Parametric correlation coefficient of fuzzy numbers

In the present paper, the researchers discuss the problem of parametric interval approximation of fuzzy numbers. It is the interval which fulfills two conditions. In the first, this interval is a continuous interval approximation operator. In the second, the parametric weighted distance between this interval and the approximated number is minimal and continuous. This interval can be used as a crisp set approximation with respect to a fuzzy quantity. These indices can be applied for comparison of fuzzy numbers namely fuzzy correlations in fuzzy environments and expert's systems. Finally, some of their applications are mentioned.


INTRODUCTION
Representing fuzzy numbers by proper intervals is an interesting and important problem.An interval representation of a fuzzy number may have many useful applications.By using such a representation, it is possible to apply in fuzzy number approaches some results derived in the field of interval number analysis.For example, it may be applied to a comparison of fuzzy numbers by using the order relations defined on the set of interval numbers.Various authors in [Grzegorzewski, 2002;Saneifard, 2009] have studied the crisp approximation of fuzzy sets.They proposed a rough theoretic definition of that crisp approximation, called the nearest interval approximation of a fuzzy set.Moreover, quite different approach to crisp approximation of fuzzy sets was applied in [Cheng et al., 1998].They proposed a rough theoretic definition of that crisp approximation, called the nearest ordinary set of a fuzzy set, and they suggested a construction of such a set.They discussed rather discrete fuzzy sets.Their approximation of the given fuzzy set is not unique.Thus, this article will not discuss this method.Having reviewed the previous interval approximation, this article proposes here a method to find the parametric interval approximation of a fuzzy number, that, it fulfills two conditions.In the first, this interval is a continuous interval approximation operator.In the second, the parametric distance between this interval and the approximated number is minimal and continuous.The main purpose of this article is for this parametric interval of a fuzzy number be used as a crisp set approximation of a fuzzy number.Some of their applications are mentioned.The paper is organized as follows: In Section 2, this article recalls some fundamental results on fuzzy numbers.In Section 3, a crisp set approximation of a fuzzy number (parametric interval approximate) is obtained.In this Section some remarks are proposed and illustrated.Applications of this approximation are in the Section 4. The paper ends with conclusions in Section 5.
. In this article symbols  denotes the minimum operators.Definition 4. A fuzzy set A is a fuzzy number if A is normal and convex on X .
Definition 5.For fuzzy set A , support function is defined as A space of all fuzzy numbers will be denoted by F , and this article recalls that Definition 6. Assume that the fuzzy number F A is represented by means of the following representation: Here, is the  -level set of the fuzzy number A .This article considers normal and convex fuzzy numbers.Therefore the  -level sets may be represented in the form of a segment, is a monotonically nondecreasing and is a monotonically non-increasing left-continuous functions.The functions Definition 7. [Nasibov, 2007] The following values constitute the weighted averaged representative and weighted width, respectively, of the fuzzy number A : and Here 1 0  c denotes an "optimism/pessimism" coefficient in conducting operations on fuzzy numbers.The function denotes the distribution density of the importance of the degrees of fuzziness, where In particular cases, it may be assumed . Definition 8.For arbitrary fuzzy numbers A and B the quantity is called the parametric distance between the fuzzy numbers A and B .
Definition 9. [Grzegorzewski, 2002] An operator , where , is a metric defined in the family of all fuzzy numbers.Definition 10. [Grzegorzewski, 2002].An interval approximation operator satisfying in condition ) (c for any , is called the continuous interval approximation operator.

Parametric interval approximation
Various authors [Cheng et al., 1998;Grzegorzewski, 2002] have studied the crisp approximation of fuzzy sets.They proposed a rough theoretic definition of that crisp approximation, called the nearest ordinary set and nearest interval approximation of a fuzzy set.In this Section, the researchers will propose another approximation called the parametric interval approximation.Let A be an arbitrary fuzzy number and be its  -cut set.This article will try to find a closed interval , which is the parametric interval to A with respect to metric . So, this article has to minimize, with respect to In order to minimize It is clear that, the parameters C L and C R which minimize Eq. ( 7) must satisfy Therefore, this article has the following equations: The parameters C L associated with the left bound and C R associated with the right bound of the parametric interval can be found by using Eq. ( 8) as follows: , therefore given by ( 9), minimize is the nearest parametric interval approximation of fuzzy number A with respect to metric p d .

Remark 3
If, in distribution density function However, this article wants to approximate a fuzzy number by a crisp interval.Thus, the researchers have to use an operator , respectively.Then . It shows that our parametric interval approximation is continuous interval approximation.In other words, if A and B are close enough, then their parametric interval approximations obtained by p d C are also close enough.

Application
In this section, the researchers introduce some of applications of the weighted interval approximation of fuzzy numbers.These indices can be applied for comparison of fuzzy numbers namely fuzzy correlations in fuzzy environments and expert's systems.

Correlation between fuzzy numbers
In many applications the correlation between fuzzy Saneifard and Hassasi 3413 numbers is of interest.Several authors have proposed different measures of correlation between membership functions, intuitionistic fuzzy sets and correlation [Carlsson et al., 2005;Bustince and Burillo, 1995].Hung and Wu [2002] defined a correlation by means of expected interval.They defined the correlation coefficient between fuzzy numbers A and B as follows: where This correlation coefficient shows not only the degree of relationship between the fuzzy numbers but also whether these fuzzy numbers are positively or negatively related.
The researchers extend (11) by interval and gives us more information compared to correlation coefficient in [Sezer, 2012;Zimmermann, 1991] and some others, which lie within   1 , 0 .It has all mentioned properties from correlation coefficient introduced by Zimmermann [1991]; the researchers review properties ) , ( B A p  as follows:

Proposition
For any fuzzy numbers A B , we have:

Proof
The proof is obvious.

Example
and The above example shows that the parametric function interacts on the correlation coefficient between two fuzzy numbers such that for large values of k this article has: G with triangular membership functions as follows:   [Wang and Li, 2007;Bustince and Burillo, 1995;Yu, 1993] have no decreasing behavior.For our method weighted function f interacts on correlation value between two fuzzy numbers.Hence, there is reasonable advantage in using the proposed formula (13).

Conclusion
In this study, the researchers introduce the nearest weighted interval approximation of a fuzzy number.A new type of correlation coefficient between fuzzy numbers is proposed and it is compared with some other methods.These indices can be used in many times series and regression models in a fuzzy environment.

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The researchers use the parametric correlation

Table 1 .
Summary of the result of comparison.
B be two triangular fuzzy numbers, whenever