Approximation Set of the Interval Set in Pawlak's Space

Zhang, Qinghua; Wang, Jin; Wang, Guoyin; Hu, Feng

doi:https://doi.org/10.1155/2014/317387

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Theory and Application on Rough Set, Fuzzy Logic, and Granular Computing

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Research Article | Open Access

Volume 2014 | Article ID 317387 | https://doi.org/10.1155/2014/317387

Approximation Set of the Interval Set in Pawlak's Space

Qinghua Zhang,^1,2Jin Wang,¹Guoyin Wang,¹and Feng Hu¹

Academic Editor: Xibei Yang

Received21 May 2014

Revised19 Jul 2014

Accepted19 Jul 2014

Published11 Aug 2014

Abstract

The interval set is a special set, which describes uncertainty of an uncertain concept or set with its two crisp boundaries named upper-bound set and lower-bound set. In this paper, the concept of similarity degree between two interval sets is defined at first, and then the similarity degrees between an interval set and its two approximations (i.e., upper approximation set () and lower approximation set ()) are presented, respectively. The disadvantages of using upper-approximation set () or lower-approximation set () as approximation sets of the uncertain set (uncertain concept) are analyzed, and a new method for looking for a better approximation set of the interval set is proposed. The conclusion that the approximation set () is an optimal approximation set of interval set is drawn and proved successfully. The change rules of () with different binary relations are analyzed in detail. Finally, a kind of crisp approximation set of the interval set is constructed. We hope this research work will promote the development of both the interval set model and granular computing theory.

1. Introduction

Since the twenty-first century, researchers have done more and more research on uncertain problems [1]. It is an important research topic on how to effectively deal with uncertain data and how to acquire more knowledge and rules from the big data. At the same time, many methods for acquiring uncertain knowledge from uncertain information systems appeared gradually. In 1965, fuzzy sets theory was proposed by Zadeh [2]. In 1982, rough sets theory was proposed by Pawlak [3]. In 1990, quotient space theory was presented by L. Zhang and B. Zhang [4]. In 1993, interval sets and interval sets algebra were presented by Yao [5, 6].

Rough set theory is a mathematical tool to handle the uncertain information, which is imprecise, inconsistent, or incomplete. The basic thought of rough set is to obtain concepts and rules through classification of relational database and discover knowledge by the classification induced by equivalence relations; then approximation sets of the target concept are obtained with many equivalence classes. Rough set is a useful tool to handle uncertain problems, as well as fuzzy set theory, probability theory, and evidence theory. Because rough set theory has novel ideas and its calculation is easy and simple, it has been an important technology in intelligent information processing [7–9]. The key issue of rough set is building a knowledge space which is a partition of the domain and is induced by an equivalence relation. In the knowledge space, two certain sets named upper approximation set and lower approximation set are used to describe the target concept as its two boundaries. If knowledge granularity in knowledge space is coarser, then the border region of described target concept is wider and approximate accuracy is relatively lower. On the contrary, if knowledge granularity in knowledge space is finer, then the border region is narrower and approximate accuracy is relatively higher.

The interval set theory is an effective method for describing ambiguous information [10–12] and can be used in uncertain reasoning as well as the rough set [13–15]. The interval set not only can be used to describe the partially known concept, but also can be used to study the approximation set of the uncertain target concept. So, the interval set is a more general model for processing the uncertain information [16]. The interval set is described by two sets named upper bound and lower bound [17]. The elements in lower bound certainly belong to target concept, and the elements in upper bound probably belong to target concept. When the boundary region has no element, the interval set degenerates into a usual set [5], while, in a certain knowledge granularity space, target concept may be uncertain. To solve this problem, in this paper, the approximate representation of interval set is discussed in detail in Pawlak’s approximation space. And then, the upper approximation set of interval set and lower approximation set of interval set are defined, respectively. The change rules of the approximation set of interval set with the different knowledge granularity in Pawlak’s approximation space are analyzed.

In this paper, an approximation set of the target concept is built in a certain knowledge space induced by many conditional attributes, and we find that this approximation set may have better similarity degree with the target concept than that of or . Therefore, an interval set is translated into a fuzzy set at first in this paper. And then, according to the different membership degrees of different elements in boundary region, an approximation set of interval set is obtained by cut-set with some threshold. And then, the decision-making rules can be obtained through the approximation set instead of in current knowledge granularity space. In addition, the change rules of similarity between a target concept and its approximation sets are analyzed in detail.

The method used is getting the approximation of interval sets with a special approximation degree. With this method, we can use certain sets to describe an interval set in Pawlak’s space. Our motivation is to get a mathematical theory model, which can be helpful to promote interval sets development in knowledge acquisition.

The rest of this paper is organized as follows. In Section 2, the related basic concepts and preliminary knowledge are reviewed. In Section 3, the concept of similarity degree between two interval sets is defined. The approximation set of interval set and 0.5-approximation set are proposed in Section 4. The change rules of similarity degree between the approximation sets and the target concept with the different knowledge granularity spaces are discussed in Section 5. This paper is concluded in Section 6.

2. Preliminaries

In order to introduce the approximation set of interval set more easily, many basic concepts will be reviewed at first.

Definition 1 (interval set [17]). An interval set is a new collection, and it is described by two sets named upper bound and lower bound. The interval set can be defined as follows. Let be a finite set which is called universal set, and then let be the power set of and let interval set be a subset of . In mathematical form, interval set is defined as . If , is a usual classical set.

In order to better explain the interval set, there is an example [17, 18] as follows. Let be all papers submitted to a conference. After being reviewed, there are 3 kinds of results. The first kind of results is the set of papers certainly accepted and represented by . The second kind of results is the set of papers that need to be further reviewed and represented by . The last kind of results is the set of papers rejected and represented by . Although every paper just can be rejected or accepted, no one knows the final result before further evaluation. Through reviewing, the set of papers accepted by the conference is described as .

Definition 2 (indiscernibility relation [4, 19]). For any attribute set , let us define one unclear binary relationship .

Definition 3 (information table of knowledge expression system [4, 20]). A knowledge expression system can be described as . is the domain, and is the set of all attributes. Subset is a set of conditional attributes, and is a set of decision-making attributes. is the set of attribute values. describes the range of attribute values where . is an information function which describes attribute values of object in .

Definition 4 (upper approximation set and lower approximation set of rough set [3]). A knowledge-expression system is described as . For any and , upper approximation set and lower approximation set of rough set on are defined as follows: where is the classification of equivalence relation on . Upper approximation set and lower approximation set of rough set on can be defined in another form as follows: where and is an equivalence class of on relation . is a set of objects which certainly belong to according to knowledge ; is a set of objects which possibly belong to according to knowledge . Let be called boundary region of target concept on relation . Let be called positive region of target concept on relation . Let be called negative region of target concept on relation . is a set of objects which just possibly belong to target concept .

Definition 5 (similarity degree between two sets [20]). Let and be two subsets of domain , which means . Defining a mapping , that is, , is the similarity degree between and , if satisfies the following conditions.(1)For any (boundedness).(2)For any (symmetry).(3)For any if and only if .

Any formula satisfying (1), (2), and (3) is a similarity degree formula between two sets. Zhang et al. [20] gave out a similarity degree formula where represents the number of elements in finite subset. Obviously, this formula satisfies (1), (2), and (3).

Definition 6 (similarity degree between two interval sets). Let be an interval set and let be also an interval set. Similarity degree between two interval sets can be defined as follows: accords with Definition 5.

Definition 7 (upper approximation set and lower approximation set of an interval set). Let be an interval set. Let be an equivalence relation on domain . Upper approximation set of this interval set is defined as . Lower approximation set of this interval set is defined as .

Figures 1 and 2 are probably helpful to understand Definition 7. In Figure 1, the outer circle standing for a set and inner circle standing for a set represent an interval set , and each block represents an equivalence class. The black region represents , and the whole colored region (black and gray region) represents . In Figure 2, the outer circle standing for a set and inner circle standing for a set represent an interval set , and each block represents an equivalence class. The black region represents , and the whole colored region (black and gray region) represents .

3. Approximation Set of an Interval Set

If stands for the upper approximation set of the interval set , then the similarity degree between and can be defined as follows: If stands for the lower approximation set of the interval set , then the similarity degree between and is defined as follows:

If the knowledge space keeps unchanged, is there a better approximation set of the target concept ? In this paper, the better approximation sets of target concept will be proposed. Let be a nonempty set of objects. Let , , and the membership degree of belonging to set is defined as Obviously, .

Definition 8 (-approximation set of set [20]). Let be a nonempty set of objects, and let knowledge space be . Let , and the membership degree belonging to set is If , then is called -approximation set of set .

Definition 9 (-approximation set of set ). Let and ; then is called -approximation set of the interval set .

Figure 3 is probably helpful to understand Definition 9. In Figure 3, the outer circle standing for a set and inner circle standing for a set represent an interval set , and each block represents an equivalence class. The black region represents , and the whole colored region (black and gray region) represents .

4. Approximation Set of an Interval Set

Lemma 10 (see [20]). Let , and be all real numbers. If , then .

Lemma 11 (see [20]). Let , and be all real numbers. In the numbers, . If , then . If , then .

In order to better understand the similarity degree between and , Theorems 12 and 13 are presented as follows.

Theorem 12. Let be a finite domain, let be an interval set on , and let be an equivalence relation on . Then, .

For example, let , , . Then, , , , , , .

And then we can have , , , .

Proof. According to Definition 6,
(1) There we first prove For all , we have . That is,
Because is an equivalence relation on , the classifications induced by can be denoted as . Then, < . Obviously, , and then let . So, . Because the intersection sets between any two elements in are empty sets, we can get that Because and the intersection set between any two elements in , is empty, we have that + . So,
Because we have . In the same way, according to and Lemma 10, we can easily get Therefore,
(2) In a similar way with (1), we can have the inequality
From (1) and (2), we have . So, Theorem 12 has been proved completely.

Theorem 12 shows that the similarity degree between an interval set and its approximation set is better than the similarity degree between and its lower approximation set .

Theorem 13. Let be a finite domain, let be an interval set on , and let be an equivalence relation on . If then .

For example, let , , . Then, , , , , , ,

And then we can have ,, .

Proof. According to Definition 6,
(1) Let , and the intersection sets between any two elements in are empty sets. Because it is obvious that . In the same way, we have . Then we have − . Because the intersection sets between any two elements in are empty sets, we have , , and ∪. Because the intersection sets between any two elements in are empty sets, and are held. So,
For according to Lemma 11, we have that is to say, Therefore, we have
(2) In a similar way with (1), we can easily obtain the conclusion that when
According to (1) and (2), the inequality is held. So, Theorem 13 has been proved successfully.

Theorem 13 shows that, under some conditions, the similarity degree between an interval set and its approximation set is better than the similarity degree between and its lower approximation set .

Theorem 14. Let be a finite domain, an interval set on , and an equivalence relation on . If , then .

For example, let ,, . Then, , , , ,, , and , .

And then we can have , , , . This example is in accordance with the theorem.

Proof. (1) For all , then , which means Because < , we can easily get . Let and < , and then we can get . To simplify the proof, let and in this paper. So, . Because the intersection sets between any two elements in are empty sets, we can easily get that + + . And and we have
And because + , according to Lemma 10, the inequality is held. Therefore, we have
(2) The inequality can be easily proved in a similar way to (1).
According to (1) and (2), the inequality is held.

Based on Theorems 13 and 14, Corollary 15 can be obtained easily as follows.

Corollary 15. Let be a finite domain, an interval set on , and an equivalence relation on . . If then .

Theorem 16. Let be a finite domain, an interval set on , and an equivalence relation on . if , then .

For example, , , . Then, , , , , , , , .

And then we can have , , . This example is in accordance with the theorem.

Proof. (1) For any , we have Because is an equivalence relation on , all the classifications induced by can be denoted by . We have and as well as . We can also get . Let , where and . So, ∪, . And because the intersection sets between any two elements in , are empty sets, we have According to ∪ and because the intersection sets between any two elements in , , are empty sets, we easily have Therefore, According to we have . According to , we have + . And based on Lemma 10, we can easily have Therefore,
(2) In the same way as (1), the inequality is held.
According to (1) and (2), the inequality is held. So, the proof of Theorem 16 has been completed successfully.

Theorems 14 and 16 show that the similarity degree between an interval set and its approximation set is a monotonically decreasing function with the parameter , and the similarity degree reaches its maximum value when .

5. The Change Rules of Similarity in Different Knowledge Granularity Spaces

In different Pawlak’s approximation spaces with different knowledge granularities, the change rules of the uncertainty of rough set are a key issue [21, 22]. Many researchers try to discover the change rules of uncertainty in rough set model [23, 24]. And we also find many change rules of uncertain concept in different knowledge spaces in our other papers [20]. In this paper, we continue to discuss the change rules of the similarity degree in Pawlak’s approximation spaces with different knowledge granularities. In this paper, we focus on discussing how the similarity degree between and changes when the granules are divided into more subgranules in Pawlak’s approximation space. In other words, it is an important issue concerning how changes with different knowledge granularities in Pawlak’s approximation space.

Let be classifications of under equivalence relation . Let be classifications of under equivalence relation . If , then (). And then, is called a refinement of , which is written as . If , then . And then, is called a strict refinement of , which is written as .

Next, we will analyze the relationship between and . Let ; in other words, for all is always satisfied, and . And then, there must be two or more granules in whose union is . To simplify the proof, we suppose that there is just only one granule which is divided into two subgranules, denoted by and in , and other granules keep unchanged.

There are 9 cases, and only 6 cases are possible.

Theorem 17. Let be a finite domain, an interval set on , and and two equivalence relations on . Let be one granule which is divided into two subgranules marked as and . If Then, .

Proof. There are 6 possible cases which will be discussed one by one as follows.(1) is contained in both positive region of and positive region of . In this case, obviously is held.(2) is contained in both positive region of and negative region of . In this case, obviously, is held.(3) is contained in both negative region of and negative region of . In this case, obviously, is held.(4) is contained in both negative region of and boundary region of . In this case, is held obviously. Next, we discuss the relationship between
Let . Let where . When is in boundary region of , we should further discuss this situation. To simplify the proof, we suppose that there is just only one granule marked as in which is divided into two subgranules marked as and in . And the other granules keep unchanged.(a)If , then .(1)If , . From the proof of Theorem 12, we know Because , , and , we have For , , which means . According to Lemma 10, we have (2)If , then Because , the case that and is impossible.(b)If , then .(1)If and , then we can easily have (2)If and , then Because , we have the following.(i)If , then we have and . Therefore, (ii)If , then . Therefore, Because according to Lemma 11, we have (iii)If and , because and , we have Because according to Lemma 11, we have
According to (a) and (b) above, we have when (5) is contained in boundary region of and positive region of . In this case, Next, we discuss the relationship between
Similar to (a) and (b) in (4), when we can get So, in the condition, we can draw a conclusion that .(6) is contained in boundary region of and boundary region of .
According to the proofs of (4) and (5), if and , we easily have .
From (1), (2), (3), (4), (5), and (6), Theorem 17 is proved successfully.

Theorem 17 shows that, under some conditions, the similarity degree between an interval set and its approximation set is a monotonically increasing function when the knowledge granules in are divided into many finer subgranules in , where is a refinement of .

6. Conclusion

With the development of uncertain artificial intelligence, the interval set theory attracts more and more researchers and gradually develops into a complete theory system. The interval set theory has been successfully applied to many fields, such as machine learning, knowledge acquisition, decision-making analysis, expert system, decision support system, inductive inference, conflict resolution, pattern recognition, fuzzy control, and medical diagnostics systems. It is an important tool of granular computing as well as the rough set which is one of the three main tools of granular computing [25, 26]. In the interval set theory, the target concept is approximately described by two certain sets, that is, the upper bound and lower bound. In other words, the essence of this theory is that we deal with the uncertain problems with crisp set theory method. Many researches have been completed on extended models of the interval set, but the theories nearly cannot present better approximation set of the interval set . In this paper, the approximation set of target concept in current knowledge space is proposed from a new viewpoint and related properties are analyzed in detail.

In this paper, the interval set is transformed into a fuzzy set at first, and then the uncertain elements in boundary region are classified by cut-set with some threshold. Next, the approximation set of the interval set is defined and the change rules of in different knowledge granularity spaces are analyzed. These researches show that is a better approximation set of than both and . Finally, a kind of crisp approximation set of interval set is proposed in this paper. These researches present a new method to describe uncertain concept from a special viewpoint, and we hope these results can promote the development of both uncertain artificial intelligence and granular computing and extend the interval set model into more application fields. It is an important research issue concerning discovering more knowledge and rules from the uncertain information [27]. The fuzzy set and the rough set have been used widely [28–32]. Recently, the interval set theory is applied to many important fields, such as software testing [33], the case generation based on interval combination [34], and incomplete information table [35–38]. In the future research, we will focus on acquiring the approximation rules from uncertain information systems based on the approximation sets of an interval set.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (nos. 61272060, 61309014, and 61379114), The Natural Science Foundation of Chongqing of China (nos. cstc2012jjA40047, cstc2012jjA40032, and cstc2013jcyjA40063), and the Science & Technology Research Program of Chongqing Education Committee of China (no. KJ130534).

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Copyright © 2014 Qinghua Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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