Correlation Measures of Dual Hesitant Fuzzy Sets

The dual hesitant fuzzy sets (DHFSs) were proposed by Zhu et al. (2012), which encompass fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets as special cases. Correlation measures analysis is an important research topic. In this paper, we define the correlation measures for dual hesitant fuzzy information and then discuss their properties in detail. One numerical example is provided to illustrate these correlationmeasures.Thenwe present a direct transfer algorithmwith respect to the problem of complex operation ofmatrix synthesis when reconstructing an equivalent correlationmatrix for clustering DHFSs. Furthermore, we prove that the direct transfer algorithm is equivalent to transfer closure algorithm, but its asymptotic time complexity and space complexity are superior to the latter. Another real world example, that is, diamond evaluation and classification, is employed to show the effectiveness of the association coefficient and the algorithm for clustering DHFSs.


Introduction
Correlation indicates how well two variables move together in a linear fashion.In other words, correlation reflects a linear relationship between two variables.It is an important measure in data analysis, in particular in decision making, medical diagnosis, pattern recognition, and other real world problems [1][2][3][4][5][6][7].Zadeh [8] introduced the concept of fuzzy sets (FSs) whose basic component is only a membership function with the nonmembership function being one minus the membership function.In fuzzy environments, Hung and Wu [9] used the concept of "expected value" to define the correlation coefficient of fuzzy numbers, which lies in [−1 , 1].Hong [10] considered the computational aspect of the   -based extension principle when the principle is applied to a correlation coefficient of - fuzzy numbers and gave the exact solution of a fuzzy correlation coefficient without programming or the aid of computer resources.Atanassov [11,12] gave a generalized form of fuzzy set, called intuitionistic fuzzy set (IFS), which is characterized by a membership function and a non-membership function.In intuitionistic fuzzy environments, Gerstenkorn and Mańko [13] defined a function measuring the correlation of IFSs and introduced a coefficient of such a correlation.Bustince and Burillo [14] introduced the concepts of correlation and correlation coefficient of interval-valued intuitionistic fuzzy sets (IVIFSs) [12].Hung [15] and Mitchell [16] derived the correlation coefficient of intuitionistic fuzzy sets from a statistical viewpoint by interpreting an intuitionistic fuzzy set as an ensemble of ordinary fuzzy sets.Hung and Wu [17] proposed a method to calculate the correlation coefficient of intuitionistic fuzzy sets by means of "centroid." Xu [18] gave a detailed survey on association analysis of intuitionistic fuzzy sets and pointed out that most existing methods deriving association coefficients cannot guarantee that the association coefficient of any two intuitionistic fuzzy sets equals one if and only if these two intuitionistic fuzzy sets are the same.Szmidt and Kacprzyk [5] discussed a concept of correlation for data represented as intuitionistic fuzzy set adopting the concepts from statistics and proposed a formula for measuring the correlation coefficient (lying in [−1, 1]) of intuitionistic fuzzy sets.Robinson and Amirtharaj [19] defined the correlation coefficient of interval vague sets lying in the interval [0, 1] and proposed a new method for computing the correlation coefficient of interval vague sets lying in the interval [−1, 1] using a-cuts over the vague degrees through statistical confidence intervals which is presented by an example.Instead of using point-based membership as in fuzzy sets, interval-based membership is used in a vague set.In [20], Robinson and Amirtharaj presented a detailed comparison between vague sets and intuitionistic fuzzy sets and defined the correlation coefficient of vague sets through simple examples.Hesitant fuzzy sets (HFSs) were originally introduced by Torra [21,22].In hesitant fuzzy environments, Chen et al. [23] derived some correlation coefficient formulas for HFSs and applied them to two real world examples by using clustering analysis under hesitant fuzzy environments.Xu and Xia [24] defined the correlation measures for hesitant fuzzy information and then discussed their properties in detail.
Recently, Dubois and Prade introduced the definition of dual hesitant fuzzy set.Dual hesitant fuzzy set can reflect human's hesitance more objectively than the other classical extensions of fuzzy set (intuitionistic fuzzy set, type-2 fuzzy set (T-2FS) [25], hesitant fuzzy set, etc.).The motivation to propose the DHFSs is that when people make a decision, they are usually hesitant and irresolute for one thing or another which makes it difficult to reach a final agreement.They further indicated that DHFSs can better deal with the situations that permit both the membership and the nonmembership of an element to a given set having a few different values, which can arise in a group decision making problem.For example, in the organization, some decision makers discuss the membership degree 0.6 and the nonmembership 0.3 of an alternative  that satisfies a criterion .Some possibly assign (0.8, 0.2), while the others assign (0.7, 0.2).No consistency is reached among these decision makers.Accordingly, the difficulty of establishing a common membership degree and a non-membership degree is not because we have a margin of error (intuitionistic fuzzy set) or some possibility distribution values (type-2 fuzzy set), but because we have a set of possible values (hesitant fuzzy set).For such a case, the satisfactory degrees can be represented by a dual hesitant fuzzy element {(0.6, 0.8, 0.7), (0.3, 0.2)}, which is obviously different from intuitionistic fuzzy number (0.8, 0.2) or (0.7, 0.2) and hesitant fuzzy number {0.6, 0.8, 0.7}.The aforementioned measures, however, cannot be used to deal with the correlation measures of dual hesitant fuzzy information.Thus, it is very necessary to develop some theories for dual hesitant fuzzy sets.However, little has been done about this issue.In this paper, we mainly discuss the correlation measures of dual hesitant fuzzy information.To do this, the remainder of the paper is organized as follows.Section 2 presents some basic concepts related to DHFSs, HFSs, and IFSs.In Section 3, we propose some correlation measures of dual hesitant fuzzy elements, obtain several important conclusions, and given an example to illustrate the correlation measures.In Section 4, we propose a direct transfer clustering algorithm based on DHFSs and then use a numerical example to illustrate our algorithm.Finally, Section 5 concludes the paper with some remarks and presents future challenges.
Definition 2 (see [21,22]).Let  be a fixed set; a hesitant fuzzy set (HFS)  on  is in terms of a function that when applied to  returns a subset of [0, 1], which can be represented as the following mathematical symbol: where ℎ  () is a set of values in [0, 1], denoting the possible membership degrees of the element  ∈  to the set .For convenience, we call ℎ  () a hesitant fuzzy element (HFE).We use ⟨, ℎ  ⟩ for all  ∈  to represent HFSs.
The correlation of the IFSs  and  is defined as [13] Then, the correlation coefficient of the IFSs  and  is defined as In [23], Chen et al. defined the correlation and correlation coefficient for HFSs as follows, respectively: where   = max{(ℎ  (  )), (ℎ  (  ))} for each   in , and (ℎ  (  )) and (ℎ  (  )) represent the number of values in ℎ  (  ) and ℎ  (  ), respectively.We will talk about   in detail in the next section.

Correlation Measures of DHFEs
In this section, we first introduce the concept of correlation and correlation coefficient for DHFSs and then propose several correlation coefficient formulas and discuss their properties.
We arrange the elements in   () = (ℎ  (),   ()) in decreasing order and let  ()  () be the th largest value in ℎ  () and  ()  () the th largest value in   ().Let  ℎ (  (  )) the number of values in ℎ  (  ) and   (  (  )) be the number of values in   (  ).For convenience, ((  )) = ( ℎ ((  )),   ((  ))).In most cases, for two DHFSs  and , ( To operate correctly, we should extend the shorter one until both of them have the same length when we compare them.In [24,27], Xu and Xia extended the shorter one by adding different values in hesitant fuzzy environments.Similarly, Torra [21] also applied this ideal to derive some correlation coefficient formulas for HFSs.In fact, we can extend the shorter one by adding any value in it.The selection of this value mainly depends on the decision makers' risk preferences.Optimists anticipate desirable outcomes and may add the maximum value, while pessimists expect unfavorable outcomes and may add the minimum value.The same situation can also be found in many existing references [13,14].
We define several correlation coefficients for DHFEs.
However, from Theorem 7, we notice that all the above correlation coefficients cannot guarantee that the correlation coefficient of any two DHFSs equals one if and only if these two DHFSs are the same.Thus, how to derive the correlation coefficients of the DHFSs satisfying this desirable property is an interesting research topic.To solve this issue, in what follows, we develop a new method to calculate the correlation coefficient of the DHFSs  and .Definition 9.For two DHFSs  and  on , the correlation coefficient of  and , denoted as  DHFS 3 (, ), is defined by where Equation ( 17) is motivated by the generalized idea provided by Xu [18].Obviously, the greater the value of  DHFS 3 (, ), the closer  to .By Definition 9, we have Theorem 10.
Usually, in practical applications, the weight of each element   ∈  should be taken into account, and, so, we present the following weighted correlation coefficient.Assume that the weight of the element   ∈  is   ( = 1, 2, . . ., ) with   ∈ [0, 1] and ∑  =1   = 1; then we extend the correlation coefficient formulas given: where Note that all these formulas satisfy the properties in Theorem 7.
In what follows, we use a medical diagnosis problem in [28,29] to illustrate the developed correlation coefficient formulas.Actually, this is also a pattern recognition problem.
Example 11.To make a proper diagnosis  = { 1 (viral fever),  2 (malaria),  3 (typhoid),  4 (stomach problem), and  5 (chest problem)} for a patient with the given values of the symptoms,  = { 1 (temperature),  2 (headache),  3 (cough),  4 (stomach pain), and  5 (chest pain)}, Xu [18] considered all possible diagnoses and symptoms as HFEs.Utilizing DHFSs can take much more information into account; the more values we obtain from patients, the greater epistemic certainty we have.So, in this paper, we use DHFEs to deal with such cases; each symptom is described by a DHFE, which is described by two sets (  ) and (  ).(  ) indicates the degree that symptoms characteristic   satisfies the considered diagnoses   and (  ) indicates the degree that the symptoms characteristic   does not satisfy the considered diagnoses   .The data are given in Table 1.The set of patients is  = {Al, Bob, Joe, Ted}.The symptoms which can be also described by DHFEs are given in Table 2.We need to seek a diagnosis for each patient.
We utilize the correlation coefficient  DHFS1 to derive a diagnosis for each patient.All the results for the considered patients are listed in Table 3. From the arguments in Table 3, we can find that Ted suffers from viral fever, Al and Joe from malaria, and Bob from stomach problem.
If we utilize the correlation coefficient formulas  DHFS2 and  DHFS3 to derive a diagnosis, then the results are listed in Tables 4 and 5, respectively.
From Tables 3-5 we know that the results obtained by different correlation coefficient formulas are different.That is because these correlation coefficient formulas are based on different linear relationships.

Clustering Method Based on Direct Transfer Algorithm for HFSs
Based on clustering algorithms for IFSs [30,31], and HFSs [23] For a confidence level , when we get that   and   are of the same type using the direct transfer algorithm, we can also have the same clustering results by the transfer closure algorithm.
( For a confidence level , when we get   and   are of the same type using the transfer closure algorithm, we can also have the same clustering results by the direct transfer algorithm, which completes the proof.
We assume  = { 1 ,  2 , . . .,   } to be a set of DHFSs, and we construct the equivalent correlation matrix  2 and then construct a -cutting matrix   = (    ) × for the transfer closure algorithm.Consequently, the running time of the transfer closure algorithm is  tca = ( 3 +  2 ); by the same arguments, the direct transfer algorithm requires  dta = ( 2 ) time on the same example.And we have established  tca = ( 2 ) space bound at least for the step of constructing the equivalent correlation matrix based on the transfer closure algorithm, while, for the transfer algorithm, it constructs a -cutting matrix   = (    ) × by setting the threshold to the confidence level  and needs  tca = () space bound.We can see that the computational complexity of both two algorithms ranges depends on the number of , and the direct transfer algorithm exhibits better behavior.
Below, we conduct experiments in order to demonstrate the effectiveness of the proposed clustering algorithm for DHFSs.
Example 21.Every diamond is a miracle of time and place and chance.Like snowflakes, no two are exactly alike.Every consumer shopping for diamonds is faced with endless diamond combinations.In addition to different diamond combinations, prices are also influenced by market supply and demand conditions, fashion trends, and so forth.While  the clustering results have much to do with the threshold; the smaller the confidence level is, the more detailed the clustering will be.

Conclusions
Dual hesitant fuzzy set, as an extension of fuzzy set, can describe the situation that people have hesitancy when they make a decision more objectively than other extensions of fuzzy set (interval-valued fuzzy set, intuitionistic fuzzy set, type-2 fuzzy set, and fuzzy multiset).In this paper, the correlation coefficients for DHFSs have been studied.Their properties have been discussed, and the differences and correlations among them have been investigated in detail.We have made the clustering analysis under dual hesitant fuzzy environments with one typical real world example.To further extend the application range of the present clustering algorithm, in particular for the case that needs to assign weights for different experts, it will be necessary to generalize the original definition of DHFSs.
Given that DHFSs are a suitable technique of denoting uncertain information that is widely encountered in daily life and the latent applications of our algorithm in the field of data