模糊软集之间的相似性度量及应用

研究模糊软集的不确定度量问题,给出模糊软集的包含度、相似度公理化定义;基于模糊蕴含算子提出新的模糊软集包含度与相似度度量方法,该方法具有一定的普遍性,在某种程度上提供不同的模糊蕴含算子就可得到不同的包含度与相似度。基于新的相似度度量方法构造了一种决策方法并应用于金融企业流动性检测中。     

广义犹豫模糊软集的相似度量及其应用

在广义模糊软集和犹豫模糊软集的基础上给出广义犹豫模糊软集的概念,并研究广义犹豫模糊软集的相似度量。首先利用三种犹豫模糊集合的包含度,构造犹豫模糊集间的相似度量公式。然后在犹豫模糊集相似度基础上给出广义犹豫模糊软集相似度量的公理化定义,并构造广义犹豫模糊软集的相似度量公式,这些公式可以计算参数集不同时两个广义犹豫模糊软集间的相似度。最后利用广义犹豫模糊软集相似度量方法构造了一种决策方法,并将这个决策方法应用于环境治理问题中。通过实例验证了所提出方法的可行性和有效性。     

区间二型模糊相似度与包含度

相似度与包含度是模糊集合理论中的两个重要概念,但对于二型模糊集合的研究还较为少见．鉴于此,提出了新的区间二型模糊相似度与包含度．首先选择了二者的公理化定义;然后基于公理化定义提出了新的计算公式,并讨论了二者的相互转换关系;最后通过实例来验证二者的性能,并将区间二型模糊相似度与Yang-Shih聚类方法相结合,用于高斯区间二型模糊集合的聚类分析,得到了合理的层次聚类树．仿真实例表明新测度具有一定的实用价值．     

区间值模糊软集的信息测度及其聚类算法

针对区间值模糊软集信息测度难以精确定义的问题,提出了区间值模糊软集的距离测度、相似度、熵、包含度、子集度的公理化定义,给出了区间值模糊软集的信息测度公式,并讨论了它们的转换关系。然后提出了一个基于相似度的聚类算法,该算法结合区间值模糊软集的特性,着重对给出评价对象的具有相似知识水平的专家进行聚类,同时讨论了算法的计算复杂度。最后通过实例说明该算法能有效地处理专家聚类问题。     

Uncertainty measures for interval type-2 fuzzy sets

Fuzziness (entropy) is a commonly used measure of uncertainty for type-1 fuzzy sets. For interval type-2 fuzzy sets (IT2 FSs), centroid, cardinality, fuzziness, variance and skewness are all measures of uncertainties. The centroid of an IT2 FS has been defined by Karnik and Mendel. In this paper, the other four concepts are defined. All definitions use a Representation Theorem for IT2 FSs. Formulas for computing the cardinality, fuzziness, variance and skewness of an IT2 FS are derived. These definitions should be useful in IT2 fuzzy logic systems design using the principles of uncertainty, and in measuring the similarity between two IT2 FSs.     

Fuzzy entropy on intuitionistic fuzzy sets

In this article we exploit the concept of probability for defining the fuzzy entropy of intuitionistic fuzzy sets (IFSs). We then propose two families of entropy measures for IFSs and also construct the axiom definition and properties. Two definitions of entropy for IFSs proposed by Burillo and Bustince in 1996 and Szmidt and Kacprzyk in 2001 are used. The first one allows us to measure the degree of intuitionism of an IFS, whereas the second one is a nonprobabilistic‐type entropy measure with a geometric interpretation of IFSs used in comparison with our proposed entropy of IFSs in the numerical comparisons. The results show that the proposed entropy measures seem to be more reliable for presenting the degree of fuzziness of an IFS. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 443–451, 2006.     

Normalized distance, similarity measure, inclusion measure and entropy of interval-valued fuzzy sets and their relationship

In this paper, we introduce an axiomatic definition of an interval-valued fuzzy sets’ inclusion measure which is different from Bustince’s [H. Bustince, Indicator of inclusion grade for interval-valued fuzzy sets, Applications to approximate reasoning based on interval-valued fuzzy sets, International Journal of Approximate Reasoning, 23 (2000) 137-209]. The relationship among the normalized distance, the similarity measure, the inclusion measure, and the entropy of interval-valued fuzzy sets is investigated in detail. Furthermore, six theorems are proposed showing how the similarity measure, the inclusion measure, and the entropy of interval-valued fuzzy sets can be deduced by the interval-valued fuzzy sets’ normalized distance based on their axiomatic definitions. Some formulas have also been put forward to calculate the similarity measure, the inclusion measure, and the entropy of interval-valued fuzzy sets.     

Inclusion measure for typical hesitant fuzzy sets,the relative similarity measure and fuzzy entropy

Typical hesitant fuzzy sets (THFSs), possessing a finite-set-valued fuzzy membership degrees called typical hesitant fuzzy elements (THFEs), is a special kind of hesitant fuzzy sets. Fuzzy inclusion relationship, as the order structure in fuzzy mathematics, plays an elementary role in the theoretical research and practical applications of fuzzy sets. In this paper, a new partial order for THFEs is defined via the disjunctive semantic meaning of a set, based on which fuzzy inclusion relationship is defined for THFSs. Furthermore, inclusion measures are defined to present the quantitative ranking of every two THFEs and THFSs and different inclusion measures are constructed. The related similarity measure, distance and fuzzy entropy of THFSs are presented and their relationship with inclusion measures are investigated. Finally, an example is given to show that the inclusion measure can be applied effectively in hesitant fuzzy multi-attribute decision making.     

Fuzzy coherence measures

Coherence measures are a tool to compare those fuzzy sets that are sensitive to their own similarity as well as to their fuzzy nature. Within this article we can find three generalizations made about the definition of coherence measures: a first one for any fuzzy set, a second one for any definition about strong negation, and a final one for an extension in those coherence measures that, as a result, do not cause a value in the unit interval, but a fuzzy set in that interval. Tools and properties are offered to create coherence measures. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 1–11, 2005.     

A comparative study of ranking methods, similarity measures and uncertainty measures for interval type-2 fuzzy sets

Ranking methods, similarity measures and uncertainty measures are very important concepts for interval type-2 fuzzy sets (IT2 FSs). So far, there is only one ranking method for such sets, whereas there are many similarity and uncertainty measures. A new ranking method and a new similarity measure for IT2 FSs are proposed in this paper. All these ranking methods, similarity measures and uncertainty measures are compared based on real survey data and then the most suitable ranking method, similarity measure and uncertainty measure that can be used in the computing with words paradigm are suggested. The results are useful in understanding the uncertainties associated with linguistic terms and hence how to use them effectively in survey design and linguistic information processing.     

A new approach to similarity and inclusion measures between general type-2 fuzzy sets

Interval type-2 fuzzy similarity and inclusion measures have been widely studied. In this paper, the axiomatic definitions of general type-2 fuzzy similarity and inclusion measures are given on the basis of interval type-2 fuzzy similarity and inclusion measures. To improve the shortcomings of the existing general type-2 fuzzy similarity and inclusion measures, we define two new general type-2 fuzzy similarity measures and two new general type-2 fuzzy inclusion measures based on $\alpha $ -plane representation theory, respectively, and discuss their related properties. Unlike some existing measures, one of the proposed similarity and inclusion measures are expressed as type-1 fuzzy sets, and therefore these definitions are consistent with the highly uncertain nature of general type-2 fuzzy sets. The theoretical proof is also given to illustrate that the proposed measures are natural extensions of the most popular type-1 fuzzy measures. In the end, the performances of the proposed similarity and inclusion measures are examined.     

Divergence and fuzziness measures

In an axiomatic way a divergence between fuzzy sets is introduced which extends the symmetric difference between crisp sets. Any fuzzy measure of the divergence between two fuzzy sets weighs their “distance”. The distance between a fuzzy set and the family of crisp sets is fuzziness measure.     

Similarity measures,penalty functions,and fuzzy entropy from new fuzzy subsethood measures

In this study, we discuss a new class of fuzzy subsethood measures between fuzzy sets. We propose a new definition of fuzzy subsethood measure as an intersection of other axiomatizations and provide two construction methods to obtain them. The advantage of this new approach is that we can construct fuzzy subsethood measures by aggregating fuzzy implication operators which may satisfy some properties widely studied in literature. We also obtain some of the classical measures such as the one defined by Goguen. The relationships with fuzzy distances, penalty functions, and similarity measures are also investigated. Finally, we provide an illustrative example which makes use of a fuzzy entropy defined by means of our fuzzy subsethood measures for choosing the best fuzzy technique for a specific problem.     

Inclusion measures of probabilistic linguistic term sets and their application in classifying cities in the Economic Zone of Chengdu Plain

The probabilistic linguistic term set is a powerful tool to express and characterize people’s cognitive complex information and thus has obtained a great development in the last several years. To better use the probabilistic linguistic term sets in decision making, information measures such as the distance measure, similarity measure, entropy measure and correlation measure should be defined. However, as an important kind of information measure, the inclusion measure has not been defined by scholars. This study aims to propose the inclusion measure for probabilistic linguistic term sets. Formulas to calculate the inclusion degrees are put forward Then, we introduce the normalized axiomatic definitions of the distance, similarity and entropy measures of probabilistic linguistic term sets to construct a unified framework of information measures for probabilistic linguistic term sets. Based on these definitions, we present the relationships and transformation functions among the distance, similarity, entropy and inclusion measures. We believe that more formulas to calculate the distance, similarity, inclusion degree and entropy can be induced based on these transformation functions. Finally, we put forward an orthogonal clustering algorithm based on the inclusion measure and use it in classifying cities in the Economic Zone of Chengdu Plain, China.     

Entropy of interval-valued fuzzy sets based on distance and its relationship with similarity measure

This article proposes a new axiomatic definition of entropy of interval-valued fuzzy sets (IVFSs) and discusses its relation with similarity measure. First, we propose an axiomatic definition of entropy for IVFS based on distance which is consistent with the axiomatic definition of entropy of a fuzzy set introduced by De Luca, Termini and Liu. Next, some formulae are derived to calculate this kind of entropy. Furthermore we investigate the relationship between entropy and similarity measure of IVFSs and prove that similarity measure can be transformed by entropy. Finally, a numerical example is given to show that the proposed entropy measures are more reasonable and reliable for representing the degree of fuzziness of an IVFS.     

Entropy, similarity measure of interval-valued intuitionistic fuzzy sets and their applications

In this paper we propose an entropy measure for interval-valued intuitionistic fuzzy sets, which generalizes three entropy measures defined independently by Szmidt, Wang and Huang, for intuitionistic fuzzy sets. We also give an approach to construct similarity measures using entropy measures for interval-valued intuitionistic fuzzy sets. In particular, the proposed entropy measure for interval-valued intuitionistic fuzzy sets can yield a similarity measure. Several illustrative examples are given to demonstrate the practicality and effectiveness of the proposed formulas. We apply the similarity measure to solve problems on pattern recognitions, multi-criteria fuzzy decision making and medical diagnosis.     

Complexity reduction and interpretability improvement for fuzzy rule systems based on simple interpretability measures and indices by bi-objective evolutionary rule selection

The aim of this paper is to develop a general post-processing methodology to reduce the complexity of data-driven linguistic fuzzy models, in order to reach simpler fuzzy models preserving enough accuracy and better fuzzy linguistic performance with respect to their initial values. This post-processing approach is based on rule selection via the formulation of a bi-objective problem with one objective focusing on accuracy and the other on interpretability. The latter is defined via the aggregation of several interpretability measures, based on the concepts of similarity and complexity of fuzzy systems and rules. In this way, a measure of the fuzzy model interpretability is given. Two neuro-fuzzy systems for providing initial fuzzy models, Fuzzy Adaptive System ART based and Neuro-Fuzzy Function Approximation and several case studies, data sets from KEEL Project Repository, are used to check this approach. Both fuzzy and neuro-fuzzy systems generate Mamdani-type fuzzy rule-based systems, each with its own particularities and complexities from the point of view of the fuzzy sets and the rule generation. Based on these systems and data sets, several fuzzy models are generated to check the performance of the proposal under different restrictions of complexity and fuzziness.     

On a similarity measure between LR‐type fuzzy numbers and its application to database acquisition

This article presents a new similarity measure for LR‐type fuzzy numbers. The proposed similarity measure is based on a defined metric between LR‐type fuzzy numbers. It is known that an exponential operation is highly useful in dealing with the classical Shannon entropy and cluster analysis. We adopted, therefore, the exponential operation on this metric. Furthermore, we analyze its properties and make numerical comparisons to several similarity measures. The results show that the proposed similarity measure can overcome the drawbacks of the existing similarity measures. We then apply it to compound attributes for handling null queries to database systems. These applications can also be widely used in fuzzy queries to databases. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 1001–1016, 2005.     

A new approach for fuzzy risk analysis based on similarity measures of generalized fuzzy numbers

In this paper, we present a new method for fuzzy risk analysis based on similarity measures between generalized fuzzy numbers. First, we present a new similarity measure between generalized fuzzy numbers. It combines the concepts of geometric distance, the perimeter and the height of generalized fuzzy numbers for calculating the degree of similarity between generalized fuzzy numbers. We also prove some properties of the proposed similarity measure. We make an experiment to use 15 sets of generalized fuzzy numbers to compare the experimental results of the proposed method with the existing similarity measures. The proposed method can overcome the drawbacks of the existing similarity measures. Based on the proposed similarity measure between generalized fuzzy numbers, we present a new fuzzy risk analysis algorithm for dealing with fuzzy risk analysis problems, where the values of the evaluating items are represented by generalized fuzzy numbers. The proposed method provides a useful way to deal with fuzzy risk analysis problems.