Shannon entropy for intuitionistic fuzzy information
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.
The intuitionistic fuzzy representation of information was proposed by Atanassov , ,  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 :
The condition (1) permits to consider the third parameter, the degree of incompleteness defined by:
In addition to parameter , we define the net truth , by:
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:
For the intuitionistic fuzzy information , it was defined the complement by:
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:
The right term represents the new distance or the new dissimilarity, namely:
Using the clasical negation it results the following similarity formula:
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:
with its equivalent form:
In the space we identify the following properties for intuitionistic fuzzy certainty:
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:
with the equivalent form in the space , :
In the space the properties for the intuitionistic fuzzy score derive from the certainty properties, namely:
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:
with the equivalent form in the space :
In the space the intuitionistic fuzzy uncertainty verifies the following conditions :
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:
It results the following values for the escort fuzzy pair :
There exist the following two inequalities:
The escort fuzzy pair can be used to extend existing results from fuzzy theory ,  to intuitionistic fuzzy one. An example could be the cardinal’s calculation of an intuitionistic fuzzy set using its escort fuzzy set . 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 fuzzy information, De Luca and Termini  extended the Shannon formula for calculating the fuzzy entropy by:
From (29) it results:
There are the next four equivalent formulae:
Using (19) it results:
Because in (33) there exists symmetry between and , it results:
We notice that:
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:
Finally it results the normalized variant for Shannon entropy, namely:
We can decompose the normalized Shannon entropy in a sum with two terms, fuzziness and incompleteness , namely:
From Jensen inequality  it results that:
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
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).
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