Fuzzy Entropy： Axiomatic Definition and Neural Networks Model

The measure of uncertainty is adopted as a measure of information. The measures of fuzziness are known as fuzzy information measures. The measure of a quantity of fuzzy information gained from a fuzzy set or fuzzy system is known as fuzzy entropy. Fuzzy entropy has been focused and studied by many researchers in various fields. In this paper, firstly, the axiomatic definition of fuzzy entropy is discussed. Then, neural networks model of fuzzy entropy is proposed, based on the computing capability of neural networks. In the end, two examples are discussed to show the efficiency of the model.     

加法与乘法下的模糊熵

由于模糊集关于普通加法与乘法运算构成的代数系统是分配格、布尔代数,这就使得模糊集的应用受到一定的限制,为了扩展模糊集的应用,本文引进了模糊集的加法与乘法运算,接着研究了模糊集关于加法与乘法的运算性质以及模糊熵,距离测度,相似性测度在集合关于加法运算,乘法运算下的一些性质,并且对加法与乘法下的模糊熵与普通模糊熵作了对比.     

直观模糊集的熵、距离测度、相似测度及它们的关系

熵、距离测度和相似测度是模糊集理论中的三个重要概念.首先系统地给出了直观模糊集的熵、距离测度和相似测度的公理化定义,并讨论了它们之间的一些基本关系.然后在距离测度公理化定义的基础上产生了一些新的直观模糊集的熵公式.     

模糊熵与距离测度的相互诱导及其应用

模糊信息论就是利用模糊数学这一工具来研究带有模糊不确定性的信息的．模糊熵和距离测度是模糊信息论中两个重要的度量方法．本文主要讨论模糊熵和距离测度之间的相互关系,由此得到几个由模糊熵诱导的距离测度公式和几个由距离测度诱导出的模糊熵公式,说明了模糊熵和距离测度是可以相互诱导的．最后,举例说明距离测度公式在模式识别中的应用．     

区间直觉模糊集的模糊熵群决策方法

针对专家权重未知、专家判断信息以区间直觉模糊集给出的多属性群决策问题,提出了一种新的模糊熵决策方法。通过定义区间直觉模糊集的模糊熵判断专家信息的模糊程度,进而确定每位专家的权重;然后计算备选方案距理想方案和负理想方案的模糊交叉熵距离,得到每个专家对方案的排序;再分别利用加权算术算子和加权几何算子集结专家的排序结果,得到专家群体对方案的排序。实例分析验证了方法的有效性。     

Some similarity measures of intuitionistic fuzzy sets and their applications to multiple attribute decision making

Atanassov (1986) defined the notion of intuitionistic fuzzy set, which is a generalization of the notion of Zadeh’ fuzzy set. In this paper, we first develop some similarity measures of intuitionistic fuzzy sets. Then, we define the notions of positive ideal intuitionistic fuzzy set and negative ideal intuitionistic fuzzy set. Finally, we apply the similarity measures to multiple attribute decision making under intuitionistic fuzzy environment.     

基于熵和关联系数的概率对偶犹豫模糊多属性决策方法

针对概率对偶犹豫模糊环境下属性权重完全未知的多属性决策问题,提出基于熵和关联系数的多属性决策方法。首先定义了概率对偶犹豫模糊熵的公理化定义和公式,然后基于概率对偶犹豫模糊集的特征信息集合和熵测度定义了概率对偶犹豫模糊集的关联系数,最后根据概率对偶犹豫模糊集的熵和关联系数构建多属性决策模型,并通过算例验证了该模型的有效性和合理性。     

Intuitionistic and interval-valued intutionistic fuzzy preference relations and their measures of similarity for the evaluation of agreement within a group

Szmidt and Kacprzyk (Lecture Notes in Artificial Intelligence 3070:388–393, 2004a) introduced a similarity measure, which takes into account not only a pure distance between intuitionistic fuzzy sets but also examines if the compared values are more similar or more dissimilar to each other. By analyzing this similarity measure, we find it somewhat inconvenient in some cases, and thus we develop a new similarity measure between intuitionistic fuzzy sets. Then we apply the developed similarity measure for consensus analysis in group decision making based on intuitionistic fuzzy preference relations, and finally further extend it to the interval-valued intuitionistic fuzzy set theory.     

Measures of fuzzy sets and measures of fuzziness

The measures presented in this paper are defined by using Weber's concept of decomposable measures m of crisp sets, having in particular the Archimedean decomposable operations in view (Section 2). Measures $\text{m}$ of fuzzy sets are introduced as integrals with respect to m. For the Archimedean cases, Weber's integral will be used as alternative to Sugeno's and Choquet's concepts (Section 3). What ‘fuzziness’ means will be described by functions of fuzziness F (another name: entropy N-functions) with respect to a negation. In addition to the types of functions of fuzziness which are induced by concave functions, we discuss also the ones which are induced by fuzzy connectives (Section 4). Now, using m for measuring the ‘importance of items’ and F for the ‘fuzziness’ of the possible values of a fuzzy set ?, m?(F ° ?) serves us as a measure of the fuzziness $\text{F}$? of ?. The concepts of De Luca and Termini, Capocelli and De Luca, Kaufmann, Knopfmacher, Loo, Gottwald, Dombi and, under the restriction to the Archimedean cases, also the concepts of Trillas and Riera and Yager turn out to be special cases (Section 5).     

基于Vague集信息的多属性群决策专家水平评判方法

针对基于Vague集信息的多属性群决策专家水平评判问题提出了两种评判方法.首先引进了基于Vague集信息的多属性群决策信息体(即决策信息体)的相关概念,通过决策信息体构造了基于Vague集信息的一致性决策矩阵及模糊熵,其次利用Vague集信息的相似度量以及Vague集信息的模糊熵两种信息不确定性度量方法,对基于Vague集信息的多属性群决策专家水平评判问题提出了两种评判方法,即统计分析方法和模糊熵分析方法,对专家的评判水平进行排序.最后,通过一个算例说明两种方法的一致性、有效性和实用性.     

关于粗糙模糊集的相似度量

基于对不确定性信息处理的背景,定义了粗糙模糊值与粗糙模糊集的相似度量,研究了它们的有关性质.     

L—Fuzzy集上的Fuzzy值函数的Fuzzy积分的收敛

本文引入了L-Fuzzy集合上的Fuzzy值函数关于Fuzzy值Fuzzy测度的Fuzzy值Fuzzy积分的概念,给出上述概念的几个等价定义,讨论其基本性质,得到一系列积分序列的收敛定理。     

基于犹豫模糊二元语义的多属性决策方法

深入研究了犹豫模糊二元语义多属性决策问题。首先利用幂均算子给出了犹豫模糊二元语义集的均值函数,并基于均匀分布概率准则和二元语义的距离测度提出了犹豫模糊二元语义集两两比较的可能度公式,进一步给出了可能度排序公式的性质。针对属性值为犹豫模糊二元语义集的多属性决策问题,提出了一种基于熵权的多属性决策方法。最后结合实际问题,验证了该方法的有效性和可行性。     

广义概率对偶犹豫模糊语言幂集结算子在多属性决策中的应用

在对偶犹豫模糊语言集、概率对偶犹豫模糊集和广义幂集结算子的基础上,研究了在概率对偶犹豫模糊语言环境下的广义幂集结算子问题。首先,给出了概率对偶犹豫模糊语言集的定义、运算规则、得分函数、精确函数、距离测度、熵。然后,定义了广义概率对偶犹豫模糊语言幂集结算子,并研究其具有的性质。其次,提出了一种决策方法来解决集结数据之间存在相互关系的概率对偶犹豫模糊语言多属性决策问题。最后,结合相关案例验证了该方法的有效性和可行性。     

基于双边犹豫模糊非均衡语言优先集成算子的应急响应预案评估方法

在应用多属性决策理论求解应急响应预案评估问题时,问题结构的复杂性往往使决策者评价信息存在高度不确定性且属性相对重要性仅能以优先级关系来表征。为此,本文首先提出了双边犹豫模糊非均衡语言集这种新型信息形式以使决策者能够灵活有效的表征复杂评价信息,并定义了运算法则、熵和距离测度;其次,基于熵测度开发了双边犹豫模糊非均衡语言优先加权集成算子,并构建了能够考虑属性优先关系的多属性决策方法;进一步针对属性相对重要性不能由定性分析获得的情况,设计了客观权重确定方法,并构建了另一种更具实际灵活性的VIKOR决策方法;最后,实例研究表明了方法的有效性与优势。     

基于Vague集的模糊多目标决策方法及应用

针对目前基于Vague集多目标决策中Vague值计算困难以及确定目标满意度的下界和不满意度的上界存在主观随意性问题.提出了一种基于Vague集的模糊多目标决策方法.利用属性数学中的属性集和属性测度理论构造目标的真隶属度函数、假隶属度函数和犹豫度函数,从而可计算出目标的Vague值;采用记分函数计算方案的多目标评分值,从而可以对方案进行排序并选择出最优方案.应用实例验证了该方法的有效性和实用性.     

基于模糊熵的直觉模糊多属性群决策方法

针对专家权重未知、专家判断信息以直觉模糊集给出的多属性群决策问题,提出了一种新的决策方法.通过定义直觉模糊集的模糊熵计算专家判断信息的模糊程度,进而确定每位专家的权重.然后定义直觉模糊集的模糊交叉熵确定备选方案距理想方案和负理想方案的距离,再根据加权算术算子集结专家的判断信息,得到方案的排序.最后,通过一个实例分析验证了方法的有效性.     

Similarity,inclusion and entropy measures between type-2 fuzzy sets based on the Sugeno integral

Similarity measures of type-2 fuzzy sets are used to indicate the similarity degree between type-2 fuzzy sets. Inclusion measures for type-2 fuzzy sets are the degrees to which a type-2 fuzzy set is a subset of another type-2 fuzzy set. The entropy of type-2 fuzzy sets is the measure of fuzziness between type-2 fuzzy sets. Although several similarity, inclusion and entropy measures for type-2 fuzzy sets have been proposed in the literatures, no one has considered the use of the Sugeno integral to define those for type-2 fuzzy sets. In this paper, new similarity, inclusion and entropy measure formulas between type-2 fuzzy sets based on the Sugeno integral are proposed. Several examples are used to present the calculation and to compare these proposed measures with several existing methods for type-2 fuzzy sets. Numerical results show that the proposed measures are more reasonable than existing measures. On the other hand, measuring the similarity between type-2 fuzzy sets is important in clustering for type-2 fuzzy data. We finally use the proposed similarity measure with a robust clustering method for clustering the patterns of type-2 fuzzy sets.     

Gaussian kernel based fuzzy rough sets: Model, uncertainty measures and applications

Kernel methods and rough sets are two general pursuits in the domain of machine learning and intelligent systems. Kernel methods map data into a higher dimensional feature space, where the resulting structure of the classification task is linearly separable; while rough sets granulate the universe with the use of relations and employ the induced knowledge granules to approximate arbitrary concepts existing in the problem at hand. Although it seems there is no connection between these two methodologies, both kernel methods and rough sets explicitly or implicitly dwell on relation matrices to represent the structure of sample information. Based on this observation, we combine these methodologies by incorporating Gaussian kernel with fuzzy rough sets and propose a Gaussian kernel approximation based fuzzy rough set model. Fuzzy T-equivalence relations constitute the fundamentals of most fuzzy rough set models. It is proven that fuzzy relations with Gaussian kernel are reflexive, symmetric and transitive. Gaussian kernels are introduced to acquire fuzzy relations between samples described by fuzzy or numeric attributes in order to carry out fuzzy rough data analysis. Moreover, we discuss information entropy to evaluate the kernel matrix and calculate the uncertainty of the approximation. Several functions are constructed for evaluating the significance of features based on kernel approximation and fuzzy entropy. Algorithms for feature ranking and reduction based on the proposed functions are designed. Results of experimental analysis are included to quantify the effectiveness of the proposed methods.     

Correspondence between fuzzy measures and classical measures

In this paper we study the question whether, given a fuzzy measure (as defined in [3] and [4]). there exists a classical measure such that the fuzzy measure of a measurable fuzzy set μ equals the classical measure of the area below the membership function of μ. The results are that in the case of finite additivity there is a one-to-one correspondence between classical measures and fuzzy measures, whereas in the case of countable additivity this result only holds for generated fuzzy σ-algebras. Finally, some connections of that problem with the existence of an extension of a fuzzy measure defined on an arbitrary fuzzy σ-algebra σ to the generated fuzzy σ-algebra $\text{σ}$ are discussed.