A three-way decision method based on hybrid data 1

Abstract

An information system as a database that represents relationships between objects and attributes is an important mathematical model in the field of artificial intelligence. Hybrid data means boolean, categorical, real-valued, set-valued data and missing data in this paper. A hybrid information system is an information system where its attribute is hybrid data. This paper proposes a three-way decision method based on hybrid data. First, the distance between two objects based on the conditional attribute set in a given hybrid information system is developed and Gaussian kernel based on this distance is acquired. Then, the fuzzy T_cos-equivalence relation, induced by this information system, is obtained by using Gaussian kernel. Next, the decision-theoretic rough set model in this hybrid information system is presented. Moreover, a three-way decision method is given by means of this decision-theoretic rough set model and inclusion degree between two fuzzy sets. Finally, an example is employed to illustrate the feasibility of the proposed method, which may provide an effective method for hybrid data analysis in real applications.

Keywords

Three-way decision hybrid data decision-theoretic rough set gaussian kernel feasibility

1 Introduction

Rough set theory, proposed by Pawlak [23], is a valid tool for dealing with uncertainty information. Based on rough set theory, an information system as a database that represents relationships between objects and attributes was presented by Pawlak [23, 24]. To date, rough sets have been widely used in many fields, such as uncertainty modeling [2 , 30], reasoning with uncertainty [36], rule extraction [1 , 33], classification and feature selection [7 , 32], these fields are associated with information systems.

In recent decades, several extensions of Pawlak rough set model have been presented according to various requirements, including decision-theoretic rough set model [37], Bayesian rough set model [37], variable precision rough set model [31] and probabilistic rough sets model [40].

On the basis of decision-theoretic rough set model, Yao [38 , 43] proposed a decision-making model which is called a three-way decision model, this model give positive region, boundary region and negative region to good semantic interpretation. More concretely, using these three regions, three-way decision is raised based on decision-theoretic rough sets where decision rules contain three kinds: the positive rule get from positive region is used to represent the acceptance of something; the boundary rule get from boundary region is used to represent further observation, or deferment decision; the negative rule get from negative region is used to represent the rejection of something. Three-way decisions well show the thinking mode of human in solving decision problems, moreover, provide a reliable theoretical foundation for decision making problems.

Three-way decisions have received much attention by scholars, as well as many excellent research contributions have been made. For example, in view of the decision risk of the loss function, Li et al. [17] put forward the corresponding three-way decision making models; taking into account the multiple source information environment; Qian et al. [26] proposed multi-granulation three-way decisions; considering different uncertainty decision measures; Liu et al. presented a series of uncertainty three-way decisions consisted of stochastic three-way decisions [15], interval three-way decisions [14] and fuzzy three-way decisions [16]; Hu et al. [3 –6] studied three-way decision spaces; Yu et al. [42, 44] developed a lot of studies on three-way decision model with clustering analysis; Li et al. [18 , 34] gave a three-way decision method in a hybrid information system with images and its application in medical diagnosis. Therefore, these existing works have greatly promoted the development of three-way decisions.

In reality, there are a lot of information systems in which the data are hybrid. To handle this phenomenon, Zeng et al. [46] studied a fuzzy rough set approach for incremental feature selection on hybrid information systems. Up to now, three-way decisions in hybrid information systems have not been studied. In this paper, we will investigate three-way decisions based on decision-theoretic rough set in a hybrid information system.

The remaining part of this paper is organized as follows. In Section 2, we recall some concepts about fuzzy relations, Pawlak’s rough sets and hybrid information systems, and explain basic thought of Bayesian theory and three-way decision. In Section 3, we construct distance between two objects based on the conditional attribute set in a hybrid information system and obtain fuzzy T_cos equivalence relation induced by this information system by means of Gaussian kernel method. In Section 4, we propose a three-way decision method based on hybrid data. In Section 5, we give an example is given to show the feasibility of the proposed method. In Section 6, we introduce “Compliance with Ethical Standards”. In Section 7, we conclude this paper and highlight the prospects for potential future development in different fields.

2 Preliminaries

In this section, we review some notions about fuzzy sets, fuzzy relations, Pawlak rough sets and hybrid information systems, and clarify basic thought of Bayesian theory and three-way decision.

Throughout this paper, U denotes a finite set called the universe, 2^U expresses the family of all subsets of U. I means the unit interval [0, 1].

Put $U = {x_{1}, x_{2}, \dots, x_{n}} .$

2.1 Fuzzy sets and fuzzy relations

Fuzzy sets are extensions of ordinary sets. A fuzzy set P in U is defined as a function assigning to each element x of U a value A (x) ∈ I and A (x) is called the membership degree of x to the fuzzy set A.

In this paper, I^U denotes the set of all fuzzy sets in U. The cardinality of A ∈ I^U can be calculated with $| A | = \sum_{i = 1}^{n} A (x_{i}) .$

If R is a fuzzy set in U × U, then R is called a fuzzy relation on U. In this paper, I^U×U denotes the set of all fuzzy relations on U.

Let R ∈ I^U×U. Then R may be represented by $M (R) = (\begin{matrix} r_{11} & r_{12} & . . . & r_{1 n} \\ r_{21} & r_{22} & . . . & r_{2 n} \\ . . . & . . . & . . . & . . . \\ r_{n 1} & r_{n 2} & . . . & r_{nn} \end{matrix}),$ where r_ij = R (x_i, x_j) ∈ I means the similarity between two objects x_i and x_j.

If M (R) is a unit matrix, then R is said to be a fuzzy identity relation, and we write as R =▵; if r_ij = 1, i, j ≤ n, then R is said to be a fuzzy universal relation, and we write as R = ω.

Let R ∈ I^U×U. For each x ∈ U, we define a fuzzy set S_R (x): $S_{R} (x) (y) = R (x, y) .$ Then S_R (x) can be viewed as the fuzzy neighborhood or the information granule of the point x [25].

Definition 2.1. ([21]) A function T : I² → I is called a t-norm, if it satisfies the following conditions:

(1) Commutativity: T (a, b) = T (b, a) ,

(2) Associativity: T (T (a, b) , c) = T (a, T (b, c)) ,

(3) Monotonicity: a ≤ c, b ⩽ d = T (a, b) ⩽ T (c, d) ,

(4) Boundary condition: T (a, 1) = a .

Example 2.2. For any x, y ∈ U, define $T_{\cos} (x, y) = (xy - \sqrt{1 - x^{2}} \sqrt{1 - y^{2}}) \lor 0 .$ Then T_cos is a t-norm.

Definition 2.3. ([46]) Let T be the t-norm. Suppose R ∈ I^U×U. Then R is a T-fuzzy equivalence relation on U if it satisfies the following conditions:

(1) Reflexivity: R (x, x) =1,

(2) Symmetry: R (x, y) = R (y, x) ,

(3) T-transitivity: T (R (x, y) , R (y, z)) ⩽ R (x, z) .

Proposition 2.4. ([20]) Suppose that f : U × U → I satisfies f (x, x) =1 for all x ∈ U . Then for any x, y, z ∈ U, $T_{\cos} (f (x, y), f (y, z)) \leq f (x, z) .$

Corollary 2.5. Given R ∈ I^U×U. If R is reflexive, then R is T_cos-transitive.

2.2 Pawlak rough sets

In this section, we briefly recall some basic concepts about Pawlak rough sets.

Given that R is an equivalence relation on U. Then R partitions the universe U into a family of disjoint subsets called equivalence classes. For an equivalence relation R, the equivalence class including x is denoted as follows, $[x]_{R} = {y \in U : xRy},$ The family of all equivalence classes of R is denoted as follows, $U / R = {[x]_{R} : x \in U} .$

Definition 2.6. ([23]) Let R be an equivalence relation on U. Then (U, R) is said to be a Pawlak approximation space. Based on (U, R), the lower approximation, upper approximation and boundary region of X ∈ 2^U, denoted by $\underline{R} (X)$ , $\bar{R} (X)$ and BN_R (X), respectively, are defined as follows: $\underline{R} (X) = {x \in U : [x]_{R} \subseteq X},$ $\bar{R} (X) = {x \in U : [x]_{R} \cap X \neq \emptyset},$ ${BN}_{R} (X) = \bar{R} (X) - \underline{R} (X) .$

Using the inclusion degree, the lower approximation, upper approximation and boundary region of X ∈ 2^U can be equivalently defined as $\underline{R} (X) = {x \in U : ID ([x]_{R}, X) = 1},$ $\bar{R} (X) = {x \in U : ID ([x]_{R}, X) > 0},$ ${BN}_{R} (X) = {x \in U : 0 < ID ([x]_{R}, X) < 1} .$

From the above definition, the objects of non-0 and non-1 membership function values in the universe are assigned to the boundary region since the equivalence relation appears to be too strict. In practice, some more diffuse classifications will make more sense. Therefore, Ziarko [45] first proposed an error-tolerance level with set inclusion [45] through a pair of threshold α and β instead of 0 and 1 where 0 ≤ β < α ≤ 1, in order to put forward the concept of variable precision rough set model which is defined as follows: $\underline{R} (X) = {x \in U : ID ([x]_{R}, X) \geq α},$ $\bar{R} (X) = {x \in U : ID ([x]_{R}, X) > β} .$

2.3 Hybrid information systems

In this section, we briefly recall some basic concepts about hybrid information systems.

Definition 2.7. ([23]) Let U be an object set and A a attribute set. Suppose that U and A are finite sets. Then the pair (U, A) is called an information system, if each attribute a ∈ A determines a information function a : U → V_a, where V_a is the set of information function values of the attribute a.

If (U, A) is an information system, and A = C ∪ D where C is a conditional attribute set and D is a decision attribute set, then (U, A) is called a decision information system.

Definition 2.8. ([23]) Let (U, A) be an information system. Then for any P ⊆ A, an equivalence relation ind (P) is defined by $(x, y) \in ind (P) \Leftrightarrow \forall a \in P, a (x) = a (y) .$

For any P ⊆ A and x ∈ U, denote $[x]_{ind (P)} = {y : (x, y) \in ind (P)},$ $U / ind (P) = {[x]_{ind (P)} : x \in U} .$

Definition 2.9. ([46]) Let (U, A) be a decision information system with A = C ∪ D. Then (U, A) is called a hybrid information system, if C = C^c ∪ C^b ∪ C^r ∪ C^s, where C^c, C^b, C^r and C^s are the categorical, boolean, real-valued and set-valued attribute set, respectively.

Example 2.10. (Continued from Example 1 in [46]) Table 1 depicts a hybrid information system (U, A) with A = C ∪ {d}.

Table 1
A hybrid information system (U, A)

Headache (a₁) Muscle pain(a₂) Temperature (a₃) Syndrome (a₄) d

x ₁ Sick Yes 40 {C, R, A} Flu

x ₂ Sick Yes 39.5 {C, R, A} Flu

x ₃ Middle ? 39 {C} Flu

x ₄ Middle Yes 36.8 {R} Rhinitis

x ₅ Middle No ? {R} Rhinitis

x ₆ No No 36.6 {R, A} Health

x ₇ No ? ? {A} Health

x ₈ No Yes 38 {C, R, A} Flu

x ₉ ? Yes 37 {R} Health

	Headache (a₁)	Muscle pain(a₂)	Temperature (a₃)	Syndrome (a₄)	d
x ₁	Sick	Yes	40	{C, R, A}	Flu
x ₂	Sick	Yes	39.5	{C, R, A}	Flu
x ₃	Middle	?	39	{C}	Flu
x ₄	Middle	Yes	36.8	{R}	Rhinitis
x ₅	Middle	No	?	{R}	Rhinitis
x ₆	No	No	36.6	{R, A}	Health
x ₇	No	?	?	{A}	Health
x ₈	No	Yes	38	{C, R, A}	Flu
x ₉	?	Yes	37	{R}	Health

2.4 Basic thought of Bayesian theory and three-way decision

In this subsection, we review basic thought of Bayesian theory and three-way decision.

Definition 2.11. ([47]) Suppose that a set function P : 2^U → I is called a probability measure, if it satisfies the following conditions:

(1) P (U) =1,

(2) If A∩ B = ∅, then P (A ∪ B) = P (A) + P (B) .

Definition 2.12. ([47]) Let P be a probability measure, A, B ∈ 2^U and P (B) >0, then conditional probability of A under condition B is defined as: $P (A | B) = \frac{P (A \cap B)}{P (B)} .$

Suppose that the state and decision set denoted by Ω = {ω₁, ω₂, ⋯ , ω_m} and A = {a₁, a₂, ⋯ , a_n}, respectively. P (ω_j| [x] _R) represents to the conditional probability of the object x in the state ω_j. λ (a_i|ω_j) expresses the cost of taking the action a_i under the state ω_j. Then the expected utility of the action a_i can be expressed as $R (a_{i} | x) = \sum_{j = 1}^{m} λ (a_{i} | ω_{j}) P (ω_{j} | [x]_{R}) .$ Thus, the overall risk of decision rules can be calculated as follows: $R = \sum_{x \in U} R (a_{i} | x) P ([x]_{R}) .$

For each object x, one can calculate the risk conditions of (a_i|x) and select the minimum risk from the conditions of action.

Based on the lower and upper approximations of Pawlak’s rough sets, for any X ∈ 2^U, positive region, boundary region and negative region on X can be divided as follows: $POS (X) = \underline{R} (X),$ $BND (X) = \bar{R} (X) - \underline{R} (X),$ $NEG (X) = U - \bar{R} (X) .$

A positive rule generated by a positive region corresponds to an acceptance decision; A negative rule generated by a negative region corresponds to a rejection decision; The boundary rules generated by the boundary region make the delay decision. That’s what we say, three-way decision model.

In three-way decision, we make use of two state sets and three action sets depicting the decision process. Two states can be denoted by Ω = {X, ¬ X}, indicating that an equivalence class [X] _R is in X and not in X, respectively. Three action sets can be described as A = {a₁, a₂, a₃}, where a₁, a₂ and a₃ indicate to acceptance of something, deferment decision and reject of something, respectively.

Given that P (X| [x]) and P (¬ X| [x]) are the condition probability of the object belongs to the state X and ¬X, respectively. For i = 1, 2, 3, λ (a_i|X) and λ (a_i| ¬ X) are denoted as the cost, for taking the action a_i of the states X and ¬X, respectively. Then, the expected loss related to take the action a_i as follows,

$R (a_{i} | X) = λ (a_{i} | X) \cdot P (X | [x]) + λ (a_{i} | \neg X) \cdot P (\neg X | [x]) .$

Through the Bayesian decision process, the decision rules with minimum-cost criterion as follows:

(P1) : x ∈ POS (X), if R (a₁| [x]) ⩽ R (a₂| [x]) and R (a₁| [x])⩽ R (a₃| [x]) ;

(B1) : x ∈ BND (X), if R (a₂| [x]) ⩽ R (a₁| [x]) and R (a₂| [x])⩽ R (a₃| [x]) ;

(N1) : x ∈ NEG (X), if R (a₃| [x]) ⩽ R (a₁| [x]) and R (a₃| [x]) ⩽ R (a₂| [x]) .

On account of Yao’s study, the minimum-risk decision rules (P), (B), (N) are able to written as:

(P1′) : x ∈ POS (X), if P (X| [x]) ⩾ α and P (X| [x])⩽ γ ;

(B1′) : x ∈ BND (X), if β⩽ P (X| [x]) ⩽ α ;

(N1′) : x ∈ NEG (X), if P (X| [x]) ⩽ β and P (X| [x]) ⩽ γ .

Where $α = \frac{λ_{12} - λ_{32}}{(λ_{31} - λ_{11}) + (λ_{12} - λ_{32})},$ $β = \frac{λ_{32} - λ_{22}}{(λ_{21} - λ_{31}) + (λ_{32} - λ_{22})},$ $γ = \frac{λ_{12} - λ_{22}}{(λ_{21} - λ_{11}) + (λ_{12} - λ_{22})},$ $λ_{i 1} = λ (a_{i} | X), λ_{i 2} = λ (a_{i} | \neg X) (i = 1, 2, 3) .$ By P (X| [x]), we can decide how to assign X into these three disjoint regions in the Bayesian decision procedure.

3 The fuzzy T_cos-equivalence relation induced by a hybrid information system

In this section, we give the fuzzy T_cos-equivalence relation induced by a hybrid information system by means of Gaussian kernel method.

3.1 The distance between two objects based on the conditional attribute set

To construct the distance between two objects based on the conditional attribute set in a hybrid information system, a novel distance between two information values should be presented.

Definition 3.1. ([46]) Let (U, A) be a hybrid information system with A = C ∪ {d}. ∀ x, y ∈ U, ∀ a ∈ C^b, the distance between a (x) and a (y) is defined as $d (a (x), a (y)) = {\begin{matrix} 0 & a (x) = a (y) \\ 1 & a (x) \neq a (y) \end{matrix} .$

For A, B ∈ 2^U, denote $A \oplus B = A \cup B - A \cap B .$ Then A ⊕ B is called the symmetric difference between A and B.

Obviously, ∣A ⊕ B ∣ = ∣ A ∪ B ∣ - ∣ A ∩ B ∣ .

Definition 3.2. Let (U, A) be a hybrid information system. ∀ x, y ∈ U, ∀ a ∈ C^c, the distance between a (x) and a (y) is defined as $d (a (x), a (y)) = \frac{| [x]_{a} \oplus [y]_{a} |}{| [x]_{a} \cup [y]_{a} |} .$

Because each boolean attribute can be viewed as a categorical attribute, thus the following proposition ensures that Definition 3.2 is reasonable.

Proposition 3.3. Let (U, A) be a hybrid information system with A = C ∪ {d}. Given a ∈ C^b. Then ∀ x, y ∈ U, $\frac{| [x]_{a} \oplus [y]_{a} |}{| [x]_{a} \cup [y]_{a} |} = {\begin{matrix} 0 & a (x) = a (y) \\ 1 & a (x) \neq a (y) \end{matrix} .$

Proof. By Definition 3.2. $d (a (x), a (y)) = \frac{| [x]_{a} \oplus [y]_{a} |}{| [x]_{a} \cup [y]_{a} |} .$

If a (x) = a (y) , we have [x] _a = [y] _a, then [x] _a ∪ [y] _a = [x] _a ∩ [y] _a = [x] _a.Thus $d (a (x), a (y)) = \frac{| [x]_{a} \oplus [y]_{a} |}{| [x]_{a} \cup [y]_{a} |}$ $= \frac{| ([x]_{a} \cup [y]_{a}) | - | ([y]_{a} \cap [x]_{a}) |}{| [x]_{a} \cup [y]_{a} |} = 0 .$ If a (x) ≠ a (y) , we have [x] _a∩ [y] _a = ∅, then $d (a (x), a (y)) = \frac{| [x]_{a} \oplus [y]_{a} |}{| [x]_{a} \cup [y]_{a} |}$ $= \frac{| ([x]_{a} \cup [y]_{a}) | - | ([y]_{a} \cap [x]_{a}) |}{| [x]_{a} \cup [y]_{a} |} = 1 .$ Thus $d (a (x), a (y)) = {\begin{matrix} 0 & a (x) = a (y) \\ 1 & a (x) \neq a (y) \end{matrix} .$

□

Definition 3.4. Let (U, A) be a hybrid information system with A = C ∪ {d}. ∀ x, y ∈ U, ∀ a ∈ C^r, the distance between a (x) and a (y) is defined as $d (a (x), a (y)) = \frac{| a (x) - a (y) |}{2 θ_{a}} .$ where θ_a is the standard deviation under the attribute a.

Definition 3.5. ([46]) Let (U, A) be a hybrid information system with A = C ∪ {d}. ∀ x, y ∈ U, ∀ a ∈ C. If a (x) = ? or a (y) = ? and x ≠ y, distance between a (x) and a (y) is defined as $d (a (x), a (y)) = 1 .$

Definition 3.6. ([46]) Let (U, A) be a hybrid information system with A = C ∪ {d}. ∀ x, y ∈ U, ∀ a ∈ C^s, the distance between a (x) and a (y) is defined as $d (a (x), a (y)) = 1 - \frac{| a (x) \cap a (y) |}{M_{a}} .$ where M_a = max {∣ a (x) ∣ : x ∈ U}.

According to the above five definitions, a new hybrid distance is defined as follows.

Definition 3.7. Let (U, A) be a hybrid information system with A = C ∪ {d}. ∀ x, y ∈ U, the distance between x and y based on the conditional attribute set is defined as $d (x, y) = \sqrt{\sum_{a \in C} d^{2} (a (x), a (y))} .$

3.2 The fuzzy T_cos-equivalence relation induced by a hybrid information system

Gaussian kernel method is an important methodology in machine learning and pattern recognition. For making data linear and simplifying classification tasks, it maps data into a higher dimensional feature space [28]. Hu et al. [8, 9] found that there are some relationships between rough sets and Gaussian kernel method, so Gaussian kernel is used to obtain fuzzy relations. In this section, we use Gaussian kernel to extract a fuzzy T_cos-equivalence relation on the object set of a given hybrid information system.

Gaussian kernel $G (x_{,} y) = \exp (- \frac{∥ x - y ∥^{2}}{2 δ^{2}})$ is using to compute the similarity between two objects x and y, where ∥x - y∥ is the Euclidean distance between two objects x and y, δ is a threshold. In this paper, pick δ ∈ (0, 1].

Obviously, G (x, y) satisfies:

(1) G (x, y) ∈ [0, 1];

(2) G (x, y) = G (y, x);

(3) G (x, x) =1.

Definition 3.8. Let (U, A) be a hybrid information system with A = C ∪ {d}. Given δ ∈ (0, 1], denote $R_{C}^{G} (δ) (x_{i}, x_{j}) = \exp (\frac{- d^{2} (x_{i}, x_{j})}{2 δ^{2}}),$ $M (R_{C}^{G} (δ)) = (R_{C}^{G} (δ) (x_{i}, x_{j}))_{n \times n} .$ Then $M (R_{C}^{G} (δ))$ is called the Gaussian kernel matric of (U, A) with respect to δ.

Theorem 3.9. Let (U, A) be a hybrid information system with A = C ∪ {d}. Given δ ∈ (0, 1]. Then $R_{C}^{G} (δ)$ is a fuzzy T_cos-equivalence relation on U.

Proof. This holds by Corollary 2.5.

Definition 3.10. Let (U, A) be a hybrid information system with A = C ∪ {d}. Given δ ∈ (0, 1]. Then $R_{C}^{G} (δ)$ is called the fuzzy T_cos-equivalence relation induced by (U, A) with respect to δ.

Example 3.11. Based on Table 1, pick $δ = \sqrt{0.3}$ , then $M (R_{C}^{G} (\sqrt{δ})) =$ Thus the fuzzy T_cos-equivalence relation $R_{C}^{G} (\sqrt{δ})$ is obtained.

An algorithm for obtaining the fuzzy T_cos-equivalence relation is designed as follows.

4 A three-way decision method based on hybrid data

In this section, we give a three-way decision method based on hybrid data.

4.1 Decision-theoretic rough set models for a hybrid information system

In this subsection, we propose the decision-theoretic rough set model based on the fuzzy T_cos-equivalence relation induced by a given hybrid information system.

Definition 4.1. ([10]) Let A, B ∈ I^U. Then the inclusion degree ID (A, B) is defined as follows: $I (A, B) = \frac{| A \cap B |}{| A |} .$

Definition 4.2. Let (U, A) be a hybrid information system with A = C ∪ {d}. Given δ ∈ (0, 1]. Suppose that $R_{C}^{G} (δ)$ is the fuzzy T_cos-equivalence relation induced by (U, A). Based on $(U, R_{C}^{G} (δ))$ , the variable precision lower and upper approximation of X ∈ U/ind (D) with respect to (α, β) can be defined as follows: ${\underline{R_{C}^{G} (δ)}}^{α} (X) = {x \in U : ID (S_{R_{C}^{G} (δ)} (x), X) \geq α},$ ${\bar{R_{C}^{G} (δ)}}^{β} (X) = {x \in U : ID (S_{R_{C}^{G} (δ)} (x), X) > β},$ where $X (x) = {\begin{matrix} 1, x \in X, \\ 0, x \notin X . \end{matrix}$

Based on the above decision-theoretic rough model, for any X ∈ 2^U, the universe U can be divided into three regions, which are called the positive region, boundary region and negative region of X, denoted by ${POS}_{R_{C}^{G} (δ)}^{(α, β)} (X)$ , ${BND}_{R_{C}^{G} (δ)}^{(α, β)} (X)$ and ${NEG}_{R_{C}^{G} (δ)}^{(α, β)} (X)$ , respectively. They are defined as follows: ${POS}_{R_{C}^{G} (δ)}^{(α, β)} (X) = {\underline{R_{C}^{G} (δ)}}^{α} (X),$ ${BND}_{R_{C}^{G} (δ)}^{(α, β)} (X) = {\bar{R_{C}^{G} (δ)}}^{β} (X) - {\underline{R_{C}^{G} (δ)}}^{α} (X),$ ${NEG}_{R_{C}^{G} (δ)}^{(α, β)} (X) = U - {\bar{R_{C}^{G} (δ)}}^{β} (X) .$

Obviously,

${BND}_{R_{C}^{G} (δ)}^{(α, β)} (X) = {x \in U : β < ID (S_{R_{C}^{G} (δ)} (x), X) < α},$

${NEG}_{R_{C}^{G} (δ)}^{(α, β)} (X) = {x \in U : ID (S_{R_{C}^{G} (δ)} (x), X) \leq β} .$

4.2 A three-way decision method in a hybrid information system

Under Bayesian risk decision criterion, the positive rule generated by the positive region corresponds to make a decision of acceptance; the negative rule generated by the negative region corresponds to make a decision of rejection; the boundary rule generated by the boundary region corresponds to make a decision of delay. That’s what we say, three-way decision method.

Based on the Bayesian decision procedure and Yao’s decision-theoretic study, three-way decision which are based on inclusion degree can be written as follows:

(P) : if $ID (S_{R_{C}^{G} (δ)} (x), X) \geq α$ , then $x \in {POS}_{R_{C}^{G} (δ)}^{(α, β)} (X)$ .

(B) : if $ID (S_{R_{C}^{G} (δ)} (x), X) \leq β$ , then $x \in {NEG}_{R_{C}^{G} (δ)}^{(α, β)} (X) .$

(N) : if $β < ID (S_{R_{C}^{G} (δ)} (x), X) < α$ , then $x \in {BND}_{R_{C}^{G} (δ)}^{(α, β)} (X)$ .

5 An illustrative example

In this section, we employ an example to illustrate the feasibility of the proposed method.

Example 5.1. (Continued from Example 2.10 and 3.11)Obviously, U/ind ({d}) = {D₁, D₂, D₃}, where $D_{1} = {x_{1}, x_{2}, x_{3}, x_{8}}, D_{2} = {x_{4}, x_{5}}, D_{3} = {x_{6}, x_{7}, x_{9}} .$

Below, we will describe in detail the process of three-way decision in a hybrid information system (U, C ∪ {d}).(1) Pick α = 0.8, β = 0.15 and $δ_{1} = \sqrt{0.3}$ .

By using Gaussian kernel, we obtain the Gaussian kernel matric of (U, A) with respect to δ₁. $R_{C}^{G} (δ_{1}) =$

Denote $R_{1} = R_{C}^{G} (δ_{1})$ . Then we can get nine fuzzy information granules on U with respect to δ₁ as follows: $S_{R_{1}} (x_{1}) = \frac{1.0000}{x_{1}} + \frac{0.7184}{x_{2}} + \frac{0.0132}{x_{3}} + \frac{0.0068}{x_{4}} + \frac{0.0032}{x_{5}} + \frac{0.0016}{x_{6}} + \frac{0.0032}{x_{7}} + \frac{0.0687}{x_{8}} + \frac{0.0092}{x_{9}}$ , $S_{R_{1}} (x_{2}) = \frac{0.7184}{x_{1}} + \frac{1.0000}{x_{2}} + \frac{0.0160}{x_{3}} + \frac{0.0143}{x_{4}} + \frac{0.0032}{x_{5}} + \frac{0.0035}{x_{6}} + \frac{0.0032}{x_{7}} + \frac{0.1069}{x_{8}} + \frac{0.0185}{x_{9}}$ , $S_{R_{1}} (x_{3}) = \frac{0.0132}{x_{1}} + \frac{0.0160}{x_{2}} + \frac{1.0000}{x_{3}} + \frac{0.0105}{x_{4}} + \frac{0.0067}{x_{5}} + \frac{0.0016}{x_{6}} + \frac{0.0013}{x_{7}} + \frac{0.0132}{x_{8}} + \frac{0.0025}{x_{9}}$ , $S_{R_{1}} (x_{4}) = \frac{0.0068}{x_{1}} + \frac{0.0143}{x_{2}} + \frac{0.0105}{x_{3}} + \frac{1.0000}{x_{4}} + \frac{0.0357}{x_{5}} + \frac{0.0168}{x_{6}} + \frac{0.0013}{x_{7}} + \frac{0.0626}{x_{8}} + \frac{0.1870}{x_{9}}$ , $S_{R_{1}} (x_{5}) = \frac{0.0032}{x_{1}} + \frac{0.0032}{x_{2}} + \frac{0.0067}{x_{3}} + \frac{0.0357}{x_{4}} + \frac{1.0000}{x_{5}} + \frac{0.0170}{x_{6}} + \frac{0.0013}{x_{7}} + \frac{0.0032}{x_{8}} + \frac{0.0067}{x_{9}}$ , $S_{R_{1}} (x_{6}) = \frac{0.0016}{x_{1}} + \frac{0.0035}{x_{2}} + \frac{0.0016}{x_{3}} + \frac{0.0168}{x_{4}} + \frac{0.0170}{x_{5}} + \frac{1.0000}{x_{6}} + \frac{0.0170}{x_{7}} + \frac{0.1090}{x_{8}} + \frac{0.0163}{x_{9}}$ , $S_{R_{1}} (x_{7}) = \frac{0.0032}{x_{1}} + \frac{0.0032}{x_{2}} + \frac{0.0013}{x_{3}} + \frac{0.0013}{x_{4}} + \frac{0.0013}{x_{5}} + \frac{0.0170}{x_{6}} + \frac{1.0000}{x_{7}} + \frac{0.0710}{x_{8}} + \frac{0.0013}{x_{9}}$ , $S_{R_{1}} (x_{8}) = \frac{0.0687}{x_{1}} + \frac{0.1069}{x_{2}} + \frac{0.0132}{x_{3}} + \frac{0.0626}{x_{4}} + \frac{0.0032}{x_{5}} + \frac{0.1090}{x_{6}} + \frac{0.0710}{x_{7}} + \frac{1.0000}{x_{8}} + \frac{0.0699}{x_{9}}$ , $S_{R_{1}} (x_{9}) = \frac{0.0092}{x_{1}} + \frac{0.0185}{x_{2}} + \frac{0.0025}{x_{3}} + \frac{0.1870}{x_{4}} + \frac{0.0067}{x_{5}} + \frac{0.0163}{x_{6}} + \frac{0.0013}{x_{7}} + \frac{0.0699}{x_{8}} + \frac{1.0000}{x_{9}} .$ Based on the inclusion degree between the fuzzy granules S_R (x_i) and D_j, we haveID (S_{R
₁} (x₁) , D₁) =0.9868, ID (S_{R
₁} (x₂) , D₁) =0.9773, ID (S_{R
₁} (x₃) , D₁) =0.9787, ID (S_{R
₁} (x₄) , D₁) =0.0706, ID (S_{R
₁} (x₅) , D₁) =0.0151, ID (S_{R
₁} (x₆) , D₁) =0.0978, ID (S_{R
₁} (x₇) , D₁) =0.0716, ID (S_{R
₁} (x₈) , D₁) =0.7902, ID (S_{R
₁} (x₉) , D₁) =0.0763 ; ID (S_{R
₁} (x₁) , D₂) =0.0055, ID (S_{R
₁} (x₂) , D₂) =0.0093, ID (S_{R
₁} (x₃) , D₂) =0.0162, ID (S_{R
₁} (x₄) , D₂) =0.7758, ID (S_{R
₁} (x₅) , D₂) =0.9617, ID (S_{R
₁} (x₆) , D₂) =0.0286, ID (S_{R
₁} (x₇) , D₂) =0.0024, ID (S_{R
₁} (x₈) , D₂) =0.0437, ID (S_{R
₁} (x₉) , D₂) =0.1477 ;ID (S_{R
₁} (x₁) , D₃) =0.0077, ID (S_{R
₁} (x₂) , D₃) =0.0134, ID (S_{R
₁} (x₃) , D₃) =0.0051, ID (S_{R
₁} (x₄) , D₃) =0.1536, ID (S_{R
₁} (x₅) , D₃) =0.0232, ID (S_{R
₁} (x₆) , D₃) =0.8736, ID (S_{R
₁} (x₇) , D₃) =0.9261, ID (S_{R
₁} (x₈) , D₃) =0.1661, ID (S_{R
₁} (x₉) , D₃) =0.7760 .Based on ID (S_{R
₁} (x_i) , D₁), we can get the lower approximation, upper approximation and boundary region of D₁, respectively. ${\underline{R}}_{1}^{α} (D_{1}) = {x_{1}, x_{2}, x_{3}}$ , ${\bar{R}}_{1}^{β} (D_{1}) = {x_{1}, x_{2}, x_{3}, x_{8}}$ , ${BND}_{R_{1}}^{(α, β)} (D_{1}) = {x_{8}}$ .Based on ID (S_{R
₁} (x_i) , D₂), we can get the lower approximation, upper approximation and boundary region of D₂, respectively. ${\underline{R}}_{1}^{α} (D_{2}) = {x_{5}}$ , ${\bar{R}}_{1}^{β} (D_{2}) = {x_{4}, x_{5}}$ , ${BND}_{R_{1}}^{(α, β)} (D_{2}) = {x_{4}}$ .Based on ID (S_{R
₁} (x_i) , D₃), we can get the lower approximation, upper approximation and boundary region of D₃, respectively. ${\underline{R}}_{1}^{α} (D_{3}) = {x_{6}, x_{7}}$ , ${\bar{R}}_{1}^{β} (D_{3}) = {x_{4}, x_{6}, x_{7}, x_{8}, x_{9}}$ , ${BND}_{R_{1}}^{(α, β)} (D_{3}) = {x_{4}, x_{8}, x_{9}}$ .Thus, the three-way decision rulers are obtained as follows:(P1) if $x \in {\underline{R}}_{1}^{α} (D_{1})$ , then x made “Flu”;(N1) if $x \in U - {\bar{R}}_{1}^{β} (D_{1})$ , then x did not make “Flu”;(B1) otherwise, x need to do further diagnosis.(P2) if $x \in {\underline{R}}_{1}^{α} (D_{2})$ , then x made “Rhinitis”;(N2) if $x \in U - {\bar{R}}_{1}^{β} (D_{2})$ , then x did not make “Rhinitis”;(B3) otherwise, x need to do further diagnosis.(P3) if $x \in {\underline{R}}_{1}^{α} (D_{3})$ , then x is healthy;(N3) if $x \in U - {\bar{R}}_{1}^{β} (D_{3})$ , then x is not healthy;(B3) otherwise, x need to do further diagnosis.(2) Suppose α = 0.8, β = 0.15 and $δ_{2} = \sqrt{0.6}$ .By using Gaussian kernel, we obtain the Gaussian kernel matric of (U, A) with respect to δ₂. $R_{C}^{G} (δ_{2}) =$

Denote $R_{2} = R_{C}^{G} (δ_{2})$ . Then we can get nine fuzzy information granules on U with respect to δ₂ as follows: $S_{R_{2}} (x_{1}) = \frac{1.0000}{x_{1}} + \frac{0.8476}{x_{2}} + \frac{0.1149}{x_{3}} + \frac{0.0822}{x_{4}} + \frac{0.0567}{x_{5}} + \frac{0.0399}{x_{6}} + \frac{0.0567}{x_{7}} + \frac{0.2621}{x_{8}} + \frac{0.0962}{x_{9}}$ , $S_{R_{2}} (x_{2}) = \frac{0.8476}{x_{1}} + \frac{1.0000}{x_{2}} + \frac{0.1264}{x_{3}} + \frac{0.1194}{x_{4}} + \frac{0.0567}{x_{5}} + \frac{0.0595}{x_{6}} + \frac{0.0567}{x_{7}} + \frac{0.1069}{x_{8}} + \frac{0.0185}{x_{9}}$ , $S_{R_{2}} (x_{3}) = \frac{0.1149}{x_{1}} + \frac{0.1264}{x_{2}} + \frac{1.0000}{x_{3}} + \frac{0.1024}{x_{4}} + \frac{0.0821}{x_{5}} + \frac{0.0396}{x_{6}} + \frac{0.0357}{x_{7}} + \frac{0.1149}{x_{8}} + \frac{0.0495}{x_{9}}$ , $S_{R_{2}} (x_{4}) = \frac{0.0822}{x_{1}} + \frac{0.1194}{x_{2}} + \frac{0.1024}{x_{3}} + \frac{1.0000}{x_{4}} + \frac{0.1889}{x_{5}} + \frac{0.1298}{x_{6}} + \frac{0.0357}{x_{7}} + \frac{0.2501}{x_{8}} + \frac{0.4324}{x_{9}}$ , $S_{R_{2}} (x_{5}) = \frac{0.0567}{x_{1}} + \frac{0.0567}{x_{2}} + \frac{0.0821}{x_{3}} + \frac{0.1889}{x_{4}} + \frac{1.0000}{x_{5}} + \frac{0.1304}{x_{6}} + \frac{0.0357}{x_{7}} + \frac{0.0567}{x_{8}} + \frac{0.0821}{x_{9}}$ , $S_{R_{2}} (x_{6}) = \frac{0.0399}{x_{1}} + \frac{0.0595}{x_{2}} + \frac{0.0396}{x_{3}} + \frac{0.1298}{x_{4}} + \frac{0.1304}{x_{5}} + \frac{1.0000}{x_{6}} + \frac{0.1304}{x_{7}} + \frac{0.3302}{x_{8}} + \frac{0.1278}{x_{9}}$ , $S_{R_{2}} (x_{7}) = \frac{0.0567}{x_{1}} + \frac{0.0567}{x_{2}} + \frac{0.0357}{x_{3}} + \frac{0.0357}{x_{4}} + \frac{0.0357}{x_{5}} + \frac{0.1304}{x_{6}} + \frac{1.0000}{x_{7}} + \frac{0.1304}{x_{8}} + \frac{0.0357}{x_{9}}$ , $S_{R_{2}} (x_{8}) = \frac{0.2621}{x_{1}} + \frac{0.3270}{x_{2}} + \frac{0.1149}{x_{3}} + \frac{0.2501}{x_{4}} + \frac{0.0567}{x_{5}} + \frac{0.3302}{x_{6}} + \frac{0.1304}{x_{7}} + \frac{1.0000}{x_{8}} + \frac{0.1278}{x_{9}}$ , $S_{R_{2}} (x_{9}) = \frac{0.0962}{x_{1}} + \frac{0.1362}{x_{2}} + \frac{0.0495}{x_{3}} + \frac{0.4324}{x_{4}} + \frac{0.0821}{x_{5}} + \frac{0.1278}{x_{6}} + \frac{0.0357}{x_{7}} + \frac{0.1278}{x_{8}} + \frac{1.0000}{x_{9}} .$ Based on the inclusion degree between the fuzzy granules S_R (x_i) and D_j, we haveID (S_{R
₂} (x₁) , D₁) =0.8702, ID (S_{R
₂} (x₂) , D₁) =0.8430, ID (S_{R
₂} (x₃) , D₁) =0.8143, ID (S_{R
₂} (x₄) , D₁) =0.2367, ID (S_{R
₂} (x₅) , D₁) =0.1493, ID (S_{R
₂} (x₆) , D₁) =0.2361, ID (S_{R
₂} (x₇) , D₁) =0.1842, ID (S_{R
₂} (x₈) , D₁) =0.6556, ID (S_{R
₂} (x₉) , D₁) =0.1962 ; ID (S_{R
₂} (x₁) , D₂) =0.0543, ID (S_{R
₂} (x₂) , D₂) =0.0645, ID (S_{R
₂} (x₃) , D₂) =0.1108, ID (S_{R
₂} (x₄) , D₂) =0.5079, ID (S_{R
₂} (x₅) , D₂) =0.7038, ID (S_{R
₂} (x₆) , D₂) =0.1309, ID (S_{R
₂} (x₇) , D₂) =0.0471, ID (S_{R
₂} (x₈) , D₂) =0.1180, ID (S_{R
₂} (x₉) , D₂) =0.2464 ;ID (S_{R
₂} (x₁) , D₃) =0.0754, ID (S_{R
₂} (x₂) , D₃) =0.0924, ID (S_{R
₂} (x₃) , D₃) =0.0749, ID (S_{R
₂} (x₄) , D₃) =0.2554, ID (S_{R
₂} (x₅) , D₃) =0.1469, ID (S_{R
₂} (x₆) , D₃) =0.6330, ID (S_{R
₂} (x₇) , D₃) =0.7687, ID (S_{R
₂} (x₈) , D₃) =0.2264, ID (S_{R
₂} (x₉) , D₃) =0.5573 .Based on ID (S_{R
₂} (x_i) , D₁), we can get the lower approximation, upper approximation and boundary region of D₁, respectively. ${\underline{R}}_{2}^{α} (D_{1}) = {x_{1}, x_{2}, x_{3}}$ , ${\bar{R}}_{2}^{β} (D_{1}) = {x_{1}, x_{2}, x_{3}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}}$ , ${BND}_{R_{2}}^{(α, β)} (D_{1}) = {x_{5}, x_{6}, x_{7}, x_{8}, x_{9}}$ .Based on ID (S_{R
₂} (x_i) , D₂), we can get the lower approximation, upper approximation and boundary region of D₂, respectively. ${\underline{R}}_{2}^{α} (D_{2}) = \emptyset$ , ${\bar{R}}_{2}^{β} (D_{2}) = {x_{4}, x_{5}, x_{9}}$ , ${BND}_{R_{2}}^{(α, β)} (D_{2}) = {x_{4}, x_{5}, x_{9}}$ .Based on ID (S_{R
₂} (x_i) , D₃), we can get the lower approximation, upper approximation and boundary region of D₃, respectively. ${\underline{R}}_{2}^{α} (D_{3}) = \emptyset$ , ${\bar{R}}_{2}^{β} (D_{3}) = {x_{4}, x_{6}, x_{7}, x_{8}, x_{9}}$ , ${BND}_{R_{2}}^{(α, β)} (D_{3}) = {x_{4}, x_{6}, x_{7}, x_{8}, x_{9}}$ .Thus, the three-way decision rulers are obtained as follows:(P1) if $x \in {\underline{R}}_{2}^{α} (D_{1})$ , then x made “Flu”;(N1) if $x \in U - {\bar{R}}_{2}^{β} (D_{1})$ , then x did not make “Flu”;(B1) otherwise, x need to do further diagnosis.(P2) if $x \in {\underline{R}}_{2}^{α} (D_{2})$ , then x made “Rhinitis”;(N2) if $x \in U - {\bar{R}}_{2}^{β} (D_{2})$ , then x did not make “Rhinitis”;(B2) otherwise, x need to do further diagnosis.(P3) if $x \in {\underline{R}}_{2}^{α} (D_{3})$ , then x is healthy;(N3) if $x \in U - {\bar{R}}_{2}^{β} (D_{3})$ , then x is not healthy;(B3) otherwise, x need to do further diagnosis.

6 Conclusions

In this paper, the distance between two objects based on the conditional attribute set in a given hybrid information system has been developed. Gaussian kernel based on this distance has been applied for obtaining the fuzzy T_cos-equivalence relation. Through the inclusion degree between two fuzzy sets, the decision-theoretic rough set model in this hybrid information system has been presented. A three-way decision method in this hybrid information system is given. An example by using this method has been employed to illustrate effectiveness of the proposed method in real applications. It shows that the proposed method shall be helpful for handling decision problems with hybrid data. In future work, we will apply other rough set models to study hybrid information systems.

Footnotes

Acknowledgments

The authors would like to thank the editors and the anonymous reviewers for their valuable comments and suggestions, which have helped immensely in improving the quality of the paper. This work is supported by National Social Science Foundation of China (16XJY015), Research Topic of Guangxi Philosophy and Social Science Planning (15BGL003), Natural Science Foundation of Guangxi (2018GXNSFAA294134), Guangxi Higher Education Institutions of China (Document No.[2019] 52).

References

Blaszczynski

, Slowinski

and Szelag

, Sequential covering rule induction algorithm for variable consistency rough set approaches, Information Sciences 181(5) (2011), 987–1002.

Cornelis

, Jensen

, Martin

G.H.

and Slezak

, Attribute selection with fuzzy decision reducts, Information Sciences 180 (2010), 209–224.

B.Q.

, Three-way decisions space and three-way decisions, Information Sciences 281 (2014), 21–52.

B.Q.

, Three-way decisions based on semi-three-way decision spaces,C383}, Information Sciences 382’{l (2017), 415–440.

B.Q.

, Three-way decision spaces based on partially ordered sets and three-way decisions based on hesitant fuzzy sets, Knowledge-Based Systems 91 (2016), 16–31.

B.Q.

, Wong

H.U.

and Yiu

K.F.C.

, On two novel types of threeway decisions in three-way decision spaces, International Journal of Approximate Reasoning 82 (2017), 285–306.

Q.H.

, Pedrycz

, Yu

D.R.

and Lang

, Selecting discrete and continuous features based on neighborhood decision error minimization, IEEE transactions on systems man and cybernetics (Part B) 40 (2010), 137–150.

Q.H.

, Xie

Z.X.

and Yu

D.R.

, Hybrid attribute reduction based on a novel fuzzy-rough model and information granulation, Pattern Recognition 40 (2007), 3509–3521.

Q.H.

, Zhang

, Chen

D.G.

, Pedrycz

and Yu

D.R.

, Gaussian kernel based fuzzy rough sets: model, uncertainty measures and applications, International Journal of Approximate Reasoning 51 (2010), 453–471.

10.

Lin

G.P.

, Liang

J.Y.

, Qian

Y.H.

and Li

J.J.

, A fuzzy multigranulation decision-theoretic approach to multi-source fuzzy information systems, Knowledge-Based Systems 91 (2016), 102–113.

11.

Jensen

and Shen

, Semantics-preserving dimensionality reduction: rough and fuzzy rough based approaches, IEEE Transactions on Snowledge and Data Engineering 16 (2004), 1457–1471.

12.

Jensen

and Shen

, New approaches to fuzzy-rough feature selection, IEEE Transactions on Fuzzy Systems 17 (2009), 824–838.

13.

Kryszkiewicz

, Rules in incomplete information systems, Information Sciences 113 (1999), 271–292.

14.

Liang

D.C.

and Liu

, Systematic studies on three-way decisions with interval-valued decision-theoretic rough sets, Information Sciences 276 (2014), 186–203.

15.

Liu

, Li

T.R.

and Liang

D.C.

, Three-way decisions in stochastic decision-theoretic rough sets, Transactions on Rough Sets XVIII 18 (2014), 110–130.

16.

Liang

D.C.

, Liu

, Pedrycz

and Hu

, Triangular fuzzy decision-theoretic rough sets, International Journal of Approximate Reasoning 54 (2013), 1087–1106.

17.

H.X.

and Zhou

X.Z.

, Risk decision making based on decisiontheoretic rough set: a three-way view decision model, Journal International Journal of Computational Intelligence Systems 4 (2011), 1–11.

18.

Z.W.

, Zhang

P.F.

, Xie

N.X.

, Zhang

G.Q.

and Wen

C.F.

, A novel three-way decision method in a hybrid information system with images and its application in medical diagnosis, Engineering Applications of Artificial Intelligence 92 (2020), 103651.

19.

Z.W.

, Xie

N.X.

, Huang

and Zhang

G.Q.

, A three-way decision method in a hybrid decision information system and its application in medical diagnosis, Artificial Intelligence Review https://doi.org/10.1007/s10462-020-09805-w.

20.

Moser

, On the-transitivity of kernels, Fuzzy Sets and Systems 157 (2006), 1787–1796.

21.

Moser

, On representing and generating kernels by fuzzy equivalence relations, Journal of Machine Learning Research 7 (2006), 2603–2630.

22.

J.S.

, Leung

and Wu

W.Z.

, An uncertainty measure in partition-based fuzzy rough sets, International Journal of General Systems 34 (2005), 77–90.

23.

Pawlak

, Rough sets, International Journal of Computer Information Science 11 (1982), 341–356.

24.

Pawlak

, Rough Sets: Theoretical aspects of reasoning about data, Kluwer Academic Publishers, Dordrecht (1991).

25.

Qian

Y.H.

, Liang

J.Y.

, Wu

W.Z.

and Dang

C.Y.

, Information granularity in fuzzy binary GrC model, IEEE Transactions on Fuzzy Systems 19(2) (2011), 253–264.

26.

Qian

Y.H.

, Zhang

, Sang

Y.L.

and Liang

J.Y.

, Multigranulation decision-theoretic rough sets, International Journal of Approximate Reasoning 55 (2014), 225–237.

27.

Rodrĺłguez

R.M.

, Martĺłnez

and Herrera

, Hesitant fuzzy linguistic term sets for decision making, IEEE Transactions on Fuzzy Systems 20(1) (2012), 109–119.

28.

Shawe-Tayor

and Cristianini

, Kernel methods for patternn analysis, Cambridge University Press, (2004).

29.

Swiniarski

R.W.

and Skowron

, Rough set methods in feature selection and recognition, Pattern Recognition Letters 24 (2003), 833–849.

30.

Slowinski

and Vanderpooten

, A generalized definition of rough approximations based on setilarity, IEEE Transactions on Snowledge and Data Engineering 12 (2000), 331–336.

31.

Slezak

and Ziarko

, The investigation of the bayesian rough set model, International Journal of Approximate Reasoning 40 (2005), 81–91.

32.

Thangavel

and Pethalakshmi

, Dimensionality reduction based on rough set theory: A review, Applied Soft Computing 9 (2009), 1–12.

33.

Wierman

M.J.

, Measuring uncertainty in rough set theory, International Journal of General Systems 28 (1999), 283–297.

34.

Wang

, Zhang

P.F.

and Li

Z.W.

, A three-way decision method based on Gaussian kernel in a hybrid information system with images: An application in medical diagnosis, Applied Soft Computing 77 (2019), 734–749.

35.

Xia

M.M.

and Xu

Z.S.

, Hesitant fuzzy information aggregation in decision making, International Journal of Approximate Reasoning 52 (2011), 395–407.

36.

Yao

Y.Y.

, Relational interpretations of neighborhood operators and rough set approximation operators, Information Sciences 111 (1998), 239–259.

37.

Yao

Y.Y.

, Information granulation and rough set approximation, International Journal of Intelligent Systems 16 (2001), 87–104.

38.

Yao

Y.Y.

, Decision-theoretic rough set models, Lecture Notes in Artificial Intelligence 4481 (2007), 1–12.

39.

Yao

Y.Y.

, Three-way decision: an interpretation of rules in rough set theory, Lecture Notes in Artificial Intelligence 5589 (2009), 642–649.

40.

Yao

Y.Y.

, Three-way decisions with probabilistic rough sets, Information Sciences 180 (2010), 341–353.

41.

Yao

Y.Y.

, The superiority of three-way decisions in probabilistic rough set models, Information Sciences 181 (2011), 1080–1096.

42.

, Liu

Z.G.

and Wang

G.Y.

, An automatic method to determine the number of clusters using decision-theoretic rough set, International Journal of Approximate Reasoning 55 (2014), 101–115.

43.

Yao

Y.Y.

and Wong

S.K.M.

, A decision theoretic framework for approximating concepts, International Journal of Man-machine Studies 37 (1992), 793–809.

44.

, Zhang

and Wang

G.Y.

, A tree-based incremental overlapping clusteringmethod using the three-way decision theory, Knowledge-Based Systems 91 (2016), 189–203.

45.

Ziarko

, Variable precision rough sets model, Journal of Computer and Systems Sciences 46(1) (1993), 39–59.

46.

Zeng

A.P.

, Li

T.R.

, Liu

, Zhang

J.B.

and Chen

H.M.

, A fuzzy rough set approach for incremental feature selection on hybrid information systems, Fuzzy Sets and Systems 258 (2015), 39–60.

47.

Zhang

W.X.

, Wu

W.Z.

, Liang

J.Y.

and Li

D.Y.

, Rough set theory and methods, Chinese Scientific Publishers, Beijing, 2001.

A three-way decision method based on hybrid data 1

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Fuzzy sets and fuzzy relations

2.2 Pawlak rough sets

2.3 Hybrid information systems

3 The fuzzy T cos -equivalence relation induced by a hybrid information system

3.1 The distance between two objects based on the conditional attribute set

3.2 The fuzzy T cos -equivalence relation induced by a hybrid information system

4 A three-way decision method based on hybrid data

4.1 Decision-theoretic rough set models for a hybrid information system

4.2 A three-way decision method in a hybrid information system

5 An illustrative example

6 Conclusions

Footnotes

Acknowledgments

References

3 The fuzzy T_cos-equivalence relation induced by a hybrid information system

3.2 The fuzzy T_cos-equivalence relation induced by a hybrid information system