A new correlation coefficient of the intuitionistic fuzzy sets and its application

Abstract

In this paper, we propose a new correlation coefficient between intuitionistic fuzzy sets. We then use this new result to compute some examples through which we find that it benefits from such an outcome with some well-known results in the literature. As in statistics with real variables, we refer to variance and covariance between two intuitionistic fuzzy sets. Then, we determined the formula for calculating the correlation coefficient based on the variance and covariance of the intuitionistic fuzzy set, the value of this correlation coefficient is in [−1,1]. Then, we develop this direction to build correlation coefficients between the interval-valued intuitionistic fuzzy sets and apply it in the pattern recognition problem.

Keywords

Intuitionistic fuzzy set interval – valued intuitionistic fuzzy set variance covariance correlation coefficient

1 Introduction

In 1986, Atanassov introduced the intuitionistic fuzzy set [1], it is a generalization of Zadeh’s fuzzy set [19]. An intuitionistic fuzzy set (IFS) consider the information both the membership function and non-membership function. After that, the interval-valued intuitionistic fuzzy set (IVIFS) was introduced by Atanassov and Gargov [3]. In which, the membership function and non-membership function are subinterval of [0, 1]. As opposed to fuzzy sets, intuitionistic fuzzy set also have broad applications for uncertain data processing such as decision making, medical diagnose, agriculture [4 , 20]. Along with similar measurements, distance measurements, correlation measurements of intuitionistic fuzzy sets and interval – valued intuitionistic fuzzy set are also studied and widely used in many areas and now it is a hot topic [5 , 22].

The concept of the correlation coefficient was the first studied by Gerstenkorn, and Mańko in 1991 [7]. In this correlation coefficient, the variance and covariance are constructed directly from the scalar product of the values of the membership function and the non-membership function, respectively, of two intuitionistic fuzzy sets. Then, the correlation coefficient between the interval – valued intuitionistic fuzzy set is introduced by the Bustince and Burillo [5] in 1995. Later, many authors have studied and developed correlations in different trends and in other spaces. In the results of the correlation studies, in the 1990 s, we find that the values of the correlation coefficient are in the range [0,1].

In the 21st century, scholars have developed new methods for building correlation coefficients for fuzzy sets. In the new results, with some studies show the value of the correlation coefficient value received in the interval [−1,1]. We can include some methods as Hung’s method [8], and the method of Liu et al. [12]. Hung’s method is based on a statistical viewpoint to calculate the correlation coefficient. In 2016, Liu et al. [12] propose the concept of the deviation of the intuitionistic fuzzy numbers and based on this concept, the authors have developed the notion of correlation coefficient between the intuitionistic fuzzy sets, after that they also extend this approach to the IVIFSs case.

However, there are some cases where the correlation coefficient according to Hung’s method or method of (Liu et al.) does not cover them. These cases can be viewed in the examples presented in this article. In those cases, if using our method, there are reasonable results. In this paper, we propose a new method to determine the correlation coefficient between the intuitionistic fuzzy sets, in which the value of the correlation coefficient computed according to our method lies in interval [−1, 1]. Then, we develop the method to construct the correlation coefficient between the interval – valued intuitionistic fuzzy sets. Final, we also provide examples to illustrate our approach, and apply it to diagnostic problems in medicine and to the problem of pattern recognition.

The rest of this paper is organized as follows. In Section 2, we recall the concept of intuitionistic fuzzy set, interval – valued intuitionistic fuzzy set, the correlation coefficients which are computed by using the methods of Gerstenkorn and Mansko [7], Hung [8], Xu [17] and Liu et al. [12]. In Section 3, we construct the new correlation coefficient between the IFSs, in this section we also give some examples that compare the results computed based on our method to other methods. In Section 4, we extend our method to determine the correlation coefficient between the IVIFSs. Finally, we present the conclusion in Section 5.

2 Preliminaries

Let X be a universal set. We have

Definition 1. [1] An intuitionistic fuzzy set on X is a defined by form $A = {(x, μ_{A} (x), ν_{A} (x)) | x \in X}$ in which μ_A (x) ∈ [0, 1] and ν_A (x) ∈ [0, 1] are the membership degree and the non-membership of the element x in X to A, respectively, and $μ_{A} (x) + ν_{A} (x) \leq 1, \forall x \in X .$

Definition 2. [2, 3] An interval – value intuitionistic fuzzy set on X is a defined by form $A = {(x, [μ_{A}^{L} (x), μ_{A}^{U} (x)], [ν_{A}^{L} (x), ν_{A}^{U} (x)]) | x \in X}$ in which $[μ_{A}^{L} (x), μ_{A}^{U} (x)] \subseteq [0, 1]$ and $[ν_{A}^{L} (x), ν_{A}^{U} (x)] \subseteq [0, 1]$ are the membership degree and the non-membership of the element x in X to A, respectively, and $μ_{A}^{U} (x) + ν_{A}^{U} (x) \leq 1, \forall x \in X$ .

Now, we recall some knowledge correlation coefficient in literal.

Given X ={ x₁, x₂, …, x_n } is a universal set. And $\begin{matrix} A = {(x_{i}, μ_{A} (x_{i}), ν_{A} (x_{i})) | x_{i} \in X}, \\ B = {(x_{i}, μ_{B} (x_{i}), ν_{B} (x_{i})) | x_{i} \in X} \end{matrix}$

are two IFSs on X.

Definition 3. [2, 12] Let A = {(x_i, μ_A (x_i), ν_A (x_i)) | x_i ∈ X} be a IFS on X. The average of A is $E (A) = ({\bar{μ}}_{A}, {\bar{ν}}_{A}) = (\frac{1}{n} \sum_{i = 1}^{n} μ_{A} (x_{i}), \frac{1}{n} \sum_{i = 1}^{n} ν_{A} (x_{i}))$

Note that, when n = 2 we have $E (A) = ({\bar{μ}}_{A}, {\bar{ν}}_{A}) = (\frac{1}{2} \sum_{i = 1}^{2} μ_{A} (x_{i}), \frac{1}{2} \sum_{i = 1}^{2} ν_{A} (x_{i}))$

That was introduced in [17] by T. Buhaesku.

The correlation coefficient of Gerstenkorn and Mansko [8]. $ρ (A, B) = \frac{C (A, B)}{\sqrt{T (A) T (B)}}$

where $\begin{matrix} C (A, B) = \sum_{i = 1}^{n} [μ_{A} (x_{i}) μ_{B} (x_{i}) + ν_{A} (x_{i}) ν_{B} (x_{i})], \\ T (A) = \sum_{i = 1}^{n} [μ_{A}^{2} (x_{i}) + ν_{A}^{2} (x_{i})], \\ T (B) = \sum_{i = 1}^{n} [μ_{B}^{2} (x_{i}) + ν_{B}^{2} (x_{i})] . \end{matrix}$

The correlation coefficient of Hung [8] $ρ (A, B) = \frac{1}{2} (ρ_{1} + ρ_{2})$ where $\begin{matrix} ρ_{1} = \frac{\sum_{i = 1}^{n} (μ_{A} (x_{i}) - \bar{μ_{A}}) (μ_{B} (x_{i}) - \bar{μ_{B}})}{\sqrt{\sum_{i = 1}^{n} (μ_{A} (x_{i}) - \bar{μ_{A}})^{2}} \sqrt{\sum_{i = 1}^{n} (μ_{B} (x_{i}) - \bar{μ_{B}})^{2}}} \\ ρ_{2} = \frac{\sum_{i = 1}^{n} (ν_{A} (x_{i}) - \bar{ν_{A}}) (ν_{B} (x_{i}) - \bar{ν_{B}})}{\sqrt{\sum_{i = 1}^{n} (ν_{A} (x_{i}) - \bar{ν_{A}})^{2}} \sqrt{\sum_{i = 1}^{n} (ν_{B} (x_{i}) - \bar{ν_{B}})^{2}}} \end{matrix}$

The correlation coefficient of Xu [18] $ρ (A, B) = \frac{1}{2 n} \sum_{i = 1}^{n} [\frac{Δ μ_{min} + Δ μ_{\max}}{Δ μ_{i} + Δ μ_{max}} + \frac{Δ ν_{min} + Δ ν_{max}}{Δ ν_{i} + Δ ν_{max}}]$ where $\begin{matrix} Δ μ_{i} = | μ_{A} (x_{i}) - μ_{B} (x_{i}) |, Δ ν_{i} = | ν_{A} (x_{i}) - ν_{B} (x_{i}) | \\ Δ μ_{max} = m ax Δ μ_{i}, Δ ν_{max} = m ax Δ ν_{i} \\ Δ μ_{min} = min Δ μ_{i}, Δ ν_{min} = min Δ ν_{i} \end{matrix}$

The correlation coefficient of Liu et al. [12] $ρ (A, B) = \frac{C (A, B)}{\sqrt{D (A) D (B)}}$ where $C (A, B) = \frac{1}{n - 1} \sum_{i = 1}^{n} d_{i} (A) d_{i} (B)$ , $D (A) = \frac{1}{n - 1} \sum_{i = 1}^{n} d_{i}^{2} (A)$ $D (B) = \frac{1}{n - 1} \sum_{i = 1}^{n} d_{i}^{2} (B)$ in which $\begin{matrix} d_{i} (A) = (μ_{A} (x_{i}) - {\bar{μ}}_{A}) - (ν_{A} (x_{i}) - {\bar{ν}}_{A}) \\ d_{i} (B) = (μ_{B} (x_{i}) - {\bar{μ}}_{B}) - (ν_{B} (x_{i}) - {\bar{ν}}_{B}) \end{matrix}$ for all i = 1, 2, …, n .

3 A new correlation coefficient of the IFSs

Let X = {x₁, x₂, …, x_n} be a finite set, A and B are two arbitrary IFSs in X. We denote d_i (A) = (μ_A (x_i) - ${\bar{μ}}_{A})$ - (ν_A (x_i) - ${\bar{ν}}_{A})$ , d_i (B) = (μ_B (x_i) - ${\bar{μ}}_{B})$ - (ν_B (x_i) - ${\bar{ν}}_{B})$ for all i = 1, 2, …, n .

Definition 4. Let A = {(x_i μ_A (x_i) ν_A (x_i)) | x_i ∈ X} be a IFS on X. The variance of A can be represented as

$\begin{matrix} D (A) & = & \frac{1}{n - 1} \sum_{i = 1}^{n} {(μ_{A} (x_{i}) - {\bar{μ}}_{A})^{2} \\ + (ν_{A} (x_{i}) - {\bar{ν}}_{A})^{2} + d_{i}^{2} (A)} \end{matrix}$ (1)

Definition 5. Let A ={(x_i, μ_A (x_i) , ν_A (x_i)) |x_i ∈ X} and B ={(x_i, μ_B (x_i) , ν_B (x_i)) |x_i ∈ X} be two IFSs on X. The covariance of A and B can be defined by

$\begin{matrix} COV (A, B) \\ = \frac{1}{n - 1} \sum_{i = 1}^{n} {(μ_{A} (x_{i}) - {\bar{μ}}_{A}) (μ_{B} (x_{i}) - {\bar{μ}}_{B}) \\ + (ν_{A} (x_{i}) - {\bar{ν}}_{A}) (ν_{B} (x_{i}) - {\bar{ν}}_{B}) + d_{i} (A) d_{i} (B)} \end{matrix}$ (2)

Proposition 1. Let A and B be two arbitrary IFSs in X. We have

COV (A, B) = COV (B, A)

COV (A, A) = D (A)

|COV (A, B) | ≤ D (A) ^0.5D (B) ^0.5

Proof.

It is easily to obtain (1), (2).

(3). According to the Cauchy – Schwarz inequality | 〈 u, v 〉 |² ≤ ∥ u ∥ ² ∥ v ∥ ² i.e. ${(\sum_{k = 1}^{m} u_{k} v_{k})}^{2} \leq (\sum_{k = 1}^{m} u_{k}^{2}) (\sum_{k = 1}^{m} v_{k}^{2})$ for all $u = (u_{1}, u_{2}, \dots, u_{m}), v = (v_{1}, v_{2}, \dots, v_{m}) \in ℝ^{m}$ .

We have $\begin{matrix} COV (A, B)^{2} = {\frac{1}{n - 1} \sum_{i = 1}^{n} [(μ_{A} (x_{i}) - {\bar{μ}}_{A}) (μ_{B} (x_{i}) - {\bar{μ}}_{B}) \\ {+ (ν_{A} (x_{i}) - {\bar{ν}}_{A}) (ν_{B} (x_{i}) - {\bar{ν}}_{B}) + d_{i} (A) d_{i} (B)]}}^{2} \\ \leq \frac{1}{n - 1} \sum_{i = 1}^{n} [(μ_{A} (x_{i}) - \bar{μ_{A}})^{2} + (ν_{A} (x_{i}) - \bar{ν_{A}})^{2} + d_{i}^{2} (A)] \\ \times \frac{1}{n - 1} \sum_{i = 1}^{n} [(μ_{B} (x_{i}) - \bar{μ_{B}})^{2} + (ν_{B} (x_{i}) - \bar{ν_{B}})^{2} + d_{i}^{2} (B)] \\ = D (A) D (B) . \end{matrix}$

So that $| COV (A, B) | \leq D (A)^{0.5} D (B)^{0.5} □$

Now, we can define the correlation coefficient of the intuitionistic fuzzy sets. Which is similar to the correlation coefficient of real number variables in statistic.

Definition 6. Let A and B be two arbitrary IFSson X. The correlation coefficient of A and B can be defined by $ρ (A, B) = \frac{COV (A, B)}{\sqrt{D (A) D (B)}}$ (3)

Remark 1. Our correlation coefficient estimates the value in [-1, 1], while many known correlation coefficients take only values in [0, 1]. This is also consistent with the correlation coefficient we often see in real-world statistics. The correlation coefficient evaluates the linear relationship between the values of the two fuzzy sets on the elements of the observed set X. When two fuzzy sets have a real linear relationship then the correlation coefficient between themare ±1.

Theorem 1. Given two IFSs A and B , then we have

ρ (A, B) = ρ (B, A)

-1 ≤ ρ (A, B) ≤1

If A = kB + b for some k > 0, then ρ (A, B) =1. Here, A = kB + b means that μ_A = kμ_B + b and ν_A = kν_B + b.

If A = kB + b for some k < 0, then ρ (A, B) = -1 .

Proof.

Straightforward.

From proposition 2, we have |COV (A, B) | ≤ D (A) ^0.5D (B) ^0.5. It means that -D (A) ^0.5D (B) ^0.5 ≤ COV (A, B) ≤ D (A) ^0.5D (B) ^0.5 Hence, we have

$- 1 \leq ρ (A, B) = \frac{COV (A, B)}{\sqrt{D (A) D (B)}} \leq 1 .$

If μ_A = kμ_B + b and ν_A = kν_B + b we have

$\begin{matrix} COV (A, B) = \frac{1}{n - 1} \sum_{i = 1}^{n} {(μ_{A} (x_{i}) - {\bar{μ}}_{A}) (μ_{B} (x_{i}) \\ - {\bar{μ}}_{B}) + (ν_{A} (x_{i}) - {\bar{ν}}_{A}) (ν_{B} (x_{i}) - {\bar{ν}}_{B}) + d_{i} (A) d_{i} (B)} \\ = \frac{1}{n - 1} \sum_{i = 1}^{n} {(k μ_{B} (x_{i}) - k {\bar{μ}}_{B}) (μ_{B} (x_{i}) - {\bar{μ}}_{B}) \\ + (k ν_{B} (x_{i}) - k {\bar{ν}}_{B}) (ν_{B} (x_{i}) - {\bar{ν}}_{B}) + {kd}_{i} (B) d_{i} (B)} \\ = \frac{k}{n - 1} \sum_{i = 1}^{n} {(μ_{B} (x_{i}) - {\bar{μ}}_{B}) (μ_{B} (x_{i}) - {\bar{μ}}_{B}) \\ + (ν_{B} (x_{i}) - {\bar{ν}}_{B}) (ν_{B} (x_{i}) - {\bar{ν}}_{B}) + d_{i} (B) d_{i} (B)} \\ = kD (B) . \end{matrix}$ and $\begin{matrix} D (A) = \frac{1}{n - 1} \sum_{i = 1}^{n} {(μ_{A} (x_{i}) - {\bar{μ}}_{A})^{2} + (ν_{A} (x_{i}) - {\bar{ν}}_{A})^{2} + d_{i}^{2} (A)} \\ = \frac{1}{n - 1} \sum_{i = 1}^{n} {(k μ_{B} (x_{i}) - k {\bar{μ}}_{B})^{2} + (k ν_{B} (x_{i}) - k {\bar{ν}}_{B})^{2} + {kd}_{i}^{2} (B)} \\ = \frac{k}{n - 1} \sum_{i = 1}^{n} {(μ_{B} (x_{i}) - {\bar{μ}}_{B})^{2} + (ν_{B} (x_{i}) - {\bar{ν}}_{B})^{2} + d_{i}^{2} (B)} \\ = k^{2} D (B) . \end{matrix}$

If k > 0 then $\begin{matrix} ρ (A, B) = \frac{COV (A, B)}{\sqrt{D (A) D (B)}} = \frac{kD (B)}{\sqrt{k^{2} D (B) D (B)}} \\ = \frac{kD (B)}{kD (B)} = 1 . \end{matrix}$

If k < 0 then

$ρ (A, B) = \frac{COV (A, B)}{\sqrt{D (A) D (B)}} = \frac{kD (B)}{- kD (B)} = - 1 . □$

Now, we consider some examples to compare our proposed correlation coefficient and some other knowledge correlation coefficient.

Example 1. In this example, let’s look at Liu’s example in [12]. Suppose that A and B are two IFSs in X ={ x₁, x₂, x₃ } where $\begin{matrix} A = {(x_{1}, 0.1, 0.2), (x_{2}, 0.2, 0.1), (x_{3}, 0.3, 0)}, \\ B = {(x_{1}, 0.3, 0), (x_{2}, 0.2, 0.2), (x_{3}, 0.1, 0.4)} . \end{matrix}$

(1) By our method $E (A) = (\frac{0.1 + 0.2 + 0.3}{3}, \frac{0.2 + 0.1 + 0}{3}) = (0.2, 0.1)$ $E (B) = (\frac{0.3 + 0.2 + 0.1}{3}, \frac{0 + 0.2 + 0.4}{3}) = (0.2, 0.2)$ $\begin{matrix} d_{1} (A) = (μ_{A} (x_{1}) - {\bar{μ}}_{A}) - (ν_{A} (x_{1}) - {\bar{ν}}_{A}) = - 0.2, \\ d_{2} (A) = (μ_{A} (x_{2}) - {\bar{μ}}_{A}) - (ν_{A} (x_{2}) - {\bar{ν}}_{A}) = 0, \\ d_{3} (A) = (μ_{A} (x_{3}) - {\bar{μ}}_{A}) - (ν_{A} (x_{3}) - {\bar{ν}}_{A}) = 0.2 \end{matrix}$ and $d_{1} (B) = 0.3, d_{2} (B) = 0, d_{3} (B) = - 0.3 .$ Hence $\begin{matrix} COV (A, B) = \frac{1}{n - 1} \sum_{i = 1}^{n} {(μ_{A} (x_{i}) - {\bar{μ}}_{A}) (μ_{B} (x_{i}) - {\bar{μ}}_{B}) \\ + (ν_{A} (x_{i}) - {\bar{ν}}_{A}) (ν_{B} (x_{i}) - {\bar{ν}}_{B}) + d_{i} (A) d_{i} (B)} \\ = \frac{1}{2} {(0.1 - 0.2) \times (0.3 - 0.2) + (0.2 - 0.1) \times (0 - 0.2) \\ + (0.2 - 0.2) \times (0.2 - 0.2) + (0.1 - 0.1) \times (0.2 - 0.2) \\ + (0.3 - 0.2) \times (0.1 - 0.2) + (0 - 0.1) \times (0.4 - 0.2) \\ + (- 0.2) \times 0.3 + 0 \times 0 + (- 0.2) \times 0.2 = - 0.09 . \end{matrix}$ and $\begin{matrix} D (A) = \frac{1}{n - 1} \sum_{i = 1}^{n} {(μ_{A} (x_{i}) - {\bar{μ}}_{A})^{2} + (ν_{A} (x_{i}) - {\bar{ν}}_{A})^{2} + d_{i}^{2} (A)} \\ = \frac{1}{2} {(0.1 - 0.2)^{2} + (0.2 - 0.1)^{2} + (- 0.2)^{2} \\ + (0.2 - 0.2)^{2} + (0.1 - 0.1)^{2} + 0^{2} \\ + (0.3 - 0.2)^{2} + (0 - 0.1)^{2} + 0 . 2^{2}} = 0.06 . \end{matrix}$ $\begin{matrix} D (B) = \frac{1}{n - 1} \sum_{i = 1}^{n} {(μ_{B} (x_{i}) - {\bar{μ}}_{B})^{2} + (ν_{B} (x_{i}) - {\bar{ν}}_{B})^{2} + d_{i}^{2} (B)} \\ = \frac{1}{2} {(0.3 - 0.2)^{2} + (0.0 - 0.2)^{2} + (0.3)^{2} \\ + (0.2 - 0.2)^{2} + (0.2 - 0.2)^{2} + 0^{2} \\ + (0.1 - 0.2)^{2} + (0.4 - 0.2)^{2} + (- 0.3)^{2}} = 0.14 \end{matrix}$

So that $ρ (A, B) = \frac{- 0.09}{\sqrt{0.06 \times 0.14}} = - 0.98198 .$

(2) By the method of Liu et al in [12] $ρ (A, B) = - 1 .$

(3) By the method in Gerstenkorn and Marko [7] $ρ (A, B) = 0.12 .$

(4) By the method in Xu [18] $ρ (A, B) ≃ 0.74 .$

(5) By the method in Hung [8] $ρ (A, B) = - 1 .$

These results are consistent, because A is the set of elements whose value increases, B is a set whose values are decreasing. But not exist k ≠ 0 such that B = kA + b, in particular μ_B = - μ_A + 0.4 and ν_B = -2ν_A + 0.4.

Example 2. Suppose that A and B are two IFSs in X ={ x₁, x₂, x₃ } where $\begin{matrix} A = {(x_{1}, 0.14, 0.17), (x_{2}, 0.14, 0.17), (x_{3}, 0.7, 0.2)} \\ B = {\begin{matrix} (x_{1}, 0.158, 0.149), (x_{2}, 0.158, 0.149), \\ (x_{3}, 0.149, 0.14) \end{matrix}} \end{matrix}$

(1) By our method $\begin{matrix} E (A) = (0.15, 0.18); E (B) = (0.155, 0.146) \\ COV (A, B) = - 0.00018; D (A) = 0.0006; \\ D (B) = 0.000054 \end{matrix}$ and $ρ (A, B) = - 1 .$

(2) By the method of Liu et al. in [12]

We can not determine the correlation coefficient according to this method.

(3) By the method in Hung [8] $ρ (A, B) = - 1 .$

In this example, the results of our method match those of the Hung’s method. In this example, the results calculated according to our method coincide with the results calculated by the Hung’s method, both methods yield a correlation coefficient ρ (A, B) = -1 . This result is reasonable, because two intuitionistic fuzzy sets A, B have a linear relation B = -0.3A + 0.2. But the method of Liu et al. in [12] does not tell us anything about thisdata.

Example 3. Suppose that A and B are two IFSs in X ={ x₁, x₂, x₃ } where $\begin{matrix} A = {(x_{1}, 0.14, 0.17), (x_{2}, 0.14, 0.17), (x_{3}, 0.17, 0.2)} \\ B = {\begin{matrix} (x_{1}, 0.27, 0.285), (x_{2}, 0.27, 0.85), \\ (x_{3}, 0.285, 0.3) \end{matrix}} \end{matrix}$

(1) By our method $\begin{matrix} E (A) = (0.15, 0.18); E (B) = (0.275, 0.29) \\ COV (A, B) = 0.0006; D (A) = 0.0012; \\ D (B) = 0.0003 \end{matrix}$

and ρ (A, B) =1 .

(2) By the method of Liu et al. in [12]

We can not determine the correlation coefficient according to this method.

(3) By the method in Hung [8] $ρ (A, B) = 1 .$

In this example, the results of our method match those of the Hung’s method. In this example, the results calculated according to our method also coincide with the results calculated by the Hung’s method, both methods yield a correlation coefficient ρ (A, B) =1 . This result is reasonable, because two intuitionistic fuzzy sets A, B have a linear relation B = 0.5A + 0.2. But the method of Liu et al. in [12] does not tell us anything about this data.

Example 4. Suppose that A and B are two IFSs in X ={ x₁, x₂, x₃ } where $\begin{matrix} A = {(x_{1}, 0, 0.1), (x_{2}, 0, 0.2), (x_{3}, 0, 0.3)} \\ B = {(x_{1}, 0.4, 0.44), (x_{2}, 0.4, 0.48), (x_{3}, 0.4, 0.52)} \end{matrix}$

(1) By our method $\begin{matrix} E (A) = (0, 0.2); E (B) = (0.4, 0.48) \\ COV (A, B) = 0.008; D (A) = 0.02; D (B) = 0.0032 \end{matrix}$ and ρ (A, B) =1 .

(2) By the method of Liu et al. in [12] $ρ (A, B) = 1 .$

(3) By the method in Hung [8]

We can not determine the correlation coefficient according to this method.

In this example, the results of our method match those of Liu et al. [12]. In this example, the results calculated according to our method also coincide with the results calculated by using the method of Liu et al, both methods yield a correlation coefficient ρ (A, B) =1 . This result is reasonable, because two intuitionistic fuzzy sets A, B have a linear relation B = 0.4A + 0.4. But the method of Hung in [8] does not tell us anything about this data.

Example 5. Suppose that A and B are two IFSs in X ={ x₁, x₂, x₃ } where $\begin{matrix} A = {(x_{1}, 0.1, 0), (x_{2}, 0.2, 0), (x_{3}, 0.3, 0)} \\ B = {(x_{1}, 0.36, 0.4), (x_{2}, 0.32, 0.4), (x_{3}, 0.28, 0.4)} \end{matrix}$

(1) By our method $\begin{matrix} E (A) = (0.2, 0); E (B) = (0.32, 0.4) \\ COV (A, B) = - 0.112; D (A) = 0.28; D (B) = 0.0448 \end{matrix}$ and ρ (A, B) = -1 .

(2) By the method of Liu et al. in [12] $ρ (A, B) = - 1 .$

(3) By the method in Hung [8]

We can not determine the correlation coefficient according to this method.

In this example, the results of our method match those of Liu et al. In this example, the results calculated according to our method also coincide with the results calculated by using the method of Liu et al, both methods yield a correlation coefficient ρ (A, B) = -1 . Because two intuitionistic fuzzy sets A, B have a linear relation B = -0.4A + 0.4. But the method of Hung in [8] does not tell us anything about this data.

Example 6. In this example, we use the our method to the medical diagnosis with data quoted in Szmidt and Kacprzyk [16]. Usage of diagnostic methods D = {Viral fever (V) , Malaria (M) , Typhoid (T) , Stomach problem (S) , Chest problem} for patients with given values of symptoms S=temperature, headache, stomach pain, cough, chest pain. In this case, the intuitionistic fuzzy set is useful to handle them. Here, for each d_k ∈ D (k = 1, 2, …, 5) is expressed in form that is an intuitionistic fuzzy set on the universal set S ={ s₁, s₂, s₃, s₄, s₅ }, see Table 1. The information of symptoms characteristic for the considered patients is given in Table 2. In which, for each patient p_i (i = 1, 2, 3, 4) is an intuitionistic fuzzy set in the universal set

Table 1

Symptoms characteristic for the considered diagnoses

	Viral fever	Malaria	Typhoid	Stomach problem	Chest problems
Temperature	(0.4,0)	(0.7,0)	(0.3,0.3)	(0.1,0.7)	(0.1,0.8)
Headache	(0.3,0.5)	(0.2,0.6)	(0.6,0.1)	(0.2,0.4)	(0,0.8)
Stomach pain	(0.1,0.7)	(0,0.9)	(0.2,0.7)	(0.8,0)	(0.2,0.8)
Cough	(0.4,0.3)	(0.7,0)	(0.2,0.6)	(0.2,0.7)	(0.2,0.8)
Chest pain	(0.1,0.7)	(0.1,0.8)	(0.1,0.9)	(0.2,0.7)	(0.8,0.1)

Table 2

Symptoms characteristic for the considered patients

	Temperature	Headache	Stomach pain	Cough	Chest pain
Al	(0.8,0.1)	(0.6,0.1)	(0.2,0.8)	(0.6,0.1)	(0.1,0.6)
Bob	(0,0.8)	(0.4,0.4)	(0.6,0.1)	(0.1,0.7)	(0.1,0.8)
Joe	(0.8,0.1)	(0.8,0.1)	(0,0.6)	(0.2,0.7)	(0,0.5)
Ted	(0.6,0.1)	(0.5,0.4)	(0.3,0.4)	(0.7,0.2)	(0.3,0.4)

$S = {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}} .$

To select the appropriate diagnostic method we calculate the correlation of each patient with the diagnostic methods. For each patient, the appropriate diagnostic method will have the highest correlation coefficient.

The correlation coefficients of a diagnosis d_k ∈ D (k = 1, 2, …, 5) for each patient p_i (i = 1, 2, 3, 4) is $ρ (p_{i}, d_{k}) = \frac{COV (p_{i}, d_{k})}{\sqrt{D (p_{i}) D (d_{k})}} .$

The computed results of correlation coefficients are listed in Table 3. From the results, we see that Al should use diagnostic methods corresponding to Viral fever, Bob use a Stomach problem, Joe use a Typhoid and Ted use a Chest problems.

Table 3

Correlation coefficients of symptoms for each patient to the considered set of possible diagnoses

	Viral fever	Malaria	Typhoid	Stomach problem	Chest problems
Al	0.2819	0.228	0.159	−0.7217	−0.7217
Bob	−0.6755	−0.7838	−0.2136	0.3025	−0.5736
Joe	0.0713	−0.0869	0.2633	−0.55	−0.6316
Ted	0.3231	0.1854	−0.1179	−0.7565	−0.7243

We also cite the results listed in [12]. These results together with the results calculated according to our method are listed in Table 4. From this table, we see that the computed result by our method is identical to the computed result by Hung’s method.

Table 4

The most possible diagnosis for each patient under different methods

	Our method	Method in	Method in	Method in	Method in	Method in
		Ref. [12]	Ref. [7]	Ref. [18]	Ref. [8]	Ref. [16]
Al	V	V	M	M	V	V
Bob	S	S	S	S	S	S
Joe	T	T	T	V	T	T
Ted	V	M	V	V	V	M

All six methods point to Bob in accordance with our S diagnosis. Our method and the other four methods indicate that Joe should use T diagnostics. Patient Al should use V, and Ted use diagnosis V.

4 The correlation coefficient of the IVIFSs

In this section, we extend our method to the IVIFSs.

Definition 7. Let X ={ x₁, x₂, …, x_n } be a universal set. Given an IVIFSs

$Ã = {(x_{i}, [μ_{\tilde{Ã}}^{L} (x_{i}), μ_{\tilde{A}}^{U} (x_{i})], [ν_{\tilde{Ã}}^{L} (x_{i}), ν_{\tilde{Ã}}^{U} (x_{i})]) | x_{i} \in X}$ The average of A is $E (Ã) = (\begin{matrix} [\frac{1}{n} \sum_{i = 1}^{n} μ_{\tilde{A}}^{L} (x_{i}), \frac{1}{n} \sum_{i = 1}^{n} μ_{\tilde{A}}^{U} (x_{i})], \\ [\frac{1}{n} \sum_{i = 1}^{n} ν_{\tilde{A}}^{L} (x_{i}), \frac{1}{n} \sum_{i = 1}^{n} ν_{\tilde{A}}^{U} (x_{i})] \end{matrix})$ (4)

We denote $μ_{\tilde{A}} (x_{i}) = \frac{μ_{\tilde{A}}^{L} (x_{i}) + μ_{\tilde{A}}^{U} (x_{i})}{2}$ , $ν_{\tilde{A}} (x_{i}) = \frac{ν_{\tilde{A}}^{L} (x_{i}) + ν_{\tilde{A}}^{U} (x_{i})}{2}$ , ${\bar{μ}}_{\tilde{A}} = \frac{1}{2} (\frac{1}{n} \sum_{i = 1}^{n} μ_{\tilde{A}}^{L} (x_{i}) + \frac{1}{n} \sum_{i = 1}^{n} μ_{\tilde{A}}^{U} (x_{i}))$ (5) and ${\bar{ν}}_{\tilde{A}} = \frac{1}{2} (\frac{1}{n} \sum_{i = 1}^{n} ν_{\tilde{A}}^{L} (x_{i}) + \frac{1}{n} \sum_{i = 1}^{n} ν_{\tilde{A}}^{U} (x_{i})) .$ (6)

We can determine variance, covariance and correlation coefficient between IVIFSs.

Definition 8. Let $\tilde{A}$ be a IVIFS on X. $\tilde{A} = {(x_{i}, [μ_{\tilde{A}}^{L} (x_{i}), μ_{\tilde{A}}^{U} (x_{i})], [ν_{\tilde{A}}^{L} (x_{i}), ν_{\tilde{A}}^{U} (x_{i})]) | x_{i} \in X}$

We denote $\begin{matrix} d_{i} (\tilde{A}) = (μ_{\tilde{A}} (x_{i}) - {\bar{μ}}_{\tilde{A}}) - (ν_{\tilde{A}} (x_{i}) - {\bar{ν}}_{\tilde{A}}), \\ d_{i} (\tilde{B}) = (μ_{\tilde{B}} (x_{i}) - {\bar{μ}}_{\tilde{B}}) - (ν_{\tilde{B}} (x_{i}) - {\bar{ν}}_{\tilde{B}}) \end{matrix}$

The variance of $\tilde{A}$ can be represented as

$\begin{matrix} D (\tilde{A}) & = & \frac{1}{n - 1} \sum_{i = 1}^{n} {(μ_{\tilde{A}} (x_{i}) - {\bar{μ}}_{\tilde{A}})^{2} \\ + (ν_{\tilde{A}} (x_{i}) - {\bar{ν}}_{\tilde{A}})^{2} + d_{i}^{2} (\tilde{A})} \end{matrix}$ (7)

Definition 9. Let $\tilde{A}$ and $\tilde{B}$ be two IVIFSs on X $\begin{matrix} \tilde{A} = {(x_{i}, [μ_{\tilde{A}}^{L} (x_{i}), μ_{\tilde{A}}^{U} (x_{i})], [ν_{\tilde{A}}^{L} (x_{i}), ν_{\tilde{A}}^{U} (x_{i})]) | x_{i} \in X} \\ \tilde{B} = {(x_{i}, [μ_{\tilde{B}}^{L} (x_{i}), μ_{\tilde{B}}^{U} (x_{i})], [ν_{\tilde{B}}^{L} (x_{i}), ν_{\tilde{B}}^{U} (x_{i})]) | x_{i} \in X} \end{matrix}$

The covariance of $\tilde{A}$ and $\tilde{B}$ can be defined by $\begin{matrix} COV (\tilde{A}, \tilde{B}) = \frac{1}{n - 1} \sum_{i = 1}^{n} {(μ_{\tilde{A}} (x_{i}) - {\bar{μ}}_{\tilde{A}}) (μ_{\tilde{B}} (x_{i}) - {\bar{μ}}_{\tilde{B}}) \\ + (ν_{\tilde{A}} (x_{i}) - {\bar{ν}}_{\tilde{A}}) (ν_{\tilde{B}} (x_{i}) - {\bar{ν}}_{\tilde{B}}) + d_{i} (\tilde{A}) d_{i} (\tilde{B})} \end{matrix}$ (8)

Proposition 2. Let $\tilde{A}$ and $\tilde{B}$ be two arbitrary IVIFSs on X. We have $\begin{matrix} (1) COV (\tilde{A}, \tilde{B}) = COV (\tilde{B}, \tilde{A}) \\ (2) COV (\tilde{A}, \tilde{A}) = D (\tilde{A}) \\ (3) | COV (\tilde{A}, \tilde{B}) | \leq D (\tilde{A})^{0.5} D (\tilde{B})^{0.5} \end{matrix}$ (9)

Proof. Similarity to proof of Proposition 1. □

Now, we can define the correlation coefficient of the interval - valued intuitionistic fuzzy sets (IVIFSs). Which is similar to the correlation coefficient of real number variables in statistic.

Definition 10. Let $\tilde{A}$ and $\tilde{B}$ be two arbitrary IVIFSs on X. The correlation coefficient of $\tilde{A}$ and $\tilde{B}$ can be defined by $ρ (\tilde{A}, \tilde{B}) = \frac{COV (\tilde{A}, \tilde{B})}{\sqrt{D (\tilde{A}) D (\tilde{B})}}$ (10)

Theorem 2. Given two IVIFSs $\tilde{A}$ and $\tilde{B}$ , then we have

$ρ (\tilde{A}, \tilde{B}) = ρ (\tilde{B}, \tilde{A})$

$- 1 \leq ρ (\tilde{A}, \tilde{B}) \leq 1$ .

If $\tilde{A} = k \tilde{B} + b$ for some k > 0, then $ρ (\tilde{A}, \tilde{B}) = 1$ . Here, $\tilde{A} = k \tilde{B} + b$ means that $μ_{\tilde{A}} = k μ_{\tilde{B}} + b$ and $ν_{\tilde{A}} = k ν_{\tilde{B}} + b$ .

If $\tilde{A} = k \tilde{B} + b$ for some k < 0, then $ρ (\tilde{A}, \tilde{B}) = - 1 .$

Proof.

(1), (2) is easy to verify.

(3). If $\tilde{A} = k \tilde{B} + b$ means that $μ_{\tilde{A}}^{L} = k μ_{\tilde{B}}^{L} + b$ , $μ_{\tilde{A}}^{U} = k μ_{\tilde{B}}^{U} + b$ and $ν_{\tilde{A}}^{L} = k ν_{\tilde{B}}^{L} + b$ , $ν_{\tilde{A}}^{U} = k ν_{\tilde{B}}^{U} + b$ . We have $\begin{matrix} μ_{\tilde{A}} (x_{i}) = \frac{μ_{\tilde{A}}^{L} (x_{i}) + μ_{\tilde{A}}^{U} (x_{i})}{2} = \frac{k μ_{\tilde{B}}^{L} (x_{i}) + b + k μ_{\tilde{B}}^{U} (x_{i}) + b}{2} \\ = \frac{k (μ_{\tilde{B}}^{L} (x_{i}) + μ_{\tilde{B}}^{U} (x_{i})) + 2 b}{2} = k μ_{\tilde{B}} (x_{i}) + b, \\ ν_{\tilde{A}} (x_{i}) = \frac{ν_{\tilde{A}}^{L} (x_{i}) + ν_{\tilde{A}}^{U} (x_{i})}{2} = \frac{k ν_{\tilde{B}}^{L} (x_{i}) + b + k ν_{\tilde{B}}^{U} (x_{i}) + b}{2} \\ = \frac{k (ν_{\tilde{B}}^{L} (x_{i}) + ν_{\tilde{B}}^{U} (x_{i})) + 2 b}{2} = k ν_{\tilde{B}} (x_{i}) + b . \end{matrix}$

So that $\begin{matrix} {\bar{μ}}_{\tilde{A}} = \frac{1}{2} (\frac{1}{n} \sum_{i = 1}^{n} μ_{\tilde{A}}^{L} (x_{i}) + \frac{1}{n} \sum_{i = 1}^{n} μ_{\tilde{A}}^{U} (x_{i})) \\ = k {\bar{μ}}_{\tilde{B}} + b, \\ {\bar{ν}}_{\tilde{A}} = \frac{1}{2} (\frac{1}{n} \sum_{i = 1}^{n} ν_{\tilde{A}}^{L} (x_{i}) + \frac{1}{n} \sum_{i = 1}^{n} ν_{\tilde{A}}^{U} (x_{i})), \\ = k {\bar{ν}}_{\tilde{B}} + b . \end{matrix}$

Table 5

Data information of minerals

	${\tilde{A}}_{1}$	${\tilde{A}}_{2}$	${\tilde{A}}_{3}$	$\tilde{A}$
x ₁	([0.72,0.74],[0.1,0.12])	([0.42,0.45],[0.38,0.40])	([0.30,0.32],[0.45,0.47])	([0.60,0.63],[0.30,0.35])
x ₂	([0.00,0.05],[0.80,0.82])	([0.65,0.67],[0.28,0.30])	([0.90,1.00],[0.00,0.00])	([0.50,0.53],[0.34,0.36])
x ₃	([0.18,0.20],[0.62,0.63])	([0.00,1.00],[0.00,0.00])	([0.18,0.20],[0.70,0.73])	([0.20,0.21],[0.68,0.70])
x ₄	([0.49,0.50],[0.35,0.37])	([0.70,0.90],[0.00,0.10])	([0.15,0.16],[0.75,0.78])	([0.20,0.22],[0.75,0.77])
x ₅	([0.01,0.02],[0.60,0.63])	([0.80,1.00],[0.00,0.00])	([0.00,0.05],[0.88,0.90])	([0.05,0.07],[0.87,0.90])
x ₆	([0.72,0.74],[0.12,0.13])	([0.90,1.00],[0.00,0.00])	([0.65,0.68],[0.25,0.30])	([0.65,0.70],[0.25,0.30])

Table 6

Correlation coefficients between ${\tilde{Ã}}_{i} (i = 1, 2, 3)$ and $\tilde{Ã}$ under different methods

	Our method	Method in	Method in	Method in	Method in	Method in
		Ref. [12]	Ref. [5]	Ref. [18]	Ref. [8]	Ref. [21]
$ρ ({\tilde{A}}_{1}, \tilde{A})$	−0.02	0.53	0.86	0.78	0.52	0.85
$ρ ({\tilde{A}}_{2}, \tilde{A})$	−0.59	−0.52	0.52	0.77	−0.53	0.51
$ρ ({\tilde{A}}_{3}, \tilde{A})$	0.17	0.81	0.94	0.84	0.81	0.94

Hence, we have $\begin{matrix} COV (\tilde{A}, \tilde{B}) = \frac{1}{n - 1} \sum_{i = 1}^{n} {(μ_{\tilde{A}} (x_{i}) - {\bar{μ}}_{\tilde{A}}) (μ_{\tilde{B}} (x_{i}) \\ - {\bar{μ}}_{\tilde{B}}) + (ν_{\tilde{A}} (x_{i}) - {\bar{ν}}_{\tilde{A}}) (ν_{\tilde{B}} (x_{i}) - {\bar{ν}}_{\tilde{B}}) + d_{i} (\tilde{A}) d_{i} (\tilde{B})} \\ = \frac{1}{n - 1} \sum_{i = 1}^{n} {(k μ_{\tilde{B}} (x_{i}) - k {\bar{μ}}_{\tilde{B}}) (μ_{\tilde{B}} (x_{i}) - {\bar{μ}}_{\tilde{B}}) \\ + (k ν_{\tilde{B}} (x_{i}) - k {\bar{ν}}_{\tilde{B}}) (ν_{\tilde{B}} (x_{i}) - {\bar{ν}}_{\tilde{B}}) + {kd}_{i} (\tilde{B}) d_{i} (\tilde{B})} \\ = \frac{k}{n - 1} \sum_{i = 1}^{n} {(μ_{\tilde{B}} (x_{i}) - {\bar{μ}}_{\tilde{B}}) (μ_{\tilde{B}} (x_{i}) - {\bar{μ}}_{\tilde{B}}) \\ + (ν_{\tilde{B}} (x_{i}) - {\bar{ν}}_{\tilde{B}}) (ν_{\tilde{B}} (x_{i}) - {\bar{ν}}_{\tilde{B}}) + d_{i} (\tilde{B}) d_{i} (\tilde{B})} \\ = kD (\tilde{B}) . \end{matrix}$ and $\begin{matrix} D (\tilde{A}) = \frac{1}{n - 1} \sum_{i = 1}^{n} {(μ_{\tilde{A}} (x_{i}) - {\bar{μ}}_{\tilde{A}})^{2} + (ν_{\tilde{A}} (x_{i}) - {\bar{ν}}_{\tilde{A}})^{2} + d_{i}^{2} (\tilde{A})} \\ = \frac{1}{n} \sum_{i = 1}^{n} [k^{2} (μ_{\tilde{B}} (x_{i}) - {\bar{μ}}_{\tilde{B}})^{2} + k^{2} (ν_{\tilde{B}} (x_{i}) - {\bar{ν}}_{\tilde{B}})^{2} + k^{2} d_{i} (\tilde{B})^{2}] \\ = k^{2} D (\tilde{B}) \end{matrix}$

If k > 0 then $ρ (\tilde{A}, \tilde{B}) = \frac{COV (\tilde{A}, \tilde{B})}{\sqrt{D (\tilde{A}) D (\tilde{B})}} = 1 .$

(4) If k < 0 then $\begin{matrix} ρ (\tilde{A}, \tilde{B}) = \frac{COV (\tilde{A}, \tilde{B})}{\sqrt{D (\tilde{A}) D (\tilde{B})}} \\ = \frac{kD (\tilde{B})}{\sqrt{k^{2} D (\tilde{B}) D (\tilde{B})}} = \frac{kD (\tilde{B})}{- kD (\tilde{B})} = - 1 . □ \end{matrix}$

Example 7. We consider a pattern recognition problem about the classification of minerals; the data was quoted from Liu et al. [12]. There are three classes of given minerals, which are expressed by the IVIFSs ${\tilde{A}}_{1}, {\tilde{A}}_{2}, {\tilde{A}}_{3}$ in the feature space X ={ x₁, x₂, x₃, x₄, x₅, x₆ }. Now, if there is a new mineral $\tilde{A}$ along with its known attribute values. Our aim is to determine which class that $\tilde{A}$ belongs to. The descriptive data information is given in Table 5.

From the result in Table 6, we can comment that the correlation coefficients between ${\tilde{A}}_{i} (i = 1, 2, 3)$ and $\tilde{A}$ as follow $ρ ({\tilde{A}}_{3}, \tilde{A}) > ρ ({\tilde{A}}_{2}, \tilde{A}) > ρ ({\tilde{A}}_{1}, \tilde{A})$ , so that we can put $\tilde{A}$ belongs to class ${\tilde{A}}_{3}$ . We also compare the result of our method to the results obtained by other methods in Refs. [5 , 21], the result using our method is identical with the results using methods in Refs. [5 , 21].

5 Conclusion

From a statistical standpoint, the domain of the correlation coefficient is [−1,1]. Not only does our research indicate that, but many other studies point out. This value domain is more significant than correlation coefficients for only [0,1], because, besides pointing out the linear relationship between two sets of data in a space of observation objects, the correlation coefficient also indicates the variability of the two sets of data. Two datasets may have the same tendency, or tend to decrease (in case of positive correlation). It is also possible that the first data set is incremented, the second data set is reduced; in contrast, the first data set is reduced, the second data set is incremented (in the case of a negative correlation). The correlation coefficient of two intuitionistic fuzzy sets should also reflect this. In addition, because of the characteristics of intuitionistic fuzzy sets, there are two functions: a membership function, a non-membership function of a intuitionistic fuzzy set by the feature (elements) of the sample space. Thus, the correlation between intuitionistic fuzzy sets has its own characteristics. As many of the authors have previously studied, we consider the correlation coefficients of intuitionistic fuzzy sets based on both membership functions and non-member functions. Our method can effectively solve some cases where a previous method was difficult. This is demonstrated by the examples we present in this article. In this article, we also apply the methods we propose in determining the appropriate diagnostic methods in medicine, and in the problem of patternrecognition.

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