A hierarchy-based similarity measure for intuitionistic fuzzy sets

Due to some unreasonable results obtained from most current similarity measures for intuitionistic fuzzy sets (IFSs), we introduce a necessary condition to obtain a stronger definition of similarity measures for IFSs, and present a new similarity measure derived from a general idea of similarity measures for concepts on a lattice. In experiments, we focus our attention on two basic directions of performance evaluation: one is how much the proposed measure is reasonable and the other is how much accuracy the measure produces when it is applied to classification problems. The experimental results show that the proposed measure is reasonable and achieves a satisfactory performance on classification problems.     

软直觉模糊集

直觉模糊集是在模糊集上增加了一个新的属性参数：非隶属度函数,成为描述“非此非彼”的“模糊概念”的工具。为了提高决策的精确性,将软集与直觉模糊集相结合,构造了一种新的数学模型,即直觉模糊软集,对其性质进行了一些讨论。     

一种新的直觉模糊集的熵

通过分析现有直觉模糊熵计算方法中存在的一些问题,基于直觉模糊集理论和熵的概念,提出一种新的直觉模糊集的熵,给出它的结构、性质,并将其与现有的几种直觉模糊集的熵进行比较,发现其计算结果介于其他几种方法的计算结果之间,因而能够对模糊问题的模糊性做出更好的界定,有助于人们对模糊问题的理解和把握。最后,通过实例验证了它的有效性。     

Clustering algorithm for intuitionistic fuzzy sets

The intuitionistic fuzzy set (IFS) theory, originated by Atanassov [K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87-96], has been used in a wide range of applications, such as logic programming, medical diagnosis, pattern recognition, and decision making, etc. However, so far there has been little investigation of the clustering techniques of IFSs. In this paper, we define the concepts of association matrix and equivalent association matrix, and introduce some methods for calculating the association coefficients of IFSs. Then, we propose a clustering algorithm for IFSs. The algorithm uses the association coefficients of IFSs to construct an association matrix, and utilizes a procedure to transform it into an equivalent association matrix. The λ-cutting matrix of the equivalent association matrix is used to cluster the given IFSs. Moreover, we extend the algorithm to cluster interval-valued intuitionistic fuzzy sets (IVIFSs), and finally, demonstrate the effectiveness of our clustering algorithm by experimental results.     

单向S-粗直觉模糊集

针对系统具有的模糊、不确定及动态特性,基于直觉模糊集和单向S-粗集理论,提出单向S-粗直觉模糊集,给出它的结构、性质、存在背景和意义解释。分析了单向S-粗直觉模糊集与Pawlak粗集,单向S-粗模糊集以及单向S-粗集等之间的关系。     

区间直觉模糊粗糙集

将模糊粗糙集推广到区间直觉模糊粗糙集,基于区间直觉模糊等价关系和两个论域之间的一般区间直觉模糊关系,给出了区间直觉模糊粗糙集模型的不同形式,并讨论了一些相关性质。     

改进的粗糙直觉模糊集

在Pawlak近似空间中,针对直觉模糊目标集合,假设在信息粒度不变的情况下,试图寻求目标集合更好的近似集。在现有的粗糙直觉模糊集的基础之上,利用直觉模糊粗糙隶属函数,采用分段函数的形式建立直觉模糊集新的下近似与上近似算子,并讨论新模型的一些基本性质。与现有的粗糙直觉模糊集相比,改进后的模型无论在近似精度方面,还是与目标集合的相似度方面,都有了较大的改善和提高。最后通过数值算例验证了所给结论的正确性。     

二型直觉模糊集

二型模糊集和直觉模糊集都具有很强的实际应用背景.二型模糊集增强了系统处理不确定性的能力,直觉模糊集为解决人们判断问题所出现的犹豫信息提供了理论依据.本文在二型模糊集和直觉模糊集的基础上,给出了二型直觉模糊集的概念,证明了二型直觉模糊集是一型模糊集、直觉模糊集、区间值模糊集、区间值直觉模糊集的广义形式,讨论了二型直觉模糊集的基本运算和二型直觉模糊关系.最后,研究了基于二型直觉模糊理论的近似推理,并实例说明了二型直觉模糊集的实际应用背景.     

Generalized aggregation operators for intuitionistic fuzzy sets

The generalized ordered weighted averaging (GOWA) operators are a new class of operators, which were introduced by Yager (Fuzzy Optim Decision Making 2004;3:93–107). However, it seems that there is no investigation on these aggregation operators to deal with intuitionistic fuzzy or interval‐valued intuitionistic fuzzy information. In this paper, we first develop some new generalized aggregation operators, such as generalized intuitionistic fuzzy weighted averaging operator, generalized intuitionistic fuzzy ordered weighted averaging operator, generalized intuitionistic fuzzy hybrid averaging operator, generalized interval‐valued intuitionistic fuzzy weighted averaging operator, generalized interval‐valued intuitionistic fuzzy ordered weighted averaging operator, generalized interval‐valued intuitionistic fuzzy hybrid average operator, which extend the GOWA operators to accommodate the environment in which the given arguments are both intuitionistic fuzzy sets that are characterized by a membership function and a nonmembership function, and interval‐valued intuitionistic fuzzy sets, whose fundamental characteristic is that the values of its membership function and nonmembership function are intervals rather than exact numbers, and study their properties. Then, we apply them to multiple attribute decision making with intuitionistic fuzzy or interval‐valued intuitionistic fuzzy information. © 2009 Wiley Periodicals, Inc.     

A robust correlation coefficient for probabilistic dual hesitant fuzzy sets and its applications

As a generalization of the hesitant fuzzy sets (HFSs) and dual HFSs (DHFSs), probabilistic dual hesitant fuzzy sets (PDHFSs) are a strong and valuable tool to represent the imprecise information by embedding both the features of HFSs and probabilistic information instantaneously. Meanwhile, a correlation coefficient is a prominent measure to measure the relationship between two sets. Motivated by these primary characteristics, it is interesting to present some information measures to the PDHFSs and hence decision-making approach based on the correlation coefficient. In this paper, we develop a method to solve the multi-criteria decision-making (MCDM) problem under PDHFS environment. For it, firstly, we define the informational energy and the covariance between the two PDHFSs and study their properties. Secondly, we develop correlation coefficients and the weighted correlation coefficients for PDHFSs. In the formulation, DHFSs are able to represent the information in terms of their respective degrees, while the assigned probabilities give more details about the level of agreeness or disagreeness. Also, some properties of the proposed measures are also studied. Thirdly, a novel algorithm is developed based on the proposed operators to solve MCDM problems. A practical example is provided to verify the developed approach and to demonstrate its practicality and feasibility. Also, a comparative analysis with several existing studies reveals the proposed method is better during solving the decision-making problems.

     

Intuitionistic fuzzy linguistic clustering algorithm based on a new correlation coefficient for intuitionistic fuzzy linguistic information

Pattern Analysis and Applications - For observations to be classified, when scoring rules are imprecise or the cost of their computation is too high, the clustering method under linguistic...     

一种新的直觉模糊集相似度测量

直觉模糊集IFS是模糊集的进一步推广,比传统的模糊集在处理模糊性和不确定性等方面更具灵活性和实用性。已有的部分相似度测量方法,不满足相似度的公理化或有些存在反直觉的情形。通过对Boran和AKay提出的直觉模糊集的相似性度量的研究,提出了一种新的改进的相似度测量公式,证明了该公式的合理性,最后通过实例说明了该方法的有效性。     

Correlation and correlation coefficient of generalized orthopair fuzzy sets

Generalized orthopair fuzzy sets are extensions of ordinary fuzzy sets by relaxing restrictions on the degrees of support for and support against. Correlation analysis is to measure the statistical relationships between two samples or variables. In this paper, we propose a function measuring the interrelation of two -rung orthopair fuzzy sets, whose range is the unit interval . First, the correlation and correlation coefficient of -rung orthopair membership grades are presented, and their basic properties are investigated. Second, these concepts are extended to -rung orthopair fuzzy sets on discrete universes. Then, we discuss their applications in cluster analysis under generalized orthopair fuzzy environments. And, a real-world problem involving the evaluation of companies is used to illustrate the detailed processes of the clustering algorithm. Finally, we introduce the correlation and correlation coefficient of -rung orthopair fuzzy sets on both bounded and unbounded continuous universes and provide some numerical examples to substantiate such arguments.     

Level sets and the representation theorem for intuitionistic fuzzy sets

We discuss the concept of a level set of a fuzzy set and the related ideas of the representation theorem and Zadeh’s extension principle. We then describe the extension of these ideas to the case of interval valued fuzzy sets (IVFS). We then recall the formal equivalence between IVFS and intuitionistic fuzzy sets (IFS). This equivalence allows us to naturally extend the concepts of level sets, representation theorem and extension principle from the domain of IVFS to the domain of IFS. What is important to note here is that in the case of these non-standard fuzzy sets, interval valued and intuitionistic, the number of distinct level sets can be greater then the number of distinct membership grades of the fuzzy set being represented. This is a result of the fact that the distinct level sets are generated by the power set of the membership grades. In particular, the minimum of each subset of membership grades provides a level set. In the case of the standard fuzzy sets the minimum of a subset of membership grades results in one of the elements in the subset. In the case of the non-standard fuzzy sets, the membership grades are not linearly ordered and hence taking the minimum of a subset of these can result in a value that was not one of the members of the subset.

On characterization of intuitionistic fuzzy rough sets based on intuitionistic fuzzy implicators

This paper presents a general framework for the study of relation-based (I,T)-intuitionistic fuzzy rough sets by using constructive and axiomatic approaches. In the constructive approach, by employing an intuitionistic fuzzy implicator I and an intuitionistic fuzzy triangle norm T, lower and upper approximations of intuitionistic fuzzy sets with respect to an intuitionistic fuzzy approximation space are first defined. Properties of (I,T)-intuitionistic fuzzy rough approximation operators are examined. The connections between special types of intuitionistic fuzzy relations and properties of intuitionistic fuzzy approximation operators are established. In the axiomatic approach, an operator-oriented characterization of (I,T)-intuitionistic fuzzy rough sets is proposed. Different axiom sets characterizing the essential properties of intuitionistic fuzzy approximation operators associated with various intuitionistic fuzzy relations are explored.     

OWA aggregation of intuitionistic fuzzy sets

We recall the concept of an intuitionistic fuzzy subset (IFS). Fundamental to an IFS is the fact that it is defined using two values, a degree of membership and degree of non-membership. The ordered weighted averaging (OWA) operator is introduced and several of its features are described. Particularly notable is the idea of the dual of an OWA operator. We next discuss the aggregation of a collection of IFS using a prescribed OWA operator. It is shown that while the aggregation of the degrees of membership is performed using the prescribed OWA operator, the aggregation of the degrees of non-membership requires use of the dual of the prescribed OWA operator. The Choquet integral aggregation operator is introduced and applied to the aggregation of IFSs. Here again the concept of the dual is needed to perform the aggregation of the degrees of non-membership. We also discuss the aggregation of IFSs using the Sugeno integral. Fundamental to this work is our realisation of the importance of the concept of the dual operators in dealing with the aggregation of IFS.     

二型直觉模糊粗糙集

将二型直觉模糊集和粗糙集理论融合,建立二型直觉模糊粗糙集模型。首先,在二型直觉模糊近似空间中,定义了一对二型直觉模糊上、下近似算子,并讨论了二型直觉模糊关系退化为普通二型模糊关系和一般等价关系时,上、下近似算子的具体变化形式。然后,将普通二型模糊集之间包含关系的定义推广到了二型直觉模糊集,在此基础上研究了二型直觉模糊上、下近似算子的一些性质。最后,定义了自反的、对称的和传递的二型直觉模糊关系,并讨论了这3种特殊的二型直觉模糊关系与近似算子的特征之间的联系。该结论进一步丰富了二型模糊集理论和粗糙集理论,为二型直觉模糊信息系统的应用奠定了良好的理论基础。     

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.     

Image fusion using intuitionistic fuzzy sets

Image fusion is the process of combining one or more images which are obtained from different environment into a single image which is more useful for further image processing tasks. Image registration and image fusion are of great importance in defence and civilian sectors, particularly for recognizing a ground/air force vehicle and medical imaging. In this paper a new way is drawn to fuse two or more images by using maximum, minimum operations in intuitionistic fuzzy sets (IFSs). IFSs are more suitable for image processing since every digital image have lot of uncertainties. In processing phase, images are reformed into intuitionistic fuzzy images (IFIs). Entropy is employed to obtain the optimum value of the parameter in membership and non-membership function. Then the resulting IFIs are disintegrated into image blocks and the corresponding blocks of the images are reunioned by finding the count of blackness and whiteness of the blocks. This paper evaluates the performance of simple averaging (AVG), principal component analysis (PCA), discrete wavelet transform (DWT), stationary wavelet transform (SWT), dual tree complex wavelet transform (DTCWT), multi-resolution singular value decomposition (MSVD), nonsubsampled contourlet transform (NSCT) and IFS (proposed method) in terms of various performance measure. The experimental and comparison results show that luminance and contrast is of great importance for image processing and prove that the proposed method is better than all other methods.     

An efficient similarity measure for intuitionistic fuzzy sets

We introduce a new methodology for measuring the degree of similarity between two intuitionistic fuzzy sets. The new method is developed on the basis of a distance defined on an interval by the use of convex combination of endpoints and also focusing on the property of min and max operators. It is shown that among the existing methods, the proposed method meets all the well-known properties of a similarity measure and has no counter-intuitive examples. The validity and applicability of the proposed similarity measure is illustrated with two examples known as pattern recognition and medical diagnosis.