Information structures in an incomplete real-valued information system 1

1.1 Research background and related works

Granular computing, presented by Zadeh [45–48], is a basic issue in knowledge representation and data mining. Its purpose is to seek for an approximation scheme, which allows us to view a phenomenon with different levels of granularity and then can effectively solve a complex problem. Information granulation, organization and causation are basic notions of granular computing. Information granule is a family of objects that are drawn together by some constraints, such as indistinguishability, similarity or functionality. The process of constructing information granules is called information granulation. It granulates a universe into a family of disjoint or overlapping information granules. Granular structure is the family of information granules where the internal structure of each information granule is visible as a sub-structure. Naturally, granular structure can be depicted as a vector consisting of information granules. Lin [11–13] and Yao [40–42] explained the importance of granular computing, this aroused people’s interest in it. Until now, the study on granular computing mainly has four methods, i.e., rough set theory [24], fuzzy set theory [44], concept lattice [23, 35] and quotient space theory [51].

Rough set theory is an effective tool to deal with uncertainty. An information system based on rough set theory was presented by Pawlak [24, 27]. Most applications of rough sets, such as uncertaintymodeling [2, 32], reasoning with uncertainty [39, 40], rule extraction [1, 36], uncertainty measure [3, 49], classification and feature selection [5, 33] are related to information systems.

In granular computing in information systems, the study of information structures is an important research topic. An equivalence relation is a special kind of similarity between two objects from a data set. Given an information system, each attribute subset determines an equivalence relation on the object set of this system. This equivalence relation partitions the object set into some disjoint classes, these classes are said to be equivalence classes. If two objects belong to the same equivalence class, then we may say that they cannot be distinguished under this equivalence relation. Thus, each equivalence class is seen as an information granule consisting of indistinguishable objects [1]. The family of all these information granules constitutes a vector, this vector is said to be an information structure in the given information system induced by this attribute subset. Actually, information structures in an information system are granular structures in the meaning of granular computing. For example, Zhang et al. [49] considered information structures in a fully fuzzy information system; Li et al. [18, 19] studied knowledge structures in a knowledge base and relationships between knowledge bases; Li et al. [17] gave information structure in distributed fc-decision information systems.

1.2 Motivation and inspiration

With the forthcoming era of information age, information acquisition, analysis and processing have become a hot research topic in the information technology world. It means that information uncertainty analysis, information fusion, attribute reduction and classification are becoming increasingly significant. Rough set theory is developed around the concept of an information system. An information system as a database that represents relationships between objects and attributes. If a information system’s information function values are real numbers but it has missing values, then it can be called an incomplete real-valued information system. So far, three-way decision in an incomplete real-valued information system has not been studied. The purpose of this paper is to use the decision-theoretic rough set model to go into the three-way decision on Gaussian kernel in an incomplete real-valued information system. Let (U, A) be an information system. Given P⊂ A. Thereupon, an equivalence relation (or indiscernibility relation) ind (P) can be defined by ( x , y ) ∈ ind ( P ) ⇔ ∀ a ∈ P , a ( x ) = a ( y ) . It is distinct that the initial learning of information systems was based on this equivalence relation (or indiscernibility relation). Let (U, A) be an incomplete real-valued information system. Given a∈ A. For x, y ∈ U, a (x) and a (y) are the information function values of the objects x and y on the attribute a, respectively. What we should especially note is that “a (x) = a (y) " is actually hard to come true. Whereupon, we can not consider ind (P) in (U, A). How do we describe the relationships between two information function values a (x) and a (y)? We can consider the relationships between them from the following aspects: For one thing, we can consider the similarity between a (x) and a (y). “a (x) = a (y) " actually shows that a (x) and a (y) are completely similar. For another thing, we can also consider the difference between a (x) and a (y). Generally speaking, we use the distance between a (x) and a (y) to reflect the differences between them. “a (x) = a (y) " actually signifies that there is no difference between a (x) and a (y), so the distance between them is 0.

In this paper, we will use the distance to describe the relationship between two information function values and spread the article according to the following ideas. First, the distance between two information function values of a given attribute in an incomplete real-valued information system is constructed. Then, the distances between two objects in a given subsystem of an incomplete real-valued information system is structured. Next, the fuzzy T _cos -equivalence relation induced by this subsystem by using Gaussian kernel method is obtained. In this way, the fuzzy neighborhood of every point can be viewed as the information granule. Information structure of this subsystem is consequently proposed.

The remaining part of this paper is organized as follows. In Section 2, we look back on some basic conceptions about fuzzy sets, fuzzy relations and incomplete real-valued information systems. In Section 3, we construct distance between two objects based on a given subsystem in an incomplete real-valued information system and obtain fuzzy T _cos equivalence relation induced by this subsystem by means of Gaussian kernel method. In Section 4, we propose some concepts of information structures in an incomplete real-valued information systems and give their properties. In Section 5, we conclude this paper and emphasize the potential of the future.

2.1 Fuzzy sets and fuzzy relations

Fuzzy sets are extensions of ordinary sets [44]. A fuzzy set P in U is defined as a function that assigns a value P (x) ∈ I to each element x in U, and P (x) is called the membership degree of x to the fuzzy set P.

In this paper, I ^U denotes the set of all fuzzy sets in U. The cardinality of P ∈ I ^U can be calculated with | P | = ∑ i = 1 n P ( 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 ) = ( r 11 r 12 . . . r 1 n r 21 r 22 . . . r 2 n . . . . . . . . . . . . r n 1 r n 2 . . . r nn ) , where r _ij  = R (x _i , x _j ) ∈ I means the degree of similarity between two objects x _i and x _j .

If M (R) = E (the identity matrix), then R is a fuzzy identity relation, and we write as R =▵; if r _ij  = 1,  i, j ≤ n, then R will be a fuzzy universal relation, and we write as R = ω.

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 ) = max { xy - 1 - x 2 1 - y 2 , 0 } . Then T _cos is a t-norm.

Definition 2.3. [[20]] 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 ) ) ≤ f ( x , z ) .

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

2.2 An incomplete real-valued information system

In this part, we recall the concept of an incomplete real-valued information system.

Definition 2.6. [[24]] Let U be an object set and A be an 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 decides an information function a : U → V _a , where V _a  = {a (x) : x ∈ U}.

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.

If (U, A) is an information system. Given P ⊆ A. Then a binary relation on U can be defined as ind ( P ) = { ( x , y ) ∈ U × U : ∀ a ∈ P , a ( x ) = a ( y ) } .

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

Definition 2.7. [[24]] Let (U, A) be an information system. Then the pair (U, A) is called an incomplete information system, if there are x ∈ U and a ∈ A such that a (x) is missing.

Definition 2.8. [[24]] Let (U, A) be an information system. Then (U, A) is called a real-valued information system, if for any x ∈ U and a ∈ A, a (x) is a real number.

Definition 2.9. [[24]] Let (U, A) be a real-valued information system. Then (U, A) is called an incomplete real-valued information system, if (U, A) is incomplete.

Example 2.10. The data of Table 1 come from the Auto mpg in UCI repository of machine learning database. They are randomly selected from this data set.

Table 1 expresses an incomplete real-valued information system (U, A), where U = {x₁, x₂, ⋯ , x₁₃}is an auto set and C = {Cylinders  (a₁) , Displacement (a₂) , Horsepower  (a₃) , Weight  (a₄) , Acceleration  (a₅) , Modelyear  (a₆) ,  Origin  (a₇)}.

3.1 Distances between two objects in an incomplete real-valued information system

To construct the distance between two objects in an incomplete real-valued information system, a novel distance function should be presented.

Definition 3.1. Let (U, A) be an incomplete real-valued information system. ∀ x, y ∈ U, ∀ a ∈ C, the distance between a (x) and a (y) is defined as

d (a (x) , a (y)) = { 0 , x = y ; 1 , x ≠ y , a ( x ) = * or a ( y ) = * ; 0 , x ≠ y , a ( x ) ≠ * , a ( y ) ≠ * , a ( x ) = a ( y ) ; | a ( x ) - a ( y ) | θ a , x ≠ y , a ( x ) ≠ * , a ( y ) ≠ * , a ( x ) ≠ a ( y ) , where θ _a  = max {a (x) : x ∈ U} - min {a (x) : x ∈ U} .

Definition 3.2. Let (U, A) be an incomplete real-valued information system. Given P ⊆ A. ∀ x, y ∈ U, the distance between x and y in the subsystem (U, P) is defined as d P ( x , y ) = ∑ a ∈ P d 2 ( a ( x ) , a ( y ) ) .

Proposition 3.3. Let (U, A) be an incomplete real-valued information system. Given P ⊆ A. Then ∀ x, y ∈ U, 0 ≤ d P ( x , y ) ≤ | P | .

Proof. By Definitions 3.1, and 3.1, ∀ a ∈ P , ∀ x , y ∈ U , 0 ≤ d ( a ( x ) , a ( y ) ) ≤ 1 .

Then ∀ x , y ∈ U , 0 ≤ ∑ a ∈ P d 2 ( a ( x ) , a ( y ) ) ≤ | P | .

Thus ∀ x , y ∈ U , 0 ≤ d P ( x , y ) ≤ | P | .

Example 3.4. (Continued Example 2.2) Calculate d _A  (x₁, x₃).

Since a₃ is a real-valued attribute, by Definition 3.1, we have d ( a 3 ( x 1 ) , a 3 ( x 3 ) ) = | a 3 ( x 1 ) - a 3 ( x 3 ) | θ a = 140 - 98 315 ≈ 0.1333 , where θ _a  = max {a₃ (x) : x ∈ U} - min {a₃ (x) : x ∈ U} .

Then d A ( x 1 , x 3 ) = ∑ a ∈ A d 2 ( x 1 , x 3 ) ≈ 1.1665 .

Example 3.5. Through the above definitions, the distance between two information function values of a given attribute in an incomplete real-valued information system are given as follows (see Tables 2-8).

Example 3.6. (Continued from Examples 2.2)

Gaussian kernel method is a significant means in machine learning and pattern recognition. In the cause of making data linear and simplifying classification tasks, it maps data into a higher dimensional feature space [30]. Hu et al. [6, 7] found that there were some relationships between rough sets and Gaussian kernel method, in consequence Gaussian kernel can be used to obtain fuzzy relations. The part of this Gaussian kernel will be used to extract a fuzzy T _cos -equivalence relation on the object set of a given incomplete real-valued information system.

Gaussian kernel G ( x , y ) = exp [ - ∥ x - y ∥ 2 2 δ 2 ] is used 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].

Evidently G (x, y) satisfies:

                        (1) G (x, y) ∈ [0, 1];

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

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

Definition 3.7. Let (U, A) be an incomplete real-valued information system. Given P ⊆ A and δ ∈ (0, 1], denote R P G ( δ ) ( x i , x j ) = exp [ - d P 2 ( x i , x j ) 2 δ 2 ] . Then R P G ( δ ) is called the fuzzy relation induced by the subsystem (U, P) with respect to δ.

Theorem 3.8. Let (U, A) be an incomplete real-valued information system. Given P ⊆ A and δ ∈ (0, 1]. Then R P G ( δ ) is a fuzzy T _cos -equivalence relation on U.

Proof. This holds by Corollary 2.1.

Example 3.9. (Continued from Examples 2.2) Pick δ = 0.6 .

Thus, the fuzzy T _cos -equivalence relation R A G ( δ ) is gained.

An algorithm for generating the fuzzy T _cos -equivalence relation R P G ( δ ) is showed as follows.

4.1 Some concepts of information structures in an incomplete real-valued information system.

Given 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 information granule of the point x [29]. According to this viewpoint, Qian et al. [29] defined the fuzzy granular structure of R as

S ( R ) = ( S R ( x 1 ) , S R ( x 2 ) , ⋯ , S R ( x n ) ) .

Based on Qian’s idea, we give the concept of information structures in an incomplete real-valued information system in the following definition.

Definition 4.1. Let (U, A) be an incomplete real-valued information system. Given P ⊆ A and δ ∈ (0, 1], denote S δ ( P ) = ( S R P G ( δ ) ( x 1 ) , S R C G ( δ ) ( x 2 ) , ⋯ , S R C G ( δ ) ( x n ) ) . Then S ^δ  (P) is called δ-information structure of the subsystem (U, P).

Example 4.2. (Continued from Examples 2.2 and 3.2) By Example 3.2,

S δ ( A ) = ( S R A G ( δ ) ( x 1 ) , S R A G ( δ ) ( x 2 ) , ⋯ , S R A G ( δ ) ( x 13 ) ) , where S R A G ( δ ) ( x 1 ) = 1.0000 x 1 + 0.0701 x 2 + 0.3218 x 3 + 0.1774 x 4 + 0.3420 x 5 + 0.1879 x 6 + 0.3179 x 7 + 0.3566 x 8 + 0.1422 x 9 + 0.2561 x 10 + 0.2952 x 11 + 0.3419 x 12 + 0.3287 x 13 , S R A G ( δ ) ( x 2 ) = 0.0701 x 1 + 1.0000 x 2 + 0.0353 x 3 + 0.2900 x 4 + 0.0491 x 5 + 0.0108 x 6 + 0.1435 x 7 + 0.0300 x 8 + 0.0077 x 9 + 0.0152 x 10 + 0.0233 x 11 + 0.0347 x 12 + 0.0282 x 13 , S R A G ( δ ) ( x 3 ) = 0.3218 x 1 + 0.0353 x 2 + 1.0000 x 3 + 0.2488 x 4 + 0.2343 x 5 + 0.6592 x 6 + 0.4741 x 7 + 0.4086 x 8 + 0.4801 x 9 + 0.4004 x 10 + 0.9111 x 11 + 0.8884 x 12 + 0.8638 x 13 , S R A G ( δ ) ( x 4 ) = 0.1774 x 1 + 0.2900 x 2 + 0.2488 x 3 + 1.0000 x 4 + 0.1225 x 5 + 0.1171 x 6 + 0.4904 x 7 + 0.1026 x 8 + 0.0744 x 9 + 0.0736 x 10 + 0.1522 x 11 + 0.1975 x 12 + 0.1867 x 13 , S R A G ( δ ) ( x 5 ) = 0.3420 x 1 + 0.0491 x 2 + 0.2343 x 3 + 0.1225 x 4 + 1.0000 x 5 + 0.2283 x 6 + 0.3695 x 7 + 0.3063 x 8 + 0.2227 x 9 + 0.2303 x 10 + 0.2631 x 11 + 0.3248 x 12 + 0.3244 x 13 , S R A G ( δ ) ( x 6 ) = 0.1879 x 1 + 0.0108 x 2 + 0.6592 x 3 + 0.1171 x 4 + 0.2283 x 5 + 1.0000 x 6 + 0.4405 x 7 + 0.3219 x 8 + 0.9281 x 9 + 0.3835 x 10 + 0.7808 x 11 + 0.7519 x 12 + 0.7300 x 13 , S R A G ( δ ) ( x 7 ) = 0.3179 x 1 + 0.1435 x 2 + 0.4741 x 3 + 0.4904 x 4 + 0.3695 x 5 + 0.4405 x 6 + 1.0000 x 7 + 0.2590 x 8 + 0.3974 x 9 + 0.1934 x 10 + 0.4734 x 11 + 0.5975 x 12 + 0.5594 x 13 , S R A G ( δ ) ( x 8 ) = 0.3566 x 1 + 0.0300 x 2 + 0.4086 x 3 + 0.1026 x 4 + 0.3063 x 5 + 0.3219 x 6 + 0.2590 x 7 + 1.0000 x 8 + 0.2578 x 9 + 0.3957 x 10 + 0.4197 x 11 + 0.4281 x 12 + 0.4118 x 13 , S R A G ( δ ) ( x 9 ) = 0.1422 x 1 + 0.0077 x 2 + 0.4801 x 3 + 0.0744 x 4 + 0.2227 x 5 + 0.9281 x 6 + 0.3974 x 7 + 0.2578 x 8 + 1.0000 x 9 + 0.3097 x 10 + 0.6501 x 11 + 0.6453 x 12 + 0.6378 x 13 , S R A G ( δ ) ( x 10 ) = 0.2561 x 1 + 0.0152 x 2 + 0.4004 x 3 + 0.0736 x 4 + 0.2303 x 5 + 0.3835 x 6 + 0.1934 x 7 + 0.3957 x 8 + 0.3097 x 9 + 1.0000 x 10 + 0.4149 x 11 + 0.3920 x 12 + 0.3819 x 13 , S R A G ( δ ) ( x 11 ) = 0.2952 x 1 + 0.0233 x 2 + 0.9111 x 3 + 0.1522 x 4 + 0.2631 x 5 + 0.7808 x 6 + 0.4734 x 7 + 0.4197 x 8 + 0.6501 x 9 + 0.4149 x 10 + 1.0000 x 11 + 0.9705 x 12 + 0.9418 x 13 , S R A G ( δ ) ( x 12 ) = 0.3419 x 1 + 0.0347 x 2 + 0.8884 x 3 + 0.1975 x 4 + 0.3248 x 5 + 0.7519 x 6 + 0.5975 x 7 + 0.4281 x 8 + 0.6453 x 9 + 0.3920 x 10 + 0.9705 x 11 + 1.0000 x 12 + 0.9758 x 13 , S R A G ( δ ) ( x 13 ) = 0.3287 x 1 + 0.0282 x 2 + 0.8638 x 3 + 0.1867 x 4 + 0.3244 x 5 + 0.7300 x 6 + 0.5594 x 7 + 0.4118 x 8 + 0.6378 x 9 + 0.3819 x 10 + 0.9418 x 11 + 0.9758 x 12 + 1.0000 x 13 , are the fuzzy information granules of x₁, x₂, ⋯ , x₁₃, respectively.

Definition 4.3. (U, A) be an incomplete real-valued information system. Given δ₁, δ₂ ∈ (0, 1] and P, Q ⊆ A. If for each i, S R P G ( δ 1 ) ( x i ) = S R Q G ( δ 2 ) ( x i ) , then S ^{δ ₁}  (P) and S ^{δ ₂}  (Q) are called to be the same. We write S ^{δ ₁}  (P) = S ^{δ ₂}  (Q).

Definition 4.4. Let (U, A) be an incomplete real-valued information system. Given δ ∈ (0, 1]. Put S δ ( U , A ) = { S δ ( P ) : P ⊆ A } Then is called δ-information structure base of (U, A).

Below, we propose dependence between information structures.

Definition 4.5. Let (U, A) be an incomplete real-valued information system. Given δ₁, δ₂ ∈ (0, 1] and P, Q ⊆ A.

(1) S ^{δ ₂}  (Q) is called to depend on S ^{δ ₁}  (P), if for each i, S R P G ( δ 1 ) ( x i ) ⊆ S R Q G ( δ 2 ) ( x i ) , we write S ^{δ ₂}  (Q) ⪯ S ^{δ ₁}  (P); S ^{δ ₂}  (Q) is called to depend strictly on S ^{δ ₁}  (P), if S ^{δ ₁}  (P) ⪯ S ^{δ ₂}  (Q) and S ^{δ ₁}  (P) ≠ S ^{δ ₂}  (Q), we write S ^{δ ₁}  (P) ≺ S ^{δ ₂}  (Q).

(2) S ^{δ ₂}  (Q) is called to depend partially on S ^{δ ₁}  (P), if there exists i, S R P G ( δ 1 ) ( x i ) ⊑ S R Q G ( δ 2 ) ( x i ) , we write S ^{δ ₁}  (P) ⊑ S ^{δ ₂}  (Q); S ^{δ ₂}  (Q) is called to depend partially strictly on S ^{δ ₁}  (P), if  S R P G ( δ 1 ) ( x i ) ⊑ S R Q G ( δ 2 ) ( x i ) and S ^{δ ₁}  (P) ≠ S ^{δ ₂}  (Q), we write S ^{δ ₁}  (P) ^{sqsubsetSδ₂} (Q).

(3) S ^{δ ₂}  (Q) is called to be independent on S ^{δ ₁}  (P), if for each i, S R P G ( δ 1 ) ( x i ) S R Q G ( δ 2 ) ( x i ) , we write S ^{δ ₁}  (P) ⋈ S ^{δ ₂}  (Q).Obviously, S ^{δ ₁}  (P) = S ^{δ ₂}  (Q)   ⇔ S ^{δ ₁}  (P)⪯ S ^{δ ₂}  (Q) ,  S ^{δ ₂}  (Q) ⪯ S ^{δ ₁}  (P) ; S ^{δ ₁}  (P)⪯ S ^{δ ₂}  (Q) ⇒ S ^{δ ₁}  (P) ⊑ S ^{δ ₂}  (Q) ; S ^{δ ₁}  (P) ≺ S ^{δ ₂}  (Q) ⇒ S ^{δ ₁}  (P) ⊏ S ^{δ ₂}  (Q) .

4.2 Properties of information structures in an incomplete real-valued information system

In this subsection, we give properties of information structures in an incomplete real-valued information system.

Theorem 4.6. Let (U, A) be an incomplete real-valued information system. Given δ₁, δ₂ ∈ (0, 1] and P, Q ⊆ A. Then S δ 1 ( P ) = S δ 2 ( Q ) ⇔ R P G ( δ 1 ) = R Q G ( δ 2 ) .

Proof. Obviously.□

Theorem 4.7. Let (U, A) be an incomplete real-valued information system. Given δ₁, δ₂ ∈ (0, 1] and P, Q ⊆ A. Then S δ 1 ( P ) ⪯ S δ 2 ( Q ) ⇔ R P G ( δ 1 ) ⊆ R Q G ( δ 2 ) .

Proof. Clearly.□

Corollary 4.8. Let (U, A) be an incomplete real-valued information system. Given δ₁, δ₂ ∈ (0, 1] and P, Q ⊆ A. Then S δ 1 ( P ) ≺ S δ 2 ( Q ) ⇔ R P G ( δ 1 ) ⊂ R Q G ( δ 2 ) .

Proof. This follows from Theorems 4.2 and 4.2.

Theorem 4.9. Let (U, A) be an incomplete real-valued information system.

(1) If 0 < δ₁ ≤ δ₂ ≤ 1, then for any P ⊆ A, S ^{δ ₁}  (P) ⪯ S ^{δ ₂}  (P);

(2) If P ⊆ Q ⊆ A, then for any δ ∈ (0, 1], S ^δ  (Q) ⪯ S ^δ  (P).

Proof. (1) Suppose 0 < δ₁ ≤ δ₂ ≤ 1. For any i, j, it is clear that exp ( - d P 2 ( x i , x j ) 2 δ 1 2 ) ≤ exp ( - d P 2 ( x i , x j ) 2 δ 2 2 ) .

Then R P G ( δ 1 ) ( x i , x j ) ≤ R P G ( δ 2 ) ( x i , x j ) .

So R P G ( δ 1 ) ⊆ R P G ( δ 2 ) .

By Theorem 4.2, S δ 1 ( P ) ⪯ S δ 2 ( P ) .

(2) By Definition 4.3, R P G ( δ ) ( x i , x j ) = exp ( - d P 2 ( x i , x j ) 2 δ 2 ) . R Q G ( δ ) ( x i , x j ) = exp ( - d Q 2 ( x i , x j ) 2 δ 2 ) .

Then R Q G ( δ ) ( x i , x j ) ≤ R P G ( δ ) ( x i , x j ) ( 1 ≤ i , j ≤ n ) .

So R Q G ( δ ) ⊆ R P G ( δ ) .

Thus, by Theorem 4.2, S δ ( Q ) ⪯ S δ ( P ) .

Corollary 4.10. Let (U, A) be an incomplete real-valued information system. Given 0 < δ₁ ≤ δ₂ ≤ 1 and P ⊆ Q ⊆ A. Then

S ^{δ ₁}  (Q) ⪯ S ^{δ ₂}  (Q) ⪯ S ^{δ ₂}  (P),  S ^{δ ₁}  (Q) ⪯ S ^{δ ₁}  (P) ⪯ S ^{δ ₂}  (P).

Proof. This holds by Theorem 4.2.

Definition 4.11.] Let (U, A) be an incomplete real-valued information system. Given δ ∈ (0, 1]. Suppose that S ^δ  (U, A) is the hybrid information structure base of (U, A). Then a mapping D : S ^δ  (U, A) × S ^δ  (U, A) → [0, 1] is called the inclusion degree on S ^δ  (U, A), if for any P, Q, L ⊆ A

(1) 0 ≤ D (S ^δ  (Q)/S ^δ  (P)) ≤1;

(2) S ^δ  (P) ⪯ S ^δ  (Q) implies D (S ^δ  (Q)/S ^δ  (P)) =1;

(3) S ^δ  (P) ⊑ S ^δ  (Q) ⊑ S ^δ  (L) implies D (S ^δ  (P)/S ^δ  (L)) ≤ D (S ^δ  (P)/S ^δ  (Q)).

Definition 4.12. Let (U, A) be an incomplete real-valued information system. For any P, Q ⊆ A, define D (S ^δ  (Q)/S ^δ  (P)) = ∑ l = 1 n | S R Q G ( δ ) ( x l ) | ∑ i = 1 n | S R Q G ( δ ) ( x i ) | χ S R Q G ( δ ) ( x l ) ( S R P G ( δ ) ( x l ) ) , where    χ S R Q G ( δ ) ( x l ) ( S R P G ( δ ) ( x l ) ) = { 1 , if S R P G ( δ ) ( x l ) ⊆ S R Q G ( δ ) ( x l ) , 0 , if S R P G ( δ ) ( x l ) ⊈ S R Q G ( δ ) ( x l ) .

Proposition 4.13. D in Definition 4.2 is the inclusion degree under Definition 4.2.

Proof. Obviously.

The following theorem shows the fact that relationships between information structures in an incomplete real-valued information system can be quantitatively described by the inclusion degree.

Theorem 4.14. Let (U, A) be an incomplete real-valued information system. Given P, Q ⊆ A and δ ∈ (0, 1]. Then

(1) S ^δ  (P) ⪯ S ^δ  (Q) ⇔ D (S ^δ  (Q)/S ^δ  (P)) =1 .

(2) S ^δ  (P) ⋈ S ^δ  (Q) ⇔ D (S ^δ  (Q)/S ^δ  (P)) =0 .

(3) S ^δ  (P) ⊑ S ^δ  (Q) ⇔0 < D (S ^δ  (Q)/S ^δ  (P)) ≤1 .

Proof. (1) “⇒" is obvious. We prove “⟸". Put | S R Q G ( δ ) ( x l ) | = q l , ∑ l = 1 n | S R Q G ( δ ) ( x l ) | = q . Then q = ∑ l = 1 n q l . Since D (S ^δ  (Q)/S ^δ  (P)) =1, we have ∑ l = 1 n q l χ S R Q G ( δ ) ( x l ) ( S R P G ( δ ) ( x l ) ) = q = ∑ l = 1 n q l . Then ∑ l = 1 n q l ( 1 - χ S R Q G ( δ ) ( x l ) ( S R P G ( δ ) ( x l ) ) ) = 0 . Thus ∀ l, 1 - χ S R Q G ( δ ) ( x l ) ( S R P G ( δ ) ( x l ) ) = 0 .

It follows that ∀ l, S R P G ( δ ) ( x l ) ⊆ S R Q G ( δ ) ( x l ) .

Hence S ^δ  (P) ⪯ S ^δ  (Q).

(2) “⇒". Since S ^δ  (P) ⋈ S ^δ  (Q), we have S R P G ( δ ) ( x l ) ⊈ S R Q G ( δ ) ( x l ) ( ∀ l ) . Then ∀ l, χ S R Q G ( δ ) ( x l ) ( S R P G ( δ ) ( x l ) ) = 0 .

Thus D (S ^δ  (Q)/S ^δ  (P)) =0.

“⟸". Since D (S ^δ  (Q)/S ^δ  (P)) =0, we obtain that ∀ l, S R Q G ( δ ) ( x l ) ( S R P G ( δ ) ( x l ) ) = 0 .

Then ∀ l, S R P G ( δ ) ( x l ) ⊈ S R Q G ( δ ) ( x l ) . Thus S ^δ  (P) ⋈ S ^δ  (Q).

(3) This follows from (1) and (2).

4.3 Information distance between two information structures

Considering separation between information structures, in this subsection, we propose the concept of information distance to differentiate two information structures in the same incomplete real-valued information system and give some of its properties.

For A, B ∈ I ^U , denote A ⊕ B = ( A - B ) ∪ ( B - A ) . Then A ⊕ B is called the symmetric difference of A and B.