# Entropy, Distance and Similarity Measures under Interval-Valued Intuitionistic Fuzzy Environment.

This paper presents new axiomatic definitions of entropy measure using concept of probability and distance for interval-valued intuitionistic fuzzy sets (IvIFSs) by considering degree of hesitancy which is consistent with the definition of entropy given by De Luca and Termini. Thereafter, we propose some entropy measures and also derived relation between distance, entropy and similarity measures for IvIFSs. Further, we checked the performance of proposed entropy and similarity measures on the basis of intuition and compared with the existing entropy and similarity measures using numerical examples. Lastly, proposed similarity measures are used to solve problems in the field of pattern recognition and medical diagnoses.

Povzetek: V prispevku so predstavljene nove aksiomatske definicije entropijske mere za intervalno intuicionisticne mehke mnozice.

Keywords: entropy measure, distance, similarity measure, interval valued intuitionistic fuzzy sets

1 Introduction

Fuzzy set theory (Zadeh, 1965) is tool that can handle uncertainty and imprecision effortlessly. Interval-valued, intuitionistic, interval-valued intuitionistic fuzzy sets (Zadeh (1975), Atanassov (1986), Atanassov & Gargov (1989)), vague sets (Gau & Buehrer, 1993) and R-fuzzy sets (Yang, Hinde, 2010) are various generalizations of Fuzzy sets (FSs). From all these generalizations IvIFSs and intuitionistic fuzzy sets (IFSs) are two conventional extensions of FSs. IvIFSs are more practical and flexible than IFSs as they are characterized by membership and non-membership degree range instead of real numbers. It makes IvIFSs more useful in dealing with real world complexities which arises due to insufficient information, lack of data, imprecise knowledge and human nature wherein range is provided instead of real numbers. Distance, entropy and similarity measures are the central arenas that are investigated by various researchers under intuitionistic and interval-valued fuzzy environment (IFE and IvFE). These measures identify the similarity or dissimilarity between two FSs. Till date, vivid entropy, distance or similarity measures are presented by various investigators. Some of these research findings are mentioned as follows: Xu (2007 a, b) introduced the concept of similarity between IvIFSs along with some distance measure. Zang et al. (2009) defined a entropy axiomatically for interval-valued fuzzy sets (IvFSs) and discussed relation between entropy and similarity measures. Xu and Yager (2009) studied preference relation and defined similarity measure under IvFE and interval-valued intuitionistic environment (IvIFE). Wei et al (2011) derived a generalized measure of entropy for IvIFSs. Cosine similarity measures for IvIFSs are defined by both Ye (2012) and Singh (2012). Sun & Liu (2012), Hu & Li (2013), Zhang et al. (2014) proposed entropy and similarity measure along with their relationship for IvIFSs. Applications of the aforesaid entropy, distance and similarity measures are for recognition of patterns, medical diagnoses, and decision making with multiple criteria and expert systems problems. However, most of these distance, similarity or entropy measures do not consider hesitancy index between IvIFSs. Hesitance index play a very important role when membership and non-membership degree do not differ much for two IvIFSs but their hesitant index does. Some of the authors, Xu (2007), Xu & Xia (2011), Wu et al. (2014) considered hesitancy into the measure of distance, similarity and entropy developed by them. Since hesitancy index also has a vital role in any decision making as it outclasses the existing methods and deals with decision process in a better way. Dammak et al. (2016) studies some possibility measures in multi-criteria decision making under IvIFE. Tiwari & Gupta(in press) proposed generalized entropy and similarity measure for IvIFS with application in decision making. Zang et al. (2016) defined some operations on IvIFSs and proposed some aggregation operators for IvIFSs w.r.t. the restricted interval Shapley function with application in multi-criteria decision making. In this paper we have developed some of the distance, entropy and similarity measures by taking all the three degrees in account and applied it to pattern recognition and medical diagnoses under IvIFE.

This work is organized in various sections. Section 2 has basic definition and operations on IvIFSs. Section 3, presents the relationship between distance and entropy measures along with example to check the performance of entropy measures on the basis of intuition. A relation between measure of entropy and similarity measure is proposed in Section 4. Further, comparison of new similarity measures with the few existing one in done. Thereafter in section 5 we applied new similarity measures to recognition of patterns and medical diagnoses. Lastly conclusion is drawn in Section 6.

2 IvIFSs along with its distance and similarity measures

This section has definitions and concepts for IvIFSs. In this paper [OMEGA] = {[x.sub.1], ...., [x.sub.n]} denotes the universe of discourse; C([OMEGA]) and IvIFSs([OMEGA]) denote all crisp sets and IvIFSs respectively in [OMEGA].

Definition 1 (Atanassov & Gargov, 1989): An IvIFS A in the finite universe [OMEGA] is defined by a triplet <[x.sub.i], [MV.sub.A] ([x.sub.i]), [NV.sub.A] ([x.sub.i]), [HV.sub.A] ([x.sub.i])> as [x.sub.i] [member of] [OMEGA] where [MV.sub.A] ([x.sub.i]) = [[MV.sub.AL] ([x.sub.i]), [MV.sub.AU] ([x.sub.i])] is called membership value interval, [NV.sub.A] ([x.sub.i]) = [[NV.sub.AL] ([x.sub.i]), [NV.sub.AU] ([x.sub.i])} is called non-membership value interval and [HV.sub.A]([x.sub.i]) = [[HV.sub.AL]([x.sub.i]), [HV.sub.AU]([x.sub.i])], [HV.sub.AL] ([x.sub.i]) = 1 - [MV.sub.AU] ([x.sub.i]) - [NV.sub.AU] ([x.sub.i]), [H.sub.AU] ([x.sub.i]) = 1 - [MV.sub.AL] ([x.sub.i]) - [NV.sub.AL]([x.sub.i]) such that 0 [less than or equal to] [MV.sub.AU]([x.sub.i]) + [NV.sub.AU] ([x.sub.i]) [less than or equal to] 1 for each [x.sub.i] [member of] A.

Liu (1992) defined distance and similarity measures for IvIFSs axiomatically which are given as follows:

Definition 2: For any two IvIFSs A and B, a real valued function D: IvIFSs([OMEGA]) X IvIFSs([OMEGA]) [right arrow] [0,1] is termed as a distance measure of IvIFSs on [OMEGA], if it satisfies the below mentioned axioms:

1. For any crisp set A, we have D(A, [bar.A]) = 1.

2. Distance between any two IvIFSs A and B is zero iff A = B.

3. Distance measure is symmetrical w.r.t to any two IvIFSs A and B.

4. For any three IvIFSs A, B and C such that A [subset or equal to] B [subset or equal to] C, we have D (A, C) [greater than or equal to] D (A, B) and D (A, C) [greater than or equal to] D(B,C).

Distance between FSs was presented by (Kacprzyk, 1997). Then its extension was proposed by Atanassov in 1999 as two dimensional distances whereas third parameter hesitancy degree in distance was introduced by Szmidt and Kacprzyk (2000) for intuitionistic fuzzy sets. Yang & Chiclana (2012) proved three dimensional distance consistency over two dimensional distances. Grzegorzewski (2004) and Park et al. (2007) gave distance measure for IvFSs and IvIFSs respectively. Here we extend the distance measures by considering hesitancy degree for IvIFSs. For any two IvIFSs A and B, we define the following measures of distance:

1) Normalized Euclidean Distance

[mathematical expression not reproducible] ... (1)

2) Normalized Hamming Distance

[mathematical expression not reproducible] ... (2)

3) Hamming Hausdorff Normalized Distance

[mathematical expression not reproducible] ... (3)

4) Hausdorff Normalized Hamming Distance

[mathematical expression not reproducible] ... (4)

5) Averaged fifth Distance Measure

[mathematical expression not reproducible] ... (5)

6) Generalized Measure of Distance, for p [greater than or equal to] 2,

[mathematical expression not reproducible] (6)

Definition 3: Let A and B be any two IvIFSs, a real valued function S: IvIFSs([OMEGA]) x IvIFSs([OMEGA]) [left arrow] [0,1] is defined as a measure of similarity for IvIFSs on [OMEGA], if it satisfies axioms mentioned below:

1. For any crisp set A, we have S (A, [bar.A]) = 0

2. Measure of similarity between any two IvIFSs is 1 iff A = B.

3. Measure of similarity is symmetric w.r.t. any two IvIFSs.

4. For any three IvIFSs A, B and C such that A [subset or equal to] B [subset or equal to] C. We have S (A, C) [less than or equal to] S (A, B) and S (A, C) [less than or equal to] S(B, C).

From axiomatic definition of distance and similarity measures it is clear that S (A, B) = 1 - D (A, B) where A and B are IvIFSs, D and S are distance and similarity measure for IvIFSs respectively.

2.1 Entropy measure for IvIFSs

In 1972, De Luca and Termini defined measure of entropy for FSs. Hung & Yang (2006) extended definition for IFSs considering hesitancy degree. The following definition for entropy is an extension of definition of entropy proposed by Hung & Yang (2006) for IvIFSs.

Definition 4: A real valued function E: IvIFSs([OMEGA]) [right arrow] [0,1] is termed as measure of entropy under IvIFE, if below mentioned axioms are satisfied:

1. E (A) = 0, [for all] A [member of] C([OMEGA]);

2. E(A) = 1, iff [M.sub.A]([x.sub.i]) = [N.sub.A] = [H.sub.A]([x.sub.i])= = [1/3, 1/3] [for all] [x.sub.i] [member of] [OMEGA];

3. E (A) [less than or equal to] E (B), if A is less fuzzy than B;

4. E (A) = E ([bar.A]), where [bar.A] is complement of A, where A, B [member of] IvIFSs([OMEGA]).

Above definition is steady with description of measure of entropy given by De Luca & Termini (1972). As it is known that complete description of an IvIFS A [member of] [OMEGA] has three degrees membership, non-membership and hesitancy with [MV.sub.AU]([x.sub.i]) + [NV.sub.AU]([x.sub.i]) + [HV.sub.AL]([x.sub.i]) = 1 and [MV.sub.AL] ([x.sub.i]) + [NV.sub.AL]([x.sub.i]) + [HV.sub.AU]([x.sub.i]) = 1 with 0 [less than or equal to] [MV.sub.AU]([x.sub.i]), [NV.sub.AU]([x.sub.i]), [HV.sub.AL]([x.sub.i]), [MV.sub.AL]([x.sub.i]), [NV.sub.AL]([x.sub.i]), [HV.sub.AU]([x.sub.i]) [less than or equal to] 1. By taking all the three in to consideration we may assume them as probability measure. Therefore the entropy is maximum when all the variables are equal (i.e. [MV.sub.AU]([x.sub.i]) = [NV.sub.AU]([x.sub.i]) = [HV.sub.AL]([x.sub.i]) = 1/3 and [MV.sub.AL]([x.sub.i]) = [NV.sub.AL]([x.sub.i]) = [HV.sub.AU] ([x.sub.i]) = 1/3) and zero (minimum) when only one variable is exists (i.e. [MV.sub.AL]([x.sub.i]) = [MV.sub.AU]([x.sub.i]) = 1, [NV.sub.AL]([x.sub.i]) = [NV.sub.AU] ([x.sub.i]) = 0, [HV.sub.AL]([x.sub.i]) = [HV.sub.AU]([x.sub.i]) = 0 or [MV.sub.AL]([x.sub.i]) = [MV.sub.AU]([x.sub.i]) = 0, [NV.sub.AL]([x.sub.i]) = [NV.sub.AU]([x.sub.i]) = 1, [HV.sub.AL] ([x.sub.i]) = [HV.sub.AU] ([x.sub.i]) = 1 or [MV.sub.AL]([x.sub.i]) = [MV.sub.AU]([x.sub.i]) = 0, [NV.sub.AL]([x.sub.i]) = [NV.sub.AU]([x.sub.i]) = 0, [HV.sub.AL]([x.sub.i]) = [HV.sub.AU]([x.sub.i]) = 1). Again we extend the definition of entropy given by Zang et al. (2014) based on distance for IvIFSs. In following definition we have considered degree of hesitation, which is not considered by other definitions of entropy.

Definition 5: A real-valued function E: IvIFSs(OMEGA]) [right arrow] [0,1] is termed as measure of entropy under IvIFE, if the following axioms are satisfied:

1. E (A) = 0, [for all] A [member of] C([OMEGA]);

2. E (A) = 1, iff all the three description of IvIFSs intervals satifies [MV.sub.A]([x.sub.i]) = [NV.sub.A]([x.sub.i]) = [HV.sub.A]([x.sub.i])= [1/3, 1/3], [for all] [x.sub.i] [member of] [OMEGA]

3. If D (A, <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>) [greater than or equal to] D(B, <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>), then E(A) [less than or equal to] E(B), [for all]A, B [member of] IvIFSs(O[MEGA]), where D is measure of distance.

4. E(A) = E([bar.A]), where [bar.A] is complement of A, where A, B [member of] IvIFSs([OMEGA]).

In the next section, we derive a relation which relates measure of distance and entropy for IvIFSs, which satisfies all the axioms of the definition of entropy.

3 Relation between measure of distance and entropy

Here, we develop a technique which obtains entropy measure for IvIFSs which satisfies the aforementioned properties.

Theorem 2: Let [D.sub.j], j= 1, ..., 6 be the above-mentioned six distance measure equations (1)-(6) between IvIFSs, then, [E.sub.j](A) = 1 - 3[D.sub.j] (A, <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>), j= 1, ..., 6 for any A [member of] IvIFSs([OMEGA]) are measure of entropy of IvIFSs.

Proof: We prove that [E.sub.j] (A), for j = 1, ..., 6 satisfies conditions given by definition 5.

Property 1): If A [member of] C([OMEGA]) [right arrow] A([x.sub.i]) = <[1,1], [0,0], [0,0]> or A([x.sub.i]) = <[0,0], [1,1], [0,0]>, [for all] [x.sub.i] [member of] [OMEGA], then for j= 1, ..., 6

[D.sub.j] (A, <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>) = 1/3

Thus, [E.sub.j] (A) = 0

Property 2): For all j = 1, ..., 6, [E.sub.j](A) = 1

[??] 1 - 3[D.sub.j] (A, <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>) = 1

[??] 3[D.sub.j] (A, <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>) = 0

[??] A = <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>

Property 3): Let A and B be any two IvIFSs and

[D.sub.j] (A, <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>) [greater than or equal to]

[D.sub.j] (B, <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>) then

1 - 3[D.sub.j] (A, <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>)

[greater than or equal to] 1 - 3[D.sub.j] (B, <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>)

[right arrow] [E.sub.j](A) [less than or equal to] [E.sub.j](B), for all j = 1,...6

Property 4) : Let A be any IvIFS then [bar.A] = {<[x.sub.i], [NV.sub.AL]([x.sub.i]), [NV.sub.AU]([x.sub.i])], [[MV.sub.AL]([x.sub.i]), [MV.sub.AU]([x.sub.i])]> /[x.sub.i] [member of] [OMEGA]}

[??] [D.sub.j] (A, <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>)

= [D.sub.j] ([bar.A], <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>)

Thus, [E.sub.j] (A) = [E.sub.j]([bar.A]), for all j= 1, ..., 6.

From theorem 2 and various distance formulas' mentioned (equation (1) to (6)), we get corresponding entropy formulas as follows:

[mathematical expression not reproducible]

To check the consistency of proposed entropy measures with the intuitionist beliefs we have used the following example.

Example: Consider two IvIFSs A = {x, <[0.2,0.2], [0.2,0.3], [0.5,0.6]),x [member of] [OMEGA]} and B = {x, <[0.2,0.3], [0.4,0.6], [0.1,0.4]),x [member of] [OMEGA]}, clearly we can see that A is more fuzzy than B. Then [E.sub.j](A) and [E.sub.j] (B) are given in Table 1 :
```Table 1 : Entropies.

Entropies    A        B

[E.sub.1]    0.5570   0.65866
[E.sub.2]    0.675    0.7
[E.sub.3]    0.6      0.525
[E.sub.4]    0.675    0.75
[E.sub.5]    0.5125   0.6
[E.sub.6]    0.7171   0.672
```

Since [E.sub.3](A) > [E.sub.3](B) and [E.sub.6](A) > [E.sub.6](B) which indicates that [E.sub.3] and [E.sub.6] are consistent with the intuition.

3.1 Comparison of existing entropy measure with proposed entropy measures

We compared performance of existing entropy measures with the proposed measures with the help of an example. Let A be an IvIFS, then

[mathematical expression not reproducible]

where [[bar.MV].sub.A]([x.sub.i]) = [MV.sub.AL]([x.sub.i]) + [tau][DELTA][MV.sub.A]([x.sub.i]), [[bar.NV].sub.A] ([x.sub.i]) = [NV.sub.AL]([x.sub.i]) + [tau][DELTA][NV.sub.A]([x.sub.i]) and [DELTA][MV.sub.A]([x.sub.i]) = [MV.sub.AU]([x.sub.i]) - [MV.sub.AL]([x.sub.i]) and [DELTA][NV.sub.A]([x.sub.i]) = [NV.sub.AU]([x.sub.i]) - [NV.sub.AL]([x.sub.i]), [tau] [member of] [0,1]

are the measures of entropies given by Zang et al. (2010), Wei et al.(2011) and Zang et al. (2011) respectively. Example: Consider the example from Sun & Liu (2012) to review the entropies for an IvIFSs [A.sub.i] which are as follows:

"[A.sub.1] = {x, <[0.1,0.2], [0.2,0.4], [0.4,0.7]>, x [member of] [OMEGA]},

[A.sub.2] = {x, <[0.2,0.2], [0.3,0.5], [0.3,0.5]>, x [member of] [OMEGA]),

[A.sub.3] = {x, <[0.2,0.4], [0,0], [0.6,0.8]),x [member of] [OMEGA]},

[A.sub.4] = {x, <[0.3,0.4], [0,0.142857], [0.4571,0.7]), x [member of] [OMEGA]},

[A.sub.5] = {x, <[0.1,0.1], [0,0.2], [0.6,0.9]>, x [member of] [OMEGA]},

[A.sub.6] = {x, <[0,0.2], [0,0.2], [0.4,1.0]>, x [member of] [OMEGA]}"

The values of [E.sub.ZJ]([A.sub.1]) = 0.5 = [E.sub.ZJ]([A.sub.2]), [E.sub.WW]([A.sub.3]) = 0.7 = [E.sub.WW]([A.sub.4]) and [E.sub.ZM]([A.sub.5]) = 0.5 = [E.sub.ZM] ([A.sub.6]). Thus, the entropies [E.sub.ZJ]([A.sub.i]), [E.sub.WW]([A.sub.i]) and [E.sub.ZM]([A.sub.i]) are unreasonable. Proposed entropies [E.sub.j], j = 1,2,.. 6 can discriminate the fuzziness of all the IvIFSs [A.sub.i], i = 1, ..., 6 given as follows and give the reasonable results given in Table 2.

Here we have proposed some entropy measures and evaluated its performance on the basis of intuitionistic belief and comparison with existing measures. In next section we propose relation between measures of entropy and similarity under IvIFE. Also, we have defined new measures of similarity and have evaluated their performance by comparing them with some existing measures.

4 Relations between measure of entropy and similarity together with new similarity measures

In this section contains definition of new measures of similarity under IvIFE and determined an important relation between entropy and similarity measure IvIFSs which is discussed as follows.

Theorem 3: Let [S.sub.j], for j = 1, .., 6 be measure of similarity of IvIFSs w.r.t. the measure of distance [D.sub.j], for j = 1, .., 6 respectively, and A be any IvIFS. Then [E.sub.j](A) = 3[S.sub.j] (A, <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>) - 2, for j = 1, ..., 6 are measures of entropy for IvIFSs.

Proof: We prove that [E.sub.j](A), for j = 1, ...,6 satisfies conditions given by definition 5.

Property 1): If A [member of] C([OMEGA]) [right arrow] A([x.sub.i]) = <[1,1], [0,0], [0,0]> or A([x.sub.i]) = <[0,0], [1,1], [0,0]>, [for all] [x.sub.i] [member of] [OMEGA], then for j= 1, ..., 6

[S.sub.j] (A, <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>) = 1 - [D.sub.j] (A, <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>) = 2/3.

Thus, [E.sub.j](A) = 0

Property 2): For all j = 1, ..., 6, [E.sub.j](A) = 1

[??]3[S.sub.j] (A, <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>) - 2 = 1 [??] [S.sub.j] (A, <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>) = 1 [??] A = <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>

Property3): Let A and B be any two IvIFSs and

[D.sub.j] (A, <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>) [greater than or equal to] [D.sub.j] (B, <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>) then 1 - [D.sub.j] (A, <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>) [less than or equal to] 1 - [D.sub.j] (B, <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>) [??] [S.sub.j] (A, <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>) [less than or equal to] [S.sub.j] (B, <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>)

[??] [E.sub.j] (A) [less than or equal to] [E.sub.j] (B), for all j = 1, ..., 6

Property 4) : Let A be any IvIFS then [mathematical expression not reproducible]

[mathematical expression not reproducible]

Thus, [E.sub.j](A) = [E.sub.j]([bar.A]), for all j = 1, ..., 6. Next, we present a conversion technique to define similarity measures established by entropy measure for IvIFSs.

Definition 6 : For any two IvIFSs A and B in [OMEGA], such that both A and B are defined by the triplet <[x.sub.i], [[MV.sub.AL]([x.sub.i]), [MV.sub.AU]([x.sub.i])], [[NV.sub.AL]([x.sub.i]), [NV.sub.AU]([x.sub.i])]> and <[x.sub.i], [[MV.sub.BL]([x.sub.i]), [MV.sub.BU]([x.sub.i])], [[NV.sub.BL]([x.sub.i]), [NV.sub.BU]([x.sub.i])]> respectively. we define an IvIFSs 0(A, B)using A and B as given below:

[mathematical expression not reproducible]

Theorem 4: E([empty set](A, B)) be a similarity measure for IvIFSs A and B, where E is an entropy.

Proof: To prove that E([empty set](A, B)) is a measure of similarity, we need to prove property given by definition 3 holds .

Property 1): If A [member of] C([OMEGA]) [right arrow] A([x.sub.i]) = <[1,1], [0,0], [0,0]> or A([x.sub.i]) = <[0,0], [1,1], [0,0]>, for any [x.sub.i] [member of] [OMEGA], then [MV.sub.[empty set](A,[bar.A])L] ([x.sub.i]) = 0 = [MV.sub.[empty set](A,[bar.A])U]([x.sub.i]); And [NV.sub.[empty set](A,[bar.A])L] ([x.sub.i]) = 1 = [NV.sub.[empty set](A,[bar.A])U]([x.sub.i]) Thus, [empty set](A,[bar.A]) = {<[x.sub.i], [0,0], [1,1], [0,0]>/[x.sub.i] [member of] [OMEGA]] [right arrow] S (A, [bar.A]) = E([empty set](A, [bar.A])) = 0

Property 2): Assume that S (A, B) = 1 [right arrow] E([empty set](A, B)) = 1

[mathematical expression not reproducible]

Property 3): [empty set] (A,B) = [empty set] (B,A) by definition of [MV.sub.[empty set](A,B)L]([x.sub.i]), [MV.sub.[empty set](A,B)U]([x.sub.i]), [NV.sub.[empty set](A,B)L]([x.sub.i]), [NV.sub.[empty set](A,B)U]([x.sub.i]) for any [x.sub.i] [member of] [OMEGA] [right arrow] E([empty set](A,B)) = E([empty set](B,A)) [??] S(A,B) = S(B,A)

Property 4): Let A, B and C be any three IvIFSs such that [mathematical expression not reproducible]

So,we have

[mathematical expression not reproducible]

Similarly, we have [mathematical expression not reproducible].

So form definition of entropy corresponding to distance function, we get E([empty set](A,C)) [less than or equal to] E([empty set](A, B)) and E([empty set](A,C)) [less than or equal to] E([empty set] (B,C)) or S([empty set](A, C)) [less than or equal to] S([empty set] (A, B)) and S([empty set] (A, C)) [less than or equal to] S([empty set] (B,C).

Corollary 1: Let E be an entropy measure for IvIFSs and [empty set] (A, B)be an IvIFS defined on two IvIFSs A and B acaccording to definition 6, then E([bar.[empty set](A, B)]) measure of similarity for A, B [member of] IvIFSs([OMEGA]).

Proof: Proof followed from the definition of complement interval-valued intuitionistic fuzzy sets and theorem 4.

Definition 7: Let A, B [member of] IvIFSs([OMEGA]), we can define IvIFS [eta](A, B)using A, B as follows:

[mathematical expression not reproducible]

Where [alpha] [member of] [1, [infinity][and [x.sub.i] [member of] [OMEGA].

Theorem 5: For any two IvIFSs A and B, E([eta](A, B)) is a measure of similarity, where E is an entropy measure.

Proof: To prove that E([eta](A, B)) is a measure of similarity, we need to prove property given by definition 3 holds.

Property 1): If A [member of] C([OMEGA]) [right arrow] A([x.sub.i]) = <[1,1], [0,0], [0,0]> or A([x.sub.i]) = <[0,0], [1,1], [0,0]>, for any [x.sub.i] [member of] [OMEGA], then [MV.sub.[empty set](A,[bar.A])L] ([x.sub.i]) = 1 = [MV.sub.[empty set](A,[bar.A])U]([x.sub.i]); and [NV.sub.[empty set](A,[bar.A])L] ([x.sub.i]) = 0 = [NV.sub.[empty set](A,[bar.A])U]([x.sub.i]) Thus, [empty set](A,[bar.A]) = {<[x.sub.i], [1,1], [0,0], [0,0])/[x.sub.i] [member of] [OMEGA]} [right arrow] S (A, [bar.A]) = E([empty set](A, [bar.A])) = 0

Property 2): Assume that S(A, B) = 1 Then E([eta](A, B)) = 1

[mathematical expression not reproducible]

and

[mathematical expression not reproducible].

Property 3): [eta](A, B) = [eta](B, A) by definition of [MV.sub.[eta](A,B)L]([x.sub.i]), [MV.sub.[eta](A,B)U]([x.sub.i]), [NV.sub.[eta](A,B)L]([x.sub.i]), [NV.sub.[eta](A,B)U]([x.sub.i]) for any [x.sub.i] [member of] [OMEGA] [right arrow]E([empty set] (A,B)) = E([empty set] (B,A)) [??]S(A, B) = S(B, A)

Property 4): Let A, B and C be any three IvIFSs such that [mathematical expression not reproducible]

So, we have from definition 7

[mathematical expression not reproducible]

Similarly, we have [right arrow] [eta](A, C) [contains or equal to] [eta](B, C) [contains or equal to] <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>. Thus

D ([eta](A, B), <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>) [less than or equal to] D([eta](A,C) <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>) and D ([eta](B,C), <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>) [less than or equal to] D([eta](A,C), <[1/3, 1/3], [1/3, 1/3], [1/3, 1/3]>).

So from definition of entropy corresponding to distance function, we get E([empty set](A, C)) [less than or equal to] E([empty set] (A, B)) and E([eta](A, C)) [less than or equal to] E([eta](B, C)) or S([eta](A, C)) [less than or equal to] S([eta](A, B)) and S([eta](A, C)) [less than or equal to] S([eta](B,C)).

Corollary 2: Let E be an entropy for IvIFSs and [eta](A, B) be an IvIFS defined on A, B [member of] IvlFSs([OMEGA]) as defined in definition 7, then E([bar.[eta](A, B)]) is measure of similarity for A, B [member of] IvIFSs([OMEGA).

Proof: Proof followed from the definition of complement of IvIFSs and theorem 5.

4.1 Weighted similarity measure

Let w = [([w.sub.1], [w.sub.2], ..., [w.sub.n]).sup.T] be the weights provided to each element [x.sub.i] [member of] [OMEGA], i = 1,2, ..., n. Then the weighted similarity measure based on the aforesaid similarity measures are defined as S (A, B) = [[summation].sup.n.sub.i=1] [w.sub.i] S(A([x.sub.i]), B([x.sub.i])), where [w.sub.i] [greater than or equal to] 0 and [[summation].sup.n.sub.i=1] [w.sub.i] = 1.

4.2 Comparison with some select measures of similarity

Here, we compare the performance of proposed measure of similarity with some of the existing similarity measures as follows.

For any two IvIFSs A and B, then some existing similarity measures are given as follows:

[mathematical expression not reproducible]

where [mathematical expression not reproducible] is proposed by Wu et al. (2014).

* [S.sub.HL](A,B) = 1 - 1/4n [[summation].sup.n.sub.i=1] [absolute value of [MV.sub.AL]([x.sub.i]) - [MV.sub.BL]([x.sub.i])] + [absolute value of [MV.sub.AU] ([x.sub.i]) - [MV.sub.BU]([x.sub.i])] + [absolute value of [NV.sub.AL]([x.sub.i]) - [NV.sub.BL] ([x.sub.i])] + [absolute value of [NV.sub.AU]([x.sub.i]) - [NV.sub.BU]([x.sub.i])] is given by Hu and Li (2013).

* [S.sub.S] (A,B) = [mathematical expression not reproducible] introduced by Singh(2012)

* [S.sub.Su] (A,B) = [mathematical expression not reproducible]

is proposed by Sun & Liu (2012).

To review the performance of similarity measures let us consider an example. Consider the following IvIFSs

A = {[x.sub.i], <[0.5,0.5], [0.5,0.5], [0,0]), [x.sub.i] [member of] [OMEGA]}, B = {[x.sub.i], <[0.3,0.4], [0.4,0.5], [0.1,0.3]>, [x.sub.i] [member of] [OMEGA]}, C = {[x.sub.i], <[0.3,0.3], [0.3,0.3], [0.4,0.4]>, [x.sub.i] [member of] [OMEGA]}, D = {[x.sub.i], <[0.6,0.6], [0.4,0.4], [0,0]>, [x.sub.i] [member of] [OMEGA]}

Intuitively, it is clear that A is more similar to D than B and C. The result corresponding to measure of similarity measures given in Table 3 :
```Table 3: Comparison of Similarity Measures.

[S.sub.W]                    0.83333   0.69308    0.8181
[S.sub.HL]                   0.9       0.8        0.9
[S.sub.S]                    0.99227   0.9        0.98058
[S.sub.Su]                   0.89655   0.8571     0.9310
[S.sub.1] ([phi])            0.7450    0.73722    0.8646
[S.sub.2] ([phi])            0.78806   0.761886   0.89594
[S.sub.3] ([phi])            0.72614   0.68377    0.84188
[S.sub.4] ([phi])            0.78806   0.74188    0.89594
[S.sub.5] ([phi])            0.68210   0.62283    0.84391
[S.sub.6] ([phi]) for p=2    0.77639   0.77141    0.87090
[S.sub.1] ([eta])            0.93205   0.89426    0.98708
[S.sub.2] ([eta])            0.95217   0.92075    0.99184
[S.sub.3] ([eta])            0.91759   0.85853    0.98418
[S.sub.4] ([eta])            0.95217   0.92075    0.99184
[S.sub.5] ([eta])            0.92825   0.88113    0.98776
[S.sub.6] ([eta]) for p=2    0.93292   0.89590    0.98709
```

From the similarity measures listed in table 3, we can see that

[S.sub.W], [S.sub.HL] and [S.sub.S] are inconsistent with intuition where as [S.sub.Su], [S.sub.j]([phi]) and [S.sub.j]([eta]), j = 1, ..., 6.

In this section we have derived a relation between entropy and similarity measure. Then we defined some similarity measures, compared its performance with existing similarity measures. In section 5 we applied proposed similarity measures to draw conclusion in pattern recognition and medical diagnoses.

5 Applications of proposed similarity measures

Here the proposed similarity measures are applied to some of the situation that deals with imperfect information.

5.1 Pattern recognition

Here we use an example of pattern recognition considered by Xu (2007) and adapted by Wei et al. (2011) and Wu et al.(2014) for classification of building material.

Example: There are four types of building materials [A.sub.i], i = 1,2,3,4 and an anonymous building material B, which is characterized by the IvIFSs defined on x = {[x.sub.1], [x.sub.2], ..., [x.sub.12]} with weighted vector

w = [(0.1, 0.05, 0.08, 0.06, 0.03, 0.07, 0.09, 0.12, 0.15, 0.07, 0.13, 0.05).sup.T]

and we have the data given as follows by Xu (2007).

"[A.sub.1] [mathematical expression not reproducible]

[A.sub.2] [mathematical expression not reproducible]

[A.sub.3] [mathematical expression not reproducible]

[A.sub.4] [mathematical expression not reproducible]

We need to identify which pattern is most similar to B using the maximum degree principle of measures of similarity between IvIFSs. Using the anticipated similarity measures defined in this paper, we get the following results given in table 4:
```Table 4: Application to pattern recognition.

[A.sub.1] B   [A.sub.2] B

[S.sub.1]([phi])             0.688569      0.803061
[S.sub.2]([phi])             0.725141      0.854747
[S.sub.3]([phi])             0.654357      0.742627
[S.sub.4]([phi])             0.725141      0.861997
[S.sub.5]([phi])             0.587711      0.789371
[S.sub.6]([phi]) for p=2     0.738413      0.80768
[S.sub.1] ([eta])            0.780631      0.773645
[S.sub.2] ([eta])            0.816725      0.811175
[S.sub.3] ([eta])            0.767465      0.749444
[S.sub.4] ([eta])            0.816725      0.811175
[S.sub.5] ([eta])            0.725088      0.716763
[S.sub.6]([eta]) for p=2     0.814325      0.801194

[A.sub.3] B   [A.sub.4] B

[S.sub.1]([phi])             0.84671       0.936548
[S.sub.2]([phi])             0.868809      0.95075
[S.sub.3]([phi])             0.818343      0.9265
[S.sub.4]([phi])             0.868809      0.95075
[S.sub.5]([phi])             0.803214      0.926125
[S.sub.6]([phi]) for p=2     0.863582      0.939988
[S.sub.1] ([eta])            0.768612      0.979413
[S.sub.2] ([eta])            0.804125      0.98595
[S.sub.3] ([eta])            0.76081       0.975
[S.sub.4] ([eta])            0.804125      0.98595
[S.sub.5] ([eta])            0.706188      0.978925
[S.sub.6]([eta]) for p=2     0.810194      0.979588
```

From the above values it is clear that B is most similar to [A.sub.4] as the value corresponding to each similarity measure is highest for [A.sub.4]. So, we can conclude that [A.sub.4] building material consistent with the specification and this result is consistent with the results presented by Wu et al.(2014).

5.2 Medical diagnoses

Many authors Wei et al. (2011), Wu et al. (2014), Singh (2012) employed IvIFSs to execute medical diagnosis in their works. Here we use the data used by Singh(2012) to do medical diagnosis using the proposed measure of similarity

Example: Let A and B be the set that represent the set of diagnoses and symptoms respectively given as A = {<[A.sub.1], Viral fever>, <[A.sub.2], Malaria>, <[A.sub.3], Typhoid>} and = {<[B.sub.1]; Temperature>, <[B.sub.2], Headache>, <[B.sub.3], Cough>} Assume the patient is represented by

[mathematical expression not reproducible]

and the weights corresponding to each attribute is equal and each diagnosis is given by the following IvIFSs

[mathematical expression not reproducible]

Using the proposed similarity measure we classify the patient P in one of the diagnoses [A.sub.1], [A.sub.2], [A.sub.3]. The results are as follows in Table 5.

From the above table 5 it is clear that patient P can be diagnosed with viral fever.

6 Conclusion

Entropy, distance and similarity measure are significant research area in fuzzy information theory as they are efficient tools to deal with uncertain and insufficient information. Here we have derived new definition of entropy based on distance measure by considering degree of hesitancy in to account and derived relation between distance, entropy and similarity measures under IvIFE. Further, we have compared the derived similarity measures with some of the existing similarity measure and instances are used to show that the derived measures are able to draw conclusion when existing measures give the same result. Thereafter, proposed measures of similarity are applied to recognition of patterns and medical diagnoses.

https://doi.org/10.31449/inf.v42i4.1303

Pratiksha Tiwari and Priti Gupta

Delhi Institute of Advanced Studies, Plot No. 6, Sector- 25, Rohini, Delhi, India

E-mail: parth12003@yahoo.co.in

Received: July 21, 2016

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```Table 2: Comparison of entropies.

[E.sub.1]     [E.sub.2]     [E.sub.3]
([A.sub.i])   ([A.sub.i])   ([A.sub.i])

[A.sub.1]   0.58167       0.625         0.45
[A.sub.2]   0.735425      0.75          0.65
[A.sub.3]   0.367544      0.4           0.3
[A.sub.4]   0.523512      0.582143      0.425
[A.sub.5]   0.27889       0.3           0.15
[A.sub.6]   0.271989      0.375         0.01

[E.sub.4]     [E.sub.5]     [E.sub.6]
([A.sub.i])   ([A.sub.i])   ([A.sub.i])

[A.sub.1]   0.675         0.4875        0.6063
[A.sub.2]   0.8           0.675         0.765479
[A.sub.3]   0.45          0.15          0.490098
[A.sub.4]   0.607143      0.398214      0.566987
[A.sub.5]   0.3           0.05          0.395848
[A.sub.6]   0.45          0.1375        0.292893

Table 5: Application to Medical diagnoses.

[A.sub.1] P   [A.sub.2] P   [A.sub.3] P

[S.sub.1] ([phi])           0.80666       0.753483      0.770859
[S.sub.2] ([phi])           0.840736      0.788069      0.808633
[S.sub.3] ([phi])           0.766473      0.703639      0.743099
[S.sub.4] ([phi])           0.840736      0.788069      0.808633
[S.sub.5] ([phi])           0.761105      0.682104      0.712949
[S.sub.6] ([phi]) for p=2   0.821222      0.776552      0.796418
[S.sub.1] ([eta])           0.916133      0.872675      0.885256
[S.sub.2] ([eta])           0.938333      0.9025        0.911667
[S.sub.3] ([eta])           0.89835       0.847525      0.865842
[S.sub.4] ([eta])           0.938333      0.9025        0.911667
[S.sub.5] ([eta])           0.9075        0.85375       0.8675
[S.sub.6] ([eta]) for p=2   0.918319      0.877512      0.891129
```
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