Correlation Measure for Interval valued Pythagorean Fuzzy Sets

In this paper, we study the concept of Interval Valued Pythagorean set [IVPS] with special cases of membership and non-membership degrees. In this IVPS set membership and non-membership degrees are satisfying the condition (uA(x)) 2 + (νA(x)) 2 ≤ 1 instead of uA(x) + νA(x) > 1. Here the correlation measure of IVPS set is an extension of correlation measure of Interval valued Pythagorean fuzzy set.


Introduction
Based on the centric method, Hanafy introduced and studied the correlation and correlation coefficient. Correlation coefficients are beneficial tools used to determine the degree of similarity between objects. The correlation coefficients in fuzzy environments have many applications in the field of medicals diagnosis and clustering etc.
Definition 2.2 Let E be an universe set. An Intuitionistic fuzzy set A on E can be defined as follows: and v A (x) are the degree of membership and degree of non-membership of the element x respectively.
3 Let X is a non-empty set (universe). A Interval valued Pythagorean fuzzy set A on X is an object of the form Definition 2.5 Let X be a non-empty set and I the unit interval [0, 1]. A Interval Valued Pythagorean fuzzy set S is an object having the form where the functions u A : X → [0, 1] and v A : X → [0, 1] denote respectively the degree of membership and degree of non-membership of each element x ∈ X to the set P and 0 ≤ (u A (x)) 2 +(v A (x)) 2 ≤ 1. 3 Interval-valued Pythagorean fuzzy set Definition 3.1 Let X be a non-empty set (universe). A set A on X is an object of the is the degree of membership, and v A (x) is the degree of non-membership. Here u A (x) and v A (x) are dependent components.
Definition 3.2 Let X be a nonempty set and I be the unit interval [0,1]. Let object The defined correlation measure between IVPS A and IVPS B satisfies the following properties As the membership, indeterminate and non-membership functions of the IVPS lies between 0 and 1, ρ(A, B), Also lies between 0 and 1. We will prove By Cauchy-Schwartz inequality Let the ρ (A, B) = 1. Then, the unite measure is possible only if which completes the proof. and ρ (A, B) to weighted correlation coefficient as follows: It is easy to verify that if ω = ( 1 n , 1 n , . . . , 1 n ) T , then equations (3) and (4) reduces to (1) and (2) respectively.
By using Cauchy-Schwarz inequality, we get It is easy to verify that if ω = ( 1 n , 1 n , · · · , 1 n ) T , then equation (6) and (5) reduces (1) and (2) respectively.     On the other hand, if we assign weights 0.10, 0.20, 0.30 and 0.40 respectively, then by applying correlation coefficient given in equations (3) and (4), we can give the following values of the correlation coefficient: Using Equations (3), we get the value of ρ (A, B).  The highest correlation measure from the Tables 3, 4, 5 and 6 give the proper medical diagnosis. Therefore, patient P 1 suffers from Viral Fever, patient P 2 suffers from Malaria and patient P 3 suffers from Dengu. Hence, we can see from the above four kinds of correlation coefficient indices that the results are same.

Conclusion
In this paper, we found the correlation measure of Interval valued Pythagorean set [IVPS] and proved some of their basic properties. Based on that the present paper, we have extended the theory of correlation coefficient to the Pythagorean in which the constraint condition of sum of membership, non-membership and indeterminacy be less. Illustrate examples have handle the situation where the existing correlation coefficient in NS environment fails. Also to deal with the situations where the elements in a set are correlative, a weighted correlation coefficients has been defined. We studied an application of correlation measure of Interval valued Pythagorean set.