D0L:10.13374.issn1001-053x.2012.08.001 第34卷第8期 北京科技大学学报 Vol.34 No.8 2012年8月 Journal of University of Science and Technology Beijing Aug.2012 基于粗糙商集的模糊隶属度集值统计算法 方宏伟2》李长洪1,2四 纪洪广1,2) 方玲玲) 1)北京科技大学金属矿山高效开采与安全教有部重点实验室，北京1000832)北京科技大学土木与环境工程学院，北京100083 3)苏州大学计算机科学与技术学院，苏州215006 区通信作者，E-mail:lch@usth.cdu.cn 摘要模糊隶属度无统一算法，定义存在分歧.根据模糊概念“内涵明确，外延不明确”的特点，定义隶属度为不同外延对内 涵的从属程度.在信息系统中，概念的外延用对象表示，内涵由属性表示，由此提出了求解隶属度的新算法：由原始统计数据 组成初始信息系统，用粗糙集理论求得其商集并构建集值信息系统：该集值信息系统对应的条件概率空间中的条件概率即为 隶属度.广义上信息系统可分为信息系统（无决策属性）和目标信息系统（有决策属性）两类.隶属度也可分为两类：第一类外 延对象为内涵属性本身值，如年龄对青年人的隶属度（信息系统）：第二类外延对象为不同于内涵属性的另一属性值，如边坡 工程安全系数对稳定状态的隶属度（目标信息系统）.计算以上两个实例，前者与己有结论作对比验证，后者与函数选择、经 典统计方法及贝叶斯公理作对比验证，可知结果可靠，算法可行. 关键词模糊集理论；隶属函数；粗糙集：信息系统：条件概率 分类号0159 Set-valued statistical algorithm for fuzzy membership based on the rough quotient set FANG Hong-wei.2》,LI Chang+-hong2回，JI Hong3-guang2，FANG Ling-mg 1)Key Laboratory of the Ministry of Education of China for High-Efficient Mining and Safety of Metal Mines,University of Science of Technology Beijing, Beijing 100083,China 2)School of Civil and Environmental Engineering,University of Science and Technology Beijing,Beijing 100083,China 3)School of Computer Science and Technology,Soochow University,Suzhou 215006,China Corresponding author,E-mail:lch@ustb.edu.cn ABSTRACT There is no uniform algorithm for fuzzy membership,and the definitions differ.According to the characteristic of the fuzzy concept "the meaning is clear and the extension is ambiguous",the membership is defined as the subordinate degree of different extensions to the connotation.In information systems,the extension of the theory of knowledge discovery is expressed by the object, and the meaning is expressed by its attributes.Based on the research results,a new algorithm for calculating the membership was proposed:the initial information systems are composed of original statistical data,and the set-valued information system is constructed by the quotient set which uses the rough set theory:in the set-valued information system,the conditional probability in the correspond- ing conditional probability space is the membership.In general,the information systems are divided into information systems without decision attributes and target information systems with decision-making attributes.The membership is also divided into two categories: firstly,the content of the extension object is the value of the property itself,such as young people to the age (information system); secondly,the extension object is different from the content attribute value of another property,such as engineering safety factor to stability (target information system).These two instances were calculated,the former is compared with the existing research results and the latter is verified by the function selection,classical statistical method and Bayesian formula;it is shown that the algorithm is feasible and the results are reliable. KEY WORDS fuzzy set theory:membership functions:rough sets:information systems:conditional probability 模糊隶属度（或隶属函数）是模糊数学应用与实践的基石，其确定（或建立）存在两个问题0：(1) 收稿日期：2011-一10-一10 基金项目：北京市重点学科共建项目：国家重点基础研究发展计划资助项目(2010CB731501,2010CB731506)第 34 卷 第 8 期 2012 年 8 月 北京科技大学学报 Journal of University of Science and Technology Beijing Vol． 34 No． 8 Aug． 2012 基于粗糙商集的模糊隶属度集值统计算法 方宏伟1，2) 李长洪1，2) ! 纪洪广1，2) 方玲玲3) 1) 北京科技大学金属矿山高效开采与安全教育部重点实验室，北京 100083 2) 北京科技大学土木与环境工程学院，北京 100083 3) 苏州大学计算机科学与技术学院，苏州 215006 !通信作者，E-mail: lch@ ustb． edu． cn 摘 要 模糊隶属度无统一算法，定义存在分歧． 根据模糊概念“内涵明确，外延不明确”的特点，定义隶属度为不同外延对内 涵的从属程度． 在信息系统中，概念的外延用对象表示，内涵由属性表示，由此提出了求解隶属度的新算法: 由原始统计数据 组成初始信息系统，用粗糙集理论求得其商集并构建集值信息系统; 该集值信息系统对应的条件概率空间中的条件概率即为 隶属度． 广义上信息系统可分为信息系统( 无决策属性) 和目标信息系统( 有决策属性) 两类． 隶属度也可分为两类: 第一类外 延对象为内涵属性本身值，如年龄对青年人的隶属度( 信息系统) ; 第二类外延对象为不同于内涵属性的另一属性值，如边坡 工程安全系数对稳定状态的隶属度( 目标信息系统) ． 计算以上两个实例，前者与已有结论作对比验证，后者与函数选择、经 典统计方法及贝叶斯公理作对比验证，可知结果可靠，算法可行． 关键词 模糊集理论; 隶属函数; 粗糙集; 信息系统; 条件概率 分类号 O159 Set-valued statistical algorithm for fuzzy membership based on the rough quotient set FANG Hong-wei 1，2) ，LI Chang-hong1，2) ! ，JI Hong-guang1，2) ，FANG Ling-ling3) 1) Key Laboratory of the Ministry of Education of China for High-Efficient Mining and Safety of Metal Mines，University of Science of Technology Beijing， Beijing 100083，China 2) School of Civil and Environmental Engineering，University of Science and Technology Beijing，Beijing 100083，China 3) School of Computer Science and Technology，Soochow University，Suzhou 215006，China !Corresponding author，E-mail: lch@ ustb． edu． cn ABSTRACT There is no uniform algorithm for fuzzy membership，and the definitions differ． According to the characteristic of the fuzzy concept“the meaning is clear and the extension is ambiguous”，the membership is defined as the subordinate degree of different extensions to the connotation． In information systems，the extension of the theory of knowledge discovery is expressed by the object， and the meaning is expressed by its attributes． Based on the research results，a new algorithm for calculating the membership was proposed: the initial information systems are composed of original statistical data，and the set-valued information system is constructed by the quotient set which uses the rough set theory; in the set-valued information system，the conditional probability in the correspond￾ing conditional probability space is the membership． In general，the information systems are divided into information systems without decision attributes and target information systems with decision-making attributes． The membership is also divided into two categories: firstly，the content of the extension object is the value of the property itself，such as young people to the age ( information system) ; secondly，the extension object is different from the content attribute value of another property，such as engineering safety factor to stability ( target information system) ． These two instances were calculated，the former is compared with the existing research results and the latter is verified by the function selection，classical statistical method and Bayesian formula; it is shown that the algorithm is feasible and the results are reliable． KEY WORDS fuzzy set theory; membership functions; rough sets; information systems; conditional probability 收稿日期: 2011--10--10 基金项目: 北京市重点学科共建项目; 国家重点基础研究发展计划资助项目( 2010CB731501，2010CB731506) 模糊隶属度( 或隶属函数) 是模糊数学应用与 实践的基石，其确定( 或建立) 存在两个问题［1］: ( 1) DOI:10.13374/j.issn1001-053x.2012.08.001