Covering-based Rough Sets Research Articles

Neighbor selection for user-based collaborative filtering using covering-based rough sets

Recommender systems (RSs) provide personalized information by learning user preferences. User-based collaborative filtering (UBCF) is a significant technique widely utilized in RSs. The traditional UBCF approach selects k-nearest neighbors from candidate neighbors comprised by all users; however, this approach cannot achieve good accuracy and coverage values simultaneously. We present a new approach using covering-based rough set theory to improve traditional UBCF in RSs. In this approach, we insert a user reduction procedure into the traditional UBCF approach. Covering reduction in covering-based rough sets is used to remove redundant users from all users. Then, k-nearest neighbors are selected from candidate neighbors comprised by the reduct-users. Our experimental results suggest that, for the sparse datasets that often occur in real RSs, the proposed approach outperforms than the traditional UBCF, and can provide satisfactory accuracy and coverage simultaneously.

On some types of fuzzy covering-based rough sets

Fuzzy coverings are a natural extension of the coverings by replacing crisp sets with fuzzy sets. Recently, an excellent introduction to the definition of a fuzzy β-covering is due to Ma and two fuzzy covering-based rough set models are presented. In this paper, by introducing some new definitions of fuzzy β-covering approximation spaces, the properties of fuzzy β-covering approximation spaces and Ma's fuzzy covering-based rough set models are studied. Furthermore, three new types of fuzzy covering-based rough set models as generalizations of Ma's models are first proposed in this paper. First, some properties of fuzzy β-covering and its fuzzy β-neighborhood family are proposed. We present a necessary and sufficient condition for fuzzy β-neighborhood family induced by a fuzzy β-covering to be equal to the fuzzy β-covering itself. Then we study the characterizations of Ma's fuzzy covering-based rough set models and give a necessary and sufficient condition for two fuzzy β-coverings to generate the same fuzzy covering lower approximation or the same fuzzy covering upper approximation. Finally, this paper proposes three new types of fuzzy covering-based rough set models by introducing a new notion of a fuzzy complementary β-neighborhood.

Dependence space of topology and its application to attribute reduction

Attribute reduction plays an important role in pattern recognition and machine learning. Covering-based rough sets, as a technique of granular computing, can be a useful tool for studying attribute reduction. Topology has a close relationship with rough sets and plays a significant role in attribute reduction in information systems. So it is meaningful to combine topology with rough sets to address the problems of attribute reduction. In this paper, we mainly discuss and address the problem of attribute reduction in incomplete information systems with dependence space induced by topological base. Firstly, we investigate the topological structure induced by covering-based rough sets and some characteristics of the topological structure are presented. Secondly, a new type of dependence space is constructed in terms of the base of topological structure, and some characteristics of the dependence space are investigated. Finally, we apply the obtained results of the space to the attribute reduction in incomplete information systems. Especially, a discernibility matrix is defined for the attribute reduction in incomplete information systems.

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.

A semantically sound approach to Pawlak rough sets and covering-based rough sets

In this paper, we discuss a semantically sound approach to covering-based rough sets. We recall and elaborate on a conceptual approach to Pawlak's rough set model, in which we consider a two-part descriptive language. The first part of the language is used to describe conjunctive concepts, while in the second part disjunctions are allowed as well. Given the language, we discuss its elementary and definable sets, and we study how the approximation operators can be seen as derived notions of the family of definable sets, which is represented by a Boolean algebra over a partition. Furthermore, we generalize the two parts of the language in order to describe concepts of covering-based rough sets. Unfortunately, the family of definable sets will no longer be represented by a Boolean algebra over a partition, but by the union-closure of a covering. Therefore, only the derived covering-based lower approximations of sets are definable for the generalized language. In addition, it is discussed how the two-part languages are used to construct decision rules, which are used in data mining and machine learning.

A fuzzy covering-based rough set model and its generalization over fuzzy lattice

In this paper, we study a new type of fuzzy covering-based rough set model by introducing the notion of fuzzy β-minimal description. We mainly address the following issues in this paper. First, we present the definition of fuzzy β-minimal description and study its properties. Then, we define a novel type of fuzzy covering-based rough set model and investigate the properties of this model. Furthermore, the axiomizations and matrix representations of the fuzzy covering lower and fuzzy covering upper approximations are the vital problems we investigate in this paper. Moreover, we explore the conditions under which two fuzzy β-coverings generate the same fuzzy covering lower or fuzzy covering upper approximations. Finally, we generalize the model to L-fuzzy covering-based rough set which is defined over fuzzy lattices. Similar to the fuzzy covering-based rough set model, we also address the issues mentioned above to the L-fuzzy covering-based rough set.

RETRACTED ARTICLE: Covering-based rough set classification system

Medical data classification is applied in intelligent medical decision support system to classify diseases into different categories. Several classification methods are commonly used in various healthcare settings. These techniques are fit for enhancing the nature of prediction, initial identification of sicknesses and disease classification. The categorization complexities in healthcare area are focused around the consequence of healthcare data investigation or depiction of medicine by the healthcare professions. This study concentrates on applying uncertainty (i.e. rough set)-based pattern classification techniques for UCI healthcare data for the diagnosis of diseases from different patients. In this study, covering-based rough set classification (i.e. proposed pattern classification approach) is applied for UCI healthcare data. Proposed CRS gives effective results than delicate pattern classifier model. The results of applying the CRS classification method to UCI healthcare data analysis are based upon a variety of disease diagnoses. The execution of the proposed covering-based rough set classification is contrasted with other approaches, such as rough set (RS)-based classification methods, Kth nearest neighbour, improved bijective soft set, support vector machine, modified soft rough set and back propagation neural network methodologies using different evaluating measures.

Covering-based rough sets and modal logics. Part I

Two conjectures on the covering-based rough set semantics for modal logics in [35] are answered. The C2 and C5 semantics give rise to the same modal system S4. There are Galois connections between C2 and C5 which lead to the covering-based semantics for the temporal logic system S4t. The P1 and C4 semantics give rise to the same modal system KTB.

An intuitionistic fuzzy graded covering rough set

Exploring rough sets from the perspective of covering represents a promising direction in rough set theory, where concepts are approximated by substituting of an equivalent relation in classical rough set theory with a covering in covering-based rough set theory. By combining intuitionistic fuzzy (IF) β-neighborhoods induced by an IF β-covering with IF rough sets, this study develops a new rough set model, which is a generalization of the β-neighborhood fuzzy covering rough sets and IF rough sets. First, we present the concepts of IF graded covering and IF graded neighborhood, namely, IF β-covering and IF β-neighborhood, respectively. We propose an IF graded approximation space on the basis of IF graded neighborhood and discuss its uncertainty measures, namely, information entropy and rough entropy. Second, we generalize an IF rough set based on IF relation to one based on the presented IF graded neighborhood and use the distance between two IF sets to characterize the latter. Third, we present the matrix computation method for the upper and lower approximations of the presented IF rough sets based on the IF graded neighborhood. Fourth, from the multi-granulation perspective, we examine the IF graded approximation space, uncertainty measures, IF rough sets, and computation methods for its reducts. Finally, we discuss several generalizations similar to the presented IF covering rough sets.

Knowledge reduction of dynamic covering decision information systems when varying covering cardinalities

In covering-based rough set theory, non-incremental approaches are time-consuming for performing knowledge reduction of dynamic covering decision information systems when the cardinalities of coverings change as a result of object immigration and emigration. Because computing approximations of sets is an important step for knowledge reduction of dynamic covering decision information systems, efficient approaches to calculating the second and sixth lower and upper approximations of sets using the type-1 and type-2 characteristic matrices, respectively, are essential. In this paper, we provide incremental approaches to computing the type-1 and type-2 characteristic matrices of dynamic coverings whose cardinalities vary with the immigration and emigration of objects. We also design incremental algorithms to compute the second and sixth lower and upper set approximations. Experimental results demonstrate that the incremental approaches effectively improve the efficiency of set approximation computation. Finally, we employ several examples to illustrate the feasibility of the incremental approaches for knowledge reduction of dynamic covering decision information systems when increasing the cardinalities of coverings.

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.

Covering-based fuzzy rough sets

Many researchers have combined rough set theory and fuzzy set theory in order to easily approach problems of imprecision and uncertainty. Covering-based rough sets are one of the important generalizations of classical rough sets. Naturally, covering-based fuzzy rough sets can be studied as a combination of covering-based rough set theory and fuzzy set theory. It is clear that Pawlak's rough set model and fuzzy rough set model are special cases of the covering-based fuzzy rough set model. This paper investigates the properties of covering-based fuzzy rough sets. In addition, operations of intersection, union and complement on covering-based fuzzy rough sets are investigated. Finally, the corresponding algebraic properties are discussed in detail.

Connectedness of graphs and its application to connected matroids through covering-based rough sets

Graph theoretical ideas are highly utilized by computer science fields especially data mining. In this field, a data structure can be designed in the form of tree. Covering is a widely used form of data representation in data mining and covering-based rough sets provide a systematic approach to this type of representation. In this paper, we study the connectedness of graphs through covering-based rough sets and apply it to connected matroids. First, we present an approach to inducing a covering by a graph, and then study the connectedness of the graph from the viewpoint of the covering approximation operators. Second, we construct a graph from a matroid, and find the matroid and the graph have the same connectedness, which makes us to use covering-based rough sets to study connected matroids. In summary, this paper provides a new approach to studying graph theory and matroid theory.

Representative-based classification through covering-based neighborhood rough sets

Considerable progress has been made in the theory of covering-based rough sets. However, there has been a lack of research on their application to classification tasks, especially for nominal data. In this paper, we propose a representative-based classification approach for nominal data using covering-based rough sets. The classifier training task considers three issues. First, we define the neighborhood of an instance. The size of the neighborhood is determined by a similarity threshold ?. Second, we determine the maximal neighborhood of each instance in the positive region by computing its minimal ? value. These neighborhoods form a covering of the positive region. Third, we employ two covering reduction techniques to select a minimal set of instances called representatives. To classify a new instance, we compute its similarity with each representative. The similarity and minimal ? of the representative determine the distance. Representatives with the minimal distance are employed to obtain the class label. Experimental results on different datasets indicate that the classifier is comparable with or better than the ID3, C4.5, NEC, and NCR algorithms.

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 approximation number function and the characterization of covering approximation space

There is renewed interest in the topic of covering-based rough sets. It is important to understand the characterization of covering approximation spaces. Recently, Zhu et al. presented the concept of an approximation number function, which can be viewed as a quantitative tool for analyzing the covering approximation spaces. We focus on this concept and show that an approximation number function can be characterized by its properties. We give the axiomatic definition of an approximation number function, and prove that there is a one-to-one correspondence between the set of all lower approximation number functions and the set of all covering approximation spaces. This suggests that the concept is fundamental to the characterization of covering approximation spaces. We apply approximation number functions to the existing studies on approximation spaces, and a series of matroidal structures are induced in an approximation space.

An integrated covering-based rough fuzzy set clustering approach for sequential data

Clustering is categorised as hard or soft in nature. Soft clusters may have fuzzy or rough boundaries. Rough clustering can help researchers to discover overlapping clusters in many applications such as web mining and text mining. Rough set approach is a very useful tool to handle the unclear and ambiguous data. As rough sets make use of the equivalence relation property, they remain rigid and it is unreliable and inefficient for real time applications where the datasets may be very large. In this paper, we provide a solution to this problem with covering-based rough set approach. Covering-based rough set is an extension of rough set approach in which the equivalence relation has been relaxed. This method is based on coverings rather than partitions. This makes it more flexible than rough sets and it is more convenient for dealing with complex applications. Clustering sequential data is one of the vital research tasks. We uses covering-based similarity measure which gives better results as compared to rough set which uses set and sequence similarity measure. In this paper, covering-based rough fuzzy set clustering approach is proposed to resolve the uncertainty of sequence data.

Rough topology on covering-based rough sets

A classical rough set theory was given by Pawlak (1982). Since then rough set theory has been investigated by many researchers. Rough set theory can be viewed as an approach to vagueness. Many scientific fields find vagueness concepts interesting; therefore, the rough set theory has important applications in fields such as data analysis and information systems. There are many works on topological and algebraic approaches to rough sets. For example, the union and intersection operations of classical rough sets were introduced by Pomykala and Pomykala (1988). Classical rough sets have been defined on partitions which is a special case of the formalism of coverings. This allows us to study covering-based rough sets. In this paper, we consider covering-based rough sets as an extension of classical rough sets studied by Bryniarski (1989). We also construct a rough topology on a given covering-based rough set. In order to do that, union and intersection operations will be extended to covering-based rough sets. In the future, our aim is to examine applications of rough topology on covering-based rough sets. This work leads us to study some properties of covering-based rough topology.

Constructing importance measure of attributes in covering decision table

In rough set theory, attributes importance measure is a crucial factor in applications of attribute reduction and feature selection. Many importance measure methodologies for discrete-valued information system or decision table have been developed. However, there are only limited studies on importance measurement for numerical-valued information system. In this paper, knowledge change-based importance measure, with the structural characteristics of fuzzy measure, is introduced to evaluate the importance of attributes in covering decision table. We first present the concept of similarity block in attribute space, based on which coverings are induced to construct the lower and upper approximation operators in covering-based rough sets. In particular, the traditional importance measure is extended to deal with covering decision table. Further, an evaluation model based on the knowledge change-based importance measure is constructed. Experiments are conducted on the public data sets from UCI, and a case study on the students’ overall evaluation is given finally. Theoretical analysis and experimental results show that the proposed importance measure is effective for evaluating the importance of attributes in covering decision table.

Closed-set lattice and modular matroid induced by covering-based rough sets

Covering is a common form of data representation, and covering-based rough sets, a technique of granular computing, provide an effective tool to deal with this type of data. However, many important problems of covering-based rough sets, such as covering reduction, are NP-hard so that most algorithms to solve them are greedy ones. Matroid theory, based on linear algebra and graph theory, provides well-established platforms for greedy algorithms. Lattice has been widely used in diverse fields, especially algorithm design, which plays an important role in covering reduction. Therefore, it is necessary to integrate covering-based rough sets with matroid and lattice. In this paper, we construct three types of matroids through covering-based rough sets and investigate their modularity. Moreover, we investigate some characteristics of these types of closed-set lattices induced by these three types of matroids and the relationships among these closed-set lattices. First, based on covering-based rough sets, three families of sets are constructed and proved to satisfy independent set axiom of matroids. So three types of matroids are induced by covering-based rough sets in this way. Second, some characteristics of these matroids, such as rank function, closure operator and closed set, are presented. Moreover, we investigate the characteristics of these closed-set lattices induced by these three types of matroids, such as modular pair, modular element. Finally, the relationships among these closed-set lattices induced by these three types of matroids are investigated. Especially, we prove that these three types of matroids induced by covering-based rough sets are all modular matroids.