# A New Correlation Coefficient forT-Spherical Fuzzy Sets and Its Application in Multicriteria Decision-Making and Pattern Recognition

Authors: Wajid Ali; Mehboob Ali; Israr Hussain; Syed Sajid Ullah; Roobaea Alroobaea; Saddam Hussain; Ahmed Binmahfoudh; Fazlullah Umar
Journal: Journal of Sensors
 (2022)

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## Abstract

The goal of this paper is to design a new correlation coefficient forT-spherical fuzzy sets (TSFSs), which can accurately measure the nature of correlation (i.e., positive and negative) as well as the degree of relationship between TSFS. In order to formulate our proposed idea, we had taken inspiration from the statistical concept of the correlation coefficient. While doing so, we firstly introduce the variance and covariance of two TSFS and then constructed our scheme using these two newly defined notions. The numerical value of our proposed correlation coefficient lies within the interval −1,+1, as it should be from a statistical point of view, whereas the existing methods cannot measure the negative correlation between TSFS, as their numerical value falls within the interval 0,1, which is not reasonable both statistically and intuitively. This aspect has also been thoroughly demonstrated using some numerical examples. The comparison results witnessed the dominance and upper hand of our proposed method over the existing definitions, with reliable and better results. In order to demonstrate the feasibility, usefulness, and practical application, we applied our proposed scheme to solve technical and scientific problems of multicriteria decision-making and pattern recognition. The numerical results show that our proposed scheme is practically suitable, technically applicable, and intuitively reasonable.


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## Body

## 1. Introduction

The notion of fuzzy sets (FS) was initially proposed by Zadeh [1] to deal with imprecise and vague situations where the membership MAx is assigned to each element [2, 3], and it is a generalization of the crisp set [4]. In crisp set theory, we consider only deterministic and precise situations that could not handle imprecise and vague situations. Alternatively, fuzzy sets provide better solutions for real-life problems. Various researchers have developed certain novel techniques related to fuzzy sets and demonstrated their application in other fields such as [5–14] that are some practical examples. In a fuzzy set, however, only the degree of membership MAx is of relevance to research, and nonmembership NAx is not an option. In light of this, Atanassov [15] developed the notion of intuitionistic fuzzy sets (IFS), which takes into account both membership and nonmembership functions under a particular condition: 0≤MAx+NAx≤1, where 0≤MAx≤1 and 0≤NAx≤1, and the term τAx=1−MAx−NAx is called degree of hesitancy. IFS has a wide range of application in many fields, e.g., decision-making [16], logic programming [17], pattern recognition [18–20] medical diagnosis [21], information retrieval [22] and cluster analysis [23], and communication [24–26]. Although the theory of IFS has been successfully applied in different fields, there are some real-life situations where human opinions involve more than two independent statements, like yes, no, abstain, and refusal. Therefore, to deal with these types of situations, Cuong and Kreinovich [27] extended Zadeh FS and Atanassov’s IFS into picture fuzzy sets (PFS) with the following condition MAx+IAx+NAx≤1, where MAx∈0,1 is the degree of positive membership, IAx∈0,1 which represents the degree of neutral membership, and NAx∈0,1 is a degree of negative membership. The proposed idea of Cuong and Kreinovich [27] has great importance and application to deal with human opinion efficiently as we have discussed earlier but in some particular scenarios, the sum of membership degrees exceeds 1, i.e., MAx+IAx+NAx>1. In these situations, IFS is insufficient to provide a satisfactory result. To address this problem, Mahmood et al. [28] developed the spherical fuzzy set (SFS) by adding new constraints: MA2x+IA2x+NA2x≤1forSFSs and MAnx+IAnx+NAnx≤1forTSFSs which is a generalization of FS, IFS, and PFS, which satisfies the sum of membership degrees is less than or equal to 1. The concept of TSFS is a recent development in fuzzy set theory and successfully applied to medical diagnosis and multiattribute decision-making. In addition, the extensions of IFS method are also found in literature abundantly such as fuzzy multisets [29], type-2 fuzzy sets [30], hesitant fuzzy sets [31], and Pythagorean fuzzy sets, [32].earson proposed the concept of correlation in 1895, and it has since become one of the most widely used indices in statistics. Correlation is a statistical term that describes how two variables are related in a linear pattern. Murthy et al. [33] introduced the concept of fuzzy set correlation for the first time in 1985. The correlation coefficient for IFS was proposed by Gerstenkorn and Manko [34] who introduced the correlation coefficient for IFS. Later on, Hong and Hwang [35] went on to extend intuitionistic fuzzy sets’ association in probability space. Mitchell [36] illustrated new formula for intuitionistic fuzzy sets and interpreted two intuitionistic fuzzy sets as ensembles of ordinary fuzzy sets. Several novel expansions, such as [23, 37–41], have also been proposed in the literature. Ullah et al. [42] recently discovered correlation coefficients for T-spherical fuzzy sets, which they used in clustering and multiattribute decision-making. Guleria and Bajaj [43] also presented a similar idea of the correlation coefficient for TSFSs and illustrated its application in pattern recognition and medical diagnosis. These newly developed ideas are applicable in many areas but their proposed method only measures [0, 1] ranging correlations and is unable to deal with negative correlation. Keeping this drawback in mind, we propose a new correlation coefficient for TSFSs that can easily address the highlighted weaknesses in the existing methods. The numerical value of our proposed correlation coefficient lies within the interval −1,+1 as it should be from a statistical point of view, whereas the existing methods cannot measure the negative correlation between TSFSs, as their numerical value falls within the interval 0,1, which is not reasonable both statistically and intuitively.The organization of this manuscript has been furnished as follows. Section2 entails basic concepts of IFS, PFS, SFS, and TSFS. It also reviews some existing ideas of the CC of TSFS. Section 3 details our proposed scheme (CC of TSFS), and it also validates and verifies our method with the help of some theorems and propositions. In Section 4, we demonstrate numerical comparisons regarding the performance of our method and the other existing methods to show the strengths and superiority of the proposed idea. Section 5 illustrates the feasibility and usefulness of our idea in the thematic areas of MCDM and PR with the help of some real-world problems. In Section 6, we finalize this manuscript with concluding remarks.


## 2. Preliminaries

This section has been dedicated to recalling some essential theories, functions, and characteristics of TSFSs as well as the coefficient of correlation to facilitate the understanding of our research study. In the entire paper, we useMx, Nx, Ix, and Λx to represent the degree of membership, nonmembership, abstinence, and refusal in a unit interval [0, 1], respectively.


### 2.1. Generalizations of Fuzzy Sets

Definition 1. (see [15]).
LetF be an intuitionistic fuzzy set (IFS) on a universe of discourse X=x1,x2,x3,⋯,xm, and then it can be algebraically defined as F=x,MFx,NFx/x∈X, with a constraint of 0≤MFx+NFx≤1, whereas H=1−MFx+NFx is known as the degree of hesitancy for an element x∈X to be belonging toF.Definition 2. (see [44]).
A mathematical expression for a PFSP defined over a universe of discourse X=x1,x2,x3,⋯,xm can be presented as J=x,MJx,IJx,NJx/x∈X, with a constraint of 0≤MJx+IJx+NJx≤1, where the degree of refusal for an element x∈X to be in J is represented as Λx=1−MJx+IJx+NJx.Definition 3. (see [45]).
The standard negation set of the PFSJ can be expressed as NJ=x,NJx,ΛJx,MJx/x∈X.Definition 4. (see [28]).
LetS be an SFS on a universe of discourse X=x1,x2,x3,⋯,xm, and then it can be mathematically defined as S=x,MSx,ISx,NSx/x∈X, with a constraint of 0≤MS2x+IS2+NS2x≤1, while the degree of refusal for an element x∈X to be belonging to S can be expressed as Λx=1−MS2x+IS2+NS2x.Definition 5. (see [28]).
For a TSFST on a universe of discourse X=x1,x2,x3,⋯,xm, its mathematical representation can be expressed as T=x,MTx,ITx,NTx/x∈X, with a constraint of 0≤MTnx+ITn+NTnx≤1. The degree of refusal for an element x∈X to be in TSFS T is defined as ΛTx=1−MTnx+ITn+NTnxn.


### 2.2. Existing Methods (CC of TSFSs)

Definition 6. (see [42]).
LetS1 and S2 be two TSFSs in a universe of discourse X=x1,x2,x3,⋯,xm, and the coefficient of correlation between TSFSs S1 and S2 is
(1)RS1,S2=∑i=1mMS1nxi×MS2nxi+IS1nxi×IS2nxi+NS1nxi×NS2nxi+ΛS1nxi×ΛS2nxi∑i=1mMS12xin+IS12xin+NS12xin+ΛS12xin×∑i=1mMS22xin+IS22xin+NS22xin+ΛS22xin,
while the correlation coefficient for weighted TSFSs is(2)RwS1,S2=∑i=1mwiMS1nxi×MS2nxi+IS1nxi×IS2nxi+NS1nxi×NS2nxi+ΛS1nxi×ΛS2nxi∑i=1mwiMS12xin+IS12xin+NS12xin+ΛS12xin×∑i=1mwiMS22xin+IS22xin+NS22xin+ΛS22xin.
Both of the above definitions (1) and (2) satisfy the following features:
(a)
RS1,S2=RS2,S1(b)
0≤RS1,S2≤1(c)
RS1,S2=1,iffS1=S2Definition 7. (see [42]).
Another form of correlation coefficient between two TSFSsS1 and S2 in X=x1,x2,x3,⋯,xm is
(3)R∗S1,S2=∑i=1mMS1nxi×MS2nxi+IS1nxi×IS2nxi+NS1nxi×NS2nxi+ΛS1nxi×ΛS2nximax∑i=1mMS12xin+IS12xin+NS12xin+ΛS12xin,∑i=1mMS22xin+IS22xin+NS22xin+ΛS22xin,
whereas its weighted form can be defined as(4)Rw∗S1,S2=∑i=1mwiMS1nxi×MS2nxi+IS1nxi×IS2nxi+NS1nxi×NS2nxi+ΛS1nxi×ΛS2nximax∑i=1mwiMS12xin+IS12xin+NS12xin+ΛS12xin,∑i=1mwiMS22xin+IS22xin+NS22xin+ΛS22xin.
Equations (3) and (4) also satisfy the following properties:
(a)
R∗S1,S2=R∗S2,S1(b)
0≤R∗S1,S2≤1(c)
R∗S1,S2=1,iffS1=S2Definition 8. (see [43]).
Coefficient of correlation between TSFSsS1 and S2 in X=x1,x2,x3,⋯,xm is defined as
(5)KS1,S2=∑i=1mMS1nxi×MS2nxi+IS1nxi×IS2nxi+NS1nxi×NS2nxi+ΛS1nxi×ΛS2nxi∑i=1mMS1nxi2+IS1nxi2+NS1nxi2+ΛS1nxi21/2×∑i=1mMS2nxi2+IS2nxi2+NS2nxi2+ΛS2nxi21/2,
while the correlation coefficient for weighted TSFSs is(6)KwS1,S2=∑i=1mwiMS1nxi×MS2nxi+IS1nxi×IS2nxi+NS1nxi×NS2nxi+ΛS1nxi×ΛS2nxi∑i=1mwiMS1nxi2+IS1nxi2+NS1nxi2+ΛS1nxi21/2×∑i=1mwiMS2nxi2+IS2nxi2+NS2nxi2+ΛS2nxi21/2.Definition 9. (see [43]).
Another definition of the correlation coefficient between two TSFSsS1 and S2 is
(7)K∗S1,S2=∑i=1mMS1nxi×MS2nxi+IS1nxi×IS2nxi+NS1nxi×NS2nxi+ΛS1nxi×ΛS2nximax∑i=1mMS12xin+IS12xin+NS12xin+ΛS12xin,∑i=1mMS22xin+IS22xin+NS22xin+ΛS22xin,
whereas its weighted form can be defined as(8)Kw∗S1,S2=∑i=1mwiMS1nxi×MS2nxi+IS1nxi×IS2nxi+NS1nxi×NS2nxi+ΛS1nxi×ΛS2nximax∑i=1mwiMS12xin+IS12xin+NS12xin+ΛS12xin,∑i=1mwiMS22xin+IS22xin+NS22xin+ΛS22xin.
Equations (5)–(8) satisfy the following properties:
(a)
KS1,S2=KS2,S1(b)
0≤KS1,S2≤1(c)
KS1,S2=1,iffS1=S2


## 2.1. Generalizations of Fuzzy Sets

Definition 1. (see [15]).
LetF be an intuitionistic fuzzy set (IFS) on a universe of discourse X=x1,x2,x3,⋯,xm, and then it can be algebraically defined as F=x,MFx,NFx/x∈X, with a constraint of 0≤MFx+NFx≤1, whereas H=1−MFx+NFx is known as the degree of hesitancy for an element x∈X to be belonging toF.Definition 2. (see [44]).
A mathematical expression for a PFSP defined over a universe of discourse X=x1,x2,x3,⋯,xm can be presented as J=x,MJx,IJx,NJx/x∈X, with a constraint of 0≤MJx+IJx+NJx≤1, where the degree of refusal for an element x∈X to be in J is represented as Λx=1−MJx+IJx+NJx.Definition 3. (see [45]).
The standard negation set of the PFSJ can be expressed as NJ=x,NJx,ΛJx,MJx/x∈X.Definition 4. (see [28]).
LetS be an SFS on a universe of discourse X=x1,x2,x3,⋯,xm, and then it can be mathematically defined as S=x,MSx,ISx,NSx/x∈X, with a constraint of 0≤MS2x+IS2+NS2x≤1, while the degree of refusal for an element x∈X to be belonging to S can be expressed as Λx=1−MS2x+IS2+NS2x.Definition 5. (see [28]).
For a TSFST on a universe of discourse X=x1,x2,x3,⋯,xm, its mathematical representation can be expressed as T=x,MTx,ITx,NTx/x∈X, with a constraint of 0≤MTnx+ITn+NTnx≤1. The degree of refusal for an element x∈X to be in TSFS T is defined as ΛTx=1−MTnx+ITn+NTnxn.


## 2.2. Existing Methods (CC of TSFSs)

Definition 6. (see [42]).
LetS1 and S2 be two TSFSs in a universe of discourse X=x1,x2,x3,⋯,xm, and the coefficient of correlation between TSFSs S1 and S2 is
(1)RS1,S2=∑i=1mMS1nxi×MS2nxi+IS1nxi×IS2nxi+NS1nxi×NS2nxi+ΛS1nxi×ΛS2nxi∑i=1mMS12xin+IS12xin+NS12xin+ΛS12xin×∑i=1mMS22xin+IS22xin+NS22xin+ΛS22xin,
while the correlation coefficient for weighted TSFSs is(2)RwS1,S2=∑i=1mwiMS1nxi×MS2nxi+IS1nxi×IS2nxi+NS1nxi×NS2nxi+ΛS1nxi×ΛS2nxi∑i=1mwiMS12xin+IS12xin+NS12xin+ΛS12xin×∑i=1mwiMS22xin+IS22xin+NS22xin+ΛS22xin.
Both of the above definitions (1) and (2) satisfy the following features:
(a)
RS1,S2=RS2,S1(b)
0≤RS1,S2≤1(c)
RS1,S2=1,iffS1=S2Definition 7. (see [42]).
Another form of correlation coefficient between two TSFSsS1 and S2 in X=x1,x2,x3,⋯,xm is
(3)R∗S1,S2=∑i=1mMS1nxi×MS2nxi+IS1nxi×IS2nxi+NS1nxi×NS2nxi+ΛS1nxi×ΛS2nximax∑i=1mMS12xin+IS12xin+NS12xin+ΛS12xin,∑i=1mMS22xin+IS22xin+NS22xin+ΛS22xin,
whereas its weighted form can be defined as(4)Rw∗S1,S2=∑i=1mwiMS1nxi×MS2nxi+IS1nxi×IS2nxi+NS1nxi×NS2nxi+ΛS1nxi×ΛS2nximax∑i=1mwiMS12xin+IS12xin+NS12xin+ΛS12xin,∑i=1mwiMS22xin+IS22xin+NS22xin+ΛS22xin.
Equations (3) and (4) also satisfy the following properties:
(a)
R∗S1,S2=R∗S2,S1(b)
0≤R∗S1,S2≤1(c)
R∗S1,S2=1,iffS1=S2Definition 8. (see [43]).
Coefficient of correlation between TSFSsS1 and S2 in X=x1,x2,x3,⋯,xm is defined as
(5)KS1,S2=∑i=1mMS1nxi×MS2nxi+IS1nxi×IS2nxi+NS1nxi×NS2nxi+ΛS1nxi×ΛS2nxi∑i=1mMS1nxi2+IS1nxi2+NS1nxi2+ΛS1nxi21/2×∑i=1mMS2nxi2+IS2nxi2+NS2nxi2+ΛS2nxi21/2,
while the correlation coefficient for weighted TSFSs is(6)KwS1,S2=∑i=1mwiMS1nxi×MS2nxi+IS1nxi×IS2nxi+NS1nxi×NS2nxi+ΛS1nxi×ΛS2nxi∑i=1mwiMS1nxi2+IS1nxi2+NS1nxi2+ΛS1nxi21/2×∑i=1mwiMS2nxi2+IS2nxi2+NS2nxi2+ΛS2nxi21/2.Definition 9. (see [43]).
Another definition of the correlation coefficient between two TSFSsS1 and S2 is
(7)K∗S1,S2=∑i=1mMS1nxi×MS2nxi+IS1nxi×IS2nxi+NS1nxi×NS2nxi+ΛS1nxi×ΛS2nximax∑i=1mMS12xin+IS12xin+NS12xin+ΛS12xin,∑i=1mMS22xin+IS22xin+NS22xin+ΛS22xin,
whereas its weighted form can be defined as(8)Kw∗S1,S2=∑i=1mwiMS1nxi×MS2nxi+IS1nxi×IS2nxi+NS1nxi×NS2nxi+ΛS1nxi×ΛS2nximax∑i=1mwiMS12xin+IS12xin+NS12xin+ΛS12xin,∑i=1mwiMS22xin+IS22xin+NS22xin+ΛS22xin.
Equations (5)–(8) satisfy the following properties:
(a)
KS1,S2=KS2,S1(b)
0≤KS1,S2≤1(c)
KS1,S2=1,iffS1=S2


## 3. Proposed New Correlation Coefficient of TSFSs

We propose a new correlation coefficient(s) of TSFSs, including both the weighted and unweighted environments, to overcome the drawbacks of the existing definitions [37, 38]. It has been formulated using a statistical theory of correlation coefficient between two phenomena, that is, the mathematical ratio between the covariance of two TSFSs and the geometric mean of their individual variances. Initially, we introduce the variances and covariance of two TSFSs and use those new definitions to construct our proposed correlation coefficient.Motivated from [46], we designed the structure of our proposed new correlation coefficient of TSFSs, demonstrated as follows.LetT1 and T2 be two TSFSs X=x1,x2,x3,⋯,xm, and we describe two new variables based on the deviations of membership grades as
(9)QiT1=MT1nxi−M¯T1nxi−IT1nxi−I¯T1nxi−NT1nxi−N¯T1nxi+ΛT1nxi−Λ¯T1nxi,QiT2=MT2nxi−M¯T2nxi−IT2nxi−I¯T2nxi−NT2nxi−N¯T2nxi+ΛT2nxi−Λ¯T2nxi,∀xi∈X,where(10)M¯T1nxi=1m∑i=1mMT1nxi,I¯T1nxi=1m∑i=1mIT1nxi,N¯T1nxi=1m∑i=1mNT1nxi,Λ¯T1nxi=1m∑i=1mΛT1nxi,M¯T2nxi=1m∑i=1mMT2nxi,I¯T2nxi=1m∑i=1mIT2nxi,N¯T2nxi=1m∑i=1mNT2nxi,andΛ¯T2nxi=1m∑i=1mΛT2nxi.Definition 10.
For two TSFSsT1=x,MT1nx,IT1nx,NT1nx/x∈X and T2=x,MT2nx,IT2nx,NT2nx/x∈X, variances in terms of new variables can be defined as
(11)VarT1=1m−1∑i=1mQi2T1,VarT2=1m−1∑i=1mQi2T2,
while their covariance is mathematically expressed as(12)CovT1,T2=1m−1∑i=1mQiT1×QiT2.Proposition 11.
ForCovT1,T2 to be the covariance of two TSFSs T1 and T2, it should satisfy the following conditions:
(13)CovT1,T2=CovT2,T1CovT1,T1=VarT1,CovT1,T1≤VarT11/2×VarT21/2Proof.
i and ii are straight forward
iii Using Cauchy-Schwarz inequality, we have
(14)∑i=1naibi2≤∑i=1nai2×∑i=1nbi2or∑i=1naibi≤∑i=1nai21/2×∑i=1nbi21/2,HenceCovT1,T22=1m−1∑i=1mQiT1×NiT22=1m−12∑i=1mQiT1×NiT22,CovT1,T22≤1m−12∑i=1mQi2T1×∑i=1mQi2T2,CovT1,T22≤1m−1∑i=1mQi2T1×1m−1∑i=1mQi2T2,CovT1,T22≤VarT1×VarT2,CovT1,T2≤VarT11/2×VarT21/2.Definition 12.
LetT1 and T2 be two TSFSs on a universe of discourse X=x1,x2,x3,⋯,xm. The correlation coefficient between T1 and T2 is defined as
(15)ρT1,T2=CovT1,T2VarT1×VarT2,ρT1,T2=∑i=1mQiT1×QiT2∑i=1mQi2T1×∑i=1mQi2T2.Theorem 13.
We have two TSFSsT1 and T2, defined over the universe of discourse X=x1,x2,x3,⋯,xm, and then the correlation coefficient between them must fulfill the following conditions:
(a)
ρT1,T2=ρT2,T1(b)
−1≤ρT1,T2≤+1(c)
ρT1,T2=1, if T1=cT2 for any constant c>0, where T1=cT2 means, MT1nx=cMT2nx∈0,1, IT1nx=cIT2nx∈0,1, NT1nx=cNT2nx∈0,1, and ΛT1nx=cΛT2nx∈0,1(d)
ρT1,T2=−1, if T1=cT2, for any constant c<0Proof.
(a)
It is straightforward(b)
From proposition11 (14), we know that CovT1,T2≤VarT11/2×VarT21/2; hence,(16)−VarT11/2×VarT21/2≤CovT1,T2.
By using these two inequalities,(17)−VarT11/2×VarT21/2≤CovT1,T2≤VarT11/2×VarT21/2,−1≤CovT1,T2VarT11/2×VarT21/2≤1−1≤ρT1,T2≤+1.(c)
Since for any constantc>0, MT1nx=cMT2nx, IT1nx=cIT2nx, NT1nx=cNT2nx, and ΛT1nx=cΛT2nx, hence(18)QiT1=MT1nxi−M¯T1nxi−IT1nxi−I¯T1nxi−NT1nxi−N¯T1nxi+ΛT1nxi−Λ¯T1nxi,QiT1=cMT2nxi−cM¯T2nxi−cIT2nxi−cI¯T2nxi−cNT2nxi−cN¯T2nxi+cΛT2nxi−cΛ¯T2nxi,QiT1=cMT2nxi−M¯T2nxi−IT2nxi−I¯T2nxi−NT2nxi−N¯T2nxi+ΛT2nxi−Λ¯T2nxi,QiT1=cQiT2.
Therefore,CovT1,T2=1/m−1∑i=1mQiT1×QiT2=1/m−1∑i=1mcQiT2×QiT2.
(19)CovT1,T2=1m−1∑i=1mcQi2T2,
CovT1,T2=cVarT2,
(20)VarT1=1m−1∑i=1mQi2T1=1m−1∑i=1mc2Qi2T2,VarT1=c2VarT2.
Hence, for any constantc>0,
(21)ρT1,T2=CovT1,T2VarT1×VarT2=kVarT2k2VarT2×VarT2,ρT1,T2=kVarT2kVarT22=kVarT2kVarT2,ρT1,T2=1.(d)
SinceA=kB, so for any constant c<0, MT1nx=−cMT2nx, IT1nx=−cIT2nx, NT1nx=−cNT2nx, and ΛT1nx=−cΛT2nx, therefore,(22)QiT1=MT1nxi−M¯T1nxi−IT1nxi−I¯T1nxi−NT1nxi−N¯T1nxi+ΛT1nxi−Λ¯T1nxi,QiT1=−cMT2nxi+cM¯T2nxi−−cIT2nxi+cI¯T2nxi−−cNT2nxi+N¯T2nxi+−cΛT2nxi+cΛ¯T2nxi,QiT1=−cMT2nxi−M¯T2nxi+cIT2nxi−I¯T2nxi+cNT2nxi−N¯T2nxi−cΛT2nxi−Λ¯T2nxi,QiT1=−cMT2nxi−M¯T2nxi−IT2nxi−I¯T2nxi−NT2nxi−N¯T2nxi+ΛT2nxi−Λ¯T2nxi,QiT1=−cQiT2.
Hence,(23)CovT1,T2=1m−1∑i=1mQiT1×QiT2=1m−1∑i=1m−cQiT2×QiT2,CovT1,T2=1m−1∑i=1m−cQi2T2,
CovT1,T2=−cVarT2, and
(24)VarT1=1m−1∑i=1mQi2T1=1m−1∑i=1mc2Qi2T2,VarT1=c2VarT2.
So, forc<0,(25)ρT1,T2=CovT1,T2VarT1×VarT2=−cVarT2c2VarT2×VarT2,ρT1,T2=−cVarT2cVarT22=−cVarT2cVarT2,ρT1,T2=−1.


### 3.1. Special Cases

(a)
Whenn=2, equation (15) reduces to the correlation coefficient of SFSs(b)
Whenn=1, equation (15) reduces to the correlation coefficient of PFSs(c)
Whenn=q and IAxi=IBxi=0, equation (15) reduces to the correlation coefficient of Q-ROFSs(d)
Whenn=2 and ΛT1xi=ΛT2xi=0, equation (15) reduces to the correlation coefficient of PyFSs(e)
Whenn=1 and ΛT1xi=ΛT2xi=0, equation (15) reduces to the correlation coefficient of IFSsWe also propose a weighted correlation coefficient between two TSFSs in the following, to deal with the practical problems and situations, with different objects having unequal importance, in which we consider their weights−VarwA×VarwB≤CovwA,B≤VarwA×VarwB keeping in view their relative importance.Definition 14.
Suppose we have two TSFSsT1 and T2 defined over a universe of discourse X=x1,x2,x3,⋯,xm, having a weight vector w=w1,w2,w3,⋯,wm for its elements xi∈X, where wi≥0 and ∑i=1mwi=1, then their weighted correlation coefficient can be defined as
(26)ρwT1,T2=CovwT1,T2VarwT1×VarwT2,ρwT1,T2=∑i=1mwiQiT1×QiT2∑i=1mwiQi2T1×∑i=1mwiQi2T2.If all of the elementsxi∈X have equal weights (i.e.w=1/m,1/m,1/m,⋯,1/m), then equation (26) can be reduced to equation (15).From the above definitions, we derive the following proposition and theorem.Proposition 15.
If we have two SFSsρwA,B=CovwA,B/VarAVarwB=aVarwB/a2VarwBVarwB and ρwA,B=aVarwB/a2VarwB2=aVarwB/aVarwB in a universe of discourse ρwA,B=1, then their weighted covariance satisfies the following conditions:
(i)
CovwA,B=CovwB,A(ii)
CovwA,A=VarwA(iii)
CovwA,B≤VarwA1/2×VarwB1/2Proof.
It can be proved like Proposition11.Theorem 16.
For two TSFSsT1 and T2 defined over X=x1,x2,x3,⋯,xm, then the weighted correlation coefficient between them must satisfy the following properties:
(a)
ρwT1,T2=ρwT2,T1(b)
−1≤ρwT1,T2≤+1(c)
ρwT1,T2=1, if T1=cT2 for any constant c>0(d)
ρwT1,T2=−1, if T1=cT2, for any constant c<0Proof.
Theorem16 can also be proved like Theorem 13.Definition 17.
LetT be a TSFS defined over a universe of discourse X=x1,x2,x3,⋯,xm, and then standard negation set for it be characterized as MNTxi=NTxi, INTxi=ΛTxi and NNTxi=MTxi for all δFmin=miniFAxi−Fxi and can be mathematically expressed as NT=x,NTx,ΛTx,MTx/x∈X, where 0≤MTnx+ITnx+NBnx≤1 and where ΛTx=1−MTnx+ITnx+NTnx is the degree of refusal for an element x∈X to be in SFSs T.


## 3.1. Special Cases

(a)
Whenn=2, equation (15) reduces to the correlation coefficient of SFSs(b)
Whenn=1, equation (15) reduces to the correlation coefficient of PFSs(c)
Whenn=q and IAxi=IBxi=0, equation (15) reduces to the correlation coefficient of Q-ROFSs(d)
Whenn=2 and ΛT1xi=ΛT2xi=0, equation (15) reduces to the correlation coefficient of PyFSs(e)
Whenn=1 and ΛT1xi=ΛT2xi=0, equation (15) reduces to the correlation coefficient of IFSsWe also propose a weighted correlation coefficient between two TSFSs in the following, to deal with the practical problems and situations, with different objects having unequal importance, in which we consider their weights−VarwA×VarwB≤CovwA,B≤VarwA×VarwB keeping in view their relative importance.Definition 14.
Suppose we have two TSFSsT1 and T2 defined over a universe of discourse X=x1,x2,x3,⋯,xm, having a weight vector w=w1,w2,w3,⋯,wm for its elements xi∈X, where wi≥0 and ∑i=1mwi=1, then their weighted correlation coefficient can be defined as
(26)ρwT1,T2=CovwT1,T2VarwT1×VarwT2,ρwT1,T2=∑i=1mwiQiT1×QiT2∑i=1mwiQi2T1×∑i=1mwiQi2T2.If all of the elementsxi∈X have equal weights (i.e.w=1/m,1/m,1/m,⋯,1/m), then equation (26) can be reduced to equation (15).From the above definitions, we derive the following proposition and theorem.Proposition 15.
If we have two SFSsρwA,B=CovwA,B/VarAVarwB=aVarwB/a2VarwBVarwB and ρwA,B=aVarwB/a2VarwB2=aVarwB/aVarwB in a universe of discourse ρwA,B=1, then their weighted covariance satisfies the following conditions:
(i)
CovwA,B=CovwB,A(ii)
CovwA,A=VarwA(iii)
CovwA,B≤VarwA1/2×VarwB1/2Proof.
It can be proved like Proposition11.Theorem 16.
For two TSFSsT1 and T2 defined over X=x1,x2,x3,⋯,xm, then the weighted correlation coefficient between them must satisfy the following properties:
(a)
ρwT1,T2=ρwT2,T1(b)
−1≤ρwT1,T2≤+1(c)
ρwT1,T2=1, if T1=cT2 for any constant c>0(d)
ρwT1,T2=−1, if T1=cT2, for any constant c<0Proof.
Theorem16 can also be proved like Theorem 13.Definition 17.
LetT be a TSFS defined over a universe of discourse X=x1,x2,x3,⋯,xm, and then standard negation set for it be characterized as MNTxi=NTxi, INTxi=ΛTxi and NNTxi=MTxi for all δFmin=miniFAxi−Fxi and can be mathematically expressed as NT=x,NTx,ΛTx,MTx/x∈X, where 0≤MTnx+ITnx+NBnx≤1 and where ΛTx=1−MTnx+ITnx+NTnx is the degree of refusal for an element x∈X to be in SFSs T.


## 4. Comparison with Existing Methods

This section has been dedicated to establishing a full comparative study of the results acquired from our suggested scheme and other current approaches [42, 43] using numerical examples to demonstrate the advantage and novelty of our concept. Using these examples and numerical results, we also intuitively showed the advantages of our proposed method to address the shortcomings of the existing methods. From a statistical as well as intuitive point of view, if two phenomena or datasets are moving in the same direction, then they have a positive relationship, while if their movement is in the opposite direction, then their relationship would be inverse. The correlation coefficient is an important instrument in statistical theory for determining the degree and nature of a link between two processes or variables.So, our definitions of correlation coefficient should accurately measure both the degree and the nature of relationship between two TSFSs.We present some numerical examples in the following to show those characteristics of our proposed scheme and its comparison with existing methods.Example 18.
LetT1 and T2 be two TSFSs in a universe of discourse X=x1,x2,x3, characterized as
(27)T1=x1,0.9,0.5,0.7,x2,0.7,0.6,0.8,x3,0.5,0.8,0.9,T2=x1,0.7,0.6,0.9,x2,0.8,0.5,0.7,x3,0.9,0.4,0.6.By using our new definition, the standard negation set of a TSFST1 is SNT1=x1,0.7,0.,0.9,x2,0.8,0.,0.7,x3,0.9,0.,0.5.The following Table1 represents the comparative results of our proposed method and the existing definitions [37, 38] of the correlation coefficient when applied to the above TSFSs.Table 1
Coefficient of correlation.
ρTi,TjExisting definitionsOur proposed idea[37, 38][37, 38]ρT1,SNT10.37980.3505-0.9865ρT1,T20.43040.4051-0.9859It is intuitively justifiable that the relationship between a TSFS and its negation set should have an inverse relation, with a negative correlation coefficient. It can be observed from the above Table1 that none of the definitions [42, 43] can accurately measure the negative relationship between TSFS T1 and its standard negation set SNT1, rather reflecting a positive relation (0.3798, 0.3505) which cannot be possible, while our proposed correlation coefficient shows that there is a negatively strong correlation (-0.9865) between TSFS T1 and its standard negation set SNT1. Similarly, it can be seen that the corresponding membership grades of TSFSs T1 and T2 are moving in an opposite direction over the various point of the universe of discourse X=x1,x2,x3. So, they should have an inverse relationship between them, with a negative correlation coefficient. It can be clearly seen from the above Table 1 that the value of our proposed correlation coefficient (-0.9859) portrays that there exists a strongly negative correlation between TSFSs T1 and T2, as it should be statistically as well as intuitively. On the other and the existing methods show a positive correlation (0.4304., 0.4051) between them and overlook their inverse relationship. Hence, it shows the superiority and upper hand of our proposed method over the existing definitions of correlation coefficient for TSFSs.Example 19.
Letw=0.22,0.33,0.45 be the weight vector for the elements of TSFSs in Example 18. After applying our proposed weighted correlation coefficient and the existing methods for weighted TSFSs [42, 43], we present the comparative performance results in the following Table 2.Table 2
Weighted correlation coefficient.
ρwTi,TjExisting definitionsOur proposed definition[42, 43][42, 43]ρwT1,SNT10.31530.2850-0.9862ρwT1,T20.37740.3473-0.9859It can also be clearly seen from Table2 that our proposed method accurately measures the negative relationship between TSFSs T1 and its standard negation set SNT1 having a weighted correlation coefficient -0.9862, rationally endorsing the strongly negative correlation between them. Contrarily, the existing definitions [42, 43] show a positive correlation (0.3153, 0.2850) between TSFSs T1 and its standard negation set SNT1, which cannot be logically possible. Moreover, there should be a negative correlation between TSFSs T1 and T1 (as explained in example 18). Table 2 shows that our method demonstrates a negative correlation (-0.9859) between T1 and T1 while the existing methods cannot measure it accurately, as they show a positive correlation coefficient (0.3774, 0.3473), which cannot be intuitively possible, and these shortcomings reduce the reliability and accuracy of the existing definitions. Furthermore, the competency of our proposed method to address these drawbacks shows its effectiveness and advantages.


## 5. Application in Multicriteria Decision-Making and Pattern Recognition

This section has been designated to illustrate the application of our proposed idea of the correlation coefficient for TSFSs in the practical domain of MCDM and pattern recognition to show its suitability and usefulness. Pattern recognition and machine learning are versatile practices that have widely discussed in different disciplines such as [19, 20, 24–26, 47, 48]: these are some examples.


### 5.1. Application in MCDM

This subsection has been planned to show the application of our proposed scheme in the field of multicriteria decision-making by using a real MCDM problem, taken from [28]. To do so, we present here the implementation procedure of our proposed scheme for dealing with an MCDM problem. Suppose there is a set of “k” alternatives defined as TSFSs A=A1,A2,⋯,Ak and a set of “m” evaluation features F=F1,F2,⋯,Fm, having weight vector w=w1,w2,⋯,wmT, 0≤w≤1, and ∑i=1mwi=1, we define the ideal alternative A∗ as a TSFS by taking maximum membership value, minimum nonmembership, and minimum indeterminacy value for each evaluation criteria across all of the given alternatives, like an ideal positive solution in TOPSIS. A∗=Fi,max1≤j≤kMAjFi,min1≤j≤kIAjFi,min1≤j≤kNAjFi/Fi∈F. By applying our proposed correlation coefficient, we calculate the relationship between ideal choice A∗ and the given alternatives. Furthermore, by using the principle of the maximum correlation coefficient, we select an alternative as the best option, which has the highest correlation coefficient with an ideal alternative A∗.Example 20.
In order to strengthen the sales and purchase department, a firm had advertised the position of manager and selected 3 candidates after conducting an initial screening of the prospective candidates, who had applied. For the selection of the best candidate, the top management of the firm had considered the following three features to evaluate the aforementioned initially short-listed candidates. (i)F1: communication skills, (ii) F2: sense of responsibility, and (iii) F3: creativity. The candidates, evaluation criteria, and their weights have been presented in the form of sets as a set of alternatives, set of evaluation criteria, and weights vector, respectively, as follows.
A=A1,A2,A3,F=F1,F2,F3, and w=0.20,0.35,0.45T. After a comprehensive evaluation of the candidates, their decision information has been presented in the following Table 3 as a T-spherical fuzzy decision matrix (TSFDM).Table 3
T-spherical fuzzy information of alternatives Ajj=1,2,3.
AlternativesF1F2F3A1<0.8, 0.5, 0.4><0.7, 0.4, 0.4><0.3, 0.5, 0.4>A2<0.9, 0.2, 0.4><0.6, 0.3, 0.2><0.4, 0.1, 0.7>A3<0.5, 0.5, 0.5><0.8, 0.2, 0.3><0.6, 0.4, 0.3>It can be clearly seen that the decision information in the above Table3 is TSFI n=3. So, we apply our proposed scheme to find a correlation coefficient between the ideal candidate A∗=0.9,0.2,0.4,0.8,0.2,0.2,0.6,0.1,0.3 and the initially short-listed three candidates. The results have been furnished in the following Table 4.Table 4
Association betweenA∗ and Ajj=1,2,3.
ρwA∗,AiA1A2A3ρwA∗,Ai0.97030.8307-0.2530It can be clearly seen that the candidateA1 has the highest degree of relationship with the ideal choice A∗ as compared to that of the other alternatives. Hence, using the principle of maximum correlation, we can say that the candidate A1 is the most suitable choice for the position of manager. This is in complete agreement with the decision, from where this example has been taken.


### 5.2. Application in Pattern Recognition

In this section, we show how we used our proposed theory to solve a pattern recognition problem and recognize an unknown pattern while taking into account the properties of several well-known patterns. We develop a relationship between the unknown sample patternP∗ and the set of known patterns P=P1,P2,⋯,Pk using our proposed correlation coefficient on the basis of a set of evaluation criteria. Taking motivation from the idea of the recognition principle [49], we developed an analogy as a degree of belongingness of the unknown pattern to any one of the known patterns to recognize its actual class as j∗=argmaxjρP∗,Pj, where ρP∗,Pj is the degree of relationship between P∗ and Pj that can be calculated using our proposed correlation coefficient. More will be the value of j∗, and more will be the closeness P∗ to the jth known pattern. A synthetic/fictitious numerical example has been given in the following to demonstrate the application of our method in pattern recognition problems.Example 21.
LetT1, T2, and T3 be three patterns which are known, and T be an unknown sample pattern, defined as follows in the form of TSFSs. These TSFSs have been defined over a space of points X=x1,x2,x3, and weights have been expressed in the form of a weight vector as w=0.31,0.36,0.33T. The characteristic values of the abovementioned patterns have been furnished in the following Table 5.Table 5
T-spherical fuzzy pattern information matrix (TSFIM).
Patternsx1x2x3T1<0.42, 0.42, 0.42><0.42, 0.42, 0.62><0.63, 0.63, 0.72>T2<0.42, 0.42, 0.62><0.52, 0.63, 0.82><0.72, 0.72, 0.82>T3<0.42, 0.52, 0.63><0.52, 0.63, 0.72><0.63, 0.72, 0.82>T<0.42, 0.42, 0.42><0.42, 0.42, 0.52><0.63, 0.63, 0.72>By applying our proposed schemen=5, we measure the relationship between the unknown sample pattern T and the other three known patterns (T1, T2, T3) to identify its belongingness. The obtained results have been presented in the following Table 4.Table6 clearly shows that the unknown sample pattern T has the highest correlation coefficient with the known pattern T1 as compared to the other known patterns T2 and T3. Hence, using the principle of maximum belongingness j∗=argmaxjρP∗,Pj, we can conclude that the unknown sample pattern T belongs to the known pattern T1.Table 6
Relationship betweenT and Tjj=1,2,3.
ρwT,TjT1T2T3ρwT,Tj0.95660.51850.9548


## 5.1. Application in MCDM

This subsection has been planned to show the application of our proposed scheme in the field of multicriteria decision-making by using a real MCDM problem, taken from [28]. To do so, we present here the implementation procedure of our proposed scheme for dealing with an MCDM problem. Suppose there is a set of “k” alternatives defined as TSFSs A=A1,A2,⋯,Ak and a set of “m” evaluation features F=F1,F2,⋯,Fm, having weight vector w=w1,w2,⋯,wmT, 0≤w≤1, and ∑i=1mwi=1, we define the ideal alternative A∗ as a TSFS by taking maximum membership value, minimum nonmembership, and minimum indeterminacy value for each evaluation criteria across all of the given alternatives, like an ideal positive solution in TOPSIS. A∗=Fi,max1≤j≤kMAjFi,min1≤j≤kIAjFi,min1≤j≤kNAjFi/Fi∈F. By applying our proposed correlation coefficient, we calculate the relationship between ideal choice A∗ and the given alternatives. Furthermore, by using the principle of the maximum correlation coefficient, we select an alternative as the best option, which has the highest correlation coefficient with an ideal alternative A∗.Example 20.
In order to strengthen the sales and purchase department, a firm had advertised the position of manager and selected 3 candidates after conducting an initial screening of the prospective candidates, who had applied. For the selection of the best candidate, the top management of the firm had considered the following three features to evaluate the aforementioned initially short-listed candidates. (i)F1: communication skills, (ii) F2: sense of responsibility, and (iii) F3: creativity. The candidates, evaluation criteria, and their weights have been presented in the form of sets as a set of alternatives, set of evaluation criteria, and weights vector, respectively, as follows.
A=A1,A2,A3,F=F1,F2,F3, and w=0.20,0.35,0.45T. After a comprehensive evaluation of the candidates, their decision information has been presented in the following Table 3 as a T-spherical fuzzy decision matrix (TSFDM).Table 3
T-spherical fuzzy information of alternatives Ajj=1,2,3.
AlternativesF1F2F3A1<0.8, 0.5, 0.4><0.7, 0.4, 0.4><0.3, 0.5, 0.4>A2<0.9, 0.2, 0.4><0.6, 0.3, 0.2><0.4, 0.1, 0.7>A3<0.5, 0.5, 0.5><0.8, 0.2, 0.3><0.6, 0.4, 0.3>It can be clearly seen that the decision information in the above Table3 is TSFI n=3. So, we apply our proposed scheme to find a correlation coefficient between the ideal candidate A∗=0.9,0.2,0.4,0.8,0.2,0.2,0.6,0.1,0.3 and the initially short-listed three candidates. The results have been furnished in the following Table 4.Table 4
Association betweenA∗ and Ajj=1,2,3.
ρwA∗,AiA1A2A3ρwA∗,Ai0.97030.8307-0.2530It can be clearly seen that the candidateA1 has the highest degree of relationship with the ideal choice A∗ as compared to that of the other alternatives. Hence, using the principle of maximum correlation, we can say that the candidate A1 is the most suitable choice for the position of manager. This is in complete agreement with the decision, from where this example has been taken.


## 5.2. Application in Pattern Recognition

In this section, we show how we used our proposed theory to solve a pattern recognition problem and recognize an unknown pattern while taking into account the properties of several well-known patterns. We develop a relationship between the unknown sample patternP∗ and the set of known patterns P=P1,P2,⋯,Pk using our proposed correlation coefficient on the basis of a set of evaluation criteria. Taking motivation from the idea of the recognition principle [49], we developed an analogy as a degree of belongingness of the unknown pattern to any one of the known patterns to recognize its actual class as j∗=argmaxjρP∗,Pj, where ρP∗,Pj is the degree of relationship between P∗ and Pj that can be calculated using our proposed correlation coefficient. More will be the value of j∗, and more will be the closeness P∗ to the jth known pattern. A synthetic/fictitious numerical example has been given in the following to demonstrate the application of our method in pattern recognition problems.Example 21.
LetT1, T2, and T3 be three patterns which are known, and T be an unknown sample pattern, defined as follows in the form of TSFSs. These TSFSs have been defined over a space of points X=x1,x2,x3, and weights have been expressed in the form of a weight vector as w=0.31,0.36,0.33T. The characteristic values of the abovementioned patterns have been furnished in the following Table 5.Table 5
T-spherical fuzzy pattern information matrix (TSFIM).
Patternsx1x2x3T1<0.42, 0.42, 0.42><0.42, 0.42, 0.62><0.63, 0.63, 0.72>T2<0.42, 0.42, 0.62><0.52, 0.63, 0.82><0.72, 0.72, 0.82>T3<0.42, 0.52, 0.63><0.52, 0.63, 0.72><0.63, 0.72, 0.82>T<0.42, 0.42, 0.42><0.42, 0.42, 0.52><0.63, 0.63, 0.72>By applying our proposed schemen=5, we measure the relationship between the unknown sample pattern T and the other three known patterns (T1, T2, T3) to identify its belongingness. The obtained results have been presented in the following Table 4.Table6 clearly shows that the unknown sample pattern T has the highest correlation coefficient with the known pattern T1 as compared to the other known patterns T2 and T3. Hence, using the principle of maximum belongingness j∗=argmaxjρP∗,Pj, we can conclude that the unknown sample pattern T belongs to the known pattern T1.Table 6
Relationship betweenT and Tjj=1,2,3.
ρwT,TjT1T2T3ρwT,Tj0.95660.51850.9548


## 6. Conclusions

The TSFS is a useful tool for displaying the degree of positive, neutral, and negative membership, as well as the information’s dependability owing to rejection membership. Major correlation coefficient approaches based on the unit interval [0, 1] have been defined in the literature. In accordance with the conventional correlation coefficient in statistics, our newly proposed technique correlation coefficient forT-spherical fuzzy sets is the best choice for dealing with positive, negative, and no correlation [-1, 1]. The mathematical derivations for the upper and lower bound of correlation have been given. To show the novelty, we compare our method with existing methods, and the results demonstrate the good aspect of the proposed idea. Furthermore, to demonstrate the feasibility, usefulness, and practical application, we applied our proposed scheme to solve technical and scientific problems of multicriteria decision-making and pattern recognition.



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*Source: 4471945-2022-07-27.xml*