Shannon entropy for intuitionistic fuzzy information

# Shannon entropy for intuitionistic fuzzy information

Vasile Patrascu
Research Center for Electronics and Information Technology

Valahia University
Targoviste România
email: patrascu.v@gmail.com
###### Abstract

The paper presents an extension of Shannon fuzzy entropy for intuitionistic fuzzy one. This extension uses a new formula for distance between two intuitionistic fuzzy pairs.

Keywords: Intuitionistic fuzzy information, escort fuzzy information, intuitionistic fuzzy entropy, Shannon entropy.

## 1 Introduction

The intuitionistic fuzzy representation of information was proposed by Atanassov [1], [2], [3] and it is defined by the pair where is the degree of truth while is the degree of falsity. Also, Atanassov considered the following condition for the intuitionistic fuzzy pair :

 μ+ν≤1 (1)

The condition (1) permits to consider the third parameter, the degree of incompleteness defined by:

 π=1−μ−ν (2)

In addition to parameter , we define the net truth , by:

 τ=μ−ν (3)

On this way, we have two systems representation of intuitionistic fuzzy information: the primary space or the explicit space and the secondary space or the implicit space. Also, there exists a supplementary condition for the secondary space, namely:

 |τ|+π≤1 (4)

Taking into account the condition (4), using the secondary space we define the third parameter [8], degree of ambiguity , by:

 α=1−|τ|−π (5)

The formulae (2) and (3) represent the transform from the primary space to the secondary space while the next two formulae represent the inverse transform:

 μ=1−π+τ2 (6)
 ν=1−π−τ2 (7)

For the intuitionistic fuzzy information , it was defined the complement by:

 ¯x=(ν,μ) (8)

After presentation of the main parameters that will be used in this approach, the next will have the following structure: section two presents a new distance for intuitionistic fuzzy information; section three presents formulae for evaluating of some feature of intuitionistic fuzzy information like certainty, score, uncertainty; section four presents the escort fuzzy information; section five presents the Shannon entropy formula for intuitionistic fuzzy information; section six presents the conclusion while the last is the references section.

## 2 A distance for intuitionistic fuzzy information

In this section we define a new distance for intuitionistic fuzzy pairs. For two intuitionistic fuzzy pairs and , we consider the distance define by:

 d(P,Q)=|μp−μq|+|νp−νq| (9)

The distance [10], [11] is a metric and considering the auxiliary point (see Figure 1), there exists the triangle inequality, namely:

 d(P,C)+d(C,Q)≥d(P,Q) (10)

because

 d(P,C)+d(C,Q)≠0 (11)

we can transform (10) into (12):

 1≥d(P,Q)d(P,C)+d(C,Q) (12)

The right term represents the new distance or the new dissimilarity, namely:

 D(P,Q)=d(P,Q)d(P,C)+d(C,Q) (13)

From (13) and (9) it results the distance formula:

 D(P,Q)=|μp−μq|+|νp−νq|2+πp+πq (14)

Using the clasical negation it results the following similarity formula:

 S(P,Q)=1−|μp−μq|+|νp−νq|2+πp+πq (15)

We notice that distance (14) and its similarity (15) take values in the interval . Also, this distance is not a metric because it does not verify the triangle inequality.

## 3 The certainty, the score and the uncertainty for intuitionistic fuzzy information

Starting from the proposed distance defined by (14), we will construct some measures for the following three features of intuitionistic fuzzy information: the certainty, the score and the uncertainty.

### 3.1 The intuitionistic fuzzy certainty

For any intuitionistic fuzzy pair we consider its complement and we define the certainty as dissimilarity between and , namely:

 g(X)=D(X,¯X) (16)

with its equivalent form:

 g(X)=|μ−ν|2−μ−ν (17)

In the space we identify the following properties for intuitionistic fuzzy certainty:

1. if and

The property (4) shows that the certainty increases with and decreases with .

From property (4) it results that because and .

### 3.2 The intuitionistic fuzzy score

From (17) came the idea to define the intuitionistic fuzzy score by:

 r(X)=μ−ν2−μ−ν (18)

with the equivalent form in the space , [6]:

 r(X)=τ1+π (19)

In the space the properties for the intuitionistic fuzzy score derive from the certainty properties, namely:

1. if and

The property (4) shows that the intuitionistic fuzzy score increases with and decreases with . From property (4) it results that because and .

### 3.3 The intuitionistic fuzzy uncertainty

Finally, we define the intuitionistic fuzzy uncertainty using the negation of the certainty:

 e(X)=1−|μ−ν|2−μ−ν (20)

with the equivalent form in the space :

 e(X)=1−|τ|1+π (21)

In the space the intuitionistic fuzzy uncertainty verifies the following conditions [6]:

1. if and

The property (4) shows that the uncertainty decreases with and increases with .

From property (4) it results that because and

## 4 The escort fuzzy information

We will associate to any intuitionistic fuzzy information a fuzzy one that we call escort fuzzy information. The escort fuzzy pair will be determined in order to preserve the score of the intuitionistic fuzzy pair . It will be obtained by solving the following system:

 ^μ+^ν=1 (22)
 ^μ−^ν=r(μ,ν) (23)

It results the following values for the escort fuzzy pair :

 ^μ=μ+π1+π (24)
 ^ν=ν+π1+π (25)

There exist the following two inequalities:

 μ+π≥^μ≥μ (26)
 ν+π≥^ν≥ν (27)

The escort fuzzy pair can be used to extend existing results from fuzzy theory [12], [13] to intuitionistic fuzzy one. An example could be the cardinal’s calculation of an intuitionistic fuzzy set using its escort fuzzy set [6]. In this paper we will use the escort pair for extending the Shannon fuzzy entropy to Shannon intuitionistic fuzzy entropy.

## 5 The Shannon entropy for intuitionistic fuzzy information

For any intuitionistic fuzzy pair , using the escort fuzzy pair we will define the Shannon entropy [9], [5] by the formula:

 ES(X)=eS(^X) (28)

For fuzzy information, De Luca and Termini [5] extended the Shannon formula for calculating the fuzzy entropy by:

 eS(μ)=−μln(μ)−(1−μ)ln(1−μ) (29)

From (29) it results:

 eS(^X)=−^μln(^μ)−^νln(^ν) (30)

From (24), (25), (28) and (30) it results the Shannon variant for intuitionistic fuzzy entropy:

 ES(X)=−μ+π1+πln(μ+π1+π)−ν+π1+πln(ν+π1+π) (31)

There are the next four equivalent formulae:

Using (6) and (7) it results:

 ES(X)=−1+τ1+π2ln⎛⎜ ⎜⎝1+τ1+π2⎞⎟ ⎟⎠−1−τ1+π2ln⎛⎜ ⎜⎝1−τ1+π2⎞⎟ ⎟⎠ (32)

Using (19) it results:

 ES(X)=−1+r2ln(1+r2)−1−r2ln(1−r2) (33)

Because in (33) there exists symmetry between and , it results:

 ES(X)=−1+|r|2ln(1+|r|2)−1−|r|2ln(1−|r|2) (34)

From (17), (18) and (34) it results:

 ES(X)=−1+g2ln(1+g2)−1−g2ln(1−g2) (35)

We notice that:

 ∂g∂|τ|=11+π (36)
 ∂g∂π=−|τ|(1+π)2 (37)
 ∂ES∂g=12ln(1−g1+g) (38)
 ∂ES∂|τ|=1211+πln(1−g1+g) (39)
 ∂ES∂π=−12|τ|(1+π)2ln(1−g1+g) (40)

because

 1−g1+g≤1 (41)

it results:

 ln(1−g1+g)≤0 (42)

From (39), (40) and (42) it results:

 ∂ES∂|τ|≤0 (43)
 ∂ES∂π≥0 (44)

As conclusion, it results that the Shannon entropy for intuitionistic fuzzy information defined by (31) verifies the condition (4) from subsection 3.3, namely it decreases with and increases with .

Also the function defined by (31) verifies the conditions (1) and (3). In order to verify the condition (2) it necessary to multiply by the well-known normalization factor:

 λ=1ln(2) (45)

Finally it results the normalized variant for Shannon entropy, namely:

 ESN(X)=−1ln(2)[μ+π1+πln(μ+π1+π)+ν+π1+πln(ν+π1+π)] (46)

We can decompose the normalized Shannon entropy in a sum with two terms, fuzziness and incompleteness , namely:

 ESN(X)=EA(X)+EU(X) (47)

where

 EA(X)=−(μ+π)ln(μ+π)+(ν+π)ln(ν+π)(1+π)ln(2) (48)
 EU(X)=ln(1+π)ln(2) (49)

From Jensen inequality [4] it results that:

 EA(X)≤−1ln(2)ln(1+π2) (50)

From (50) it results that is maximum for while from (49) it results that is maximum for . Thus, it is highlighted that the uncertainty of intuitionistic fuzzy information has two sources: fuzziness (or ambiguity) that is the similarity of the pairs with and incompleteness (or ignorance) that is the similarity of the pairs with

## 6 Conclusion

In this paper, we presented a new formula for calculating the distance and similarity of intuitionistic fuzzy information. Then, we constructed measures for information features like score, certainty and uncertainty. Also, a new concept was introduced, namely fuzzy information escort. Then, using the fuzzy information escort, Shannon’s formula for intuitionistic fuzzy information was extended. It should be underlined that Shannon’s entropy for intuitionistic fuzzy information verifies the four defining conditions of intuitionistic fuzzy uncertainty. The measures of its two components were also identified: fuzziness (ambiguity) and incompleteness (ignorance).

## References

1. Atanassov, K. (1986) Intutitionistic fuzzy sets, Fuzzy Sets Syst., 20, 87-96.

2. Atanassov, K. (1999) Intuitionistic Fuzzy Sets: Theory and Applications. Studies in Fuzziness and Soft Computing, vol 35, Physica-Verlag, Heidelberg.

3. Atanassov, K. (2012) On Intuitionistic Fuzzy Sets Theory , Springer, Berlin.

4. Jensen, J.L.W.V. (1906) Sur les fonctions convexes et les inegalites entre les valeurs moyennes, Acta Mathematica. 30(1), 175-193.

5. De Luca, A., Termini, S. (1972) A definition of nonprobabilistic entropy in the setting of fuzzy theory. Information and Control 20, 301-312.

6. Patrascu, V. (2010) Cardinality and Entropy for Bifuzzy Sets, Proceedings of the 13 International Conference on Information Processing and Management of Uncertainty IPMU 2010, 28 Jun-02 Jul 2010, Dortmund, Germany, Part I, CCIS 80, 656-665.

7. Patrascu, V. (2012) Fuzzy membership function construction based on multi-valued evaluation, Proceedings of 10 International Conference FLINS, 26-29 August 20102, Istanbul, Turkey, Uncertainty Modeling in Knowledge Engineering and Decision Making, 756-761.

8. Patrascu, V. (2016) Refined Neutrosophic Information Based on Truth, Falsity, Ignorance, Contradiction and Hesitation, Neutrosophic Sets and Systems, Vol. 11, 57-66.

9. Shannon, C. E. (1948) A mathematical theory of communication, Bell System Tech., J. 27, 379-423.

10. Website.com,

11. Zadeh, L. (1965) Fuzzy sets, Inform and Control,8, 338-353.

12. Zadeh, L. (1965) Fuzzy sets and systems, Proc. Symp. On Systems Theory, Polytechnic Institute of Brooklyn, New York, 29-37.

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