Sequence of inequalities among fuzzy mean difference divergence measures and their applications

This paper presents a sequence of fuzzy mean difference divergence measures. The validity of these fuzzy mean difference divergence measures is proved axiomatically. In addition, it introduces a sequence of inequalities among some of these fuzzy mean difference divergence measures. The applications of proposed fuzzy mean difference divergence measures in the context of pattern recognition have been presented using a numerical example. It is shown that the proposed fuzzy mean difference divergence measures are well suited to use with linguistic variables. Finally, on establishing inequalities, we find that our proposed measures are computationally much more efficient.


Introduction
was first to use the word "entropy" to measure an uncertain degree of the randomness in a probability distribution. Entropy as a measure of fuzziness was first introduced by Zadeh (1968). There is an intrinsic similarity between two equations however Shannon entropy measures the average uncertainty in bits associated with the prediction of outcomes in a random experiment, whereas the entropy of fuzzy set describes the degree of fuzziness in a fuzzy set. The concept of fuzzy sets proposed by Zadeh (1968) has proven useful in the context of pattern recognition, image processing, speech recognition, bioinformatics, fuzzy aircraft control, feature selection, decision making, etc.
Entropy, as a very important notion for measuring fuzziness degree or uncertain information in fuzzy set theory, has received a great attention. For example, Kullback and Leibler (1951) obtained the measure of directed divergence between two probability distributions. Bhandari and Pal (1993) presented some axioms to describe the measure of directed divergence between fuzzy sets, which is proposed corresponding to Kullback and Leibler (1951) measure of directed divergence. Thereafter, many other researchers have studied the fuzzy divergence measures in different ways and provide their application in different areas. In 1999, Fan and Xie introduced the divergence measure based on exponential operation and studied its relation with divergence measure introduced in Bhandari and Pal (1993). Montes et al. (2002) studied the special classes of divergence measures and used the link between fuzzy and probabilistic uncertainty. Parkash et al. (2006) proposed two fuzzy divergence measures corresponding to Ferreri (1980) probabilistic measure of directed divergence. Ghosh et al. (2010) gave the application of Bhandari and Pal (1993) divergence measure in the area of automated leukocyte recognition. Bhatia and Singh (2012) proposed the fuzzy divergence measure corresponding to Taneja (2008) Arithmetic-geometric divergence measure.
In the recent years, many authors have introduced various divergence measures between fuzzy sets. We introduce a sequence of fuzzy mean difference divergence measures and established the inequalities among them to explore the fuzzy inequalities. The advantage of establishing the inequalities is to make the computational work much simpler. The technique of inequalities provides a better comparison among fuzzy mean divergence measures.

Preliminaries on fuzzy divergence measures
Fuzziness, a feature of uncertainty, results from the lack of sharp difference of being or not being a member of the set, i.e., the boundaries of the set under consideration are not sharply defined. A fuzzy set A defined on a universe of discourse X is given as Zadeh (1965): where μ A : X → [0, 1] is the membership function of A. The membership value μ A (x) describes the degree of the belongingness of x ∈ X in A. When μ A (x) is valued in {0, 1}, it is the characteristic function of a crisp (non-fuzzy) set. Zadeh (1965) gave some notions related to fuzzy sets, one of them which we shall need in our discussion, is as follows: Compliment of a fuzzy set A: The measure of information defined by Shannon (1948) is given by Taking into consideration the concept of fuzzy sets, De Luca and Termini (1972) introduced the measure of fuzzy entropy corresponding to Shannon's entropy given in (2) as Kullback and Leibler (1951) obtained the measure of directed divergence of probability distribution P = (p 1 , p 2 , … p n ) from probability distribution Q = (q 1 , q 2 , … q n ) as Measure of fuzzy divergence between two fuzzy sets gives the difference between two fuzzy sets and this measure of distance/difference between two fuzzy sets is called the fuzzy divergence measure. Bhandari and Pal (1993) introduced the measure of fuzzy directed divergence corresponding to (4) as The fuzzy mean divergence measures corresponding to seven geometrical mean measures given in Taneja (2012) are presented in Table 1.
We have the following Lemma in fuzzy context corresponding to the Lemma of Taneja (2005): Lemma 1: Let f : I ⊂ R + → R be a convex and differentiable function satisfying f 1 2 À Á ¼ 0.
(c) Convexity: Now we shall prove the condition of convexity of measures (6) -(23) with the help of Lemma 1.
For simplicity, Let us write Let us take μ A = z ⇒ μ B = 1 − z. So, corresponding to measures (6) -(23) we have the following generating functions: Now in all the cases from (24) -(41), we can easily check that . The first and second order derivatives of the functions (24) -(41) are as follows: We see that in the entire cases second order derivative are positive and satisfies f ′

Inequalities among fuzzy mean difference divergence measures
Theorem 2: The fuzzy mean difference divergence measures defined in (6) -(23) admit the following inequalities: i.e., we have the following inequalities: Proof: The proof of the above theorem is based on Lemma 2 and is given in parts in the following propositions.

This gives
Applying Lemma 2 for the difference of fuzzy means D CG (A, B) and D RG (A, B) and using (71), we get D CG ≤ 9 5 D RG : Proposition 13: We have D RG ≤ 5D AN Proof: Let us consider the function This gives Applying Lemma 2 for the difference of fuzzy means D RG (A, B) and D AN (A, B) and using (72), we get D RG ≤5D AN :

Application of fuzzy mean difference divergence measures to pattern recognition
We now present the application of the proposed fuzzy mean difference divergence measures in the context of pattern recognition. Next, an example related to pattern recognition is given to demonstrate the results obtained by the fuzzy mean difference divergence measures (6) -(23).

Numerical example
We now establish that the proposed fuzzy mean difference divergence measures (6) -(23) are reliable in applications with compound linguistic variables.
Example: Let F = {(x, μ F (x))/x ∈ X} be a fuzzy set in X. Tomar and Ohlan (2014) defined for any positive real number n, from the operation of power of a fuzzy set: F n = {(x, [μ F (x)] n )/x ∈ X}.
The proposed fuzzy mean difference divergence measures are used to calculate the degree of divergence/distance between these fuzzy sets. The divergence/distance values have been calculated by Eqs. (6) -(23) between different fuzzy sets. The comparative results are summarized in Table 3. For convenience, we use the notation * (i) in Table 3 to present the divergence/distance value computed from equation i. The following abbreviated notions are used in Table 3.
From the numerical results presented in Table 3, we see that the proposed fuzzy mean difference divergence measures (6) -(23) satisfy the requirement (73) -(76). Therefore, the proposed fuzzy mean difference divergence measures are consistent in the application with compound linguistic measures.

Conclusion
To sum up, we present a sequence of fuzzy mean difference divergence measures. We also establish a sequence of inequalities among some of the proposed fuzzy mean difference divergence measures. An application of the proposed divergence measures in the field of pattern recognition is established. A numerical example is used to present the consistency of these divergence measures in application with compound linguistic variables. Numerical results show that the fuzzy mean difference divergence measures are much simpler with the difference of the means involved.