Hybrid multi-granulation rough sets of variable precision based on tolerance

Abstract

Hybrid multi-granularity approach is a data processing mode in rough set theory. Taking the incomplete information system as the research object, variable precision rough set method and hybrid multi-granulation rough set method is fused. And this paper proposes a hybrid multi-granulation rough sets based on variable precision tolerance relations and discusses two other hybrid multi-granulation rough sets of variable precision based on similarity relation and limited tolerance relation. Basic properties of hybrid multi-granularity rough set of variable precision are discussed, which provides a new approach to deal with the incomplete information system.

Keywords

Hybrid multi-granulation rough sets hybrid multi-granulation rough sets model of variable precision tolerance rough set variable precision rough sets

1 Introduction

Polish scholar Z. Pawlak proposed rough set theory [1 –4], providing a new model to deal with uncertainty information, which is also used for feature selection, knowledge discovery, uncertain reasoning, decision-making assessment and other fields. At present, many scholars have weakened indiscernible relations and constructed expanding rough set models based on tolerance relations, similar relations, limited tolerance relations, etc. On the basis of tolerance rough set model, along with the variable precision introduced, this paper establishes a variable precision rough set model based on tolerance relation in incomplete information system. In this paper, research of problem-solving in two or more different granularities is carried out, a cluster of equivalence relations, rather than one equivalence relation are used to approximate the target concept, considering that a cluster of divisions has both disjunctive relations and conjunctive relations, the hybrid multi-granularity rough set is put forward. Hybrid multi-granulation rough sets model of variable precision based on tolerance relations is constructed in incomplete information system.

This paper is divided into five parts. In the first part the concepts of the variable precision rough sets, tolerance relations and hybrid multi-granularity rough sets are reviewed. In the second part hybrid multi-granularity rough sets model of variable precision based on tolerance relations is presented, and the corresponding properties are listed. In the third part examples of hybrid multi-granularity rough set model of variable precision based on tolerance relations is shown. In the fourth part discusses two hybrid multi-granulation rough sets of variable precision based on similarity relation and limited tolerance relation. In the fifth part the conclusions and the directions for future research are drawn.

2 Basic concept

2.1 Variable precision rough set

One limitation of the classical Pawlak rough set theory is that it can only process absolutely correct or affirmative classifications, thus it can’t solve problems which contain some degree of ‘include’ and ‘belong’. To overcome this limitation of classical rough set, Ziarko proposed variable precision rough set model in 1993 [5].

Definition 1. Let U be a finite set of non-empty domain, ∀X, Y ⊆ U, and $e (X, Y) = {\begin{matrix} 1 - \frac{| X \cap Y |}{| X |}, & | X | \neq 0 \\ 0, & | X | = 0 \end{matrix}$ (1)

Wherein: |X| represents the cardinality of the set X and e (X, Y) denotes the relative misclassification rate of the set X to the set Y.

Let 0 ≤ β < 0.5, if e (X, Y) ≤ β, then X is mostly contained within Y at the error of, denoted as X ⊆^βY.

Definition 2. Given information system IS = (U, AT), for X ⊆ U, A ⊆ AT, 0 ≤ β < 0.5, the lower and upper approximations of X with respect to A are defined as: $\underline{A_{β}} (X) = {x \in U : e ([x]_{A}, X) \leq β}$ (2) $\bar{A_{β}} (X) = {x \in U : e ([x]_{A}, X) < 1 - β}$ (3)

Rough set model procured by $[\underline{A_{β}} (X), \bar{A_{β}} (X)]$ is called variable precision rough set model.

2.2 Tolerance relation of incomplete information system

The most important idea in the tolerance relations rough set model is to replace the missing attribute values by ‘*’ in the information table and ‘*’ may be any value. Therefore, for a decision information system IS = (U, AT ∪ {d}), where AT is the condition attributes, d is the decision attributes, in the case of any unknown attribute values is missing, Kryszkiewicz proposed the definition of tolerance relation [6].

Definition 3. Given incomplete decision system IS = (U, AT ∪ {d}), as for the property subset A ⊆ AT, null values are indicated by ‘*’, tolerance relation TOL (A) of A is defined as: $\begin{matrix} TOL (A) = {(x, y) \in U^{2} : \forall a \in A, \\ a (x) = a (y) \lor a (x) = * \lor a (y) = *} \end{matrix}$ (4)

Definition 4. Given incomplete decision system IS = (U, AT ∪ {d}), A ⊆ AT, X ⊆ U, in tolerance relation TOL (A), the lower and upper approximations of X are defined as [7]: $\underline{A_{T}} (X) = {x \in U : {TOL}_{A} (x) \subseteq X}$ (5) $\bar{A_{T}} (X) = {x \in U : {TOL}_{A} (x) \cap X \neq φ}$ (6)

Where in: TOL_A (x) = {y ∈ U : (x, y) ∈ TOL(A)} represents the tolerance class of object x with respect to set A.

2.3 Hybrid multi-granularity rough set model based on tolerance relation

In incomplete information system based on tolerance relation, the scholars of our country have proposed multi-granularity rough set model. From the perspective of granular computing [8 –10], multi-granularity rough set model derives a single grain structure is from a set of binary relations, which then extends to multiple grain structure. In this model, knowledge of multiple grain structure is used to approximate the concepts [11 –17].

Definition 5. Given incomplete decision system IS = (U, AT ∪ {d}), Let A₁, A₂, … , A_m be m attributes subsets of AT, in tolerance relations TOL (A₁) , TOL (A₂) , …, TOL (A_m) the optimistic multi-granular lower and upper approximations of X are defined as: $\begin{matrix} \underline{\sum_{i = 1}^{m} A_{i}^{o}} (X) = {x \in U : {TOL}_{A_{1}} (x) \subseteq X \\ \lor {TOL}_{A_{2}} (x) \subseteq X \lor \dots \lor {TOL}_{A_{m}} (x) \subseteq X} \end{matrix}$ (7) $\bar{\sum_{i = 1}^{m} A_{i}^{o}} (X) = \sim \underline{\sum_{i = 1}^{m} A_{i}^{o}} (\sim X)$ (8)

Definition 6. Given incomplete decision system IS = (U, AT ∪ {d}), Let A₁, A₂, …, A_m be m attributes subsets of AT, in tolerance relations TOL (A₁) , TOL (A₂) , …, TOL (A_m), the pessimistic multi-granular lower and upper approximations of X are defined as [18]: $\begin{matrix} \underline{\sum_{i = 1}^{m} A_{i}^{p}} (X) = {x \in U : {TOL}_{A_{1}} (x) \subseteq X \\ \lor {TOL}_{A_{2}} (x) \subseteq X \lor \dots \lor {TOL}_{A_{m}} (x) \subseteq X} \end{matrix}$ (9) $\bar{\sum_{i = 1}^{m} A_{i}^{p}} (X) = \sim \underline{\sum_{i = 1}^{m} A_{i}^{p}} (\sim X)$ (10)

Definition 7. Given incomplete decision system IS = (U, AT ∪ {d}), Let A₁, A₂, …, A_m be m attributes subsets of AT, and {TOL (A₁) , TOL (A₂) , …, TOL (A_m)} be tolerance relations.

If $\land_{i = 1}^{m_{1}} {TOL}_{1 A_{i}}, \land_{i = 1}^{m_{2}} {TOL}_{2 A_{i}}, \dots, \land_{i = 1}^{m_{J}} {TOL}_{{JA}_{i}}$ are J multi-granularity conjunctive spaces based on tolerance relations, then $\lor_{j = 1}^{J}$ $(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})$ is a hybrid multi-granularity space based on tolerance relations.

If $\lor_{i = 1}^{m_{1}} {TOL}_{1 A_{i}}, \lor_{i = 1}^{m_{2}} {TOL}_{2 A_{i}}, \dots, \lor_{i = 1}^{m_{J}} {TOL}_{{JA}_{i}}$ are J multi-granularity disjunctive spaces based on tolerance relations, then $\land_{j = 1}^{J}$ $(\lor_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})$ is a hybrid multi-granularity space based on tolerance relations.

In addition, depends on the property of the disjunctive operation and the conjunctive operation, it is worth noticing that a hybrid multi-granularity space can be turned into a multi-granularity space, which is constructed by the conjunction operation of several disjunction multi-granularity spaces. That means $\lor_{j = 1}^{J} (\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})$ and $\land_{j = 1}^{J} (\lor_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})$ can be converted to each other. Therefore, the definition of multi-granularity rough set just need to use one type of the hybrid multi-granularity spaces. This paper adopts hybrid multi-granularity space $\lor_{j = 1}^{J} (\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})$ to define hybrid multi-granularity rough set.

Definition 8. Given incomplete decision system IS = (U, AT ∪ {d}), Let A₁, A₂, …, A_m be mattributes subsets of AT, and {TOL (A₁) , TOL (A₂) , …, TOL (A_m)} be tolerance relations, the lower and upper approximations of the hybrid multi-granularity rough sets based on tolerance relations of X are defined as: $\begin{matrix} \underline{\lor_{j = 1}^{J} (\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{I}})} (X) = {x \in U : j \in {1, 2, \dots, J}, \\ i \in {1, 2, \dots, m_{j}}, [TOL]_{{jA}_{i}} \subseteq X} \end{matrix}$ (11) $\begin{matrix} \bar{\lor_{j = 1}^{J} (\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})} (X) = \sim \underline{\lor_{j = 1}^{J} (\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})} (\sim X) \end{matrix}$ (12)

3 Hybrid multi-granulation rough sets of variable precision based on tolerance relations

3.1 The definition of hybrid multi-granulation rough sets of variable precision based on tolerance relations

Definition 9. Given incomplete decision system IS = (U, AT ∪ {d}), Let A₁, A₂, … , A_m be m attributes subsets of AT, and {TOL (A₁) , TOL (A₂) , …, TOL (A_m)} be tolerance relations, the lower and upper approximations of the hybrid multi-granularity rough sets of variable precision based on variable precision tolerance relations of X are defined as: $\begin{matrix} \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{I}})}_{β}} (X) = {x \in U : j \in {1, 2, \dots, J}, \\ i \in {1, 2, \dots, m_{j}}, e ([TOL]_{1 A_{i}}, X) \leq β \\ \lor e ([TOL]_{2 A_{i}}, X) \leq β \lor \dots \lor e (TOL)_{{JA}_{i}}, X) \leq β} \end{matrix}$ (13) $\begin{matrix} \bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X) = \sim \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (\sim X) \end{matrix}$ (14)

As for the multi-granularity rough set of variable precision of X, its boundary set can be obtained through the upper and lower approximation set. $\begin{matrix} {Bn}_{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X) = \bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X) \\ - \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X) \end{matrix}$ (15)

Theorem 1.Given incomplete decision systemIS = (U, AT ∪ {d}), LetA₁, A₂, …, A_mbemattributes subsets ofAT, and {TOL (A₁) , TOL (A₂) , …, TOL (A_m)} be tolerance relations, ∀X ⊆ U $\begin{matrix} \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{I}})}_{β}} (X) = {x \in U : j \in {1, 2, \dots, J}, \\ i \in {1, 2, \dots, m_{j}}, e ([TOL]_{1 A_{i}}, X) < 1 - β \\ \lor e ([TOL]_{2 A_{i}}, X) < 1 - β \\ \lor \dots \lor e (TOL)_{{JA}_{i}}, X) < 1 - β} \end{matrix}$ (16)

Proof: From definition 9 it is known that $\begin{matrix} x \in \bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X) \Leftrightarrow \\ x \notin \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (\sim X) \Leftrightarrow \\ e ([TOL]_{1 A_{i}}, \sim X) > β \lor \\ e ([TOL]_{2 A_{i}}, \sim X) > β \lor \dots \lor \\ e ([TOL]_{JAi}, \sim X) > β \Leftrightarrow \\ 1 - \frac{| [TOL]_{1 A_{i}} \cap (\sim X) |}{| [TOL]_{1 A_{i}} |} > β \lor \dots \lor \\ 1 - \frac{| [TOL]_{{JA}_{i}} \cap (\sim X) |}{| [TOL]_{{JA}_{i}} |} > β \Leftrightarrow \\ 1 - \frac{| [TOL]_{1 A_{i}} - (\sim X) |}{| [TOL]_{1 A_{i}} |} > β \lor \dots \lor \\ 1 - \frac{| [TOL]_{{JA}_{i}} - (\sim X) |}{| [TOL]_{{JA}_{i}} |} > β \Leftrightarrow \\ 1 - 1 + \frac{| [TOL]_{1 A_{i}} \cap (\sim X) |}{| [TOL]_{1 A_{i}} |} > β \lor \dots \lor \\ 1 - 1 + \frac{| [TOL]_{{JA}_{i}} \cap (\sim X) |}{| [TOL]_{{JA}_{i}} |} > β \Leftrightarrow \\ e ([TOL]_{1 A_{i}}, X) < 1 - β \lor \dots \lor \\ e ([TOL]_{JAi}, X) < 1 - β \end{matrix}$

3.2 The properties of hybrid multi-granulation rough sets of variable precision based on tolerance relations

Theorem 2. Given incomplete decision system, IS = (U, AT ∪ {d}), Let A₁, A₂, …, A_m be m attributes subsets of AT, and {TOL (A₁) , TOL (A₂) , …, TOL (A_m)} be tolerance relations. The properties of the lower approximations $\underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X)$ and upper approximations $\bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X)$ are as follows:

$\begin{matrix} (1) \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X) \subseteq X \\ \subseteq \bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X); \end{matrix}$

$\begin{matrix} (2) \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (φ) = φ = \bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (φ), \\ \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (U) = U = \bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (U); \end{matrix}$

$\begin{matrix} (3) \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (\sim X) = \bar{\sim \lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X), \\ \bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (\sim X) = \sim \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X); \end{matrix}$

(4) $\begin{matrix} X \subseteq Y \Rightarrow \\ \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X) \subseteq \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β} (Y),} \\ \bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (Y) \subseteq \bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X); \end{matrix}$

$\begin{matrix} (5) \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (\underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X)) \\ = \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X), \\ \bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (\bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X)) \\ = \bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X); \end{matrix}$

$\begin{matrix} (6) \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X) = \cup_{j = 1}^{J} (\cap_{i = 1}^{m_{j}} \underline{{TOL}_{{jA}_{i}}^{β}} (X)), \\ \bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X) = \cup_{j = 1}^{J} (\cap_{i = 1}^{m_{j}} \bar{{TOL}_{{jA}_{i}}^{β}} (X)); \end{matrix}$

(7) $\begin{matrix} β_{1} \geq β_{2} \Rightarrow \\ \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β_{1}}} (X) \supseteq \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β_{2}}} (X) \Rightarrow \\ \bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β_{1}}} (X) \subseteq \bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β_{2}}} (X); \end{matrix}$

Proof:

(1) $\forall x \in \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X)$ , a conjunctive multi-granularity space must exists according to definition 7, ∀i = 1, 2, …, m_jand [TOL] _{jA
_i} ⊆ X. Since [TOL] _{jA
_i} is the tolerance relation cluster of x, so x ⊆ X and $\underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X) \subseteq X$ .

Similarly, it is easy to prove that $X \subseteq \bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X)$ .

(2) ∀x ∉ φ, there is ∀x ∈ U. For any conjunctive multi-granularity space $\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}}$ , and ∀i = 1, 2, …, m_j, since A is the tolerance relation of x, so that [TOL] _{jA
_i} ≠ φ ⇒ [TOL] _{jA
_i} ⊄ φ, which means $x \notin \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (φ)$ .

Similarly, it is easy to prove that $\bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (φ) = φ$ and $\underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}}$ $(U) = U = \bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (U)$ .

(3) The conclusion is clearly drawn according to the Definition 7, so the proof is omitted.

(4) $\forall x \in \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X)$ , a conjunctive multi-granularity space must exists according to Definition 7, ∀i = 1, 2, …, m_j and [TOL] _{jA
_i} ⊆ X. [TOL] _{jA
_i} ⊆ Ycan be derived from the condition X ⊆ Y, which represents that $x \in \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (Y)$ and $\underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}}$ $(X) \subseteq \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (Y) .$

Similarly, it is easy to prove that $\bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (Y) \subseteq \bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X)$ .

(5) $\underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X) \subseteq X$ can be gained according to property (1), along with $\underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (\underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X)) \subseteq \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X)$ can be obtained from property (4), therefore we just need to prove that $\underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (\underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X)) \supseteq \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X)$ , according to Definition 7 a conjunctive multi-granularity space must exists, in which ∀i = 1, 2, …, m_j, and [TOL] _{jA
_i} ⊆ X. ∀y ∈ [TOL] _{jA
_i}, [TOL] _{jA
_i} ⊆ Y exists, thus $y \in \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X)$ is established, so that $[TOL]_{{jA}_{i}} \subseteq \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X)$ and then $\underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (\underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X)) \supseteq \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X)$ .

Similarly, it is easy to prove that $\bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (\bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X)) = \bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X)$ .

(6) ∀x ∈ U, $x \in \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X) \Leftrightarrow \exists j \in {1, 2, \dots, J}, \forall i \in {1, 2, \dots, m_{j}}, [TOL]_{{jA}_{i}} \subseteq X$ $\Leftrightarrow \exists j \in {1, 2, \dots, J}, x \in (\cap_{i = 1}^{m_{j}} \underline{{TOL}_{{jA}_{i}}^{β}} (X) \Leftrightarrow x \in$ $\underline{\cup_{j = 1}^{J} (\cap_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}}^{β})} (X)$ .

Similarly, it is easy to prove that $\bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β}} (X) = \cup_{j = 1}^{J} (\cap_{i = 1}^{m_{j}} \bar{{TOL}_{{jA}_{i}}^{β}} (X))$ .

(7) For $\underline{\forall x \in \lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β_{2}}} (X)$ according to Definition 9, ∃A_i ∈ {A₁, A₂, …, A_m} which brings e ([TOL] _{JA
_i}, X) ≤ β₂. And causeβ₁ ≥ β₂, e ([TOL] _{JA
_i}, X) ≤ β₁ which means $\underline{\forall x \in \lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β_{2}}} (X)$ . Thus it is provablethat $\underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β_{1}}} (X) \supseteq$ $\underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β_{2}}}$ (X).

Similarly, it is easy to prove that $\bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β_{1}}} (X) \subseteq \bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})}_{β_{2}}}$ (X).

Theorem 2 represents some basic properties of multi-granularity rough set of hybrid variable precision. Property (1) describes the shrinkage property of the lower approximation and the expansion property of the upper approximation. Property (2) describes the normality property and remainder normality property of approximate set. Property (3) describes the duality of approximate set. Property (4) describes the monotonic property of approximate set. Property (5) describes the idempotence of approximate set. Property (6) describes the relations between multi-granularity rough set and single particle rough set. Property (7) illustrates that the different values corresponding variable precision hybrid multi-granularity relationship between lower and upper approximations [19 –24].

4 The example analysis

A given incomplete decision system IS = {U, AT ∪ {d}} is shown in Table 1, in which the set of objects U = {x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈, x₉, x₁₀} and the condition attribute set AT = {a₁₁, a₁₂, a₂₁, a₂₂}, β = 0.35, d denotes the decision attribute.

In Table 1, U/{d} = {D₁, D₂}, and D₁ = {x₁, x₂, x₆, x₈, x₉}, D₂ = {x₃, x₄, x₅, x₇, x₁₀}. $\begin{matrix} U / a_{11} = {{x_{1}, x_{7}, x_{9}}, {x_{2}, x_{3}, x_{4}, x_{5}, x_{10}}, {x_{8}}}, \\ U / a_{12} = {{x_{1}, x_{2}, x_{9}}, {x_{3}, x_{4}, x_{5}, x_{10}}, {x_{6}, x_{7}, x_{8}}}, \\ U / a_{21} = {{x_{1}, x_{2}}, {x_{2}, x_{3}, x_{4}, x_{5}, x_{6}}, {x_{7}, x_{8}, x_{9}}}, \\ U / a_{22} = {{x_{1}, x_{2}, x_{6}, x_{7}}, {x_{2}, x_{3}, x_{4}, x_{10}}, {x_{8}, x_{9}}} . \end{matrix}$

According to Definition 8 $\begin{matrix} \underline{\lor_{j = 1}^{J} (\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})} (X) = {x \in U : j \in {1, 2, \dots, J}, \\ i \in {1, 2, \dots, m_{j}}, [TOL]_{{jA}_{i}} \subseteq X} \\ \bar{\lor_{j = 1}^{J} (\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})} (X) = \sim \underline{\lor_{j = 1}^{J} (\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})} (\sim X); \end{matrix}$

We can get that $\begin{matrix} \underline{\lor_{j = 1}^{2} (\land_{i = 1}^{m_{j}} a_{ji})} (D_{1}) = φ \\ \underline{\lor_{j = 1}^{2} (\land_{i = 1}^{m_{j}} a_{ji})} (D_{2}) = φ \\ \bar{\lor_{j = 1}^{2} (\land_{i = 1}^{m_{j}} a_{ji})} (D_{1}) = U \\ \bar{\lor_{j = 1}^{2} (\land_{i = 1}^{m_{j}} a_{ji})} (D_{2}) = U \end{matrix}$

According to Definition 9 $\begin{matrix} \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{I}})}_{β}} (X) = {x \in U : j \in {1, 2, \dots, J}, \\ i \in {1, 2, \dots, m_{j}}, e ([TOL]_{1 A_{i}}, X) \leq β \\ \lor e ([TOL]_{2 A_{i}}, X) \leq β \lor \dots \lor e (TOL)_{{JA}_{i}}, X) \leq β} \\ \bar{\lor_{j = 1}^{J} (\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})} (X) = \sim \underline{\lor_{j = 1}^{J} (\land_{i = 1}^{m_{j}} {TOL}_{{jA}_{i}})} (\sim X) \end{matrix}$

We can get that $\begin{matrix} \underline{\lor_{j = 1}^{2} {(\land_{i = 1}^{m_{j}} a_{ji})}_{β}} (D_{1}) = {x_{1}, x_{2}, x_{7}, x_{8}, x_{9}} \\ {\underline{\lor_{j = 1}^{2} (\land_{i = 1}^{m_{j}} a_{ji})}}_{β} (D_{2}) = {x_{3}, x_{4}, x_{5}, x_{10}} \\ {\bar{\lor_{j = 1}^{2} (\land_{i = 1}^{m_{j}} a_{ji})}}_{β} (D_{1}) = {x_{1}, x_{2}, x_{6}, x_{7}, x_{8}, x_{9}} \\ {\bar{\lor_{j = 1}^{2} (\land_{i = 1}^{m_{j}} a_{ji})}}_{β} (D_{2}) = {x_{3}, x_{4}, x_{5}, x_{6}, x_{10}} \end{matrix}$

By way of example analysis shows, we got bigger lower approximation and smaller upper approximation, improve the accuracy of approximation. It gives new ideas and methods for incompletes system.

5 The discussion about hybrid multi-granularity rough sets of variable precision based on other relations in incomplete decision system

5.1 Hybrid multi-granularity rough sets of variable precision based on similarity relation

Definition 10. Given incomplete decision system IS = (U, AT ∪ {d}), Let A₁, A₂, …, A_m be m attributes subsets of AT, and {SIM (A₁) , SIM (A₂) , …, SIM (A_m)} be similarity relations, the lower and upper approximations of the hybrid multi-granularity rough sets of variable precision based on variable precision similarity relation of X are defined as:

$\begin{matrix} \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {SIM}_{{jA}_{i}})}_{β}} (X) = {x \in U : j \in {1, 2, \dots, J}, \\ i \in {1, 2, \dots, m_{j}}, e ([SIM]_{1 A_{i}}, X) \leq β \\ \lor e ([SIM]_{2 A_{i}}, X) \leq β \lor \dots \lor e ([SIM]_{{JA}_{i}}, X) \leq β} \end{matrix}$ (17)

$\begin{matrix} \bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {SIM}_{{jA}_{i}})}_{β}} (X) = \sim \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {SIM}_{{jA}_{i}})}_{β}} (\sim X) \end{matrix}$ (18) As for the multi-granularity rough set of variable precision of X, its boundary set can be obtained through the upper and lower approximation set. $\begin{matrix} {Bn}_{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {SIM}_{{jA}_{i}})}_{β}} (X) = \bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {SIM}_{{jA}_{i}})}_{β}} (X) \\ - \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {SIM}_{{jA}_{i}})}_{β}} (X) \end{matrix}$ (19)

Theorem 3.Given incomplete decision systemIS = (U, AT ∪ {d}), LetA₁, A₂, …, A_mbemattributes subsets ofAT, ∀X ⊆ U

$\begin{matrix} \bar{\lor_{j = 1}^{J} (\land_{i = 1}^{m_{j}} {SIM}_{{jA}_{i}})_{β}} (X) = {x \in U : j \in {1, 2, \dots, J}, \\ i \in {1, 2, \dots, m_{j}}, e ([SIM]_{1 A_{i}}, X) < 1 - β \\ \lor e ([SIM]_{2 A_{i}}, X) < 1 - β \lor \dots \lor \\ e ([SIM]_{{JA}_{i}}, X) < 1 - β} \end{matrix}$ (20)

Proof: $\begin{matrix} x \in \bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {SIM}_{{jA}_{i}})}_{β}} (X) \Leftrightarrow \\ x \notin \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {SIM}_{{jA}_{i}})}_{β}} (\sim X) \Leftrightarrow \end{matrix}$ $\begin{matrix} e ([SIM]_{1 A_{i}}, \sim X) > β \lor \dots \lor e ([SIM]_{{JA}_{i}}, \sim X) > β} \Leftrightarrow \\ 1 - \frac{| [SIM]_{1 A_{i}} \cap (\sim X) |}{| [SIM]_{1 A_{i}} |} > β \lor \dots \lor \\ 1 - \frac{| [SIM]_{{JA}_{i}} \cap (\sim X) |}{| [SIM]_{{JA}_{i}} |} > β \Leftrightarrow \\ 1 - \frac{| [SIM]_{1 A_{i}} - [SIM]_{1 A_{i}} \cap X |}{| [SIM]_{1 A_{i}} |} > β \lor \dots \lor \\ 1 - \frac{| [SIM]_{{JA}_{i}} - [SIM]_{{JA}_{i}} \cap X |}{| [SIM]_{{JA}_{i}} |} > β \Leftrightarrow \\ 1 - 1 + \frac{| [SIM]_{1 A_{i}} \cap X |}{| [SIM]_{1 A_{i}} |} > β \lor \dots \lor \\ 1 - 1 + \frac{| [SIM]_{{JA}_{i}} \cap X |}{| [SIM]_{{JA}_{i}} |} > β \Leftrightarrow \\ e ([SIM]_{1 A_{i}}, X) < 1 - β \lor \\ e ([SIM]_{2 A_{i}}, X) < 1 - β \lor \dots \lor \\ e ([SIM]_{JAi}, X) < 1 - β \end{matrix}$

5.2 Hybrid multi-granularity rough sets of variable precision based on limited tolerance relation

Definition 11. Given incomplete decision system IS = (U, AT ∪ {d}), Let A₁, A₂, …, A_m be m attributes subsets of AT, and LTOL (A₁) , LTOL (A₂) , …, LTOL (A_m) be lim-ited tolerance relations, the lower and upper approximations of the hybrid multi-granularity rough sets of variable precision based on variable precision limited tolerance relation of X are defined as: $\begin{matrix} \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {LTOL}_{{jA}_{i}})}_{β}} (X) = {x \in U : j \in {1, 2, \dots, J}, \\ i \in {1, 2, \dots, m_{j}}, e ([LTOL]_{1 A_{i}}, X) \leq β \\ \lor e ([LTOL]_{2 A_{i}}, X) \leq β \lor \dots \lor e ([LTOL]_{{JA}_{i}}, X) \leq β} \end{matrix}$ (21) $\bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {LTOL}_{{jA}_{i}})}_{β}} (X) = \sim \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {LTOL}_{{jA}_{i}})}_{β}} (\sim X)$ (22)

As for the multi-granularity rough set of variable precision of X, its boundary set can be obtained through the upper and lower approximation set. $\begin{matrix} {Bn}_{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {LTOL}_{{jA}_{i}})}_{β}} (X) = \bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {LTOL}_{{jA}_{i}})}_{β}} (X) \\ - \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {LTOL}_{{jA}_{i}})}_{β}} (X) \end{matrix}$ (23)

Theorem 4.Given incomplete decision systemIS = (U, AT ∪ {d}), Let A₁, A₂, …, A_mbemattributes subsets ofAT,∀X ⊆ U $\begin{matrix} \bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {LTOL}_{{jA}_{i}})}_{β}} (X) = {x \in U : j \in {1, 2, \dots, J}, \\ i \in {1, 2, \dots, m_{j}}, e ([LTOL]_{1 A_{i}}, X) < 1 - β \\ \lor e ([LTOL]_{2 A_{i}}, X) < 1 - β \\ \lor \dots \lor e ([LTOL]_{{JA}_{i}}, X) < 1 - β} \end{matrix}$ (24)Proof: $\begin{matrix} x \in \bar{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {LTOL}_{{jA}_{i}})}_{β}} (X) \Leftrightarrow \\ x \notin \underline{\lor_{j = 1}^{J} {(\land_{i = 1}^{m_{j}} {LTOL}_{{jA}_{i}})}_{β}} (\sim X) \Leftrightarrow \\ e ([LTOL]_{1 A_{i}}, \sim X) > β \lor \dots \lor e ([LTOL]_{{JA}_{i}}, \sim X) > β} \Leftrightarrow \\ 1 - \frac{| [LTOL]_{1 A_{i}} \cap (\sim X) |}{| [LTOL]_{1 A_{i}} |} > β \lor \dots \lor \\ 1 - \frac{| [LTOL]_{{JA}_{i}} \cap (\sim X) |}{| [LTOL]_{{JA}_{i}} |} > β \Leftrightarrow \\ 1 - \frac{| [LTOL]_{1 A_{i}} - [LTOL]_{1 A_{i}} \cap X |}{| [LTOL]_{1 A_{i}} |} > β \lor \dots \lor \\ 1 - \frac{| [LTOL]_{{JA}_{i}} - [LTOL]_{{JA}_{i}} \cap X |}{| [LTOL]_{{JA}_{i}} |} > β \Leftrightarrow \\ 1 - 1 + \frac{| [LTOL]_{1 A_{i}} \cap X |}{| [LTOL]_{1 A_{i}} |} > β \lor \dots \lor \\ 1 - 1 + \frac{| [LTOL]_{{JA}_{i}} \cap X |}{} | [LTOL]_{{JA}_{i}} | > β \Leftrightarrow \\ e ([LTOL]_{1 A_{i}}, X) < 1 - β \lor \dots \lor e ([LTOL]_{JAi}, X) < 1 - β \end{matrix}$

6 Conclusion

Hybrid multi-granularity space is constructed considering both Disjunctive relation and Conjunctive relation. In this paper, the misclassification rate method was used to extend multi-granularity rough set to incomplete system based on tolerance relation, the hybrid multi granularity rough set model variable precision based on tolerance relation is put forward. Owing to the variable precision method, the hybrid multi-granularity rough set model has a bigger lower approximation and a smaller upper approximation than classical rough sets and improves the approximate accuracy. Besides, this paper discusses about hybrid multi-granulation rough sets of variable precision in incomplete decision system based on similarity relation and limited tolerance relation.

Footnotes

Acknowledgments

This work is supported by the Fundamental Research Funds for the Central Universities (lzujbky-2012-43). The authors would like to thank the reviewers for their constructive comments and suggestions.

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