Similarity Measures Of Fuzzy Sets Research Articles

Similarity Measures for Fractional Orthotriple Fuzzy Sets Using Cosine and Cotangent Functions and Their Application in Accident Emergency Response

The fractional orthotriple fuzzy set (FOFS) is more generalized than the spherical fuzzy set (SFS) and picture fuzzy set (PFS) to cope with awkward and complex information in fuzzy set (FS) theory. The FOFS is a more powerful technique with respect to the existing drawbacks because of its conditions, i.e., the sum of the f powers of positive, neutral, and negative grades is bounded to [0,1]. With the advantages of the FOFS, in this paper, we study the basic definitions and some existing similarity measures (SMs) of intuitionistic fuzzy sets (IFSs), PFSs, Pythagorean fuzzy sets (PyFSs) and SFSs. The existing approaches have certain limitations and cannot be applied to problems that are in the form of FOFSs. The goal of this paper is to propose the idea of some new SMs including cosine SMs for FOFSs, SMs for FOFSs based on the cosine function, and SMs for FOFSs based on the cotangent function. Further, some weighted SMs (WSMs) are also proposed for which the weight of the attributes is considered. Then, we apply these SMs and WSMs to the pattern recognition problem. Finally, the comparative study of the new SMs for FOFSs is established with existing SMs, and also, some advantages of the proposed work are discussed.

Similarity measure of multiple sets and its application to pattern recognition

Multiple set is a newborn member of the family of generalized sets, which can model uncertainty together with multiplicity. It has the power to handle numerous uncertain features of objects in a multiple way. Multiple set theory has the edge over the well established fuzzy set theory by its capability to handle uncertainty and multiplicity simultaneously. Similarity measure of fuzzy sets is well addressed in literature and has found prominent applications in various domains. As multiple set is an efficient generalization of fuzzy set, the concept and theory of similarity measure can be extended to multiple set theory and can be developed probable applications in various real-life problems. This paper introduces the concept of similarity measure of multiple sets and proposes two different similarity measures of multiple sets and investigates their properties. Finally, this work substantiates application of the concept of similarity measure of multiple sets to pattern recognition. A numerical illustration demonstrates the effectiveness of the proposed technique to this application.

Bi-parametric distance and similarity measures of picture fuzzy sets and their applications in medical diagnosis

The concept of picture fuzzy sets (PFS) is a generalization of ordinary fuzzy sets and intuitionistic fuzzy sets, which is characterized by positive membership, neutral membership, and negative membership functions. Keeping in mind the importance of similarity measures and applications in data mining, medical diagnosis, decision making, and pattern recognition, several studies have been proposed in the literature. Some of those, however, cannot satisfy the axioms of similarity and provide counter-intuitive cases. In this paper, we propose new similarity measures for PFSs based on two parameters t and p, where t identifies the level of uncertainty and p is the Lp norm. The properties of the bi-parametric similarity and distance measures are discussed. We provide some counterexamples for existing similarity measures in the literature and show how our proposed similarity measure is important and applicable to the pattern recognition problems. In the end, we provide an application of a proposed similarity measure for medical diagnosis.

Cosine Similarity Measure of Complex Fuzzy Sets and Robustness of Complex Fuzzy Connectives

Complex fuzzy set (CFS), as a generalization of fuzzy set (FS), is characterized by complex-valued membership degrees. By considering the complex-valued membership degree as a vector in the complex unit disk, we introduce the cosine similarity measures between CFSs. Then, we investigate some invariance properties of the cosine similarity measure. Finally, the cosine similarity measure is applied to measure the robustness of complex fuzzy connectives and complex fuzzy inference.

New Similarity Measures for Dual Hesitant Fuzzy Sets and Their Application

Dual hesitant fuzzy sets (DHFSs) are powerful and efficient to express hesitant preferred and non-preferred information simultaneously. This paper focuses on similarity measures for DHFSs. To do this, it first analyzes the limitations of previous similarity measures for DHFSs. Then, several new dual hesitant fuzzy similarity measures are defined that can avoid the issues of previous ones. To discriminate the importance of decision-making criteria, several weighted similarity measures are further defined in views of additive and 2-additive measures. When the weighting information is not exactly known, optimization methods for determining additive and 2-additive measures are built, respectively. Furthermore, a method for multi-criteria decision-making based on new weighted similarity measures is developed. Finally, two numerical examples are provided to show the utilization of the new method and compare with previous methods.

Generalized dice similarity measures for q-rung orthopair fuzzy sets with applications

Recently, Yager has established that the notion of q-rung orthopair fuzzy set (q-ROFS) is more accomplished than pythagorean fuzzy set (PyFS) and intuitionistic fuzzy set (IFS) to cope with awkward and complicated information in real decision theory. This notion works with yes-, no- and refusal-type fuzzy information. The constraint of q-ROFS is that the sum of n-power of the truth grade and the n-power of the falsity grade is bounded to unit interval. Generalized dice similarity measures are complimentary concepts quantifying the difference and closeness of q-ROFSs. In this paper, we suggested a number of novel dice similarity measures (DSMs) in the surroundings of the q-ROFS, and we examined some prevailing dice similarity measures and their limitations. In addition, we took the DSMs broad view to some globalized dice similarity measures (GDSMs), and we examined some of their particular cases. We employed the novel suggested GDSMs to the best selections of items on identification problems, and we analyzed their acquired consequences. There is a development of novel work in which many situations are evaluated, and from this perspective, the suggested work is changed into already prevailing work. This study also examines the merits of novel DSMs and the limitations for DSMs of IFSs and PyFSs. The comparison between established measures with existing measures is explored and their graphical interpretations are also discussed to show the reliability and effectiveness of the explored measures.

Complex q-Rung Orthopair Fuzzy Variation Co-efficient Similarity Measures and their Approach in Medical Diagnosis and Pattern Recognition

The q-rung orthopair fuzzy set (q-ROFS) as a generalization of fuzzy set (FS), is characterized by membership and non-membership, and the sum of their q-powers is restricted to [0,1]. In this manuscript, a new Complex q-rung orthoapir fuzzy set (Cq-ROFS) is proposed by combining q-ROFS and complex fuzzy set (CFS). Cq-ROFS is a better way to process uncertain and imprecise information in decision making, which is characterized by complex-valued membership and complex-valued non-membership. First, some fundamental operational laws, score function and accuracy function, and comparison method are proposed. Further, because the vector similarity measures (VSMs) play a key role in statistics, physics and engineering sciences, some VSMs called Jaccard similarity measures (JSMs), dice similarity measures (DSMs) and cosine similarity measures (CSMs) for Cq-ROFSs and interval-valued complex q-rung orthopair fuzzy sets (IVCq-ROFSs) are investigated. Moreover, the hybrid vector similarity measures (HVSMs) called variation co-efficient similarity measures (VCSMs) for Cq-ROFSs and IVCq-ROFSs are also proposed and their properties are discussed. Finally, in order to demonstrate the feasibility of the investigated HVSMs, the existing similarity measures about complex Pythagorean fuzzy sets (CPFSs) and complex intuitionistic fuzzy sets (CIFSs) are compared with the proposed methods by numerical examples of medical diagnosis and pattern recognition.

The general formula of entropy and similarity measures for hesitant fuzzy linguistic term sets

In order to quantitatively characterize the uncertainty of hesitant fuzzy linguistic information, the entropy and similarity measures of hesitant fuzzy linguistic term sets are studied. First, the axiomatic definitions of entropy and similarity measures of hesitant fuzzy linguistic term sets are given, and general formulas of entropy and similarity measures for hesitant fuzzy linguistic term sets are proposed. Moreover, the algorithms for generating entropy and similarity of hesitant fuzzy linguistic term set are given. Then the relationship between entropy and similarity of hesitant fuzzy linguistic term set is studied, and the general formula of entropy based on similarity is proposed. These lay the theoretical foundation for flexible selection of entropy and similarity in multi-attribute decision making.

Distance and Similarity Measures for Spherical Fuzzy Sets and Their Applications in Selecting Mega Projects

A new condition on positive membership, neutral membership, and negative membership functions give us the successful extension of picture fuzzy set and Pythagorean fuzzy set and called spherical fuzzy sets ( SFS ) . This extends the domain of positive membership, neutral membership, and negative membership functions. Keeping in mind the importance of similarity measure and application in data mining, medical diagnosis, decision making, and pattern recognition, several studies on similarity measures have been proposed in the literature. Some of those, however, cannot satisfy the axioms of similarity and provide counter-intuitive cases. In this paper, we proposed the set-theoretic similarity and distance measures. We provide some counterexamples for already proposed similarity measures in the literature and shows that how our proposed method is important and applicable to the pattern recognition problems. In the end, we provide an application of a proposed similarity measure for selecting mega projects in under developed countries.

Interval-valued fuzzy reasoning method based on similarity measure

In this paper, a new similarity measure of interval-valued fuzzy sets is introduced. Meanwhile, interval-valued fuzzy reasoning method based on similarity measure for Fuzzy Modus Ponens and Fuzzy Modus Tollens are proposed, and the solutions of interval-valued fuzzy reasoning method based on similarity measure are given. Moreover, the robustness of interval-valued fuzzy reasoning method based on similarity measure is discussed. Finally, general interval-valued fuzzy reasoning method based on similarity measure is proposed.

Knowledge measure of hesitant fuzzy set and its application in multi-attribute decision-making

The fuzzy knowledge measure is considered as a dual measure of fuzzy entropy. In this work, we introduce an axiomatic framework to define a hesitant fuzzy knowledge measure (HF-knowledge measure) and investigate hesitant fuzzy entropy (HF-entropy) and HF-knowledge measure from the viewpoint of duality. We provide a characterization result to obtain a class of the HF-knowledge measure. We also obtain an HF-knowledge measure from similarity and dissimilarity measures of hesitant fuzzy sets. Here, we introduce an HF-knowledge measure and show its effectiveness with the help of an illustrative example from the viewpoint of linguistic hedges. We apply the proposed HF-knowledge measure to multiple-attribute decision-making (MADM) problem by utilizing the TOPSIS method and justify its advantage over existing HF-entropies.

Exponential and non-Exponential Based Generalized Similarity Measures for Complex Hesitant Fuzzy Sets with Applications

The purpose of this manuscript is to explore the notion of a complex hesitant fuzzy set (CHFS), as a generalization of the hesitant fuzzy set (HFS) and complex fuzzy set (CFS) to cope with the uncertain and complicated information in the real-world decision. CHFS contains truth grades in the form of a subset of the unit disc in the complex plane. The operational laws of the explored notion are also described. Further, the exponential based generalized similarity measures, without exponential based generalized similarity measures, and their important characteristics are also explored. These similarity measures are applied in the environment of pattern recognition and medical diagnosis to evaluate the proficiency and feasibility of the established measures. We also solved some numerical examples using the established measures. To examine the reliability and validity of the proposed measures by comparing it with existing measures. The advantages, comparative analysis, and graphical representation of the explored measures and existing measures are also discussed in detail.

Similarity Measures of T-Spherical Fuzzy Sets Based on the Cosine Function and Their Applications in Pattern Recognition

In this manuscript, nine similarity measures of T-spherical fuzzy set (TSFS) considering the membership degree, the hesitancy degree, the non-membership degree and the refusal degree are developed according to the cosine function. Besides, the generalizations of existing similarity measures are the similarity measures of TSFS proposed in this paper, which indicates the breadth and novelty of the proposed similarity measures. More importantly, the nine similarity measures of TSFSs are applied to pattern recognition. Then, we make a comparative study, that is, we apply the nine similarity measures of TSFSs developed in this manuscript to picture fuzzy environment, and the results obtained are consistent with the previous results. This application make the problem of building material recognition better solved in the real world. Finally, two numerical examples show the validity of the proposed similarity measure between TSFSs.

New similarity measures for Pythagorean fuzzy sets with applications

New similarity measures for Pythagorean fuzzy sets with applications

Similarity Measure of Hesitant Fuzzy Sets Based on Implication Function and Clustering Analysis

Hesitant fuzzy set (HFS) permits several possible values as the membership degree of an element to a set to express the decision makers’ hesitance. Since its appearance, HFS has been extensively applied in multi-attribute decision making, group decision making and evaluation process. Considering that the similarity measure of hesitant fuzzy sets (HFSs) is an important index in intelligent system, and the implication function can describe many subtle differences which is very suitable for dealing with hesitant fuzzy information. In this paper, we merge implication function with HFS to investigate the similarity measure of HFSs, propose some new formulas to calculate the similarity measures of HFSs which are different from the existing similarity measures of HFSs based on the distance measure, and do some comparison analysis. Meanwhile, we introduce the union and intersection operations of HFSs, the hesitant fuzzy similar relation and the hesitant fuzzy equivalent relation, and develop the hesitant fuzzy clustering algorithm. Finally, three numerical examples are used to illustrate the effectiveness and validation of our proposed method.

New similarity measures for Pythagorean fuzzy sets with applications

The concept of Pythagorean fuzzy sets (PFSs) is pertinent in finding reliable solution to decision-making problems, because of its unique nature of indeterminacy. Pythagorean fuzzy set is characterised by membership degree, non-membership degree, and indeterminate degree in such a way that the sum of the square of each of the parameters is one. The objective of this paper is to present some new similarity measures for PFSs by incorporating the conventional parameters that describe PFSs, with applications to some real-life decision-making problems. Furthermore, an illustrative example is used to establish the applicability and validity of the proposed similarity measures and compare the results with the existing comparable similarity measures to show the effectiveness of the proposed similarity measures. While analysing the reliability of the proposed similarity measures in comparison to analogous similarity measures for PFSs in literature, we discover that the proposed similarity measures, especially, s4 yields the most reasonable measure. Finally, we apply s4 to decision-making problems such as career placement, medical diagnosis, and electioneering process. Additional applications of these new similarity measures could be exploited in decision making of real-life problems embedded with uncertainty such as in multi-criteria decision-making (MCDM) and multi-attribute decision-making (MADM), respectively.

A New Methodology for Computing Semantic Relatedness: Modified Latent Semantic Analysis by Fuzzy Formal Concept Analysis

Semantic similarity is used to perceive the meaning of words in textual data and having several applications in the field of Computational linguistic and Natural language processing. Semantic similarity and semantic relatedness are interchangeable terms, in Semantic relatedness all types of semantic relationships are considered i.e. (has, part-of, contains etc.) whereas in semantic similarity only “is-a” type of relationship is used. Computing accurate similarity improves the efficiency of query processing and understanding of textual data more proficiently. In this work, a novel hybrid approach of corpus based and knowledge based method is proposed approach to compute the semantic relatedness among words. In this hybrid method, Latent semantic analysis is used to find the concepts and their correlation value and mapping is done for different words with the help of fuzzy formal concept analysis.To compute semantic relatedness on different words a fuzzy set similarity measure is used.The proposed approach has been evaluated on solar domain and attains improved results as compared to other baseline measures. This method can be used for any doamin as it is a generalized approach for similarity computation.

Contrast Similarity Measures of Fuzzy Sets

The paper proposes a new class of fuzzy set similarity measures taking in to account the proximity of membership values to the border values 0 and 1. These similarity measures take values in [0,1] and generalize the crisp weak equality relation of fuzzy sets considered in the theory of fuzzy sets. We present the method constructing a contrast similarity measure using a bipolar function symmetric with respect to 0.5. As a particular case, we consider the similarity measure defined by the contrast intensification operation considered by Lotfi Zadeh.

Similarity measures of picture fuzzy sets based on entropy and their application in MCDM

An extension of fuzzy sets (Zadeh in Inf Control 8:338–353, 1965) and intuitionistic fuzzy sets (Atanassov in Fuzzy Sets Syst 20(1):87–96, 1986) is called picture fuzzy set (PFS). It is a useful tool to deal with uncertain and inconsistent information. The distance, entropy and similarity measures play a critical role in information theory. The similarity measures of picture fuzzy sets caused by entropy have been studied and given interesting results. In this paper, we first introduced the concept of entropy measure of PFS. At the same time, we also investigated some similarity measures induced by entropy measures and applied to propose the multi-criteria decision-making problem for selecting suppliers.

New distance and similarity measures for hesitant fuzzy soft sets

The hesitant fuzzy soft set (HFSS), as a combination of hesitant fuzzy and soft sets, is regarded as a useful tool for dealing with the uncertainty and ambiguity of real-world problems. In HFSSs, each element is defined in terms of several parameters with arbitrary membership degrees. In addition, distance and similarity measures are considered as the important tools in different areas such as pattern recognition, clustering, medical diagnosis, and the like. For this purpose, the present study aimed to evaluate the distance and similarity measures for HFSSs by using well-known Hamming, Euclidean, and Minkowski distance measures. Further, some examples were used to demonstrate that these measures fail to perform well in some applications. Accordingly, new distance and similarity measures were proposed by considering a hesitance index for HFSSs and the effect of considering hesitance index was shown by using an example of pattern recognition. Finally, the application of the proposed measures and hesitance index was investigated in the clustering and decision-making problem, respectively. In conclusion, the use of the proposed measures in clustering and hesitance index in decision-making can provide better and more reasonable results.