A dissimilarity Jensen–Shannon divergence measure for intuitionistic fuzzy sets

The need of suitable divergence measures arise as they play an important role in discrimination of two probability distributions. The present communication is devoted to the introduction of one such divergence measure using Jensen inequality and Shannon entropy and its validation. Also, a new dissimilarity measure based on the proposed divergence measure is introduced. Besides establishing validation, some of its major properties are also studied. Further, a new multiple attribute decision making method based on a proposed dissimilarity measure is introduced and is thoroughly explained with the help of an illustrated example. The paper is summed up with an application of the proposed dissimilarity measure in pattern recognition.     

Combination of sources of evidence with different discounting factors based on a new dissimilarity measure

The sources of evidence may have different reliability and importance in real applications for decision making. The estimation of the discounting (weighting) factors when the prior knowledge is unknown have been regularly studied until recently. In the past, the determination of the weighting factors focused only on reliability discounting rule and it was mainly dependent on the dissimilarity measure between basic belief assignments (bba's) represented by an evidential distance. Nevertheless, it is very difficult to characterize efficiently the dissimilarity only through an evidential distance. Thus, both a distance and a conflict coefficient based on probabilistic transformations BetP are proposed to characterize the dissimilarity. The distance represents the difference between bba's, whereas the conflict coefficient reveals the divergence degree of the hypotheses that two belief functions strongly support. These two aspects of dissimilarity are complementary in a certain sense, and their fusion is used as the dissimilarity measure. Then, a new estimation method of weighting factors is presented by using the proposed dissimilarity measure. In the evaluation of weight of a source, both its dissimilarity with other sources and their weighting factors are considered. The weighting factors can be applied in the both importance and reliability discounting rules, but the selection of the adapted discounting rule should depend on the actual application. Simple numerical examples are given to illustrate the interest of the proposed approach.     

Divergence measure of Pythagorean fuzzy sets and its application in medical diagnosis

The Pythagorean fuzzy set (PFS) which is an extension of intuitionistic fuzzy set, is more capable of expressing and handling the uncertainty under uncertain environments, so that it was broadly applied in a variety of fields. Whereas, how to measure PFSs’ distance appropriately is still an open issue. It is well known that the square root of Jensen–Shannon divergence is a true metric in the probability distribution space which is a useful measure of distance. On account of this point, a novel divergence measure between PFSs is proposed by taking advantage of the Jensen–Shannon divergence in this paper, called as PFSJS distance. This is the first work to consider the divergence of PFSs for measuring the discrepancy of data from the perspective of the relative entropy. The new PFSJS distance measure has some desirable merits, in which it meets the distance measurement axiom and can better indicate the discrimination degree of PFSs. Then, numerical examples demonstrate that the PFSJS distance can avoid generating counter-intuitive results which is more feasible, reasonable and superior than existing distance measures. Additionally, a new algorithm based on the PFSJS distance measure is designed to solve the problems of medical diagnosis. By comparing the different methods in the medical diagnosis application, it is found that the new algorithm is as efficient as the other methods. These results prove that the proposed method is practical in dealing with the medical diagnosis problems.     

Multivalued type dissimilarity measure and concept of mutual dissimilarity value for clustering symbolic patterns

A successful attempt in exploring a dissimilarity measure which captures the reality is made in this paper. The proposed measure unlike other measures (Pattern Recognition 24(6) (1991) 567; Pattern Recognition Lett. 16 (1995) 647; Pattern Recognition 28(8) (1995) 1277; IEEE Trans. Syst. Man Cybern. 24(4) (1994)) is multivalued and non-symmetric. The concept of mutual dissimilarity value is introduced to make the existing conventional clustering algorithms work on the proposed unconventional dissimilarity measure.     

Pairwise-adaptive dissimilarity measure for document clustering

This paper introduces a novel pairwise-adaptive dissimilarity measure for large high dimensional document datasets that improves the unsupervised clustering quality and speed compared to the original cosine dissimilarity measure. This measure dynamically selects a number of important features of the compared pair of document vectors. Two approaches for selecting the number of features in the application of the measure are discussed. The proposed feature selection process makes this dissimilarity measure especially applicable in large, high dimensional document collections. Its performance is validated on several test sets originating from standardized datasets. The dissimilarity measure is compared to the well-known cosine dissimilarity measure using the average F-measures of the hierarchical agglomerative clustering result. This new dissimilarity measure results in an improved clustering result obtained with a lower required computational time.     

模糊集上局部散度测度的几种构造法

散度测度是度量模糊集的重要指标。首先给出了有限论域X上模糊集上散度测度和局部散度测度的定义,进而给出了局部散度测度的几个等价性条件,重点研究了局部散度测度的几种构造方法。     

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.     

基于熵的模糊信息测度研究

模糊信息测度(Fuzzy Information Measures,FIM)是度量两个模糊集之间相似性大小的一种量度,在模式识别、机器学习、聚类分析等研究中,起着重要的作用.文中对模糊测度进行了分析,研究了基于熵的模糊信息测度理论:首先,概述了模糊测度理论,指出了其优缺点;其次,基于信息熵理论,研究了模糊熵理论,建立了模糊熵公理化体系,讨论了各种模糊熵,在此基础上,提出了模糊绝对熵测度、模糊相对熵测度等模糊熵测度;最后,基于交互熵理论,建立了模糊交互熵理论,进而提出了模糊交互熵测度.这些测度理论,不仅丰富与发展了 FIM理论,而且为模式识别、机器学习、聚类分析等理论与应用研究提供了新的研究方法.     

A novel TODIM-VIKOR approach based on entropy and Jensen–Tsalli divergence measure for picture fuzzy sets in a decision-making problem

The picture fuzzy set (PFS) has grown huge attention in the research area of uncertain information from the last few years. Information measures have been widely studied in various fuzzy environments. Therefore, in this paper, we study the entropy and divergence measures under the picture fuzzy environment. First, the paper introduced a new entropy measure to measure the fuzziness degree associated with a PFS. An example is established to show the capabilities of the proposed entropy measure. Second, the paper defines a new Jensen–Tsalli divergence measure for PFS to evaluate the information of discrimination between two PFS. We also discuss several properties of entropy and divergence measures in detail. Then we present a new method, based on proposed entropy and divergence measure, to determine the objective weights of experts for multicriteria group decision making with picture fuzzy information. The final criteria weights are obtained by combining subjective and objective weights for more reliable weightage of evaluation criteria. By using this comprehensive weight-determination technique, the proposed method can effectively reduce the unreasonable impact of the extreme evaluation data on the evaluation results. Further, a new multi-criteria decision-making approach is developed based on the combining concepts of the TODIM and VIKOR method under the picture fuzzy environment. We used TODIM to obtain the overall dominance degree which considers the bounded rationality of decision makers and VIKOR is used to obtain the compromise ranking of alternatives. Lastly, an application of the proposed integrated model is demonstrated to verify the feasibility and usefulness and the outcomes of the proposed model are compared with the outcomes of the existing approaches to indicate its validity. This integrated method can effectively reduce the distortion of decision information and provide extraordinary evaluation results. The proposed approach is used in detecting the major issues due to which a company is facing such breakdowns.     

Average divergence distance as a statistical discrimination measure for hidden Markov models

This paper proposes and evaluates a new statistical discrimination measure for hidden Markov models (HMMs) extending the notion of divergence, a measure of average discrimination information originally defined for two probability density functions. Similar distance measures have been proposed for the case of HMMs, but those have focused primarily on the stationary behavior of the models. However, in speech recognition applications, the transient aspects of the models have a principal role in the discrimination process and, consequently, capturing this information is crucial in the formulation of any discrimination indicator. This paper proposes the notion of average divergence distance (ADD) as a statistical discrimination measure between two HMMs, considering the transient behavior of these models. This paper provides an analytical formulation of the proposed discrimination measure, a justification of its definition based on the Viterbi decoding approach, and a formal proof that this quantity is well defined for a left-to-right HMM topology with a final nonemitting state, a standard model for basic acoustic units in automatic speech recognition (ASR) systems. Using experiments based on this discrimination measure, it is shown that ADD provides a coherent way to evaluate the discrimination dissimilarity between acoustic models.     

Similarity–divergence intuitionistic fuzzy decision using particle swarm optimization

Many existing intuitionistic fuzzy (IF) decision methods focus on a reasonable ranking for alternatives under unknown weight information. Traditionally, the weight information is usually determined from a multiobjective optimization model based on real-valued measures such as IF distance or similarity measures, which may lose divergence information. In this paper, we propose one new type of optimization model for determining the weights based on a fuzzy measure called the similarity–divergence measure (S–D measure). First, we develop similarity and divergence measures of IF sets respectively, and a 2-tuple consisting of similarity and divergence is defined as a S–D measure. This measure is further proven to be an IF similarity degree and has practical semantics of similarity and divergence features in human’s cognition. Second, we utilize such measure to calculate fuzzy similarities of each alternative and construct a nonlinear optimization model to determine the weights. Third, we design an algorithm for solving the model with the aid of particle swarm optimization and thus develop an IF decision method. Finally, two examples are given to demonstrate our method and then it is compared with existing methods to explain its effectiveness and superiority.     

A novel similarity/dissimilarity measure for intuitionistic fuzzy sets and its application in pattern recognition

Among the most interesting measures in intuitionistic fuzzy sets (IFSs) theory, the similarity measure is an essential tool to compare and determine degree of similarity between IFSs. Although there exist many similarity measures for IFSs, most of them cannot satisfy the axioms of similarity measure or provide reasonable results. In this paper, a novel knowledge-based similarity/dissimilarity measure between IFSs is proposed. Firstly, we define a new knowledge measure of information conveyed by the IFS and prove some properties of the proposed knowledge measure. Based on the proposed knowledge measure of IFSs, we construct a novel similarity/dissimilarity measure between IFSs and prove some properties of the proposed similarity measure. Then we use some illustrative examples to show that the proposed measures, though simple in concept and calculus, can overcome the drawbacks of the existing measures. Finally, we apply the proposed similarity/dissimilarity measure between IFSs in the pattern recognition problems to demonstrate that the proposed measure is the most reliable to deal with the pattern recognition problem in comparison with the existing similarity measures.     

Some new distance measures for type-2 fuzzy sets and distance measure based ranking for group decision making problems

In this paper, we propose some distance measures between type-2 fuzzy sets, and also a new family of utmost distance measures are presented. Several properties of different proposed distance measures have been introduced. Also, we have introduced a new ranking method for the ordering of type-2 fuzzy sets based on the proposed distance measure. The proposed ranking method satisfies the reasonable properties for the ordering of fuzzy quantities. Some properties such as robustness, order relation have been presented. Limitations of existing ranking methods have been studied. Further for practical use, a new method for selecting the best alternative, for group decision making problems is proposed. This method is illustrated with a numerical example.     

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.     

Fuzzy divergence, probability measure of fuzzy events and image thresholding

A new measure called divergence between two fuzzy sets is introduced along with a few properties. Its application to clustering problems is indicated and applied to an object extraction problem. A tailored version of the probability measure of a fuzzy event is also used for image segmentation. Both parametric and non-parametric probability distributions are considered in this regard.     

A new evaluation measure using compression dissimilarity on text summarization

Evaluation of automatic text summarization is a challenging task due to the difficulty of calculating similarity of two texts. In this paper, we define a new dissimilarity measure – compression dissimilarity to compute the dissimilarity between documents. Then we propose a new automatic evaluating method based on compression dissimilarity. The proposed method is a completely “black box” and does not need preprocessing steps. Experiments show that compression dissimilarity could clearly distinct automatic summaries from human summaries. Compression dissimilarity evaluating measure could evaluate an automatic summary by comparing with high-quality human summaries, or comparing with its original document. The evaluating results are highly correlated with human assessments, and the correlation between compression dissimilarity of summaries and compression dissimilarity of documents can serve as a meaningful measure to evaluate the consistency of an automatic text summarization system.     

Multiple attribute group decision-making based on order-α divergence and entropy measures under q-rung orthopair fuzzy environment

The q-rung orthopair fuzzy set (^qROPFS), proposed by Yager, is a more effective and proficient tool to represent uncertain or vague information in real-life situations. Divergence and entropy are two important measures, which have been extensively studied in different information environments, including fuzzy, intuitionistic fuzzy, interval-valued fuzzy, and Pythagorean fuzzy. In the present communication, we study the divergence and entropy measures under the q-rung orthopair fuzzy environment. First, the work defines two new order-α divergence measures for ^qROPFSs to quantify the information of discrimination between two ^qROPFSs. We also examine several mathematical properties associated with order-α ^qROPF divergence measures in detail. Second, the paper introduces two new parametric entropy functions called “order-α ^qROPF entropy measures” to measure the degree of fuzziness associated with a ^qROPFS. We show that the proposed order-α divergence and entropy measures include several existing divergence and entropy measures as their particular cases. Further, the paper develops a new decision-making approach to solve multiple attribute group decision-making problems under the ^qROPF environment where the information about the attribute weights is completely unknown or partially known. Finally, an example of selecting the best enterprise resource planning system is provided to illustrate the decision-making steps and effectiveness of the proposed approach.     

一种新的直觉模糊集距离及其在决策中的应用

首先针对直觉模糊集距离中是否包含直觉模糊集通过隶属度、非隶属度以及犹豫度这三种信息,以及直觉模糊集距离是否满足相应距离度量的条件对其进行了详细分析,发现现有方法都是直接将犹豫度直接引入到直觉模糊集距离中,从而会产生不一致性。鉴于此,定义了一种新的直觉模糊集距离度量方法,其不仅考虑隶属度和非隶属度信息,同时还考虑犹豫度对隶属度和非隶属度的分配,从而间接地将犹豫度也引入到直觉模糊集距离中。其次,证明了所提距离度量满足距离度量条件,并结合实例将其与现有距离度量方法进行比较分析,说明了新方法的合理性。最后,将所提出方法应用于多准则模糊决策中,进一步说明了新方法的有效性和可行性。     

An improved similarity measure for generalized fuzzy numbers and its application to fuzzy risk analysis

This paper presents an improved method to compute the degree of similarity between generalized trapezoidal fuzzy numbers. The proposed similarity measure contains many features of fuzzy numbers such as geometric distance, center of gravity (COG), area, perimeter, and height. The previous methods are criticized via presenting some examples. In addition, the performance of the proposed methods is compared by the existing similarity measures using twenty different sets of generalized trapezoidal fuzzy numbers. Furthermore, the proposed method is used for fuzzy risk analysis based on similarity measures. Finally, an example is introduced to illustrate the fuzzy risk analysis.     

A new method for solving dual hesitant fuzzy assignment problems with restrictions based on similarity measure

Zhu et al. (2012) proposed dual hesitant fuzzy set as an extension of hesitant fuzzy sets which encompass fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets as a special case. Dual hesitant fuzzy sets consist of two parts, that is, the membership and nonmembership degrees, which are represented by two sets of possible values. Therefore, in accordance with the practical demand these sets are more flexible, and provides much more information about the situation. In this paper, the axiom definition of a similarity measure between dual hesitant fuzzy sets is introduced. A new similarity measure considering membership and nonmembership degrees of dual hesitant fuzzy sets has been presented and also it is shown that the corresponding distance measures can be obtained from the proposed similarity measures. To check the effectiveness, the proposed similarity measure is applied in a bidirectional approximate reasoning systems. Mathematical formulation of dual hesitant fuzzy assignment problem with restrictions is presented. Two algorithms based on the proposed similarity measure, are developed to finds the optimal solution of dual hesitant fuzzy assignment problem with restrictions. Finally, the proposed method is illustrated by numerical examples.