An Improved Correlation Coefficient of Intuitionistic Fuzzy Sets

3.1 An Improved Correlation Coefficient of the IFSs

In the following, we define an improved correlation coefficient of the IFSs.

Definition 8: Let A₁, A₂∈IFS(X), and X={x₁, x₂, …, x_n} be a finite universe of discourse, then we define

as a correlation coefficient of the IFSs A₁ and A₂, where

αi=c−Δμi−Δμmaxc−Δμmin−Δμmax, βi=c−Δνi−Δνmaxc−Δνmin−Δνmax (c>2, i=1, 2, ⋯, n),Δμi=|μA1(xi)−μA2(xi)|,   Δνi=|νA1(xi)−νA2(xi)|,Δμmin=mini{|μA1(xi)−μA2(xi)|},   Δνmin=mini{|νA1(xi)−νA2(xi)|},Δμmax=maxi{|μA1(xi)−μA2(xi)|},   Δνmax=maxi{|νA1(xi)−νA2(xi)|}.

Remark 2: Note that the condition “c>2” ensures that both α_i and β_i belong to (0, 1). Hence, the correlation coefficient ρ^*(A₁, A₂) satisfies the condition 0≤ρ^*(A₁, A₂)≤1. If some α_i and β_i are <0, it will cause ρ^*(A₁, A₂)<0. We can see example 2 as follows.

Example 2: Let A₁ and A₂ be two IFSs in X={x₁, x₂} given by

A1={〈x1, 0.9, 0.1〉, 〈x2, 0.7, 0.2〉};A2={〈x1, 0.1, 0.8〉, 〈x2, 0.6, 0.4〉}.

By Eq. (3), we can have α₁=−6, α₂=1, β₁=−4, β₂=1 when c=1, then we can calculate that ρ₁(A₁, A₂) =−0.175<0.

We present some properties of the improved correlation coefficient in the following.

Theorem 2:The correlation coefficient ρ^*(A₁, A₂) satisfies the following properties:

1. ρ^*(A₁, A₂)=ρ^*(A₂, A₁);

2. 0≤ρ^*(A₁, A₂)≤1;

3. A₁=A₂⇔ρ^*(A₁, A₂)=1.

Proof

1. Obviously.

2. Since 0<α_i≤1, 0<β_i≤1 and 0≤1−Δμ_i≤1, 0≤1−Δν_i≤1, then

   0≤αi(1−Δμi)+βi(1−Δνi)≤2 (i=1, 2, ⋯, n),

   and thus, by Eq. (3), we know that 0≤ρ^*(A₁, A₂)≤1.

3. “⇒” If A₁=A₂, it implies that: μA1(xi)=μA2(xi), νA1(xi)=νA2(xi) for all x_i∈X (i=1, 2, …, n), then

   Δμi=Δμmin=Δμmax=0, Δνi=Δνmin=Δνmax=0.

Thus, ρ^*(A₁, A₂)=1.

“⇐” If ρ^*(A₁, A₂)=1, by the fact that

0≤αi(1−Δμi)≤1, 0≤βi(1−Δνi)≤1,

we have

αi(1−Δμi)+βi(1−Δνi)=2 and αi(1−Δμi)=βi(1−Δνi)=1 (i=1, 2, ⋯, n).

Since

0<αi≤1,   0<βi≤1,   0≤1−Δμi≤1,   0≤1−Δνi≤1,

then we obtain

αi=βi=1, 1−Δμi=1, 1−Δνi=1, i.e., Δμi=Δνi=0.

Thus, μA1(xi)=μA2(xi), νA1(xi)=νA2(xi) for all x_i∈X, i.e., A₁=A₂.

Remark 3: We may draw the conclusion from Theorem 2 that the improved correlation coefficient overcomes the limitations mentioned above of Xu’s correlation coefficients.

First of all, the improved correlation coefficient avoids emerging the 0/0 type quotient, which is more reasonable in mathematical logic.

In addition, under the condition A₁≠A₂, the improved correlation coefficient ρ^*(A₁, A₂) is certainly less than one by the third property of Theorem 3, which is coincident with our intuition. For instance, in Example 1, A₁≠A₂ and ρ^*(A₁, A₂)=0.9<1. Furthermore, the improved correlation coefficient can guarantee that the correlation coefficient of any two IFSs equals 1 if and only if these two IFSs are the same. Consequently, it can be applied in medical diagnosis and clustering effectively, which will be discussed in Section 4.

In many situations, the weight of every element x_i∈X should be taken into account. For example, in the multiple attribute decision-making problems, each attribute usually has different importance and, thus, needs to be assigned a different weight. As a result, we further extend Formula (3).

Definition 9: Let A₁, A₂∈IFS(X), and X={x₁, x₂, …, x_n} be a finite universe of discourse, then we define

as a correlation coefficient of the IFSs A₁ and A₂, where

αi=c−Δμi−Δμmaxc−Δμmin−Δμmax,  βi=c−Δνi−Δνmaxc−Δνmin−Δνmax (c>2),

the weight vector of x_i (i=1, 2, …, n) is ω=(ω₁, ω₂, …, ω_n)^T, ω_i≥0<mspace;(i=1, 2, …, n) and ∑i=1nωi=1. The value of ω_i can be determined by several methods, such as statistical distribution, analytic hierarchy process, coefficient of variation method, and so on. It can also be determined according to the experts’ opinions. Especially, if ω=(1n, 1n, ⋯, 1n)T, Formula (4) reduces to Formula (3).

3.2 The Improved Correlation Coefficient of the IVIFSs

In the following, we generalize the idea of Definition 8 to the IVIFS theory.

Definition 10: Let Ã₁, Ã₂∈IVIFS(X), and X={x₁, x₂, …, x_n} be a finite universe of discourse, then we define

as a correlation coefficient of IVIFSs Ã₁ and Ã₂, where

γi=c−Δμ˜iL−Δμ˜maxLc−Δμ˜minL−Δμ˜maxL,   ζi=c−Δμ˜iU−Δμ˜maxUc−Δμ˜minU−Δμ˜maxU,φi=c−Δν˜iL−Δν˜maxLc−Δν˜minL−Δν˜maxL,   ψi=c−Δν˜iU−Δν˜maxUc−Δν˜minU−Δν˜maxU(c>2),Δμ˜iL=|μ˜A˜1L(xi)−μ˜A˜2L(xi)|,   Δν˜iL=|ν˜A˜1L(xi)−ν˜A˜2L(xi)|,Δμ˜iU=|μ˜A˜1U(xi)−μ˜A˜2U(xi)|,   Δν˜iU=|ν˜A˜1U(xi)−ν˜A˜2U(xi)| (i=1, 2, ⋯, n),Δμ˜minL=mini{|μ˜A˜1L(xi)−μ˜A˜2L(xi)|}, Δμ˜minU=mini{|μ˜A˜1U(xi)−μ˜A˜2U(xi)|},Δμ˜maxL=maxi{|μ˜A˜1L(xi)−μ˜A˜2L(xi)|}, Δμ˜maxU=maxi{|μ˜A˜1U(xi)−μ˜A˜2U(xi)|},Δν˜minL=mini{|ν˜A˜1L(xi)−ν˜A˜2L(xi)|}, Δν˜minU=mini{|ν˜A˜1U(xi)−ν˜A˜2U(xi)|},Δν˜maxL=maxi{|ν˜A˜1L(xi)−ν˜A˜2L(xi)|}, Δν˜maxU=maxi{|ν˜A˜1U(xi)−ν˜A˜2U(xi)|}.

Equation (5) also can be generalized to more general cases:

where ω=(ω₁, ω₂, …, ω_n)^T, ω_i ≥ 0 (i=1, 2, …, n) is the weight of x_i (i=1, 2, …, n) and ∑i=1nωi=1. The value of ω_i can be determined by the same way which was mentioned in Definition 9. Especially, if ω=(1n, 1n, ⋯, 1n)T, Formula (6) reduces to Formula (5).

Theorem 3:The correlation coefficient ρ2∗(A˜1, A˜2) satisfies the following properties:

1. ρ2∗(A˜1, A˜2)=ρ2∗(A˜2, A˜1);

2. 0≤ρ2∗(A˜1, A˜2)≤1;

3. A˜1=A˜2⇔ρ2∗(A˜1, A˜2)=1.

Proof

Obviously.

Since

0<γi<1, 0<ζ≤1, 0<φi≤1, 0<ψi≤1

and

0≤1−Δμ˜iL≤1, 0≤1−Δμ˜iU≤1, 0≤1−Δν˜iL≤1, 0≤1−Δν˜iU≤1,

then

0≤γi(1−Δμ˜iL)+ζi(1−Δμ˜iU)+φi(1−Δν˜iL)+ψi(1−Δν˜iU)≤4

where i=1, 2, …, n. Thus, by Eq. (5), we know that 0≤ρ2∗(A˜1, A˜2)≤1.

“⇒” If Ã₁=Ã₂, it implies that: μ˜A˜1L(xi)=μ˜A˜2L(xi), μ˜A˜1U(xi)=μ˜A˜2U(xi), ν˜A˜1L(xi)=ν˜A˜2L(xi), ν˜A˜1U(xi)=ν˜A˜2U(xi) for all x_i∈X, then

Δμ˜iL=Δμ˜minL=Δμ˜maxL=0,   Δμ˜iU=Δμ˜minU=Δμ˜maxU=0,Δν˜iL=Δν˜minL=Δν˜maxL=0,   Δν˜iU=Δν˜minU=Δν˜maxU=0.

Thus, ρ2∗(A˜1, A˜2)=1.

“⇐” If ρ2∗(A˜1, A˜2)=1, by the fact that

0≤γi(1−Δμ˜iL)≤1, 0≤ζi(1−Δμ˜iU)≤1,0≤φi(1−Δν˜iL)≤1, 0≤ψi(1−Δν˜iU)≤1,

we have

γi(1−Δμ˜iL)+ζi(1−Δμ˜iU)+φi(1−Δν˜iL)+ψi(1−Δν˜iU)=4

and

γi(1−Δμ˜iL)=ζi(1−Δμ˜iU)=φi(1−Δν˜iL)=ψi(1−Δν˜iU)=1.

Since

0<γi<1, 0<ζ≤1, 0<φi≤1, 0<ψi≤1

and

0≤1−Δμ˜iL≤1,   0≤1−Δμ˜iU≤1,   0≤1−Δν˜iL≤1,   0≤1−Δν˜iU≤1,

then, we obtain

1−Δμ˜iL=1−Δμ˜iU=1−Δν˜iL=1−Δν˜iU=1,

i.e.,

Δμ˜iL=Δμ˜iU=Δν˜iL=Δν˜iU=0.

Thus,

μ˜A˜1L(xi)=μ˜A˜2L(xi),   μ˜A˜1U(xi)=μ˜A˜2U(xi),ν˜A˜1L(xi)=ν˜A˜2L(xi),   ν˜A˜1U(xi)=ν˜A˜2U(xi),

for all x_i∈X, i.e., Ã₁=Ã₂.

4.1 The Application in Medical Diagnosis

First, for the set of diagnosis A={A₁, A₂, …, A_n}, the set of symptoms C={C₁, C₂, …, C_t}, and the set of patients B={B₁, B₂, …, B_m}, suppose the weight of C_k is ω_k (k=1, 2, …, t), then the diagnosis steps are as follows:

• Step 1: Based on the experience of experts, each symptom is described by a pair of parameters (μ, ν), i.e., the membership μ and the non-membership ν.

• Step 2: For every patient B_j (j=1, 2, …, m), according to Formula (4), we can calculate the correlation coefficient ρ1∗(Ai, Bj), (i=1, 2, ⋯, n), where A_i (i=1, 2, …, n) are the diagnosis results.

• Step 3: For every patient B_j (j=1, 2, …, m), select the diagnosis result Ai0 in A most close to the patient B_j, i.e., ρ1∗(Ai0, Bj)=max{ρ1∗(Ai, Bj)|Ai∈A}. Hence, it can be asserted that the diagnosis result of patient B_j is Ai0.

Example 3 ([14]): To make a proper diagnosis A for a patient with the given values of symptoms C, a medical knowledge base is necessary that involves elements described in terms of IFSs. The set of symptoms is

C={C1, C2, C3, C4, C5}={Temperature, Headache, Stomach pain, Cough, Chest pain},

the set of diagnosis is

A={A1, A2, A3, A4, A5}={Viral fever, Malaria, Typhoid, Stomach problem, Chest problem}

and the set of patients is

B={B1, B2, B3, B4}={Al, Bob, Joe, Ted}.

ω=(ω₁, ω₂, …, ω₅)^T is the weight vector of symptoms, let ω1=ω2=⋯=ω5=15. The data are given in Table 1 – each symptom is described by a pair of parameters (μ, ν), i.e., the membership μ and the non-membership ν. The symptoms are given in Table 2. We need to seek a diagnosis for each patient B_j (j=1, 2, 3, 4).

First, construct IFSs:

A1={〈C1, 0.4, 0.0〉, 〈C2, 0.3, 0.5〉, 〈C3, 0.1, 0.7〉, 〈C4, 0.4, 0.3〉, 〈C5, 0.1, 0.7〉},A2={〈C1, 0.7, 0.0〉, 〈C2, 0.2, 0.6〉, 〈C3, 0.0, 0.9〉, 〈C4, 0.7, 0.0〉, 〈C5, 0.1, 0.8〉},A3={〈C1, 0.3, 0.3〉, 〈C2, 0.6, 0.1〉, 〈C3, 0.2, 0.7〉, 〈C4, 0.2, 0.6〉, 〈C5, 0.1, 0.9〉},A4={〈C1, 0.1, 0.7〉, 〈C2, 0.2, 0.4〉, 〈C3, 0.8, 0.0〉, 〈C4, 0.2, 0.7〉, 〈C5, 0.2, 0.7〉},A5={〈C1, 0.1, 0.8〉, 〈C2, 0.0, 0.8〉, 〈C3, 0.2, 0.8〉, 〈C4, 0.2, 0.8〉, 〈C5, 0.8, 0.1〉},B1={〈C1, 0.8, 0.1〉, 〈C2, 0.6, 0.1〉, 〈C3, 0.2, 0.8〉, 〈C4, 0.6, 0.1〉, 〈C5, 0.1, 0.6〉},B2={〈C1, 0.0, 0.8〉, 〈C2, 0.4, 0.4〉, 〈C3, 0.6, 0.1〉, 〈C4, 0.1, 0.7〉, 〈C5, 0.1, 0.8〉},B3={〈C1, 0.8, 0.1〉, 〈C2, 0.8, 0.1〉, 〈C3, 0.0, 0.6〉, 〈C4, 0.2, 0.7〉, 〈C5, 0.0, 0.5〉},B4={〈C1, 0.6, 0.1〉, 〈C2, 0.5, 0.4〉, 〈C3, 0.3, 0.4〉, 〈C4, 0.7, 0.2〉, 〈C5, 0.3, 0.4〉}.

Then, we utilize the improved correlation coefficient to derive a diagnosis for each patient B_j (j=1, 2, 3, 4). Let c=3 in Formula (4). Then, all the diagnosis results for the considered patients are listed in Table 3.

Finally, from the arguments in Table 3, we derive a proper diagnosis as follows:

Al developed malaria; Bob had some problems with his stomach; Joe got typhoid; Ted got viral fever.

Based on the correlation coefficient (1), Xu’s diagnosis results [19] (Al suffered from malaria, Bob from a stomach problem, and both Joe and Ted from viral fever) are different from the results obtained by the improved correlation coefficient (4). Because the improved correlation coefficient has a wider scope of application and can solve the problems of the existing correlation coefficient (mentioned in Remark 2), the diagnosis results obtained by the improved correlation coefficient (4) are more reasonable. Furthermore, Vlachos and Sergiadis [16] made diagnosis utilizing the symmetric discrimination information measure D_IFS(A, B) and pointed out that the diagnosis results of D_IFS(A, B) were more effective than the results of distance-based methods and the similarity-dissimilarity measure. The diagnosis results obtained by D_IFS(A, B) were as follows: Al got viral fever, Bob had some problems with his stomach, Joe got typhoid, and Ted got viral fever, which are also different from those by the improved correlation coefficient (4). The differences are because that the results derived using D_IFS(A, B) are prone to the influence of unfair arguments with too high or too low values, while the improved correlation coefficient (4) can relieve the influence of these unfair arguments by emphasizing the role of the considered arguments as a whole [21].

4.2 The Application in Clustering

In the following, we will show that the improved correlation coefficient (4) is more effective than correlation coefficients (1) and (2) in clustering.

For the object sets A_i (i=1, 2, …, m), suppose that the weight vector of x_i (i=1, 2, …, n) is ω=(ω₁, ω₂, …, ω_n)^T, then we discuss the clustering steps:

• Step 1: Utilize Eq. (4) to calculate the correlation coefficient ρ1∗(Ai, Aj) between object sets A_i and A_j (i, j=1, 2, …, m), and construct association matrix C˙=(ρ˙ij)m×m, where ρ˙ij=ρ1∗(Ai, Aj) (i, j=1, 2, ⋯, m).

• Step 2: Obtain equivalence association matrix C˙¯=(ρ˙¯ij)m×m by finite times compositions of C˙=(ρ˙ij)m×m and construct λ-cutting matrix C˙¯λ=(λρ¯ij)m×n of the equivalence association matrix C˙¯.

• Step 3: For the λ-cutting matrix C˙¯λ, if all the elements of the ith row or column are the same with the jth row or column, we assert object sets A_i and A_j are in the same class.

Example 4 ([28]): A car market is going to classify five different cars. Every car has six evaluation factors: (1) G₁ − fuel consumption; (2) G₂ − degree of friction; (3 )G₃ − the price; (4) G₄ − degree of comfort; (5) G₅ − design; (6) G₆ − security. The information of every car under each evaluation factor is represented by intuitionistic fuzzy numbers, which are shown in Table 4.

If the weight vector of the element x_i (i=1, 2, …, 6) is ω=(16, 16, ⋯, 16)T, let c=3, then we can classify the cars by the improved correlation coefficient ρ1∗(Ai, Aj).

First, we utilize the improved correlation coefficient ρ1∗(Ai, Aj) to calculate the correlation between each pair of IFSs A_i and A_j (i, j=1, 2, …, 5), and construct an association matrix:

C=[10.84220.82030.79810.68850.842210.86340.81090.80960.82030.863410.80470.75350.79810.81090.804710.70730.68850.80960.75350.70731]

Then, we obtain an equivalent association matrix by finite times compositions of C. For

C2=[10.84220.84220.81090.80960.842210.86340.81090.80960.84220.863410.81090.80960.81090.81090.810910.80960.80960.80960.80960.80961]⊈C

C4=[10.84220.84220.81090.80960.842210.86340.81090.80960.84220.863410.81090.80960.81090.81090.810910.80960.80960.80960.80960.80961]=C2,

we get the equivalent association matrix C², denoted by C˜.

Finally, due to the fact that the confidence level λ has a close relationship with the elements of the equivalent association matrix C˜, we give a detailed sensitivity analysis with respect to the confidence level λ and get all the possible classifications of the five cars A_i (i=1, 2, 3, 4, 5).

1. if 0≤λ≤0.8096, C˜λ=[1111111111111111111111111]

   then the cars are of the same type: {A₁, A₂, …, A₅};

2. if 0.8096<λ≤0.8109, C˜λ=[1111011110111101111000001]

   then the cars are classified into the following two types: {A₁, A₂, A₃, A₄}, {A₅};

3. if 0.8109<λ≤0.8422, C˜λ=[1110011100111000001000001]

   then the cars are classified into the following three types: {A₁, A₂, A₃}, {A₄}, {A₅};

4. if 0.8422<λ≤0.8634, C˜λ=[1000001100011000001000001]

   then the cars are classified into the following four types: {A₁}, {A₂, A₃}, {A₄}, {A₅};

5. if 0.8634<λ≤1, C˜λ=[1000001000001000001000001]

   then the cars are classified into the following five types: {A₁}, {A₂}, {A₃}, {A₄}, {A₅}.

From the above analysis, we know that the improved correlation coefficient (4) can be applied in the clustering of IFSs effectively. Moreover, there are five situations based on the association matrix, which is constructed by the improved correlation coefficient (4), while only three situations were obtained in Ref. [28]. It can be seen that the improved correlation coefficient has higher accuracy in clustering.