Abstract
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.
Highlights
Shannon (1948) was first to use the word “entropy” to measure an uncertain degree of the randomness in a probability distribution
We introduce a sequence of fuzzy mean difference divergence measures and established the inequalities among them to explore the fuzzy inequalities
DRG≤5DAN : Application of fuzzy mean difference divergence measures to pattern recognition We present the application of the proposed fuzzy mean difference divergence measures in the context of pattern recognition
Summary
Shannon (1948) was first to use the word “entropy” to measure an uncertain degree of the randomness in a probability distribution. 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. Bhandari and Pal (1993) introduced the measure of fuzzy directed divergence corresponding to (4) as. Lemma 1: Let f : I ⊂ R+ → R be a convex and differentiable function satisfying f. 1⁄4 Xn 24sffiÀffiffiμffiffi2AffiffiffiðffiffixffiffiiffiÞffiffiffiþffiffiffiffiffiμffiffi2BffiffiffiðffiffixffiffiiffiÞffiffiÁffiffi þ sffiÀffiffiðffiffi1ffiffi−ffiffiffiμffiffiAffiffiffiðffiffixffiffiiffiÞffiffiÞffiffi2ffiffiffiþffiffiffiffiffiðffi1ffiffiffi−ffiffiffiμffiffiBffiffiffiðffixffiffiffiiffiÞffiffiÞffiffi2ffiffiÁffi i1⁄41 i Xn þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μAðxiÞμBðxiÞ 3 þ μBðxi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ð1−μAðxiÞÞð1−μBðxiÞÞ ð13Þ f DSGðA; BÞ−GðA; BÞ 2sffiÀffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÁffiffi 4 μ2AðxiÞ þ μ2BðxiÞ sffiÀffiffiðffiffi1ffiffi−ffiffiffiμffiffiAffiffiffiðffiffixffiffiiffiÞffiffiÞffiffi2ffiffiffiþffiffiffiffiffiðffi1ffiffiffi−ffiffiffiμffiffiBffiffiffiðffixffiffiffiiffiÞffiffiÞffiffi2ffiÁffiffi i1⁄41 g "qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#. Theorem 1: All the proposed measures (6) - (23) are valid measures of fuzzy mean difference directed divergence. Theorem 2: The fuzzy mean difference divergence measures defined in (6) - (23) admit the following inequalities:. Proof: The proof of the above theorem is based on Lemma 2 and is given in parts in the following propositions
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