Granularity In Information Systems Research Articles

A novel incremental attribute reduction approach for incomplete decision systems

With the increasing diversification and complexity of information, it is vital to mine effective knowledge from information systems. In order to extract information rapidly, we investigate attribute reduction within the framework of dynamic incomplete decision systems. Firstly, we introduce positive knowledge granularity concept which is a novel measurement on information granularity in information systems, and further give the calculation method of core attributes based on positive knowledge granularity. Then, two incremental attribute reduction algorithms are presented for incomplete decision systems with multiple objects added and deleted on the basis of positive knowledge granularity. Furthermore, we adopt some numerical examples to illustrate the effectiveness and rationality of the proposed algorithms. In addition, time complexity of the two algorithms are conducted to demonstrate their advantages. Finally, we extract five datasets from UCI database and successfully run the algorithms to obtain corresponding reduction results.

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.

On characterizations of a pair of covering-based approximation operators

Granular computing is an emerging computing paradigm of information processing. Rough set theory is a kind of models of granular computing and is used to deal with the vagueness and granularity in information systems. Covering-based rough set theory is one of the most important extensions of the classical Pawlak rough set theory. In the covering-based rough set theory, the covering lower and upper approximation operators are two basic concepts. The axiomatic characterizations of covering-based approximation operators guarantee the existence of coverings reproducing the operators, so the rough set axiomatic system is the foundation of the covering-based rough set theory. In this paper, we present a new covering upper approximation operator and explore the basic properties of the covering upper approximation operator to find an axiomatic set for characterizing the covering upper approximation operator. We also discuss the relationships between the covering upper approximation operator and three other covering upper approximation operators, compare the covering upper approximation operator with the other covering upper approximation operators by defining precision degrees and rough degrees, and present the necessary and sufficient conditions for the covering upper approximation operators to be identical. The results not only are beneficial to enrich the kinds of the covering rough sets, but also have theoretical and actual significance to covering rough sets.

Roughness and fuzziness in quantales

Rough set theory and fuzzy set theory are two useful mathematical tools for dealing with uncertainty and granularity in information systems. Motivated by the studies of roughness and fuzziness in algebraic systems and partially ordered sets such as semigroups, rings and lattices, in this paper we introduce first the notions of fuzzy (prime, semi-prime, primary) ideals of quantales and investigate their properties. Several characterizations of such ideals are presented. Then, we introduce the concepts of weak fuzzy prime ideals and strong fuzzy prime ideals of quantales, and establish relationships between these ideals and fuzzy prime ideals of quantales. By applying rough set theory to fuzzy ideals of quantales, we furthermore define rough fuzzy (prime, semi-prime, primary) ideals of quantales, generalizing Yang and Xu’s work on quantales to the fuzzy environment. Finally, relationships between the upper (resp. lower) rough fuzzy ideals of quantales and the upper (resp. lower) approximations of their homomorphic images are also discussed.

A variable precision covering-based rough set model based on functions.

Classical rough set theory is a technique of granular computing for handling the uncertainty, vagueness, and granularity in information systems. Covering-based rough sets are proposed to generalize this theory for dealing with covering data. By introducing a concept of misclassification rate functions, an extended variable precision covering-based rough set model is proposed in this paper. In addition, we define the f-lower and f-upper approximations in terms of neighborhoods in the extended model and study their properties. Particularly, two coverings with the same reductions are proved to generate the same f-lower and f-upper approximations. Finally, we discuss the relationships between the new model and some other variable precision rough set models.

Approximation operators on complete completely distributive lattices

Rough set, a tool for data mining, deals with the vagueness and granularity in information systems. In 2006, Chen et al. initiated the study of rough approximations on a complete completely distributive lattice (CCD lattice for short) and brought generalizations of rough sets into a unified framework. In this paper, we discuss the approximation operators on a CCD lattice. Based on the concept of neighborhood, three kinds of upper approximation operators and a kind of lower approximation operator are constructed. Basic properties of lower and upper approximation operators are examined. Furthermore, the relationships among these operators are analyzed.

Applications of Matrices to a Matroidal Structure of Rough Sets

Rough sets provide an efficient tool for dealing with the vagueness and granularity in information systems. They are widely used in attribute reduction in data mining. There are many optimization issues in attribute reduction. Matroids generalize the linear independence in vector spaces and are widely used in optimization. Therefore, it is necessary to integrate rough sets and matroids. In this paper, we apply matrices to a matroidal structure of rough sets through three sides, which are characteristics, operations, and axioms. Firstly, a matroid is induced by an equivalence relation, and the matroid is a representable matroid whose representable matrix is a matrix representation of the equivalence relation. Then some characteristics of the matroid are presented through the representable matrix mainly. Secondly, contraction and restriction operations are applied to the matroid through the representable matrix and approximation operators of rough sets. Finally, two axioms of circuit incidence matrices of 2-circuit matroids are obtained, where 2-circuit matroids are proposed based on the characteristics of the matroid. In a word, these results show an interesting view to investigate the combination between rough sets and matroids through matrices.

Covering-Based Rough Sets on Eulerian Matroids

Rough set theory is an efficient and essential tool for dealing with vagueness and granularity in information systems. Covering-based rough set theory is proposed as a significant generalization of classical rough sets. Matroid theory is a vital structure with high applicability and borrows extensively from linear algebra and graph theory. In this paper, one type of covering-based approximations is studied from the viewpoint of Eulerian matroids. First, we explore the circuits of an Eulerian matroid from the perspective of coverings. Second, this type of covering-based approximations is represented by the circuits of Eulerian matroids. Moreover, the conditions under which the covering-based upper approximation operator is the closure operator of a matroid are presented. Finally, a matroidal structure of covering-based rough sets is constructed. These results show many potential connections between covering-based rough sets and matroids.

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.

Roughness in quantales

Rough set theory is a useful mathematical tool for dealing with the uncertainty and granularity in information systems. In this paper, rough approximations are introduced into quantales, a kind of partially ordered algebraic structure with an associative binary multiplication. The notions of (upper, lower) rough (prime, semi-prime, primary) ideals of quantales are proposed and verified to be the extended notions of usual (prime, semi-prime, primary) ideals of quantales respectively. Global order properties of all lower (upper) rough ideals of a quantale are also investigated. The rough radical of an upper rough ideal in a quantale is defined and the prime radical theorem of rings is generalized to upper rough ideals of quantales. Some results about homomorphic images of rough ideals and rough prime ideals of semigroups and rings are also generalized and improved in the field of quantales.

Model of Covering Rough Sets in the Machine Intelligence

Rough set theory has been proposed by Pawlak as a useful tool for dealing with the vagueness and granularity in information systems. Classical rough set theory is based on equivalence relation. The covering rough sets are an improvement of Pawlak rough set to deal with complex practical problems which the latter one can not handle. This paper studies covering-based generalized rough sets. In this setting, we investigate common properties of classical lower and upper approximation operations hold for the covering-based lower and upper approximation operations and relationships among some type of covering 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.

On minimization of axiom sets characterizing covering-based approximation operators

Rough set theory was proposed by Pawlak to deal with the vagueness and granularity in information systems. The classical relation-based Pawlak rough set theory has been extended to covering-based generalized rough set theory. The rough set axiom system is the foundation of the covering-based generalized rough set theory, because the axiomatic characterizations of covering-based approximation operators guarantee the existence of coverings reproducing the operators. In this paper, the equivalent characterizations for the independent axiom sets of four types of covering-based generalized rough sets are investigated, and more refined axiom sets are presented.

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.

On Granularity in Information Systems Based on Binary Relation

In this paper, some important issues of granularity are discussed mainly in information systems (ISs) based on binary relation. Firstly, the vector representation method of knowledge granules is proposed in an infor-mation system based on binary relation to eliminate limitations of set representation method. Secondly, operators among knowledge granularity are introduced and some important properties of them are studied carefully. Thirdly, distance between two knowledge granules is established and granular space is constructed based on it. Fourthly, axiomatic definition of knowledge granularity is investigated, and one can find that some existed knowledge granularities are special cases under the definition. In addition, as an application of knowledge granular space, an example is employed to validate some results in our work.

The Topological Properties of the Approximation Operators on A CCD Lattice

Rough sets, a tool for data mining, deal with the vagueness and granularity in information systems. Rough approximations on a complete completely distributive lattice(CCD lattice for short) and brings generalizations of rough sets into a unified framework are discussed in [3]. This paper is devoted to the discussion of the relationship between approximations and topologies on a CCD lattice. It is proved that the set of all upper approximations (or of lower approximations) with respect to a partition consists of a clopen topology; and conversely, a clopen topology which obey disjoint axiom can be induced by approximations. Furthermore, the axiomatic characterizations of upper and lower approximations are presented.

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.

Generalized rough sets based on relations

Rough set theory has been proposed by Pawlak as a tool for dealing with the vagueness and granularity in information systems. The core concepts of classical rough sets are lower and upper approximations based on equivalence relations. This paper studies arbitrary binary relation based generalized rough sets. In this setting, a binary relation can generate a lower approximation operation and an upper approximation operation, but some of common properties of classical lower and upper approximation operations are no longer satisfied. We investigate conditions for a relation under which these properties hold for the relation based lower and upper approximation operations. This paper also explores the relationships between the lower or the upper approximation operation generated by the intersection of two binary relations and those generated by these two binary relations, respectively. Through these relationships, we prove that two different binary relations will certainly generate two different lower approximation operations and two different upper approximation operations.

Topological approaches to covering 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 topological properties of this type of rough sets, study the interdependency between the lower and the upper approximation operations, and establish the conditions under which two coverings generate the same lower approximation operation and the same upper approximation operation. Lastly, axiomatic systems for the lower approximation operation and the upper approximation operation are constructed.