Measures of uncertainty for an approximation space 1

Abstract

An approximation space is one basic concept in rough set theory. This paper investigates measures of uncertainty for an approximation space from granular computing viewpoint. Granular structures of an approximation spaces are first described by means of set vectors. Then, relationships between granular structures of an approximation spaces are studied from the two aspects of dependence and separation. Next, properties of granular structures of an approximation space are given. Furthermore, as an application for granular structures of an approximation spaces, measuring uncertainty of approximation spaces is investigated. Finally, one example is employed to illustrate features of the proposed measures for uncertainty of approximation spaces. These results will be helpful for understanding the essence of uncertainty for an approximation space.

Keywords

Rough set theory Approximation space Granular computing Granular structure Dependence Uncertainty Measure

1. Introduction

Rough set theory, presented by Pawlak [15], is a mathematical tool to deal with uncertainty and can be considered as the generalization of classical set theory. It has been successfully applied to intelligent systems, expert systems, knowledge discovery, pattern recognition, machine learning, signal analysis, image processing, inductive reasoning, decision analysis and many other fields [3 , 15–17].

The basic structure of rough set theory is an approximation space. Based on it, lower and upper approximations can be induced. Using these approximations, knowledge hidden in information systems may be revealed and expressed in the form of decision rules. A key notion in Pawlak rough set model is equivalence relations. The equivalence classes are the building blocks for the construction of these approximations.

Granular computing proposed by Zadeh is an important tool in artificial intelligence [25 –27]. Its key issues are granulation (or information granulation), organization and causation. Zadeh pointed out “Granulation involves decomposition of whole into parts, organization involves integration of parts into whole, and causation involves association of causes and effects". The aim of granular computing is to explore an approximation scheme, which can effectively solve a complex problem. It allows us to view a phenomenon with different levels of granularity. Now, granular computing has been applied in various fields such as data mining, data clustering, machine learning, artificial intelligence, approximate reasoning and knowledge discovery.

A granule (or an information granule) is a primitive concept in granular computing, which is a clump consisting of objects drawn together by similarity, indistinguishability and proximity of functionality [12 , 24]. It may be interpreted as one of the numerous small particles forming a larger unit. Granulation of objects leads to a family of information granules. A granular structure is a mathematical structure of the family of information granule from a data set, where the internal structure of each information granule is visible and the coaction between information granules are tested by the visible structures [4 –6].

The concept of entropy is originated from energetics. It can be used to measure out-of-order degree of a system. The entropy of a system, proposed by Shannon [21], gives a measure of uncertainty of a system. It has been applied in diverse fields as a useful mechanism for evaluating uncertainty in various modes. Some scholars have applied the extension of entropy and its variants to rough sets. For example, D $\ddot{u}$ ntsch et al. [2] proposed information entropy and three kinds of conditional entropies in rough sets for predicting a decision attribute; Beaubouef et al. [1] presented a method measuring uncertainty of rough sets and rough relation databases; Wierman et al. [22] addressed granulation measure to measure uncertainty of information; Yao et al. [24] gave a granularity measure from the angle of granulation; Liang et al. [12, 14] studied several measures on knowledge in incomplete and complete information systems; Liang et al. [13] proposed the information entropy, rough entropy and knowledge granulation in rough set theory; Xie et al. [23] studied new measures of uncertainty for an interval-valued information system; Zhang et al. [28] considered uncertainty measures in a fully fuzzy information system.

The aim of this paper is to investigate uncertainty measurement for an approximation space by using its granular structures.

The remaining part of this paper is organized as follows. In Section 2, some basic notions on approximation spaces and rough set theory are recalled. In Section 3, the concept of granular structures is introduced and dependence between granular structures are proposed. In Sections 4, some tools for measuring uncertainty of an approximation space are introduced and an illustrative examples is given. In Sections 5, a numerical experiment is given to show features of the proposed measures. Section 6 summarizes this paper.

2. Preliminaries

In this section, we recall some basic concepts about approximation spaces and rough set theory.

Throughout this paper, U denotes a non-empty finite set called the universe, 2^U denotes the family of all subsets of U and |X| denotes the cardinality of X ∈ 2^U.

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

Recall that R is a binary relation on U whenever R ⊆ U × U. If (x, y) ∈ R, then we denote it by xRy.

Let R be a binary relation on U. Then R is called

(1) reflexive, if xRx for any x ∈ U.

(2) symmetric, if xRy implies yRx for any x, y ∈ U.

(3) transitive, if xRy and yRz imply xRz for any x, y, z ∈ U.

Let R be a binary relation on U. Then R is called an equivalence relation on U, if R is reflexive, symmetric and transitive.

In this paper, U = {x₁, x₂, ⋯ , x_n}, $R^{*} (U)$ denotes the family of all equivalence relations on U.

Given $R \in R^{*} (U)$ . If R = U × U, then R is called a universal relation on U; if R = {(x, x) : x ∈ U}, then R is said to be an identity relation on U.

Let $R \in R^{*} (U)$ . Then the pair (U, R) is called an approximation space. Based on (U, R), one can define the following two rough approximations:

$\underline{R} (X) = {x \in U : [x]_{R} \subseteq X},$

$\bar{R} (X) = {x \in U : [x]_{R} \cap X \neq \emptyset} .$

Then $\underline{R} (X)$ and $\bar{R} (X)$ are called the lower and upper approximation of X, respectively.

3. Granular structures of approximation spaces

In this section, we give granular structures of approximation spaces and study dependence between them.

Definition 3.1. Let (U, R) be an approximation space. Then G (R) = ([x₁] _R, [x₂] _R, ⋯⋯ , [x_n] _R) is called the granular structure of (U, R).

Example 3.2. G (δ) = (U, U, ⋯⋯ , U),

G (▵) = ({x₁} , {x₂} , ⋯⋯ , {x_n}).

Definition 3.3. Let (U, P) and (U, Q) be two approximation spaces. Then G (P) and G (Q) are called to be the same, if for each i, [x_i] _P = [x_i] _Q. We write G (P) = G (Q).

The following definition depicts relationships between granular structures of approximation spaces from two aspects.

Definition 3.4. Let (U, P) and (U, Q) be two approximation spaces.

(1) G (Q) is called to depend on G (P), if for each i, [x_i] _P ⊆ [x_i] _Q, we write G (P) ⪯ G (Q); G (Q) is called to depend strictly on G (P), if G (P) ⪯ G (Q) and G (P) = G (Q), we write G (P) ≺ G (Q).

(2) G (Q) is called to depend partially on G (P), if there exists i, [x_i] _P ⊆ [x_i] _Q, we write G (P) ⊑ G (Q); G (Q) is called to depend partially strictly on G (P), if G (P) ⊑ G (Q) and G (P) = G (Q), we write G (P) ⊏ G (Q).

(3) G (Q) is called to be independent of G (P), if for each i, [x_i] _P ⊈ [x_i] _Q. We write G (P) ⋈ G (Q).

Obviously,

G (P) = G (Q) ⇔ G (P) ⪯ G (Q), G (Q) ⪯ G (P),

G (P) ⪯ G (Q) ⇒ G (P) ⊑ G (Q).

G (P) notsqsubseteqG (Q) ⇔ G (P) ⊑ G (Q).

Theorem 3.5. Let (U, P) and (U, Q) be two approximation spaces. Then the following are equivalent:

(1) G (P) = G (Q);

(2) U/P = U/Q;

(3) P = Q.

Proof. This is obvious. □

Definition 3.6. [[29]] Let $A = {X_{1}, X_{2}, \dots \dots, X_{kitsc}}$ and $B = {Y_{1}, Y_{2}, \dots \dots, Y_{litsc}}$ be two partitions on U.

(1) $H (A) = \sum_{i = 1}^{k} p (X_{i}) p (U - X_{i})$ is called the information amount of $A$ , where $p (X_{i}) = \frac{∣ X_{i} ∣}{∣ U ∣}$ means the probability that the element of U belongs to X_i.

(2) $H (B / A) = \sum_{i = 1}^{k} \sum_{j = 1}^{l} p (X_{i} \cap Y_{j}) p (X_{i} - Y_{j})$ is called the condition information amount of $B$ with respect to $A$ .

The following theorem quantitatively depict the dependence of granular structures by the condition information amount.

Theorem 3.7. Let (U, P) and (U, Q) be two approximation spaces. Then the following are equivalent:

(1) G (P) ⪯ G (Q);

(2) U/P refines U/Q, i.e., for each A ∈ U/P, there exists B ∈ U/Q such that A ⊆ B;

(3) P ⊆ Q;

(4) H ((U/Q)/(U/P)) =0.

Proof. (1) ⇔ (2) ⇔ (3) are obvious.

(3) ⇒ (4). Denote

U/P = {X₁, X₂, ⋯⋯ , X_k} ,

U/Q = {Y₁, Y₂, ⋯⋯ , Y_l} .

∀ i, since U/P refines U/Q, we have X_i ⊆ Y_{j
_i} for some j_i ≤ l. Then X_i - Y_{j
_i} =∅.

∀ j ≠ j_i, Y_j∩ Y_{j
_i} = ∅. Then X_i∩ Y_j = ∅.

Thus ∀ i, j, X_i - Y_j =∅ or X_i∩ Y_j = ∅. This implies $p (X_{i} \cap Y_{j}) p (X_{i} - Y_{j}) = 0 .$

Hence H ((U/Q)/(U/P)) =0.

(4) ⇒ (2). Since H ((U/Q)/(U/P)) =0, we have ∀ i, j, X_i - Y_j =∅ or X_i∩ Y_j = ∅. So ∀ i, j, X_i ⊆ Y_j or X_i∩ Y_j = ∅.

∀ i, $X_{i} = ⋃_{j = 1}^{l} (X_{i} \cap Y_{j})$ . Since X_i≠ ∅, we have X_i∩ Y_{j
_i} ≠ ∅ for some j_i ≤ l. Then X_i ⊆ Y_{j
_i}.

Thus U/P refines U/Q. □

For $R \in R^{*}$ , denote $σ (U / R) = {⋃_{x \in X} [x]_{R} | X \in 2^{U}} .$

Theorem 3.8. Let (U, P) and (U, Q) be two approximation spaces. Denote $U / Q = {D_{1}, D_{2}, \dots, D_{r}} .$ Then the following are equivalent:

(1) G (P) ⪯ G (Q);

(2) for each j, D_j ∈ σ (U/P);

(3) for each j, $\underline{P} (D_{j}) = D_{j}$ ;

(4) $⋃_{j = 1}^{r} \underline{P} (D_{j}) = U$ ;

Proof. (1) ⇒ (2). ∀ x ∈ D_j, we have D_j = [x] _Q. since G (P) ⪯ G (Q), [x] _P ⊆ [x] _Q. Then {x} ⊆ [x] _P ⊆ D_j. So

$D_{j} = ⋃_{x \in D_{j}} {x} \subseteq ⋃_{x \in D_{j}} [x]_{P} \subseteq D_{j} .$

Thus D_j = ⋃ _{x∈D_j} [x] _P ∈ σ (U/P).

(2) ⇒ (3). Since D_j ∈ σ (U/R), D_j = ⋃ _x∈X [x] _P for some X ∈ 2^U.

Thus $\underline{P} (D_{j}) = \underline{P} (⋃_{x \in X} [x]_{P}) = ⋃_{x \in X} [x]_{P} = D_{j}$ .

(3) ⇒ (4) is obvious.

(4) ⇒ (1). ∀ i ≤ n, ∀ y ∈ [x_i] _P. Since $⋃_{j = 1}^{r} \underline{P} (D_{j})$ = U, $x_{i} \in \underline{P} (D_{j})$ for some j ≤ r. Then [x_i] _P ⊆ D_j. Denote D_j = [x^*] _Q. x_i ∈ D_j implies [x_i] _Q = [x^*] _Q. Then y ∈ [x_i] _Q. So [x_i] _P ⊆ [x_i] _Q. This shows G (P) ⊏ G (Q).□ In this paper, denote $G (U) = {G (R) : R \in R^{*} (U)} .$

Definition 3.9. [[29]] Let D : G (U) × G (U) → [0, 1] be a mapping. Then D is called the inclusion degree on G (U), if

(1) 0 ≤ D (G (Q)/G (P)) ≤1;

(2) G (P) ⪯ G (Q) implies D (G (Q)/G (P)) =1;

(3) G (P) ⊑ G (Q) ⊑ G (R) implies D (G (P)/G (R)) ≤ D (G (P)/G (Q)).

Example 3.10. For any $P, Q \in R^{*} (U)$ , define $D (G (Q) / G (P)) = \sum_{l = 1}^{n} \frac{| {[x_{l}]}_{Q} |}{\sum_{i = 1}^{n} | {[x_{i}]}_{Q} |} χ_{{[x_{l}]}_{Q}} ([x_{l}]_{P}),$ where $χ_{[x_{l}]_{Q}} ([x_{l}]_{P}) = {\begin{matrix} 1, & if [x_{l}]_{P} \subseteq [x_{l}]_{Q}, \\ 0, & if [x_{l}]_{P} ⊈ [x_{l}]_{Q} . \end{matrix}$

It is easy to prove that D is the inclusion degree on G (U).

The following theorem shows that relationships between granular structures in an approximation space can be quantitatively described by the inclusion degree.

Theorem 3.11. Let $(U, R)$ be an approximation space and $P, Q \in R$ . Then

(1) G (P) ⪯ G (Q) ⇔ D (G (Q)/G (P)) =1 .

(2) G (P) ⋈ G (Q) ⇔ D (G (Q)/G (P)) =0 .

(3) G (P) ⊑ G (Q) ⇔0 < D (G (Q)/G (P)) ≤1 .

Proof. (1) “⇒" is obvious.

“⇐". Put $| [x_{l}]_{Q} | = q_{l}, \sum_{i = 1}^{n} = q | [x_{i}]_{Q} | .$ Then $q = \sum_{i = 1}^{n} q_{1}$ . Since D (G (Q)/G (P)) =1, we have $\sum_{l = 1}^{n} q_{l} χ_{{[x_{l}]}_{Q}} ([x_{l}]_{P}) = q = \sum_{i = 1}^{n} q_{1} .$ $\sum_{l = 1}^{n} q_{l} (1 - χ_{[x_{l}]_{Q}} ([x_{l}]_{P})) = 0 .$ Thus for each l, $1 - χ_{[x_{l}]_{Q}} ([x_{l}]_{P}) = 0 .$ It follows that for each l, [x_l] _P ⊆ [x_l] _Q.

Hence G (P) ⊏ G (Q).

(2) “⇒". Since G (P) ⋈ G (Q), we have ∀ l, [x_l] _P ⊈ [x_l] _Q. Then for each l, $χ_{[x_{l}]_{Q}} ([x_{l}]_{P}) = 0 .$

Thus D (G (Q)/G (P)) =0.

“⇐". Since D (G (Q)/G (P)) =0, we obtain that for each l, $χ_{[x_{l}]_{Q}} ([x_{l}]_{P}) = 0 .$

Then ∀ l, [x_l] _P ⊈ [x_l] _Q. Thus G (P) ⋈ G (Q).

(3) This holds by (2). □

4. Some tools for measuring uncertainty of approximation spaces

In this section, we investigate measuring uncertainty of approximation spaces.

4.1. Granularity measures of approximation spaces

We first propose axiom definition of information granulation of approximation spaces in the following definition.

Definition 4.1. Suppose that $G : R^{*} (U) \to (- \infty, + \infty)$ is a function. Then G is referred to as a information granulation function on $R^{*} (U)$ , if G satisfies the following conditions:

(1) Non-negativity: $\forall P \in R^{*} (U)$ , G (P) ≥0;

(2) Invariability: $\forall P, Q \in R^{*} (U)$ , if (U, P) = (U, Q), then G (P) = G (Q);

(3) Monotonicity: $\forall P, Q \in R^{*} (U)$ , if (U, P) ≺ (U, Q), then G (P) < G (Q). Moreover, for $P \in R^{*} (U)$ , G (P) is referred to as information granulation of the approximation space (U, P).

Similar to Definition 2 in [12], the information granulation of a given approximation space is given in the following definition.

Definition 4.2. Suppose that (U, P) is an approximation space. Knowledge granulation of (U, P) is defined as $G (P) = \frac{1}{n^{2}} \sum_{i = 1}^{m} | X_{i} |^{2},$ where U/P = {X₁, X₂, ⋯ , X_m}.

Proposition 4.3. Let (U, P) be an approximation space. Then $G (P) = \frac{1}{n^{2}} \sum_{i = 1}^{n} | {[x_{i}]}_{P} | .$

Proof. Denote U/P = {X₁, X₂, ⋯ , X_m} .

Suppose X_i = {x_i1, x_i2, ⋯ , x_{is
_i}}, |X_i| = s_i. Then $\sum_{i = 1}^{m} s_{i} = n, X_{i} = [x_{i 1}]_{P} = [x_{i 2}]_{P} = \dots = {[x_{i s_{i}}]}_{P} .$ So $| X_{i} | = | [x_{i 1}]_{P} | = | [x_{i 2}]_{P} | = \dots = | [x_{{is}_{i}}]_{P} | = s_{i} .$ Thus, ∀ i, ${| X_{i} |}^{2} = s_{i} | X_{i} | = \sum_{k = 1}^{s_{i}} | {[x_{i k}]}_{P} | .$

Hence

$\begin{matrix} G (P) & = \frac{1}{n^{2}} \sum_{i = 1}^{m} {| X_{i} |}^{2} = \frac{1}{n^{2}} \sum_{i = 1}^{m} \sum_{k = 1}^{s_{i}} | [x_{i k}] p | \\ = \frac{1}{n^{2}} \sum_{i = 1}^{n} | {[x_{i}]}_{P} | \sum_{n}^{i = 1} . \end{matrix}$

□

Proposition 4.4. Let (U, P) be an approximation space. Then $\frac{1}{n} \leq G (P) \leq 1 .$ Moreover, if P is an identity relation on U, then G (P) achieves the minimum value $\frac{1}{n}$ ; if P is a universal relation on U, then G (P) achieves the maximum value 1.

Proof. Since ∀ i, 1 ≤ | [x_i] _P| ≤ n, $n \leq \sum_{i = 1}^{n} | [x_{i}] p | \leq n^{2} .$

By Proposition 4.3, $\frac{1}{n} \leq G (P) \leq 1 .$

If P is an identity relation on U, then ∀ i, | [x_i] _P|=1. So $G (P) = \frac{1}{n}$ .

If P is a universal relation on U, then ∀ i, | [x_i] _P| = n. So G (P) =1.□

Theorem 4.5. Let (U, P) and (U, Q) be two approximation spaces. If (U, P) ≺ (U, Q), then G (P) < G (Q).

Proof. By Proposition 4.3,

$G (P) = \frac{1}{n^{2}} \sum_{i = 1}^{n} | {[x_{i}]}_{P} |,$

$G (Q) = \frac{1}{n^{2}} \sum_{i = 1}^{n} | {[x_{i}]}_{Q} | .$

Since (U, P) ≺ (U, Q), we have ∀ i, [x_i] _P ⊆ [x_i] _Q and ∃ j, [x_j] _P ⊊ x_i] _Q.

Then, ∀ i, | [x_i] _P| ≤ | [x_i] _Q| and ∃ j, 1 ≤ | [x_j] _P| < | [x_j] _Q| .

Hence G (P) < G (Q) . □

Theorem 4.6. G in Definition 4.2 is an information granulation function under Definition 4.1.

Proof. (1) Obviously, “non-negativity" holds.

(2) Given $P, Q \in R^{*} (U)$ . If (U, P) = (U, Q), then ∀ i, [x_i] _P = [x_i] _Q.

By Proposition 4.3, G (P) = G (Q).

(3) “Monotonicity" follows from Theorem 4.5.

□

4.2. Entropy measures of approximation spaces

In physics, entropy is often used to measure out-of-order degree of a system. The bigger the entropy value is, the higher out-of-order of a system will be. Shannon [21] applied the concept of entropy in physics to information theory for measuring uncertainty of a system.

Similar to Definition 8 in [12], the information entropy of a given approximation space is defined as follows.

Definition 4.7. Suppose that (U, P) is an approximation space. Knowledge entropy of (U, P) is defined as $H (P) = - \sum_{i = 1}^{m} \frac{| X_{i} |}{n} \log_{2} \frac{| X_{i} |}{n},$ where U/P = {X₁, X₂, ⋯ , X_m}.

Proposition 4.8. Let (U, P) be an approximation space. Then $H (P) = - \sum_{i = 1}^{n} \frac{1}{n} \log_{2} \frac{| {[x_{i}]}_{P} |}{n} .$

Proof. Denote U/P = {X₁, X₂, ⋯ , X_m} .

Suppose X_i = {x_i1, x_i2, ⋯ , x_{is
_i}}, |X_i| = s_i. Then $\sum_{i = 1}^{m} s_{i} = n, X_{i} = [x_{i 1}]_{P} = [x_{i 2}]_{P} = \dots = {[x_{i s_{i}}]}_{P} .$ So $| X_{i} | = | [x_{i 1}]_{P} | = | [x_{i 2}]_{P} | = \dots = | [x_{{is}_{i}}]_{P} | = s_{i} .$ Thus, ∀ i, $\begin{matrix} \frac{| X_{i} |}{n} {log}_{2} \frac{| X_{i} |}{n} \\ = s_{i} \frac{1}{n} {log}_{2} \frac{| X_{i} |}{n} = \sum_{k = 1}^{s_{i}} \frac{1}{n} {log}_{2} \frac{| [x_{ik}]_{P} |}{n} . \end{matrix}$

Hence $\begin{matrix} H (P) & = - \sum_{i = 1}^{m} \frac{| X_{i} |}{n} {log}_{2} \frac{| X_{i} |}{n} \\ = - \sum_{i = 1}^{m} \sum_{k = 1}^{s_{i}} \frac{1}{n} {log}_{2} \frac{| [x_{ik}]_{P} |}{n} \\ = - \sum_{i = 1}^{n} \frac{1}{n} {log}_{2} \frac{| [x_{i}]_{P} |}{n} . \end{matrix}$

□

Theorem 4.9. Let (U, P) and (U, Q) be two approximation spaces. If (U, P) ≺ (U, Q), then H (Q) < H (P).

Proof. By Proposition 4.8.

$H (P) = - \sum_{i = 1}^{n} \frac{1}{n} \log_{2} \frac{| {[x_{i}]}_{P} |}{n},$

$H (Q) = - \sum_{i = 1}^{n} \frac{1}{n} \log_{2} \frac{| {[x_{i}]}_{Q} |}{n} .$

It should be noted that (U, P) ≺ (U, Q). Then, similar to the proof of Theorem 4.5, ∀ i, 1 ≤ | [x_i] _P| ≤ | [x_i] _Q| and ∃ j, $1 \leq | [x_{j}]_{P} | < | [x_{j}]_{Q} | .$

Then, ∀ i, $- {log}_{2} \frac{| [x_{i}]_{P} |}{n} = {log}_{2} \frac{n}{| [x_{i}]_{P} |} \geq {log}_{2} \frac{n}{| [x_{i}]_{Q} |} = - {log}_{2}$ $\frac{| [x_{i}]_{Q} |}{n},$ and ∃ j, $- {log}_{2} \frac{| [x_{j}]_{P} |}{n} = {log}_{2} \frac{n}{| [x_{j}]_{P} |} > {log}_{2} \frac{n}{| [x_{j}]_{Q} |} = - {log}_{2} \frac{| [x_{j}]_{Q} |}{n}$ .

Hence H (Q) < H (P) . □

Rough entropy, introduced by Yao [24], is used to measure granularity of a given partition. Similar to Definition 6 in [12], the rough entropy of a given approximation space is proposed in the following definition.

Definition 4.10. Let (U, P) be an approximation space. Rough entropy of (U, P) is defined as $E_{r} (P) = - \sum_{i = 1}^{m} \frac{| X_{i} |}{n} \log_{2} \frac{1}{| X_{i} |},$ where U/P = {X₁, X₂, ⋯ , X_m}.

Proposition 4.11. Let (U, P) be an approximation space. Then $E_{r} (P) = - \sum_{i = 1}^{n} \frac{1}{n} \log_{2} \frac{1}{| {[x_{i}]}_{P} |} .$

Proof. Denote $U / P = {X_{1}, X_{2}, \dots, X_{m}} .$

Suppose X_i = {x_i1, x_i2, ⋯ , x_{is
_i}}, |X_i| = s_i. Then $\sum_{i = 1}^{m} s_{i} = n, X_{i} = [x_{i 1}]_{P} = [x_{i 2}]_{P} = \dots = [x_{{is}_{i}}]_{P} .$ So $| X_{i} | = | [x_{i 1}]_{P} | = | [x_{i 2}]_{P} | = \dots = | [x_{{is}_{i}}]_{P} | = s_{i} .$ Thus, ∀ i, $| X_{i} | {log}_{2} \frac{1}{| X_{i} |} = s_{i} {log}_{2} \frac{1}{| X_{i} |} = \sum_{k = 1}^{s_{i}} {log}_{2} \frac{1}{| [x_{ik}]_{P} |} .$

Hence $\begin{array}{l} E_{r} (P) & = - \sum_{i = 1}^{m} \frac{| X_{i} |}{n} \log_{2} \frac{1}{| X_{i} |} = - \frac{1}{n} \sum_{i = 1}^{m} | X_{i} | \log_{2} \frac{1}{| X_{i} |} \\ = - \frac{1}{n} \sum_{i = 1}^{m} \sum_{k = 1}^{s_{i}} \log_{2} \frac{1}{| [X_{i k}] P |} = - \frac{1}{n} \sum_{i = 1}^{n} \log_{2} \frac{1}{| [X_{i}] P |} \\ - \sum_{i = 1}^{n} \frac{1}{n} \log_{2} \frac{1}{| [x_{i}] P |} \end{array}$

□

Proposition 4.12. Let (U, P) be an approximation space. Then $0 \leq E_{r} (P) \leq {log}_{2} n .$ Moreover, if P is an identity relation on U, then E_r (P) achieves the minimum value 0; if P is a universal relation on U, then E_r (P) achieves the maximum value log ₂n.

Proof. Since ∀ i, 1 ≤ | [x_i] _P| ≤ n, 0 ≤ - log ₂ $\frac{1}{| {[x_{i}]}_{P} |} = \log_{2} | {[x_{i}]}_{P} | \leq \log_{2} n 0 \leq - \sum_{i = 1}^{n} \log_{2} \frac{1}{| {[x_{i}]}_{P} |} \leq n \log_{2} n$ .

By Proposition 4.8, $0 \leq E_{r} (P) \leq {log}_{2} n .$

If P is an identity relation on U, then ∀ i, | [x_i] _P|=1. So E_r (P) =0.

If P is a universal relation on U, then ∀ i, | [x_i] _P| = n. So E_r (P) = log ₂n.□

Theorem 4.13. Let (U, P) and (U, Q) be two approximation spaces. If (U, P) ≺ (U, Q), then E_r (P) < E_r (Q).

Proof. By Proposition 4.11,

$E_{r} (P) = - \sum_{i = 1}^{n} \frac{1}{n} \log_{2} \frac{1}{| {[x_{i}]}_{P} |},$

$E_{r} (Q) = - \sum_{i = 1}^{n} \frac{1}{n} \log_{2} \frac{1}{| {[x_{i}]}_{Q} |} .$

It should be noted that (U, P) ≺ (U, Q). Then, similar to the proof of Theorem 4.5, ∀ i, 1 ≤ | [x_i] _P| ≤ | [x_i] _Q| and ∃ j, $1 \leq | [x_{j}]_{P} | < | [x_{j}]_{Q} | .$

Then ∀ i, $- {log}_{2} \frac{1}{| [x_{i}]_{P} |} = {log}_{2} | [x_{i}]_{P} | \leq {log}_{2} | [x_{i}]_{Q} | = - {log}_{2} \frac{1}{| [x_{i}]_{Q} |},$ and ∃ j, $- {log}_{2} \frac{1}{| [x_{j}]_{P} |} = {log}_{2} | [x_{j}]_{P} | < {log}_{2} | [x_{j}]_{Q} | = - {log}_{2} \frac{1}{| [x_{j}]_{Q} |} .$

Hence E_r (P) < E_r (Q) . □

Theorem 4.14. E_r in Definition 4.10 is an information granulation function under Definition 4.1.

Proof. (1) Obviously, “non-negativity" holds.

(2) Given $P, Q \in R^{*} (U)$ . If (U, P) = (U, Q), then ∀ i, [x_i] _P = [x_i] _Q. By Proposition 4.11, E_r (P) = E_r (Q).

(3) “Monotonicity" follows from Theorem 4.13.

□

4.3. Information amounts of approximation spaces

Similar to Definition 10 in [12], the information amount of a given approximation space is proposed in the following definition.

Definition 4.15. Let (U, P) be an approximation space. Information amount of (U, P) is defined as $E (P) = \sum_{i = 1}^{m} \frac{| X_{i} |}{n} \frac{| U - X_{i} |}{n},$ where U/P = {X₁, X₂, ⋯ , X_m}, $\frac{| X_{i} |}{n}$ (resp. $\frac{| U - X_{i} |}{n}$ ) represents the probability of X_i (resp. U - X_i) within the universe U.

Proposition 4.16. Let (U, P) be an approximation space. Then $E (P) = \sum_{i = 1}^{n} \frac{1}{n} (1 - \frac{| {[x_{i}]}_{P} |}{n}) .$

Proof. Denote U/P = {X₁, X₂, ⋯ , X_m} . Suppose X_i = {x_i1, x_i2, ⋯ , x_{is
_i}}, |X_i| = s_i. Then $\sum_{i = 1}^{m} s_{i} = n, X_{i} = [x_{i 1}]_{P} = [x_{i 2}]_{P} = \dots = [x_{{is}_{i}}]_{P} .$ So $| X_{i} | = | [x_{i 1}]_{P} | = | [x_{i 2}]_{P} | = \dots = | [x_{{is}_{i}}]_{P} | = s_{i} .$ Since {X₁, X₂, ⋯ , X_m} is a partition on U, ∀ i, we have $U - X_{i} = (⋃_{k = 1}^{i - 1} X_{k}) ⋃ (⋃_{k = i + 1}^{m} X_{k}) .$ Then $\begin{matrix} | U - X_{i} | & = & \sum_{k = 1}^{i - 1} | X_{k} | + \sum_{k = i + 1}^{m} X_{k} \\ = & | U | - | X_{i} | = n - | X_{i} | . \end{matrix}$

Thus, ∀ i, $| X_{i} | | U - X_{i} | = s_{i} (n - | X_{i} |) = \sum_{k = 1}^{s_{i}} (n - | [x_{ik}]_{P} |) .$

Hence $\begin{matrix} E (P) & = \sum_{i = 1}^{m} \frac{| X_{i} |}{n} \frac{| U - X_{i} |}{n} = \sum_{i = 1}^{m} \sum_{k = 1}^{s_{i}} \frac{n - | {[x_{i k}]}_{P} |}{n^{2}} \\ = \sum_{i = 1}^{n} \frac{n - | {[x_{i}]}_{P} |}{n^{2}} = \sum_{i = 1}^{n} \frac{1}{n} (1 - \frac{| {[x_{i}]}_{P} |}{n}) . \end{matrix}$

□

Theorem 4.17. Let (U, P) and (U, Q) be two approximation spaces. If (U, P) ≺ (U, Q), then E (Q) < E (P).

Proof. By Proposition 4.16,

$E (P) = \sum_{i = 1}^{n} \frac{1}{n} (1 - \frac{| {[x_{i}]}_{P} |}{n})$ ,

$E (Q) = \sum_{i = 1}^{n} \frac{1}{n} (1 - \frac{| {[x_{i}]}_{Q} |}{n}) .$

It should be noted that (U, P) ≺ (U, Q). Then, similar to the proof of Theorem 4.5, ∀ i, 1 ≤ | [x_i] _P| ≤ | [x_i] _Q| and ∃ j, $1 \leq | [x_{j}]_{P} | < | [x_{j}]_{Q} | .$

Hence E (Q) < E (P). □

4.4. Some properties

Theorem 4.18. Let (U, P) be an approximation space. Then $G (P) + E (P) = 1 .$

Proof. Denote U/P = {X₁, X₂, ⋯ , X_m} .

Suppose X_i = {x_i1, x_i2, ⋯ , x_{is
_i}}, |X_i| = s_i. Then $\sum_{i = 1}^{m} s_{i} = n$ .

Since {X₁, X₂, ⋯ , X_m} is a partition on U, ∀ i, we have $U - X_{i} = {(\cup_{k = 1}^{i - 1} X_{k})}^{} \cup (\cup_{k = 1}^{i - 1} X_{k}) .$ Then $\begin{matrix} | U - X_{i} | & = & \sum_{k = 1}^{i - 1} | X_{k} | + \sum_{k = i + 1}^{m} | X_{k} | \\ = & | U | - | X_{i} | = n - | X_{i} | . \end{matrix}$

Then $\begin{matrix} E (P) & = \sum_{i = 1}^{m} \frac{| X_{i} |}{n} \frac{| U - X_{i} |}{n} = \sum_{i = 1}^{m} \frac{s_{i} (n - s_{i})}{n^{2}} \\ = \sum_{i = 1}^{m} \frac{s_{i}}{n} - \sum_{i = 1}^{m} \frac{s_{i}^{2}}{n^{2}} = 1 - \sum_{i = 1}^{m} \frac{| X_{i} |^{2}}{n^{2}} \\ = 1 - G (P) . \end{matrix}$

Thus G (P) + E (P) =1. □

Corollary 4.19. Let (U, P) be an approximation space. Then $0 \leq E (P) \leq 1 - \frac{1}{n} .$

Proof. By Proposition 4.4, $\frac{1}{n} \leq G (P) \leq 1$ .

By Theorem 4.18, E (P) =1 - G (P).

Thus $0 \leq E (P) \leq 1 - \frac{1}{n}$ . □

Theorem 4.20. Let (U, P) be an approximation space. Then $E_{r} (P) + H (P) = \log_{2} n .$

Proof. Denote U/P = {X₁, X₂, ⋯ , X_m} .

Suppose X_i = {x_i1, x_i2, ⋯ , x_{is
_i}}, |X_i| = s_i. Then $\sum_{i = 1}^{m} s_{i} = n$ .

Thus

$\begin{matrix} E_{r} (P) + H (P) \\ = - \sum_{i = 1}^{m} \frac{| X_{i} |}{n} {log}_{2} \frac{1}{| X_{i} |} - \sum_{i = 1}^{m} \frac{| X_{i} |}{n} {log}_{2} \frac{| X_{i} |}{n} \\ = - \sum_{i = 1}^{m} \frac{s_{i}}{n} {log}_{2} \frac{1}{s_{i}} - \sum_{i = 1}^{m} \frac{s_{i}}{n} {log}_{2} \frac{s_{i}}{n} \\ = - \sum_{i = 1}^{m} \frac{s_{i}}{n} ({log}_{2} 1 - {log}_{2} s_{i} + {log}_{2} s_{i} - {log}_{2} n) \\ = - \sum_{i = 1}^{m} \frac{s_{i}}{n} ({log}_{2} 1 - {log}_{2} n) = - \sum_{i = 1}^{m} \frac{s_{i}}{n} (- {log}_{2} n) \\ = {log}_{2} n . \end{matrix}$

□

Corollary 4.21. Let (U, P) be an approximation space. Then $0 \leq H (P) \leq \log_{2} n .$

Proof. By Proposition 4.12, 0 ≤ E_r (P) ≤ log ₂n.

By Theorem 4.20, H (P) = log₂ n - E_r (P)

Thus 0 ≤ H (P) ≤ log₂ n. □

5. A numerical experiment example

Below, a numerical experiment example is given to show features of the proposed measures.

Example 5.1. We construct a numerical experiment on the postoperative patient data set that comes from UCI Repository of machine learning databases, which is shown as Table 1. The experiment data in Table 1 is a portion randomly selected from the postoperative patient data set. Table 1 can be expressed as the information system (U, A) of postoperative patients where U = {x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈, x₉, x₁₀} is the set of postoperative patients and A = {internal temperature, surface temperature, oxygen saturation, last measurement of blood pressure, stability of patient’s surface temperature, stability of patient’s core temperature, stability of patient’s blood pressure, patient’s perceived comfort at discharge, discharge decision} = {a₁, a₂, a₃, a₄, a₅, a₆, a₇, a₈, a₉} is the set of attributes.

Table 1
The information system (U, A) of postoperative patients

U a ₁ a ₂ a ₃ a ₄ a ₅ a ₆ a ₇ a ₈ a ₉

x ₁ mid low excellent mid stable stable stable 15 A

x ₂ mid high excellent high stable stable stable 10 S

x ₃ high low excellent high stable stable mod-stable 10 A

x ₄ mid low good high stable unstable mod-stable 15 A

x ₅ mid mid excellent high stable stable stable 10 A

x ₆ high low good mid stable stable unstable 15 S

x ₇ mid low excellent high stable stable mod-stable 5 S

x ₈ high mid excellent mid unstable unstable stable 10 S

x ₉ mid high good mid stable stable stable 10 S

x ₁₀ mid low excellent mid unstable stable mod-stable 10 S

U	a ₁	a ₂	a ₃	a ₄	a ₅	a ₆	a ₇	a ₈	a ₉
x ₁	mid	low	excellent	mid	stable	stable	stable	15	A
x ₂	mid	high	excellent	high	stable	stable	stable	10	S
x ₃	high	low	excellent	high	stable	stable	mod-stable	10	A
x ₄	mid	low	good	high	stable	unstable	mod-stable	15	A
x ₅	mid	mid	excellent	high	stable	stable	stable	10	A
x ₆	high	low	good	mid	stable	stable	unstable	15	S
x ₇	mid	low	excellent	high	stable	stable	mod-stable	5	S
x ₈	high	mid	excellent	mid	unstable	unstable	stable	10	S
x ₉	mid	high	good	mid	stable	stable	stable	10	S
x ₁₀	mid	low	excellent	mid	unstable	stable	mod-stable	10	S

Denote

P_i = ind ({a₁, a₂, ⋯ , a_i}) (i = 1, 2, ⋯ , 9) .

Then (U, P_i) is the approximation space. And

U/P₁ = {{x₁, x₂, x₄, x₅, x₇, x₉, x₁₀} , {x₃, x₆, x₈}},

U/P₂ = {{x₁, x₄, x₇, x₁₀} , {x₂, x₉} , {x₃, x₆} , {x₅} , {x₈}},

U/P₃ = {{x₁, x₇, x₁₀} , {x₂} , {x₃} , {x₄} , {x₅} , {x₆} , {x₈} , {x₉}},

U/P₄ = {{x₁, x₁₀} , {x₂} , {x₃} , {x₄} , {x₅} , {x₆} , {x₇} , {x₈} , {x₉}},

U/P₅ = {{x₁} , {x₂} , {x₃} , {x₄} , {x₅} , {x₆} , {x₇} , {x₈} , {x₉} , {x₁₀}},

U/P₆ = {{x₁} , {x₂} , {x₃} , {x₄} , {x₅} , {x₆} , {x₇} , {x₈} , {x₉} , {x₁₀}},

U/P₇ = {{x₁} , {x₂} , {x₃} , {x₄} , {x₅} , {x₆} , {x₇} , {x₈} , {x₉} , {x₁₀}},

U/P₈ = {{x₁} , {x₂} , {x₃} , {x₄} , {x₅} , {x₆} , {x₇} , {x₈} , {x₉} , {x₁₀}},

U/P₉ = {{x₁} , {x₂} , {x₃} , {x₄} , {x₅} , {x₆} , {x₇} , {x₈} , {x₉} , {x₁₀}}. By Proposition 3.3,

$G (P_{1}) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{P_{1}} | = 0.58$ , $G (P_{2}) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{P_{2}} | = 0.26$ , $G (P_{3}) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{P_{3}} | = 0.16$ , $G (P_{4}) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{P_{4}} | = 0.12$ , $G (P_{5}) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{P_{5}} | = 0.1$ , $G (P_{6}) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{P_{6}} | = 0.1$ , $G (P_{7}) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{P_{7}} | = 0.1$ , $G (P_{8}) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{P_{8}} | = 0.1$ , $G (P_{9}) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{P_{9}} | = 0.1$ .

By Proposition 3.8,

$H (P_{1}) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{P_{1}} |}{10} \approx 0.882$ , $H (P_{2}) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{P_{2}} |}{10} \approx 2.122$ , $H (P_{3}) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{P_{3}} |}{10} \approx 2.847$ , $H (P_{4}) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{P_{4}} |}{10} \approx 3.122$ , $H (P_{5}) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{P_{5}} |}{10} \approx 3.322$ , $H (P_{6}) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{P_{6}} |}{10} \approx 3.322$ , $H (P_{7}) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{P_{7}} |}{10} \approx 3.322$ , $H (P_{8}) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{P_{8}} |}{10} \approx 3.322$ , $H (P_{9}) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{P_{9}} |}{10} \approx 3.322$ . By Proposition 3.11,

$E_{r} (P_{1}) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{P_{1}} |} \approx 2.44$ , $E_{r} (P_{2}) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{P_{2}} |} = 1.2$ , $E_{r} (P_{3}) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{P_{3}} |} \approx 0.476$ , $E_{r} (P_{4}) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{P_{4}} |} = 0.2$ , $E_{r} (P_{5}) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{P_{5}} |} = 0$ ,

$E_{r} (P_{6}) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{P_{6}} |} = 0$ , $E_{r} (P_{7}) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{P_{7}} |} = 0$ , $E_{r} (P_{8}) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{P_{8}} |} = 0$ , $E_{r} (P_{9}) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{P_{9}} |} = 0$ .

By Proposition 3.16,

$E (P_{1}) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{P_{1}} |}{10}) = 0.42$ , $E (P_{2}) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{P_{2}} |}{10}) = 0.74$ , $E (P_{3}) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{P_{3}} |}{10}) = 0.84$ , $E (P_{4}) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{P_{4}} |}{10}) = 0.88$ , $E (P_{5}) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{P_{5}} |}{10}) = 0.9$ , $E (P_{6}) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{P_{6}} |}{10}) = 0.9$ , $E (P_{7}) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{P_{7}} |}{10}) = 0.9$ , $E (P_{8}) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{P_{8}} |}{10}) = 0.9$ , $E (P_{9}) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{P_{9}} |}{10}) = 0.9$ .

Fig. 1

Measuring uncertainty of approximation.

The results of uncertainty measures are shown in Fig. 1. It can be seen the fact that with the attribute subset B ⊆ A growth, the information granulation and rough entropy of (U, {ind {a} : a ∈ B}) are both monotonically decreasing. Meanwhile, the information amount and information entropy of the approximation space (U, ind (B)) are both monotonically increasing with the attribute subset B ⊆ A growth. That means that information granulation, rough entropy, information amount and information entropy can be applied to measuring uncertainty of a given approximation space.

6. Conclusions

In this paper, granular structures of approximation spaces have been introduced. Dependence between granular structures has been discussed. Some properties of granular structures have been given. As an application for granular structures of approximation spaces, measures of uncertainty for approximation spaces have been investigated. These results will be very helpful for understanding the essence of uncertainty for an approximation space. In the future, we will consider applications of the proposed measures.

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