On fuzzy context of tri-parameterized generalized entropy measure linked to conditional description

Introduction

Applications of information theory [18,20,21,22] and, more recently, fuzzy set theory [11,12,13] have been found in many different domains. In this paper, we assessment, such as fuzzy entropy. Since fuzzy entropy can be approached in a variety of ways, we go over each one independently. An overview of information theory and fuzzy logic is also given.

Probability Entropy

The fuzzy set theory was discovered in 1965 by L. A. Zadeh [14,15,26] , who defined an entropy for a fuzzy event as the uncertainty associated with that event. This entropy is called a weighted Shannon entropy, in which the membership values are considered as weights. In his work, Zadeh extended the Shannon entropy to be applied as a fuzzy entropy on a fuzzy subset A for the finite set 𝑋 = {𝑥 1 , 𝑥 2 , … … . , 𝑥 𝑛 } with respect to the probability distribution 𝑃 = {𝑝 1 , 𝑝 2 , … . . , 𝑝 𝑛 } [24,25] .

𝐻 = -∑ 𝜇 𝐴 𝑛 𝑖=1

(𝑥 𝑖 )𝑝 𝑖 log 𝑝 𝑖 Where p represents A's membership function and 𝑝 𝑖 is the likelihood that 𝑥 𝑖 will occur. This scenario exhibits the three categories of uncertainty: ambiguity, randomness, and vagueness, which translates to fuzziness and randomness.

Our Result

This encourages us to establish a fuzzy entropy function with a form that is more like to Shannon's probabilistic entropy. In light of this, we increase our to provide a fuzzy counterpart of Shannon's entropy, use the probability-Membership Density Function [3,6,9] analogy. A discrete random variable 𝑋 = {𝑥 1 , 𝑥 2 , … … . 𝑥 𝑛 }with a probability mass function 𝑃 = 𝑝 1 , 𝑝 2 , … … … 𝑝 𝑛 is examined, where 𝑝 𝑖 represents the probability (𝑋 = 𝑥 𝑖 ). Given X, the probabilistic Shannon's entropy is as follows:

Where an entropy function that is "probabilistic" is indicated by the letter P in the superscript.

The prior probability pi represents previous knowledge.

concerning the prevalence of 𝑥 𝑖 . As a result, log (

) [16,17,23] provides an uncertainty metric known as the information gain related to 𝑥 𝑖 . Shannon's probabilistic entropy is calculated by adding together these information gains and weighing them according to their respective probabilities. Using the Shannon entropy [8,9,19] as a foundation, several generalized entropies have been proposed in the literature and have applications in a variety of fields, including computing, information processing, statistics, economics, etc. Agrawal and Shannon entropy given by

After analyzing the standard Shannon information entropy, Pal and Pal [8,9] proposed an information entropy known as the exponential entropy, which is given by

In accordance with (2), Kvalseth presented and examined the generalized exponential entropy of order-α, which is provided by

Using a discrete probabilistic framework, first attempt to quantify the uncertain aspects of a fuzzy event. The definition of the entropy of fuzzy set A with regard to (𝑋, 𝑃) seems to have been made by Zadeh [20] .

De Luca and Termini [2,7,10] defined fuzzy entropy for a fuzzy set A corresponding to (1) as

Afterwards, Bhandari and Pal [1,4,5] conducted a survey on fuzzy set information measures and provided several novel fuzzy entropy [10] measurements. According to (2), they've recommended taking the following action:

Pal and Pal defined exponential fuzzy entropy for a fuzzy set corresponding (3) as

Some set operation can be defined as follows: De Luca and Termini's [2] fuzzy entropy measure quantifies the degree of equivalence between the fuzzy values 𝜇 𝐴 (𝑥 1 ), 𝜇 𝐴 (𝑥 2 ), 𝜇 𝐴 (𝑥 3 ), … … , 𝜇 𝐴 (𝑥 𝑛 ), that is, the 𝐻(𝐴) has a maximum value of 𝑛 log 2 when all fuzzy values are equal, that is, when each 𝜇 𝐴 (𝑥 𝑖 ) = 1 2

, The greater the equality among, the bigger the value of 𝐻(𝐴). Consequently, we have 𝐻(𝐹) = 𝑛 log 2, where the most fuzzy distribution is

It was given some formulas of entropy, based on which, we propose the following fuzzy entropy measures

Bhandari & Pal introduced two entropy measures on fuzzy set as

Later on exponential entropy for fuzzy set is also proposed by Verma and Sharma [17] as

New Proposed Exponential Entropy Measure Using the preceding notions as a guide, we provide exponential fuzzy entropy measures as

Claim: The measurement specify 𝐸(𝐴) on fuzzy set A is a valid entropy measure.

𝑬 𝟏: Sharpness

To Prove 𝐸(𝐴) = 0 for 𝜇 𝐴 (𝑥 𝑖 ) = 0 𝑜𝑟 1 First we suppose that 𝐸(𝐴) = 0

⇒ 𝜇 𝐴 (𝑥 𝑖 ) [𝑒 (𝜇 𝐴 (𝑥 𝑖 )) 𝛼 -𝑒 (𝜇 𝐴 (𝑥 𝑖 )) 𝛽 -𝑒 (𝜇 𝐴 (𝑥 𝑖 )) 𝛾 ] + 1

= 𝑒 [1-𝜇 𝐴 (𝑥 𝑖 )] 𝛼 (𝜇 𝐴 (𝑥 𝑖 ) -1) -𝑒 [1-𝜇 𝐴 (𝑥 𝑖 )] 𝛽 (𝜇 𝐴 (𝑥 𝑖 )-1)-𝑒 [1-𝜇 𝐴 (𝑥 𝑖 )] 𝛾 (𝜇 𝐴 (𝑥 𝑖 ) -1)

⇒ [𝑒 (𝜇 𝐴 (𝑥 𝑖 )) 𝛼 -𝑒 (𝜇 𝐴 (𝑥 𝑖 )) 𝛽 -𝑒 (𝜇 𝐴 (𝑥 𝑖 )

This is true if 𝜇 𝐴 (𝑥 𝑖 ) = 0 𝑜𝑟 1 Conversely, we assume that If 𝜇 𝐴 (𝑥 𝑖 ) = 0 𝑜𝑟 1, from equation ( 5) ∑ {[𝑒 (𝜇 𝐴 (𝑥 𝑖 )) 𝛼 + 𝜇 𝐴 (𝑥 𝑖 )𝛼(𝑒) (𝜇 𝐴 (𝑥 𝑖 )) 𝛼-1 -(𝑒) (1-𝜇 𝐴 (𝑥 𝑖 )) 𝛼-1 + [0 -𝜇 𝐴 (𝑥 𝑖 )𝛼(𝑒

Case-I: Let 0 ≤ 𝜇 𝐴 (𝑥 𝑖 ) < 0.5, then we have

𝜕𝜇 𝐴 (𝑥 𝑖 ) > 0∀𝛼, 𝛽 > 0, This gives that 𝐸(𝐴) is increasing function. Case-II: Let 0.5 < 𝜇 𝐴 (𝑥 𝑖 ) ≤ 1, then we have 𝜕𝐸(𝐴) 𝜕𝜇 𝐴 (𝑥 𝑖 ) < 0∀𝛼, 𝛽 > 0, This gives that 𝐸(𝐴) is decreasing function. Case-III: Let 𝜇 𝐴 (𝑥 𝑖 ) = 0.5, then we have

It is evident from above graph that 𝐸(𝐴) is a concave function, with a global maximum at 𝜇 𝐴 (𝑥 𝑖 ) = 0.5. Therefore, 𝐸(𝐴)is the greatest if the most fuzzy set is A

(𝑬 𝟑 ): Resolution

To demonstrate that 𝐸(𝐴 * ) ≤ 𝐸(𝐴)here 𝐴 * is a sharpened form of A. We know from earlier axioms that 𝐸(𝐴) is a decreasing function for 0.5 < 𝜇 𝐴 (𝑥 𝑖 ) ≤ 1 and an increasing function for 0 ≤ 𝜇 𝐴 (𝑥 𝑖 ) < 0.5, Using the idea above.

𝜇 𝐴 * (𝑥 𝑖 ) ≤ 𝜇 𝐴 (𝑥 𝑖 ) ⇒ 𝐸(𝐴 * ) ≤ 𝐸(𝐴) (7) 𝜇 𝐴 (𝑥 𝑖 ) ≤ 𝜇 𝐴 * (𝑥 𝑖 ) ⇒ 𝐸(𝐴) ≤ 𝐸(𝐴 * ) (8) From ( 7) and ( 8)

⇒ 𝐸(𝐴 * ) ≤ 𝐸(𝐴) in both cases.

(𝑬 𝟒 ): Symmetry

Assuming that 𝐴 𝐶 is A's complement, we may use the membership function to take 𝜇 𝐴 (𝑥 𝑖 ) = 1 -𝜇 𝐴 (𝑥 𝑖 ),Then obviously

This demonstrates the validity of 𝐸(𝐴) be as an entropy measure on a fuzzy set.

Few characteristics of entropy measures

Assume that 𝐹𝑆(𝑋) represents the family of all fuzzy sets of universe X, where FS represents fuzzy sets and 𝜁, 𝜉, 𝜒 ∈ 𝐹𝑆(𝑋) is given by

Then some set operations can be defined as follows: The following are some significant characteristics of the exponential entropy measure:

Claim

For Fuzzy set 𝜻, 𝝃, 𝝌 ∈ 𝑭𝑺(𝑿) prove that

Where 𝜇 𝜁 (𝑥), 𝜇 𝜉 (𝑥), 𝜇 𝜒 be the fuzzy membership function of 𝜁, 𝜉, 𝜒 respectively.

Now, From equation ( 9), we have

𝛽 -[𝑒 𝑋 𝜃 =𝑋 (12) From equation ( 11) and ( 12), we get 𝐸((𝜁 ∪ 𝜉) ∪ χ) = 𝐸(𝜁 ∪ (𝜉 ∪ 𝜒))

(II) From the above result and from equation (10), we easily see that 𝐸((𝜁 ∩ 𝜉) ∩ χ) = 𝐸(𝜁 ∩ (𝜉 ∩ 𝜒))

Conclusion

An exponential parametric entropy measure on fuzzy sets is presented in this paper. When the appropriate value of parameters is substituted, the proposed entropy measure, which reduces to the current entropy measure, is significant and valid. These entropy metrics' significant and intriguing characteristics have also been examined. So, In terms of applicability, we may say that the defined entropy measure is more adaptable.

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