Covering-based Approximation Operators Research Articles

Properties of approximation operators over 1-neighborhood systems from the perspective of special granules

As generalizations of Pawlak-neighborhood systems, 1-neighborhood systems with symmetry or transitivity are closely related to both partition spaces and covering spaces. In this article, we analyze the properties of a single covering-based approximation operator on symmetric or transitive 1-neighborhood systems. We also investigate the relationships between different covering-based approximation operators on them. Theoretically, we illuminate some necessary and sufficient conditions for 1-neighborhood systems being symmetric, transitive, or partitions with one or two approximation operators. To reduce potential computation complexity owing to these equivalent characterizations, objects dealt by approximation operators in this work are three particular kinds of granules, namely, points of universes, elements of 1-neighborhood systems, and cores of 1-neighborhood systems. As experimental results indicate, this study outdoes some related works in terms of computational efficiency, establishing the advantages of computing on these granules. Furthermore, our research has resulted in a solution to a problem posed by Yun et al. (Axiomatization and conditions for neighborhoods in a covering to form a partition. Information Sciences 181(2011)1735–1740).

Accuracy of approximation operators during covering evolutions

We express our concern over two issues in this paper: one is which characters are foundational for multitudinous covering-based rough models as generalizations of Pawlak's rough sets, i.e., which properties of covering-based rough sets guarantee the consistency of them and Pawlak's over partitions; the other is how to utilize this equivalence in simplifying diverse covering-based approximation operators. We demonstrate that covering-based rough models are equivalent to Pawlak's on partitions if they satisfy granule selection principles, which are weaker than a combination of contraction, extension, monotonicity, addition and granularity. In order to take advantage of their equivalence, we illustrate a method named “covering evolution” to change granules from given coverings to corresponding partitions for covering-based approximation operators. The evolutions can be divided into three steps: at the beginning, coverings are transformed into 1-neighborhood systems based on some quintessential neighborhood operators, then in the middle, more-refined 1-neighborhood systems are built by using “cores” from general topology, and at last, core systems (in fact, partitions) are extracted from these more-refined 1-neighborhood systems. We lay a strong emphasis on the variations in accuracy of six representative covering-based approximation operators during three evolutions, two of which preserve the accuracy of approximation operators. The investigation carried on covering evolutions helps us to establish the corresponding mapping relationship from covering spaces to partition spaces directly, and therefore provides a convenient method for making choice of covering-based approximation operators as they are consistent with the classical Pawlak's rough sets over partitions.

Axiomatization of covering-based approximation operators generated by general or irreducible coverings

The axiomatization problems for generalized approximation operators are finding logical characters of these operators. Solving these problems is important not only for conceptual understanding generalized approximation operators, but also for theoretic discussing the properties of them. In recent years, many researchers explored and developed the axiomatic approach of generalized rough set theory and a lot of articles about axiomatizations of approximation operators were published. However, there are numbers of axiomatization problems for covering-based approximation operators are still open. In this article, we give axiomatic systems for several types of covering-based approximation operators. All the axiomatic systems given by us are original and one of our results answers an open problem raised by Zhu and Wang in 2007. We also discuss axiomatization problems for those covering-based approximation operators generated by irreducible coverings. We prove that for most types of operators, the axiomatic systems generated by irreducible coverings and by general coverings are the same. On the other hand, we show that for one type of operators, the axiomatic systems generated by irreducible coverings and by general coverings are different, and we give an axiomatic system for this type of operators generated by irreducible coverings. By several examples, we show that for each axiomatic system presented in this paper, conditions are independent of each other.

Dicovering approximation spaces and definability

In this work, we define a category Cap of covering approximation spaces whose morphisms are functions satisfying a refinement property. We give the relations among Cap, and the category Top of topological spaces and continuous functions, and the category Rere of reflexive approximation spaces and the relation preserving functions. Further, we discuss the textural versions diCap, dfDitop and diRere of these categories. Then we study the definability in Cap with respect to five covering-based approximation operators. In particular, it is observed that via the morphisms of Cap, we may get more information about the subsets of the universe.

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.

Matrix representations and interdependency on L-fuzzy covering-based approximation operators

In this paper, we study the matrix representations and interdependency of three pairs of L-fuzzy covering-based approximation operators. Matrices play an important role in machine learning and data mining, such as the Laplacian matrix of a graph in clustering analysis and matrix linear transformation in feature selection based on principle component analysis. Therefore, the matrix representations of the three pairs of L-fuzzy covering-based approximation operators are important problems which should be solved. Moreover, we give some necessary and sufficient conditions under which two L-fuzzy coverings to generate the same L-fuzzy covering-based approximation operators.

Notes on covering-based rough sets from topological point of view: Relationships with general framework of dual approximation operators

In the recent article “On some types of covering rough sets from topological points of view” [14], the author develops a topological approach to covering-based rough sets. In this context, a number of corresponding approximation operators are introduced, their inclusion relationships are verified, and various conditions under which the operators coincide are proven.On the other hand, a lot of effort has recently been dedicated by several authors to study covering-based approximation operators within a general framework of dual approximation operators [2,3,7–10,12].In this note, we study correspondences between the framework of Zhao and the framework established in [2]. In particular, we evaluate how the newly introduced topological approximation operators relate to existing ones in terms of equalities and partial order relations.

The axiomatic characterizations on L-fuzzy covering-based approximation operators

Axiomatic characterization is the foundation of L-fuzzy rough set theory: the axiom sets of approximation operators guarantee the existence of L-fuzzy relations or L-fuzzy coverings that reproduce the approximation operators. Axiomatic characterizations of approximation operators based on L-fuzzy coverings have not been fully explored, although those based on L-fuzzy relations have been studied thoroughly. Focusing on three pairs of widely used L-fuzzy covering-based approximation operators, we establish an axiom set for each of them, and their independence is examined. It should be noted that the axiom set of each L-fuzzy covering-based approximation operator is different from its crisp counterpart, with an either new or stronger axiom included in the L-fuzzy version.

The characterizations of upper approximation operators based on special coverings

Abstract In this paper, we discuss the approximation operators ${\overline {apr} _{NS}}$ and ${\overline {apr} _S}$ which are based on NS(U) and S. We not only obtain some properties of NS(U) and S, but also give examples to show some special properties. We also study sufficient and necessary conditions when they become closure operators. In addition, we give general and topological characterizations of the covering for two types of covering-based upper approximation operators being closure operators.

Neighborhood operators for covering-based rough sets

Covering-based rough sets are important generalizations of the classical rough sets of Pawlak. A common way to shape lower and upper approximations within this framework is by means of a neighborhood operator. In this article, we study 24 such neighborhood operators that can be derived from a single covering. We verify equalities between them, reducing the original collection to 13 different neighborhood operators. For the latter, we establish a partial order, showing which operators yield smaller or greater neighborhoods than others.Six of the considered neighborhood operators result in new covering-based rough set approximation operators. We study how these new approximation operators relate to existing ones in terms of partial order relations, i.e., whether the generated approximations are in general greater, smaller or incomparable. Finally, we discuss the connection between the covering-based approximation operators and relation-based approximation operators, another prominent generalization of Pawlak’s rough sets.

A rough set approach to the characterization of transversal matroids

Rough sets are efficient for data pre-processing during data mining. However, some important problems such as attribute reduction in rough sets are NP-hard and the algorithms required to solve them are mostly greedy ones. The transversal matroid is an important part of matroid theory, which provides well-established platforms for greedy algorithms. In this study, we investigate transversal matroids using the rough set approach. First, we construct a covering induced by a family of subsets and we propose the approximation operators and upper approximation number based on this covering. We present a sufficient condition under which a subset is a partial transversal, and also a necessary condition. Furthermore, we characterize the transversal matroid with the covering-based approximation operator and construct some types of circuits. Second, we explore the relationships between closure operators in transversal matroids and upper approximation operators based on the covering induced by a family of subsets. Finally, we study two types of axiomatic characterizations of the covering approximation operators based on the set theory and matroid theory, respectively. These results provide more methods for investigating the combination of transversal matroids with rough sets. • construct a covering relative to transversal matroids, establish covering rough sets. • characterize transversal matroids with covering rough set. • construct two types of circuits of transversal matroids by using covering rough sets. • propose conditions under which a subset is a partial transversal by using rough sets. • build axiomatic characterizations of approximation operators from two viewpoints.

Covering-based rough fuzzy sets and binary relation

Rough set theory is a powerful tool for dealing with uncertainty, granularity, and incompleteness of knowledge in information systems. In this paper we study covering-based rough fuzzy sets in which a fuzzy set can be approximated by the intersection of some elements in a covering of the universe of discourse. Some properties of the covering-based fuzzy lower and upper approximation operators are examined. We present the conditions under which two coverings generate the same covering-based fuzzy lower and upper approximation. We approximate fuzzy sets based on a binary relation and its properties are introduced. Finally, we establish the equivalency between rough fuzzy sets generated by a covering and rough fuzzy sets generated by a binary relation.

Condition for covering-based upper approximation operators to be closure operators of matroids

Covering-based rough set theory is a useful tool to deal with inexact, uncertain or vague knowledge in information systems. In order to broaden the application and theoretical areas of rough sets and matroids, some authors have combined them from many different viewpoints, such as circuits, rank function and spanning sets. In this paper, we study the relationship between five types of covering-based upper approximation operators and closure operators of matroids. On one hand, comparing those characterisations of five types of covering-based upper approximation operators with the characterisations of closure operators of matroids, we discuss the condition under which five types of covering-based upper approximation operators form closure operators of matroids and get some results. For example, the necessary and sufficient conditions for the second type of covering-based upper approximation operator to satisfy idempotence is obtained. On the other hand, by using the characterisations of unary coverings, close friends, neighbourhoods, indiscernible neighbourhoods and reduct, we present necessary and sufficient conditions for five types of covering-based upper approximation operators to be closure operators of 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.

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.