A new measure of Fuzzy Entropy and Fuzzy Divergence

A new measure of Fuzzy Entropy and Fuzzy Divergence

K. Balasubramanian1

1. Department of Mathematics, Kongunadu College of Engineering and Technology,

Thottiam-621215.Trichy, TamilNadu

S.Subramaniam2

2. Department of Mathematics, PRIST University, Tanjore. TamilNadu

Abstract A new measure of Fuzzy Entropy and Fuzzy Divergence are obtained and particular case of four important and some other properties of the proposed measure.

H (P)

1 (1 )

logn

i 1

p(x ) , 1,

i

0 (2)

Keywords Fuzzy Set, Fuzzy Entropy, Measure of fuzzy information.

Pal and Pal [8, 9] analyzed the classical Shannon information entropy and proposed a information entropy called exponential entropy given by

INTRODUCTION

The notion of fuzzy sets was proposed to tackling problems in which indeniteness arising from a sort of intrinsic ambiguity plays a fundamental role. Fuzziness, a feature of

E(P)

n

i 1

p(x )e1 pxi 1

(3)

i

uncertainty, results from the lack of sharp distinction of the boundary of a set, i.e., an individual is neither denitely a

These authors point out that, the exponential entropy has an

advantage over Shannons entropy. For the uniform

member of the set nor denitely not a member of it. new parametric generalized exponential entropy is proposed.

probability distribution P 1

n

, 1 ,

n

1 …………….. 1

n

n

This paper is organized as follows: some basic definition related to probability and fuzzy set theory are discussed. a new fuzzy entropy measure called, exponential fuzzy entropy of order- is proposed and verifies the axiomatic

exponential entropy has a xed upper bound

lim E 1 , 1 , 1 …………….. 1 (e 1)

n n n n n

(4)

requirements. some properties of the proposed measure are studied and limiting cases also discussed here and our conclusions are presented in

PRELIMINARIES

In this section we present some basic concepts related to

which is not the case for Shannons entropy.

Corresponding to (2), Kvalseth [6] introduced and studied generalized exponential entropy of order-, given by

n

i

p(x )e1 pxi 1

probability theory and fuzzy sets which will be needed in the following analysis. First, let us cover probabilistic part

E (P) i 1 , 0

(5)

of the preliminaries. Let

Definition 1: Let X x1 ,……..xn be a discrete universe

n 1 2 n i i 1,

P ( p , p ,……p ) : p 0. if n p n 2

of discourse. A fuzzy set A on X is characterized by a

membership function A (x) : X [0, 1].The value

i1

(x) of A at x X stands for the degree of membership

be the set of n-complete probability distribution.

For any probability distribution

P ( p , p ,……p ) . Shannon entropy [11] is

A

of x in A.

Definition 2: A fuzzy set

A* is called a sharpened version

1 2 n n

of fuzzy set A if the following conditions are satised:

defined as

n

A* (x) A (x) , if A (x) 0.5;i and

H (P) p(xi ) log p(xi )

A

i 1

(1)

* (x) A (x) , if A (x) 0.5;i

Definition 3: Let FS(X) denote the family of all FSs of

Various generalized entropies have been introduced in the

universe X, assume A, B FS (X )

is given by

literature, taking the Shannon entropy as basic and have found applications in various disciplines such as economics, statistics, information processing and computing etc.

Generalizations of Shannons entropy started with Renyis` entropy [10] of order-, given by

A x, (x) x X ; B x, (x) x X ; then some set operations can be defined as follows:

A

B

A

A

i. AC x,1 (x), (x) x X

A

B

1. A B x, min (x), (x) x X

   A

   B

2. A B x, max (x), (x) x X First attempt to quantity the uncertainly associated with a fuzzy event in the context of discrete probabilistic frame work appears to have been made by Zadeh [14],who defined the

EXPONENTIAL FUZZY ENTROPY OF ORDER

We proceed with the following formal definition: Definition 4: Let A be the fuzzy set A fuzzy set dened on discrete universe of discourse X x1 ,……..xn having

entropy of fuzzy set A with respect to ( X , P) as

n

H ( A, P) A (xi ) p(xi )log p(xi )

i 1

(6)

the membership values A (xi ),i 1,2……..n .We dene

the exponential fuzzy entropy of order- corre- sponding to (5), as

10.5 A i

1 ( x )

De Luca and Termini [2] introduced a set of four axioms

A (xi )e

A i

are widely accepted as criterion for defining any fuzzy entropy. In fuzzy set theory, the fuzzy entropy is a measure

E ( A) 1 n 1 (x ) , 0 (10)

n e 1

of fuzziness which expresses the amount of average ambiguity or difficulty in making a decision whether an

i 1

e

11 A ( x )

element belongs to a set or not. A measure of fuzziness in a fuzzy set should have at least the following axioms:

PI (Sharpness): H(A) is minimum iff A is crisp set i.e.

A (xi ) 0 or 1i.

P2 (Maximality): H(A) is maximum iff A is most fuzzy set

1

Theorem:1 The measure (10) satisfy measure of fuzzy entropy.

Proof: Symmetry follows from the definition. We prove the properties (1) to (3) are satisfied by (10).

PI (Sharpness): First let E ( A) 0 , then

2

(x )e1 ( x ) 1

(x )e11A ( x )

1

(11)

i.e. A (xi ) i.

P3 (Resolution): H ( A*) H ( A) where

A* is a

A i

A i A i

sharpened version of A.

if 0, (11) will hold when either A (xi ) 0 or

P4 (Symmetry):

H ( A) H ( AC ) where

AC is the

A (xi ) 1i 1,2,3,…..n.

complement set of A.

Since A (xi ) and 1 A (xi )

gives same degree of

Conversely, if A is a crisp set , then either A (xi ) 0 or

A (xi ) 1i 1,2,3,…..n. it gives

fuzzy entropy for a fuzzy set A corresponding to(1) as

(x )e1 ( x ) 1

(x )e11A ( x )

1

(12)

fuzziness, therefore, De Luca and Termini [2] dened

1 n

(x ) log

(x )

A i

A i A i

H ( A)

A i A i

(7)

(ie).E ( A) 0 . Hence E ( A) iff A is crisp set.

n i 1 1 A (xi )log1 A (xi )

Later on Bhandari and pal [1] made a survey on information measures on fuzzy sets and gave some new measures of fuzzy entropy. Corresponding to (2) they have

suggested the following measure:

P2 (Maximality): Let

E ( A) A (xi )

Where

n

(13)

1 ( x )

1 A (xi )e

A i

1 n

A (xi )

10.5

11 ( x ) , 0

(14)

H (P)

log (x ) 1

(x ) ,

ne

1i 1 1 (x )e A

(1 ) A i

A i A i

i 1

1, 0

(8)

Differentiating (14) w.r.t A (xi ) we get,

1 ( x ) 11 A ( x )

Pal and Pal [8,9] defined exponential fuzzy entropy for a fuzzy set corresponding (3) as

f (x )

e A i e

1

1 ( x )

A i

(x )e1 A ( xi )

,

(14)

1 n A (xi )e

A i

(x )

A i

ne10.5 1 A i

E( A)

i

(9)

1 (x ) e11 A ( x )

n e 1 i 1 1 A

(x ) e 1

A ( xi )

1

A i

Throughout this paper, we denote the set of all fuzzy sets on X by FS(X).

In the next section we propose generalized fuzzy entropy measure corresponding to (4),called exponential fuzzy

Let 0 A (xi ) 0.5, then

Thus f A (xi ) is a concave function which has a global

entropy of order- and verify the axiomatic requirements.

maximum at A (xi ) 0.5. Hence

iff A is the most fuzzy set.

E ( A) is maximum

(i.e.)

A (xi ) 0.5.i 1,2,….n.

P3 (Resolution): Since H ( A) is increasing function of

A (xi ) in the range 0,0.5and the decreasing function of A (xi ) in the range 0,0.5therefore,

A* i A i

(x ) (x ) E A* E A in 0,0.5 &

A* i A i

(x ) (x ) E A* E A in 0,0.5.

hence E A* E A.

P4 (Symmetry): It is clearly from the definition,

E A E AC . Hence provethe theorem.

1. PROPERTIES OF EXPONENTIAL FUZZY ENTROPY OF ORDER-

   The measure of exponential fuzzy entropy of order- has the following properties:

   REFERENCES

   1. D. Bhandari and N. R. Pal, Some new information measures for fuzzy sets, Information Sciences, 67,204-228,1993.

   2. A. Deluca and S. Termini, A denition of Non-Probabilistic entropy in the Setting of Fuzzy Set Theory, Information and Control, 20, 301-312, 1971.

   3. J. L. Fan and Y. L. Ma, Some new fuzzy entropy formulas, Fuzzy sets and Systems, 128(2),277-284, 2002.

   4. C. H. Hwang and M. S.Yang, On entropy of fuzzy sets, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 16(4), 519-527, 2008.

   5. D. S. Hooda, On generalized measures of fuzzy entropy, Mathematica Slovaca, 54(3), 315-325, 2004.

   6. T. O. Kvalseth, On Exponential Entropies, IEEE International Conference on Systems, Man, and Cybernetics, 4, 2822-2826, 2000.

   7. J. N. Kapur, Measures of Fuzzy Information. Mathematical Sciences Trust Society, New Delhi, 1997.

   8. N. R. Pal and S. K. Pal, Object-background segmentation using new denitions of entropy, IEE Proceedings, 136, 284-295, 1989.

   9. N. R. Pal and S. K. Pal, Entropy: a new denitions and its applications, IEEE Transactions on systems, Man and Cybernetics, 21(5), 1260-1270, 1999.

      Theorem 2: For A, B FS (X ),

   10. A. Renyi,`On measures of entropy and information, Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, CA: University of California Press,547-

   E A B E

   A B E

   A E

   B.

   561,1961.

   Pr oof :Let

   X x x X , A (xi ) B (xi ) 15

   X x x X , A (xi ) B (xi ) 16

   where A (x) & B (x) be the functions of A& B respec.

   fuzzy membership

   E ( A B)

   1

   e10.5 1

   A B (xi )e

   1

   ( xi )

   A B

   11

   ( x ) 1

   (17)

   1 A B (xi )e

   A B i

   1 ( x )

   B

   (xi )e

   B i

   1 X I X 1

   (x )e 1 1 B ( x I )

   1

   B i

   (18)

   ne1 0.5 1

   (x )e1 ( x )

   A i

   AB i

   i

   X I X

   1 A

   (x )e 1 1 A ( x I )

   1

   Lemma:

   For any A FS ( X ), and ACof fuzzy set A.

   E A E AC E A AC E A AC .

2. CONCLUSIONS

This work introduces a Fuzzy Entropy and Fuzzy Divergence measure is exponential fuzzy entropy of order- in the setting of fuzzy set theory.Some properties of this measure have been also studied.This measure generalizes Pal and Pal [9] exponential entropy and De-Luca and Termini [2] logarithmic entropy. Introduction of parameter- provides new exibility and wider application of exponential fuzzy entropy to different situations.