#### Abstract

In the paper proposed two new types of the multiple granulation rough set models, where a target concept is approximated from two different kinds of views by using the equivalence classes induced by multiple granulations. A number of important properties of the two types of MGRS are investigated. From the properties, it can be found that Pawlak’s and Qian’s rough set models are special instances of those of our MGRS. Moreover, several important measures are presented in two types of MGRS, such as rough measure and quality of approximation. Furthermore, the relationship and difference are discussed carefully among Pawlak’s rough set, Qian’s MGRS, and two new types of MGRS. In order to illustrate our multiple granulations rough set model, some examples are considered, which are helpful for applying this theory in practical issues. One can get that the paper is meaningful both in the theory and in application for the issue of knowledge reduction in complex information systems.

#### 1. Introduction

Rough set theory proposed by Pawlak [1–3] is an extension of the classical set theory and can be regarded as a soft computing tool to handle imprecision, vagueness, and uncertainty in data analysis. The theory has found its successive applications in the fields of pattern recognition [4], medical diagnosis [5], data mining [6–8], conflict analysis [9], algebra [10–12], and so on. Recently, the theory has generated a great deal of interest among more and more researchers.

The classical rough set theory is based upon the classification mechanism, from which the classification can be viewed as an equivalence relation, and knowledge granule induced by the equivalence relation can be viewed as a partition of the universe of discourse. In rough set theory, two classical sets, so-called lower and upper approximations or Pawlak’s rough approximations, a constructed and any subset of a universe of discourse can be expressed by them. In framework based on rough set theory, an attribute set is viewed as a granular space, which partitions the universe into some knowledge granules or elemental concepts. Partition, granulation, and approximation are the methods widely used in human reasoning [13, 14]. Rough set methodology presents a novel paradigm to deal with uncertainty and has been applied to feature selection [15], knowledge reduction [16–19], rule extraction [20–23], uncertainty reasoning [24, 25], and granular computing [26–31].

In practice, due to the existence of uncertainty and complexity of particular problem, the problem would not be settled perfectly by means of classical rough set. Therefore, it is vital to generalize the classical rough set model. To overcome this limitation, classical rough sets have been accomplished to several interesting and meaningful general models in recent years, which include rough set model based on tolerance relations [32], rough set model based on neighborhood operators [33], Bayesian rough set model [34], fuzzy rough set model [35], rough fuzzy set model [35], and fuzzy probabilistic rough set model [3, 9, 19, 36–45].

On the other hand, information granules have played a significant role in human cognitive processes. Information granules refer to pieces, classes, and groups divided in accordance with characteristics and performances of complex information in the process of human understanding, reasoning, and decision making. Such information processing is called the information granulation. Zadeh firstly proposed and discussed the issue of fuzzy information granulation [46] in 1979. Then, the basic idea of information granulation has been applied to many fields, such as theory of rough sets [1, 2, 47], fuzzy sets [14, 48], and evidence theories [49], and a growing number of scholars are concerned about the discipline. In 1985, Hobbs proposed the concept of granularity [50]. Zadeh firstly presented the concept of granular computing [46] in the period from 1996 to 1997. At this time, granular computing has played a more and more important role in soft computing, knowledge discovery, and data mining, and ones have achieved a large amount of excellent results [19, 27, 30, 51–58].

However, in essence, the approximations in the past approaches are still based on a singleton granulation induced from an indiscernibility relation, which can be applied to knowledge representation in distributive systems and groups of intelligent agents. In view of granular computing, an equivalence relation on the universe can be regarded as a granulation, and a partition on the universe can be regarded as a granulation space [56, 59]. Hence, the classical rough set theory is based on a single granulation (only one equivalence relation). Note that any attribute set can induce a certain equivalence relation in an information system. In the literature, to more widely apply the rough set theory in practical applications, Qian et al. [60] extended Pawlak’s single-granulation rough set model to a multi granulation rough set model (MGRS), where the set approximations are defined by multiple equivalence relations on the universe.

The main objective of this paper is to extend Pawlak’s single-granulation rough set model and Qian’s multigranulation rough set model (MGRS) to two new types of multiple granulation rough set model, where the set approximations are defined by using multiple equivalence relations on the universe. The rest of the paper is organized as follows. Some preliminary concepts in Pawlak’s rough set theory and main concepts in Qian’s MGRS are briefly reviewed in Section 2. In Sections 3 and 4, for an information system, based on multiple equivalence relations, two new types of multiple granulation rough set model are obtained, respectively, where a target concept is approximated from two different kinds of views by using the equivalence classes induced by multiple granulations. And a number of important properties of the two types of MGRS are investigated. It is shown that some of the properties of Pawlak’s and Qian’s rough set theory are special instances of those of our MGRS. Several important measures are presented in two types of MGRS, such as rough measure, quality of approximation. In Section 5, the relationship and difference are discussed among Pawlak’s rough set, Qian’s MGRS, and two new types of MGRS. Furthermore, to illustrate our multiple granulations rough set model, some examples are presented, which are helpful for applying this theory in practical issues. And finally, the paper is concluded by a summary and outlook for further research in Section 6.

#### 2. Preliminaries

The following recalls necessary concepts and preliminaries required in the sequel of our work. Detailed description of the theory can be found in the literatures [1–3, 19, 60].

##### 2.1. Pawlak’s Rough Set

The notion of information system (sometimes called data tables, attribute-value systems, knowledge representation systems, etc.) provides a convenient tool for the representation of objects in terms of their attribute values.

An information system is an ordered quadruple , where (i) is a nonempty finite set of objects, (ii) is a nonempty finite set of attributes, (iii) and is a domain of attribute , (iv) is a function such that , for every , , called an information function.

An information system with the decision is a special case of information systems in which, among the attributes, we distinguish the ones called decision attribute. The other attributes are called condition attributes. Therefore, and , where sets and are the condition attributes and the decision attribute, respectively.

Let be an information system, for , and denote then is reflexive, symmetric, and transitive. So it is an equivalence relation on .

Moreover, denote then is called the equivalence class of , and is called the equivalence class set of . For equivalence classes and equivalence relations, the following properties hold. (i)If , then . (ii)If , then . (iii)If , then .

For any subset and in information system , the Pawlak’s lower and upper approximations of with respect to equivalence relation could be defined as follows (see [1–3, 19]): The area of uncertainty or boundary is defined as

To measure the imprecision and roughness of a rough set, Pawlak recommended that has the ratio which is called the rough measure of by equivalence .

Furthermore, for an information system with the decision and , a frequently applied measure for the situation is the quality of approximation of by , also called the degree of dependency. It is defined as where , and .

##### 2.2. Qian’s Multiple Granulation Rough Set

In the rough set model MGRS, unlike Pawlak’s rough set theory, a target concept is approximated through multiple partitions induced by multiple equivalence relations [60]. In the following, we recall the relevant concepts about MGRS, and the description in detail can be found in [60].

Suppose that is an information system, and are -attribute subsets. A lower approximation and an upper approximation of are related to by the following:

Similarly, the boundary region in MGRS can be expressed as Figure 1 shows the difference between Pawlak’s rough set model and the MGRS model.

From the figure, we can find that the bias region is the lower approximation of a set obtained by a single granulation , which are expressed by the equivalence classed in the quotient set , and the shadow region is the lower approximation of induced by two granulations , which are characterized by the equivalence classes in quotient set and the quotient set .

#### 3. The First Type of Multiple Granulation Rough Set

In this section, we will propose the first type of multiple granulation rough set.

##### 3.1. The First Type of Two Granulation Rough Set

We first discuss the first type of two granulation approximations of a target set by using two equivalence relations in an information system.

*Definition 1. *Let be an information system and . The operators and are defined as follows. For all ,
We call them the first type of two granulation lower and upper approximation operators, and we call and the first type of two granulation lower approximation set and upper approximation of , respectively.

Moreover, if , we say that is the first type of rough set with respect to the two granulation spaces and . Otherwise, we say that is the first type of a definable set with respect to the two granulation spaces and .

The area of uncertainty or boundary region of this rough set is defined as

It can be found that the first two granulation rough set will be Pawlak’s rough set when two granulation spaces and satisfy . To describe conveniently in our context, we express the first type of two granulation rough set by using the 1st TGRS.

In Qian’s MGRS, the upper approximation set is characterized by the complementary set of the lower approximation of the complementary target concept , which is not consistent with Pawlak’s rough set. However, by the previous definition, it can be seen that the 1st TGRS lower and upper approximations are consistent with Pawlak’s rough set. Furthermore, ones can find that the 1st TGRS lower and upper approximations are defined through using the equivalence classes induced by multiequivalence relations in an information system, whereas Pawlak’s lower and upper approximations are represented via those derived by only on equivalence relation.

One can understand the first two granulation rough set and show the difference between the Qian’s MGRS and Pawlak’s rough set through Figure 2.

Just from Definition 1, we can obtain the following properties in the 1st TGRS in an information system.

Proposition 2. *Let be an information system, and . Then the following properties hold.*(FL_{1}) (*Contraction*),(FU_{1}) (*Extension*),(FL_{2}) (*Duality*),(FU_{2}) (*Duality*),(FL_{3}) (*Normality*),(FU_{3}) (*Normality*),(FL_{4}) (*Conormality*),(FU_{4}) (*Conormality*),(FL_{5}) (*Commutativity*),(FU_{5}) (*Commutativity*).

*Proof. *It is obvious that all terms hold when . When , the proposition can be proved as follows.

() For any , it can be known that or by Definition 1. However, and . So we can have . Hence, .

() For any , we have and . So and , that is to say, . Hence, .

() For any , then
Hence, .

() By (), we have . So it can be obtain that .

() From (), we have . besides, it is well known that . So, .

() If , then there must exist a . So we can find that and . Obviously, this is a contradiction. Thus, .

() .

() .

() and () can be proved directly by Definition 1.

In order to discover the relationship between the 1st TGRS approximations of a single set and the 1st TGRS approximations of two sets described on the universe, the following properties are given.

Proposition 3. *Let be an information system, and . Then the following properties hold.*(FL_{6}) (*L-Multiplication*),(FU_{6}) (*L-Addition*),(FL_{7}) (*Granularity*),(FU_{7}) (*Granularity*),(FL_{8}) (*U-Addition*),(FU_{8}) (*U-Multiplication*).

*Proof. *It is obvious that all terms hold when or . When and , the proposition can be proved as follows.

() For any , we have that or by Definition 1. Then, it can be obtained that and hold at the same time or and hold at the same time. So, not only or hold, but or hold at the same time. That is to say that and , that is, .

Hence, .

() For any , we have or . Then and hold at the same time or and hold at the same time. So, not only hold, but hold. That is to say .

Hence, .

() Since , one can have . Then, . Besides, it can be found that by (). So, we can obtain that , that is to say that .

Thus, .

() Since , one can have . Then, . In addition, it can be found that by (). So, we can obtain that , that is to say that .

Thus, .

() Since and , by () it can be obtained that
So, we have .

() Since and , by () it can be obtained that
So, we have .

The proposition was proved.

Here, we employ an example to illustrate the previos concepts and properties with respect to the 1st TGRS.

*Example 4. *Table 1 depicts an information system containing some information about an emporium investment project. *Locus, investment,* and *population density *are the conditional attributes of the systems, whereas *Decision* is the decision attribute. In the sequel, , and will stand for locus, investment, population density and , respectively. The domains are as follows: B—Bad, C—Common, G—Good, —Better}, H—High, L—Low}, B—Big, S—Small, M—Medium}, and Y—Yes, N—No}.

From the table, we can find that
And, if we take , then by computing we have

However, the lower approximation and upper approximation of based on Pawlak’s rough set are

Obviously, one can check the following properties.

##### 3.2. The First Type of Multiple Granulation Rough Set

In this subsection, we will consider the first type of multiple granulation approximations of a target set by using multiple equivalence relations in an information system.

*Definition 5. *Let be an information system, be attribute subsets (), and be equivalence relations, respectively. The operators and are defined as follows. For all ,
where “” means “some”, and “” means “all.” We call them the first type of multiple granulation lower and upper approximation operators, and we call and the first type of multiple granulation lower approximation set and upper approximation of , respectively.

Moreover, if , we say that is the first type of rough set with respect to multiple granulation spaces . Otherwise, we say that is the first type of definable set with respect to these multiple granulation spaces.

Similarly, the area of uncertainty or boundary region of this rough set is defined as

To describe conveniently the ideas in our context, we express the first type of multiple granulation rough set by using the 1st MGRS. Moreover, one can obtain the following properties of the 1st MGRS approximations.

Proposition 6. *Let be an information system, , , and . Then the following properties hold.*(FL_{1}) (*Contraction*),(FU_{1}) (*Extension*),(FL_{2}) (*Duality*),(FU_{2}) (*Duality*),(FL_{3}) (*Normality*),(FU_{3}) (*Normality*),(FL_{4}) (*Conormality*),(FU_{4}) (*Conormality*),(FL_{5}) (*Commutativity*),(FU_{5}) (*Commutativity*).

*Proof. *The proof of these items is similar to Proposition 2.

Proposition 7. *Let be an information system, , and . Then the following properties hold.*(FL_{6}) (*L-Multiplication*),(FU_{6}) (*L-Addition*),(FL_{7}) (*Granularity*),(FU_{7}) (*Granularity*),(FL_{8}) (*U-Addition*),(FU_{8}) (*U-Multiplication*).

*Proof. *The proof of these items is similar to Proposition 3.

Next, we will investigate several elementary measures in the 1st MGRS and their properties.

Uncertainty of a set (category) is due to the existence of a borderline region. The bigger the borderline region of a set is, the lower the accuracy of the set is (and vice versa). To more precisely express this idea, we introduce the accuracy measure to the 1st MGRS as follows.

*Definition 8. *Let be an information system, , , and . The 1st rough measure of by is defined as
where .

From the definitions, one can derive the following properties.

Proposition 9. *Let be an information system, , , and . Then
*

*Proof. *By Corollary 32, we have
And by Corollary 40, we have
So, the following hold:
Hence, by the Definition 8, we have

The proof was completed.

*Example 10 (continued from Example 4). * Computing the 1st rough measures of by using the results in Example 4, it follows that
Clearly, it follows from the earlier computation that

Note that the rough measure of a target concept defined by using multiple granulations is always much better than that defined by using a single granulation, which is suitable for more precisely characterizing a target concept and problem solving according to user requirements.

*Definition 11. *Let be a decision table, , , and, be all decision classes induced by decision attribute . Approximation quality of by , called the 1st degree of dependence, is defined as

This measure can be used to evaluate the deterministic part of the rough set description of by counting those objects which can be reclassified as blocks of with the knowledge given by . Moreover, we have the following properties with respect to the above definition.

Proposition 12. *Let be a decision table, , , and let be all decision classes induced by decision attribute . Then
*

*Proof. *For every , by Corollaries 32 and 40, we have

So,
Hence, by Definition 11, we have

The proof was completed.

*Example 13 (continued from Example 4). *Computing the degree of dependence of .

From Table 1, we can have and

Moreover, the following can be computed by Table 1 and the results of Example 4,

So, we have

Hence, it can be found that

#### 4. The Second Type of Multiple Granulation Rough Set

In this section, we will consider another multiple granulation rough set.

##### 4.1. The Second Type of Two Granulation Rough Set

We first discuss the second type of two granulation approximations of a target set by using two equivalence relations in an information system.

*Definition 14. *Let be an information system and . The operators and are defined as follows. For all ,
We call them the second type of two granulation lower and upper approximation operators, and we call and the second type of two granulation lower approximation set and upper approximation of , respectively.

Moreover, if , we say that is the second type of rough set with respect to two granulation spaces and . Otherwise, we say that is the second type of definable set with respect to two granulation spaces and .

The area of uncertainty or boundary region of this rough set is defined as

It can be found that the second two granulation rough set will be Pawlak’s rough set when two granulation spaces and satisfy . To describe conveniently in our context, we express the second type of two granulation rough set by using the 2nd TGRS.

By the previos definition, it can be seen that the 2nd TGRS lower and upper approximations are consistent with Pawlak’s rough set. Furthermore, one can find that the 2nd TGRS lower and upper approximations are defined through using the equivalence classes induced by multi equivalence relations in an information system, whereas Pawlak’s lower and upper approximations are represented via those derived by only one equivalence relation. And the 2nd TGRS lower and upper approximations are dual with the 2nd TGRS lower and upper approximations.

One can understand the second two granulation rough set and show the difference between the 1st TGRS and Pawlak’s rough set through Figure 3.

Just from Definition 14, we can obtain the following properties in the 2nd TGRS in an information system.

Proposition 15. *Let be an information system, and . Then the following properties hold.*(SL_{1}) (*Contraction*),(SU_{1}) (*Extension*),(SL_{2}) (*Duality*),(SU_{2}) (*Duality*),(SL_{3}) (*Normality*),(SU_{3}) (*Normality*),(SL_{4}) (*Conormality*),(SU_{4}) (*Conormality*),(SL_{5}) (*Commutativity*),(SU_{5}) (*Commutativity*).

*Proof. *It is obvious that all terms hold when . When , the proposition can be proved as follows.

() For any , it can be known that and by Definition 14. However, and . So we can have . Hence, .

() For any , we have and . So and , which imply that . Hence, .

() For any , then
Hence, .

() By (), we have . So it can be obtained that .

() From (), we have . Besides, it is well known that . So, .

() If , then there must exist a . So, we can find that or . Obviously, this is a contradiction. Thus, .

() .

() .

() and () can be proved directly by Definition 14.

In order to discover the relationship between the 2nd TGRS approximations of a single set and the 2nd TGRS approximations of two sets described on the universe, the following properties are given.

Proposition 16. *Let be an information system, and . Then the following properties hold.*(SL_{6}) (*L-Multiplication*),(SU_{6}) (*L-Addition*),(SL_{7}) (*Granularity*),(SU_{7}) (*Granularity*),(SL_{8}) (*U-Addition*),(SU_{8}) (*U-Multiplication*).

*Proof. *It is obvious that all terms hold when or . When and , the proposition can be proved as follows.

() For any , by Definition 18 we have
Hence, .

() For any , by Definition 18 we have
Hence, .

() Since , one can have . Then, . Besides, it can be found that by (). So, we can obtain that , that is to say that .

() Since , one can have . Then, . Besides, it can be found that by (). So, we can obtain that , that is to say that .

() Since and , by () it can be obtained that
So, we have .

() Since and , by () it can be obtained that
So, we have .

The proposition was proved.

*Example 17 (continued from Example 4). *In Example 4, we have known that
And, if we take again, then by computing we have

However, the lower approximation and upper approximation of based on Pawlak’s rough set were obtained in Example 4 as follows:

So, one can check the following properties.

##### 4.2. The Second Type of Multiple Granulation Rough Set

In this subsection, we will consider the second type of multiple granulation approximations of a target set by using multiple equivalence relations in an information system.

*Definition 18. *Let be an information system, attribute subsets (), and equivalence relations, respectively. The operators and are defined as follows. For all ,
where “” means “some” and “” means “all.” We call them the second type of multiple granulation lower and upper approximation operators, and we call and the second type of multiple granulation lower approximation set and upper approximation of , respectively.

Moreover, if , we say that is the second type of rough set with respect to multiple granulation spaces . Otherwise, we say that is the second type of definable set with respect to these multiple granulation spaces.

Similarly, the area of uncertainty or boundary region of this rough set is defined as

To describe conveniently in our context, we express the second type of multiple granulation rough set by using the 2nd MGRS. Moreover, one can obtain the following properties of the 2nd MGRS approximations.

Proposition 19. *Let be an information system, , , and . Then the following properties hold.
*

*Proof. *The proof of these items is similar to Proposition 15.

Proposition 20. *Let be an information system, , , and . Then the following properties hold. *

*Proof. *The proof of these items is similar to Proposition 16.

Next, we will investigate several elementary measures in the 2nd MGRS and their properties.

Similarly, we introduce the accuracy measure to the 2nd MGRS as follows.

*Definition 21. *Let be an information system, , and . The 2nd rough measure of by is defined as
where .

From the definitions, one can derive the following properties.

Proposition 22. *Let be an information system, , , and . Then
*

*Proof. *By Corollary 34, we have
And, we have known that
So, the following holds: