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Relationship between similarity measure and entropy of interval valued fuzzy sets
Fuzzy Sets and Systems 157 (2006) 1477 – 1484 www.elsevier.com/locate/fss
Relationship between similarity measure and entropy of interval valued fuzzy sets夡 Wenyi Zenga, b,∗ , Hongxing Lia a School of Mathematical Sciences, Beijing Normal University, Beijing 100875, PR China b Department of Medical Epidemiology and Biostatistics, Karolinska Institutet, Stockholm, SE171 77, Sweden
Received 3 March 2005; received in revised form 24 November 2005; accepted 26 November 2005 Available online 20 December 2005
Abstract In this paper, we introduce concepts of entropy of interval valued fuzzy set which is different from Bustince and Burillo [Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets, Fuzzy Sets and Systems 78 (1996) 305–316] and similarity measure of interval valued fuzzy sets, discuss their relationship between similarity measure and entropy of interval valued fuzzy sets in detail, prove three theorems that similarity measure and entropy of interval valued fuzzy sets can be transformed by each other based on their axiomatic definitions and put forward some formulas to calculate entropy and similarity measure of interval valued fuzzy sets. © 2005 Elsevier B.V. All rights reserved. Keywords: Interval valued fuzzy set; Fuzzy set; Entropy; Similarity measure
1. Introduction Since fuzzy set was introduced by Zadeh [19], many new approaches and theories treating imprecision and uncertainty have been proposed. Some of theories, like intuitionistic fuzzy set theory introduced by Atanassov [1,2], are extensions of the classic fuzzy theory. Another, well-known generalization of an ordinary fuzzy set is interval valued fuzzy set. Interval valued fuzzy set was first introduced by Zadeh [20], after that, some authors investigated the topic and obtained some meaningful conclusions. For example, Gorzalczany [9] in approximate reasoning, Turksen [15] in normal forms, Deschrijver, etc. [7] interval valued fuzzy relations and implication and so on. Moreover, some authors pointed out that there is a strong connection between intuitionistic fuzzy sets and interval valued fuzzy sets, for more details we refer the readers to [4,7,16]. Entropy and similarity measure of fuzzy sets are two important topics in fuzzy set theory. Entropy of fuzzy set describes the fuzziness degree of fuzzy set. Many scholars have studied it from different points of view. For example, in 1972, De Luca and Termini [6] introduced some axioms to describe the fuzziness degree of fuzzy set. Kaufmann [11] proposed a method to measure the fuzziness degree of fuzzy set by a metric distance between its membership function and the membership function of its nearest crisp set. Another way given by Yager [18] was to view the fuzziness degree 夡 Supported by National Natural Science Foundation of China (60474023), Science and Technology Key Project Fund of Ministry of Education (03184), and the Major State Basic Research Development Program of China (2002CB312200). ∗ Corresponding author. Department of Mathematics, Beijing Normal University, Beijing 100875, PR China. Tel.: +86 10 62209376. E-mail address: [email protected] (W. Zeng).
0165-0114/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2005.11.020
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of fuzzy set in terms of a lack of distinction between the fuzzy set and its complement. On the other hand, similarity measure of fuzzy sets indicates the similarity degree of fuzzy sets and was extensively applied in many fields such as fuzzy clustering, image processing, fuzzy reasoning and fuzzy neutral network [5,13,17]. Based on these two kinds of important measures, Liu [12] investigated entropy, distance measure and similarity measure of fuzzy sets and their relations. Fan [8] studied distance measure and fuzzy entropy induced by distance measure. Some authors have investigated interval valued fuzzy set and its some relevant topics, for example, in 2004, Grzegorzewski [10] studied distance between interval valued fuzzy sets based on the Hausdroff metric, Burillo and Bustince [3] and Szmidt and Kacprzyk [14] researched entropy of interval valued fuzzy set from different point of views, respectively. In 1996, Burillo and Bustince [3] introduced the concept of entropy of interval valued fuzzy set and intuitionistic fuzzy set, Szmidt and Kacprzyk [14] used a different approach from Burillo and Bustince [3] to propose an entropy measure for intuitionistic fuzzy set. In this paper, we want to study the relationship between entropy and similarity measure of interval valued fuzzy sets, give three theorems that entropy and similarity measure of interval valued fuzzy sets can be transformed by each other based on their axiomatic definitions and put forward some formulas to calculate entropy and similarity measure of interval valued fuzzy sets. The rest of our work is organized as follows. In Section 2, we recall some basic notions of interval valued fuzzy set. In Section 3, we introduce the concepts of entropy and similarity measure of interval valued fuzzy sets. In Section 4, we investigate relationship between similarity measure and entropy of interval valued fuzzy sets and prove three theorems that similarity measure and entropy of interval valued fuzzy sets can be transformed by each other based on their axiomatic definitions. The final section is conclusion. 2. Some notions Throughout this paper, we write X to denote the discourse set, IVFSs stands for the set of all interval valued fuzzy subsets in X, F(X) and P(X) stand for the set of all fuzzy sets and crisp sets in X, respectively. A expresses an interval valued fuzzy set, the operation “c” is the complement of interval valued fuzzy set or fuzzy set in X, ∅ stands for the empty set. Let I = [0, 1] and [I ] be the set of all closed subintervals of the interval [0, 1]. Then, according to Zadeh’s extension principle [19], we can popularize these operations such as ∨, ∧ and c to [I ], thus, ([I ], ∨, ∧, c) is a complete lattice with a minimal element 0¯ = [0, 0] and a maximal element 1¯ = [1, 1]. Furthermore, let a¯ = [a − , a + ], b¯ = [b− , b+ ], ¯ then we have, a¯ = b¯ ⇐⇒ a − = b− , a + = b+ , a¯ b¯ ⇐⇒ a − b− , a + b+ and a¯ < b¯ ⇐⇒ a¯ b¯ and a¯ = b. Considering [I ] is dense, therefore, ([I ], ∨, ∧, c) is a superior soft algebra. We call a mapping: A : X −→ [I ] an interval valued fuzzy set in X. For every A ∈ IVFSs and x ∈ X, A(x) = [A− (x), A+ (x)] is called the degree of membership of an element x to A, then fuzzy sets A− : X → [0, 1] and A+ : X → [0, 1] are called a low fuzzy set of A and a upper fuzzy set of A, respectively. For simplicity, we denote A = [A− , A+ ]. Therefore, some operations such as ∪, ∩, c can be introduced into IVFSs, thus (IVFSs, ∪, ∩, c) is a superior soft algebra. If A, B ∈ IVFSs, then the following operations can be found in Zeng [21]. A ⊆ B iff ∀x ∈ X, A− (x)B − (x) and A+ (x)B + (x), A = B iff ∀x ∈ X, A− (x) = B − (x) and A+ (x) = B + (x), Ac = [(A+ )c , (A− )c ]. 3. Entropy of interval valued fuzzy set In 2001, Szmidt and Kacprzyk [14] extended De Luca and Termini [6] axioms for fuzzy set to introduce entropy of intuitionistic fuzzy set. Based on this view point of Szmidt and Kacprzyk [14], we introduce the concept of entropy of interval valued fuzzy set which is different from Bustince and Burillo [3]. Definition 1. A real function E : IVFSs−→ [0, 1] is called an entropy on IVFSs, if E satisfies the following properties: (P1) E(A) = 0 iff A is a crisp set; (P2) E(A) = 1 iff A− (x) + A+ (x) = 1;
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(P3) E(A)E(B) if A is less fuzzy than B, i.e., A− (x) B − (x) and A+ (x) B + (x) for B − (x) + B + (x) 1 or A− (x)B − (x) and A+ (x)B + (x) for B − (x) + B + (x) 1; (P4) E(A) = E(Ac ). Then we can give the following formulas to calculate entropy of interval valued fuzzy set A: n
1 − |A (xi ) + A+ (xi ) − 1|, n i=1 n 1 (A− (xi ) + A+ (xi ) − 1)2 , E2 (A) = 1 − n i=1 b 1 |A− (x) + A+ (x) − 1| dx, E3 (A) = 1 − b−a a b − (A (x) ∧ (1 − A+ (x))) dx E4 (A) = ab , − + a (A (x) ∨ (1 − A (x))) dx E1 (A) = 1 −
(1)
(2) (3) (4)
where the integral in Eqs. (3) and (4), is Lebesgue integral. Definition 2. A real function N : IVFSs×IVFSs→ [0, 1] is called similarity measure of interval valued fuzzy sets, if N satisfies the following properties: (N1) N(A, Ac ) = 0 if A is a crisp set; (N2) N (A, B) = 1 ⇐⇒ A = B; (N3) N (A, B) = N(B, A); (N4) for all A, B, C ∈IFSs, if A ⊆ B ⊆ C, then N (A, C)N (A, B), N (A, C) N (B, C). Obviously, the axiomatic definition of similarity measure and entropy of interval valued fuzzy sets are extended from fuzzy set theory. Specially, if interval valued fuzzy sets A and B become fuzzy sets, then N (A, B) is similarity measure of fuzzy sets. Based on this point of view, then we have the following theorem. Theorem 1. N (A− , (A+ )c ) is entropy of interval valued fuzzy set A, where fuzzy sets (A+ )c is the complement of fuzzy set A+ . Proof. (P1) If A is a crisp set, then for every x ∈ X, we have A− (x) = A+ (x) = 1 or A− (x) = A+ (x) = 0, it means that A− and A+ are crisp sets and A− = A+ , therefore, N (A− , (A+ )c ) = 0. (P2) Known by the definitions of similarity measure of fuzzy sets, we have, N(A− , (A+ )c ) = 1 ⇐⇒ A− = (A+ )c ⇐⇒ A− (x) + A+ (x) = 1. (P3) Since A− (x)B − (x) and A+ (x)B + (x) for B − (x) + B + (x) 1 implies A− (x) B − (x) (B + )c (x) (A+ )c (x). Namely, A− ⊆ B − ⊆ (B + )c ⊆ (A+ )c , known by the definition of similarity measure of fuzzy sets, we have N(A− , (A+ )c )N (A− , (B + )c ) N (B − , (B + )c ). With the same reason, when A− (x)B − (x) and A+ (x) B + (x) for B − (x)+B + (x) 1, we can prove N (A− , (A+ )c ) N(B − , (B + )c ). (P4) Because Ac = [(A+ )c , (A− )c ], and known by the definition of similarity measure of fuzzy sets, we have N(A− , (A+ )c ) = N ((A+ )c , A− ) = N ((A+ )c , ((A− )c )c ). Hence, we complete the proof of Theorem 1. For X = {x1 , x2 , . . . , xn }, A , B ∈ F(X), since both n
1 N1 (A , B ) = 1 − |A (xi ) − B (xi )| n
i=1
(5)
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n 1 (A (xi ) − B (xi ))2 N2 (A , B ) = 1 − n
(6)
i=1
are similarity measures of fuzzy sets A and B , known by Theorem 1, it is easy to see that Eqs. (1)–(4) are entropy of interval valued fuzzy set A. 4. Relationship between similarity measure and entropy of interval valued fuzzy sets Considering that the real functions of similarity measure and entropy of interval valued fuzzy sets are not unique, in this section, we will discuss the relationship between similarity measure and entropy of interval valued fuzzy sets based on their axiomatic definitions. First, we propose a transform method of setting up similarity measure of interval valued fuzzy sets based on entropy of interval valued fuzzy set. For interval valued fuzzy sets A and B, we define f − (A, B) ∈ IVFSs, for every x ∈ X, f − (A, B)(x) =
1 + min(|A− (x) − B − (x)|, |A+ (x) − B + (x)|) , 2
f + (A, B)(x) =
1 + max(|A− (x) − B − (x)|, |A+ (x) − B + (x)|) 2
then we have the following theorem. Theorem 2. Suppose E be entropy of interval valued fuzzy set, for A, B ∈ IVFSs, then E(f (A, B)) is similarity measure of interval valued fuzzy sets A and B. Proof. (N1) If A is a crisp set, then for every x ∈ X, we have A− (x) = A+ (x) = 1 or A− (x) = A+ (x) = 0, it means that for every x ∈ X, we have |A− (x) − (A+ )c (x)| = 1 and |A+ (x) − (A− )c (x)| = 1. Therefore, f − (A, Ac )(x) = f + (A, Ac ) = 1, it shows f (A, Ac ) = X is a crisp set, so, E(f (A, Ac )) = 0. (N2) Known by the definition of entropy of interval valued fuzzy set, E(f (A, B)) = 1 ⇐⇒ f − (A, B)(x) + f + (A, B)(x) = 1 ⇐⇒ |A− (x) − B − (x)| = 0, |A+ (x) − B + (x)| = 0 ⇐⇒ A− (x) = B − (x), A+ (x) = B + (x) ⇐⇒ A = B. (N3) Known by the definition of f (A, B), f (A, B) = f (B, A) is obvious , therefore, E(f (A, B)) = E(f (B, A)). (N4) Since A⊆B⊆C, then for every x∈X, we have A− (x) B − (x) C − (x) and A+ (x) B + (x) C + (x). Thus, we can get |A− (x) − C − (x)| |A− (x) − B − (x)|,
|A+ (x) − C + (x)| |A+ (x) − B + (x)|.
Further, we have min(|A− (x) − C − (x)|, |A+ (x) − C + (x)|) min(|A− (x) − B − (x)|, |A+ (x) − B + (x)|) and max(|A− (x) − C − (x)|, |A+ (x) − C + (x)|) max(|A− (x) − B − (x)|, |A+ (x) − B + (x)|). Therefore, we have f − (A, C)(x) f − (A, B)(x), f + (A, C)(x)f + (A, B)(x), and known by the definition of f (A, B), f − (A, B)(x) + f + (A, B)(x)1, so E(f (A, C))E(f (A, B)).
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With the same reason, we can prove E(f (A, C)) E(f (B, C)). Hence, we complete the proof of Theorem 2. Corollary 1. Suppose E be entropy of interval valued fuzzy set, f (A, B) is defined as above, then E((f (A, B))c ) is similarity measure between interval valued fuzzy sets A and B. For interval valued fuzzy sets A and B, we define g(A, B) ∈ IVFSs, for every x ∈ X and p > 0, g − (A, B)(x) =
1 + min(|A− (x) − B − (x)|p , |A+ (x) − B + (x)|p ) , 2
g + (A, B)(x) =
1 + max(|A− (x) − B − (x)|p , |A+ (x) − B + (x)|p ) 2
then we have Corollary 2. Suppose E be entropy of interval valued fuzzy set, then both E(g(A, B)) and E((g(A, B))c ) are similarity measure between interval valued fuzzy sets A and B. Example 1. When X = {x1 , x2 , . . . , xn }, A, B ∈ IVFSs, and n
E(A) = 1 −
1 − |A (xi ) + A+ (xi ) − 1| n i=1
then n
E(f (A, B)) = 1 −
1 − (|A (xi ) − B − (xi )| + |A+ (xi ) − B + (xi )|) 2n
(7)
i=1
is similarity measure of interval valued fuzzy sets A and B. Example 2. When X = [a, b], A, B ∈ IVFSs, and b 1 E(A) = 1 − |A− (x) + A+ (x) − 1| dx b−a a then 1 E(f (A, B)) = 1 − 2(b − a)
b a
(|A− (x) − B − (x)| + |A+ (x) − B + (x)|) dx
(8)
is similarity measure of interval valued fuzzy sets A and B, where the integral is Lebesgue integral. Next, we propose another transform method of setting up entropy of interval valued fuzzy set based on similarity measure of interval valued fuzzy sets. For interval valued fuzzy set A, we define m(A), n(A) ∈ IVFSs, for every x ∈ X, m− (A)(x) =
1 + (A− (x) + A+ (x) − 1)4 , 2
n− (A)(x) =
1 − |A− (x) + A+ (x) − 1| , 2
m+ (A)(x) = n+ (A)(x) =
1 + |A− (x) + A+ (x) − 1| 2
1 − (A− (x) + A+ (x) − 1)2 2
then we have the following theorem. Theorem 3. Suppose N be similarity measure of interval valued fuzzy sets, A ∈ IVFSs, then N (m(A), n(A)) is entropy of interval valued fuzzy set A.
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Proof. (P1) If A is a crisp set, then for every x ∈ X, we have A− (x) = A+ (x) = 1 or A− (x) = A+ (x) = 0. Thus, for every x ∈ X, we can get |A− (x) + A+ (x) − 1| = 1, it means that m− (A)(x) = m+ (A)(x) = 1, n− (A)(x) = n+ (A)(x) = 0. It shows that m(A) = X and n(A) = ∅ are crisp sets, therefore, N (m(A), n(A)) = 0. (P2) Known by the definitions of m(A) and n(A), m(A) and n(A) are interval valued fuzzy sets, thus, N (m(A), n(A)) = 1 ⇐⇒ m(A) = n(A) ⇐⇒ A− (x) + A+ (x) = 1. (P3) Since A− (x)B − (x) and A+ (x)B + (x) for B − (x) + B + (x) 1 implies A− (x) B − (x) (B + (x))c (A+ (x))c . Thus, we can get |A+ (x) + A− (x) − 1||B + (x) + B − (x) − 1|. It means that n(A) ⊆ n(B) ⊆ m(B) ⊆ m(A), so we have N (m(A), n(A))N (m(B), n(A)) N (m(B), n(B)). With the same reason, when A− (x)B − (x) and A+ (x) B + (x) for B − (x) + B + (x) 1, we also have N (m(A), n(A)) N (m(B), n(B)). (P4) Known by the definitions of m(A) and n(A), we have m(A) = m(Ac ), n(A) = n(Ac ), therefore, N (m(A), n(A)) = N (m(Ac ), n(Ac )). Hence, we complete the proof of Theorem 3. Corollary 3. Suppose N be similarity measure of interval valued fuzzy sets, m(A) and n(A) are defined as above, then N((m(A))c , (n(A))c ) is entropy of interval valued fuzzy set A. Example 3. When X = {x1 , x2 , . . . , xn }, A ∈ IVFSs, and n
N(A, B) = 1 −
1 − (|A (xi ) − B − (xi )| + |A+ (xi ) − B + (xi )|) 2n i=1
then n
N (m(A), n(A)) = 1 −
1 (2|A− (xi ) + A+ (xi ) − 1| + (A− (xi ) + A+ (xi ) − 1)2 4n i=1
+(A− (xi ) + A+ (xi ) − 1)4 )
(9)
is entropy of interval valued fuzzy set A. Example 4. When X = [a, b], A ∈ IVFSs, and b 1 N(A, B) = 1 − (|A− (x) − B − (x)| + |A+ (x) − B + (x)|) dx 2(b − a) a then N (m(A), n(A)) 1 =1− 4(b − a) b × (2|A− (x) + A+ (x) − 1| + (A− (x) + A+ (x) − 1)2 + (A− (x) + A+ (x) − 1)4 ) dx
(10)
a
is entropy of interval valued fuzzy set A, where the integral is Lebesgue integral. Theorem 4. Suppose N be similarity measure of interval valued fuzzy sets, A ∈ IVFSs, then N (A, Ac ) is entropy of interval valued fuzzy set A. Proof. (P1) If A is a crisp set, then known by the definition of similarity measure of interval valued fuzzy sets, we have N(A, Ac ) = 0. (P2) Known by the definition of similarity measure of interval valued fuzzy sets, we have, N (A, Ac ) = 1 ⇐⇒ A = Ac ⇐⇒ A− (x) + A+ (x) = 1.
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(P3) Since A− (x)B − (x) and A+ (x)B + (x) for B − (x) + B + (x) 1 implies A− (x) B − (x) (B + (x))c (A+ (x))c . It means that we have A− ⊆ B − ⊆ (B + )c ⊆ (A+ )c . Therefore, known by the definition of similarity measure of interval valued fuzzy sets, we have N (A, Ac ) N (B, Ac ) N (B, B c ). With the same reason, when A− (x)B − (x) and A+ (x) B + (x) for B − (x) + B + (x) 1, we have N (A, Ac ) N(A, B c )N (B, B c ). (P4) N (A, Ac ) = N(Ac , A) is obvious. Hence, we complete the proof of Theorem 4. Example 5. When X = {x1 , x2 , . . . , xn }, A, B ∈ IVFSs, and n
N(A, B) = 1 −
1 − (|A (xi ) − B − (xi )| + |A+ (xi ) − B + (xi )|) 2n i=1
then n
1 − |A (xi ) + A+ (xi ) − 1| = E1 (A) N(A, A ) = 1 − n c
i=1
is entropy of interval valued fuzzy set A. Example 6. When X = [a, b], A, B ∈ IVFSs, and b 1 N(A, B) = 1 − (|A− (x) − B − (x)| + |A+ (x) − B + (x)|) dx 2(b − a) a then 1 N(A, A ) = 1 − b−a c
b a
|A− (x) + A+ (x) − 1| dx = E3 (A)
is entropy of interval valued fuzzy set A, where the integral is integral. 5. Conclusion In this paper, we introduce the concept of entropy of interval valued fuzzy set different from Bustince and Burillo [3] which is extended from entropy of fuzzy set and an axiomatic definition on similarity measure of interval valued fuzzy sets, give a kind of method to describe entropy of interval valued fuzzy set based on its similarity measure, discuss their relationship between similarity measure and entropy of interval valued fuzzy sets in detail, prove three theorems that similarity measure and entropy of interval valued fuzzy sets can be transformed by each other based on their axiomatic definitions and put forward some formulas to calculate entropy and similarity measure of interval valued fuzzy sets. These conclusions can be applied in many fields such as pattern recognition, image processing and fuzzy reasoning. Acknowledgements The authors wish to express their gratitude to the referees for their kind suggestions and helpful comments in revising the paper. References [1] [2] [3] [4]
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Relationship between similarity measure and entropy of interval valued fuzzy sets夡 Wenyi Zenga, b,∗ , Hongxing Lia a School of Mathematical Sciences, Beijing Normal University, Beijing 100875, PR China b Department of Medical Epidemiology and Biostatistics, Karolinska Institutet, Stockholm, SE171 77, Sweden
Received 3 March 2005; received in revised form 24 November 2005; accepted 26 November 2005 Available online 20 December 2005
Abstract In this paper, we introduce concepts of entropy of interval valued fuzzy set which is different from Bustince and Burillo [Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets, Fuzzy Sets and Systems 78 (1996) 305–316] and similarity measure of interval valued fuzzy sets, discuss their relationship between similarity measure and entropy of interval valued fuzzy sets in detail, prove three theorems that similarity measure and entropy of interval valued fuzzy sets can be transformed by each other based on their axiomatic definitions and put forward some formulas to calculate entropy and similarity measure of interval valued fuzzy sets. © 2005 Elsevier B.V. All rights reserved. Keywords: Interval valued fuzzy set; Fuzzy set; Entropy; Similarity measure
1. Introduction Since fuzzy set was introduced by Zadeh [19], many new approaches and theories treating imprecision and uncertainty have been proposed. Some of theories, like intuitionistic fuzzy set theory introduced by Atanassov [1,2], are extensions of the classic fuzzy theory. Another, well-known generalization of an ordinary fuzzy set is interval valued fuzzy set. Interval valued fuzzy set was first introduced by Zadeh [20], after that, some authors investigated the topic and obtained some meaningful conclusions. For example, Gorzalczany [9] in approximate reasoning, Turksen [15] in normal forms, Deschrijver, etc. [7] interval valued fuzzy relations and implication and so on. Moreover, some authors pointed out that there is a strong connection between intuitionistic fuzzy sets and interval valued fuzzy sets, for more details we refer the readers to [4,7,16]. Entropy and similarity measure of fuzzy sets are two important topics in fuzzy set theory. Entropy of fuzzy set describes the fuzziness degree of fuzzy set. Many scholars have studied it from different points of view. For example, in 1972, De Luca and Termini [6] introduced some axioms to describe the fuzziness degree of fuzzy set. Kaufmann [11] proposed a method to measure the fuzziness degree of fuzzy set by a metric distance between its membership function and the membership function of its nearest crisp set. Another way given by Yager [18] was to view the fuzziness degree 夡 Supported by National Natural Science Foundation of China (60474023), Science and Technology Key Project Fund of Ministry of Education (03184), and the Major State Basic Research Development Program of China (2002CB312200). ∗ Corresponding author. Department of Mathematics, Beijing Normal University, Beijing 100875, PR China. Tel.: +86 10 62209376. E-mail address: [email protected] (W. Zeng).
0165-0114/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2005.11.020
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of fuzzy set in terms of a lack of distinction between the fuzzy set and its complement. On the other hand, similarity measure of fuzzy sets indicates the similarity degree of fuzzy sets and was extensively applied in many fields such as fuzzy clustering, image processing, fuzzy reasoning and fuzzy neutral network [5,13,17]. Based on these two kinds of important measures, Liu [12] investigated entropy, distance measure and similarity measure of fuzzy sets and their relations. Fan [8] studied distance measure and fuzzy entropy induced by distance measure. Some authors have investigated interval valued fuzzy set and its some relevant topics, for example, in 2004, Grzegorzewski [10] studied distance between interval valued fuzzy sets based on the Hausdroff metric, Burillo and Bustince [3] and Szmidt and Kacprzyk [14] researched entropy of interval valued fuzzy set from different point of views, respectively. In 1996, Burillo and Bustince [3] introduced the concept of entropy of interval valued fuzzy set and intuitionistic fuzzy set, Szmidt and Kacprzyk [14] used a different approach from Burillo and Bustince [3] to propose an entropy measure for intuitionistic fuzzy set. In this paper, we want to study the relationship between entropy and similarity measure of interval valued fuzzy sets, give three theorems that entropy and similarity measure of interval valued fuzzy sets can be transformed by each other based on their axiomatic definitions and put forward some formulas to calculate entropy and similarity measure of interval valued fuzzy sets. The rest of our work is organized as follows. In Section 2, we recall some basic notions of interval valued fuzzy set. In Section 3, we introduce the concepts of entropy and similarity measure of interval valued fuzzy sets. In Section 4, we investigate relationship between similarity measure and entropy of interval valued fuzzy sets and prove three theorems that similarity measure and entropy of interval valued fuzzy sets can be transformed by each other based on their axiomatic definitions. The final section is conclusion. 2. Some notions Throughout this paper, we write X to denote the discourse set, IVFSs stands for the set of all interval valued fuzzy subsets in X, F(X) and P(X) stand for the set of all fuzzy sets and crisp sets in X, respectively. A expresses an interval valued fuzzy set, the operation “c” is the complement of interval valued fuzzy set or fuzzy set in X, ∅ stands for the empty set. Let I = [0, 1] and [I ] be the set of all closed subintervals of the interval [0, 1]. Then, according to Zadeh’s extension principle [19], we can popularize these operations such as ∨, ∧ and c to [I ], thus, ([I ], ∨, ∧, c) is a complete lattice with a minimal element 0¯ = [0, 0] and a maximal element 1¯ = [1, 1]. Furthermore, let a¯ = [a − , a + ], b¯ = [b− , b+ ], ¯ then we have, a¯ = b¯ ⇐⇒ a − = b− , a + = b+ , a¯ b¯ ⇐⇒ a − b− , a + b+ and a¯ < b¯ ⇐⇒ a¯ b¯ and a¯ = b. Considering [I ] is dense, therefore, ([I ], ∨, ∧, c) is a superior soft algebra. We call a mapping: A : X −→ [I ] an interval valued fuzzy set in X. For every A ∈ IVFSs and x ∈ X, A(x) = [A− (x), A+ (x)] is called the degree of membership of an element x to A, then fuzzy sets A− : X → [0, 1] and A+ : X → [0, 1] are called a low fuzzy set of A and a upper fuzzy set of A, respectively. For simplicity, we denote A = [A− , A+ ]. Therefore, some operations such as ∪, ∩, c can be introduced into IVFSs, thus (IVFSs, ∪, ∩, c) is a superior soft algebra. If A, B ∈ IVFSs, then the following operations can be found in Zeng [21]. A ⊆ B iff ∀x ∈ X, A− (x)B − (x) and A+ (x)B + (x), A = B iff ∀x ∈ X, A− (x) = B − (x) and A+ (x) = B + (x), Ac = [(A+ )c , (A− )c ]. 3. Entropy of interval valued fuzzy set In 2001, Szmidt and Kacprzyk [14] extended De Luca and Termini [6] axioms for fuzzy set to introduce entropy of intuitionistic fuzzy set. Based on this view point of Szmidt and Kacprzyk [14], we introduce the concept of entropy of interval valued fuzzy set which is different from Bustince and Burillo [3]. Definition 1. A real function E : IVFSs−→ [0, 1] is called an entropy on IVFSs, if E satisfies the following properties: (P1) E(A) = 0 iff A is a crisp set; (P2) E(A) = 1 iff A− (x) + A+ (x) = 1;
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(P3) E(A)E(B) if A is less fuzzy than B, i.e., A− (x) B − (x) and A+ (x) B + (x) for B − (x) + B + (x) 1 or A− (x)B − (x) and A+ (x)B + (x) for B − (x) + B + (x) 1; (P4) E(A) = E(Ac ). Then we can give the following formulas to calculate entropy of interval valued fuzzy set A: n
1 − |A (xi ) + A+ (xi ) − 1|, n i=1 n 1 (A− (xi ) + A+ (xi ) − 1)2 , E2 (A) = 1 − n i=1 b 1 |A− (x) + A+ (x) − 1| dx, E3 (A) = 1 − b−a a b − (A (x) ∧ (1 − A+ (x))) dx E4 (A) = ab , − + a (A (x) ∨ (1 − A (x))) dx E1 (A) = 1 −
(1)
(2) (3) (4)
where the integral in Eqs. (3) and (4), is Lebesgue integral. Definition 2. A real function N : IVFSs×IVFSs→ [0, 1] is called similarity measure of interval valued fuzzy sets, if N satisfies the following properties: (N1) N(A, Ac ) = 0 if A is a crisp set; (N2) N (A, B) = 1 ⇐⇒ A = B; (N3) N (A, B) = N(B, A); (N4) for all A, B, C ∈IFSs, if A ⊆ B ⊆ C, then N (A, C)N (A, B), N (A, C) N (B, C). Obviously, the axiomatic definition of similarity measure and entropy of interval valued fuzzy sets are extended from fuzzy set theory. Specially, if interval valued fuzzy sets A and B become fuzzy sets, then N (A, B) is similarity measure of fuzzy sets. Based on this point of view, then we have the following theorem. Theorem 1. N (A− , (A+ )c ) is entropy of interval valued fuzzy set A, where fuzzy sets (A+ )c is the complement of fuzzy set A+ . Proof. (P1) If A is a crisp set, then for every x ∈ X, we have A− (x) = A+ (x) = 1 or A− (x) = A+ (x) = 0, it means that A− and A+ are crisp sets and A− = A+ , therefore, N (A− , (A+ )c ) = 0. (P2) Known by the definitions of similarity measure of fuzzy sets, we have, N(A− , (A+ )c ) = 1 ⇐⇒ A− = (A+ )c ⇐⇒ A− (x) + A+ (x) = 1. (P3) Since A− (x)B − (x) and A+ (x)B + (x) for B − (x) + B + (x) 1 implies A− (x) B − (x) (B + )c (x) (A+ )c (x). Namely, A− ⊆ B − ⊆ (B + )c ⊆ (A+ )c , known by the definition of similarity measure of fuzzy sets, we have N(A− , (A+ )c )N (A− , (B + )c ) N (B − , (B + )c ). With the same reason, when A− (x)B − (x) and A+ (x) B + (x) for B − (x)+B + (x) 1, we can prove N (A− , (A+ )c ) N(B − , (B + )c ). (P4) Because Ac = [(A+ )c , (A− )c ], and known by the definition of similarity measure of fuzzy sets, we have N(A− , (A+ )c ) = N ((A+ )c , A− ) = N ((A+ )c , ((A− )c )c ). Hence, we complete the proof of Theorem 1. For X = {x1 , x2 , . . . , xn }, A , B ∈ F(X), since both n
1 N1 (A , B ) = 1 − |A (xi ) − B (xi )| n
i=1
(5)
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n 1 (A (xi ) − B (xi ))2 N2 (A , B ) = 1 − n
(6)
i=1
are similarity measures of fuzzy sets A and B , known by Theorem 1, it is easy to see that Eqs. (1)–(4) are entropy of interval valued fuzzy set A. 4. Relationship between similarity measure and entropy of interval valued fuzzy sets Considering that the real functions of similarity measure and entropy of interval valued fuzzy sets are not unique, in this section, we will discuss the relationship between similarity measure and entropy of interval valued fuzzy sets based on their axiomatic definitions. First, we propose a transform method of setting up similarity measure of interval valued fuzzy sets based on entropy of interval valued fuzzy set. For interval valued fuzzy sets A and B, we define f − (A, B) ∈ IVFSs, for every x ∈ X, f − (A, B)(x) =
1 + min(|A− (x) − B − (x)|, |A+ (x) − B + (x)|) , 2
f + (A, B)(x) =
1 + max(|A− (x) − B − (x)|, |A+ (x) − B + (x)|) 2
then we have the following theorem. Theorem 2. Suppose E be entropy of interval valued fuzzy set, for A, B ∈ IVFSs, then E(f (A, B)) is similarity measure of interval valued fuzzy sets A and B. Proof. (N1) If A is a crisp set, then for every x ∈ X, we have A− (x) = A+ (x) = 1 or A− (x) = A+ (x) = 0, it means that for every x ∈ X, we have |A− (x) − (A+ )c (x)| = 1 and |A+ (x) − (A− )c (x)| = 1. Therefore, f − (A, Ac )(x) = f + (A, Ac ) = 1, it shows f (A, Ac ) = X is a crisp set, so, E(f (A, Ac )) = 0. (N2) Known by the definition of entropy of interval valued fuzzy set, E(f (A, B)) = 1 ⇐⇒ f − (A, B)(x) + f + (A, B)(x) = 1 ⇐⇒ |A− (x) − B − (x)| = 0, |A+ (x) − B + (x)| = 0 ⇐⇒ A− (x) = B − (x), A+ (x) = B + (x) ⇐⇒ A = B. (N3) Known by the definition of f (A, B), f (A, B) = f (B, A) is obvious , therefore, E(f (A, B)) = E(f (B, A)). (N4) Since A⊆B⊆C, then for every x∈X, we have A− (x) B − (x) C − (x) and A+ (x) B + (x) C + (x). Thus, we can get |A− (x) − C − (x)| |A− (x) − B − (x)|,
|A+ (x) − C + (x)| |A+ (x) − B + (x)|.
Further, we have min(|A− (x) − C − (x)|, |A+ (x) − C + (x)|) min(|A− (x) − B − (x)|, |A+ (x) − B + (x)|) and max(|A− (x) − C − (x)|, |A+ (x) − C + (x)|) max(|A− (x) − B − (x)|, |A+ (x) − B + (x)|). Therefore, we have f − (A, C)(x) f − (A, B)(x), f + (A, C)(x)f + (A, B)(x), and known by the definition of f (A, B), f − (A, B)(x) + f + (A, B)(x)1, so E(f (A, C))E(f (A, B)).
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With the same reason, we can prove E(f (A, C)) E(f (B, C)). Hence, we complete the proof of Theorem 2. Corollary 1. Suppose E be entropy of interval valued fuzzy set, f (A, B) is defined as above, then E((f (A, B))c ) is similarity measure between interval valued fuzzy sets A and B. For interval valued fuzzy sets A and B, we define g(A, B) ∈ IVFSs, for every x ∈ X and p > 0, g − (A, B)(x) =
1 + min(|A− (x) − B − (x)|p , |A+ (x) − B + (x)|p ) , 2
g + (A, B)(x) =
1 + max(|A− (x) − B − (x)|p , |A+ (x) − B + (x)|p ) 2
then we have Corollary 2. Suppose E be entropy of interval valued fuzzy set, then both E(g(A, B)) and E((g(A, B))c ) are similarity measure between interval valued fuzzy sets A and B. Example 1. When X = {x1 , x2 , . . . , xn }, A, B ∈ IVFSs, and n
E(A) = 1 −
1 − |A (xi ) + A+ (xi ) − 1| n i=1
then n
E(f (A, B)) = 1 −
1 − (|A (xi ) − B − (xi )| + |A+ (xi ) − B + (xi )|) 2n
(7)
i=1
is similarity measure of interval valued fuzzy sets A and B. Example 2. When X = [a, b], A, B ∈ IVFSs, and b 1 E(A) = 1 − |A− (x) + A+ (x) − 1| dx b−a a then 1 E(f (A, B)) = 1 − 2(b − a)
b a
(|A− (x) − B − (x)| + |A+ (x) − B + (x)|) dx
(8)
is similarity measure of interval valued fuzzy sets A and B, where the integral is Lebesgue integral. Next, we propose another transform method of setting up entropy of interval valued fuzzy set based on similarity measure of interval valued fuzzy sets. For interval valued fuzzy set A, we define m(A), n(A) ∈ IVFSs, for every x ∈ X, m− (A)(x) =
1 + (A− (x) + A+ (x) − 1)4 , 2
n− (A)(x) =
1 − |A− (x) + A+ (x) − 1| , 2
m+ (A)(x) = n+ (A)(x) =
1 + |A− (x) + A+ (x) − 1| 2
1 − (A− (x) + A+ (x) − 1)2 2
then we have the following theorem. Theorem 3. Suppose N be similarity measure of interval valued fuzzy sets, A ∈ IVFSs, then N (m(A), n(A)) is entropy of interval valued fuzzy set A.
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Proof. (P1) If A is a crisp set, then for every x ∈ X, we have A− (x) = A+ (x) = 1 or A− (x) = A+ (x) = 0. Thus, for every x ∈ X, we can get |A− (x) + A+ (x) − 1| = 1, it means that m− (A)(x) = m+ (A)(x) = 1, n− (A)(x) = n+ (A)(x) = 0. It shows that m(A) = X and n(A) = ∅ are crisp sets, therefore, N (m(A), n(A)) = 0. (P2) Known by the definitions of m(A) and n(A), m(A) and n(A) are interval valued fuzzy sets, thus, N (m(A), n(A)) = 1 ⇐⇒ m(A) = n(A) ⇐⇒ A− (x) + A+ (x) = 1. (P3) Since A− (x)B − (x) and A+ (x)B + (x) for B − (x) + B + (x) 1 implies A− (x) B − (x) (B + (x))c (A+ (x))c . Thus, we can get |A+ (x) + A− (x) − 1||B + (x) + B − (x) − 1|. It means that n(A) ⊆ n(B) ⊆ m(B) ⊆ m(A), so we have N (m(A), n(A))N (m(B), n(A)) N (m(B), n(B)). With the same reason, when A− (x)B − (x) and A+ (x) B + (x) for B − (x) + B + (x) 1, we also have N (m(A), n(A)) N (m(B), n(B)). (P4) Known by the definitions of m(A) and n(A), we have m(A) = m(Ac ), n(A) = n(Ac ), therefore, N (m(A), n(A)) = N (m(Ac ), n(Ac )). Hence, we complete the proof of Theorem 3. Corollary 3. Suppose N be similarity measure of interval valued fuzzy sets, m(A) and n(A) are defined as above, then N((m(A))c , (n(A))c ) is entropy of interval valued fuzzy set A. Example 3. When X = {x1 , x2 , . . . , xn }, A ∈ IVFSs, and n
N(A, B) = 1 −
1 − (|A (xi ) − B − (xi )| + |A+ (xi ) − B + (xi )|) 2n i=1
then n
N (m(A), n(A)) = 1 −
1 (2|A− (xi ) + A+ (xi ) − 1| + (A− (xi ) + A+ (xi ) − 1)2 4n i=1
+(A− (xi ) + A+ (xi ) − 1)4 )
(9)
is entropy of interval valued fuzzy set A. Example 4. When X = [a, b], A ∈ IVFSs, and b 1 N(A, B) = 1 − (|A− (x) − B − (x)| + |A+ (x) − B + (x)|) dx 2(b − a) a then N (m(A), n(A)) 1 =1− 4(b − a) b × (2|A− (x) + A+ (x) − 1| + (A− (x) + A+ (x) − 1)2 + (A− (x) + A+ (x) − 1)4 ) dx
(10)
a
is entropy of interval valued fuzzy set A, where the integral is Lebesgue integral. Theorem 4. Suppose N be similarity measure of interval valued fuzzy sets, A ∈ IVFSs, then N (A, Ac ) is entropy of interval valued fuzzy set A. Proof. (P1) If A is a crisp set, then known by the definition of similarity measure of interval valued fuzzy sets, we have N(A, Ac ) = 0. (P2) Known by the definition of similarity measure of interval valued fuzzy sets, we have, N (A, Ac ) = 1 ⇐⇒ A = Ac ⇐⇒ A− (x) + A+ (x) = 1.
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(P3) Since A− (x)B − (x) and A+ (x)B + (x) for B − (x) + B + (x) 1 implies A− (x) B − (x) (B + (x))c (A+ (x))c . It means that we have A− ⊆ B − ⊆ (B + )c ⊆ (A+ )c . Therefore, known by the definition of similarity measure of interval valued fuzzy sets, we have N (A, Ac ) N (B, Ac ) N (B, B c ). With the same reason, when A− (x)B − (x) and A+ (x) B + (x) for B − (x) + B + (x) 1, we have N (A, Ac ) N(A, B c )N (B, B c ). (P4) N (A, Ac ) = N(Ac , A) is obvious. Hence, we complete the proof of Theorem 4. Example 5. When X = {x1 , x2 , . . . , xn }, A, B ∈ IVFSs, and n
N(A, B) = 1 −
1 − (|A (xi ) − B − (xi )| + |A+ (xi ) − B + (xi )|) 2n i=1
then n
1 − |A (xi ) + A+ (xi ) − 1| = E1 (A) N(A, A ) = 1 − n c
i=1
is entropy of interval valued fuzzy set A. Example 6. When X = [a, b], A, B ∈ IVFSs, and b 1 N(A, B) = 1 − (|A− (x) − B − (x)| + |A+ (x) − B + (x)|) dx 2(b − a) a then 1 N(A, A ) = 1 − b−a c
b a
|A− (x) + A+ (x) − 1| dx = E3 (A)
is entropy of interval valued fuzzy set A, where the integral is integral. 5. Conclusion In this paper, we introduce the concept of entropy of interval valued fuzzy set different from Bustince and Burillo [3] which is extended from entropy of fuzzy set and an axiomatic definition on similarity measure of interval valued fuzzy sets, give a kind of method to describe entropy of interval valued fuzzy set based on its similarity measure, discuss their relationship between similarity measure and entropy of interval valued fuzzy sets in detail, prove three theorems that similarity measure and entropy of interval valued fuzzy sets can be transformed by each other based on their axiomatic definitions and put forward some formulas to calculate entropy and similarity measure of interval valued fuzzy sets. These conclusions can be applied in many fields such as pattern recognition, image processing and fuzzy reasoning. Acknowledgements The authors wish to express their gratitude to the referees for their kind suggestions and helpful comments in revising the paper. References [1] [2] [3] [4]
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