Correlation Coefficient of Interval Neutrosophic Set

In this paper we introduce for the first time the concept of correlation coefficients of interval valued neutrosophic set (INS for short). Respective numerical examples are presented.


Introduction
Neutrosophy was pioneered by Smarandache [1]. It is a branch of philosophy which studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra [2]. Neutrosophic set theory is a powerful formal framework which generalizes the concept of the classic set, fuzzy set [3], interval-valued fuzzy set [4], intuitionistic fuzzy set [5], interval-valued intuitionistic fuzzy set [6], and so on. Neutrosophy introduces a new concept called <NeutA> which represents indeterminacy with respect to <A>. It can solve certain problems that cannot be solved by fuzzy logic. For example, a paper is sent to two reviewers, one says it is 90% acceptable and another says it is 90% unacceptable. But the two reviewers may have different backgrounds. One is an expert, and another is a new comer in this field. The impacts on the final decision of the paper by the two reviewers should be different, even though they give the same grade level of the acceptance. There are many similar problems, such as weather forecasting, stock price prediction, and political elections containing indeterminate conditions that fuzzy set theory does not handle well. This theory deals with imprecise and vague situations where exact analysis is either difficult or impossible. After the pioneering work of Smarandache. In 2005, Wang et al. [7] introduced the notion of interval neutrosophic set (INS) which is a particular case of the neutrosophic set (NS) that can be described by a membership interval, a non-membership interval, and an indeterminate interval, thus the NS is flexible and practical, and the NS provides a more reasonable mathematical framework to deal with indeterminate and inconsistent information. The theories of both neutrosophic set and interval neutrosophic set have achieved great success in various areas such as medical diagnosis [8], database [9,10], topology [11], image processing [12,13,14], and decision making problem [15]. Although several distance measures, similarity measures, and correlation measure of neutrosophic sets have been recently presented in [16,17], there is a rare investigation on correlation of interval neutrosophic sets. It is very common in statistical analysis of data to finding the correlation between variables or attributes, where the correlation coefficient is defined on ordinary crisp sets, fuzzy sets [18], intuitionistic fuzzy sets [19,20,21], and neutrosophic set [16,17] respectively. In this paper we first discuss and derive a formula for the correlation coefficient defined on the domain of interval neutrosophic sets. The paper unfolds as follows. The next section briefly introduces some definitions related to the method. Section III presents the correlation and weighted correlation coefficient of the interval neutrosophic set. Conclusions appear in the last section.

Preliminaries
In this section, we mainly recall some notions related to neutrosophic sets, and interval neutrosophic sets relevant to the present work. See especially [1,7,17] for further details and background.

Definition ([1]
). Let U be an universe of discourse; then the neutrosophic set A is an object having the form A = {< x: , , >,x ∈ U}, where the functions T,I,F : U→] − 0,1 + [ define respectively the degree of membership, the degree of indeterminacy, and the degree of nonmembership of the element x ∈ U to the set A with the condition: (1) From philosophical point of view, the neutrosophic set takes the value from real standard or nonstandard subsets of ] − 0,1 + [.So instead of ] − 0,1 + [ we need to take the interval [0,1] for technical applications, because ] − 0,1 + [will be difficult to apply in the real applications such as in scientific and engineering problems.  Let = <0, 1 ,1> and = <1, 0 ,0>

Correlation Coefficient of Neutrosophic Set ([17]).
Let A and B be two neutrosophic sets in the universe of discourse X = { , , …, }.
The correlation coefficient of A and B is given by where the correlation of two NSs A and B is given by And the informational energy of two NSs A and B are given by Respectively, the correlation coefficient of two neutrosophic sets A and B satisfies the following properties: (1) 0 R(A,B) 1

Correlation of Two Interval Neutrosophic Sets
In this section, following the correlation between two neutrosophic sets defined by A. A. Salama in [17], we extend this definition to interval neutrosophic sets. If we have a random non-crisp set, with a triple membership form, for each of two interval neutrosophic sets, we get the interest in comparing the degree of their relationship. We check if there is any linear relationship between the two interval neutrosophic sets; thus we need a formula for the sample correlation coefficient of two interval neutrosophic sets in order to find the relationship between them.

3.1.Definition
Assume that two interval neutrosophic sets A and B in the universe of discourse X = {x 1 , x 2 , x 3 , …, x n } are denoted by A= , X, and express the so-called informational energy of the interval neutrosophic sets A and B respectively.

Remark 2:
For the sake of simplicity we shall use the symbols: For the correlation of interval neutrosophic set, the following proposition is immediate from the definition.

Proposition
For A, B in the universe of discourse X={x 1 ,x 2 ,x 3 ,…,x n } the correlation of interval neutrosophic set have the following properties:
We will prove that . From the Schwartz inequality, we obtain

Weighted Correlation Coefficient of Interval Neutrosophic Sets
In order to investigate the difference of importance considered in the elements in the universe of discourse, we need to take the weights of the elements (i = 1, 2, 3, …, n). In the following we develop a weighted correlation coefficient between the interval neutrosophic sets as follows: If w = { }, equation (34) is reduced to the correlation coefficient (12); it is easy to check that the weighted correlation coefficient between INSs A and B also satisfies the properties: (1) (2) = (36)

Numerical Illustration.
In this section we present, an example to depict the method defined above, where the data is represented by an interval neutrosophic sets. It shows that the interval neutrosophic sets A and B have a good positively correlation.

Conclusion:
In this paper we introduced a method to calculate the correlation coefficient of two interval neutrosophic sets.