# Research on Novel Correlation Coefficient of Neutrosophic Cubic Sets and Its Applications

Authors: Huiling Xue; Minrong Yu; Chunfang Chen
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

Single-valued neutrosophic cubic set is a good tool to solve the vague and uncertain problems because it contains more information. The article first gives the correlation coefficient of single-valued neutrosophic cubic sets. Then, a decision method is proposed, and an application in pattern recognition is considered. Finally, examples are given to explain the feasibility of this method. At the same time, the comparative analysis shows the superiority of this method.


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## Body

## 1. Introduction

In 1965, Zadeh [1] defined the fuzzy sets (FSs). Let X={x1,x2,…,xn} be a universe, a fuzzy set A on X is described by membership function μA(x), which satisfies every element in X corresponding to a real number in the interval [0,1], and the value of μA(x) at x represents grade of membership of x∈A. Therefore, the nearer the value of μA(x) to 1, the higher the grade of membership of x∈A. The membership is the degree to which every element in X belongs to A. Example, let X={155,170,178,188} denote the height of boys, A is a set of all boys with a high feature, and the membership degree may be μA(155)=0, μA(170)=0.5, μA(178)=0.9, and μA(188)=1. That is, the decision maker thinks 188 is absolutely tall, so the membership value is naturally 1, while the membership value of 170 may be 0.5, indicating that the probability of 170 belonging to set A may be 0.5. Subsequently, some other FSs are proposed [2–7]. In order to contain more information to describe the fuzziness and uncertainty of the decision maker, cubic sets (CSs) were proposed by Jun et al. [8]. After that, some scholars put forward many other related definitions by using the CSs [9–15]. Since FSs and IFSs only consider partial information, therefore, Smarandache [16] introduced neutrosophic sets (NSs), and their values located in the ]-0,1+[, the nonstandard interval; these three values are completely independent. NSs can solve the problem of uncertainty quite well. Neutrosophic sets are also widely used in many different fields [17–24]. Wang et al. [25] introduced the interval neutrosophic sets (INSs). It is the generalization of FSs, IVFSs, and NSs. Because the nonstandard interval of NSs cannot be used to solve many specific events, Smarandache [26] proposed the single-valued neutrosophic sets (SVNSs), and it is applied to many fields [27–41]. Later, Ye [42] introduced the single-valued neutrosophic hesitant fuzzy sets (SVNHFSs); SVNHFSs are more widely available than other fuzzy sets.Alia et al. [43] also proposed the concept of neutrosophic cubic sets (NCSs) and defined some notions. They proposed the concept of NCSs by extending the concept of CSs to NSs, and it has been widely used in many fields [44–49]. By investigating some properties of NCSs, Alia et al. [50] considered distance measures of NCSs. The NCS contains much more information than the general NSs, so it is possible to show more practical value and effectiveness in MADM. At present, there are many researchers on single-valued neutrosophic cubic sets (SVNCSs), as we know, the application of its correlation coefficient has not been studied in SVNCSs environment. However, correlation is an important index and widely used in many places [51–55]. Since the correlation was used in the fuzzy environment [56–58], then the correlation coefficient was proposed and studied in different fuzzy environments [3, 4, 59–65]. Based on its importance, this article considers the correlation coefficient of SVNCSs.The structure of this article is as follows: Section2 introduces basic concepts of CSs, NSs, and NCSs. The related concepts of NCSs are proposed in Section 3. In Section 4, a decision method is given, and the feasibility and superiority of the method are illustrated by comparative analysis. In Section 5, an algorithm in pattern recognition is given, and the feasibility and superiority of the modified algorithm are illustrated by comparison and analysis. In Section 6, the paper ends with a conclusion.


## 2. Preliminaries

Let us start with some concepts, including fuzzy numbers, interval valued fuzzy sets (IVFSs), neutrosophic sets (NSs), interval neutrosophic sets (INSs), single-valued neutrosophic hesitant fuzzy sets (SVNHFSs), and neutrosophic cubic sets (NCSs).Definition 1 (see [66]).
LetX={x1,x2,…,xn} be a universe; A is a fuzzy set defined on X. μA is a mapping from X to [0,1]. The element xi belongs to the membership function of A, which is expressed as {μA(x1),μA(x2),…,μA(xn)}. And the fuzzy number can be described as(1)μAX=μAx1+μAx2+⋯+μAxn,where + denotes the join symbol and does not represent the meaning of addition.Fuzzy numbers also have some other definitions [67], such as triangular fuzzy numbers, trapezoidal fuzzy numbers, and ordering fuzzy numbers.Definition 2 (see [2]).
LetX be a set, then an IVFS A on X has the form(2)A=x,A-x,A+x∣x∈X,in which A-(x) and A+(x) are the upper and lower limits of the membership degree x∈X where 0≤A-(x)+A+(x)≤1, respectively.Definition 3 (see [68]).
LetX be a universe. The form of NSs λ is (3)λ=x,Tx,Ix,Fx∣x∈X,where T, I, F:X→]-0,1+[ define the degree of Truth, Indeterminacy, and Falsehood of x∈X to λ and(4)0-≤Tx+Ix+Fx≤3+.Definition 4 (see [25]).
LetX be a set. INSs A in X satisfies AT(x), AI(x), AF(x)⊆[0,1]. (5)A=x,AT-x,AT+x,AI-x,AI+x,AF-x,AF+x∣x∈X.Definition 5 (see [26]).
LetX={x1,x2,…,xn} be a universe; the form of the definition of the SVNSs is (6)A=x,TAx,IAx,FAx∣x∈X,where TA,IA,FA:X→[0,1], with 0≤TA(x)+IA(x)+FA(x)≤3.Definition 6 (see [39]).
LetX={x1,x2,…,xn} be a set, then, SVNHFS N on X is defined as (7)N=xi,hNxi,lNxi,gNxi∣xi∈X,in which hN(xi), lN(xi), and gN(xi) are three sets, (8)hNxi=γN1xi,γN2xi,…,γNlhxi,lNxi=δN1xi,δN2xi,…,δNllxi,gNxi=ηN1xi,ηN2xi,…,ηNlgxi,with 0≤γ,δ,η≤1 and 0≤γ++δ++η+≤3, where γ∈hN(xi), δ∈lN(xi), η∈gN(xi), and γ+∈hN+(xi)=Uη∈gN(xi) max{η} for xi∈X. Generally, the n(xi)={hN(xi),lN(xi),gN(xi)} is named a single-valued neutrosophic hesitant fuzzy element (SVNHFE). Denoted by n={h,l,g} such that h={γN1,γN2,…,γNlh}, l={δN1,δN2,…,δNll}, and g={ηN1,ηN2,…,ηNlg}, where lh, ll, and lg are the number of values in hN(xi), lN(xi), and gN(xi), respectively.Definition 7 (see [43]).
LetX be a universe; the NCS I is defined as(9)I=x,Ax,λx∣x∈X,where A is an INS and λ is a NS. A NCS can be denoted as I=A,λ.Because the HFEs are usually disorder, therefore, for convenience of calculation we need to arrange all elements ofh(x) in a certain order. And for two HFEs, the numbers of elements are usually different. Chen et al. [60] have defined l(h(x)) as the number of elements in h(x). For two HFEs hA and hB, let l=max⁡{lh(A),lh(B)}, where lh(A) and lh(B) are the numbers of values in hA and hB, respectively. When lh(A)≠lh(B), for convenience of calculation we need to add some values to make them have common length. And [26, 50] have given the regulation; they have the same length as the longer one.Based on the correlation coefficient of HFSs, Sahin [41] proposed the correlation coefficient of SVNHFSs.Definition 8 (see [41]).
LetA be a SVNHFS on universe X={x1,x2,…,xn}; the informational energy of A is defined as(10)ESVNHFSA=∑i=1n1ki∑s=1kiγAσs2xi+1pi∑z=1piδAσz2xi+1li∑t=1liηAσt2xi.Definition 9 (see [41]).
LetA, B be two SVNHFSs on X={x1,x2,…,xn}; the correlation between A and B is defined as(11)CSVNHFSA,B=∑i=1n1ki∑s=1kiγAσsxiγBσsxi+1pi∑z=1piδAσzxiδBσzxi+1li∑t=1liηAσtxiηBσtxi.And we have
(1)CSVNHFS(A,A)=ESVNHFS(A).
(2)CSVNHFS(A,B)=CSVNHFS(B,A).Definition 10 (see [41]).
LetA, B be two SVNHFSs on X={x1,x2,…,xn}; the correlation coefficient between A and B is defined as(12)ρSVNHFSA,B=CSVNHFSA,BCSVNHFSA,ACSVNHFSB,B=∑i=1n1/ki∑s=1kiγAσsxiγBσsxi+1/pi∑z=1piδAσzxiδBσzxi+1/li∑t=1liηAσtxiηBσtxi)∑i=1n1/ki∑s=1kiγAσs2xi+1/pi∑z=1piδAσz2xi+1/li∑t=1liηAσt2xi∑i=1n1/ki∑s=1kiγBσs2xi+1/pi∑z=1piδBσz2xi+1/li∑t=1liηBσt2xi.Theorem 11 (see [41]).
For two SVNHFSsA and B, (12) satisfies
(1)ρSVNHFS(A,B)=ρSVNHFS(B,A);
(2)0≤ρSVNHFS(A,B)≤1;
(3)ρSVNHFS(A,B)=1, ifA=B.Generally, the weights of different elements are different, and also the results will be different. Therefore, it is important to take into account the weights ofxi(i=1,2,…,n). So the weighted correlation coefficients of SVNHFSs are given.Definition 12 (see [41]).
Letω=(ω1,ω2,…,ωn)T be the weighting vector of xi(i=1,2,…,n) with ωi≥0 and ∑i=1nωi=1; the weighted correlation coefficient is as follows: (13)ρSVNHFSwA,B=CSVNHFSwA,BCSVNHFSwA,ACSVNHFSwB,B=∑i=1nωi1/ki∑s=1kiγAσsxiγBσsxi+1/pi∑z=1piδAσzxiδBσzxi+1/li∑t=1liηAσtxiηBσtxi)∑i=1nωi1/ki∑s=1kiγAσs2xi+1/pi∑z=1piδAσz2xi+1/li∑t=1liηAσt2xi·∑i=1nωi1/ki∑s=1kiγBσs2xi+1/pi∑z=1piδBσz2xi+1/li∑t=1liηBσt2xi.
The upper form satisfies
(1)ρSVNHFSw(A,B)=ρSVNHFSw(B,A);
(2)0≤ρSVNHFSw(A,B)≤1;
(3)ρSVNHFSw(A,B)=1, ifA=B.


## 3. Correlation Coefficient of Single-Valued Neutrosophic Cubic Sets

Mumtaz Ali [50] introduced the NCSs and also gave some theorems. In this paper, according to the correlation coefficient mentioned above, we give the correlation coefficient of SVNCSs.Definition 13.
LetI be SVNCS on X={x1,x2,…,xn}, denoted by I={x,A(x),λ(x)∣x∈X}, then the informational energy of I is defined as (14)ESVNCSI=∑i=1n13Txi2+Ixi2+Fxi2+16T-xi2+I-xi2+F-xi2+T+xi2+I+xi2+F+xi2.Definition 14.
LetI1 and I2 be two SVNCSs on X={x1,x2,…,xn}, denoted by I1={x,A1x,λ1x∣x∈X} and I2={x,A2x,λ2x∣x∈X}. Then, the correlation between I1 and I2 is defined as(15)CSVNCSI1,I2=∑i=1n13T1xiT2xi+I1xiI2xi+F1xiF2xi+16T1-xiT2-xi+I1-xiI2-xi+F1-xiF2-xi+T1+xiT2+xi+I1+xiI2+xi+F1+xiF2+xiDefinition 15.
LetI1 and I2 be two SVNCSs on X={x1,x2,…,xn}, then the correlation coefficient between I1 and I2 is defined as (16)ρSVNCSI1,I2=CSVNCSI1,I2CSVNCSI1,I1CSVNCSI2,I2=∑i=1n13T1xiT2xi+I1xiI2xi+F1xiF2xi+16T1-xiT2-xi+I1-xiI2-xi+F1-xiF2-xi+T1+xiT2+xi+I1+xiI2+xi+F1+xiF2+xi∑i=1n13T1xi2+I1xi2+F1xi2+16T1-xi2+I1-xi2+F1-xi2+T1+xi2+I1+xi2+F1+xi2·∑i=1n13T2xi2+I2xi2+F2xi2+16T2-xi2+I2-xi2+F2-xi2+T2+xi2+I2+xi2+F2+xi2-1.Theorem 16.
LetI1 and I2 be two SVNCSs; (16) satisfies the following properties:
(1)ρSVNCS(I1,I2)=ρSVNCS(I2,I1);
(2)0≤ρSVNCS(I1,I2)≤1;
(3)ρSVNCS(I1,I2)=1, ifI1=I2.Proof.
(1) The proof of (1) and (3) is obvious, so we do not give a detailed proof.
(2) The inequality0≤ρSVNCS(I1,I2) is evident. Then, we prove ρSVNCS(I1,I2)≤1.
With the help of Cauchy-Schwarz inequality,(x1y1+x2y2+⋯+xnyn)2≤(x12+x22+⋯+xn2)(y12+y22+⋯+yn2), in which (x1,x2,…,xn)∈Rn and (y1,y2,…,yn)∈Rn, the following inequality can be obtained:(17)∑i=1n13T1xiT2xi+I1xiI2xi+F1xiF2xi+16T1-xiT2-xi+I1-xiI2-xi+F1-xiF2-xi+T1+xiT2+xi+I1+xiI2+xi+F1+xiF2+xi≤∑i=1n13T1xi2+I1xi2+F1xi2+16T1-xi2+I1-xi2+F1-xi2+T1+xi2+I1+xi2+F1+xi2·∑i=1n13T2xi2+I2xi2+F2xi2+16T2-xi2+I2-xi2+F2-xi2+T2+xi2+I2+xi2+F2+xi2
It is obvious that we can seeCSVNCS(I1,I2)≤CSVNCS(I1,I1)·CSVNCS(I2,I2); we can get 0≤ρSVNCS(I1,I2)≤1. So, the theorem has proved completely.In a general way, when decision makers give different weight value to each element in the universe, the result of decision may be different. Therefore, it is particularly important to consider the weight of element when making decision. Now, let us give the weighted correlation coefficient of SVNCSs.Definition 17.
Letω=ω1,ω2,…,ωnT be the weighting vector of xi(i=1,2,…,n), where ωi≥0 and ∑i=1nωi=1; the weighted correlation coefficient is defined as(18)ρSVNCSwI1,I2=CSVNCSwI1,I2CSVNCSwI1,I1CSVNCSwI2,I2=∑i=1nωi13T1xiT2xi+I1xiI2xi+F1xiF2xi+16(T1-xiT2-xi+I1-xiI2-xi+F1-xiF2-xi+T1+xiT2+xi+I1+xiI2+xi+F1+xiF2+xi∑i=1nωi13T1xi2+I1xi2+F1xi2)+16T1-xi2+I1-xi2+F1-xi2+T1+xi2+I1+xi2+F1+xi2·∑i=1nωi13T2xi2+I2xi2+F2xi2+16T2-xi2+I2-xi2+F2-xi2+T2+xi2+I2+xi2+F2+xi2-1Specially, whenωi=1/n(i=1,2,…,n), (18) reduced to (16), and (18) satisfies the following properties:(1)ρSVNCSw(I1,I2)=ρSVNCSw(I2,I1);(2)0≤ρSVNCSw(I1,I2)≤1;(3)ρSVNCSw(I1,I2)=1, ifI1=I2.Proof.
The proof process of the theorem is identical.In the next two parts, we give the decision method and an algorithm via the correlation coefficient of SVNCSs, respectively. The superiority of the proposed methods is illustrated through comparative analysis.


## 4. Decision-Making Method Based on the Correlation Coefficient of Single-Valued Neutrosophic Cubic Sets

In this section, weighted correlation coefficient is used to select the best alternative in MADM with SVNCS environment.In MADM problems, letI={I1,I2,…,Im} be a set of alternatives; λ=(λ1,λ2,…,λn) be a set of attributes; ω=(ω1,ω2,…,ωn)T be the weighting vector of xi(i=1,2,…,n) with ωi≥0 and ∑i=1nωi=1. Ik(k=1,2,…,m) are evaluation results.I k j = ( T k j - , T k j + , I k j - , I k j + , F k j - , F k j + , T k j , I k j , F k j ) ( k = 1,2 , … , m ; j = 1,2 , ⋯ ⋯ , n ), where Tkj-,Ikj-,Fkj-,Tkj+,Ikj+,Fkj+⊆[0,1] and Tkj,Ikj,Fkj∈[0,1]. Therefore, A=(Ikj)m×n is the decision matrix.In MADM problems, we can use the ideal alternative to select the best one in all alternatives. In general, the ideal scheme does not exist, but it can provide theoretical support for decision makers. Jun Ye [46] put forward a method to construct ideal plan, and we can use this idea to construct a relatively ideal plan. The best alternative is selected according to the degree of correlation between known and ideal solution. The selection process is as follows.Step 1.
Construct the neutrosophic cubic decision matrix based on decision information.Step 2.
Set up an ideal schemeI∗={I1∗,I2∗,…,In∗}(19)Ij∗=max⁡Tkj-,max⁡Tkj+,min⁡Ikj-,min⁡Ikj+,min⁡Fkj-,min⁡Fkj+,max⁡Tkj,min⁡Ikj,min⁡FkjIt represents the benefit attribute (20)Ij∗=min⁡Tkj-,min⁡Tkj+,max⁡Ikj-,max⁡Ikj+,max⁡Fkj-,max⁡Fkj+,min⁡Tkj,max⁡Ikj,max⁡FkjIt represents the cost attribute.Step 3.
Calculate weight correlation coefficient betweenIk(k=1,2,…,m) and Ij∗ by using (18).Step 4.
Rank all of the weighted correlation coefficient of SVNCSs and select the best one.
In the following, we give an example from Lu [46] to illustrate the effectiveness of the method.Example 18.
According to [46], we can get the following data: (21)I1=0.5,0.6,0.1,0.3,0.2,0.4,0.6,0.2,0.30.5,0.6,0.1,0.3,0.2,0.4,0.6,0.2,0.30.6,0.8,0.2,0.3,0.1,0.2,0.7,0.2,0.1,I2=0.6,0.8,0.1,0.2,0.2,0.3,0.7,0.1,0.20.6,0.7,0.1,0.2,0.2,0.3,0.6,0.1,0.20.6,0.7,0.3,0.4,0.1,0.2,0.7,0.4,0.1,I3=0.4,0.6,0.2,0.3,0.1,0.3,0.6,0.2,0.20.5,0.6,0.2,0.3,0.3,0.4,0.6,0.3,0.40.5,0.7,0.2,0.3,0.3,0.4,0.6,0.2,0.3,I4=0.7,0.8,0.1,0.2,0.1,0.2,0.8,0.1,0.20.6,0.7,0.1,0.2,0.1,0.3,0.7,0.1,0.20.6,0.7,0.3,0.4,0.2,0.3,0.7,0.3,0.2,Step 1.
According to the aforementioned four alternatives, construct the neutrosophic cubic decision matrix A:(22)A=a11a12a13a21a22a23a31a32a33a41a42a43.where(23)a11=0.5,0.6,0.1,0.3,0.2,0.4,0.6,0.2,0.3;a12=0.5,0.6,0.1,0.3,0.2,0.4,0.6,0.2,0.3;a13=0.6,0.8,0.2,0.3,0.1,0.2,0.7,0.2,0.1;a21=0.6,0.8,0.1,0.2,0.2,0.3,0.7,0.1,0.2;a22=0.6,0.7,0.1,0.2,0.2,0.3,0.6,0.1,0.2;a23=0.6,0.7,0.3,0.4,0.1,0.2,0.7,0.4,0.1;a31=0.4,0.6,0.2,0.3,0.1,0.3,0.6,0.2,0.2;a32=0.5,0.6,0.2,0.3,0.3,0.4,0.6,0.3,0.4;a33=0.5,0.7,0.2,0.3,0.3,0.4,0.6,0.2,0.3;a41=0.7,0.8,0.1,0.2,0.1,0.2,0.8,0.1,0.2;a42=0.6,0.7,0.1,0.2,0.1,0.3,0.7,0.1,0.2;a43=0.6,0.7,0.3,0.4,0.2,0.3,0.7,0.3,0.2.Step 2.
Set up an ideal scheme:(24)I∗=0.7,0.8,0.1,0.2,0.1,0.2,0.8,0.1,0.20.6,0.7,0.1,0.2,0.1,0.3,0.7,0.1,0.20.5,0.7,0.3,0.4,0.3,0.4,0.6,0.4,0.3.Step 3.
Calculate weight correlation coefficient:
ρ S V N C S w ( I ∗ , I 1 ) = 0.9542, ρSVNCSw(I∗,I2)=0.9822, ρSVNCSw(I∗,I3)=0.6291, ρSVNCSw(I∗,I4)=0.9944.Step 4.
Rank all of the weighted correlation coefficient of SVNCSs:ρSVNCSw(I,I4)≥ρSVNCSw(I,I2)≥ρSVNCSw(I,I1)≥ρSVNCSw(I,I3) and thus, the alternative I4 was selected as the best alternative.Comparative Analysis. From three cosine measures of SVNCSs from Lu [46] and the correlation coefficient of SVNCSs, we have the result in Table 1.Table 1
The valuesI∗ with Ii</caption> and ranking orders.
Value Ranking Order The Best Alternative ρSVNCSw(I∗,Ii) 0.9542,0.9822,0.6291,0.9944 ρ 4 > ρ 2 > ρ 1 > ρ 3 I 4  Sω1(I∗,Ii) 0.9564,0.9855,0.9596,0.9945 S 4 > S 2 > S 3 > S 1 I 4  Sω2(I∗,Ii) 0.9769,0.9944,0.9795,0.9972 S 4 > S 2 > S 3 > S 1 I 4  Sω3(I∗,Ii) 0.9892,0.9959,0.9897,0.9989 S 4 > S 2 > S 3 > S 1 I 4From the results of Table1, we note that the rank S4(I,I4)≥S2(I,I2)≥S3(I,I3)≥S1(I,I1) is changed to ρSVNCSw(I,I4)≥ρSVNCSw(I,I2)≥ρSVNCSw(I,I1)≥ρSVNCSw(I,I3). What is more, comparing the results of Lu [46], we can see that all the weighted cosine measure values are quite close; therefore, it is difficult to convince such a ranking result. And the values vary from 0.9564 to 0.9945 of Sω1, and we note that Sω1(I,I1) with Sω1(I,I3) and Sω1(I,I2) with Sω1(I,I4) are very close, which makes it difficult to make clear judgments about all alternatives in decision-making. Sω2 varies from 0.9769 to 0.9972; moreover, the values of Sω2(I,I1) with Sω2(I,I3) and Sω2(I,I2) with Sω2(I,I4) are basically not any gaps, which can be seen approximately equal, so that the results of the decision making may be suspected. And Sω3 varies from 0.9892 to 0.9989, the difference between Sω3(I,I1) with Sω3(I,I3) is only 0.0005, and the difference between Sω3(I,I2) with Sω3(I,I4) is 0.003, so it is difficult to make the most appropriate decision for decision makers, perhaps this is not a desirable decision method.Obviously, these cosine measure values cannot clearly discriminate the difference of different alternatives. Conversely, the result of this paper lies from 0.6291 to 0.9944; the distinction between different values is relatively large. Moreover, the minimum cell division in the four numbers is also 0.0122, which is more convincing than that article [46]; thus, we can easily select the best alternative. Furthermore, from the table, we can see that several methods are basically consistent with the evaluation results of the fourth schemes. But it is difficult to make a relatively clear and reliable judgment between the alternative 1 and the alternative 3 with the previous cosine measurement method. However, the method of this paper gives a clear comparison between them. As shown in Figure 1, we can clearly see the result ρSVNCSw(I,I4)≥ρSVNCSw(I,I2)≥ρSVNCSw(I,I1)≥ρSVNCSw(I,I3).Figure 1
Correlation coefficient and cosine measures of alternatives with ideal alternative.Compared with Lu [46] method, we find the method of this paper more effective and superior.


## 5. Applications in Pattern Recognition Problems

In this section, we propose a recognition method based on the correlation coefficient of SVNCSs, which is derived from [50]. According to the maximum correlation principle in mathematical statistics, we assume that if the correlation coefficient of ideal pattern with sample pattern is greater than or equal to 0.7, we consider that the sample model belongs to a group of ideal model. The algorithm is as follows.Algorithm 19.
Step 1. Construct an ideal SVNCS I∗ on X.
Step 2. Construct SVNCSs Ii, i=1,2,3⋯,n as the sample pattern that is recognized.
Step 3. Calculate the correlation coefficient ρSVNCSs(I∗,Ii), i=1,2,…,n.
Step  4. If ρSVNCSs(I∗,Ii)≥0.7, the Ii belongs to the ideal pattern I∗ and if ρSVNCSs(I∗,Ii)<0.7, the Ii does not belong to the ideal pattern I∗.In the following, we give two examples to illustrate the utility of the correlation coefficient of SVNCSs in pattern recognition.Example 20.
This example is adapted from [50]; three sample patterns Ii(i=1,2,3) and the ideal pattern I∗ are given as(25)I1=0.2,0.5,0.4,0.5,0.3,0.5,0.5,0.6,0.40.7,0.7,0.1,0.3,0.1,0.3,0.2,0.2,0.20.6,0.8,0.5,0.6,0.3,0.4,0.3,0.1,0.7,I2=0.3,0.7,0.3,0.5,0.3,0.9,0.2,0.5,0.20.6,0.7,0.1,0.8,0.2,0.3,0.1,0.5,0.70.6,0.9,0.9,1.0,0.3,0.4,0.7,0.1,0.7,I3=0.8,0.9,0.1,0.2,0.8,0.9,0.9,0.8,0.90.1,0.2,0.8,0.9,0.3,0.9,0.9,0.8,0.90.5,0.9,0.1,1.0,0.4,0.7,0.9,0.9,0.1,I∗=0.2,0.4,0.3,0.5,0.3,0.5,0.1,0.2,0.40.5,0.7,0.0,0.5,0.2,0.3,0.1,0.2,0.20.6,0.8,0.0,0.1,0.3,0.4,0.3,0.1,0.7.
Calculate the correlation coefficient ofρSVNCS(I∗,Ii), i=1,2,3.
ρ S V N C S ( I ∗ , I 1 ) = 0.93, ρSVNCS(I∗,I2)=0.83, ρSVNCS(I∗,I3)=0.63.
SinceρSVNCS(I∗,I1)≥0.7, ρSVNCS(I∗,I2)≥0.7, and ρSVNCS(I∗,I3)<0.7, we consider the samples I1 and I2 belong to ideal pattern, and I3 does not belong to the ideal pattern.
We notice that the result of Example20 is consistent with this paper. However, the result of this article is more superior.
Next, we will give another example to illustrate the approach of this article.Example 21.
The sample patternsIi(i=1,2) and the ideal pattern I∗ are given as follows:(26)I1=0.8,0.9,0.1,0.2,0.8,0.9,0.9,0.8,0.90.1,0.2,0.8,0.9,0.3,0.9,0.9,0.8,0.90.5,0.9,0.1,1.0,0.4,0.7,0.9,0.9,0.1,I2=0.8,0.9,0.1,0.2,0.8,0.9,0.8,0.8,0.90.1,0.2,0.8,0.9,0.7,0.9,0.3,0.5,0.70.7,0.9,0.5,1.0,0.5,0.7,0.7,0.9,0.1,I∗=0.2,0.4,0.3,0.5,0.3,0.5,0.1,0.2,0.40.5,0.7,0.0,0.5,0.2,0.3,0.1,0.2,0.20.6,0.8,0.0,0.1,0.3,0.4,0.3,0.1,0.7.Using the method of [50], we can get d(I1,I∗)=0.51, d(I2,I∗)=0.49, so the sample pattern I1 does not belong to the ideal pattern, and I2 belongs to the ideal pattern. However, using the method of this paper we will get result ρSVNCS(I∗,I1)=0.63, and ρSVNCS(I∗,I2)=0.66; therefore, according to the principle of maximum correlation, we can judge that I1 does not belong to the ideal pattern, and I2 also does not belong to the ideal pattern.Comparative Analysis. In the pattern recognition problems, the results of Example 21 are different. Using the method of [50], the recognition standard is 0.5. According to the minimum distance measures principle (principle of maximum similarity measures), because d(I,I1)=0.51, and d(I,I2)=0.49 are very close to 0.5, so the judgment may have certain deviation; it is difficult to make accurate identification for some distance measures near 0.5. That is to say, the recognition result will be very fuzzy; therefore, this standard does not have its superiority in this case. However, the method of this paper can clearly identify the relationship between the sample pattern and the ideal pattern, and it effectively avoids the ambiguity of the result (the correlation is greater than or equal to 0.7).Therefore, using the principle of maximum correlation is more persuasive and more intuitive to accept.


## 6. Conclusion

SVNCSs are a generalization of SVNSs, INSs, and CSs. We start to introduce some concepts of correlation coefficient of SVNCSs and discuss the properties. We also propose the decision method by using the weighted correlation coefficient of SVNCSs and give applications of correlation coefficient of SVNCSs in pattern recognition. Finally we give examples to explain the effectiveness of the method and give comparative analysis to show the superiority of the method.



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*Source: 7453025-2019-01-15.xml*