Grey relational analysis between hesitant fuzzy sets with applications to pattern recognition

doi:10.1016/j.eswa.2017.09.048

Expert Systems with Applications

Volume 92, February 2018, Pages 521-532

https://doi.org/10.1016/j.eswa.2017.09.048 Get rights and content

Highlights

•
Analyze the drawbacks of the existing HFSs fuzzy measures.
•
Apply the grey relational analysis to the HFSs for the first time.
•
Define the difference and slope concept of the HFSs.
•
Propose the HFSs, HFSs slope and HFSs synthetic grey relational degree.
•
Apply grey relational analysis to the HFSs pattern recognition problems.

Abstract

Hesitant Fuzzy Sets (HFSs) is an important tool to deal with uncertain and vague information. There have been lots of fuzzy measures for it from different views. However, these fuzzy measures are more or less inappropriate in the applications. The distance and similarity measures only consider the closeness of the HFSs, while the correlation coefficients only consider the linear fashion. They are merely one side of the HFSs intrinsic fuzzy measures. Therefore, in this paper, we apply the grey relational analysis to the HFSs for the first time and define the HFSs grey relational degree to express the closeness. Furthermore, we creatively propose the difference and slope concept of the HFSs. Based on the difference and slope we define the HFSs slope grey relational degree to represent the linear fashion. Sequentially, combining the HFSs grey relational degree and HFSs slope grey relational degree together, we construct the HFSs synthetic grey relational degree, which takes both the closeness and the linear fashion into consideration. With the help of the proposed HFSs synthetic grey relational degree we propose the hesitant fuzzy grey relational recognition methodology. Finally, we apply the HFSs synthetic grey relational degree to deal with the pattern recognition problems. Compared with some examples, the performance of the proposed HFSs synthetic grey relational degree outperforms the existing HFSs fuzzy measures in the accuracy and integrity.

Introduction

In 2009, Torra (Torra and Narukawa, 2009, Torra, 2010) originally introduced the hesitant fuzzy sets (HFSs), which is one of the most efficient decision making techniques to deal with imprecise and vague information. Despite of the youth of HFSs, it shows its advantages over the traditional fuzzy set. The HFSs permits the membership having a set of possible values in [0, 1], allows the membership degree that an element to a set presented by several possible values. It can express the hesitant information more comprehensively than other extensions of fuzzy set. Since its proposal, it has been applied in such the real world problems as decision-making Chen et al., 2013a, Chen et al., 2015; Qian, Wang, and Feng (2013); Liao, Xu, and Xia (2014); Xia and Xu (2011); Xia, Xu, and Chen (2013); Xu and Xia (2012); Xu and Zhang (2013), cluster analysis (Zhang & Xu, 2015), pattern recognition (Zeng, Li, & Yin, 2016) and linguistic computing (Rodríguez, Martínez, & Herrera, 2012; Liao, Xu, & Zeng, 2015b). However, we realize that the existing applications of HFSs in decision making and pattern recognition are mainly based on the distance, similarity measures and correlation coefficients of HFSs, none of them refer to the grey relational analysis of the HFSs.

Actually, the grey relational analysis of the fuzzy sets takes an important occupation in the fuzzy measure field. It measures the closeness of two fuzzy sets just like the distance and similarity measure. Many researchers have focused on the grey relational analysis of fuzzy sets and proposed several approaches to solve the decision making and pattern recognition problems. Yang and John (2012) extended the standard grey number to the generalized grey number and investigated the degree of greyness for grey sets. In addition, Yang and Hinde (2010) initially proposed the concept of the R-fuzzy set and proved that a grey set is a special case of R-fuzzy sets. Yang also made some other contributions to the grey theory with Prof. Sifeng Liu, Prof. Robert John and Forrest (Yang, Liu, & John, 2014; Liu, Forrest, & Yang, 2012), which is greatly helpful to improve the perception of the grey theory. Kong, Wang and Wu (2011) presented a new algorithm based on grey relational analysis to discuss fuzzy soft set decision problems. Zhang, Liu, and Zhai (2011) developed an extended grey relational analysis method for solving MCDM problems with interval-valued triangular fuzzy numbers and unknown information on criterion weights. Zhang, Jin, and Liu (2013) proposed a grey relational projection method for the MADM problems with intuitionistic trapezoidal fuzzy number attribute. Kuo and Liang (2011) presented an effective fuzzy MCDM method combining concepts of VIKOR and grey relational analysis to evaluate service quality of Northeast-Asian international airports by conducting customer surveys. Wei, 2010, Wei, 2011a established an optimization model based on the basic ideal of traditional grey relational analysis (GRA) method to investigate the multiple attribute decision-making problems with intuitionistic fuzzy information. Wei (2011b) also developed a method for multiple attribute group decision making problems with 2-tuple linguistic information based on the traditional idea of grey relational analysis. Later he (Wei, 2011c) extended the grey relational analysis to the dynamic hybrid multiple attribute decision making problems, which the decision information is expressed in real numbers, interval numbers or linguistic labels. Guo (2013) also proposed a combined GRA for intuitionistic fuzzy group decision-making approach to hybrid multiple attribute group decision making. Liu, You, Fan, and Lin (2014) proposed an improved grey relational analysis method for the risk evaluation in Failure mode and effects analysis. Tang (2015); Li, Wen, and Xie (2015) combined grey relational analysis and Dempster-Shafer theory of evidence to propose a novel fuzzy soft set approach in decision making respectively. Li, Hipel, and Dang (2015) put forward an enhanced grey clustering analysis method based on accumulation sequences using grey relational analysis to specify hierarchies of clusters in panel data. Moreover, Khuman, Yang, and John (2016a); Khuman (2016b) proposed the significance measure for R-fuzzy set that enabled the use of the degree of grey incidence to analyze the R-fuzzy set, which is meaningful to encapsulate uncertainty. He also improved the grey model forecasting method, extended the grey relational analysis to the natural language processing and discussed some of the intrinsic differences between fuzzy and grey in his papers (Khuman et al., 2013, Khuman et al., 2014, Khuman et al., 2015, Khuman et al., 2016c), which improve the use of the grey systems in the fuzzy domain. As mentioned above, there have been so many grey relational analysis methods for fuzzy sets such as the triangular fuzzy numbers, linguistic information and intuitionistic fuzzy sets in the existing literatures. However, as for as we know, there are little literatures (Li & Wei, 2014) about grey relational analysis for hesitant fuzzy sets up to now.

The present fuzzy measures for hesitant fuzzy sets mainly focus on the distance, similarity measures and correlation coefficients. Xu and Xia (2011a) proposed a variety of distance and similarity measures for hesitant fuzzy sets. Afterwards, they (Xu & Xia, 2011b) extended the distance measures and first proposed the correlation measures for hesitant fuzzy information. Li, Zeng, and Zhao (2015a) pointed out some drawbacks of the existing distance measures by counterexamples and presented some new distance measures between hesitant fuzzy sets based on the hesitance degree of hesitant fuzzy elements. Zeng et al. (2016) also introduced some distance and similarity measures by taking into account the hesitance degree of hesitant fuzzy sets in pattern recognition process. Chen, Xu, and Xia (2013b) extended Xu and Xia's correlation coefficients (Xu & Xia, 2011b) from hesitant fuzzy element (HFE) into hesitant fuzzy sets (HFSs) and proposed some correlation coefficients of hesitant fuzzy sets and interval-valued hesitant fuzzy sets for clustering analysis. Besides, Liao, Xu, and Zeng (2015a) proposed a novel correlation coefficient between hesitant fuzzy sets to the medical diagnosis and clustering problems. Afterwards, they (Liao, Xu, Zeng, & Merigó, 2015c) extended their notion to the correlation coefficients of HFLTSs. However, we should point out that these fuzzy measures for hesitant fuzzy sets are more or less unreasonable in the applications of the HFSs. The distance and similarity measures can calculate the closeness of two HFSs, while they can not measure the tendency of the variation for HFSs. Although the correlation coefficients can calculate the linear relationship of two HFSs, they can not measure the closeness of HFSs. Therefore, the applications of the HFSs calculated by the present fuzzy measures are controversial to an extent. They are only one aspect of the real fuzzy measures, especially the correlation coefficients used in the pattern recognition problems. Under this condition, the closeness of HFSs ought to take a more important position than the linear relationship.

Consequently, the purpose of this paper is to improve the existing fuzzy measures of the HFSs and originally propose some novel fuzzy measures for HFSs based on grey relational analysis. The novelties of this paper are mainly in the six aspects: (1) Elaborate the limitations of the existing fuzzy measures of the HFSs, especially the correlation coefficients of HFSs. (2) Define the grey relational coefficient and grey relational degree of the HFSs for the first time. (3) The proposal of the difference and slope of the HFSs to describe the tendency of the variation for HFSs. (4) Define the grey relational coefficient and grey relational degree for the slope of the HFSs. (5) Combine the HFSs grey relational degree and HFSs slope grey relational degree together to construct a synthetic grey relational degree for the HFSs. (6) The applications of the proposed grey relational analysis for the HFSs to the pattern recognition problems with hesitant fuzzy information.

The remainder of this paper is organized as follows: Section 2 briefly reviews the concepts of HFSs and the existing fuzzy measures between them, especially the distance measure and correlation coefficients. We also debate the drawbacks of these fuzzy measures detailedly and review the grey relational analysis theory in this section. In Section 3, we define the grey relational coefficient and grey relational degree for HFSs for the first time and propose the definition of the difference and slope of the HFSs. Subsequently, we define the grey relational coefficient and grey relational degree for the slope of the HFSs. Furthermore, we combine both the two grey relational degree for the HFSs to form a synthetic grey relational degree. In Section 4, we develop a hesitant fuzzy multi-criteria pattern recognition method based on the proposed grey relational analysis between HFSs. In Section 5, we apply the proposed grey relational degree for the HFSs to the practical pattern recognition problems and numerical examples to illustrate its validity and feasibility in detail. Finally, the paper ends with some concluding remarks and future challenges in Section 6.

Section snippets

Preliminaries

In this section, we recall the hesitant fuzzy sets, some necessary distance, similarity measures and correlation coefficients between HFSs and give a detailed debate of their drawbacks. At the end of this section, we provide the grey relational theory.

Grey relational analysis between HFSs

In this section, we apply grey relational theory to the HFSs domain and define the HFSs grey relational coefficient and grey relational degree for the first time. Thereafter, in order to deal with the pattern recognition problems, we improve the HFSs grey relational coefficient and grey relational degree to the HFSs synthetic grey relational degree, which takes the considerations of both the closeness and linear fashion of the HFSs simultaneously.

The proposed HFSs synthetic grey relational degree to pattern recognition problems with hesitant fuzzy information

In this section, we apply the proposed HFSs synthetic grey relational degree to pattern recognition problems with hesitant fuzzy information, and propose the hesitant fuzzy grey relational recognition methodology.

Suppose that a hesitant fuzzy pattern recognition problem has m unknown patterns A_i(i = 1, 2, ⋅⋅⋅, m), each pattern has n hesitant fuzzy criteria C_j(j = 1, 2, ⋅⋅⋅, n), denote $h_{A_{i}} (C_{j}) = {γ_{A_{i} 1}, γ_{A_{i} 2}, \dots, γ_{A_{i} k}, \dots, γ_{A_{i} l_{i j}}}$ as the hesitant fuzzy information of the pattern A_i on the criteria C_j,

Pattern recognition applications

In this section, we apply the proposed HFSs synthetic grey relational methodology to deal with pattern recognition problems and compare it with the existing correlation coefficients measures.

Example 1

We consider two cases represented by HFSs on X = {x₁,x₂,x₃} as follows:

Case 1:

A = { < x₁,{0.2,0.4} > , < x₂,{0.1,0.3,0.4} > , < x₃,{0.1,0.2,0.4,0.5} > },

B = { < x₁,{0.4,0.8} > , < x₂,{0.2,0.6,0.8} > , < x₃,{0.2,0.4,0.8,1.0} > }.

Case 2:

A = { < x₁,{0.2,0.4} > , < x₂,{0.3,0.4,0.5} > , < x₃

Conclusion

In this paper, we focus on the pattern recognition problems with hesitant fuzzy information. We analyze the restriction of the existing HFSs fuzzy measures, such as the distance, similarity and correlation coefficient measures. Then we extend the grey relational analysis to the HFSs fuzzy measure and introduce the HFSs grey relational degree to represent the closeness of the HFSs for the first time. Furthermore, we also creatively propose the difference and slope concept of the HFSs. Based on

Acknowledgments

The authors are very grateful to the anonymous reviewers and the editor for their valuable comments and suggestions in improving this paper. This work is supported by the Major Research Plan of the National Natural Science Foundation of China (Grant no. 91538201), the National Natural Science Foundation of China (Grant nos. 61671463, 61571454), the Natural Science Foundation of Shandong Province (Grant no. ZR2017BG014) and the Program for New Century Excellent Talents in University (Grant no.

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Grey relational analysis between hesitant fuzzy sets with applications to pattern recognition

Highlights

Abstract

Introduction

Section snippets

Preliminaries

Grey relational analysis between HFSs

The proposed HFSs synthetic grey relational degree to pattern recognition problems with hesitant fuzzy information

Pattern recognition applications

Conclusion

Acknowledgments

Knowledge-Based Systems

Applied Mathematical Modelling

Information Sciences

Expert Systems with Applications

Journal of Grey System

Journal of Computational and Applied Mathematics

Expert Systems with Applications

Information Sciences

Expert Systems with Applications

Artificial Intelligence in Medicine

Knowledge-Based Systems

Knowledge-Based Systems

Expert Systems with Applications

Knowledge-Based Systems

Applied Soft Computing

Knowledge-Based Systems

Expert Systems with Applications

Expert Systems with Applications

Knowledge-Based Systems

International Journal of Approximate Reasoning

Information Sciences

Pattern Recognition Letters

Computers & Industrial Engineering