模糊集理论的新拓展及其应用研究
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摘要
自1965年美国著名控制论专家L.Zadeh教授提出Fuzzy集理论以来,模糊理论在许多领域得到了成功的运用。Fuzzy集是经典集合论的扩充和发展,随着模糊知识处理技术的发展,近年来多种新的拓展模糊概念相继被引入,如直觉模糊集、Grey集、L ?模糊集、区间值模糊集、Vague集等。对这些拓展模糊集合之间的相互关系、拓展模糊集合之间的相互转化进行了详细分析,并给出了变换算子。
在各种拓展模糊集合理论中Vague集(直觉模糊集)的研究最具有代表性,着重分析了Vague集的性质,给出了一种Vague集的三维表示法,并讨论了在三维表示下Vague集的运算规则及其性质。Vague集是Fuzzy集理论的自然扩展,依据投票模型,在考虑中立者的思想倾向的基础上,提出三种Vague集向Fuzzy集的转化方法。模糊熵是模糊集模糊性度量的一个重要工具,传统的Vague集的模糊性度量仅仅考虑了元素的模糊性,而没有考虑隶属度的模糊性,因此存在许多不合理之处。对元素的模糊性和元素隶属度的模糊性二者进行综合考虑,引入了犹豫强度和冲突强度的概念,定义了一种新的Vague集的模糊熵,实例表明该模糊熵的定义是十分合理的。并且通过投票模型,在考虑投票模型中中立者的思想倾向变化的基础上,提出了连续Vague集的概念。
以Vague集为代表,通过考察模糊理论的发展历程,根据隶属函数值域的特征对模糊集及各种拓展模糊集重新进行分类,将经典集合和Fuzzy集归类为点值集合,Vague集等拓展模糊集合归类为区间值集合。区间值集合虽然比点值集合能更准确地刻画模糊信息,但是它的缺陷是人们对元素隶属度在区间内部的分布状况一无所知,为了解决这个问题,提出了正态模糊集合模型。一个正态模糊集合建立了论域中元素到一个正态分布函数簇的映射,它很直观生动地刻画出了元素对集合的隶属度的分布情况。对正态模糊集合的交、并、补等运算性质进行了详细研究,并给出了正态模糊集合的模糊性度量和相似性度量方法。正态模糊集合是模糊集理论新的推广,详细讨论了Fuzzy集、Vague集和正态模糊集三者之间的相互关系,并给出了具体的转化方法。提出了基于正态模糊集合的Vague集的相似度量,实例表明该相似度量方法不但优于现存的各种相似度量方法,而且适用于语言变量。
模糊集理论的提出为软划分提供了有力的分析工具,于是模糊聚类逐步成为了研究的热点。模糊聚类分析包括两个重要的研究内容:模式的相似性度量和聚类算法。正态模糊集合为模式的相似性度量提供了新的方法,由于正态模糊集和Vague集的天然联系,通过正态模糊集合能很容易地给出一种新的区间值数据的模糊聚类算法,实例表明该算法能得出较满意的聚类效果。
Since the theory of fuzzy sets was proposed by professor L.Zadeh, it has been successfully applied in many fields. Fuzzy set is the extension of classical crisp set. In recent years, many new fuzzy sets extensions have been proposed along with the development of fuzzy set theory, such as Intuitionistic Fuzzy Sets, Grey Sets, L-Fuzzy Sets, Interval-valued Fuzzy Sets and Vague Sets, etc., and the relationships among these fuzzy set extensions are analyzed in detail.
Vague set (Intuitionistic Fuzzy Sets) is the most representative theory among those fuzzy set extensions. A kind of three dimension expression of Vague Sets was given; the operation formula and properties were discussed. Based on the consideration of the tendency of neutrals in the vote model, three transformations from Vague Sets to Fuzzy Sets were proposed. Fuzzy entropy is an important method to measure the fuzzy degree of Vague Sets, but the traditional fuzzy entropy of Vague Sets are not reasonable because they only considered the fuzziness of elements and ignore the fuzziness of subject degree. We consider the fuzziness of elements and subject degree together, the concepts of hesitate degree and conflict degree are introduced, then, a new kind of fuzzy entropy of vague sets is presented, examples show that this kind of fuzzy entropy definition is very reasonable. Finally, the definition of continuous vague sets was proposed.
Based on the characteristic of the values of the subjecting functions, after investigating the development process of fuzzy theory, we reclassify the reclassical crisp set and Fuzzy Set as point value set; put Vague Set and other fuzzy set extensions into interval valued sets. Even the interval valued set can describe the fuzzy information very well, but is can not show the distribution of the subject degree. In order to solve this problem, we propose the concept of Normal Distribution Fuzzy Set. The Normal Distribution Fuzzy Set establish a map from a discussed domain X to a set of normal distribution functions, the every normal distribution function can directly describe the distribution and subject degree of the corresponding element. The properties of union、 intersection and complementation about the Normal Distribution Fuzzy Set are discussed. The fuzziness degree of Normal Distribution Fuzzy Set is proposed. Normal Distribution Fuzzy Set is the extension of Vague Set, then, the relationships and mutual transformation among Fuzzy Set、Vague Set and Normal Distribution Fuzzy Set are discussed. The similarity measure between Vague Sets based on normal distribution is given. Examples show that this method is better than any existed similarity measures; furthermore, this method is suit to be used in linguistic variables.
The fuzzy theory provides a powerful tool for soft classification. It has been used to deal with cluster problems since it was presented. Fuzzy cluster analysis includes two main contents: the similarity measure between two patterns and the cluster algorithm. Base on the spontaneous relationship between Normal Distribution Fuzzy Sets and Vague Set, Normal Dstribution Fzzy Sets provide a new similarity measure method for patterns and a new kind of cluster algorithm of interval values. It has been proved can gain satisfied result easily.
在各种拓展模糊集合理论中Vague集(直觉模糊集)的研究最具有代表性,着重分析了Vague集的性质,给出了一种Vague集的三维表示法,并讨论了在三维表示下Vague集的运算规则及其性质。Vague集是Fuzzy集理论的自然扩展,依据投票模型,在考虑中立者的思想倾向的基础上,提出三种Vague集向Fuzzy集的转化方法。模糊熵是模糊集模糊性度量的一个重要工具,传统的Vague集的模糊性度量仅仅考虑了元素的模糊性,而没有考虑隶属度的模糊性,因此存在许多不合理之处。对元素的模糊性和元素隶属度的模糊性二者进行综合考虑,引入了犹豫强度和冲突强度的概念,定义了一种新的Vague集的模糊熵,实例表明该模糊熵的定义是十分合理的。并且通过投票模型,在考虑投票模型中中立者的思想倾向变化的基础上,提出了连续Vague集的概念。
以Vague集为代表,通过考察模糊理论的发展历程,根据隶属函数值域的特征对模糊集及各种拓展模糊集重新进行分类,将经典集合和Fuzzy集归类为点值集合,Vague集等拓展模糊集合归类为区间值集合。区间值集合虽然比点值集合能更准确地刻画模糊信息,但是它的缺陷是人们对元素隶属度在区间内部的分布状况一无所知,为了解决这个问题,提出了正态模糊集合模型。一个正态模糊集合建立了论域中元素到一个正态分布函数簇的映射,它很直观生动地刻画出了元素对集合的隶属度的分布情况。对正态模糊集合的交、并、补等运算性质进行了详细研究,并给出了正态模糊集合的模糊性度量和相似性度量方法。正态模糊集合是模糊集理论新的推广,详细讨论了Fuzzy集、Vague集和正态模糊集三者之间的相互关系,并给出了具体的转化方法。提出了基于正态模糊集合的Vague集的相似度量,实例表明该相似度量方法不但优于现存的各种相似度量方法,而且适用于语言变量。
模糊集理论的提出为软划分提供了有力的分析工具,于是模糊聚类逐步成为了研究的热点。模糊聚类分析包括两个重要的研究内容:模式的相似性度量和聚类算法。正态模糊集合为模式的相似性度量提供了新的方法,由于正态模糊集和Vague集的天然联系,通过正态模糊集合能很容易地给出一种新的区间值数据的模糊聚类算法,实例表明该算法能得出较满意的聚类效果。
Since the theory of fuzzy sets was proposed by professor L.Zadeh, it has been successfully applied in many fields. Fuzzy set is the extension of classical crisp set. In recent years, many new fuzzy sets extensions have been proposed along with the development of fuzzy set theory, such as Intuitionistic Fuzzy Sets, Grey Sets, L-Fuzzy Sets, Interval-valued Fuzzy Sets and Vague Sets, etc., and the relationships among these fuzzy set extensions are analyzed in detail.
Vague set (Intuitionistic Fuzzy Sets) is the most representative theory among those fuzzy set extensions. A kind of three dimension expression of Vague Sets was given; the operation formula and properties were discussed. Based on the consideration of the tendency of neutrals in the vote model, three transformations from Vague Sets to Fuzzy Sets were proposed. Fuzzy entropy is an important method to measure the fuzzy degree of Vague Sets, but the traditional fuzzy entropy of Vague Sets are not reasonable because they only considered the fuzziness of elements and ignore the fuzziness of subject degree. We consider the fuzziness of elements and subject degree together, the concepts of hesitate degree and conflict degree are introduced, then, a new kind of fuzzy entropy of vague sets is presented, examples show that this kind of fuzzy entropy definition is very reasonable. Finally, the definition of continuous vague sets was proposed.
Based on the characteristic of the values of the subjecting functions, after investigating the development process of fuzzy theory, we reclassify the reclassical crisp set and Fuzzy Set as point value set; put Vague Set and other fuzzy set extensions into interval valued sets. Even the interval valued set can describe the fuzzy information very well, but is can not show the distribution of the subject degree. In order to solve this problem, we propose the concept of Normal Distribution Fuzzy Set. The Normal Distribution Fuzzy Set establish a map from a discussed domain X to a set of normal distribution functions, the every normal distribution function can directly describe the distribution and subject degree of the corresponding element. The properties of union、 intersection and complementation about the Normal Distribution Fuzzy Set are discussed. The fuzziness degree of Normal Distribution Fuzzy Set is proposed. Normal Distribution Fuzzy Set is the extension of Vague Set, then, the relationships and mutual transformation among Fuzzy Set、Vague Set and Normal Distribution Fuzzy Set are discussed. The similarity measure between Vague Sets based on normal distribution is given. Examples show that this method is better than any existed similarity measures; furthermore, this method is suit to be used in linguistic variables.
The fuzzy theory provides a powerful tool for soft classification. It has been used to deal with cluster problems since it was presented. Fuzzy cluster analysis includes two main contents: the similarity measure between two patterns and the cluster algorithm. Base on the spontaneous relationship between Normal Distribution Fuzzy Sets and Vague Set, Normal Dstribution Fzzy Sets provide a new similarity measure method for patterns and a new kind of cluster algorithm of interval values. It has been proved can gain satisfied result easily.
引文
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