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Feature Article: a New Similarity Measure for Vague Sets 15 14 Feature Article: Jingli Lu, Xiaowei Yan, Dingrong Yuan and Zhangyan Xu A New Similarity Measure for Vague Sets Jingli Lu, Xiaowei Yan, Dingrong Yuan, Zhangyan Xu Abstract--Similarity measure is one of important, effective and their similarity should not be 1. In reference [5] Dug Hun Hong widely-used methods in data processing and analysis. As vague set put forward another similarity measure MH, according to the theory has become a promising representation of fuzzy concepts, in formulae of MH, we can obtain the similarity of vague values [0.3, this paper we present a similarity measure approach for better 0.7] and [0.4, 0.6] is M ([0.3,0.7],[0.4,0.6]) = 0.9 , in the same understanding the relationship between two vague sets in applica- H tions. Compared to existing similarity measures, our approach is far model, we can get M H ([0.3,0.6],[0.4,0.7]) = 0.9 . In a voting more reasonable, practical yet useful in measuring the similarity model, the vague value [0.3, 0.7] can be interpreted as: “the vote between vague sets. for a resolution is 3 in favor, 3 against and 4 abstentions”; [0.4, Index Terms-- Fuzzy sets, Vague sets, Similarity measure 0.6] can be interpreted as: “the vote for a resolution is 4 in favor, 4 against and 2 abstention”. [0.3, 0.6] and [0.4, 0.7] can have I. INTRODUCTION similar interpretation. Intuitively, [0.3, 0.7] and [0.4, 0.6] may be more similar than [0.3, 0.6] and [0.4, 0.7]. Therefore sometimes In the classical set theory introduced by Cantor, a German the results of M model are not accordant with our intuition. mathematician, values of elements in a set are only one of 0 and H After analyzing most existing vague sets and vague values simi- 1. That is, for any element, there are only two possibilities: in or larity measures, we find out that almost each measure has its not in the set. Therefore, the theory cannot handle the data with defect. In section 2.2, we will illustrate them with more examples. ambiguity and uncertainty. Then we have to make a choice according to the applications. Zadeh proposed fuzzy theory in 1965 [1]. The most important A more reasonable approach is proposed to measure similarity in feature of a fuzzy set is that fuzzy set A is a class of objects that this paper, after analyzing existing methods. satisfy a certain (or several) property. Each object x has a mem- The remaining of this paper is organized as follows. In Section bership degree of A, denoted as µ (x). This membership function A 2, several methods of similarity measure for vague set are dis- has the following characteristics: The single degree contains the cussed. An improved similarity measure method and its proper- evidences for both supporting and opposing x. It cannot only ties are given in Section 3. Section 4 concludes this paper. represent one of the two evidences, but it cannot represent both at the same time too. II. PRELIMINARIES In order to deal with this problem, Gau and Buehrer proposed the concept of vague set in 1993 [2], by replacing the value of an A. Vague set element in a set with a sub-interval of [0, 1]. Namely, a true- In this section, we review some basic definitions of vague values membership function tv(x) and a false-membership function fv(x) and vague sets from [2], [3], [4]. are used to describe the boundaries of membership degree. These Definition 1 Vague Sets [2]: Let X be a space of points (ob- two boundaries form a sub-interval [tv(x), 1 – fv(x)] of [0, 1]. The jects), with a generic element of X denoted by x. A vague set V in vague set theory improves description of the objective real world, X is characterized by a truth-membership function t and a false- becoming a promising tool to deal with inexact, uncertain or v vague knowledge. Many researchers have applies this theory to membership function fv . tv is a lower bound on the grade of many situations, such as fuzzy control, decision-making, knowl- membership of x derived from the evidence for x, and fv is a edge discovery and fault diagnosis. And the tool has presented lower bound on the negation of x derived from the evidence more challenging than that with fuzzy sets theory in applications. against x, t and f both associate a real number in the interval In intelligent activities, it is often needed to compare and cou- v v ple between two fuzzy concepts. That is, we need to check [0,1] with each point in X, where tv + f v ≤ 1 . That is whether two knowledge patterns are identical or approximately same, to find out functional dependence relations between con- tv : X → [0,1] ; fv : X → [0,1] cepts in a data mining system. Many measure methods have been This approach bounds the grade of membership of x to a sub- proposed to measure the similarity between two vague sets (val- interval [t (x),1− f (x)] of [0,1] ues). Each of them is given from different side, having its own v v counterexamples. Such as Shyi-Ming Chen proposed a similarity When X is continuous, a vague set V can be written as measure MC in [3], whereas from the MC model we can gain the V = ∫ [tV (x),1− fV (x)] / x , x ∈ X . similarity of vague values [0.5, 0.5] and [0, 1] is 1, obviously X When X is discrete, a vague set V can be written as Department of Computer Science, Guangxi Normal University, Gulin, 541004, P.C.China. n [email protected]; [email protected]. V = ∑[tV (xi ),1− fV (xi )]/ xi , xi ∈ X . i=1 November 2005 Vol.6 No.2 IEEE Intelligent Informatics Bulletin Feature Article: A New Similarity Measure for Vague Sets 15 Definition 2: Let x and y be two vague values, where 0.8]). According to our intuition, pair of ([0.4, 0.8], [0.5, 0.8]) is more similar than pair of ([0.4, 0.8], [0.5, 0.7]), but in the model x = [t x ,1 − f x ] and y = [t y ,1 − f y ] . If t x = t y and of ML, the two pairs of vague values have the same similarity. f x = f y , then the vague values x and y are called equal (i.e., The MO model also reflects the equal concerns between the difference of true-membership degrees and the difference of [t x ,1 − f x ] = [t y ,1 − f y ] ). false-membership degrees. But similar to the MH model, the MO Definition 3: Let A and B be vague sets of the universe of dis- model does not consider whether the differences are positive or course U, U ={u1 ,u2 ,Lun }, where negative. The above methods of similarity measure can be used to solve A = [t (u ),1− f (u )] / u +[t (u ),1− f (u )] / u A 1 A 1 1 A 2 A 2 2 the problem of how to determine the similarity between two +L+[t A (un ),1− f A (un )] / un vague values in a certain extent. But each of them focuses on B = [t (u ),1− f (u )]/u +[t (u ),1− f (u )]/u B 1 B 1 1 B 2 B 2 2 different aspects. There are three factors which affect the similar- +L+[tB (un ),1− fB (un )]/un ity of vague values: true-membership function tx, false- membership function f , and 1 – t – f . The reason there are If ∀i , [t A (ui ),1− f A (ui )] = [tB (ui ),1− f B (ui )] , then the vague x x x many counterexamples under the measures of the M , M and M sets A and B are called equal, where 1 ≤ i ≤ n . H L O models is that the weights of |tx – ty|, | fx – fy| and |(ty + fy) – (tx + B. Research into similarity measure fx)| in the above methods are constants. Its explicit characteristic Currently, there have been many similarity measurements for is that it is not considered whether the difference is positive or vague set (value). Suppose that X = [tx, 1 – fx] and Y = [ty, 1 – fy] negative. Based on this idea, a new weighted and variable simi- are two vague values over the discourse universe U. Let S(x) = tx larity measure is proposed. It can considerably reduce the possi- – fx,, S(y) = ty – fy, the MC, MH, ML and MO models are defined bility of similarity coincidence. respectively in [3], [5], [6] and [7] as follows, | S(x) − S(y) | III. MEASURING THE SIMILARITY BETWEEN VAGUE SETS M (x, y) = 1− C 2 This section constructs a new approach for measuring the (1) | (t x − t y ) − ( f x − f y ) | similarity between vague sets, analyzes the properties and illus- = 1− trates the use by examples. 2 | t − t | + | f − f | A. A New Similarity Measure M (x, y) = 1− x y x y (2) H 2 We first give an example. Assume that there are four candi- | S(x) − S(y) | | t x − t y | + | f x − f y | dates A, B, C, D, and ten voters. One voter supports A, one op- M L (x, y) = 1− − poses A; two support B, one opposes B; seven support C, one 4 4 (3) opposes C ; eight support D, and one opposes D.
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