Three-way decision for incomplete real-valued data

Abstract

An information system (IS) is a database that expresses relationships between objects and attributes. An IS with decision attributes is said to be a decision information system (DIS). An incomplete real-valued decision information system (IRVDIS) is a DIS based on incomplete real-valued data. This paper studies three-way decision (3WD) for incomplete real-valued data and its application. In the first place, the distance between two objects on the basis of the conditional attribute set in an IRVDIS is constructed. In the next place, the fuzzy T_cos-equivalence relation on the object set of an IRVDIS is received by means of Gaussian kernel. After that, the decision-theoretic rough set model for an IRVDIS is presented. Furthermore, the 3WD method is proposed based on this model. Lastly, to illustrate the feasibility of the proposed method, an application of the proposed method is given. It is worth mentioning that levels of risk may be determined by thresholds that can be directly acquired according to risk preference of different decision-makers, as well as the decision rule for each decision class under different levels of risk is showed in tabular forms.

Keywords

3WD IRVDIS Method Decision-theoretic rough set Gaussian kernel Inclusion degree Auto diagnostic

1 Introduction

1.1 Research background and related works

Rough set theory (RST) brought forward by Pawlak [14, 15]. RST is a tool to describe incomplete information and fuzzy information. It can analyze the fuzzy information or incomplete information and find the potential relationship in the implicit expectation, so as to reveal the underlying rules. Rough set is based on equivalence relation division to study the uncertainty of knowledge, and the concept of uncertainty is described by two definite sets, the upper approximation set and the lower approximation set. The “upper approximation” and “lower approximation” are used to describe the possibility and the certainty respectively, and the “boundary region” is used to describe the uncertainty. When it comes to decision problems, roughness (or precision) can be used to describe the performance of the extracted decision rules. As a data analysis and processing theory, RST has been widely and successfully applied in machine learning, information security, Internet of things, cloud computing, biological information processing, decision support and analysis and other fields.

The approximation of classical rough sets is defined based on the qualitative relationship between concepts (that is, contains or intersects not null). It does not take into account the extent to which concepts intersect, and is therefore not applicable to many practical problems. In order to solve the problem that Pawlak rough set model is too strict and lacks fault tolerance, various probabilistic rough set extension models have been proposed. Further research shows that the 3WDs are not limited to rough sets, but a more general and effective decision and information processing model. In 1990, Yao et al. [30] proposed the decision rough set model, extending 0.5-probability rough set model that is put forward by Pawlak et al. [17]. The main starting point of decision rough set model is to use conditional probability to define the degree of intersection of concepts and two thresholds to define the upper and lower approximation sets of probability.

Under the decision-theoretic rough set model, Yao [25–27 , 29] brought out a decision-making model. Since the 3WDs were put forward, they have attracted wide attention from scholars at home and abroad. A lot of progress has been made in the theory, model and application of the three decision making. For example, Aranda-Corral et al. [1] put forward a model of 3WDs for knowledge harnessing; Li al. [28] applied 3WD to complex network analysis; Lang [4] investigated general conflict analysis model based on 3WD; Luo et al. [5] researched 3WD under incomplete information in RST; Li et al. [10, 11] presented respectively 3WD methods in a hybrid information system and a hybrid information system with images, and applied these 3WD methods to medical diagnosis; Luo et al. [6] researched 3WD with incomplete information based on similarity and satisfiability; Li et al. [9] studied 3WD on two universes; Shen et al. [21] considered 3WD based blocking reduction models in hierarchical classification; Zhang et al. [33] brought forward a novel sequential 3WDs model based on penalty function; Zhang et al. [36] investigated 3WDs of rough vague sets from the perspective of fuzziness; Wang et al. [23] proposed a 3WD method based on Gaussian kernel in a hybrid information system with images.

1.2 Motivation and inspiration

An information system (IS) is a database which expresses relationships between objects and attributes. RST is developed around an IS. An incomplete real-valued decision information system (IRVDIS) is an IS whose information values are real numbers and incomplete. So far, 3WD (3WD) in an IRVDIS has not been studied. The purpose of this paper is to use decision-theoretic rough set model in an IRVDIS to go into 3WD.

The remaining sections of this article are designed below. In Section 2, we recall fuzzy relations, Pawlak’s rough sets and IRVDISs. In Section 3, we give distance between two objects based on the conditional attribute set in an IRVDIS and construct the fuzzy T_cos-equivalence relations by using Gaussian kernel. In Section 4, we propose information structures in an IRVDIS. In Section 5, we present a 3WD method based on decision-theoretic rough set model in an IRVDIS. In Section 6, we put forward an application to show the feasibility of the proposed method. In Section 7, we conclude this paper.

2 Preliminaries

We first review some basic notions of fuzzy relations, Pawlak rough sets and IRVDISs.

Throughout this paper, U signifies a non-empty finite set, 2^U implies the collection of all subsets of U, |S| means the cardinality of S ∈ 2^U and I denotes [0, 1].

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

2.1 Fuzzy relations

A fuzzy set F in U is defined as a function F : U → I, where F (u) is said to be the membership degree of u to F.

In this paper, I^U denotes the set of all fuzzy sets in U. The cardinality of F ∈ I^U can be calculated with

$| F | = \sum_{i = 1}^{n} F (u_{i}) .$

If R ∈ I^U×U, then R is said to be a fuzzy relation on U.

Let R ∈ I^U×U. Then R may be represented by

$M (R) = (r_{ij})_{n \times n},$ where r_ij = R (u_i, u_j) ∈ I expresses the degree of similarity between u_i and u_j.

R is said to be identity, if M (R) = E (here E means an identity matrix), and written as R =▵; R is said to be universal, if ∀ i, j, r_ij = 1, and written as R = ω.

Let R ∈ I^U×U. ∀ u ∈ U, define

$S_{R} (u) (v) = R (u, v) .$ Then S_R (u) may be seen as the information granule of u [18].

Obviously, $S_{R} (u_{i}) = \frac{r_{i 1}}{u_{1}} + \frac{r_{i 2}}{u_{2}} + \dots + \frac{r_{in}}{u_{n}},$ where r_ij = R (u_i, u_j).

Definition 2.1.([13]) Given a function T : I² → I. Considering the following conditions:

(1) Commutativity: T (m, n) = T (n, m) ;

(2) Associativity: T (T (m, n) , p) = T (m, T (n, p)) ;

(3) Monotonicity: m≤ p, n ⩽ q = T (m, n) ⩽ T (p, q) ;

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

If T satisfies (1), (2), (3) and (4), then it is said to be a t-norm.

Example 2.2. ∀ u, v ∈ U, define $T_{\cos} (u, v) = \max {uv - \sqrt{1 - u^{2}} \sqrt{1 - v^{2}}, 0} .$ Then T_cos is a t-norm.

Definition 2.3. ([32]) Suppose that T is a t-norm and R ∈ I^U×U. Considering the following conditions:

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

(2) Symmetry: R (u, v) = R (v, u) ,

(3) T-transitivity: T (R (u, v) , R (v, w)) ⩽ R (u, w) .

If R satisfies (1), (2) and (3), then it is said to be a T-fuzzy equivalence relation on U.

Proposition 2.4. ([12]) Let f : U × U → I be a map. If f satisfies f (u, u) =1, ∀ u ∈ U, then ∀ u, v, w ∈ U, $T_{\cos} (f (u, v), f (v, w)) \leq f (u, w) .$

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

2.2 Pawlak rough sets

Suppose that R is an equivalence relation on U. Denote $[u]_{R} = {v \in U : uRv},$ $U / R = {[u]_{R} : u \in U} .$

Definition 2.6. ([16]) Let R be an equivalence relation on U. Then the lower approximation, upper approximation and boundary region of S ∈ 2^U are defined as ${\underline{apr}}_{R} (S) = {u \in U : [u]_{R} \subseteq S},$ ${\bar{apr}}_{R} (S) = {u \in U : [u]_{R} \cap S \neq \emptyset},$ ${BND}_{R} (S) = {\bar{apr}}_{R} (S) - {\underline{apr}}_{R} (S) .$

Using ordinary inclusion measure, ${\underline{apr}}_{R} (S)$ , ${\bar{apr}}_{R} (S)$ and BND_R (S) can be equivalently defined as ${\underline{apr}}_{R} (S) = {u \in U : I ([u]_{R}, S) = 1},$ ${\bar{apr}}_{R} (S) = {u \in U : I ([u]_{R}, S) > 0},$ ${BND}_{R} (S) = {u \in U : 0 < I ([u]_{R}, S) < 1} .$

From the above definition, since the equivalence relation appears to be too strict, the objects whose membership function values are non-0 and non-1 in the universe are assigned to the boundary region. In reality, some more diffuse classifications will be more meaningful. Consequently, Ziarko [31] first introduced an error-tolerance level with set inclusion [35] through a pair of threshold α and β instead of 0 and 1 where 0 ≤ β < α ≤ 1. Variable precision rough set model is defined as ${\underline{apr}}_{R} (S) = {u \in U : I ([u]_{R}, S) \geq α},$ ${\bar{apr}}_{R} (S) = {u \in U : I ([u]_{R}, S) > β} .$

2.3 An IRVDIS

In this part, we recall the concept of an IRVDIS.

Definition 2.7.([14]) Let U be an object set and A be an attribute set. Then (U, A) is said to be an information system (IS), if ∀ a ∈ A decides an information function a : U → V_a, where V_a = {a (u) : u ∈ U}.

Suppose that (U, A) is an IS. Then (U, A) is said to be a decision IS, if A = C ∪ D, C is a conditional attribute set and D is a decision attribute set.

Given that (U, A) is an IS and P ⊆ A. Define $ind (P) = {(u, v) \in U \times U : \forall a \in P, a (u) = a (v)} .$

Apparently, ind (P) is an equivalence relation on U and ind (P) = ⋂ _a∈Pind ({a}) .

Definition 2.8.([14]) Let (U, A) be an IS. Then (U, A) is said to be an incomplete IS (IIS), if ∃ u ∈ U and a ∈ A such that a (u) is missing.

Given that (U, A) is an IIS and P ⊆ A. Define

sim (P) = {(u, v) ∈ U × U : ∀ a ∈ P, a (u) = a (v) or a (u) = * or a (v) = * , where * is a missing value.Clearly, sim (P) is a tolerance relation on U and sim (P) = ⋂ _a∈Psim ({a}) .

Let (U, A) be an IIS. ∀ a ∈ A, denote $V_{a}^{*} = V_{a} - {a (u) : a (u) = *} .$

Definition 2.9.([7]) Let (U, A) be an IS. Then (U, A) is said to be a real-valued IS (RVIS), if ∀ u ∈ U and a ∈ A, a (u) is a real number.

Definition 2.10.([7]) Let (U, A) be a RVIS. Then (U, A) is said to be an incomplete real-valued IS (IRVIS), if (U, A) is incomplete.

Definition 2.11.([7]) Let (U, A) be an IRVIS. Then (U, A) is said to be an IRVDIS, if (U, A) is a decision IS.

Example 2.12. The data of Table 1 comes from the Auto mpg in UCI.

Table 1
An IRVDIS

U a ₁ a ₂ a ₃ a ₄ a ₅ a ₆ a ₇ d

u ₁ 23.6 4 140 * 2905 14.3 80 1

u ₂ 15.5 8 400 190 4325 12.2 77 1

u ₃ 35.7 4 98 80 1915 14.4 79 1

u ₄ 18 6 232 100 3288 15.5 71 1

u ₅ 23 4 151 * 3035 20.5 82 1

u ₆ 44.3 4 90 48 2085 21.7 80 2

u ₇ 25.4 5 183 77 3530 20.1 79 2

u ₈ 34.5 4 100 * 2320 15.8 81 2

u ₉ 44 4 97 52 2130 24.6 82 2

u ₁₀ 40.9 4 85 * 1835 17.3 80 2

u ₁₁ 38 4 91 67 1995 16.2 82 3

u ₁₂ 34 4 108 70 2245 16.9 82 3

u ₁₃ 31 4 91 68 1970 17.6 82 3

U	a ₁	a ₂	a ₃	a ₄	a ₅	a ₆	a ₇	d
u ₁	23.6	4	140	*	2905	14.3	80	1
u ₂	15.5	8	400	190	4325	12.2	77	1
u ₃	35.7	4	98	80	1915	14.4	79	1
u ₄	18	6	232	100	3288	15.5	71	1
u ₅	23	4	151	*	3035	20.5	82	1
u ₆	44.3	4	90	48	2085	21.7	80	2
u ₇	25.4	5	183	77	3530	20.1	79	2
u ₈	34.5	4	100	*	2320	15.8	81	2
u ₉	44	4	97	52	2130	24.6	82	2
u ₁₀	40.9	4	85	*	1835	17.3	80	2
u ₁₁	38	4	91	67	1995	16.2	82	3
u ₁₂	34	4	108	70	2245	16.9	82	3
u ₁₃	31	4	91	68	1970	17.6	82	3

Table 1 means an IRVDIS (U, C ∪ {d}), where

$U = {u_{1}, u_{2}, \dots, u_{13}}$ is an auto set, C = {Cylinders (a₁), Displacement (a₂), Horsepower (a₃), Weight (a₄), Acceleration (a₅), Modelyear (a₆) , Origin (a₇)} is a conditional attribute set and d denotes a decision attribute.

3 The fuzzy T_cos-equivalence relation in an IIVIS

In this section, the fuzzy T_cos-equivalence relation in an IIVIS is constructed.

3.1 The distance between two objects in an IRVDIS

The distance between two objects in an IRVDIS is presented as below.

Definition 3.1. Suppose that (U, C ∪ {d}) is an IRVDIS. ∀ u, v ∈ U, ∀ a ∈ C, the distance between a (u) and a (v) is defined as $d (a (u), a (v)) = {\begin{matrix} 0 & u = v \\ 1 & u \neq v, a (u) = * or a (v) = * \\ 0 & u \neq v, a (u) \neq *, a (v) \neq *, a (u) = a (v) \\ \frac{| a (u) - a (v) |}{θ_{a}} & u \neq v, a (u) \neq *, a (v) \neq *, a (u) \neq a (v) \end{matrix}$ where θ_a = max {a (u) : u ∈ U} - min {a (u) : u ∈ U} .

Definition 3.2. Suppose that (U, C ∪ {d}) is an IRVDIS. ∀ u, v ∈ U, the distance between u and v based on C is defined as $d_{C} (u, v) = \sqrt{\sum_{a \in C} d^{2} (a (u), a (v)))} .$ Morover, $d_{C} = (d_{C} (u_{i}, u_{j}))_{n \times n}$ is said to be the distance matrix based on C in (U, C ∪ {d}).

Proposition 3.3. Suppose that (U, C ∪ {d}) is an IRVDIS. Then ∀ u, v ∈ U, $0 \leq d_{C} (u, v) \leq \sqrt{| C |} .$

Proof. By Definitions 3.1, and 3.1, $\forall a \in C, \forall u, v \in U, 0 \leq d (a (u), a (v)) \leq 1 .$

Then $\forall u, v \in U, 0 \leq \sum_{a \in C} d^{2} (a (u), a (v)) \leq | C | .$

Thus $\forall u, v \in U, 0 \leq d_{C} (u, v) \leq \sqrt{| C |} .$ □

Example 3.4. (Continued from Table 1 in Example 2.12)

Calculate the distance d_C (u₁, u₃) between u₁ and u₃.

Since a₃ is a real-valued attribute, by Definition 3.1, we have $\begin{matrix} d (a_{3} (u_{1}), a_{3} (u_{3})) & = \frac{| a_{3} (u_{1}) - a_{3} (u_{3}) |}{θ_{a}} \\ = \frac{140 - 98}{315} \approx 0.1333 . \end{matrix}$

Then

d_C (u₁, u₃) $\begin{matrix} = \sqrt{\sum_{a \in C} d^{2} (u_{1}, u_{3})} \\ = \sqrt{0 . 4201^{2} + 0 . 0000^{2} + 0 . 1333^{2} + 1 . 0000^{2} + 0 . 5703^{2} + 0 . 0081^{2} + 0 . 0909^{2}} \\ \approx 1.1665 . \end{matrix}$

Example 3.5. By the above definitions, the distance between two information values of a given attribute in an IRVDIS are shown in Tables 2 –8.

Table 2
The distance between a₁ (u_i) and a₁ (u_j) (i, j = 1, 2, …, 13)

d (a₁ (u_i) , a₁ (u_j)) u ₁ u ₂ u ₃ u ₄ u ₅ u ₆ u ₇ u ₈ u ₉ u ₁₀ u ₁₁ u ₁₂ u ₁₃

u ₁ 0.0000 0.2813 0.4201 0.1944 0.0208 0.7187 0.0625 0.3785 0.7083 0.6007 0.5000 0.3611 0.2569

u ₂ 0.2813 0.0000 0.7014 0.0868 0.2604 1.0000 0.3438 0.6597 0.9896 0.8819 0.7813 0.6424 0.5382

u ₃ 0.4201 0.7014 0.0000 0.6146 0.4410 0.2986 0.3576 0.0417 0.2882 0.1806 0.0799 0.0590 0.1632

u ₄ 0.1944 0.0868 0.6146 0.0000 0.1736 0.9132 0.2569 0.5729 0.9028 0.7951 0.6944 0.5556 0.4514

u ₅ 0.0208 0.2604 0.4410 0.1736 0.0000 0.7396 0.0833 0.3993 0.7292 0.6215 0.5208 0.3819 0.2778

u ₆ 0.7187 1.0000 0.2986 0.9132 0.7396 0.0000 0.6562 0.3403 0.0104 0.1181 0.2187 0.3576 0.4618

u ₇ 0.0625 0.3438 0.3576 0.2569 0.0833 0.6562 0.0000 0.3160 0.6458 0.5382 0.4375 0.2986 0.1944

u ₈ 0.3785 0.6597 0.0417 0.5729 0.3993 0.3403 0.3160 0.0000 0.3299 0.2222 0.1215 0.0174 0.1215

u ₉ 0.7083 0.9896 0.2882 0.9028 0.7292 0.0104 0.6458 0.3299 0.0000 0.1076 0.2083 0.3472 0.4514

u ₁₀ 0.6007 0.8819 0.1806 0.7951 0.6215 0.1181 0.5382 0.2222 0.1076 0.0000 0.1007 0.2396 0.3438

u ₁₁ 0.5000 0.7813 0.0799 0.6944 0.5208 0.2187 0.4375 0.1215 0.2083 0.1007 0.0000 0.1389 0.2431

u ₁₂ 0.3611 0.6424 0.0590 0.5556 0.3819 0.3576 0.2986 0.0174 0.3472 0.2396 0.1389 0.0000 0.1042

u ₁₃ 0.2569 0.5382 0.1632 0.4514 0.2778 0.4618 0.1944 0.1215 0.4514 0.3438 0.2431 0.1042 0.0000

d (a₁ (u_i) , a₁ (u_j))	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆	u ₇	u ₈	u ₉	u ₁₀	u ₁₁	u ₁₂	u ₁₃
u ₁	0.0000	0.2813	0.4201	0.1944	0.0208	0.7187	0.0625	0.3785	0.7083	0.6007	0.5000	0.3611	0.2569
u ₂	0.2813	0.0000	0.7014	0.0868	0.2604	1.0000	0.3438	0.6597	0.9896	0.8819	0.7813	0.6424	0.5382
u ₃	0.4201	0.7014	0.0000	0.6146	0.4410	0.2986	0.3576	0.0417	0.2882	0.1806	0.0799	0.0590	0.1632
u ₄	0.1944	0.0868	0.6146	0.0000	0.1736	0.9132	0.2569	0.5729	0.9028	0.7951	0.6944	0.5556	0.4514
u ₅	0.0208	0.2604	0.4410	0.1736	0.0000	0.7396	0.0833	0.3993	0.7292	0.6215	0.5208	0.3819	0.2778
u ₆	0.7187	1.0000	0.2986	0.9132	0.7396	0.0000	0.6562	0.3403	0.0104	0.1181	0.2187	0.3576	0.4618
u ₇	0.0625	0.3438	0.3576	0.2569	0.0833	0.6562	0.0000	0.3160	0.6458	0.5382	0.4375	0.2986	0.1944
u ₈	0.3785	0.6597	0.0417	0.5729	0.3993	0.3403	0.3160	0.0000	0.3299	0.2222	0.1215	0.0174	0.1215
u ₉	0.7083	0.9896	0.2882	0.9028	0.7292	0.0104	0.6458	0.3299	0.0000	0.1076	0.2083	0.3472	0.4514
u ₁₀	0.6007	0.8819	0.1806	0.7951	0.6215	0.1181	0.5382	0.2222	0.1076	0.0000	0.1007	0.2396	0.3438
u ₁₁	0.5000	0.7813	0.0799	0.6944	0.5208	0.2187	0.4375	0.1215	0.2083	0.1007	0.0000	0.1389	0.2431
u ₁₂	0.3611	0.6424	0.0590	0.5556	0.3819	0.3576	0.2986	0.0174	0.3472	0.2396	0.1389	0.0000	0.1042
u ₁₃	0.2569	0.5382	0.1632	0.4514	0.2778	0.4618	0.1944	0.1215	0.4514	0.3438	0.2431	0.1042	0.0000

Table 3

The distance between two a₂ (u_i) and a₂ (u_j) (i, j = 1, 2, …, 13)

d (a₂ (u_i) , a₂ (u_j))	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆	u ₇	u ₈	u ₉	u ₁₀	u ₁₁	u ₁₂	u ₁₃
u ₁	0.0000	1.0000	0.0000	0.5000	0.0000	0.0000	0.2500	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
u ₂	1.0000	0.0000	1.0000	0.5000	1.0000	1.0000	0.7500	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
u ₃	0.0000	1.0000	0.0000	0.5000	0.0000	0.0000	0.2500	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
u ₄	0.5000	0.5000	0.5000	0.0000	0.5000	0.5000	0.2500	0.5000	0.5000	0.5000	0.5000	0.5000	0.5000
u ₅	0.0000	1.0000	0.0000	0.5000	0.0000	0.0000	0.2500	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
u ₆	0.0000	1.0000	0.0000	0.5000	0.0000	0.0000	0.2500	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
u ₇	0.2500	0.7500	0.2500	0.2500	0.2500	0.2500	0.0000	0.2500	0.2500	0.2500	0.2500	0.2500	0.2500
u ₈	0.0000	1.0000	0.0000	0.5000	0.0000	0.0000	0.2500	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
u ₉	0.0000	1.0000	0.0000	0.5000	0.0000	0.0000	0.2500	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
u ₁₀	0.0000	1.0000	0.0000	0.5000	0.0000	0.0000	0.2500	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
u ₁₁	0.0000	1.0000	0.0000	0.5000	0.0000	0.0000	0.2500	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
u ₁₂	0.0000	1.0000	0.0000	0.5000	0.0000	0.0000	0.2500	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
u ₁₃	0.0000	1.0000	0.0000	0.5000	0.0000	0.0000	0.2500	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000

Table 4

The distance between a₃ (u_i) and a₃ (u_j) (i, j = 1, 2, …, 13)

d (a₃ (u_i) , a₃ (u_j))	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆	u ₇	u ₈	u ₉	u ₁₀	u ₁₁	u ₁₂	u ₁₃
u ₁	0.0000	0.8254	0.1333	0.2921	0.0349	0.1587	0.1365	0.1270	0.1365	0.1746	0.1556	0.1016	0.1556
u ₂	0.8254	0.0000	0.9587	0.5333	0.7905	0.9841	0.6889	0.9524	0.9619	1.0000	0.9810	0.9270	0.9810
u ₃	0.1333	0.9587	0.0000	0.4254	0.1683	0.0254	0.2698	0.0063	0.0032	0.0413	0.0222	0.0317	0.0222
u ₄	0.2921	0.5333	0.4254	0.0000	0.2571	0.4508	0.1556	0.4190	0.4286	0.4667	0.4476	0.3937	0.4476
u ₅	0.0349	0.7905	0.1683	0.2571	0.0000	0.1937	0.1016	0.1619	0.1714	0.2095	0.1905	0.1365	0.1905
u ₆	0.1587	0.9841	0.0254	0.4508	0.1937	0.0000	0.2952	0.0317	0.0222	0.0159	0.0032	0.0571	0.0032
u ₇	0.1365	0.6889	0.2698	0.1556	0.1016	0.2952	0.0000	0.2635	0.2730	0.3111	0.2921	0.2381	0.2921
u ₈	0.1270	0.9524	0.0063	0.4190	0.1619	0.0317	0.2635	0.0000	0.0095	0.0476	0.0286	0.0254	0.0286
u ₉	0.1365	0.9619	0.0032	0.4286	0.1714	0.0222	0.2730	0.0095	0.0000	0.0381	0.0190	0.0349	0.0190
u ₁₀	0.1746	1.0000	0.0413	0.4667	0.2095	0.0159	0.3111	0.0476	0.0381	0.0000	0.0190	0.0730	0.0190
u ₁₁	0.1556	0.9810	0.0222	0.4476	0.1905	0.0032	0.2921	0.0286	0.0190	0.0190	0.0000	0.0540	0.0000
u ₁₂	0.1016	0.9270	0.0317	0.3937	0.1365	0.0571	0.2381	0.0254	0.0349	0.0730	0.0540	0.0000	0.0540
u ₁₃	0.1556	0.9810	0.0222	0.4476	0.1905	0.0032	0.2921	0.0286	0.0190	0.0190	0.0000	0.0540	0.0000

Table 5

The distance between a₄ (u_i) and a₄ (u_j) (i, j = 1, 2, …, 13)

d (a₄ (u_i) , a₄ (u_j))	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆	u ₇	u ₈	u ₉	u ₁₀	u ₁₁	u ₁₂	u ₁₃
u ₁	0.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
u ₂	1.0000	0.0000	0.7746	0.6338	1.0000	1.0000	0.7958	1.0000	0.9718	1.0000	0.8662	0.8451	0.8592
u ₃	1.0000	0.7746	0.0000	0.1408	1.0000	0.2254	0.0211	1.0000	0.1972	1.0000	0.0915	0.0704	0.0845
u ₄	1.0000	0.6338	0.1408	0.0000	1.0000	0.3662	0.1620	1.0000	0.3380	1.0000	0.2324	0.2113	0.2254
u ₅	1.0000	1.0000	1.0000	1.0000	0.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
u ₆	1.0000	1.0000	0.2254	0.3662	1.0000	0.0000	0.2042	1.0000	0.0282	1.0000	0.1338	0.1549	0.1408
u ₇	1.0000	0.7958	0.0211	0.1620	1.0000	0.2042	0.0000	1.0000	0.1761	1.0000	0.0704	0.0493	0.0634
u ₈	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	0.0000	1.0000	1.0000	1.0000	1.0000	1.0000
u ₉	1.0000	0.9718	0.1972	0.3380	1.0000	0.0282	0.1761	1.0000	0.0000	1.0000	0.1056	0.1268	0.1127
u ₁₀	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	0.0000	1.0000	1.0000	1.0000
u ₁₁	1.0000	0.8662	0.0915	0.2324	1.0000	0.1338	0.0704	1.0000	0.1056	1.0000	0.0000	0.0211	0.0070
u ₁₂	1.0000	0.8451	0.0704	0.2113	1.0000	0.1549	0.0493	1.0000	0.1268	1.0000	0.0211	0.0000	0.0141
u ₁₃	1.0000	0.8592	0.0845	0.2254	1.0000	0.1408	0.0634	1.0000	0.1127	1.0000	0.0070	0.0141	0.0000

Table 6

The distance between a₅ (u_i) and a₅ (u_j) (i, j = 1, 2, …, 13)

d (a₅ (u_i) , a₅ (u_j))	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆	u ₇	u ₈	u ₉	u ₁₀	u ₁₁	u ₁₂	u ₁₃
u ₁	0.0000	0.5703	0.3976	0.1538	0.0522	0.3293	0.2510	0.2349	0.3112	0.4297	0.3655	0.2651	0.3755
u ₂	0.5703	0.0000	0.9679	0.4165	0.5181	0.8996	0.3193	0.8052	0.8815	1.0000	0.9357	0.8353	0.9458
u ₃	0.3976	0.9679	0.0000	0.5514	0.4498	0.0683	0.6486	0.1627	0.0863	0.0321	0.0321	0.1325	0.0221
u ₄	0.1538	0.4165	0.5514	0.0000	0.1016	0.4831	0.0972	0.3888	0.4651	0.5835	0.5193	0.4189	0.5293
u ₅	0.0522	0.5181	0.4498	0.1016	0.0000	0.3815	0.1988	0.2871	0.3635	0.4819	0.4177	0.3173	0.4277
u ₆	0.3293	0.8996	0.0683	0.4831	0.3815	0.0000	0.5803	0.0944	0.0181	0.1004	0.0361	0.0643	0.0462
u ₇	0.2510	0.3193	0.6486	0.0972	0.1988	0.5803	0.0000	0.4859	0.5622	0.6807	0.6165	0.5161	0.6265
u ₈	0.2349	0.8052	0.1627	0.3888	0.2871	0.0944	0.4859	0.0000	0.0763	0.1948	0.1305	0.0301	0.1406
u ₉	0.3112	0.8815	0.0863	0.4651	0.3635	0.0181	0.5622	0.0763	0.0000	0.1185	0.0542	0.0462	0.0643
u ₁₀	0.4297	1.0000	0.0321	0.5835	0.4819	0.1004	0.6807	0.1948	0.1185	0.0000	0.0643	0.1647	0.0542
u ₁₁	0.3655	0.9357	0.0321	0.5193	0.4177	0.0361	0.6165	0.1305	0.0542	0.0643	0.0000	0.1004	0.0100
u ₁₂	0.2651	0.8353	0.1325	0.4189	0.3173	0.0643	0.5161	0.0301	0.0462	0.1647	0.1004	0.0000	0.1104
u ₁₃	0.3755	0.9458	0.0221	0.5293	0.4277	0.0462	0.6265	0.1406	0.0643	0.0542	0.0100	0.1104	0.0000

Table 7

The distance between a₆ (u_i) and a₆ (u_j) (i, j = 1, 2, …, 13)

d (a₆ (u_i) , a₆ (u_j))	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆	u ₇	u ₈	u ₉	u ₁₀	u ₁₁	u ₁₂	u ₁₃
u ₁	0.0000	0.1694	0.0081	0.0968	0.5000	0.5968	0.4677	0.1210	0.8306	0.2419	0.1532	0.2097	0.2661
u ₂	0.1694	0.0000	0.1774	0.2661	0.6694	0.7661	0.6371	0.2903	1.0000	0.4113	0.3226	0.3790	0.4355
u ₃	0.0081	0.1774	0.0000	0.0887	0.4919	0.5887	0.4597	0.1129	0.8226	0.2339	0.1452	0.2016	0.2581
u ₄	0.0968	0.2661	0.0887	0.0000	0.4032	0.5000	0.3710	0.0242	0.7339	0.1452	0.0565	0.1129	0.1694
u ₅	0.5000	0.6694	0.4919	0.4032	0.0000	0.0968	0.0323	0.3790	0.3306	0.2581	0.3468	0.2903	0.2339
u ₆	0.5968	0.7661	0.5887	0.5000	0.0968	0.0000	0.1290	0.4758	0.2339	0.3548	0.4435	0.3871	0.3306
u ₇	0.4677	0.6371	0.4597	0.3710	0.0323	0.1290	0.0000	0.3468	0.3629	0.2258	0.3145	0.2581	0.2016
u ₈	0.1210	0.2903	0.1129	0.0242	0.3790	0.4758	0.3468	0.0000	0.7097	0.1210	0.0323	0.0887	0.1452
u ₉	0.8306	1.0000	0.8226	0.7339	0.3306	0.2339	0.3629	0.7097	0.0000	0.5887	0.6774	0.6210	0.5645
u ₁₀	0.2419	0.4113	0.2339	0.1452	0.2581	0.3548	0.2258	0.1210	0.5887	0.0000	0.0887	0.0323	0.0242
u ₁₁	0.1532	0.3226	0.1452	0.0565	0.3468	0.4435	0.3145	0.0323	0.6774	0.0887	0.0000	0.0565	0.1129
u ₁₂	0.2097	0.3790	0.2016	0.1129	0.2903	0.3871	0.2581	0.0887	0.6210	0.0323	0.0565	0.0000	0.0565
u ₁₃	0.2661	0.4355	0.2581	0.1694	0.2339	0.3306	0.2016	0.1452	0.5645	0.0242	0.1129	0.0565	0.0000

Table 8

The distance between a₇ (u_i) and a₇ (u_j) (i, j = 1, 2, …, 13)

d (a₇ (u_i) , a₇ (u_j))	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆	u ₇	u ₈	u ₉	u ₁₀	u ₁₁	u ₁₂	u ₁₃
u ₁	0.0000	0.2727	0.0909	0.8182	0.1818	0.0000	0.0909	0.0909	0.1818	0.0000	0.1818	0.1818	0.1818
u ₂	0.2727	0.0000	0.1818	0.5455	0.4545	0.2727	0.1818	0.3636	0.4545	0.2727	0.4545	0.4545	0.4545
u ₃	0.0909	0.1818	0.0000	0.7273	0.2727	0.0909	0.0000	0.1818	0.2727	0.0909	0.2727	0.2727	0.2727
u ₄	0.8182	0.5455	0.7273	0.0000	1.0000	0.8182	0.7273	0.9091	1.0000	0.8182	1.0000	1.0000	1.0000
u ₅	0.1818	0.4545	0.2727	1.0000	0.0000	0.1818	0.2727	0.0909	0.0000	0.1818	0.0000	0.0000	0.0000
u ₆	0.0000	0.2727	0.0909	0.8182	0.1818	0.0000	0.0909	0.0909	0.1818	0.0000	0.1818	0.1818	0.1818
u ₇	0.0909	0.1818	0.0000	0.7273	0.2727	0.0909	0.0000	0.1818	0.2727	0.0909	0.2727	0.2727	0.2727
u ₈	0.0909	0.3636	0.1818	0.9091	0.0909	0.0909	0.1818	0.0000	0.0909	0.0909	0.0909	0.0909	0.0909
u ₉	0.1818	0.4545	0.2727	1.0000	0.0000	0.1818	0.2727	0.0909	0.0000	0.1818	0.0000	0.00000	0.0000
u ₁₀	0.0000	0.2727	0.0909	0.8182	0.1818	0.0000	0.0909	0.0909	0.1818	0.00000	0.1818	0.1818	0.1818
u ₁₁	0.1818	0.4545	0.2727	1.0000	0.0000	0.1818	0.2727	0.0909	0.0000	0.1818	0.0000	0.0000	0.0000
u ₁₂	0.1818	0.4545	0.2727	1.0000	0.0000	0.1818	0.2727	0.0909	0.0000	0.1818	0.0000	0.0000	0.0000
u ₁₃	0.1818	0.4545	0.2727	1.0000	0.0000	0.1818	0.2727	0.0909	0.0000	0.1818	0.0000	0.0000	0.0000

Example 3.6. (Continued from Examples 2.12)

By Definition 3.1, d_C in (U, C ∪ {d}) is shown as below.

3.2 The fuzzy T_cos-equivalence relation induced by an IRVDIS

In this part, the fuzzy T_cos-equivalence relation induced by an IRVDIS by means of Gaussian kernel method is given.

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

Evidently, G (u, v) satisfies:

(1) G (u, v) ∈ [0, 1];

(2) G (u, v) = G (v, u);

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

Definition 3.7. Suppose that (U, C ∪ {d}) is an IRVDIS. Given δ ∈ (0, 1], denote $R_{C}^{G} (δ) (u_{i}, u_{j}) = \exp [- \frac{d_{C}^{2} (u_{i}, u_{j})}{2 δ^{2}}],$ $M (R_{C}^{G} (δ)) = (R_{C}^{G} (δ) (u_{i}, u_{j}))_{n \times n} .$ Then $M (R_{C}^{G} (δ))$ is said to be the Gaussian kernel matric of (U, C ∪ {d}) with respect to δ.

Theorem 3.8. Suppose that (U, C ∪ {d}) is an IRVDIS. Given δ ∈ (0, 1]. Then $R_{C}^{G} (δ)$ is a T_cos-equivalence relation on U.

Proof. This holds by Corollary 2.5. □

Definition 3.9. Suppose that (U, C ∪ {d}) is an IRVDIS. Given δ ∈ (0, 1]. Then $R_{C}^{G} (δ)$ is said to be the fuzzy T_cos-equivalence relation induced by (U, C ∪ {d}) with respect to δ.

Example 3.10. (Continued from Examples 2.12) Pick $δ_{1} = \sqrt{0.3}$ , $δ_{2} = \sqrt{0.6}$ , $δ_{3} = \sqrt{0.8}$ . Then

Thus, the fuzzy T_cos-equivalence relation $R_{C}^{G} (δ_{i})$ is gained (i = 1, 2, 3).

An algorithm for arising the fuzzy T_cos-equivalence relation $R_{C}^{G} (δ)$ is given as below.

In Algorithm 1, we first find the distance between two information values through a double loop according to Definition 3.1, and then use the obtained distance between the information values to calculate the distance between two objects based on C according to Definition 3.2, finally, according to the specified threshold, use the distance between the two objects to find the similarity between the two objects in an IRVDIS. Therefore, the time complexity of Algorithm 1 is related to the number of object and attributes of an IRVDIS. For a dimensionality of |C| × |U|, because the complexity of computing the distance between two information value under an attribute in an IRVDIS is O (|U| × (|U|-1)), so the overall time complexity of Algorithm 1 is O (|C||U|²).

4 Information structures in an IRVDIS

In this category, information structures in an IRVDIS are considered.

Why do we study its information structures in IRVDIS? The reason is that its information structures contribute to knowledge discovery from this IRVDIS.

Given R ∈ I^U×U. Qian et al. [18] defined $S (R) = (S_{R} (u_{1}), S_{R} (u_{2}), \dots, S_{R} (u_{n})) .$ as the fuzzy granular structure of R.

Definition 4.1. Suppose that (U, C ∪ {d}) is an IRVDIS. Given δ ∈ (0, 1], denote $S (C, δ) = (S_{R_{C}^{G} (δ)} (u_{1}), S_{R_{C}^{G} (δ)} (u_{2}), \dots, S_{R_{C}^{G} (δ)} (u_{n})) .$ Then S (C, δ) is said to be the information structure of (U, C ∪ {d}) with respect to C and δ.

Example 4.2. (Continued from Examples 2.12 and 3.10) $\begin{matrix} S (C, δ_{1}) = \\ (S_{R_{C}^{G} (δ_{1})} (u_{1}), S_{R_{C}^{G} (δ_{1})} (u_{2}), \dots, S_{R_{C}^{G} (δ_{1})} (u_{13}), \end{matrix}$ where $S_{R_{C}^{G} (δ_{1})} (u_{1}) = \frac{1.0000}{u_{1}} + \frac{0.0049}{u_{2}} + \frac{0.1035}{u_{3}} + \frac{0.0315}{u_{4}} + \frac{0.1170}{u_{5}} + \frac{0.0353}{u_{6}} + \frac{0.1011}{u_{7}} + \frac{0.1271}{u_{8}} + \frac{0.0202}{u_{9}} + \frac{0.0656}{u_{10}} + \frac{0.0871}{u_{11}} + \frac{0.1169}{u_{12}} + \frac{0.1080}{u_{13}},$ $S_{R_{C}^{G} (δ_{1})} (u_{2}) = \frac{0.0049}{u_{1}} + \frac{1.0000}{u_{2}} + \frac{0.0012}{u_{3}} + \frac{0.0841}{u_{4}} + \frac{0.0024}{u_{5}} + \frac{0.0001}{u_{6}} + \frac{0.0206}{u_{7}} + \frac{0.0009}{u_{8}} + \frac{0.0001}{u_{9}} + \frac{0.0002}{u_{10}} + \frac{0.0005}{u_{11}} + \frac{0.0012}{u_{12}} + \frac{0.0008}{u_{13}},$ $S_{R_{C}^{G} (δ_{1})} (u_{3}) = \frac{0.1035}{u_{1}} + \frac{0.0012}{u_{2}} + \frac{1.0000}{u_{3}} + \frac{0.0619}{u_{4}} + \frac{0.0549}{u_{5}} + \frac{0.4345}{u_{6}} + \frac{0.2248}{u_{7}} + \frac{0.1669}{u_{8}} + \frac{0.2305}{u_{9}} + \frac{0.1603}{u_{10}} + \frac{0.8301}{u_{11}} + \frac{0.7892}{u_{12}} + \frac{0.7461}{u_{13}},$ $S_{R_{C}^{G} (δ_{1})} (u_{4}) = \frac{0.0315}{u_{1}} + \frac{0.0841}{u_{2}} + \frac{0.0619}{u_{3}} + \frac{1.0000}{u_{4}} + \frac{0.0150}{u_{5}} + \frac{0.0137}{u_{6}} + \frac{0.2405}{u_{7}} + \frac{0.0105}{u_{8}} + \frac{0.0055}{u_{9}} + \frac{0.0054}{u_{10}} + \frac{0.0231}{u_{11}} + \frac{0.0390}{u_{12}} + \frac{0.0349}{u_{13}},$ $S_{R_{C}^{G} (δ_{1})} (u_{5}) = \frac{0.1170}{u_{1}} + \frac{0.0024}{u_{2}} + \frac{0.0549}{u_{3}} + \frac{0.0150}{u_{4}} + \frac{1.0000}{u_{5}} + \frac{0.0521}{u_{6}} + \frac{0.1365}{u_{7}} + \frac{0.9340}{u_{8}} + \frac{0.0496}{u_{9}} + \frac{0.0530}{u_{10}} + \frac{0.0692}{u_{11}} + \frac{0.1055}{u_{12}} + \frac{0.1055}{u_{13}},$ $S_{R_{C}^{G} (δ_{1})} (u_{6}) = \frac{0.0353}{u_{1}} + \frac{0.0001}{u_{2}} + \frac{0.4345}{u_{3}} + \frac{0.0137}{u_{4}} + \frac{0.0521}{u_{5}} + \frac{1.0000}{u_{6}} + \frac{0.1941}{u_{7}} + \frac{0.1036}{u_{8}} + \frac{0.8615}{u_{9}} + \frac{0.1471}{u_{10}} + \frac{0.6097}{u_{11}} + \frac{0.5653}{u_{12}} + \frac{0.5329}{u_{13}},$ $S_{R_{C}^{G} (δ_{1})} (u_{7}) = \frac{0.1011}{u_{1}} + \frac{0.0206}{u_{2}} + \frac{0.2248}{u_{3}} + \frac{0.2405}{u_{4}} + \frac{0.1365}{u_{5}} + \frac{0.1941}{u_{6}} + \frac{1.0000}{u_{7}} + \frac{0.0671}{u_{8}} + \frac{0.1579}{u_{9}} + \frac{0.0374}{u_{10}} + \frac{0.2241}{u_{11}} + \frac{0.3570}{u_{12}} + \frac{0.3129}{u_{13}},$ $S_{R_{C}^{G} (δ_{1})} (u_{8}) = \frac{0.1271}{u_{1}} + \frac{0.0009}{u_{2}} + \frac{0.1669}{u_{3}} + \frac{0.0105}{u_{4}} + \frac{0.9340}{u_{5}} + \frac{0.1036}{u_{6}} + \frac{0.0671}{u_{7}} + \frac{1.0000}{u_{8}} + \frac{0.0665}{u_{9}} + \frac{0.1566}{u_{10}} + \frac{0.1761}{u_{11}} + \frac{0.1833}{u_{12}} + \frac{0.1696}{u_{13}},$ $S_{R_{C}^{G} (δ_{1})} (u_{9}) = \frac{0.0202}{u_{1}} + \frac{0.0001}{u_{2}} + \frac{0.2305}{u_{3}} + \frac{0.0055}{u_{4}} + \frac{0.0496}{u_{5}} + \frac{0.8615}{u_{6}} + \frac{0.1579}{u_{7}} + \frac{0.0665}{u_{8}} + \frac{1.0000}{u_{9}} + \frac{0.0959}{u_{10}} + \frac{0.4226}{u_{11}} + \frac{0.4165}{u_{12}} + \frac{0.4068}{u_{13}},$ $S_{R_{C}^{G} (δ_{1})} (u_{10}) = \frac{0.0656}{u_{1}} + \frac{0.0002}{u_{2}} + \frac{0.1603}{u_{3}} + \frac{0.0054}{u_{4}} + \frac{0.0530}{u_{5}} + \frac{0.1471}{u_{6}} + \frac{0.0374}{u_{7}} + \frac{0.1566}{u_{8}} + \frac{0.0959}{u_{9}} + \frac{1.0000}{u_{10}} + \frac{0.1722}{u_{11}} + \frac{0.1536}{u_{12}} + \frac{0.1458}{u_{13}},$ $S_{R_{C}^{G} (δ_{1})} (u_{11}) = \frac{0.0871}{u_{1}} + \frac{0.0005}{u_{2}} + \frac{0.8301}{u_{3}} + \frac{0.0231}{u_{4}} + \frac{0.0692}{u_{5}} + \frac{0.6097}{u_{6}} + \frac{0.2241}{u_{7}} + \frac{0.1761}{u_{8}} + \frac{0.4226}{u_{9}} + \frac{0.1722}{u_{10}} + \frac{1.0000}{u_{11}} + \frac{0.9419}{u_{12}} + \frac{0.8870}{u_{13}},$ $S_{R_{C}^{G} (δ_{1})} (u_{12}) = \frac{0.1169}{u_{1}} + \frac{0.0012}{u_{2}} + \frac{0.7892}{u_{3}} + \frac{0.0390}{u_{4}} + \frac{0.1055}{u_{5}} + \frac{0.5653}{u_{6}} + \frac{0.3570}{u_{7}} + \frac{0.1833}{u_{8}} + \frac{0.4165}{u_{9}} + \frac{0.1536}{u_{10}} + \frac{0.9419}{u_{11}} + \frac{1.0000}{u_{12}} + \frac{0.9523}{u_{13}},$ $S_{R_{C}^{G} (δ_{1})} (u_{13}) = \frac{0.1080}{u_{1}} + \frac{0.0008}{u_{2}} + \frac{0.7461}{u_{3}} + \frac{0.0349}{u_{4}} + \frac{0.1052}{u_{5}} + \frac{0.5329}{u_{6}} + \frac{0.3129}{u_{7}} + \frac{0.1696}{u_{8}} + \frac{0.4068}{u_{9}} + \frac{0.1458}{u_{10}} + \frac{0.8870}{u_{11}} + \frac{0.9523}{u_{12}} + \frac{1.0000}{u_{13}} .$ $\begin{matrix} S (C, δ_{2}) = \\ (S_{R_{C}^{G} (δ_{2})} (u_{1}), S_{R_{C}^{G} (δ_{2})} (u_{2}), \dots, S_{R_{C}^{G} (δ_{2})} (u_{13}), \end{matrix}$ where

$S_{R_{C}^{G} (δ_{2})} (u_{1}) = \frac{1.0000}{u_{1}} + \frac{0.0701}{u_{2}} + \frac{0.3218}{u_{3}} + \frac{0.1774}{u_{4}} + \frac{0.3420}{u_{5}} + \frac{0.1879}{u_{6}} + \frac{0.3179}{u_{7}} + \frac{0.3566}{u_{8}} + \frac{0.1422}{u_{9}} + \frac{0.2561}{u_{10}} + \frac{0.2952}{u_{11}} + \frac{0.3419}{u_{12}} + \frac{0.3287}{u_{13}},$ $S_{R_{C}^{G} (δ_{2})} (u_{2}) = \frac{0.0701}{u_{1}} + \frac{1.0000}{u_{2}} + \frac{0.0353}{u_{3}} + \frac{0.2900}{u_{4}} + \frac{0.0491}{u_{5}} + \frac{0.0108}{u_{6}} + \frac{0.1435}{u_{7}} + \frac{0.0300}{u_{8}} + \frac{0.0077}{u_{9}} + \frac{0.0152}{u_{10}} + \frac{0.0233}{u_{11}} + \frac{0.0347}{u_{12}} + \frac{0.0282}{u_{13}},$ $S_{R_{C}^{G} (δ_{2})} (u_{3}) = \frac{0.3218}{u_{1}} + \frac{0.0353}{u_{2}} + \frac{1.0000}{u_{3}} + \frac{0.2488}{u_{4}} + \frac{0.2343}{u_{5}} + \frac{0.6592}{u_{6}} + \frac{0.4741}{u_{7}} + \frac{0.4086}{u_{8}} + \frac{0.4801}{u_{9}} + \frac{0.4004}{u_{10}} + \frac{0.9111}{u_{11}} + \frac{0.8884}{u_{12}} + \frac{0.8638}{u_{13}},$ $S_{R_{C}^{G} (δ_{2})} (u_{4}) = \frac{0.1774}{u_{1}} + \frac{0.2900}{u_{2}} + \frac{0.2488}{u_{3}} + \frac{1.0000}{u_{4}} + \frac{0.1225}{u_{5}} + \frac{0.1171}{u_{6}} + \frac{0.4904}{u_{7}} + \frac{0.1026}{u_{8}} + \frac{0.0744}{u_{9}} + \frac{0.0736}{u_{10}} + \frac{0.1522}{u_{11}} + \frac{0.1975}{u_{12}} + \frac{0.1867}{u_{13}},$ $S_{R_{C}^{G} (δ_{2})} (u_{5}) = \frac{0.3420}{u_{1}} + \frac{0.0491}{u_{2}} + \frac{0.2343}{u_{3}} + \frac{0.1225}{u_{4}} + \frac{1.0000}{u_{5}} + \frac{0.2283}{u_{6}} + \frac{0.3695}{u_{7}} + \frac{0.3063}{u_{8}} + \frac{0.2227}{u_{9}} + \frac{0.2303}{u_{10}} + \frac{0.2631}{u_{11}} + \frac{0.3248}{u_{12}} + \frac{0.3244}{u_{13}},$ $S_{R_{C}^{G} (δ_{2})} (u_{6}) = \frac{0.1879}{u_{1}} + \frac{0.0108}{u_{2}} + \frac{0.6592}{u_{3}} + \frac{0.1171}{u_{4}} + \frac{0.2283}{u_{5}} + \frac{1.0000}{u_{6}} + \frac{0.4405}{u_{7}} + \frac{0.3219}{u_{8}} + \frac{0.9281}{u_{9}} + \frac{0.3835}{u_{10}} + \frac{0.7808}{u_{11}} + \frac{0.7519}{u_{12}} + \frac{0.7300}{u_{13}},$ $S_{R_{C}^{G} (δ_{2})} (u_{7}) = \frac{0.3179}{u_{1}} + \frac{0.1435}{u_{2}} + \frac{0.4741}{u_{3}} + \frac{0.4904}{u_{4}} + \frac{0.3695}{u_{5}} + \frac{0.4405}{u_{6}} + \frac{1.0000}{u_{7}} + \frac{0.2590}{u_{8}} + \frac{0.3974}{u_{9}} + \frac{0.1934}{u_{10}} + \frac{0.4734}{u_{11}} + \frac{0.5975}{u_{12}} + \frac{0.5594}{u_{13}},$ $S_{R_{C}^{G} (δ_{2})} (u_{8}) = \frac{0.3566}{u_{1}} + \frac{0.0300}{u_{2}} + \frac{0.4086}{u_{3}} + \frac{0.1026}{u_{4}} + \frac{0.3063}{u_{5}} + \frac{0.3219}{u_{6}} + \frac{0.2590}{u_{7}} + \frac{1.0000}{u_{8}} + \frac{0.2578}{u_{9}} + \frac{0.3957}{u_{10}} + \frac{0.4197}{u_{11}} + \frac{0.4281}{u_{12}} + \frac{0.4118}{u_{13}},$ $S_{R_{C}^{G} (δ_{2})} (u_{9}) = \frac{0.1422}{u_{1}} + \frac{0.0077}{u_{2}} + \frac{0.4801}{u_{3}} + \frac{0.0744}{u_{4}} + \frac{0.2227}{u_{5}} + \frac{0.9281}{u_{6}} + \frac{0.3974}{u_{7}} + \frac{0.2578}{u_{8}} + \frac{1.0000}{u_{9}} + \frac{0.3097}{u_{10}} + \frac{0.6501}{u_{11}} + \frac{0.6453}{u_{12}} + \frac{0.6378}{u_{13}},$ $S_{R_{C}^{G} (δ_{2})} (u_{10}) = \frac{0.2561}{u_{1}} + \frac{0.0152}{u_{2}} + \frac{0.4004}{u_{3}} + \frac{0.0736}{u_{4}} + \frac{0.2303}{u_{5}} + \frac{0.3835}{u_{6}} + \frac{0.1934}{u_{7}} + \frac{0.3957}{u_{8}} + \frac{0.3097}{u_{9}} + \frac{1.0000}{u_{10}} + \frac{0.4149}{u_{11}} + \frac{0.3920}{u_{12}} + \frac{0.3819}{u_{13}},$ $S_{R_{C}^{G} (δ_{2})} (u_{11}) = \frac{0.2952}{u_{1}} + \frac{0.0233}{u_{2}} + \frac{0.9111}{u_{3}} + \frac{0.1522}{u_{4}} + \frac{0.2631}{u_{5}} + \frac{0.7808}{u_{6}} + \frac{0.4734}{u_{7}} + \frac{0.4197}{u_{8}} + \frac{0.6501}{u_{9}} + \frac{0.4149}{u_{10}} + \frac{1.0000}{u_{11}} + \frac{0.9705}{u_{12}} + \frac{0.9418}{u_{13}},$ $S_{R_{C}^{G} (δ_{2})} (u_{12}) = \frac{0.3419}{u_{1}} + \frac{0.0347}{u_{2}} + \frac{0.8884}{u_{3}} + \frac{0.1975}{u_{4}} + \frac{0.3248}{u_{5}} + \frac{0.7519}{u_{6}} + \frac{0.5975}{u_{7}} + \frac{0.4281}{u_{8}} + \frac{0.6453}{u_{9}} + \frac{0.3920}{u_{10}} + \frac{0.9705}{u_{11}} + \frac{1.0000}{u_{12}} + \frac{0.9758}{u_{13}},$ $S_{R_{C}^{G} (δ_{2})} (u_{13}) = \frac{0.3287}{u_{1}} + \frac{0.0282}{u_{2}} + \frac{0.8638}{u_{3}} + \frac{0.1867}{u_{4}} + \frac{0.3244}{u_{5}} + \frac{0.7300}{u_{6}} + \frac{0.5594}{u_{7}} + \frac{0.4118}{u_{8}} + \frac{0.6378}{u_{9}} + \frac{0.3819}{u_{10}} + \frac{0.9418}{u_{11}} + \frac{0.9758}{u_{12}} + \frac{1.0000}{u_{13}} .$ $\begin{matrix} S (C, δ_{3}) = \\ (S_{R_{C}^{G} (δ_{3})} (u_{1}), S_{R_{C}^{G} (δ_{3})} (u_{2}), \dots, S_{R_{C}^{G} (δ_{3})} (u_{13}), \end{matrix}$ where $S_{R_{C}^{G} (δ_{3})} (u_{1}) = \frac{1.0000}{u_{1}} + \frac{0.1363}{u_{2}} + \frac{0.4272}{u_{3}} + \frac{0.2733}{u_{4}} + \frac{0.4473}{u_{5}} + \frac{0.2854}{u_{6}} + \frac{0.4234}{u_{7}} + \frac{0.4614}{u_{8}} + \frac{0.2316}{u_{9}} + \frac{0.3600}{u_{10}} + \frac{0.4004}{u_{11}} + \frac{0.4471}{u_{12}} + \frac{0.4341}{u_{13}},$ $S_{R_{C}^{G} (δ_{3})} (u_{2}) = \frac{0.1363}{u_{1}} + \frac{1.0000}{u_{2}} + \frac{0.0814}{u_{3}} + \frac{0.3952}{u_{4}} + \frac{0.1044}{u_{5}} + \frac{0.0334}{u_{6}} + \frac{0.2332}{u_{7}} + \frac{0.0721}{u_{8}} + \frac{0.0261}{u_{9}} + \frac{0.0434}{u_{10}} + \frac{0.0597}{u_{11}} + \frac{0.0804}{u_{12}} + \frac{0.0689}{u_{13}},$ $S_{R_{C}^{G} (δ_{3})} (u_{3}) = \frac{0.4272}{u_{1}} + \frac{0.0814}{u_{2}} + \frac{1.0000}{u_{3}} + \frac{0.3523}{u_{4}} + \frac{0.3367}{u_{5}} + \frac{0.7316}{u_{6}} + \frac{0.5713}{u_{7}} + \frac{0.5111}{u_{8}} + \frac{0.5768}{u_{9}} + \frac{0.5034}{u_{10}} + \frac{0.9325}{u_{11}} + \frac{0.9150}{u_{12}} + \frac{0.8960}{u_{13}},$ $S_{R_{C}^{G} (δ_{3})} (u_{4}) = \frac{0.2733}{u_{1}} + \frac{0.3952}{u_{2}} + \frac{0.3523}{u_{3}} + \frac{1.0000}{u_{4}} + \frac{0.2071}{u_{5}} + \frac{0.2001}{u_{6}} + \frac{0.5861}{u_{7}} + \frac{0.1813}{u_{8}} + \frac{0.1425}{u_{9}} + \frac{0.1413}{u_{10}} + \frac{0.2436}{u_{11}} + \frac{0.2963}{u_{12}} + \frac{0.2841}{u_{13}},$ $S_{R_{C}^{G} (δ_{3})} (u_{5}) = \frac{0.4473}{u_{1}} + \frac{0.1044}{u_{2}} + \frac{0.3367}{u_{3}} + \frac{0.2071}{u_{4}} + \frac{1.0000}{u_{5}} + \frac{0.3303}{u_{6}} + \frac{0.4739}{u_{7}} + \frac{0.4117}{u_{8}} + \frac{0.3241}{u_{9}} + \frac{0.3325}{u_{10}} + \frac{0.3674}{u_{11}} + \frac{0.4302}{u_{12}} + \frac{0.4298}{u_{13}},$ $S_{R_{C}^{G} (δ_{3})} (u_{6}) = \frac{0.2854}{u_{1}} + \frac{0.0334}{u_{2}} + \frac{0.7316}{u_{3}} + \frac{0.2001}{u_{4}} + \frac{0.3303}{u_{5}} + \frac{1.0000}{u_{6}} + \frac{0.5407}{u_{7}} + \frac{0.4273}{u_{8}} + \frac{0.9456}{u_{9}} + \frac{0.4873}{u_{10}} + \frac{0.8307}{u_{11}} + \frac{0.8074}{u_{12}} + \frac{0.7898}{u_{13}},$ $S_{R_{C}^{G} (δ_{3})} (u_{7}) = \frac{0.4234}{u_{1}} + \frac{0.2332}{u_{2}} + \frac{0.5713}{u_{3}} + \frac{0.5861}{u_{4}} + \frac{0.4739}{u_{5}} + \frac{0.5407}{u_{6}} + \frac{1.0000}{u_{7}} + \frac{0.3630}{u_{8}} + \frac{0.5005}{u_{9}} + \frac{0.2916}{u_{10}} + \frac{0.5707}{u_{11}} + \frac{0.6796}{u_{12}} + \frac{0.6468}{u_{13}},$ $S_{R_{C}^{G} (δ_{3})} (u_{8}) = \frac{0.4614}{u_{1}} + \frac{0.0721}{u_{2}} + \frac{0.5111}{u_{3}} + \frac{0.1813}{u_{4}} + \frac{0.4117}{u_{5}} + \frac{0.4273}{u_{6}} + \frac{0.3630}{u_{7}} + \frac{1.0000}{u_{8}} + \frac{0.3618}{u_{9}} + \frac{0.4989}{u_{10}} + \frac{0.5214}{u_{11}} + \frac{0.5293}{u_{12}} + \frac{0.5141}{u_{13}},$ $S_{R_{C}^{G} (δ_{3})} (u_{9}) = \frac{0.2316}{u_{1}} + \frac{0.0261}{u_{2}} + \frac{0.5768}{u_{3}} + \frac{0.1425}{u_{4}} + \frac{0.3241}{u_{5}} + \frac{0.9456}{u_{6}} + \frac{0.5005}{u_{7}} + \frac{0.3618}{u_{8}} + \frac{1.0000}{u_{9}} + \frac{0.4151}{u_{10}} + \frac{0.7240}{u_{11}} + \frac{0.7200}{u_{12}} + \frac{0.7137}{u_{13}},$ $S_{R_{C}^{G} (δ_{3})} (u_{10}) = \frac{0.3600}{u_{1}} + \frac{0.0434}{u_{2}} + \frac{0.5034}{u_{3}} + \frac{0.1413}{u_{4}} + \frac{0.3325}{u_{5}} + \frac{0.4873}{u_{6}} + \frac{0.2916}{u_{7}} + \frac{0.4989}{u_{8}} + \frac{0.4151}{u_{9}} + \frac{1.0000}{u_{10}} + \frac{0.5170}{u_{11}} + \frac{0.4954}{u_{12}} + \frac{0.4858}{u_{13}},$ $S_{R_{C}^{G} (δ_{3})} (u_{11}) = \frac{0.4004}{u_{1}} + \frac{0.0597}{u_{2}} + \frac{0.9325}{u_{3}} + \frac{0.2436}{u_{4}} + \frac{0.3674}{u_{5}} + \frac{0.8307}{u_{6}} + \frac{0.5707}{u_{7}} + \frac{0.5214}{u_{8}} + \frac{0.7240}{u_{9}} + \frac{0.5170}{u_{10}} + \frac{1.0000}{u_{11}} + \frac{0.9778}{u_{12}} + \frac{0.9560}{u_{13}},$ $S_{R_{C}^{G} (δ_{3})} (u_{12}) = \frac{0.4471}{u_{1}} + \frac{0.0804}{u_{2}} + \frac{0.9150}{u_{3}} + \frac{0.2963}{u_{4}} + \frac{0.4302}{u_{5}} + \frac{0.8074}{u_{6}} + \frac{0.6796}{u_{7}} + \frac{0.5293}{u_{8}} + \frac{0.7200}{u_{9}} + \frac{0.4954}{u_{10}} + \frac{0.9778}{u_{11}} + \frac{1.0000}{u_{12}} + \frac{0.9818}{u_{13}},$ $S_{R_{C}^{G} (δ_{3})} (u_{13}) = \frac{0.4341}{u_{1}} + \frac{0.0689}{u_{2}} + \frac{0.8960}{u_{3}} + \frac{0.2841}{u_{4}} + \frac{0.4298}{u_{5}} + \frac{0.7898}{u_{6}} + \frac{0.6468}{u_{7}} + \frac{0.5141}{u_{8}} + \frac{0.7137}{u_{9}} + \frac{0.4858}{u_{10}} + \frac{0.9560}{u_{11}} + \frac{0.9818}{u_{12}} + \frac{1.0000}{u_{13}} .$

5 A 3WD method based on gaussian kernel in an IRVDIS

In this section, a 3WD method based on Gaussian kernel in an IRVDIS is proposed.

5.1 A summary of 3WD methods

In this portion, the knowledge of 3WD is reviewed.

∀ S ∈ 2^U, U can be divided into three regions: $POS (S) = {\underline{apr}}_{R} (S),$ $BND (S) = {\bar{apr}}_{R} (S) - {\underline{apr}}_{R} (S),$ $NEG (S) = U - {\bar{apr}}_{R} (S) .$

Under Bayesian risk decision criterion, POS (S) generates the positive rule: acceptance; NEG (S) generates the negative rule: rejection; BND (S) generates the boundary rule: delay. That’s the 3WD method.

Definition 5.1.([34]) Let P : 2^U → I be a set function. Considering the following conditions:

(1) P (U) =1 .

(2) If E∩ F = ∅, then P (E ∪ F) = P (E) + P (F) .

If P satisfies (1) and (2), then it is said to be a probability measure.

Definition 5.2. ([34]) Given that P is a probability measure, E, F ∈ 2^U and P (F) >0. Then conditional probability of E under the condition F is defined as $P (E | F) = \frac{P (E \cap F)}{P (F)} .$

By a probability measure, we have $POS (S) = {u \in U : P (S | [u]_{R}) = 1},$ $BND (S) = {u \in U : 0 < P (S | [u]_{R}) < 1},$ $NEG (S) = {u \in U : P (S | [u]_{R}) = 0} .$

Decision-theoretic rough set model is derived from Bayesian theory. In this model, we use two state sets Ω = {X, ¬ X} and three action sets to depict the decision process, where Ω = {X, ¬ X} measns an element in X and not in X, respectively. A set of three decision sets $A = {P, B, N}$ is given, where P, B and N decide u ∈ POS (S), u ∈ BND (S) and u ∈ NEG (S), respectively. The loss function that can be seen as the risk or cost of three actions in two states is given by the following 3 × 2 matrix:

	X (P)	¬X (N)
a _P	λ _PP	λ _PN
a _B	λ _BP	λ _BN
a _N	λ _NP	λ _NN

In this matrix, λ_PP, λ_BP and λ_NP denote the losses caused by taking actions a_P, a_B and a_N, respectively, when an object belongs to X. Similarly, λ_PN, λ_BN and λ_NN denote the losses incurred by taking actions a_P, a_B and a_N when the object belongs to ¬X (N). The expected losses (conditional risks) of taking three different actions on objects in [u] _R are:

$R (a_{P} | [u]_{R}) = λ_{PP} P (S | [u]_{R}) + λ_{PN} P (\neg S | [u]_{R}),$ $R (a_{B} | [u]_{R}) = λ_{BP} P (S | [u]_{R}) + λ_{BN} P (\neg S | [u]_{R}),$ $R (a_{N} | [u]_{R}) = λ_{NP} P (S | [u]_{R}) + λ_{NN} P (\neg S | [u]_{R}) .$

The Bayesian decision procedure gives the minimum-risk decision rule as below.

(P) If R (a_P| [u] _R) ⩽ R (a_B| [u] _R) and R (a_P| [u] _R) ⩽R (a_N| [u] _R), then u ∈ POS (S);

(B) If R (a_B| [u] _R) ⩽ R (a_P| [u] _R) and R (a_B| [u] _R) ⩽R (a_N| [u] _R), then u ∈ BND (S);

(N) If R (a_N| [u] _R) ⩽ R (a_P| [u] _R) and R (a_N| [u] _R) ⩽R (a_B| [u] _R), then u ∈ NEG (S).

Apparently, P (S| [u]) + P (¬ S| [u]) =1 . Liu et al. [8] found that the rule only relied on the conditional probability P (S| [u]) and the loss functions λ_•• (• = P, B, N). On account of Yao’s study [27], considering a reasonable semantic interpretation of loss functions with λ_PP ≤ λ_BP < λ_NP and λ_NN ≤ λ_BN < λ_PN, he simplified the rule only based on the conditional probability P (S| [u]) and the loss function λ_••. Then the minimum-risk decision rule (P), (B), (N) is able to written as:

(P1) If P (S| [u] _R) ⩾ α, then u ∈ POS (S);

(B1) If β < P (S| [u] _R) < α, then u ∈ BND (S);

(N1) If P (S| [u] _R) ⩽ β, then u ∈ NEG (S). Where $α = \frac{λ_{PN} - λ_{BN}}{(λ_{PN} - λ_{BN}) + (λ_{BP} - λ_{PP})},$ $β = \frac{λ_{BN} - λ_{NN}}{(λ_{BN} - λ_{NN}) + (λ_{NP} - λ_{BP})} .$ with 0 ≤ α ≤ β ≤ 1 .

Thus, the (α, β)-probabilistic positive region, (α, β)-probabilistic boundary region and (α, β)-probabilistic negative region of S ∈ 2^U that are given by rules (P1), (B1), and (N1) can be redefined as below. ${POS}^{(α, β)} (S) = {u \in U : P (S | [u]_{R}) \geq α},$ ${BND}^{(α, β)} (S) = {u \in U : β < P (S | [u]_{R}) < α},$ ${NEG}^{(α, β)} (S) = {u \in U : P (S | [u]_{R}) \leq β} .$

5.2 The decision-theoretic rough set model in an IRVDIS

In this subsection, the decision-theoretic rough set model in an IRVDIS is proposed.

First, the inclusion measure between two fuzzy sets is recalled.

Definition 5.3. ([2]) Let E, F ∈ I^U. Then the inclusion measure I (E, F) is defined as $I (E, F) = \frac{| E \cap F |}{| F |},$

If E, F ∈ 2^U, then $I (E, F) = P (E | F) .$

Below, the decision-theoretic rough set model in an IRVDIS are defined as below, where the conditional probability is replaced by the inclusion measure between two fuzzy sets.

Definition 5.4. Suppose that (U, C ∪ {d}) is an IRVDIS. Given δ ∈ (0, 1], as well as α and β with 0 ≤ β < α ≤ 1. Suppose that $R_{C}^{G} (δ)$ is the fuzzy T_cos-equivalence relation induced by (U, C ∪ {d}) with respect to δ. Based on $(U, R_{C}^{G} (δ))$ , α-lower approximation ${\underline{{app}_{C}^{G} (δ)}}^{α} (D)$ and β-upper approximation ${\bar{{app}_{C}^{G} (δ)}}^{β}$ of D ∈ U/ind ({d}) are defined as below. ${\underline{{app}_{C}^{G} (δ)}}^{α} (D) = {u \in U : I (S_{R_{C}^{G} (δ)} (u), D) \geq α},$ ${\bar{{app}_{C}^{G} (δ)}}^{β} (D) = {u \in U : I (S_{R_{C}^{G} (δ)} (u), D) > β},$ where $D (u) = {\begin{matrix} 1, x \in D, \\ 0, x \notin D . \end{matrix}$

Denote $U / ind ({d}) = {D_{1}, D_{2}, \dots, D_{l}} .$

Theorem 5.5. Suppose that (U, C ∪ {d}) is an IRVDIS. Given δ ∈ [0, 1]. Then the following properties hold.(1) If i ≠ j, then for each α ∈ (0, 1], ${\underline{{app}_{C}^{G} (δ)}}^{α} (D_{i}) \cap {\underline{{app}_{C}^{G} (δ)}}^{α} (D_{j}) = \emptyset$ .(2) If i ≠ j, then for each β ∈ (0, 1], ${\bar{{app}_{C}^{G} (δ)}}^{β} (D_{i}) \cap {\bar{{app}_{C}^{G} (δ)}}^{β} (D_{j}) = \emptyset$ .(3) If α ∈ (0.5, 1], then for each i, ${\underline{{app}_{C}^{G} (δ)}}^{α} (D_{i}) \subseteq {\bar{(R_{C}^{G} (δ))}}^{1 - α} (D_{i})$ .(4) If α₁ ≤ α₂, β₁ ≤ β₂, then for each i, ${\underline{(R_{C}^{G} (δ))}}^{α_{2}} (D_{i}) \subseteq {\underline{(R_{C}^{G} (δ))}}^{α_{1}} (D_{i}), {\bar{(R_{C}^{G} (δ))}}^{β_{2}} (D_{i}) \subseteq {\bar{(R_{C}^{G} (δ))}}^{β_{1}} (D_{i}) .$

Proof. Obviously. □ (α, β)-positive region, (α, β)-boundary region and (α, β)-negative region of D ∈ 2^U can be defined as below. ${POS}_{R_{C}^{G} (δ)}^{(α, β)} (D) = {\underline{{app}_{C}^{G} (δ)}}^{α} (D),$ ${BND}_{R_{C}^{G} (δ)}^{(α, β)} (D) = {\bar{{app}_{C}^{G} (δ)}}^{β} (D) - {\underline{{app}_{C}^{G} (δ)}}^{α} (D),$ ${NEG}_{R_{C}^{G} (δ)}^{(α, β)} (D) = U - {\bar{{app}_{C}^{G} (δ)}}^{β} (D) .$

Thus ${POS}_{R_{C}^{G} (δ)}^{(α, β)} (D) = {u \in U : I (S_{R_{C}^{G} (δ)} (u), D) \geq α},$ ${BND}_{R_{C}^{G} (δ)}^{(α, β)} (D) = {u \in U : β < I (S_{R_{C}^{G} (δ)} (u), D) < α},$ ${NEG}_{R_{C}^{G} (δ)}^{(α, β)} (D) = {u \in U : I (S_{R_{C}^{G} (δ)} (u), D) \leq β} .$

5.3 A 3WD method based on Gaussian kernel in an IRVDIS

In this part, a 3WD method is proposed based on Gaussian kernel in an IRVDIS by using the decision-theoretic rough set model, as well as its algorithm is given.

Suppose that (U, C ∪ {d}) is an IRVDIS. Given that α, β is a pair threshold with 0 ≤ β < α ≤ 1, δ ∈ (0, 1]. Based on the Bayesian decision procedure and Yao’s decision-theoretic study, the decision rule of D ∈ U/ind ({d}) can be written as below.

(P) : If $I (S_{R_{C}^{G} (δ)} (u), D) \geq α$ , then $u \in {POS}_{R_{C}^{G} (δ)}^{(α, β)} (D)$ , so we make a decision of acceptance;

(N) : If $I (S_{R_{C}^{G} (δ)} (u), D) \leq β$ , then $u \in {NEG}_{R_{C}^{G} (δ)}^{(α, β)} (D)$ , so we make a decision of rejection;

(B) : If $β < I (S_{R_{C}^{G} (δ)} (u), D) < α$ , then $u \in {BND}_{R_{C}^{G} (δ)}^{(α, β)} (D)$ , so we make a decision of delay.

The detailed step-wise procedure as an algorithm of 3WD in an IRVDIS is given as below.

6 An application in auto diagnostic

In this section, an example of auto diagnostic is employed to clarify an application of the proposed method.

The detailed step-wise procedure as an algorithm of 3WD in an IRVDIS is given as below.

Auto diagnostic is the process of detecting the working ability or condition of an auto. In our following discussions, we researcch the problem on diagnosis of low configuration (large consumption of fuel and the traction control system is incomplete) with three-way decision. The standards of judging a low configuration auto depend on many factors (such as Cylinders, Displacement, Horsepower, Weight, Acceleration, Model year, Origin etc.). The IRVDIS in Table 1 is used to explain the process of auto diagnostic. In Table 1, U = {u₁, u₂, ⋯ , u₁₃}

represents thirteen autos; C = {a₁, a₂, ⋯ , a₇} represents seven different auto configurations; d denotes the decision attribute. However, sometimes we may misjudge an auto’s conditions only according to theses configurations. This requires a more accurate analysis. λ_•• expresses the loss function when one takes a certain action with it’s corresponding state. Specifically, the set of two states is given by Ω = {X, ¬ X}, which indicates whether an auto has a low configuration(large consumption of fuel and the traction control system is incomplete). λ_PP, λ_BP and λ_NP signify the losses incurred for improving the configuration, further observation and do not improve(that means high configuration) when an auto is poorly configured(large consumption of fuel and the traction control system is incomplete), respectively. λ_PN, λ_BN and λ_NN express the losses incurred for taking the same actions when an auto is highly configured, respectively. The corresponding loss functions for each low configuration are carefully examined by technicians.

First, the low congfiguration auto’s condition is preliminarily judged by Table 1.U/ind ({d}) = {D₁, D₂, D₃}, whereD₁ = {u₁, u₂, u₃, u₄, u₅} is a set of autos that have high configurations,

D₂ = {u₆, u₇, u₈, u₉, u₁₀} is a set of autos may large consumption of fuel or the traction control system is incomplete,D₃ = {u₁₁, u₁₂, u₁₃} is a set of autos that are low configuration.Pick $δ_{1} = \sqrt{0.3}$ , $δ_{2} = \sqrt{0.6}$ , $δ_{3} = \sqrt{0.8}$ . Denote $R_{1} = R_{C}^{G} (δ_{1}), R_{2} = R_{C}^{G} (δ_{2}), R_{3} = R_{C}^{G} (δ_{3}) .$

Then, by Example 4, the information structure of (U, C ∪ {d}) is proposed as below (i = 1, 2, 3):S (C, δ_i) = (S_{R
_i} (u₁) , S_{R
_i} (u₂) , S_{R
_i} (u₃) , S_{R
_i} (u₄) , S_{R
_i} (u₅) , S_{R
_i} (u₆) , S_{R
_i} (u₇) , S_{R
_i} (u₈) , S_{R
_i} (u₉) , S_{R
_i} (u₁₀) , S_{R
_i} (u₁₁) , S_{R
_i} (u₁₂) , S_{R
_i} (u₁₃)), where

S_{R
_i} (u₁) , S_{R
_i} (u₂) , S_{R
_i} (u₃) , S_{R
_i} (u₄) , S_{R
_i} (u₅) , S_{R
_i} (u₆) , S_{R
_i} (u₇) , S_{R
_i} (u₈) , S_{R
_i} (u₉) , S_{R
_i} (u₁₀) , S_{R
_i} (u₁₁) , S_{R
_i} (u₁₂) , S_{R
_i} (u₁₃) are thirteen fuzzy information granules.Next, we compute I (S_{R
_i} (u_j) , D_k) (i = 1, 2, 3 ; j = 1, 2, ⋯ , 13 ; k = 1, 2, 3) (see Tables 9 –11).

Table 9
Computing I (S_{R
₁} (u) , D_i) (i = 1, 2, 3)

U I (S_{R
₁} (u) , D₁) I (S_{R
₁} (u) , D₂) I (S_{R
₁} (u) , D₃)

u ₁ 0.6552 0.1821 0.1627

u ₂ 0.9782 0.0196 0.0022

u ₃ 0.2540 0.2533 0.4924

u ₄ 0.7619 0.1761 0.0507

u ₅ 0.4414 0.6206 0.1039

u ₆ 0.0965 0.5957 0.3077

u ₇ 0.2354 0.4738 0.2908

u ₈ 0.3919 0.4408 0.1673

u ₉ 0.0819 0.5844 0.3337

u ₁₀ 0.0819 0.6552 0.2150

u ₁₁ 0.1855 0.2948 0.5197

u ₁₂ 0.1871 0.2981 0.5148

u ₁₃ 0.1842 0.2902 0.5256

U	I (S_{R ₁} (u) , D₁)	I (S_{R ₁} (u) , D₂)	I (S_{R ₁} (u) , D₃)
u ₁	0.6552	0.1821	0.1627
u ₂	0.9782	0.0196	0.0022
u ₃	0.2540	0.2533	0.4924
u ₄	0.7619	0.1761	0.0507
u ₅	0.4414	0.6206	0.1039
u ₆	0.0965	0.5957	0.3077
u ₇	0.2354	0.4738	0.2908
u ₈	0.3919	0.4408	0.1673
u ₉	0.0819	0.5844	0.3337
u ₁₀	0.0819	0.6552	0.2150
u ₁₁	0.1855	0.2948	0.5197
u ₁₂	0.1871	0.2981	0.5148
u ₁₃	0.1842	0.2902	0.5256

Table 10

Computing I (S_{R
₂} (u) , D_i) (i = 1, 2, 3)

U	I (S_{R ₂} (u) , D₁)	I (S_{R ₂} (u) , D₂)	I (S_{R ₂} (u) , D₃)
u ₁	0.4619	0.3047	0.4007
u ₂	0.8312	0.1192	0.0496
u ₃	0.2657	0.3498	0.3845
u ₄	0.5177	0.2654	0.1659
u ₅	0.4351	0.3378	0.2271
u ₆	0.2133	0.4700	0.3460
u ₇	0.3325	0.4007	0.2852
u ₈	0.2563	0.4756	0.2681
u ₉	0.1947	0.5928	0.3360
u ₁₀	0.2194	0.5133	0.2673
u ₁₁	0.2254	0.3754	0.3991
u ₁₂	0.2368	0.3729	0.3903
u ₁₃	0.2350	0.3692	0.3959

Table 11

Computing I (S_{R
₃} (u) , D_i) (i = 1, 2, 3)

U	I (S_{R ₃} (u) , D₁)	I (S_{R ₃} (u) , D₂)	I (S_{R ₃} (u) , D₃)
u ₁	0.4287	0.3307	0.2406
u ₂	0.7356	0.1749	0.0895
u ₃	0.2805	0.3694	0.3501
u ₄	0.5177	0.2908	0.1915
u ₅	0.4033	0.3604	0.2362
u ₆	0.2133	0.4590	0.4590
u ₇	0.3325	0.3918	0.2757
u ₈	0.2798	0.4529	0.2673
u ₉	0.1947	0.4824	0.3229
u ₁₀	0.2478	0.4833	0.2689
u ₁₁	0.2473	0.3905	0.3621
u ₁₂	0.2586	0.3870	0.3544
u ₁₃	0.2576	0.3841	0.3582

Ultimately, we give the 3WD rules (see Tables 12 –14).

Table 12

The 3WD rules of D₁

(α, β)	R_i	Region	0.1	0.2	0.3	0.4
0.6	R ₁	${POS}_{R_{1}}^{(α, β)} (D_{1})$	{u₁, u₂, u₄}	{u₁, u₂, u₄}	{u₁, u₂, u₄}	{u₁, u₂, u₄}
		${BND}_{R_{1}}^{(α, β)} (D_{1})$	U-{u₁, u₂, u₄, u₆, u₉, u₁₀}	{u₃, u₅, u₇, u₈}	{u₅, u₈}	{u₅}
		${NEG}_{R_{1}}^{(α, β)} (D_{1})$	{u₆, u₉, u₁₀}	{u₆, u₉, u₁₀, u₁₁, u₁₂, u₁₃}	U-{u₁, u₂, u₄, u₅, u₈}	U-{u₁, u₂, u₄, u₅}
	R ₂	${POS}_{R_{2}}^{(α, β)} (D_{1})$	{u₂}	{u₂}	{u₂}	{u₂}
		${BND}_{R_{2}}^{(α, β)} (D_{1})$	U-{u₂}	U-{u₂, u₉}	{u₁, u₄, u₅, u₇}	{u₁, u₄, u₅}
		${NEG}_{R_{2}}^{(α, β)} (D_{1})$	∅	{u₉}	{u₃, u₆, u₈, u₉, u₁₀, u₁₁, u₁₂, u₁₃}	U-{u₁, u₂, u₄, u₅}
	R ₃	${POS}_{R_{3}}^{(α, β)} (D_{1})$	{u₂}	{u₂}	{u₂}	{u₂}
		${BND}_{R_{3}}^{(α, β)} (D_{1})$	U-{u₂}	U-{u₂, u₉}	{u₁, u₄, u₅, u₇}	{u₁, u₄, u₅}
		${NEG}_{R_{3}}^{(α, β)} (D_{1})$	∅	{u₉}	{u₃, u₆, u₈, u₉, u₁₀, u₁₁, u₁₂, u₁₃}	U-{u₁, u₂, u₄, u₅}
0.7	R ₁	${POS}_{R_{1}}^{(α, β)} (D_{1})$	{u₂, u₄}	{u₂, u₄}	{u₂, u₄}	{u₂, u₄}
		${BND}_{R_{1}}^{(α, β)} (D_{1})$	U-{u₂, u₄, u₆, u₉, u₁₀}	{u₁, u₃, u₅, u₇, u₈}	{u₁, u₅, u₈}	{u₁, u₅}
		${NEG}_{R_{1}}^{(α, β)} (D_{1})$	{u₆, u₉, u₁₀}	{u₆, u₉, u₁₀, u₁₁, u₁₂, u₁₃}	U-{u₁, u₂, u₄, u₅, u₈}	U-{u₁, u₂, u₄, u₅}
	R ₂	${POS}_{R_{2}}^{(α, β)} (D_{1})$	{u₂}	{u₂}	{u₂}	{u₂}
		${BND}_{R_{2}}^{(α, β)} (D_{1})$	U-{u₂}	U-{u₂, u₉}	{u₁, u₄, u₅, u₇}	{u₁, u₄, u₅}
		${NEG}_{R_{2}}^{(α, β)} (D_{1})$	∅	{u₉}	{u₃, u₆, u₈, u₉, u₁₀, u₁₁, u₁₂, u₁₃}	U-{u₁, u₂, u₄, u₅}
	R ₃	${POS}_{R_{3}}^{(α, β)} (D_{1})$	{u₂}	{u₂}	{u₂}	{u₂}
		${BND}_{R_{3}}^{(α, β)} (D_{1})$	U-{u₂}	U-{u₂, u₉}	{u₁, u₄, u₅, u₇}	{u₁, u₄, u₅}
		${NEG}_{R_{3}}^{(α, β)} (D_{1})$	∅	{u₉}	{u₃, u₆, u₈, u₉, u₁₀, u₁₁, u₁₂, u₁₃}	U-{u₁, u₂, u₄, u₅}
0.8	R ₁	${POS}_{R_{1}}^{(α, β)} (D_{1})$	{u₂}	{u₂}	{u₂}	{u₂}
		${BND}_{R_{1}}^{(α, β)} (D_{1})$	U-{u₂, u₆, u₉, u₁₀}	{u₁, u₃, u₄, u₅, u₇, u₈}	{u₁, u₄, u₅, u₈}	{u₁, u₄, u₅}
		${NEG}_{R_{1}}^{(α, β)} (D_{1})$	{u₆, u₉, u₁₀}	{u₆, u₉, u₁₀, u₁₁, u₁₂, u₁₃}	U-{u₁, u₂, u₄, u₅, u₈}	U-{u₁, u₂, u₄, u₅}
	R ₂	${POS}_{R_{2}}^{(α, β)} (D_{1})$	{u₂}	{u₂}	{u₂}	{u₂}
		${BND}_{R_{2}}^{(α, β)} (D_{1})$	U-{u₂}	U-{u₂, u₉}	{u₁, u₄, u₅, u₇}	{u₁, u₄, u₅}
		${NEG}_{R_{2}}^{(α, β)} (D_{1})$	∅	{u₉}	{u₃, u₆, u₈, u₉, u₁₀, u₁₁, u₁₂, u₁₃}	U-{u₁, u₂, u₄, u₅}
	R ₃	${POS}_{R_{3}}^{(α, β)} (D_{1})$	∅	∅	∅	∅
		${BND}_{R_{3}}^{(α, β)} (D_{1})$	U	U-{u₉}	{u₁, u₂, u₄, u₅, u₇}	{u₁, u₂, u₄, u₅}
		${NEG}_{R_{3}}^{(α, β)} (D_{1})$	∅	{u₉}	{u₃, u₆, u₈, u₉, u₁₀, u₁₁, u₁₂, u₁₃}	U-{u₁, u₂, u₄, u₅}
0.9	R ₁	${POS}_{R_{1}}^{(α, β)} (D_{1})$	{u₂}	{u₂}	{u₂}	{u₂}
		${BND}_{R_{1}}^{(α, β)} (D_{1})$	U-{u₂, u₆, u₉, u₁₀}	{u₁, u₃, u₄, u₅, u₇, u₈}	{u₁, u₄, u₅, u₈}	{u₁, u₄, u₅}
		${NEG}_{R_{1}}^{(α, β)} (D_{1})$	{u₆, u₉, u₁₀}	{u₆, u₉, u₁₀, u₁₁, u₁₂, u₁₃}	U-{u₁, u₂, u₄, u₅, u₈}	U-{u₁, u₂, u₄, u₅}
	R ₂	${POS}_{R_{2}}^{(α, β)} (D_{1})$	∅	∅	∅	∅
		${BND}_{R_{2}}^{(α, β)} (D_{1})$	U	U-{u₉}	{u₁, u₂, u₄, u₅, u₇}	{u₁, u₂, u₄, u₅}
		${NEG}_{R_{2}}^{(α, β)} (D_{1})$	∅	{u₉}	{u₃, u₆, u₈, u₉, u₁₀, u₁₁, u₁₂, u₁₃}	U-{u₁, u₂, u₄, u₅}
	R ₃	${POS}_{R_{3}}^{(α, β)} (D_{1})$	∅	∅	∅	∅
		${BND}_{R_{3}}^{(α, β)} (D_{1})$	U	U-{u₉}	{u₁, u₂, u₄, u₅, u₇}	{u₁, u₂, u₄, u₅}
		${NEG}_{R_{3}}^{(α, β)} (D_{1})$	∅	{u₉}	{u₃, u₆, u₈, u₉, u₁₀, u₁₁, u₁₂, u₁₃}	U-{u₁, u₂, u₄, u₅}

Table 13

The 3WD rules of D₂

(α, β)	R_i	Region	0.1	0.2	0.3	0.4
0.6	R ₁	${POS}_{R_{1}}^{(α, β)} (D_{2})$	{u₅, u₁₀}	{u₅, u₁₀}	{u₅, u₁₀}	{u₅, u₁₀}
		${BND}_{R_{1}}^{(α, β)} (D_{1})$	U-{u₂, u₅, u₁₀}	U-{u₁, u₂, u₄, u₅, u₁₀}	{u₆, u₇, u₈, u₉}	{u₆, u₇, u₈, u₉}
		${NEG}_{R_{1}}^{(α, β)} (D_{2})$	{u₂}	{u₁, u₂, u₄}	U-{u₅, u₆, u₇, u₈, u₉, u₁₀}	U-{u₅, u₆, u₇, u₈, u₉, u₁₀}
	R ₂	${POS}_{R_{2}}^{(α, β)} (D_{2})$	∅	∅	∅	∅
		${BND}_{R_{2}}^{(α, β)} (D_{2})$	U	U-{u₂}	U-{u₂, u₄}	{u₆, u₇, u₈, u₉, u₁₀}
		${NEG}_{R_{2}}^{(α, β)} (D_{2})$	∅	{u₂}	{u₂, u₄}	U-{u₆, u₇, u₈, u₉, u₁₀}
	R ₃	${POS}_{R_{3}}^{(α, β)} (D_{2})$	∅	∅	∅	∅
		${BND}_{R_{3}}^{(α, β)} (D_{2})$	U	U-{u₂}	U-{u₂, u₄}	{u₆, u₈, u₉, u₁₀}
		${NEG}_{R_{3}}^{(α, β)} (D_{2})$	∅	{u₂}	{u₂, u₄}	U-{u₆, u₈, u₉, u₁₀}
0.7	R ₁	${POS}_{R_{1}}^{(α, β)} (D_{2})$	∅	∅	∅	∅
		${BND}_{R_{1}}^{(α, β)} (D_{2})$	U-{u₂}	U-{u₁, u₂, u₄}	{u₅, u₆, u₇, u₈, u₉, u₁₀}	{u₅, u₆, u₇, u₈, u₉, u₁₀}
		${NEG}_{R_{1}}^{(α, β)} (D_{2})$	{u₂}	{u₁, u₂, u₄}	U-{u₅, u₆, u₇, u₈, u₉, u₁₀}	U-{u₅, u₆, u₇, u₈, u₉, u₁₀}
	R ₂	${POS}_{R_{2}}^{(α, β)} (D_{2})$	∅	∅	∅	∅
		${BND}_{R_{2}}^{(α, β)} (D_{2})$	U	U-{u₂}	U-{u₂, u₄}	{u₆, u₇, u₈, u₉, u₁₀}
		${NEG}_{R_{2}}^{(α, β)} (D_{2})$	∅	{u₂}	{u₂, u₄}	U-{u₆, u₇, u₈, u₉, u₁₀}
	R ₃	${POS}_{R_{3}}^{(α, β)} (D_{2})$	∅	∅	∅	∅
		${BND}_{R_{3}}^{(α, β)} (D_{2})$	U	U-{u₂}	U-{u₂, u₄}	{u₆, u₈, u₉, u₁₀}
		${NEG}_{R_{3}}^{(α, β)} (D_{2})$	∅	{u₂}	{u₂, u₄}	U-{u₆, u₈, u₉, u₁₀}
0.8	R ₁	${POS}_{R_{1}}^{(α, β)} (D_{2})$	∅	∅	∅	∅
		${BND}_{R_{1}}^{(α, β)} (D_{2})$	U-{u₂}	U-{u₁, u₂, u₄}	{u₅, u₆, u₇, u₈, u₉, u₁₀}	{u₅, u₆, u₇, u₈, u₉, u₁₀}
		${NEG}_{R_{1}}^{(α, β)} (D_{2})$	{u₂}	{u₁, u₂, u₄}	U-{u₅, u₆, u₇, u₈, u₉, u₁₀}	U-{u₅, u₆, u₇, u₈, u₉, u₁₀}
	R ₂	${POS}_{R_{2}}^{(α, β)} (D_{2})$	∅	∅	∅	∅
		${BND}_{R_{2}}^{(α, β)} (D_{2})$	U	U-{u₂}	U-{u₂, u₄}	{u₆, u₇, u₈, u₉, u₁₀}
		${NEG}_{R_{2}}^{(α, β)} (D_{2})$	∅	{u₂}	{u₂, u₄}	U-{u₆, u₇, u₈, u₉, u₁₀}
	R ₃	${POS}_{R_{3}}^{(α, β)} (D_{2})$	∅	∅	∅	∅
		${BND}_{R_{3}}^{(α, β)} (D_{2})$	U	U-{u₂}	U-{u₂, u₄}	{u₆, u₈, u₉, u₁₀}
		${NEG}_{R_{3}}^{(α, β)} (D_{2})$	∅	{u₂}	{u₂, u₄}	U-{u₆, u₈, u₉, u₁₀}
0.9	R ₁	${POS}_{R_{1}}^{(α, β)} (D_{2})$	∅	∅	∅	∅
		${BND}_{R_{1}}^{(α, β)} (D_{2})$	U-{u₂}	U-{u₁, u₂, u₄}	{u₅, u₆, u₇, u₈, u₉, u₁₀}	{u₅, u₆, u₇, u₈, u₉, u₁₀}
		${NEG}_{R_{1}}^{(α, β)} (D_{2})$	{u₂}	{u₁, u₂, u₄}	U-{u₅, u₆, u₇, u₈, u₉, u₁₀}	U-{u₅, u₆, u₇, u₈, u₉, u₁₀}
	R ₂	${POS}_{R_{2}}^{(α, β)} (D_{2})$	∅	∅	∅	∅
		${BND}_{R_{2}}^{(α, β)} (D_{2})$	U	U-{u₂}	U-{u₂, u₄}	{u₆, u₇, u₈, u₉, u₁₀}
		${NEG}_{R_{2}}^{(α, β)} (D_{2})$	∅	{u₂}	{u₂, u₄}	U-{u₆, u₇, u₈, u₉, u₁₀}
	R ₃	${POS}_{R_{3}}^{(α, β)} (D_{2})$	∅	∅	∅	∅
		${BND}_{R_{3}}^{(α, β)} (D_{2})$	U	U-{u₂}	U-{u₂, u₄}	{u₆, u₈, u₉, u₁₀}
		${NEG}_{R_{3}}^{(α, β)} (D_{2})$	∅	{u₂}	{u₂, u₄}	U-{u₆, u₈, u₉, u₁₀}

Table 14

The 3WD rules of D₃

(α, β)	R_i	Region	0.1	0.2	0.3	0.4
0.6	R ₁	${POS}_{R_{1}}^{(α, β)} (D_{3})$	∅	∅	∅	∅
		${BND}_{R_{1}}^{(α, β)} (D_{3})$	U-{u₂, u₄}	U-{u₁, u₂, u₄, u₅, u₈}	U-{u₁, u₂, u₄, u₅, u₇, u₈, u₁₀}	U-{u₁, u₂, u₄, u₅, u₆, u₇, u₈, u₉, u₁₀}
		${NEG}_{R_{1}}^{(α, β)} (D_{3})$	{u₂, u₄}	{u₁, u₂, u₄, u₅, u₈}	{u₁, u₂, u₄, u₅, u₇, u₈, u₁₀}	{u₁, u₂, u₄, u₅, u₆, u₇, u₈, u₉, u₁₀}
	R ₂	${POS}_{R_{2}}^{(α, β)} (D_{3})$	∅	∅	∅	∅
		${BND}_{R_{2}}^{(α, β)} (D_{3})$	U-{u₂}	U-{u₂, u₄}	U-{u₂, u₄, u₅, u₇, u₈, u₁₀}	{u₁}
		${NEG}_{R_{2}}^{(α, β)} (D_{3})$	{u₂}	{u₂, u₄}	{u₂, u₄, u₅, u₇, u₈, u₁₀}	U-{u₁}
	R ₃	${POS}_{R_{3}}^{(α, β)} (D_{3})$	∅	∅	∅	∅
		${BND}_{R_{3}}^{(α, β)} (D_{3})$	U-{u₂}	U-{u₂, u₄}	U-{u₁, u₂, u₄, u₅, u₇, u₈, u₁₀}	{u₆}
		${NEG}_{R_{3}}^{(α, β)} (D_{3})$	{u₂}	{u₂, u₄}	{u₁, u₂, u₄, u₅, u₇, u₈, u₁₀}	U-{u₆}
0.7	R ₁	${POS}_{R_{1}}^{(α, β)} (D_{3})$	∅	∅	∅	∅
		${BND}_{R_{1}}^{(α, β)} (D_{3})$	U-{u₂, u₄}	U-{u₁, u₂, u₄, u₅, u₈}	U-{u₁, u₂, u₄, u₅, u₇, u₈, u₁₀}	U-{u₁, u₂, u₄, u₅, u₆, u₇, u₈, u₉, u₁₀}
		${NEG}_{R_{1}}^{(α, β)} (D_{3})$	{u₂, u₄}	{u₁, u₂, u₄, u₅, u₈}	{u₁, u₂, u₄, u₅, u₇, u₈, u₁₀}	{u₁, u₂, u₄, u₅, u₆, u₇, u₈, u₉, u₁₀}
	R ₂	${POS}_{R_{2}}^{(α, β)} (D_{3})$	∅	∅	∅	∅
		${BND}_{R_{2}}^{(α, β)} (D_{3})$	U-{u₂}	U-{u₂, u₄}	U-{u₂, u₄, u₅, u₇, u₈, u₁₀}	{u₁}
		${NEG}_{R_{2}}^{(α, β)} (D_{3})$	{u₂}	{u₂, u₄}	{u₂, u₄, u₅, u₇, u₈, u₁₀}	U-{u₁}
	R ₃	${POS}_{R_{3}}^{(α, β)} (D_{3})$	∅	∅	∅	∅
		${BND}_{R_{3}}^{(α, β)} (D_{3})$	U-{u₂}	U-{u₂, u₄}	U-{u₁, u₂, u₄, u₅, u₇, u₈, u₁₀}	{u₆}
		${NEG}_{R_{3}}^{(α, β)} (D_{3})$	{u₂}	{u₂, u₄}	{u₁, u₂, u₄, u₅, u₇, u₈, u₁₀}	U-{u₆}
0.8	R ₁	${POS}_{R_{1}}^{(α, β)} (D_{3})$	∅	∅	∅	∅
		${BND}_{R_{1}}^{(α, β)} (D_{3})$	U-{u₂, u₄}	U-{u₁, u₂, u₄, u₅, u₈}	U-{u₁, u₂, u₄, u₅, u₇, u₈, u₁₀}	U-{u₁, u₂, u₄, u₅, u₆, u₇, u₈, u₉, u₁₀}
		${NEG}_{R_{1}}^{(α, β)} (D_{3})$	{u₂, u₄}	{u₁, u₂, u₄, u₅, u₈}	{u₁, u₂, u₄, u₅, u₇, u₈, u₁₀}	{u₁, u₂, u₄, u₅, u₆, u₇, u₈, u₉, u₁₀}
	R ₂	${POS}_{R_{2}}^{(α, β)} (D_{3})$	∅	∅	∅	∅
		${BND}_{R_{2}}^{(α, β)} (D_{3})$	U-{u₂}	U-{u₂, u₄}	U-{u₂, u₄, u₅, u₇, u₈, u₁₀}	{u₁}
		${NEG}_{R_{2}}^{(α, β)} (D_{3})$	{u₂}	{u₂, u₄}	{u₂, u₄, u₅, u₇, u₈, u₁₀}	U-{u₁}
	R ₃	${POS}_{R_{3}}^{(α, β)} (D_{3})$	∅	∅	∅	∅
		${BND}_{R_{3}}^{(α, β)} (D_{3})$	U-{u₂}	U-{u₂, u₄}	U-{u₁, u₂, u₄, u₅, u₇, u₈, u₁₀}	{u₆}
		${NEG}_{R_{3}}^{(α, β)} (D_{3})$	{u₂}	{u₂, u₄}	{u₁, u₂, u₄, u₅, u₇, u₈, u₁₀}	U-{u₆}
0.9	R ₁	${POS}_{R_{1}}^{(α, β)} (D_{3})$	∅	∅	∅	∅
		${BND}_{R_{1}}^{(α, β)} (D_{3})$	U-{u₂, u₄}	U-{u₁, u₂, u₄, u₅, u₈}	U-{u₁, u₂, u₄, u₅, u₇, u₈, u₁₀}	U-{u₁, u₂, u₄, u₅, u₆, u₇, u₈, u₉, u₁₀}
		${NEG}_{R_{1}}^{(α, β)} (D_{3})$	{u₂, u₄}	{u₁, u₂, u₄, u₅, u₈}	{u₁, u₂, u₄, u₅, u₇, u₈, u₁₀}	{u₁, u₂, u₄, u₅, u₆, u₇, u₈, u₉, u₁₀}
	R ₂	${POS}_{R_{2}}^{(α, β)} (D_{3})$	∅	∅	∅	∅
		${BND}_{R_{2}}^{(α, β)} (D_{3})$	U-{u₂}	U-{u₂, u₄}	U-{u₂, u₄, u₅, u₇, u₈, u₁₀}	{u₁}
		${NEG}_{R_{2}}^{(α, β)} (D_{3})$	{u₂}	{u₂, u₄}	{u₂, u₄, u₅, u₇, u₈, u₁₀}	U-{u₁}
	R ₃	${POS}_{R_{3}}^{(α, β)} (D_{3})$	∅	∅	∅	∅
		${BND}_{R_{3}}^{(α, β)} (D_{3})$	U-{u₂}	U-{u₂, u₄}	U-{u₁, u₂, u₄, u₅, u₇, u₈, u₁₀}	{u₆}
		${NEG}_{R_{3}}^{(α, β)} (D_{3})$	{u₂}	{u₂, u₄}	{u₁, u₂, u₄, u₅, u₇, u₈, u₁₀}	U-{u₆}

λ_•• signifies the loss function when we take a certain action with it’s corresponding state.

In this paper, the corresponding loss functions for each low configuration auto are carefully estimated by technicians is considered. Nevertheless, the process can’t be controlled as it is subjective and uncertain. To make better decisions, different decision rules are compared when α and β change (0 ≤ β < α).

Below, instructions are given by three cases.(1) Pick $δ_{1} = \sqrt{0.3}$ , α = 0.6, β = 0.3. Then ${POS}_{R_{1}}^{(0.6, 0.3)} (D_{1}) = {u_{1}, u_{2}, u_{4}},$ ${BND}_{R_{1}}^{(0.6, 0.3)} (D_{1}) = {u_{5}, u_{8}},$ ${NEG}_{R_{1}}^{(0.6, 0.3)} (D_{1}) = {u_{3}, u_{6}, u_{7}, u_{9}, u_{10}, u_{11}, u_{12}, u_{13}} .$

Thus, the decision rule of D₁ is proposed as below.(P₁) If $u \in {POS}_{R_{1}}^{(0.6, 0.3)} (D_{1})$ , then u is highly configured;(B₁) If $u \in {BND}_{R_{1}}^{(0.6, 0.3)} (D_{1})$ , then u needs to be further detected;(N₁) If $u \in {NEG}_{R_{1}}^{(0.6, 0.3)} (D_{1})$ , then u is poorly configured.Table 12 gives the decision rules of D₁ when δ, α and β change.(2) Pick $δ_{2} = \sqrt{0.6}$ , α = 0.7, β = 0.3. Then ${POS}_{R_{2}}^{(0.7, 0.3)} (D_{2}) = \emptyset,$ ${BND}_{R_{2}}^{(0.7, 0.3)} (D_{2}) = {u_{1}, u_{3}, u_{5}, u_{6}, u_{7},$ $u_{8}, u_{9}, u_{10}, u_{11}, u_{12}, u_{13}},$ ${NEG}_{R_{2}}^{(0.7, 0.3)} (D_{2}) = {u_{2}, u_{4}} .$

Thus, the decision rule of D₂ is proposed as below.(P₂) If $u \in {POS}_{R_{2}}^{(0.7, 0.3)} (D_{2})$ , then u has large consumption of fuel;(B₂) If $u \in {BND}_{R_{2}}^{(0.7, 0.3)} (D_{2})$ , then u needs to be further examined;(N₂) If $u \in {NEG}_{R_{2}}^{(0.7, 0.3)} (D_{2})$ , then u does not have large consumption of fuel.Table 13 gives the decision rules of D₂ when δ, α and β change.(3) Pick $δ_{3} = \sqrt{0.8}$ , α = 0.6, β = 0.4. Then ${POS}_{R_{3}}^{(0.6, 0.4)} (D_{3}) = \emptyset,$ ${BND}_{R_{3}}^{(0.6, 0.4)} (D_{3}) = {u_{6}},$ ${NEG}_{R_{3}}^{(0.6, 0.4)} (D_{3}) = {u_{1}, u_{2}, \dots, u_{13}} .$

Thus, the decision rule of D₃ is proposed as below.(P₃) If $u \in {POS}_{R_{3}}^{(0.6, 0.4)} (D_{3})$ , then u whose the traction control system is incomplete;(B₃) If $u \in {BND}_{R_{3}}^{(0.6, 0.4)} (D_{3})$ , then u needs to be further inspected;(N₃) If $u \in {NEG}_{R_{3}}^{(0.6, 0.4)} (D_{3})$ , then u whose the traction control system is not incomplete.Table 14 gives the decision rules of D₃ when δ, α and β change.If ${POS}_{R_{i}}^{(α, β)} (D) = \emptyset (i = 1, 2, 3)$ , then the decision rule has no research value.In Tables 9 –11, the parameter δ ∈ (0, 1] dominates the similarity between two objects based on Gaussian kernel in an IRVDIS. The value of δ presents the tightness of the fuzzy T_cos-equivalence relation, which is decided by the IS itself. However, a pair of the parameters α and β describes the risk attitude of a decision maker. In general, our method provides the two scenarios together and presets a man-machine viewpoint on 3WD. The decision makers can directly acquire the corresponding decision rules with different values of α and β in Tables 12 –14.

7 Conclusions

This paper has studied 3WD for incomplete real-valued data. Gaussian kernel has been used to extract the fuzzy T_cos-equivalence relation on the object set of an IRVDIS. The decision-theoretic rough set model in an IRVDIS has been given. A 3WD method has been presented based on Gaussian kernel in an IRVDIS by using this rough set model. An example of auto diagnostic has been employed to illustrate the proposed method’s feasibility. These results show that proposed method will contribute to disposing incomplete real-valued data. This study can be applied in evaluation and measurement of intellectual property value. In future work, we will apply other rough set methods to study an IRVDIS.

Footnotes

Acknowledgments

This work was supported by National Social Science Fund’s Major Research Special Project (18VHQ013), China-ASEAN Institute for Innovation Governance and Intellectual Property Research (2019ZCY04), Collaborative Innovation Center for Integration of Terrestrial and Marine Economies (2019YB22), Natural Science Foundation of Guangxi (2018GXNSFAA294134), Guangxi Science and Technology Program (2017AD23056), and Funding of high-level innovation team and excellent scholar program of Guangxi universities (Document No. [2019] 52).

References

Aranda-Corral

G.A.

, Borrego-Dĺłaz

and Galĺćn-Pĺćez

, A model of three-way decisions for Knowledge Harnessing, International Journal of Approximate Reasoning 120 (2020), 184–202.

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.

Kryszkiewicz

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

Lang

G.M.

, A general conflict analysis model based on threeway decision, International Journal of Machine Learning and Cybernetics 11 (2020), 1083–1094.

Luo

J.F.

, Fujita

, Yao

Y.Y.

and Qin

K.Y.

, On modeling similarity and three-way decision under incomplete information in rough set theory, Knowledge-Based Systems 191 (2020), 105251.

Luo

J.F.

, Hu

M.J.

and Qin

K.Y.

, Three-way decision with incomplete information based on similarity and satisfiability, International Journal of Approximate Reasoning 120 (2020), 151–183.

Liang

D.C.

and Liu

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

Liu

, Liang

D.C.

and Wang

C.C.

, A novel three-way decision model based on incomplete information system, Knowledge-Based Systems 91 (2016), 32–45.

J.N.

, Sun

Q.Q.

, Chen

H.M.

and Yi

H.J.

, Three-way decision on two universes, Information Sciences 515 (2020), 263–279.

10.

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.

11.

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.

12.

Moser

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

13.

Moser

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

14.

Pawlak

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

15.

Pawlak

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

16.

Pawlak

, Rough sets, International Journal of Computer and Information Sciences 11(5) (1982), 341–356.

17.

Pawlak

, Wong

S.K.M.

and Ziarko

, Rough sets: Probabilistic versus deterministic approach, International Journal of Man-Machine Studies 29(1) (1988), 81–95.

18.

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.

19.

Shawe-Tayor

and Cristianini

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

20.

Swiniarski

R.W.

and Skowron

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

21.

Shen

W.Y.

, Wei

Z.H.

, Li

Q.W.

, Zhang

H.Y.

and Miao

D.Q.

, Threeway decisions based blocking reduction models in hierarchical classification, Information Sciences 523 (2020), 63–76.

22.

Wierman

M.J.

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

23.

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 Journal 77 (2019), 734–749.

24.

Yao

Y.Y.

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

25.

Yao

Y.Y.

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

26.

Yao

Y.Y.

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

27.

Yao

Y.Y.

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

28.

Yang

and Li

J.H.

, Complex network analysis of three-way decision researches, International Journal of Machine Learning and Cybernetics 11 (2020), 973–987.

29.

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.

30.

Yao

Y.Y.

, Wong

S.K.M.

and Lingras

, Adecision-theoretic rough set model/ Ras Z W, Zernankova M, Emrich M L eds. Method-ologies for Intelligent Systems, 5. New York: North-Holland, (1990).

31.

Ziarko

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

32.

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.

33.

Zhang

Q.H.

, Pang

G.H.

and Wang

G.Y.

, A novel sequential three-way decisions model based on penalty function, Knowledge-Based Systems 192 (2020), 105350.

34.

Zhang

W.X.

, Wu

W.Z.

, Liang

J.Y.

and Li

D.Y.

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

35.

Zhang

H.Y.

and Zhang

W.X.

, Hybrid monotonic inclusion measure and its use in measuring similarity and distance between fuzzy sets, Fuzzy Sets Systems 160 (2009), 107–118.

36.

Zhang

Q.H.

, Zhao

, Yang

and Wang

G.Y.

, Three-way decisions of rough vague sets from the perspective of fuzziness, Information Sciences 523 (2020), 111–132.