Covering Rough Set Research Articles

Relations arising from coverings and their topological structures

Covering rough sets are an important extension of Pawlak rough sets. This paper studies the relations arising from coverings and their topological structures. Every covering induces a reflexive and transitive relation. We represent the approximate pairs proposed by Ma (2015) [20] with different combinations of a relation and its inverse. Based on this representation, we give the relationship among approximate pairs. We also consider the topological structures induced by these lower approximations and establish the relationship among these topologies. The results show that the approximate pairs can be precisely characterized by a relation and its inverse.

Fast algorithms of attribute reduction for covering decision systems with minimal elements in discernibility matrix

Covering rough sets, which generalize traditional rough sets by considering coverings instead of partitions, are introduced to deal with set-valued, missing-valued or real-valued data sets. For decision systems with such kinds of data sets, attribute reduction with covering rough sets aims to delete superfluous attributes, and fast algorithms of finding reducts are clearly meaningful for practical problems. In the existing study of attribute reduction with covering rough sets, the approach of discernibility matrix is the theoretical foundation. However, it always shares heavy computation load and large store space because of finding and storing all elements in the discernibility matrix. In this paper, we find that only minimal elements in the discernibility matrix are sufficient to find reducts. This fact motivates us in this paper to develop algorithms to find reducts by only employing minimal elements without computing other elements in the discernibility matrix. We first define the relative discernible relation of covering to characterize the relationship between minimal elements in the discernibility matrix and particular sample pairs in the covering decision system. By employing this relative discernible relation, we then develop algorithms to search the minimal elements in the discernibility matrix and find reducts for the covering decision system. Finally, experimental comparisons with other existing algorithms of covering rough sets on several data sets demonstrate that the proposed algorithms in this paper can greatly reduce the running time of finding reducts.

Special types of coverings and axiomatization of rough sets based on partial orders

Covering rough sets are a generalization of Pawlak rough sets, in which a partition of the universal set induced by an equivalence relation is replaced by a covering. In this paper, covering rough sets are transformed into generalized rough sets induced by binary relations. The paper discusses three theoretical topics. First, we consider a special type of covering in which the neighborhoods form a reduction of the covering, and we obtain necessary and sufficient conditions for neighborhoods in a covering form a reduction of the covering. Second, we study another special type of covering, and give conditions for the covering lower and upper approximations to be dual to each other. Finally, we give an axiomatic system that characterizes the lower and upper approximations of rough sets based on a partial order.

On measurements of covering rough sets based on granules and evidence theory

Covering rough sets generalize classical rough sets by relaxing partitions of the universe to coverings. The existing research of covering rough sets mainly concentrates on comparing of different kinds of approximation operators and designing algorithms for attribute reductions of decision systems. Less efforts have been made in measurements of covering rough sets. To enrich this field, in this paper we investigate the measurements of covering rough sets based on granules and evidence theory, which can be used to solve more complex practical problems. Firstly, it is shown that the most common sixteen pairs of covering approximation operators can be grouped into five pairs of dual covering approximation operators and six pairs of non-dual covering approximation operators according to definitions based on element, granule, and subsystem. Secondly, we define the rough degree and precision for the purpose of characterizing the uncertainty of these summarized covering approximation operators under a unified framework. The underlying uncertainty of covering rough sets results from the roughness of granulation of coverings. Finally, we investigate the relationships between the covering approximation operators and evidence theory. It shows that some kinds of covering approximation operators can be characterized by belief and plausibility functions, while others do not share this property. The sufficient conditions under which covering approximation operators can be measured by belief and plausibility functions are further explored, and the corresponding counterexamples are given to illustrate those that cannot be measured. Based on above discussions, we develop a basic framework of numerical characterization on measurements of covering rough sets with granules and evidence theory.

Fast approach to knowledge acquisition in covering information systems using matrix operations

Covering rough set theory provides an effective approach to dealing with uncertainty in data analysis. Knowledge acquisition is a main issue in covering rough set theory. However, the original rough set methods are still expensive for this issue in terms of time consumption. To further improvement, we propose fast approaches to knowledge acquisition in covering information systems by employing novel matrix operations. Firstly, several matrix operations are introduced to compute set approximations and reducts of a covering information system. Then, based on the proposed matrix operations, the knowledge acquisition algorithms are designed. In the end, experiments are conducted to illustrate that the new algorithms can dramatically reduce the time consumptions for computing set approximations and reducts of a covering information system, and the larger the scale of a data set is, the better the new algorithms perform.

Uncertainty Measurement for Covering Rough Sets

Covering rough set theory is an important generalization of traditional rough set theory. So far, the studies on covering generalized rough sets mainly focus on constructing different types of approximation operations. However, little attention has been paid to uncertainty measurement in covering cases. In this paper, a new kind of partial order is proposed for coverings which is used to evaluate the uncertainty measures. Consequently, we study uncertain measures like roughness measure, accuracy measure, entropy and granularity for covering rough set models which are defined by neighborhoods and friends. Some theoretical results are obtained and investigated.

Evidence-theory-based numerical algorithms of attribute reduction with neighborhood-covering rough sets

Covering rough sets generalize traditional rough sets by considering coverings of the universe instead of partitions, and neighborhood-covering rough sets have been demonstrated to be a reasonable selection for attribute reduction with covering rough sets. In this paper, numerical algorithms of attribute reduction with neighborhood-covering rough sets are developed by using evidence theory. We firstly employ belief and plausibility functions to measure lower and upper approximations in neighborhood-covering rough sets, and then, the attribute reductions of covering information systems and decision systems are characterized by these respective functions. The concepts of the significance and the relative significance of coverings are also developed to design algorithms for finding reducts. Based on these discussions, connections between neighborhood-covering rough sets and evidence theory are set up to establish a basic framework of numerical characterizations of attribute reduction with these sets.

On some types of neighborhood-related covering rough sets

Covering rough sets are natural extensions of the classical rough sets by relaxing the partitions to coverings. Recently, the concept of neighborhood has been applied to define different types of covering rough sets. In this paper, by introducing a new notion of complementary neighborhood, we consider some types of neighborhood-related covering rough sets, two of which are firstly defined. We first show some basic properties of the complementary neighborhood. We then explore the relationships between the considered covering rough sets and investigate the properties of them. It is interesting that the set of all the lower and upper approximations belonging to the considered types of covering rough sets, equipped with the binary relation of inclusion ⊆, constructs a lattice. Finally, we also discuss the topological importance of the complementary neighborhood and investigate the topological properties of the lower and upper approximation operators.

On the structure of definable sets in covering approximation spaces

Covering rough sets, a generalization of the classical rough sets, are main research topics of rough set theory. Various covering rough set models have been proposed. In this paper, ten important types of covering rough set models are first reviewed. The algebraic structures of definable sets, inner definable sets and outer definable sets in these covering rough sets are then investigated. Based on the concept of definable sets, we further explore relations among the ten covering rough sets. Finally, the conditions for neighborhood \(\{N(x):x\in U\}\) to form a partition of the universe U are discussed, and an open problem proposed by Yun et al. is answered.

A Granular Reduction Algorithm Based on Covering Rough Sets

The granular reduction is to delete dispensable elements from a covering. It is an efficient method to reduce granular structures and get rid of the redundant information from information systems. In this paper, we develop an algorithm based on discernability matrixes to compute all the granular reducts of covering rough sets. Moreover, a discernibility matrix is simplified to the minimal format. In addition, a heuristic algorithm is proposed as well such that a granular reduct is generated rapidly.

Attribute Reduction in Formal Contexts: A Covering Rough Set Approach

This paper proposes an approach to attribute reduction in formal contexts via a covering rough set theory. The notions of reducible attributes and irreducible attributes of a formal context are first introduced and their properties are examined. Judgment theorems for determining all attribute reducts in the formal context are then obtained. According to the attribute reducts, all attributes of the formal context are further classified into three types and the characteristic of each type is characterized by the properties of irreducible classes of the formal context. Finally, by using the discernibility attribute sets, a method of distinguishing the reducible attributes and the irreducible attributes in formal contexts is presented.