Representing complex vague soft set by quaternion numbers

Abstract

Based on the quaternion system, we give a new representation of the complex vague soft set, and related logical operations. This new representation contains more information than before. Three quaternion distance measures are proposed and a decision model is established. The disease diagnosis of breast cancer is applied to the model to reflect the superiority of the model. By comparing the diagnostic errors under the different distance measures, the most suitable distance measure for this dataset is selected.

Keywords

Quaternion vague soft set complex vague soft set

1 Introduction

In real life, there are numerous uncertain data. However, when we apply these data to make decisions only through classical theory, we sometimes fail to achieve our goals. Therefore, many mathematicians began to study uncertain data. Over time, a set of systematic fuzzy theory has been gradually formed to solve these problems, such as Intuitive fuzzy set theory, Rough set theory and so on.

In 1999, in order to deal with uncertainties, Molodtsov [1] introduced the concept of soft set as a new mathematical tool. Since then, the soft set theory has received extensive attention. The concept and basic properties of soft set theory were presented in [1, 2]. Maji et al. [3] were the first one to introduce the concept of fuzzy soft sets, and Majumdar and Samanta [4] further generalized it. Yang et al. [5] introduced the notion of interval-valued fuzzy soft set. Xiao et al. [6] introduced the notion of exclusive disjunctive soft sets. Maji et al. [7, 8] gave the concept of intuitionistic fuzzy soft sets. Wang and Qu [9] introduced several different measures and their properties of vague soft sets.

The theory of vague sets was first proposed by Gau and Buehrer [10]. A vague set is defined by a closed interval formed by a combination of membership function and non-membership function. In fact, vague set is an extension of fuzzy set [9]. The basic concepts and some applications of vague set theory and its extensions can be found in [10 –18]. As mentioned in [19], vague sets are equivalent to intuitionistic fuzzy sets (IFS). So we conclude that the equivalence relation still holds after combining the above two sets with the concept of soft set [9, 11].

In 2011, Tamir et al. [20] gave the representation of the pure complex grade of membership in the polar coordinate system and the Cartesian coordinate system respectively, and proposed a new concept of complex fuzzy class (CFC). Other researchers had studied more related concepts on this basis. Among them, Selvachandran et al. [21] proposed the concept of complex vague soft set (CVSS) based on the polar representation form of complex membership function. In 2016, Tamir et al. [22] combined the degrees of membership with non-membership in IFS by the tool of complex numbers, and proposed several important properties. As an extension of the complex number system in the form of Cartesian coordinates, the quaternion system expands the dimension of the space from two to four, which can contain more information. Based on the advantages of the quaternion system, Ngan et al. [23] combined the system with IFSs to include more fuzzy information.

Inspired by the previous paragraph, in this paper, we combine the advantages of the quaternion system with CVSS, and introduce a new representation of complex vague soft sets with the help of the quaternion system. Then we propose three different distance measures, and give a new decision model to apply. Compared to the representation of VSS, new representation extands the dimension of fuzzy information. Compared to the previous CVSS representation, the main contributions of using quaternion representation are: (1) this representation combines the advantages of the quaternion system and provides a compact representation of CVSS; (2) it combines membership and non-membership functions together for a more comprehensive analysis to make the final result more accurate. Finally, the practicality of the model in medicine can be seen through examples in medical diagnosis.

2 Preliminaries

In this section, we will show several basic notions which are necessary for this paper. They are presented as follows:

Definition 2.1. [10] A vague set over a universe U can be expressed as

A ={ (x, [t_A (x) , 1 - f_A (x)]) ∣ x ∈ U }, i.e. A (x) = [t_A (x) , 1 - f_A (x)], where ∀x ∈ U, 0 ⩽ t_A (x) + f_A (x) ⩽1. Then t_A (x), f_A (x) are called the degree of membership and non-membership of the element x to vague set A, respectively.

Definition 2.2. [10] Let A and B be two vague sets over a universe U. For ∀x ∈ U, the union, intersection and complement of vague sets are defined as follows:

$\begin{matrix} A \cup B (x) & = [t_{A} (x) \lor t_{B} (x), 1 - (f_{A} (x) \land f_{B} (x))] \\ A \cap B (x) & = [t_{A} (x) \land t_{B} (x), 1 - (f_{A} (x) \lor f_{B} (x))] \\ A^{c} (x) & = [f_{A} (x), 1 - t_{A} (x)] . \end{matrix}$

Definition 2.3. [1] Let P (U) be the power set of a universe U, E be a set of parameters and A ⊆ E. F is a mapping given by F : A → P (U). Then the pair (F, A) is called a soft set over U.

Definition 2.4. [24] Let V (U) be a set of all vague sets over a universe U, E be a set of parameters and A ⊆ E. F : A → V (U). Then the pair (F, A) is called a vague soft set (VSS) over U.

To illustrate the idea, let us consider the example as following:

Example 2.1. Let U ={ x₁, x₂, x₃ } be a set of three cars under consideration to purchase. E ={ e₁, e₂, e₃ } = { clean, large, cheap } is a set of parameters. The vague soft set (F, E) shows how attractive these cars are to decision maker. Let

$\begin{matrix} F_{e_{1}} & = {(x_{1}, [0.4, 0.6]), (x_{2}, [0.6, 0.9]), (x_{3}, [0.9, 1])} \\ F_{e_{2}} & = {(x_{1}, [0.3, 0.5]), (x_{2}, [0.5, 0.8]), (x_{3}, [0.7, 0.9])} \\ F_{e_{3}} & = {(x_{1}, [0.5, 0.8]), (x_{2}, [0.7, 0.8]), (x_{3}, [0.6, 0.8])} \end{matrix}$ Then the parameterized family {F_{e
_i}∣ i = 1, 2, 3 } of vague sets over U is a vague soft set (F, E).

Definition 2.5. [24] Let (F, A), (G, B) be two VSSs over a universe U. If A ⊆ B and for ∀a ∈ A, F (a) ⊆ G (a). Then (F, A) is called a vague soft subset of (G, B), denoted by (F, A) ⊆ (G, B).

Definition 2.6. [24] Let (F, A), (G, B) be two VSSs over a universe U. If (F, A) ⊆ (G, B) and (G, B) ⊆ (F, A), i.e. A = B and for ∀a ∈ A, ∀ x ∈ U, t_{F
_a} (x) = t_{G
_a} (x) , 1 - f_{F
_a} (x) =1 - f_{G
_a} (x), then we call (F, A) is vague soft equal to (G, B), denoted by (F, A) = (G, B).

Definition 2.7. [24] The complement of the vague soft set (F, A) is defined by (F, A) ^c = (F^c, A), where $t_{F_{α}^{c}} (x) = f_{F_{α}} (x), 1 - f_{F_{α}^{c}} (x) = 1 - t_{F_{α}} (x), \forall α \in A, x \in U$ .

Definition 2.8. [2, 24] Let (F, A) and (G, B) be two vague soft sets over a universe U. The union of (F, A) and (G, B) is a vague soft set (P, C) over U, where C = A ∪ B and for ∀a ∈ C,

$P_{a} (x) = {\begin{matrix} F_{a} (x) & a \in A - B, \\ G_{a} (x) & a \in B - A, \\ F_{a} \cup G_{a} (x) & a \in A \cap B . \end{matrix} x \in U .$ We denote it by (F, A) ∪ (G, B) = (P, C).

Definition 2.9. [2, 24] Let (F, A) and (G, B) be two vague soft sets over a universe U. The intersection of (F, A) and (G, B) is a vague soft set (H, C) over U, where C = A ∪ B and for ∀a ∈ C,

$H_{a} (x) = {\begin{matrix} F_{a} (x) & a \in A - B, \\ G_{a} (x) & a \in B - A, \\ F_{a} \cap G_{a} (x) & a \in A \cap B . \end{matrix} x \in U .$ We denote it by (F, A) ∩ (G, B) = (H, C).

Definition 2.10. [20] A complex fuzzy class Γ over X can be defined as:

$Γ = {(E, x, μ_{Γ} (E, x)) : E \in 2^{x}, x \in X},$ where 2^x is the power-set of X. The pure complex membership degree is shown as

$μ_{Γ} (E, x) = μ_{r} (E) + j μ_{i} (x),$ where μ_r (E) , μ_i (x) ∈ [0, 1].

Definition 2.11. [22] Let A be an IFS characterized by the complex function z = μ + jv, where μ, v : X → [0, 1] are the functions of membership and non-membership, respectively. And μ, v satisfies μ + v ∈ [0, 1]. Then A can be represented as:

$A = {(x, z) ∣ x \in X, z = μ (x) + jv (x)}$

3 New representation of complex vague soft set based on quaternion numbers

Based on the above basic concepts, in this chapter, firstly, we rewrite the definition and the property of the vague soft set by complex function in Cartesian coordinates. Secondly, we combine the quaternion system in 4-dimensional space, introduced by Hamilton in 1943 [25], with the concept of complex vague soft sets to propose a new definition of CVSSs and give a new order relation. Finally, the advantages of the new expressions introduced are demonstrated through concrete examples.

Definition 3.1. Let (F, A) be a VSS over a universe U. By using complex function, then (F, A) can be represented by the collection of approximations of the form:

(F, A) = { (x, z_{F
_a}) : z_{F
_a} = t_{F
_a} (x) + j (1 - f_{F
_a} (x)) , x∈ U, a ∈ A }, $j = \sqrt{- 1}$ . And for ∀x ∈ U, ∀a ∈ A, t_{F
_a} (x) , f_{F
_a} (x) , t_{F
_a} (x) + f_{F
_a} (x) ∈ [0, 1] holds.

Definition 3.2. Let (F, A) and (G, B) be two vague soft sets over a universe U. (F, A) is defined to be a vague soft subset of (G, B) iff the following holds:

A ⊆ B,

∀a ∈ A, z_{F
_a} ≤ z_{G
_a} ⇔ t_{F
_a} (x) ≤ t_{G
_a} (x) , f_{F
_a}(x) ≥ f_{G
_a} (x) , ∀ x ∈ U.

This relation is denoted by (F, A) ⊆ (G, B). Particularly, if both (F, A) ⊆ (G, B) and (G, B) ⊆ (F, A) hold, then we call (F, A) and (G, B) are vague soft equal, denoted by (F, A) = (G, B).

Definition 3.3. Let (F_Q, A) be a complex vague soft set over a universe U. Then, (F_Q, A) can be defined by a quaternion function Q_{F
_a} = q_{F
_a} = α_{F
_a} + iβ_{F
_a} + j (1 - ω_{F
_a}) + kγ_{F
_a}, where i, j, k are complex roots, i² = j² = k² = ijk = -1. Here, for every parameter a ∈ A, α_{F
_a}, β_{F
_a}, ω_{F
_a}, and γ_{F
_a} are the functions of real membership, imaginary membership, real non-membership, and imaginary non-membership, respectively. For ∀x ∈ U, a ∈ A, the functions α_{F
_a}, β_{F
_a}, ω_{F
_a} and γ_{F
_a} satisfy the following conditions:

α_{F
_a} (x) , β_{F
_a} (x) , ω_{F
_a} (x) , γ_{F
_a} (x) ∈ [0, 1],

α_{F
_a} (x) + β_{F
_a} (x) ≤1,

ω_{F
_a} (x) + γ_{F
_a} (x) ≤1,

α_{F
_a} (x) + ω_{F
_a} (x) ≤1,

β_{F
_a} (x) + γ_{F
_a} (x) ≤1,

The values α_{F
_a} (x) , β_{F
_a} (x) , ω_{F
_a} (x), and γ_{F
_a} (x) are the degrees of real membership, imaginary membership, real non-membership and imaginary non-membership, respectively, of x ∈ U, a ∈ A. Then, (F_Q, A) is called the complex vague soft set based on quaternion function(CVSS-Q), denoted as follows:

$\begin{matrix} (F_{Q}, A) = {F_{Q} (a) : a \in A} = \\ {(x, Q_{F_{a}} (x)) : x \in U, a \in A} \end{matrix}$ Q_{F
_a} = q_{F
_a} is called the characteristic quaternion function of CVSS-Q. And it is also written as follows:

$\begin{matrix} q_{F_{a}} = (α_{F_{a}} + i β_{F_{a}}) + j (1 - (ω_{F_{a}} \\ + i γ_{F_{a}})) = t_{F_{a}} + j (1 - f_{F_{a}}) \end{matrix}$ here, t_{F
_a} = α_{F
_a} + iβ_{F
_a} is the complex function of membership, and f_{F
_a} = ω_{F
_a} + iγ_{F
_a} is the complex function of non-membership. For each x ∈ U, a ∈ A, the values t_{_{F
_a}} (x), f_{_{F
_a}} (x) are the complex membership and complex non-membership degrees of x ∈ F_Q (a) , a ∈ A, respectively.

It can be seen that in q_{F
_a}, if β_{F
_a} = γ_{F
_a} = 0, then we can get q_{F
_a} = α_{F
_a} + j (1 - ω_{F
_a}) = t_{_{F
_a}} + j (1 - f_{_{F
_a}}), where α_{F
_a} (x) , ω_{F
_a} (x) , α_{F
_a} (x) + ω_{F
_a} (x) ∈ [0, 1]. Hence, the representation of VSSs in Definition3.1 is a special case of the proposed model.

Here, we take an example to check the condition "t_{F
_a} + f_{F
_a} ∈ [0, 1] " of Definition3.1 for the introduced sets, i.e., the inequalities of Definition3.3.

Example 3.1. Let A ={ a₁, a₂, a₃ }, U ={ x₁, x₂, x₃ } and (F_Q, A) be a CVSS-Q over U:

$\begin{matrix} q_{F_{a_{1}}} & = (\frac{0.4 + 0.6 j}{x_{1}}, \frac{0.6 + 0.9 j}{x_{2}}, \\ \frac{0.9 + 0.1 i + 1 j + 0.9 k}{x_{3}}) \\ q_{F_{a_{2}}} & = (\frac{0.3 + 0.1 i + 0.5 j + 0.3 k}{x_{1}}, \\ \frac{0.5 + 0.4 i + 0.8 j + 0.2 k}{x_{2}}, \frac{0.7 + 0.9 j + 0.8 k}{x_{3}}) \\ q_{F_{a_{3}}} & = (\frac{0.5 + 0.8 j}{x_{1}}, \frac{0.7 + 0.8 j}{x_{2}}, \frac{0.6 + 0.8 j}{x_{3}}) \end{matrix}$

Take q_{F
_{a
₁}} (x₃) for example. Due to 0.9 + 0.1 = 1 ≤1, 0 + 0.9 = 0.9 ≤ 1, we say that (x₃, F_Q (a₁)) satisfies the conditions as follows: α_{F
_a} (x) + β_{F
_a} (x) ≤1, ω_{F
_a} (x) + γ_{F
_a} (x) ≤1, α_{F
_a} (x) + ω_{F
_a} (x) ≤1, β_{F
_a} (x) + γ_{F
_a} (x) ≤1. Similarly, we can conclude that for other a_i ∈ A, x_i ∈ U, the conditions are still satisfied. So (F_Q, A) indeed satisfies the conditions of a complex vague soft set in Definition3.3.

Definition 3.4. Let (F_Q, A) be a CVSS-Q over a universe U. For ∀a ∈ A, the set $Q_{F_{a}}^{*}$ is defined by:

$\begin{matrix} Q_{F_{a}}^{*} = {q_{F_{a}} = (α_{F_{a}}, β_{F_{a}}, ω_{F_{a}}, γ_{F_{a}}) \\ = α_{F_{a}} + i β_{F_{a}} + j (1 - ω_{F_{a}}) + k γ_{F_{a}} | α_{F_{a}}, \\ β_{F_{a}}, ω_{F_{a}}, γ_{F_{a}}, α_{F_{a}} + β_{F_{a}}, ω_{F_{a}} + γ_{F_{a}}, α_{F_{a}} \\ + ω_{F_{a}}, β_{F_{a}} + γ_{F_{a}} \in [0, 1]} \end{matrix}$ Hereinafter assumming that if $q_{F_{a}}^{t} \in Q_{F_{a}}^{*}$ then $q_{F_{a}}^{t}$ can be denoted by $q_{F_{a}}^{t} = (α_{F_{a}}^{t}, β_{F_{a}}^{t}, ω_{F_{a}}^{t}, γ_{F_{a}}^{t}), (t = 1, 2, . . .) .$ For all a ∈ A, the order relation on $Q_{F_{a}}^{*}$ is defined as follow:

$\begin{matrix} q_{F_{a}}^{1} \leq q_{F_{a}}^{2} \Leftrightarrow α_{F_{a}}^{1} \leq α_{F_{a}}^{2}, β_{F_{a}}^{1} \\ \geq β_{F_{a}}^{2}, ω_{F_{a}}^{1} \geq ω_{F_{a}}^{2}, γ_{F_{a}}^{1} \leq γ_{F_{a}}^{2} \end{matrix}$ The units of $Q_{F_{a}}^{*}$ are denoted by $1_{Q_{F_{a}}^{*}} = (1, 0, 0, 1)$ and $0_{Q_{F_{a}}^{*}} = (0, 1, 1, 0) .$

Example 3.2. Let (F_Q, A) be the CVSS-Q defined in Example 3.1. We can re-written it as follows:

$\begin{matrix} q_{F_{a_{1}}} & = (\frac{(0.4, 0, 0.4, 0)}{x_{1}}, \frac{(0.6, 0, 0.1, 0)}{x_{2}}, \\ \frac{(0.9, 0.1, 0, 0.9)}{x_{3}}) \\ q_{F_{a_{2}}} & = (\frac{(0.3, 0.1, 0.5, 0.3)}{x_{1}}, \frac{(0.5, 0.4, 0.2, 0.2)}{x_{2}}, \\ \frac{(0.7, 0, 0.1, 0.8)}{x_{3}}) \\ q_{F_{a_{3}}} & = (\frac{(0.5, 0, 0.2, 0)}{x_{1}}, \frac{(0.7, 0, 0.2, 0)}{x_{2}}, \\ \frac{(0.6, 0, 0.2, 0)}{x_{3}}) \end{matrix}$

Definition 3.5. Let $Q_{F_{a}}^{t} = q_{F_{a}}^{t} = α_{F_{a}}^{t} + i β_{F_{a}}^{t} + j (1 - ω_{F_{a}}^{t}) + k γ_{F_{a}}^{t} \in Q_{F_{a}}^{*}$ and (F_{Q
_t}, A) be the CVSSs-Q on U, where $α_{F_{a}}^{t}, β_{F_{a}}^{t}, ω_{F_{a}}^{t}, γ_{F_{a}}^{t}, (t = 1; 2)$ are the functions of real membership, imaginary membership, real non-membership and imaginary non-membership degrees under the parameter a ∈ A, respectively.

The (F_{Q
₁}, A) is defined to be a subset of (F_{Q
₂}, A) if and only if for ∀a ∈ A, $q_{F_{a}}^{1} \leq q_{F_{a}}^{2}$ holds. We denote it by (F_{Q
₁}, A) ⊆ (F_{Q
₂}, A).

Remark 3.1. Compared to the representation of two degrees, the proposed quaternion model extends the expressive power of VSSs to four degrees.

Let (F, A), (G, B) be two CVSSs-Q over a universe U. If A ⊆ B and for ∀a ∈ A, q_{F
_a} ≤ q_{G
_a} holds (as defined in Definition3.5), then we get z_{F
_a} ≤ z_{G
_a} (as defined in Definition3.2), where z_{F
_a} = α_{F
_a} + j (1 - ω_{F
_a}), z_{G
_a} = α_{G
_a} + j (1 - ω_{G
_a}). Therefore, the proposed quaternion representation is able to better evaluate the relationship between VSSs.

In order to show the superiority of quaternion representation over the complex one, here we give an example.

Example 3.3. Let (F, A), (G, A) be two vague soft sets over a universe U. Let A ={ e₁, e₂ } and for ∀x ∈ U,

$\begin{matrix} q_{F_{e_{1}}} (x) & = 0.4 + 0.2 i + 0.5 j + 0.1 k; \\ q_{F_{e_{2}}} (x) & = 0.3 + 0.5 i + 0.6 j + 0.3 k; \\ q_{G_{e_{1}}} (x) & = 0.4 + 0.1 i + 0.5 j + 0.3 k; \\ q_{G_{e_{2}}} (x) & = 0.3 + 0.2 i + 0.6 j + 0.5 k; \end{matrix}$

From Definition3.4, 3.5, we have (F_Q, A) ⊆ (G_Q, A). But from Definition3.2, we have (F, A) = (G, A).

From the above example, it can be seen that in the Cartesian coordinate system, the conclusion obtained under complex number representation is not as accurate as that under quaternion representation. Therefore, the new representation proposed in this article contains more fuzzy information and leads to a more accurate conclusion.

4 Some algebraic operations

In this chapter, we give new logical operations on CVSSs-Q and analyze their properties.

Definition 4.1. The negation on $Q_{F_{a}}^{*}$ is a non-increasing function $Q_{F_{a}}^{*} \to Q_{F_{a}}^{*}$ that satisfies:

$N (1_{Q_{F_{a}}^{*}}) = 0_{Q_{F_{a}}^{*}}, N (0_{Q_{F_{a}}^{*}}) = 1_{Q_{F_{a}}^{*}} .$ A negation N is called an involution if $N (N (q_{F_{a}})) = q_{F_{a}}, \forall q_{F_{a}} \in Q_{F_{a}}^{*}$ .

Proposition 4.1. The following operation is an involutive negation on $Q_{F_{a}}^{*}$ , called the standard negation on $Q_{F_{a}}^{*}$ :

$\begin{matrix} N_{s} (q_{F_{a}}) = (ω_{F_{a}}, γ_{F_{a}}, α_{F_{a}}, β_{F_{a}}) = \\ ω_{F_{a}} + i γ_{F_{a}} + j (1 - α_{F_{a}}) + k β_{F_{a}}, \forall q_{F_{a}} \in Q_{F_{a}}^{*} \end{matrix}$

Proof. Firstly, from $q_{F_{a}} \in Q_{F_{a}}^{*}$ , we can get: α_{F
_a} + β_{F
_a}, ω_{F
_a} + γ_{F
_a}, α_{F
_a} + ω_{F
_a}, β_{F
_a} + γ_{F
_a} ∈ [0, 1]. Hence, we can obtain $N (q_{F_{a}}) \in Q_{F_{a}}^{*}$ . Secondly, we can easily find that $N_{s} (1_{Q_{F_{a}}^{*}}) = 0_{Q_{F_{a}}^{*}}, N_{s} (0_{Q_{F_{a}}^{*}}) = 1_{Q_{F_{a}}^{*}}$ . Now, consider $q_{F_{a}}^{1} \leq q_{F_{a}}^{2}$ , from definition we can get $α_{F_{a}}^{1} \leq α_{F_{a}}^{2}, β_{F_{a}}^{1} \geq β_{F_{a}}^{2}, ω_{F_{a}}^{1} \geq ω_{F_{a}}^{2}, γ_{F_{a}}^{1} \leq γ_{F_{a}}^{2}$ , therefore $N (q_{F_{a}}^{1}) \geq N (q_{F_{a}}^{2})$ . Finally, we can observe that $N_{s} (N_{s} (q_{F_{a}})) = q_{F_{a}}, \forall q_{F_{a}} \in Q_{F_{a}}^{*}$ . □

Definition 4.2. Let (F_Q, A) be a CVSS-Q over a universe U, with $Q_{F_{a}} = q_{F_{a}} = α_{F_{a}} + i β_{F_{a}} + j (1 - ω_{F_{a}}) + k γ_{F_{a}} \in Q_{F_{a}}^{*}$ . The negation (the complement) of (F_Q, A), denoted as $(F_{Q}, A)^{N} = (F_{Q}^{N}, A)$ , is the set represented by the negation (the complement) of Q_{F
_a}, i.e.

$Q_{F_{a}}^{N} = N (q_{F_{a}})$ where N is a negation on $Q_{F_{a}}^{*}$ .

Remark 4.1. From Definition 4.2, we have $Q_{F_{a}}^{N_{s}} = N_{s} (q_{F_{a}}) = t_{F_{a}}^{c} + j (1 - f_{F_{a}})^{c} = t_{F_{a}}^{c} + j (1 - f_{F_{a}}^{c})$ , where $t_{F_{a}}^{c} = ω_{F_{a}} + i γ_{F_{a}}$ and $f_{F_{a}}^{c} = α_{F_{a}} + i β_{F_{a}}$ , $(F_{Q}, A)^{N} = (F_{Q}^{N}, A)$ , respectively. Furthermore, from Proposition 4.1, we can conclude that ((F_Q, A) ^{N
_s}) ^{N
_s} = (F_Q, A).

Definition 4.3. Let (F_Q, A) , (G_Q, B) be two CVSSs-Q over a universe U, the union of which is a CVSS-Q (P_Q, C), where C = A ∪ B. Then ∀a ∈ C, (P_Q, C) can be characterized by the function as follows:

$q_{P_{a}} = {\begin{matrix} q_{F_{a}} & a \in A - B, \\ q_{G_{a}} & a \in B - A, \\ q_{F_{a}} \lor q_{G_{a}} & a \in A \cap B . \end{matrix} \forall x \in U .$ where q_{F
_a} ∨ q_{G
_a} = (α_{F
_a} ∨ α_{G
_a}, β_{F
_a} ∧ β_{G
_a}, ω_{F
_a} ∧ ω_{G
_a}, γ_{F
_a} ∨ γ_{G
_a}). We denote it by (F_Q, A) ∪ (G_Q, B) = (P_Q, C).

Definition 4.4. Suppose that (F_Q, A) , (G_Q, B) are two CVSSs-Q over a universe U, the intersection of which is a CVSS-Q (H_Q, C), where C = A ∪ B. Then ∀a ∈ C, (H_Q, C) can be characterized by the quaternion function as follows:

$q_{H_{a}} = {\begin{matrix} q_{F_{a}} & a \in A - B, \\ q_{G_{a}} & a \in B - A, \\ q_{F_{a}} \land q_{G_{a}} & a \in A \cap B . \end{matrix} \forall x \in U .$ where q_{F
_a} ∧ q_{G
_a} = (α_{F
_a} ∧ α_{G
_a}, β_{F
_a} ∨ β_{G
_a}, ω_{F
_a} ∨ ω_{G
_a}, γ_{F
_a} ∧ γ_{G
_a}). We denote it by (F_Q, A) ∩ (G_Q, B) = (H_Q, C).

Example 4.1. Let (F_Q, A) and (G_Q, B) be two CVSSs-Q over the same universe U. (F_Q, A) is defined as the Example 3.2, and (G_Q, B) is defined as follows: B ={ a₁, b₁ },

$\begin{matrix} q_{G_{a_{1}}} & = (\frac{(0.4, 0.2, 0.5, 0.1)}{x_{1}}, \frac{(0.6, 0, 0.1, 0)}{x_{2}}, \frac{(0.3, 0.5, 0.4, 0.3)}{x_{3}}) \\ q_{G_{b_{1}}} & = (\frac{(0.4, 0.2, 0.5, 0.1)}{x_{1}}, \frac{(0.6, 0, 0.1, 0)}{x_{2}}, \frac{(0.3, 0.5, 0.6, 0.3)}{x_{3}}) \end{matrix}$ Thus, for (P_Q, C) = (F_Q, A) ∪ (G_Q, B), (H_Q, C) = (F_Q, A) ∩ (G_Q, B), we have C = A∪ B = { a₁, a₂, a₃, b₁ }. Thus, A - B ={ a₂, a₃ } , B - A = { b₁ } , A ∩ B = { a₁ }.

The (P_Q, C) , (H_Q, C) are characterized by the quaternion function as:

In A - B,

$\begin{matrix} q_{P_{a_{2}}} = q_{H_{a_{2}}} = q_{F_{a_{2}}} & = (\frac{(0.3, 0.1, 0.5, 0.3)}{x_{1}}, \frac{(0.5, 0.4, 0.2, 0.2)}{x_{2}}, \frac{(0.7, 0, 0.1, 0.8)}{x_{3}}) \\ q_{P_{a_{3}}} = q_{H_{a_{3}}} = q_{F_{a_{3}}} & = (\frac{(0.5, 0, 0.2, 0)}{x_{1}}, \frac{(0.7, 0, 0.2, 0)}{x_{2}}, \frac{(0.6, 0, 0.2, 0)}{x_{3}}) \end{matrix}$

In B - A,

$q_{P_{b_{1}}} = q_{H_{b_{1}}} = q_{G_{b_{1}}} = (\frac{(0.4, 0.2, 0.5, 0.1)}{x_{1}}, \frac{(0.6, 0, 0.1, 0)}{x_{2}}, \frac{(0.3, 0.5, 0.6, 0.3)}{x_{3}})$

In A ∩ B,

$\begin{matrix} q_{P_{a_{1}}} & = q_{F_{a_{1}}} \lor q_{G_{a_{1}}} \\ = (\frac{(0.4 \lor 0.4, 0 \land 0.2, 0.4 \land 0.5, 0 \lor 0.1)}{x_{1}}, \frac{(0.6 \lor 0.6, 0 \land 0, 0.1 \land 0.1, 0 \lor 0)}{x_{2}}, \frac{(0.9 \lor 0.3, 0.1 \land 0.5, 0 \land 0.4, 0.9 \lor 0.3)}{x_{3}}) \\ = (\frac{(0.4, 0, 0.4, 0.1)}{x_{1}}, \frac{(0.6, 0, 0.1, 0)}{x_{2}}, \frac{(0.9, 0.1, 0, 0.9)}{x_{3}}) \\ q_{H_{a_{1}}} & = q_{F_{a_{1}}} \land q_{G_{a_{1}}} \\ = (\frac{(0.4 \land 0.4, 0 \lor 0.2, 0.4 \lor 0.5, 0 \land 0.1)}{x_{1}}, \frac{(0.6 \land 0.6, 0 \lor 0, 0.1 \lor 0.1, 0 \land 0)}{x_{2}}, \frac{(0.9 \land 0.3, 0.1 \lor 0.5, 0 \lor 0.4, 0.9 \land 0.3)}{x_{3}}) \\ = (\frac{(0.4, 0.2, 0.5, 0)}{x_{1}}, \frac{(0.6, 0, 0.1, 0)}{x_{2}}, \frac{(0.3, 0.5, 0.4, 0.3)}{x_{3}}) \end{matrix}$

Remark 4.2. We can easily find that the union and the intersection of two CVSSs-Q over U are still CVSSs-Q.

Proposition 4.2. Let (F_{Q
₁}, A₁), (F_{Q
₂}, A₂), (F_{Q
₃}, A₃) be three CVSSs-Q over a universe U, a ∈ A_t, $Q_{F_{a}}^{t} = q_{F_{a}}^{t} \in Q_{F_{a}}^{*} (t = 1, 2, 3)$ . For "∪", "∩" and negation "N" on $Q_{F_{a}}^{*}$ ,

(reflectivity property)

(F_Q
₁, A₁) ∪ (F_Q
₁, A₁) = (F_Q
₁, A₁),

(F_Q
₁, A₁) ∩ (F_Q
₁, A₁) = (F_Q
₁, A₁).

(commutative property)

(F_Q
₁, A₁) ∪ (F_Q
₂, A₂) = (F_Q
₂, A₂) ∪ (F_Q
₁, A₁),

(F_Q
₁, A₁) ∩ (F_Q
₂, A₂) = (F_Q
₂, A₂) ∩ (F_Q
₁, A₁).

(associative property)

((F_Q
₁, A₁) ∪ (F_Q
₂, A₂)) ∪ (F_Q
₃, A₃) = (F_Q
₁, A₁) ∪ ((F_Q
₂, A₂) ∪ (F_Q
₃, A₃)),

((F_Q
₁, A₁) ∩ (F_Q
₂, A₂)) ∩ (F_Q
₃, A₃) = (F_Q
₁, A₁) ∩ ((F_Q
₂, A₂) ∩ (F_Q
₃, A₃)).

(distributive property)

((F_Q
₁, A₁) ∩ (F_Q
₂, A₂)) ∪ (F_Q
₃, A₃) = ((F_Q
₁, A₁) ∪ (F_Q
₃, A₃)) ∩ ((F_Q
₂, A₂) ∪ (F_Q
₃, A₃)),

((F_Q
₁, A₁) ∪ (F_Q
₂, A₂)) ∩ (F_Q
₃, A₃) = ((F_Q
₁, A₁) ∩ (F_Q
₃, A₃)) ∪ ((F_Q
₂, A₂) ∩ (F_Q
₃, A₃)).

(negative property)

((F_Q
₁, A₁) ∪ (F_Q
₂, A₂)) ^N = (F_Q
₁, A₁) ^N ∩ (F_Q
₂, A₂) ^N,

((F_Q
₁, A₁) ∩ (F_Q
₂, A₂)) ^N = (F_Q
₁, A₁) ^N ∪ (F_Q
₂, A₂) ^N.

Proof. The properties (1-4) are deduced from the properties of "∪", "∩" and negation "N" on $Q_{F_{a}}^{*}$ . As for the property (5), we can prove it as follows:

(1) for ∀a ∈ A₁ ∩ A₂,

$N_{s} (q_{F_{a}}^{1} \lor q_{F_{a}}^{2}) = (ω_{F_{a}^{1}} \land ω_{F_{a}^{2}}, γ_{F_{a}^{1}} \lor γ_{F_{a}^{2}}, α_{F_{a}^{1}} \lor α_{F_{a}^{2}}, β_{F_{a}^{1}} \land β_{F_{a}^{2}})$ ,

$\begin{matrix} N_{s} (q_{F_{a}}^{1}) \land N_{s} (q_{F_{a}}^{2}) & = (ω_{F_{a}^{1}}, γ_{F_{a}^{1}}, α_{F_{a}^{1}}, β_{F_{a}^{1}}) \land (ω_{F_{a}^{2}}, γ_{F_{a}^{2}}, α_{F_{a}^{2}}, β_{F_{a}^{2}}) \\ = (ω_{F_{a}^{1}} \land ω_{F_{a}^{2}}, γ_{F_{a}^{1}} \lor γ_{F_{a}^{2}}, α_{F_{a}^{1}} \lor α_{F_{a}^{2}}, β_{F_{a}^{1}} \land β_{F_{a}^{2}}) \end{matrix}$

So we can get $N_{s} (q_{F_{a}}^{1} \lor q_{F_{a}}^{2}) = N_{s} (q_{F_{a}}^{1}) \land N_{s} (q_{F_{a}}^{2})$ .

(2) for all a ∈ A₁ - A₂ and a ∈ A₂ - A₁, the equation still holds, obviously.

Thus, we can conclude that the property (5) holds.□

5 Quaternion distance measure

In this chapter, we propose and study a quaternion-based Euclidean distance measure and a new order relation in CVSS-Q.

Definition 5.1. Let d : CVSS - Q (U) × CVSS - Q (U) → [0, 1] be a mapping, (F_Q, A) , (G_Q, A) , (P_Q, A) ∈ CVSSs - Q (U) and for ∀a ∈ A, $q_{F_{a}} \in Q_{F_{a}}^{*}$ , $q_{G_{a}} \in Q_{G_{a}}^{*}$ . d ((F_Q, A) , (G_Q, A)) is called the degree of distance between (F_Q, A) and (G_Q, A) if it satisfies the following conditions:

d ((F_Q, A) , (G_Q, A)) = d ((G_Q, A) , (F_Q, A));

d ((F_Q, A) , (G_Q, A)) ∈ [0, 1];

$d ((F_{Q}, A), (G_{Q}, A)) = 1 \Leftrightarrow \forall a \in A, x \in U, q_{F_{a}} = 0_{Q_{F_{a}}^{*}}, q_{G_{a}} = 1_{Q_{G_{a}}^{*}}$ or $q_{F_{a}} = 1_{Q_{F_{a}}^{*}}, q_{G_{a}} = 0_{Q_{G_{a}}^{*}}$ ;

d ((F_Q, A) , (G_Q, A)) =0 ⇔ (F_Q, A) = (G_Q, A);

(F_Q, A) ⊆ (G_Q, A) ⊆ (P_Q, A) ⇒ d ((F_Q, A) , (P_Q, A)) ⩾ max(d ((F_Q, A) , (G_Q, A)) , d ((G_Q, A) , (P_Q, A))).

Definition 5.2. Let U ={ x₁, x₂, …, x_n } be a universal set of elements and A ={ a₁, a₂, …, a_m } be a universal set of parameters, $q_{F_{a}} \in Q_{F_{a}}^{*}$ , $q_{G_{a}} \in Q_{G_{a}}^{*}$ . Define the quaternion distance measures between two CVSSs-Q (F_Q, A) and (G_Q, A) as follows:

The Euclidean quaternion distance:

$d_{E} ((F_{Q}, A), (G_{Q}, A)) = \sqrt{\frac{1}{4 mn} \sum_{j = 1}^{n} \sum_{i = 1}^{m} [\begin{matrix} (α_{F_{a_{i}}} (x_{j}) - α_{G_{a_{i}}} (x_{j}))^{2} + (β_{F_{a_{i}}} (x_{j}) - β_{G_{a_{i}}} (x_{j}))^{2} \\ + (ω_{F_{a_{i}}} (x_{j}) - ω_{G_{a_{i}}} (x_{j}))^{2} + (γ_{F_{a_{i}}} (x_{j}) - γ_{G_{a_{i}}} (x_{j}))^{2} \end{matrix}]}$

The Hamming quaternion distance:

$d_{v} ((F_{Q}, A), (G_{Q}, A)) = \frac{1}{4 mn} \sum_{j = 1}^{n} \sum_{i = 1}^{m} [\begin{matrix} | α_{F_{a_{i}}} (x_{j}) - α_{G_{a_{i}}} (x_{j}) | + | β_{F_{a_{i}}} (x_{j}) - β_{G_{a_{i}}} (x_{j}) | \\ + | ω_{F_{a_{i}}} (x_{j}) - ω_{G_{a_{i}}} (x_{j}) | + | γ_{F_{a_{i}}} (x_{j}) - γ_{G_{a_{i}}} (x_{j}) | \end{matrix}]$

The Hausdorff quaternion measure:

$d_{H} ((F_{Q}, A), (G_{Q}, A)) = \frac{1}{mn} \sum_{j = 1}^{n} \sum_{i = 1}^{m} max {\begin{matrix} | α_{F_{a_{i}}} (x_{j}) - α_{G_{a_{i}}} (x_{j}) |, | β_{F_{a_{i}}} (x_{j}) - β_{G_{a_{i}}} (x_{j}) |, \\ | ω_{F_{a_{i}}} (x_{j}) - ω_{G_{a_{i}}} (x_{j}) |, | γ_{F_{a_{i}}} (x_{j}) - γ_{G_{a_{i}}} (x_{j}) | \end{matrix}}$

Theorem 5.1. All the quaternion distance measures given in the Definition 5.2 are the quaternion distance functions of CVSSs-Q.

Proof. Let’s take the proof of the Euclidean quaternion distance measure as an example to illustrate.

(1) From the definition of d_E ((F_Q, A) , (G_Q, A)), we can get d_E ((F_Q, A) , (G_Q, A)) = d_E ((G_Q, A) , (F_Q, A)).

(2) From the definition of q_{F
_a}, q_{G
_a}, we know that for all a_i ∈ A, x_j ∈ U, α, β, γ, ω ∈ [0, 1]. Thus, (α_{F
_{a
_i}} (x_j) - α_{G
_{a
_i}} (x_j)) ², (β_{F
_{a
_i}} (x_j) - β_{G
_{a
_i}} (x_j)) ², (ω_{F
_{a
_i}} (x_j) - ω_{G
_{a
_i}} (x_j)) ², (γ_{F
_{a
_i}} (x_j) - γ_{G
_{a
_i}} (x_j)) ² ∈ [0, 1]. So we can conclude that d_E ((F_Q, A) , (G_Q, A)) ∈ [0, 1] .

(3) Similar to (2), we can prove that $d_{E} ((F_{Q}, A), (G_{Q}, A)) = 1 \Leftrightarrow \forall a_{i} \in A, x_{j} \in U, q_{F_{a}} = 0_{Q_{F_{a}}^{*}}, q_{G_{a}} = 1_{Q_{G_{a}}^{*}}$ or $q_{F_{a}} = 1_{Q_{F_{a}}^{*}}, q_{G_{a}} = 0_{Q_{G_{a}}^{*}}$ .

(4) Because d_E ((F_Q, A) , (G_Q, A)) =0 ⇔ ∀ a_i ∈ A, x_j ∈ U, α_{F
_{a
_i}} - α_{G
_{a
_i}} = 0, β_{F
_{a
_i}} - β_{G
_{a
_i}} = 0, γ_{F
_{a
_i}} - γ_{G
_{a
_i}} = 0, ω_{F
_{a
_i}} - ω_{G
_{a
_i}} = 0 . Thus, for all a_i ∈ A, x_j ∈ U, we have q_{F
_{a
_i}} (x_j) = q_{G
_{a
_i}} (x_j), which is equivalent to (F_Q, A) = (G_Q, A).

(5) By the Definition 3.5, we know that (F_Q, A) ⊆ (G_Q, A) ⊆ (P_Q, A) implies for all a_i ∈ A, q_{F
_{a
_i}} ≤ q_{G
_{a
_i}} ≤ q_{P
_{a
_i}}. Combining Definition 3.4 and the expression of the Euclidian quaternion distance measure between two CVSSs-Q, we can easily conclude that d_E ((F_Q, A) , (P_Q, A)) ⩾ max(d_E ((F_Q, A) , (G_Q, A)) , d_E ((G_Q, A) , (P_Q, A))) .

Similarly, we can obtain proofs of the other two distance measures. □

Example 5.1. Let A ={ a₁, a₂ } be a parameter set and (F_Q, A), (G_Q, A) be two CVSSs-Q over U ={ x₁, x₂, x₃ }. Let

$\begin{matrix} q_{F_{a_{1}}} & = (\frac{(0.3, 0.1, 0.5, 0.3)}{x_{1}}, \frac{(0.5, 0.4, 0.2, 0.2)}{x_{2}}, \frac{(0.7, 0, 0.1, 0.8)}{x_{3}}) \\ q_{F_{a_{2}}} & = (\frac{(0.5, 0, 0.2, 0)}{x_{1}}, \frac{(0.7, 0, 0.2, 0)}{x_{2}}, \frac{(0.6, 0, 0.2, 0)}{x_{3}}) \\ q_{G_{a_{1}}} & = (\frac{(0.4, 0.2, 0.5, 0.1)}{x_{1}}, \frac{(0.6, 0, 0.1, 0)}{x_{2}}, \frac{(0.3, 0.5, 0.4, 0.3)}{x_{3}}) \\ q_{G_{a_{2}}} & = (\frac{(0.4, 0.2, 0.5, 0.1)}{x_{1}}, \frac{(0.6, 0, 0.1, 0)}{x_{2}}, \frac{(0.3, 0.5, 0.6, 0.3)}{x_{3}}) \end{matrix}$ Here m = 2, n = 3. Then after calculating, we have d_E ((F_Q, A) , (G_Q, A)) ≈0.273.

6 Decision making model based on Quaternion distance measures

Based on the CVSS being represented by quaternion numbers and four quaternion distance measures proposed above, this chapter propose a new decision making model. Here we define the quaternion decision model as QDM.

A complete set of medical disease decision-making model is divided into the following parts:

1. Collecting patient data:

(1) Encoding the patient;

(2) Each group of patient data is processed by CVSS-Q.

2. Constructing disease diagnostic criteria on CVSS-Q through the training dataset.

3. Calculating the distance between patient dataset and disease diagnosis criteria, and then give medical diagnoses.

The QDM model is performed in C-QDM. The detailed description is as follows:

1. Quaternionification: Consider indicators I_j (j = 1, 2, . . . , n) of a disease D and the data corresponding to m groups of patients p_i (i = 1, 2, . . . , m). The dataset encoded in Cartesian form of quaternion numbers is shown as follows:

	I ₁	...	I _j	...	I _n	Class-Y
p ₁	(α₁₁, β₁₁, ω₁₁, γ₁₁)	...	(α_1j, β_1j, ω_1j, γ_1j)	...	(α_1n, β_1n, ω_1n, γ_1n)	y ₁
...	...	...	...	...	...	...
p _i	(α_i1, β_i1, ω_i1, γ_i1)	...	(α_ij, β_ij, ω_ij, γ_ij)	...	(α_in, β_in, ω_in, γ_in)	y _i
...	...	...	...	...	...	...
p _m	(α_m1, β_m1, ω_m1, γ_m1)	...	(α_mj, β_mj, ω_mj, γ_ml)	...	(α_mn, β_mn, ω_mn, γ_mn)	y _m

The (α_ij, β_ij, ω_ij, γ_ij) = (α_j (p_i) , β_j (p_i) , ω_j (p_i) , γ_j (p_i)) and y_i represent the data of patient p_i corresponding to the indicator I_j and the final diagnostic result of p_i, respectively. The α_ij, β_ij, ω_ij, γ_ij are calculated as follows:

$α_{j} (x) = {\begin{matrix} 0 x < a_{j} \\ \frac{x - a_{j}}{b_{j} - a_{j}} a_{j} \leq x \leq b_{j}, \\ 1 x > b_{j} \end{matrix}$

$β_{j} (x) = {\begin{matrix} 1, x < a_{j}^{'} \\ \frac{b_{j}^{'} - x}{b_{j}^{'} - a_{j}^{'}}, a_{j}^{'} \leq x \leq b_{j}^{'}, \\ 0, x > b_{j}^{'} \end{matrix}$

and

$ω_{j} (x) = {\begin{matrix} 1, x < a_{j}^{'} \\ {(\frac{b_{j}^{'} - x}{b_{j}^{'} - a_{j}^{'}})}^{2}, a_{j}^{'} \leq x \leq b_{j}^{'}, \\ 0, x > b_{j}^{'} \end{matrix}$

$γ_{j} (x) = {\begin{matrix} 0, x < a_{j} \\ {(\frac{x - a_{j}}{b_{j} - a_{j}})}^{2}, a_{j} \leq x \leq b_{j}, \\ 1, x > b_{j} \end{matrix}$

where $a_{j}^{'} \leq a_{j}$ , $b_{j}^{'} \leq b_{j}$ .

2. Trainig Process: Assuming that there has a set of training dataset, which contains s + t groups of data. For every indicator I_j (j = 1, 2, . . . , n), the judgment standard of it can be expressed as (α_j, β_j, ω_j, γ_j), which can be calculated by the following:

$f (x) = \frac{| M [xY] - M [x] M [Y] |}{\sqrt{M [x^{2}] - M [x]^{2}} \sqrt{M [Y^{2}] - M {[Y]}^{2}}}$

α_j = f (α_ij), β_j = min{ 1 - α_j, f (β_ij) }, ω_j = min{ 1 - α_j, f (ω_ij) }, γ_j = min {1 - β_j, 1 - ω_j, f (γ_ij)}

where $M [X] = \frac{1}{s} \sum_{i = 1}^{s} x_{i}$ , $X = {x_{i}}_{i = 1}^{s}$ .

3. Testing process: Let dataset of patient p_i be P_i, and the diagnosis criteria of the disease D be D_*. Then $D_{*} = {(α_{j}, β_{j}, ω_{j}, γ_{j})}_{j = 1}^{n}$ and the distance between P_i and D_* can be shown as:

$d_{i} = d_{E} ((P_{i}, I), (D_{*}, I)) = \sqrt{\frac{1}{4 n} \sum_{j = 1}^{n} (α_{ij} - α_{j})^{2} + (β_{ij} - β_{j})^{2} + (ω_{ij} - ω_{j})^{2} + (γ_{ij} - γ_{j})^{2}}$

If the degree of fit with disease D is divided into the following stages:

$y_{i} = {\begin{matrix} 1, if d_{i} < e; \\ 0, if d_{i} \geq e \end{matrix}$

where e ∈ [0, 1] is the best threshold selected by the training process.

Remark 6.1. In addition to the Euclidean distance measure proposed in this model, the other two distance measures we proposed in Section V can also be applied to this model,where:

$d_{v} ((P_{i}, I), (D_{*}, I)) = \frac{1}{4 n} \sum_{j = 1}^{n} [| α_{ij} - α_{j} | + | β_{ij} - β_{j} | + | ω_{ij} - ω_{j} | + | γ_{ij} - γ_{j} |]$

$d_{H} ((P_{i}, I), (D_{*}, I)) = \frac{1}{n} \sum_{j = 1}^{n} max {| α_{ij} - α_{j} |, | β_{ij} - β_{j} |, | ω_{ij} - ω_{j} |, | γ_{ij} - γ_{j} |}$

Example 6.1. Suppose the indicator set of the cold is I ={ temperature, cough, sorethroat, headache, runnynose }. Now, we provide 4 sets of data for these 5 indicators. And after quaternization and calculating, the patient dataset is shown as Table 1 and the disease diagnosis criteria is as follows:

Table 1
Patient dataset of the cold

Temperature Cough Sore throat Headache Runny nose Class of Cold

p ₁ (0.8, 0, 0.1, 0.6) (0.6, 0, 0.1, 0.4) (0.2, 0.8, 0.8, 0) (0.6, 0, 0.1, 0.4) (0.1, 0.8, 0.6, 0) 0

p ₂ (0, 0.9, 0.8, 0) (0.4, 0.6, 0.4, 0.2) (0.6, 0, 0.1, 0.4) (0.1, 0.8, 0.7, 0) (0.1, 0.8, 0.8, 0) 0

p ₃ (0.8, 0, 0.1, 0.6) (0.8, 0, 0.1, 0.6) (0, 0.8, 0.6, 0) (0.2, 0.8, 0.7, 0) (0, 0.7, 0.5, 0) 0

p ₄ (0.6, 0, 0.1, 0.4) (0.5, 0.5, 0.4, 0.3) (0.3, 0.6, 0.4, 0.1) (0.7, 0, 0.2, 0.5) (0.3, 0.6, 0.4, 0.1) 1

Cold (0.1, 0.3, 0.3, 0) (0.3, 0.5, 0.6, 0.3) (0.1, 0.1, 0.2, 0.1) (0.7, 0.3, 0.3, 0.7) (0.9, 0.1, 0.1, 0.9)

	Temperature	Cough	Sore throat	Headache	Runny nose	Class of Cold
p ₁	(0.8, 0, 0.1, 0.6)	(0.6, 0, 0.1, 0.4)	(0.2, 0.8, 0.8, 0)	(0.6, 0, 0.1, 0.4)	(0.1, 0.8, 0.6, 0)	0
p ₂	(0, 0.9, 0.8, 0)	(0.4, 0.6, 0.4, 0.2)	(0.6, 0, 0.1, 0.4)	(0.1, 0.8, 0.7, 0)	(0.1, 0.8, 0.8, 0)	0
p ₃	(0.8, 0, 0.1, 0.6)	(0.8, 0, 0.1, 0.6)	(0, 0.8, 0.6, 0)	(0.2, 0.8, 0.7, 0)	(0, 0.7, 0.5, 0)	0
p ₄	(0.6, 0, 0.1, 0.4)	(0.5, 0.5, 0.4, 0.3)	(0.3, 0.6, 0.4, 0.1)	(0.7, 0, 0.2, 0.5)	(0.3, 0.6, 0.4, 0.1)	1
Cold	(0.1, 0.3, 0.3, 0)	(0.3, 0.5, 0.6, 0.3)	(0.1, 0.1, 0.2, 0.1)	(0.7, 0.3, 0.3, 0.7)	(0.9, 0.1, 0.1, 0.9)

The class of the cold is defined as follows:

$y_{i} = {\begin{matrix} 1, if d_{i} < 0.48; \\ 0, if d_{i} \geq 0.48 \end{matrix}$

According to the quaternion Euclidean distance formula of the model, we can calculate as follows:

$d_{1} = \sqrt{\frac{1}{4 * 5} (0.98 + 0.6 + 0.87 + 0.23 + 2.19)} = 0.4935$ ; d₂ = 0.4868; d₃ = 0.5376; d₄ = 0.3486.

Then, from the model, we can give the final diagnosis: the patient p₄ has a cold but others not. This is consistent with the actual diagnostic results in the table.

However, if we calculate the Euclidean distance measure of vague soft set, we can get:

$d_{1}^{'} = \sqrt{\frac{1}{2 * 5} (0.53 + 0.34 + 0.37 + 0.05 + 0.89)} = 0.4669$ ; $d_{2}^{'} = 0.4712; d_{3}^{'} = 0.5079; d_{4}^{'} = 0.3017$ .

Thus, we can give the final diagnosis: the patient p₁, p₂, p₄ has a cold, which contradicts the actual diagnostic results in the table.

7 Experiments

In this section, we validate the decision model proposed inSection 6 with real-world real patient data. In addition to the Euclidean distance measure used in the model, we replace the other two distance measures into the instances and apply them to real cases, resulting in three sets of diagnostic results. The three diagnosis results are compared, and the optimal solutions under different circumstances are proposed.

The real patient data we use is Breast Cancer Wisconsin (Original) Dataset (BCWD) from the University of California (UCI) Machine Learning Repository (https ://archive . ics . uci . edu/ml/index . php).

7.1 Experimental process

The description of BCWD is shown as below:

Datasets	Number of classes	Number of attributes	Number of elements	Number of unavailable distances
BCWD	2	9	683	16

Here we delete the instances containing "?" and divide the remaining instances into two groups,applying into training process and testing process, respectively. Then the experimental process is specifically divided into the following steps:

step1. Use quaternion function to quaternize total dataset.

step2. Construct disease diagnostic criteria by calculating the training dataset.

step3. Calculate the distance between the training dataset and criteria.

step4. Give the value of e, according to the distance and the class attributes.

step5. Calculate the distance between testing dataset and criteria.

step6. Give medical diagnosis by comparing the results in previous step and e. And derive accuracy.

step7. Replace the distance measure in the model with the other two measures and repeat the above steps.

7.2 Experimental results

We program the training and testing process of the dataset in python software, and the program finally output the similar distance for each patient and disease. Then the comparison between the obtained results and the actual diagnostic results are done in Excel. The test results for different distance measures in the new model are obtained. In addition, we append the distance measures of vague soft set for comparison. According to the testing results, we draw the graph as shown in Fig. 1. The first column of each group of graphs represents the real number of diagnoses in the testing dataset; the second to fourth columns represent the numbers of diagnoses under the d_E, d_v, d_H distance measures, respectively. The fifth to seventh columns represent the numbers of diagnoses under the Euclidean distance measure( $d_{E}^{'}$ ), Hamming distance measure( $d_{v}^{'}$ ) and Hausdorff distance measure( $d_{H}^{'}$ )of vague soft set, respectively.

Fig. 1

Testing Results.

From the figure, we can easily find that all the distance measures under quaternion system give much higher accuracy in disease diagnosis than those under complex system. After calculating, we find that the accuracy of d_E, d_v, d_H are 90.2%, 90.6%, 5.5%, while the accuracy of $d_{E}^{'}, d_{v}^{'}, d_{H}^{'}$ are 76.5%, 76.7%, 31.8%. Furthermore, let the result under the measure d_E, d_v, d_H, $d_{E}^{'}$ , $d_{v}^{'}$ , $d_{H}^{'}$ be the case 1-6. Then, we can see that based on the quaternion representation, the results of Case 1-2 are closer to the real situation than the results based on complex representation (Case 4-5) corresponding to the same type of distance metric. Although the accuracy of Case 6 is higher than Case 3, it considers that all patients are unhealthy. The high accuracy of this situation is because the number of patients in this data set happens to be high. Therefore, from the perspective of not letting go of any possibility of illness, Case 6 is better. More generally, from the point of view of correct diagnosis, using the first two distance measures under the quaternion representation works better.

Remark 7.1. The inaccuracy of disease diagnosis of this model in different datasets may have the following reasons: the error of the original dataset, the error of retaining decimals, the error caused by different boundaries when Quaternionification, the error caused by the value of e.

8 Conclusion

Beginning from our conviction in the importance of the concepts of VSS. In this article, we first use the complex number function to represent the VSS, and then by using quaternion function, we introduce a new representation method of CVSSs. In this way, we can simply express the degree of real membership, imaginary membership, real non-membership and imaginary non-membership of the initial elements under each parameter. Based on the above concepts, we give some logic and operations on CVSS-Q. Then, we give three quaternion distance measures between two CVSSs-Q and apply to decision models, behaving more excellent than before. The model can also be applied in many other important areas, such as decision making, pattern recognition, cluster analysis, fuzzy control, comprehensive decision analysis, etc.

However, there are still many imperfections in the content proposed in the article: there are still many distance measures under quaternion, and the accuracy of this model needs to be improved. Therefore, in future research, I will focus on these points.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Footnotes

Acknowledgements

The works described in this paper are supported by the National Natural Science Foundation of China under Grant nos.11501444,11726019.

References

Molodtsov

, Soft set theory-first results, Computers & Mathematics with Applications 37(4-5) (1999), 19–31.

Maji

P.K.

, Biswas

and Roy

A.R.

, Soft set theory, Computers & Mathematics with Applications 45(4-5) (2003), 555–562.

Maji

P.K.

, Biswas

and Roy

A.R.

, Fuzzy soft sets, Journal ofFuzzy Mathematics 9(3) (2001), 589–602.

Majumdar

and Samanta

S.K.

, Generalised fuzzy soft sets, Computers & Mathematics with Applications 59(4) (2010), 1425–1432.

Yang

, Lin

T.Y.

, Yang

, et al., Combination of interval-valued fuzzy set and soft set, Computers & Mathematics with Applications 58(3) (2009), 521–527.

Xiao

, Gong

, Xia

, et al., Exclusive disjunctive soft sets, Computers & Mathematics with Applications 59(6) (2010), 2128–2137.

Maji

P.K.

, Biswas

and Roy

A.R.

, Intuitionistic fuzzy soft sets, Journal of Fuzzy Mathematics 9(3) (2001), 677–692.

Maji

P.K.

, Roy

A.R.

and Biswas

, On intuitionistic fuzzy softsets, Journal of Fuzzy Mathematics 12(3) (2004), 669–684.

Wang

and Qu

, Entropy, similarity measure and distance measureof vague soft sets and their relations, Information Sciences 244 (2013), 92–106.

10.

Gau

W.L.

and Buehrer

D.J.

, Vague sets, IEEE Transactions onSystems, Man, and Cybernetics 23(2) (1993), 610–614.

11.

Wang

, Some properties of entropy of vague soft sets and itsapplications, Journal of Intelligent & Fuzzy Systems 29(4) (2015), 1443–1452.

12.

Chen

S.M.

, Similarity measures between vague sets and betweenelements, IEEE Transactions on Systems, Man, and Cybernetics,Part B (Cybernetics) 27(1) (1997), 153–158.

13.

Chen

S.M.

, et al., Analyzing fuzzy system reliability using vagueset theory, International Journal of Applied Science and Engineering 1(1) (2003), 82–88.

14.

Hong

D.H.

and Choi

C.H.

, Multicriteria fuzzy decision-making problems based on vague set theory, Fuzzy Sets and Systems 114(1) (2000), 103–113.

15.

, Using an improved measure function of vague sets for multicriteria fuzzy decisionmaking, Expert Systems with Applications 37(6) (2010), 4706–4709.

16.

Zhang

, Zhang

, Lai

K.K.

, et al., An novel approach to supplier selection based on vague sets group decision, Expert Systems with Applications 36(5) (2009), 9557–9563.

17.

Kumar

, Yadav

S.P.

, Kumar

, et al., Fuzzy reliability of amarine power plant using interval valued vague sets, International Journal of Applied Science and Engineering 4(1) (2006), 71–82.

18.

Kumar

, Yadav

S.P.

and Kumar

, Fuzzy system reliability analysisusing Tω (the weakest t-norm) based arithmetic operations onL-R type interval valued vague sets, International Journal ofQuality & Reliability Management 24(8) (2007), 846–860.

19.

Bustince

and Burillo

, Vague sets are intuitionistic fuzzysets, Fuzzy Sets and Systems 79(3) (1996), 403–405.

20.

Tamir

D.E.

, Lin

and Kandel

, A new interpretation of complexmembership grade, International Journal of Intelligent Systems 26 (2011), 285–312.

21.

Selvachandran

, Maji

P.K.

, Abed

I.E.

, et al., Complex vague softsets and its distance measures, Journal of Intelligent & Fuzzy Systems 31(1) (2016), 55–68.

22.

Tamir

D.E.

, Ali

, Rishe

N.D.

, et al., Complex number representation of intuitionistic fuzzy sets, World Conference on Soft Computing, USA, Berkeley, (2016), 108-113.

23.

Ngan

R.T.

, Ali

, Tamir

D.E.

, et al., Representing complex intuitionistic fuzzy set by quaternion numbers and applications to decision making, Applied Soft Computing 87 (2020), 105961.

24.

, Ma

, Wang

, et al., Vague soft sets and their properties, Computers & Mathematics with Applications 59(2) (2010), 787–794.

25.

Hamilton XI

W.R.

, On quaternions; or on a new system of imaginariesin algebra, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 33(219) (1848), 58–60.