A New Formula for Fuzzy Entropy of Vague Sets

Jiguo Zhang Institute of Information Management and Decision,

Hohai University, Changzhou, Jiangsu 213022, China

zhangjg@hhuc.edu.cn

Lei Qiu State Key Laboratory of Hydrology-Water Resources

and Hydraulic Engineering, Hohai University Nanjing, Jiangsu 210098, China

qiulei@hhu.edu.cn

Abstract-- A new distance measure is defined for vague sets,

and then the corresponding measure formula for the fuzzy

entropy of vague sets is proposed. Finally, comparison of

the existing formulae and our one shows that our formula

is more reasonable.

Keywords-- Fuzzy Entropy; Vague Sets; Fuzzy Sets

I. INTRODUCTION

Since the inception of the theory of fuzzy sets [1], its evolution and application have mushroomed in a range of areas. Gau and Gargov [2] introduced vague sets (VS) as generalization of fuzzy sets (FS). Investigators have been increasingly concerned with the vague set theory, especially the conversion from vague sets to fuzzy sets and the process of inversion, and their fuzzy entropy.

Many investigations have been devoted to the fuzzy entropy of vague sets. Different definitions of fuzzy entropy have been proposed by Zhang and Jiang [3---7]. Analysis of these formulae shows that some information will be lost when calculating the fuzzy entropy of vague sets. In order to overcome this problem, a new measure is established, which is related to the formula for conversion from vague sets to fuzzy sets, and the uncertainty of vague sets. For the sake of discussion later, in section 2, we will briefly review the definition of vague sets and transforming them into fuzzy sets. In section 3 we will establish a distance formula between vague sets, and then a new practical fuzzy entropy ____________ Supported by the Public-interest Industry Project of Ministry of Water Resources under Grant 200801027

measure will be derived based on the conversion of vague sets and fuzzy entropy calculation, set up by Kosko [8], in a relation to a measure of fuzziness with fuzzy sets. Finally, several entropy measures mentioned in this paper will be compared through illustration.

Throughout this paper, X is the universal set; VSs(X) is the class of all vague sets of X; FSs(X) is the class of all fuzzy sets of X; Let FA be a fuzzy set transformed from a vague set.

II. VAGUE SETS

Definition 1: Let X be a space of objects, with a generic element of X donated by x. A vague set A in X is characterized by a true-membership function )(xt A and a false-membership function )(xf A , as

{ }XxxfxtA AA ∈><= )(),( ; that is,

]1 ,0[:)( →Xxt A ]1 ,0[:)( →XxfA

where 1)()( ≤+ xfxt AA . )(xt A is the lower bound of

A on the grade of membership of x derived from the evidence for x. )(xf A is a lower bound of A on the

grade of membership of x derived from the evidence against x. )()(1)( xfxtx AAA −−=π is called unknown-degree of x in A and )](1 ),([ xfxt AA − is the

vague value of x in A, noted A(x). About transforming vague sets into fuzzy set, Zhang and Lin[9] proposed the following formulae using principles of information diffusion[10]

2010 International Conference on Artificial Intelligence and Computational Intelligence

978-0-7695-4225-6/10 $26.00 © 2010 IEEE

DOI 10.1109/AICI.2010.223

495

2010 International Conference on Artificial Intelligence and Computational Intelligence

978-0-7695-4225-6/10 $26.00 © 2010 IEEE

DOI 10.1109/AICI.2010.223

495

)()(1)(1

)()( xxxf

xtx AA

AAFA π

πμ

+−

+=

(1)

)()(1)(1

)()( xxxt

xfx AA

AAAF π

πμ

+−

+= (2)

where AF is complementary set of fuzzy set FA

III. FUZZY ENTROPY OF VAGUE SETS

Based upon De Luca [12] axioms for fuzzy sets to give an entropy measure we put forward an entropy definition for vague sets.

Definition 2. A real function VE : VSs(X ) → [0,1] is called a fuzzy entropy on VSs( X ), if it satisfies the following properties:

(q1). 0)( =AVE iff A is a non-fuzzy set, that is, a

crisp set; (q2). 1)( =AVE iff )()(, xfxtXx AA =∈∀ ; (q3). If BA ⊆ , that is when )()( xfxt BB ≤ ,

)()( xtxt BA ≤ , )()( xfxf BA ≥ is correct; when )()( xfxt BB ≥ , )()( xtxt BA ≥ , )()( xfxf BA ≤ is correct,

then )()( BVEAVE ≤ ; and

(q4). )()( AVEAVE = .

Owing to the existing fuzziness of unknown-degree )(xAπ and the fuzziness of uncertainty [the

true-membership degree )(xt A , the false-membership degree )(xf A ], the fuzzy entropy corresponding to fuzzy

sets is difficult to determine. Let us concentrate on the CDEΔ in Fig. 2. According to Szmidt and Kacprzyk [11], a vague set A can be denoted ),,( AAA ft π , so a

non-fuzzy set (a crisp set) corresponds to the point C (the element fully belongs to it as )0,0 ,1(),,( =AAA ft π )

and at point D (the element fully belongs to it as )0,1 ,0(),,( =AAA ft π ). Points C and D representing a

crisp set have the degree of fuzziness equal to 0. Furthermore, point E has maximal fuzziness which equals to 1.

Figure. 1 Vague set and its fuzziness diagram

The following fuzzy entropy formula of a fuzzy set was proposed in Kosko [16]:

),(),(

)(far

near

AAdAAd

FE = (3)

Following Kosko [16], the fuzzy entropy of a fuzzy set can be affirmed by the ratio of nearest non-fuzzy set

nearA distance to the furthest non-fuzzy set farA

distance. Now, a new distance formula is derived. Following

conversion formula (1), a vague set )](1 ),([)( xfxtxA AA −= is transformed into a fuzzy set

FA whose membership function is )(xFAμ . Consider

this vague set A as a new sign using a binary group, i.e., { })(),()( xxxA AFA πμ= (4)

Assume that A and B are vague sets. Then the distance between them is defined as

)()()()(),( xxxxBAd BAFBFA ππμμ −+−= (5)

As shown in Fig. 2, Points C and D are two crisp sets, according to formula (4):

{ }0 ,1)( =xC { }0 ,0)( =xD

F D(0,1,0)

A

a b

C(1,0,0)

E(0,0,1)

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Figure. 2 (a) A case when point C is the nearest non-fuzzy

neighbor,

point D is the farthest non-fuzzy neighbor of A. point C is the farthest non-fuzzy neighbor of A.

For the situation in Fig. 2(a), because vague sets A has )()( xfxt AA > , the corresponding fuzzy sets should

satisfy

)()( xx AFFA μμ >

and 2/1)( >xFAμ

Hence, the nearest and furthest crisp sets are C and D, respectively. According to theory of [16], the fuzzy entropy is

)()()()(

0)(0)(0)(1)(

)(xxxx

xxxx

AVEFAA

AFA

AFA

AFA

μπμπ

πμπμ

++

=−+−−+−

= (6)

Figure. 2(b) A case when point D is the nearest non-fuzzy

neighbor,

For the situation in Fig. 3(b), because vague set A has )()( xfxt AA < , the corresponding fuzzy set should

satisfy

)()( xx AFFA μμ < and 2/1)( <xFAμ

Hence, the nearest and furthest crisp sets are D and C, respectively, according to theory of [16] again, the fuzzy entropy is

)()()()(

0)(1)(0)(0)(

)(xxxx

xxxx

AVEAFA

FAA

AFA

AFA

μπμπ

πμπμ

++=

−+−−+−

= (7)

Combining with (6) and (7), we get { }{ })(),(max)(

)(),(min)()(

xxxxxx

AVEAFFAA

AFFAA

μμπμμπ

++

=

(8) Let A be the vague set on the spaces of objections,

which is made of n elements. Then the fuzzy entropy is { }{ }∑

= ++

=n

i AFFAA

AFFAA

xxxxxx

nAVE

1 )(),(max)()(),(min)(1)(

μμπμμπ

(9)

Expressed by the true or false membership combined with (1) and (2), formula (8) changes to

{ } { }

{ } { }∑= −−

+++

−−+

++=

n

iAA

A

AAAA

AAA

AAAA

tfft

tfft

nAVE

1 1,1max1

,max

1,1min1

,min1)(

πππ

πππ

(10)

Hence, the fuzzy entropy of vague sets A can now be written as:

{ }{ }∑

= −−−−=

n

i AA

AAZS xtxf

xtxfn

AVE1 )(1 ),(1max

)(1 ),(1min1)( (11)

IV. REMARKS AND COMPARISION

Another operative formula for fuzzy entropy of vague sets was put forward by Zhang and Jiang [13]:

∑=

=n

i iAiA

iAiAZ xfxt

xfxtn

AVE1 )}( ),(max{

)}( ),(min{1)( (12)

This formula has shortcomings. When 0)( =xt A ,

F

E(0,0,1)

D(0,1,0) C(1,0,0)

A a b

F D(0,1,0) C(1,0,0)

A a

b

E(0,0,1)

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whatever the value of )(xfA , the conclusion

is 0)( =AVE , which is obviously unreasonable.

Example 5: Let A1 and B1 be vague sets with a single element x, ]1.0 ,0[)(1 =xA and ]9.0 ,0[)(1 =xB ,

although the unknown-degree degree of B1 is 9 times of A1, according to formula (12), 0)()( == BVEAVE ZZ .

Formula (12) does not consider the uncertainty of unknown-degree )(xπ . Therefore, formula (12) is not

the appropriate operative formula for fuzzy entropy of vague sets.

Example 8: For vague sets from examples 5-7, we can get more reasonable results than from the ones mentioned above

{ }{ }

{ }{ } 1.0

9.01 ,01max9.01 ,01min

)(1 ),(1max)(1 ),(1min

)(

11

11

1

=−−−−=

−−−−

=xfxtxfxt

AVE

AA

AA

ZS

{ }{ }

{ }{ } 9.0

1.01 ,01max1.01 ,01min

)(1 ),(1max)(1 ),(1min

)(

11

11

1

=−−−−=

−−−−

=xfxtxfxt

BVE

BB

BB

ZS

V. COCLUSION

Compared to other formulae for conversion of vague sets to fuzzy sets, the conversion formula deduced using the principle information diffusion theory has the characteristics of being an objective, intuitionistic, and simple operative and exclusive membership function. Furthermore, it satisfies all of the proposed constraint conditions and provides a valid, practical method for vague sets. In addition, a new fuzzy entropy formula has a simple format and is convenient to operate in terms of Kosko’s definition of fuzzy entropy.

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on System, Man and Cybernetics 23 (2), 1993, pp. 610-614.

[3] Qiansheng Zhang, Shengyi Jiang, A note on information entropy measures for vague sets and its

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[4] P. Fan, J. Liang,, T. Li, new fuzzy entropy of vague sets, Computer engineering and applications(chinese), 43(13), 2007, pp.179-181

[5] G. Huang, Y. Liu, On the fuzzy entropy of vague sets, Computer engineering and applications(chinese), 33, 2005, pp.48-50

[6] F. Li, A. Lu, L, Cai, Fuzzy entropy of vague sets and its construction method, Computer application and software(chinese), 2, 2002, pp. 10-12

[7] G, huang, Y, Liu, The fuzzy entropy of vague sets based on non-fuzzy sets, Computer application and software(chinese), 6(22), 2005, pp.16-17

[8] B. Kosko, Fuzzy Entropy and Conditioning, Information Sciences 40, 1986, pp.165-174.

[9] J. Zhang, W. Lin, Annotation of Conversion Formula from Vague Sets to Fuzzy Sets, Fuzzy Systems and Mathematics(to be published)

[10] C. Huang, Principle of information diffusion, Fuzzy Sets and System 91, 1997, pp. 69-90

[11] Eulalia Szmidt, Janusz Kacprzky, Entropy for intuitionistic fuzzy sets, Fuzzy Sets and System 118, 2001, pp. 467-477.

[12] A. De Luca, S. Termini, A definition of a non-probabilistic entropy in the setting of fuzzy sets theory, Information and Control 20, 1972, pp.301-312

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