模糊近似空间中模糊粗糙集的新定义及应用
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摘要
在很多实际的系统中均不同程度的存在着不确定性，因此，如何处理这些信息显得尤为重要，智能信息处理越来越受到广大学者的关注，成为当前信息科学与应用研究的一个热点。人们要求计算机像人脑一样，能自行识别和处理客观世界中的不确定问题。
模糊集理论是美国控制论专家扎德于1965年提出的一种处理非精确的现象的数学工具，该工具着眼于集合的模糊性，利用隶属度的概念来表示集合的模糊程度。
粗糙集理论是由波兰教授帕拉克于20世纪80年代初提出的一种研究不完备，不确定知识库和数据的表达，学习，归纳的数学工具。该数学工具着眼于集合的粗糙程度，借助于不可分辨思想来刻画集合的粗糙程度。
既然模糊集理论和粗糙集理论都可运用观察，测试数据表达知识，进行推理，因此，自然要寻求这两者的结合，即作为粗糙集描述近似空间中的知识是模糊概念时，就有模糊粗糙集理论的产生。这样我们可以利用粗糙集的概念考虑模糊集合的粗近似问题，以及利用模糊集的理论研究模糊粗糙集的模糊性。
本文在简要回忆一下模糊集和粗糙集的基本概念后，以粗糙集理论为背景，在模糊近似空间中给出了模糊粗糙集的另一种定义，讨论了该定义与文献定义的模糊粗糙集的区别，论证了本文所定义的模糊粗糙集的近似精度大于文献定义的模糊粗糙集的近似精度；利用模糊集的有关理论定义了模糊粗糙变换，讨论了模糊粗糙变换的性质，并在此基础上给出了模糊粗糙集的扩张定理；在模糊粗糙集概念的基础上，从概率论的出发点来研究模糊粗糙集，首次提出了概率模糊料糙集的概念，讨论了概率模糊粗糙近似算子的性质。研究了Bayes决策与概率模糊粗糙近似的关系，给出其在医疗诊断方面的具体应用。
To some extent, uncertainty exists in most of real systems. Therefore, it is very important how to deal with these kinds of information, and intelligent information proceeding is getting more and more concerned by scholars and becomes a very hot point in theory and application of information science. People need computer to automatically recognize and manage indefinite phenomena in impersonal world. like human cerebra.
Fuzzy set theory, put forward by American cybernetics expert Zadeh, is mathematician tool to conduct the inaccurate phenomena. It focuses on the blurring of sets, and explains the extent of inaccuracy of sets using the concept of membership function.
Rough set theory, put forward by professor Pawlak, in the early of 20 century 80's, is a method to study the expression and leaning of incomplete or uncertain knowledge base. The method focuses on the roughness of sets, uses no-division concept to express the roughness of sets.
Naturally a combination of these two theories can be searched since fuzzy set theory and rough set theory can be used to express knowledge through observing and testing datum. That is, the fuzzy rough set theory comes into being. When the knowledge described in approximate unite is fuzzy notion. Under such circumstance we may analyze rough approximation problems by rough set theory, and study roughness of fuzzy rough set by fuzzy set theory.
In this article, after recalling the basis concepts of fuzzy set and rough set , a
new definition of fuzzy rough set is offered, and the deference between the definition and that of given in document is discussed, the approximate accuracy is larger than that of given in document. Using fuzzy set theory, fuzzy rough conversion is defined and its properties are discussed, and the extension theorem of fuzzy rough set is given. Based on fuzzy rough set theory and probability theory, a kind of probability fuzzy rough set is given first, moreover, it's properties are given and its application in medical diagnose is discussed.
模糊集理论是美国控制论专家扎德于1965年提出的一种处理非精确的现象的数学工具，该工具着眼于集合的模糊性，利用隶属度的概念来表示集合的模糊程度。
粗糙集理论是由波兰教授帕拉克于20世纪80年代初提出的一种研究不完备，不确定知识库和数据的表达，学习，归纳的数学工具。该数学工具着眼于集合的粗糙程度，借助于不可分辨思想来刻画集合的粗糙程度。
既然模糊集理论和粗糙集理论都可运用观察，测试数据表达知识，进行推理，因此，自然要寻求这两者的结合，即作为粗糙集描述近似空间中的知识是模糊概念时，就有模糊粗糙集理论的产生。这样我们可以利用粗糙集的概念考虑模糊集合的粗近似问题，以及利用模糊集的理论研究模糊粗糙集的模糊性。
本文在简要回忆一下模糊集和粗糙集的基本概念后，以粗糙集理论为背景，在模糊近似空间中给出了模糊粗糙集的另一种定义，讨论了该定义与文献定义的模糊粗糙集的区别，论证了本文所定义的模糊粗糙集的近似精度大于文献定义的模糊粗糙集的近似精度；利用模糊集的有关理论定义了模糊粗糙变换，讨论了模糊粗糙变换的性质，并在此基础上给出了模糊粗糙集的扩张定理；在模糊粗糙集概念的基础上，从概率论的出发点来研究模糊粗糙集，首次提出了概率模糊料糙集的概念，讨论了概率模糊粗糙近似算子的性质。研究了Bayes决策与概率模糊粗糙近似的关系，给出其在医疗诊断方面的具体应用。
To some extent, uncertainty exists in most of real systems. Therefore, it is very important how to deal with these kinds of information, and intelligent information proceeding is getting more and more concerned by scholars and becomes a very hot point in theory and application of information science. People need computer to automatically recognize and manage indefinite phenomena in impersonal world. like human cerebra.
Fuzzy set theory, put forward by American cybernetics expert Zadeh, is mathematician tool to conduct the inaccurate phenomena. It focuses on the blurring of sets, and explains the extent of inaccuracy of sets using the concept of membership function.
Rough set theory, put forward by professor Pawlak, in the early of 20 century 80's, is a method to study the expression and leaning of incomplete or uncertain knowledge base. The method focuses on the roughness of sets, uses no-division concept to express the roughness of sets.
Naturally a combination of these two theories can be searched since fuzzy set theory and rough set theory can be used to express knowledge through observing and testing datum. That is, the fuzzy rough set theory comes into being. When the knowledge described in approximate unite is fuzzy notion. Under such circumstance we may analyze rough approximation problems by rough set theory, and study roughness of fuzzy rough set by fuzzy set theory.
In this article, after recalling the basis concepts of fuzzy set and rough set , a
new definition of fuzzy rough set is offered, and the deference between the definition and that of given in document is discussed, the approximate accuracy is larger than that of given in document. Using fuzzy set theory, fuzzy rough conversion is defined and its properties are discussed, and the extension theorem of fuzzy rough set is given. Based on fuzzy rough set theory and probability theory, a kind of probability fuzzy rough set is given first, moreover, it's properties are given and its application in medical diagnose is discussed.
引文
[1] Beaubouef T, Petry F, Arora G.. Information-theoretic measures of uncertainty for rough sets and rough relational databases[J]. Information Science 1998, 1.9:185-195
[2] Chan C C.A rough set approach to attribute generalization in data mining[J].Journal of Information Science 1998, 107:169-176
[3] 施恩伟，林谦 模糊集合论基础[M] 成都：西南交通大学出版社1992：61．
[4] Chemieleqski M R,Grzymala-Busse J W.Global discretization of continous attributes as preprocessing for machine leaning[J]. International Journal of Approximation Reasoning
[5] Jagielska I, Matthews C, Whitfort T, An Inwestigation into the application of neural networks. fuzzy logic, genetic algorithma, and rough sets to automated knowledge acquisition for classification problems[J]. Neurocomputing 1999, 24:37-54
[6] Nanda S and Majumdar S. Fuzzy rough sets. Fuzzy Sets and Systems 45 (1992): 157-160.
[7] Lingras y. a rough logic formalism for fuzzy controllers:a hard and soft computing view[J]. Journal of the American society for information science 1998, 49(5):415-422
[8] Mesherry D. Knowledge Discovery by Inspection[J]. Decision Support System, 1997,21:43-47
[9] Nakamura A. A Rough Logic based on incomplete information and its application[J], International Journal of Approximate Reasoning, 1996, 15:367-378
[10] Pawlak Z, Rough Sets[J]. International Journal of Computer and Information Science 1982,11:341-356
[11] Pawlak Z Rough sets: theoretical aspects of reasoning about data[A], Boston: Kluwer Academic Publishers 1991
[12] Pawlak Z Rough sets approach to knowledge-based decision support[J], European Journal of Operational Research 1997, 99:48-57
[13] Pomerol J C. Artificial Intelligence and Human Decision Making[J] European Journal of Operational Research 1997, 99:3-25
[14] Vakarelov D.A modal logic for similarity relations in Pawlak knowledge representation systems[J]. Fundamental Information 1991, 15:61-79
[15] 程昕，莫智文 模糊粗糙集的分解定理及表现定理[J] 四川师范大学学报 24(2001)111-113
[16] Yao Y Y. Two views of the theory of rough sets in finite universes[J]. International Journal of Approximation Reasoning 1996, 15:291-31
[17] 刘旺金，何家儒 模糊数学导论[M] 成都：四川教育出版社 1992：35～60
[18] Pawlak Z. Rough sets and fuzzy sets[J]. Fuzzy Sets and Systems 1985, (17):99～102
[19] 张文修、吴伟志等 粗糙集理论与方法[M] 科学出版社 2000；
[20] 王国胤编著 粗集理论与知识获取。西安：西安交通大学出版社。2001，5。
[21] 陈挺 决策分析[M] 科学出版社 1987；
[22] 苗夺谦、王玉 粗糙集理论中知识粗糙性与信息熵关系的讨论[J]。模式识别与人工智能，1998，34-40
[23] 苗夺谦、王玉 粗糙集理论中的概念与运算的信息表示[J]。软件学报 1999，10：113-116
[24] 程昕，莫智文 模糊粗糙集及粗糙模糊集的模糊度[J] 模糊系统与数学，2001，15：15-17
[25] 施恩伟 Rough集上不易辨识关系的特征性质[A]。科学通报 17，1990
[26] 朱林、何建敏、常松 粗集与神经网络相结合的股票价格预测模型[M]。中国管理科学，2002，10。
[27] 唐建国 粗集理论处理不完备信息的可信度分析[J]，控制与决策 2002．2．235-238
[28] 郭庆 基于截集的模糊粗糙集及粗糙度 昆明理工大学硕士研究生学位论文；(15)；
[29] 张文修、吴伟志 基于随机集的粗糙集模型(Π) 西安交通大学学报，2001，35(4)：425～429
[30] 刘贵龙 模糊近似空间的模糊粗糙集[M] 模糊系统与数学 2002，16，75-77
[31] 李兵，吴孟达 信息系统中一种基于属性相对重要度的启发式约简算法[M] 模糊系统与数学，2002，16，258-261
[32] 张文修，吴伟志 粗糙集理论介绍与研究综述[J] 模糊系统与数学 2000，4，1-10
[33] 施恩伟 Rough 集理论研究状况[J] 云南师范大学学报 1994，3，7-10
[34] 常犁云，王国胤等一种基于粗糙集的属性约简及规则提取方法[J]，软件学报 1999，11，1206-1211
[35] 王玉、苗夺谦、周育健。关于Rough Set理论与应用的综述[J]。模式识别与人工智能1996，9，327-334
[36] 李兵，吴孟达 粗集研究中的模糊集方法[J] 模糊系统与数学 2002，2，69-73
[37] 孙有发等 基于集统计的模糊神经网络专家系统及应用[J] 模糊系统与数学 2001，12，97-101
[38] 陈遵得，粗集神经网络智能系统及其应用[J] 模式识别与人工智能 1999，1，1-5
[39] 苏健，高济 基于元信息的粗集规则增式生成方法[J] 模式识别与人工智能 2001，4，428-433
[40] 苗夺谦、王玉 基于粗集的多变量决策树构造方法[J] 软件学报 1997，6，425-430
[2] Chan C C.A rough set approach to attribute generalization in data mining[J].Journal of Information Science 1998, 107:169-176
[3] 施恩伟，林谦 模糊集合论基础[M] 成都：西南交通大学出版社1992：61．
[4] Chemieleqski M R,Grzymala-Busse J W.Global discretization of continous attributes as preprocessing for machine leaning[J]. International Journal of Approximation Reasoning
[5] Jagielska I, Matthews C, Whitfort T, An Inwestigation into the application of neural networks. fuzzy logic, genetic algorithma, and rough sets to automated knowledge acquisition for classification problems[J]. Neurocomputing 1999, 24:37-54
[6] Nanda S and Majumdar S. Fuzzy rough sets. Fuzzy Sets and Systems 45 (1992): 157-160.
[7] Lingras y. a rough logic formalism for fuzzy controllers:a hard and soft computing view[J]. Journal of the American society for information science 1998, 49(5):415-422
[8] Mesherry D. Knowledge Discovery by Inspection[J]. Decision Support System, 1997,21:43-47
[9] Nakamura A. A Rough Logic based on incomplete information and its application[J], International Journal of Approximate Reasoning, 1996, 15:367-378
[10] Pawlak Z, Rough Sets[J]. International Journal of Computer and Information Science 1982,11:341-356
[11] Pawlak Z Rough sets: theoretical aspects of reasoning about data[A], Boston: Kluwer Academic Publishers 1991
[12] Pawlak Z Rough sets approach to knowledge-based decision support[J], European Journal of Operational Research 1997, 99:48-57
[13] Pomerol J C. Artificial Intelligence and Human Decision Making[J] European Journal of Operational Research 1997, 99:3-25
[14] Vakarelov D.A modal logic for similarity relations in Pawlak knowledge representation systems[J]. Fundamental Information 1991, 15:61-79
[15] 程昕，莫智文 模糊粗糙集的分解定理及表现定理[J] 四川师范大学学报 24(2001)111-113
[16] Yao Y Y. Two views of the theory of rough sets in finite universes[J]. International Journal of Approximation Reasoning 1996, 15:291-31
[17] 刘旺金，何家儒 模糊数学导论[M] 成都：四川教育出版社 1992：35～60
[18] Pawlak Z. Rough sets and fuzzy sets[J]. Fuzzy Sets and Systems 1985, (17):99～102
[19] 张文修、吴伟志等 粗糙集理论与方法[M] 科学出版社 2000；
[20] 王国胤编著 粗集理论与知识获取。西安：西安交通大学出版社。2001，5。
[21] 陈挺 决策分析[M] 科学出版社 1987；
[22] 苗夺谦、王玉 粗糙集理论中知识粗糙性与信息熵关系的讨论[J]。模式识别与人工智能，1998，34-40
[23] 苗夺谦、王玉 粗糙集理论中的概念与运算的信息表示[J]。软件学报 1999，10：113-116
[24] 程昕，莫智文 模糊粗糙集及粗糙模糊集的模糊度[J] 模糊系统与数学，2001，15：15-17
[25] 施恩伟 Rough集上不易辨识关系的特征性质[A]。科学通报 17，1990
[26] 朱林、何建敏、常松 粗集与神经网络相结合的股票价格预测模型[M]。中国管理科学，2002，10。
[27] 唐建国 粗集理论处理不完备信息的可信度分析[J]，控制与决策 2002．2．235-238
[28] 郭庆 基于截集的模糊粗糙集及粗糙度 昆明理工大学硕士研究生学位论文；(15)；
[29] 张文修、吴伟志 基于随机集的粗糙集模型(Π) 西安交通大学学报，2001，35(4)：425～429
[30] 刘贵龙 模糊近似空间的模糊粗糙集[M] 模糊系统与数学 2002，16，75-77
[31] 李兵，吴孟达 信息系统中一种基于属性相对重要度的启发式约简算法[M] 模糊系统与数学，2002，16，258-261
[32] 张文修，吴伟志 粗糙集理论介绍与研究综述[J] 模糊系统与数学 2000，4，1-10
[33] 施恩伟 Rough 集理论研究状况[J] 云南师范大学学报 1994，3，7-10
[34] 常犁云，王国胤等一种基于粗糙集的属性约简及规则提取方法[J]，软件学报 1999，11，1206-1211
[35] 王玉、苗夺谦、周育健。关于Rough Set理论与应用的综述[J]。模式识别与人工智能1996，9，327-334
[36] 李兵，吴孟达 粗集研究中的模糊集方法[J] 模糊系统与数学 2002，2，69-73
[37] 孙有发等 基于集统计的模糊神经网络专家系统及应用[J] 模糊系统与数学 2001，12，97-101
[38] 陈遵得，粗集神经网络智能系统及其应用[J] 模式识别与人工智能 1999，1，1-5
[39] 苏健，高济 基于元信息的粗集规则增式生成方法[J] 模式识别与人工智能 2001，4，428-433
[40] 苗夺谦、王玉 基于粗集的多变量决策树构造方法[J] 软件学报 1997，6，425-430