Similarity measures of intuitionistic fuzzy sets based on Hausdorff distance

doi:10.1016/j.patrec.2004.06.006

Pattern Recognition Letters

Volume 25, Issue 14, 15 October 2004, Pages 1603-1611

https://doi.org/10.1016/j.patrec.2004.06.006 Get rights and content

Abstract

This paper presents a new method for similarity measures between intuitionistic fuzzy sets (IFSs). We will present a method to calculate the distance between IFSs on the basis of the Hausdorff distance. We will then use this distance to generate a new similarity measure to calculate the degree of similarity between IFSs. Finally we will prove some properties of the proposed similarity measure and use several examples to compare the proposed similarity measure with existing methods. Numerical results show that the proposed similarity measure is much simpler than existing methods and is well suited to use with linguistic variables.

Introduction

The notion of defining intuitionistic fuzzy sets (IFSs) for fuzzy set generalizations, introduced by Atanassov (1986), has proven interesting and useful in various application areas. Since this fuzzy set generalization can present the degrees of membership and non-membership with a degree of hesitancy, the knowledge and semantic representation becomes more meaningful and applicable (cf. Atanassov, 1986, Atanassov, 1994, Atanassov, 1999). These IFSs have been widely studied and applied in a variety of areas such as logic programming (cf. Atanassov and Gargov, 1990; Atanassov and Georgeiv, 1993), decision making problems (Szmidt and Kacprzyk, 1996) and in medical diagnostics (De et al., 2001), etc.

In many applications, the similarity between IFSs is very important. Recently, Li and Cheng (2002) discussed similarity measures on IFSs and showed how these measures may be used in pattern recognition problems. However, Li and Cheng's similarity measures may not be effective in some cases. To overcome the drawbacks of Li and Cheng's methods, Liang and Shi (2003) proposed several new similarity measures and also discussed the relationships between these measures. Numerical comparisons showed that Liang and Shi's similarity measures are more reasoned than Li and Cheng's. On the other hand, Mitchell (2003) interpreted IFSs as ensembles of ordered fuzzy sets from a statistical viewpoint to modify Li and Cheng's methods.

In this study, we present a new method to calculate the degree of similarity between IFSs based on the Hausdorff distance concept. Numerical results show that the proposed similarity measure is much simpler than existing methods and is well suited to use with linguistic variables. The organization of this paper is as follows: in Section 2, we review some similarity measures of IFSs (cf. Li and Cheng, 2002; Liang and Shi, 2003; Mitchell, 2003), and the Hausdorff distance concept. In Section 3, we present a method to calculate the distance between IFSs on the basis of the Hausdorff distance. We then use this distance to generate a new similarity measure to calculate the degree of similarity between IFSs. We will also prove some properties of the proposed similarity measure. In Section 4, we compare the proposed similarity measure with existing methods. Conclusions are made in Section 5.

Section snippets

Intuitionistic fuzzy set

An IFS $A$ in X is given by Atanassov (1986) as $A ={(x,μ_{Ã} (x),ν_{Ã} (x)) | x∈X},$ where $μ_{Ã} (x):X→[0,1] ν_{Ã} (x):X→[0,1]$ with the condition $0⩽μ_{Ã} (x)+ν_{Ã} (x)⩽1 ∀x∈X.$ The numbers $μ_{Ã} (x)$ and $ν_{Ã} (x)$ denote the degree of membership and non-membership of x to $A$ , respectively. In this paper, we denote IFSs(X) as the set of all IFSs in X. In the following, we present some basic operations on IFSs which will be needed in our next discussion.

Definition 2.1

If $A$ and $B$ are two IFSs of the set X, then

(i)
$A ⊆ B$ if and only if ∀x∈X, $μ_{Ã} (x)⩽μ_{B̃} (x)$ and

Similarity measures between intuitionistic fuzzy sets

Distance is an important concept in science and engineering. In the following, we define a distance between two IFSs based on the Hausdorff distance. Let $A$ and $B$ be two IFSs in X={x₁,x₂,…,x_n} and let $I_{Ã} (x_{i})$ and $I_{B̃} (x_{i})$ be subintervals on [0,1] denoted by the following: $I_{Ã} (x_{i})=[μ_{Ã} (x_{i}),1−ν_{Ã} (x_{i})], I_{B̃} (x_{i})=[μ_{B̃} (x_{i}),1−ν_{B̃} (x_{i})], i=1,2,…,n.$ Let $H(I_{Ã} (x_{i}),I_{B̃} (x_{i}))$ be the Hausdorff distance between $I_{Ã} (x_{i})$ and $I_{B̃} (x_{i})$ . We then can define the distance $d_{H} (A, B)$ between $A$ and $B$ as follows: $d_{H} (A, B)= 1 n ∑_{i=1}^{n} H(I_{A}$

Numerical examples

In this section, we present several examples to compare the proposed similarity measure with existing similarity measures. For convenience, we consider p=1 and ω_i=1/3, i=1,2,3 in similarity measures S_d^p, S_e^p, S_s^p, S_h^p and S_mod.

Example 1

Liang and Shi, 2003

Assume that there are three patterns denoted with IFSs in X={x₁,x₂,x₃}. The three patterns are denoted as follows: $A_{1} ={(x_{1},0.3,0.3),(x_{2},0.2,0.2),(x_{3},0.1,0.1)};$ $A_{2} ={(x_{1},0.2,0.2),(x_{2},0.2,0.2),(x_{3},0.2,0.2)};$ $A_{3} ={(x_{1},0.4,0.4),(x_{2},0.4,0.4),(x_{3},0.4,0.4)}.$ Assume that a sample $B ={(x_{1}$

Conclusions

In this paper, we presented a new method for similarity measures between IFSs. First, we gave a method to calculate the distance of IFSs on the basis of the Hausdorff distance. We then use this distance to generate a new similarity measure to calculate the degree of similarity between IFSs. We not only proved some properties of the proposed similarity measure, we also used several examples to make comparisons between the proposed similarity measure and existing methods. Based on the results, we

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Similarity measures of intuitionistic fuzzy sets based on Hausdorff distance

Abstract

Introduction

Section snippets

Intuitionistic fuzzy set

Similarity measures between intuitionistic fuzzy sets

Numerical examples

Liang and Shi, 2003

Conclusions

Fuzzy Sets Systems

Fuzzy Sets Systems

Fuzzy Sets Systems

Fuzzy Sets Systems

Fuzzy Sets Systems

Pattern Recognition Lett

Pattern Recognition Lett

Inform. Sci

Informat. Process. Manage

Pattern Recognition