绍;
第二章介绍了粗糙集理论的基本知识及其特点;
第三章回顾群论的基本知识;
第四章研究了粗糙集模型的推广问题包括粗糙幂集、双论域模型以及S—粗集副集的代数性质讨论;
Rough set theory is not only a new mathematical tool dealing with vagueness and uncertainty but also a new and effective soft computing method.It was been widely used in the area of machine learning,knowledge discovery,decision anaysis,artificial intelligence,date mining,pattern recognition,ect.At some time,some articles that have been studied about the theory combined pure mathematics with rough sets have been emerged,and some new mathematical notions,such as rough ideal insemigroups,rough invariant subgroups,rough groupsand rough subgroups,homomorphism and isomorphism of rough groups,are introduced.Certainly,with integration of rough structure and algebra structure,topology structure,order structure and the other structure,some new vital mathematical branches will be emerged. In addition,extention of the premitive model is a main current direction of study,The chief way are by the way of structural and algebral.In this paper,the main work is to extended the premitive model of rough sets;Accorting to the Rough structure and algebra structure,the main work is to study the application of rough set theory on algebra system-group,so that rough algebra system is better perfectly.The general process as follows:1. Upgraded their universes to their power sets, the concept of rough power sets will be raised firstly and point out several related properties; The premitive model of rough sets is extended by the way of structural,The model of rough set on the double universe is advanced and point out several related properties; discussed the algebraic property of assistant sets of single-direction S-rough sets in approximate space.2. Discussed the algebra system of the rough set, introduces such concepts as rough-fuzzy subgroup, gives several relations properties; upgraded their universes to their power sets, the concept of rough hypergroup will be raised firstly and point out several related properties;discussed the application of rough sets theory on decision analysis, Using measures of
本文在现有的Pawlak粗糙集模型基础上,对模型进行推广;根据粗糙结构和代数结构及已有的研究成果的基础上,更进一步地研究了粗糙集理论在代数系统一群上的应用,以此建立较为完善的粗糙代数系统。主要结果如下:
1、把Pawlak粗糙集模型中的论域U向其幂集上提升,提出粗糙幂集的概念;把Pawlak粗糙集模型中单一论域的模型推广为双论域的情形;研究了S-粗集模型的代数性质。
2、在粗糙集代数系统中,引入粗糙模糊群的概念,讨论了模糊群与环上的粗糙集特性;在粗糙集类中引入超代数运算,提出群上的粗糙幂群的概念,并由此讨论它的一些性质;介绍粗糙集理论在决策分析中的应用,利用粗糙集间的相似度量,讨论决策条件与结论匹配的一类决策问题。
论文分为六章:
第一章对粗糙集理论的研究现状和发展前景进行介绍;
第二章介绍了粗糙集理论的基本知识及其特点;
第三章回顾群论的基本知识;
第四章研究了粗糙集模型的推广问题包括粗糙幂集、双论域模型以及S—粗集副集的代数性质讨论;
Rough set theory is not only a new mathematical tool dealing with vagueness and uncertainty but also a new and effective soft computing method.It was been widely used in the area of machine learning,knowledge discovery,decision anaysis,artificial intelligence,date mining,pattern recognition,ect.At some time,some articles that have been studied about the theory combined pure mathematics with rough sets have been emerged,and some new mathematical notions,such as rough ideal insemigroups,rough invariant subgroups,rough groupsand rough subgroups,homomorphism and isomorphism of rough groups,are introduced.Certainly,with integration of rough structure and algebra structure,topology structure,order structure and the other structure,some new vital mathematical branches will be emerged. In addition,extention of the premitive model is a main current direction of study,The chief way are by the way of structural and algebral.In this paper,the main work is to extended the premitive model of rough sets;Accorting to the Rough structure and algebra structure,the main work is to study the application of rough set theory on algebra system-group,so that rough algebra system is better perfectly.The general process as follows:1. Upgraded their universes to their power sets, the concept of rough power sets will be raised firstly and point out several related properties; The premitive model of rough sets is extended by the way of structural,The model of rough set on the double universe is advanced and point out several related properties; discussed the algebraic property of assistant sets of single-direction S-rough sets in approximate space.2. Discussed the algebra system of the rough set, introduces such concepts as rough-fuzzy subgroup, gives several relations properties; upgraded their universes to their power sets, the concept of rough hypergroup will be raised firstly and point out several related properties;discussed the application of rough sets theory on decision analysis, Using measures of
引文
[1] Z.Pawlak.Rough Sets[j].international Journal of computer and information seiences,1982(11):341-356.
[2] Dubios D.Prade H.Twofold fuzzy sets and rough sets-somes assues in knowledge reprsentation.Fuzzy sets and System, 1987,23:3～18
[3] Z.Pawlak.Rough Sets-Theoretical Aspects of Reasoning about Data[M]. Kluwer Academic Publishers, Dordreche, 1991
[4] Chan C C. A rough set approach to attribute generalization in date mining. Jounal of Internation Sciences, 1998,107:169～176
[5] Lingras P J,Yao Y Y.Date mining using extensions of the rough set model. Jounal of American society for Information science,1998,49(5):415～422
[6] Mc Sherry D. Knowledge discorery by inspection.Ddecion support system, 1997,21:43-47
[7] Z.Pawlak.Rough Set approach to konwlege-based dection support. European Jounal of operational research,1997,99:48～57
[8] Pomerol JC.Artificial intelligence and human decision making.European Journal of Operation Rearch, 1997,99:3～25
[9] Yao Y Y, Lingras P.Internations of belief functions in the theory of rough sets[J]. internation Sciences, 1998, 104: 81～106
[10] Yao Y Y, Lin T Y. Generalization of rough sets using modal logic. Intelligent Automation and Soft computing, 1996, 2: 103～120
[11] Yao Y Y. Reational interpretations of neighborhood operators and rough set approximation. Information sciences, 1998, 111: 239～259
[12] Nand S,Majumdar S.Fuzzy rough sets.Fuzzy Sets and System, 1992,45:157～160
[13] Bodianova S.approximation of fuzzy conceps in decision making.Fuzzy Sets and System,1997,51:147～209
[14] Dubois D,Prade H.Rough fuzzy sets and fuzzy rough sets.International Journal of General Systems,1990,17:191~209
[15] Kuncheva L I.Fuzzy rough sets:Application to feture selection.Fuzzy Sets and System,1992,51:147~153
[16]Morsi N N,Yakout M M.Axiomatic for fuzzy rough sets. Fuzzy Sets and System,1998,100:327~342
[17]Pawlak Z,Wong S K M,Ziarko W.Rough sets:probabilistic versus deter- ministi approach.Internation Journal of Man-Machine studies,1998,29:81~ 95
[18] Yao Y Y.A decision theoretic framework for approximating concepts. Internation Journal of Man-Machine studies, 1992,37:793~809
[19] Ziarko W.Variable precision rough set model. Journal ofComputer and System Sciences,1993,46:39~59
[20] Wong S K,Ziarko W.Comparision of the probabilistic approximate classification and the fuzzy set model.Fuzzy sets and Systems, 1987,21:357 ~362
[21] Banerjee M,Pal S K.Rouhgness of fuzzy set. Information Sciences. 1996,93:235~246
[22] Beaubouef T,Petry F,Arora G.Information-hteoretic measures of uncertainty for rough setsand rough relational databases.Information Sciences, 1998,109:185~195
[23] Yao Y Y.Tow views of the theory of rough sets in finite universes. International Journal of Approximate Reasioning,1996,15:291~317
[24] Jagielska I,Mathews C,Whitfort T.An investigation into the application of neural newworks,fuzzy logic,genetic algorithms,and rough sets to automated knowledge acquisitional for classification problems. Neurocomputing,1999,24:37~54
[25] Lin T Y.A rough logic formalisn foe fuzzy controller:A hard and soft conputing view. International Journal of Approximate Reasioning,1996,15: 395~414
[26] Nakamura A.A rough logic based on incomplete information and its application. International Journal of Approximate Reasioning, 1996, 15: 367～378
[27] Kuroki N.Rough ideals in semigroups.Information Sciences, 1997, 1000: 139～163
[28] R. Biswas and S. Nanda. Rough groups and rough subgroup.Bull.Polish, Acad. Sci. Math, 1994, 42: 251～254
[29] 刘键勤.粗糙集理论及其最新进展.计算机技术与自动化,1998,17:43～48
[30] 刘清,黄兆华,姚力文.Rough集理论:现状与前景.计算机科学,1997,24:1～5
[31] 张文修,吴伟志.粗糙集理论介绍和研究综述.模糊系统与数学,2000,14:1～12
[32] 胡可云,陆玉昌,石纯一.粗糙集理论及其应用进展.清华大学学报,2001,1:64～68
[33] 曾黄麟.粗糙集理论及其应用—关于数据推理的新方法.重庆:重庆大学出版社,1998
[34] 王国胤.Rough集理论与知识获取.西安:西安交通出版社,2001
[35] 张文修,吴伟志,梁吉业,李德玉.粗糙集理论与方法.北京:科学出版社,1998
[36] 刘清.Rough集及Rough推理.北京:科学出版社,2001
[37] 苗夺谦,王珏.粗糙集理论中知识粗糙性与信息熵关系的讨论.模式识别与人工智能,1998,11:34～40
[38] 苗夺谦,王珏.粗糙集理论概念与运算信息表示.软件学报,1999,10:113～116
[39] 王珏,苗夺谦,周育健.关于Rough Sets理论与应用的综述.模式识别与人工智能,1996,9:337～334
[40] 李莉.基于可变精度粗集模型的增量式归纳学习.计算机科学,1999,26:55～58
[41] 李德玉,苗夺谦.粗糙半群的性质.计算机科学,2001,28(5.专刊):42～43
[42] 韩素青,胡桂荣.粗糙陪集、粗糙不变子群.计算机科学,2001,28(5.专
[2] Dubios D.Prade H.Twofold fuzzy sets and rough sets-somes assues in knowledge reprsentation.Fuzzy sets and System, 1987,23:3～18
[3] Z.Pawlak.Rough Sets-Theoretical Aspects of Reasoning about Data[M]. Kluwer Academic Publishers, Dordreche, 1991
[4] Chan C C. A rough set approach to attribute generalization in date mining. Jounal of Internation Sciences, 1998,107:169～176
[5] Lingras P J,Yao Y Y.Date mining using extensions of the rough set model. Jounal of American society for Information science,1998,49(5):415～422
[6] Mc Sherry D. Knowledge discorery by inspection.Ddecion support system, 1997,21:43-47
[7] Z.Pawlak.Rough Set approach to konwlege-based dection support. European Jounal of operational research,1997,99:48～57
[8] Pomerol JC.Artificial intelligence and human decision making.European Journal of Operation Rearch, 1997,99:3～25
[9] Yao Y Y, Lingras P.Internations of belief functions in the theory of rough sets[J]. internation Sciences, 1998, 104: 81～106
[10] Yao Y Y, Lin T Y. Generalization of rough sets using modal logic. Intelligent Automation and Soft computing, 1996, 2: 103～120
[11] Yao Y Y. Reational interpretations of neighborhood operators and rough set approximation. Information sciences, 1998, 111: 239～259
[12] Nand S,Majumdar S.Fuzzy rough sets.Fuzzy Sets and System, 1992,45:157～160
[13] Bodianova S.approximation of fuzzy conceps in decision making.Fuzzy Sets and System,1997,51:147～209
[14] Dubois D,Prade H.Rough fuzzy sets and fuzzy rough sets.International Journal of General Systems,1990,17:191~209
[15] Kuncheva L I.Fuzzy rough sets:Application to feture selection.Fuzzy Sets and System,1992,51:147~153
[16]Morsi N N,Yakout M M.Axiomatic for fuzzy rough sets. Fuzzy Sets and System,1998,100:327~342
[17]Pawlak Z,Wong S K M,Ziarko W.Rough sets:probabilistic versus deter- ministi approach.Internation Journal of Man-Machine studies,1998,29:81~ 95
[18] Yao Y Y.A decision theoretic framework for approximating concepts. Internation Journal of Man-Machine studies, 1992,37:793~809
[19] Ziarko W.Variable precision rough set model. Journal ofComputer and System Sciences,1993,46:39~59
[20] Wong S K,Ziarko W.Comparision of the probabilistic approximate classification and the fuzzy set model.Fuzzy sets and Systems, 1987,21:357 ~362
[21] Banerjee M,Pal S K.Rouhgness of fuzzy set. Information Sciences. 1996,93:235~246
[22] Beaubouef T,Petry F,Arora G.Information-hteoretic measures of uncertainty for rough setsand rough relational databases.Information Sciences, 1998,109:185~195
[23] Yao Y Y.Tow views of the theory of rough sets in finite universes. International Journal of Approximate Reasioning,1996,15:291~317
[24] Jagielska I,Mathews C,Whitfort T.An investigation into the application of neural newworks,fuzzy logic,genetic algorithms,and rough sets to automated knowledge acquisitional for classification problems. Neurocomputing,1999,24:37~54
[25] Lin T Y.A rough logic formalisn foe fuzzy controller:A hard and soft conputing view. International Journal of Approximate Reasioning,1996,15: 395~414
[26] Nakamura A.A rough logic based on incomplete information and its application. International Journal of Approximate Reasioning, 1996, 15: 367～378
[27] Kuroki N.Rough ideals in semigroups.Information Sciences, 1997, 1000: 139～163
[28] R. Biswas and S. Nanda. Rough groups and rough subgroup.Bull.Polish, Acad. Sci. Math, 1994, 42: 251～254
[29] 刘键勤.粗糙集理论及其最新进展.计算机技术与自动化,1998,17:43～48
[30] 刘清,黄兆华,姚力文.Rough集理论:现状与前景.计算机科学,1997,24:1～5
[31] 张文修,吴伟志.粗糙集理论介绍和研究综述.模糊系统与数学,2000,14:1～12
[32] 胡可云,陆玉昌,石纯一.粗糙集理论及其应用进展.清华大学学报,2001,1:64～68
[33] 曾黄麟.粗糙集理论及其应用—关于数据推理的新方法.重庆:重庆大学出版社,1998
[34] 王国胤.Rough集理论与知识获取.西安:西安交通出版社,2001
[35] 张文修,吴伟志,梁吉业,李德玉.粗糙集理论与方法.北京:科学出版社,1998
[36] 刘清.Rough集及Rough推理.北京:科学出版社,2001
[37] 苗夺谦,王珏.粗糙集理论中知识粗糙性与信息熵关系的讨论.模式识别与人工智能,1998,11:34～40
[38] 苗夺谦,王珏.粗糙集理论概念与运算信息表示.软件学报,1999,10:113～116
[39] 王珏,苗夺谦,周育健.关于Rough Sets理论与应用的综述.模式识别与人工智能,1996,9:337～334
[40] 李莉.基于可变精度粗集模型的增量式归纳学习.计算机科学,1999,26:55～58
[41] 李德玉,苗夺谦.粗糙半群的性质.计算机科学,2001,28(5.专刊):42～43
[42] 韩素青,胡桂荣.粗糙陪集、粗糙不变子群.计算机科学,2001,28(5.专