Types Of Covering-based Rough Sets Research Articles

Certain types of coverings based rough sets with application

Covering-based rough sets are important generalizations of the classical rough sets of Pawlak. In this paper, by means of j-neighborhoods, complementary j-neighborhoods and j-adhesions, we build some new different types of j-covering approximations based rough sets and study related properties. Also, we explore the relationships between the considered j-covering approximations and investigate the properties of them. Using different neighborhoods, some different general topologies are generated as topologies induced from a binary relation. Finally, an interesting application of the new types of covering-based rough sets to the rheumatic fever is given.

Characterizations and applications of parametric covering-based rough sets

As a technique for granular computing, rough sets deal with the vagueness and granularity in information systems. Covering-based rough sets are natural extensions of the classical rough sets by relaxing the partitions to coverings and have been applied in many fields. However, many vital issues in covering-based rough sets, including attribute reduction, are NP-hard and therefore the algorithms for addressing them are usually greedy. Hence, it is necessary and helpful to generalize the covering-based rough sets from different viewpoints. In this paper, a new type of covering-based rough sets, named parametric covering-based rough sets, are proposed and some properties and applications of the parametric covering-based rough sets are investigated. First, a concept of inclusion degree is introduced into covering-based rough set theory to explore some properties of the parametric covering approximation space. Second, the parametric covering-based rough sets are established on the basis of inclusion degree. Moreover, some properties of the parametric covering-based rough sets are introduced. Third, it is found that the calculations of corresponding parametric covering-based lower and upper approximations can be converted into the operations of matrices, which makes the calculations convenient. Finally, a simple application of the parametric covering-based rough sets to network security is introduced.

The rough membership functions on four types of covering-based rough sets and their applications

Pawlak’s rough membership functions not only give numerical characterizations of Pawlak’s rough set approximations, but also establishes the relationships between Pawlak’s rough sets and fuzzy sets or probabilistic rough sets. However, it is noteworthy that Pawlak’s rough membership functions have limitations when handling incomplete data that exist widely in the real world. As will be shown in this paper, one way to overcome this is to construct rough membership functions for covering-based rough sets. In this paper, we first use an example in evidence-based medicine to illustrate how to use Pawlak’s rough membership function on numerically characterizing decisions under the circumstances where data are complete. Then, we construct covering-based rough membership functions for four types of covering-based rough sets which were examined by Zhu and Wang (in IEEE Transactions on Knowledge and Data Engineering 19(8)(2007) 1131-1144 and Information Sciences 201(2012) 80-92), and use them to characterize these covering-based rough sets numerically. Finally, we present theoretical backgrounds for these covering-based rough membership functions, and illustrate how to apply them on numerically characterizing decisions under the circumstances where data are incomplete.

On twelve types of covering-based rough sets.

Covering approximation spaces are a generalization of equivalence-based rough set theories. In this paper, we will consider twelve types of covering based approximation operators by combining four types of covering lower approximation operators and three types of covering upper approximation operators. Then, we will study the properties of these new pairs and show they have most of the common properties among existing covering approximation pairs. Finally, the relation between these new pairs is studied.

Relationships between generalized rough sets based on covering and reflexive neighborhood system

Rough set theory is an important tool to deal with uncertainty, granularity and incompleteness of knowledge. The covering-based rough sets is one of the most important extensions of the classical Pawlak rough sets. In covering-based rough set theory, many types of rough sets were already defined in the literature. To find out the relationships among different types of covering-based rough sets and give an axiomatic definition is a very important issue. The generalized rough sets in neighborhood system is another important extension of the classical Pawlak rough sets. In this paper, we investigate relationships between covering-based rough sets and generalized rough sets in neighborhood system. Then, twenty-three types of the element based, the granule based and the subsystem based definitions of covering approximation operators are unified under the framework of generalized approximation operators in neighborhood systems. Moreover, properties of the generalized approximation operators in neighborhood systems are presented.

The relationship among different covering approximations

This paper studies covering-based rough sets by means of characteristic functions of sets. We consider covering-based rough sets from both constructive and axiomatic approaches. The relationship among four types of covering-based rough sets is discussed. We also outline the topologies induced by different covering approximations.

Properties of two types of covering-based rough sets

Rough set theory is a tool to deal with the vagueness and granularity in information systems. The core concepts of classical rough sets are lower and upper approximations based on equivalence relations, or partitions. Recently, some sufficient and necessary conditions were given in Zhu (Proceedings of the first international workshop granular computing and brain informatics (Gr BI’06), IEEE international conference on web intelligence (WI 06), pp. 494–497, 2006), Zhu and Wang (Proceedings of the IEEE international conference on data mining (ICDM’06) workshop foundation of data mining and novel techniques in high dimensional structural and unstructured data, pp. 407–411, 2006), (IEEE Trans Knowl Data Eng, 19, pp 1131–1144, 2007, Proceedings of the sixth international conference on machine learning and cybernetics, 33), under which a lower approximation operator and an upper approximation operator satisfy certain classical properties. In this paper, we give a counterexample to show that the condition given in Theorem 13 in (Proceedings of the first international workshop granular computing and brain informatics (Gr BI’06), IEEE international conference on web intelligence (WI 06), pp. 494–497, 2006) is not necessary. The correct formulation is stated. We also give other conditions for a covering, under which certain classical properties hold for the second and third types of covering-based lower and upper approximation operators.

Relationships between covering-based rough sets and relation-based rough sets

Rough set theory is an important technique to deal with vagueness and granularity in information systems. In rough set theory, relation-based rough sets and covering-based rough sets are two important extensions of the classical rough sets. This paper investigates relationships between relation-based rough sets and covering-based rough sets. Firstly, we unify five types of covering-based rough sets under a framework of a type of relation-based rough sets, and establish the equivalences between four types of covering-based rough sets and the type of relation-based rough sets, respectively. Then, within the proposed framework, conditions required for two different coverings to generate the same approximation operators are investigated. We also present some sufficient and necessary conditions for different types of covering-based rough sets to be equal. Furthermore, two examples are taken to illustrate applications of the discussion about relationships between covering-based rough sets and relation-based rough sets.

The fourth type of covering-based rough sets

As a technique for granular computing, rough sets deal with the vagueness and granularity in information systems. Covering-based rough sets have been proposed to generalize this theory for wider application. Three types of covering-based rough sets have been studied for different situations. To make the theory more complete, this paper proposes a fourth type of covering-based rough sets. Compared with the existing ones, the new type shows its special characteristic in the interdependency between its lower and upper approximations. We carry out a systematical study of this new theory. First, we discuss basic properties such as normality, contraction, and monotone. Then we investigate the conditions for this type of covering-based rough sets to satisfy the properties of Pawlak’s rough sets and study the interdependency between the lower and upper approximation operations. In addition, axiomatic systems for the lower and upper approximation operations are established. Lastly, we address the relationships between this type of covering-based rough sets and the three existing ones.

Topological Approaches to the Second Type of Covering-Based Rough Sets

Rough sets, a tool for data mining, deal with the vagueness and granularity in information systems. This paper studies covering-based rough sets from the topological view. We explore the relationship between the relative closure and the second type of covering upper approximation. The major contributions of this paper are that we use the definition of the relative closure and the relative interior to discuss the conditions under which the relative operators satisfy certain classical properties. The theorems we get generalize some of the results in Zhu’s paper.

Relationship between generalized rough sets based on binary relation and covering

Rough set theory is a powerful tool for dealing with uncertainty, granularity, and incompleteness of knowledge in information systems. This paper systematically studies a type of generalized rough sets based on covering and the relationship between this type of covering-based rough sets and the generalized rough sets based on binary relation. Firstly, we present basic concepts and properties of this kind of rough sets. Then we investigate the relationships between this type of generalized rough sets and other five types of covering-based rough sets. The major contribution in this paper is that we establish the equivalency between this type of covering-based rough sets and a type of binary relation based rough sets. Through existing results in binary relation based rough sets, we present axiomatic systems for this type of covering-based lower and upper approximation operations. In addition, we explore the relationships among several important concepts such as minimal description, reduction, representative covering, exact covering, and unary covering in covering-based rough sets. Investigation of this type of covering-based will benefit to our understanding of other types of rough sets based on covering and binary relation.

On Three Types of Covering-Based Rough Sets

Rough set theory is a useful tool for data mining. It is based on equivalence relations and has been extended to covering-based generalized rough set. This paper studies three kinds of covering generalized rough sets for dealing with the vagueness and granularity in information systems. First, we examine the properties of approximation operations generated by a covering in comparison with those of the Pawlak's rough sets. Then, we propose concepts and conditions for two coverings to generate an identical lower approximation operation and an identical upper approximation operation. After the discussion on the interdependency of covering lower and upper approximation operations, we address the axiomization issue of covering lower and upper approximation operations. In addition, we study the relationships between the covering lower approximation and the interior operator and also the relationships between the covering upper approximation and the closure operator. Finally, this paper explores the relationships among these three types of covering rough sets.