Pair Of Approximation Operators Research Articles

Covering based multi-granulation rough fuzzy sets with applications to feature selection

Feature selection acts as an important preprocessing method to reduce redundant information. In order to effectively evaluate the classification information hidden in a given attribute subset, a novel rough set model is put forth via integrating covering based rough fuzzy sets with multi-granulation rough sets. In view of this, fuzzy β-neighborhood is employed to describe the information representation and knowledge fusion of covering families, by which a pair of approximation operators are formulated and a new multi-granulation rough fuzzy set model is introduced. The generalized model gives a unified perspective for existing rough set models. We then investigate the axiomatic characterizations by view of optimism and pessimism. Finally, the data reduction is processed from the point of keeping the discrimination ability. The multi-granulation significance function of a candidate attribute in term of fuzzy decisions is defined, using which a greedy algorithm is developed for multi-granulation feature selection. Experiments on twelve different types of datasets show that our model is efficient and superior to three popular algorithms in terms of reduction rate and classification performance.

On numerical characterizations of the topological reduction of incomplete information systems based on evidence theory

Abstract Knowledge reduction of information systems is one of the most important parts of rough set theory in real-world applications. Based on the connections between the rough set theory and the theory of topology, a kind of topological reduction of incomplete information systems is discussed. In this study, the topological reduction of incomplete information systems is characterized by belief and plausibility functions from evidence theory. First, we present that a topological space induced by a pair of approximation operators in an incomplete information system is pseudo-discrete, which deduces a partition. Then, the topological reduction is characterized by the belief and plausibility function values of the sets in the partition. A topological reduction algorithm for computing the topological reducts in incomplete information systems is also proposed based on evidence theory, and its efficiency is examined by an example. Moreover, relationships among the concepts of topological reduct, classical reduct, belief reduct, and plausibility reduct of an incomplete information system are presented.

Boundary region-based variable precision covering rough set models

As an essential way of knowledge representation and knowledge acquisition in uncertainty theory, variable precision rough set (VPRS) has been widely considered by researchers. Firstly, the corresponding CVPRS model via boundary region is investigated based on an existing covering-based rough set model. Its algebraic structures and properties are systematically studied. This model mainly depicts a pair of boundary operators and a pair of approximation operators with a certain threshold from the perspective of single knowledge granularity. And it not only relaxes the complete inclusion relation but also retains the basic properties of the classical rough set. Secondly, an attribute reduction approach is given for a covering-based decision information systemusing the proposed CVPRS model, which keeps the lower or upper approximation unchanged. The different performances of boundary operators and their related three indices in these reduction methods are compared. The results show that boundary operators and distribution functions have outstanding advantages in these applications. Finally, an extraction method of necessity rules and possibility rules corresponding to decision classes are established. The validity and security of the above methods are verified theoretically. Numerical experiments further show that the proposed models and methods have a certain application value.

Topological structures of a type of granule based covering rough sets

Rough set theory is one of important models of granular computing. Lower and upper approximation operators are two important basic concepts in rough set theory. The classical Pawlak approximation operators are based on partition and have been extended to covering approximation operators. Covering is one of the fundamental concepts in the topological theory, then topological methods are useful for studying the properties of covering approximation operators. This paper presents topological properties of a type of granular based covering approximation operators, which contains seven pairs of approximation operators. Then, topologies are induced naturally by the seven pairs of covering approximation operators, and the topologies are just the families of all definable subsets about the covering approximation operators. Binary relations are defined from the covering to present topological properties of the topological spaces, which are proved to be equivalence relations. Moreover, connectedness, countability, separation property and Lindel?f property of the topological spaces are discussed. The results are not only beneficial to obtain more properties of the pairs of covering approximation operators, but also have theoretical and actual significance to general topology.

Algebraic structures related to nilpotent minimum algebras and rough sets1

Equipping NM-algebras with a Brouwer-like negation ∼, we introduce a kind of algebra under the name BZNM-algebra in the present paper and investigate its algebraic properties in detail. By means of the Brouwer-like negation ∼ and the internal Kleene negation ¬ in NM-algebras, two types of modal ope rators μ and ν on BZNM-algebras are defined. For any element a of BZNM-algebra, μ (a) and ν (a) turn out to be the best approximation of a from the bottom and the top, respectively, with Boolean skeleton serving as the collection of sharp elements. It is also shown that these two modal operators turn to have an S5 behavior. Additionally, various types of sharp elements with respect to the negation operators including ∼, ¬, ♭ (an anti-intuitionistic negation) are defined, and the relationship between these sets and that of ⊗-idempotent (⊕-idempotent) elements are also investigated. It is then proved that the proposed BZNM-algebras are indeed equivalent to NM▵ algebras under a suitable transformation of operators. Furthermore, a special kind of BZNM-algebras, i.e., BZNM3-algebras, is investigated and its equivalent characterizations are given. A new type of rough approximation operators are proposed, and it is shown that such a pair of approximation operators is equivalent to the defined operators μ and ν. Finally, two interesting examples of BZNM-algebras are presented.

On some types of covering rough sets from topological points of view

The concept of coverings is one of the fundamental concepts in topological spaces and plays a big part in the study of topological problems. This motivates the research of covering rough sets from topological points of view. From topological points of view, we can get a good insight into the essence of covering rough sets and make our discussions concise and profound. In this paper, we first construct a type of topology called the topology induced by the covering on a covering approximation space. This notion is indeed in the core of this paper. Then we use it to define the concepts of neighborhoods, closures, connected spaces, and components. Drawing on these concepts, we define several pairs of approximation operators. We not only investigate the relationships among them, but also give clear explanations of the concepts discussed in this paper. For a given covering approximation space, we can use the topology induced by the covering to investigate the topological properties of the space such as separation, connectedness, etc. Finally, a diagram is presented to show that the collection of all the lower and upper approximations considered in this paper constructs a lattice in terms of the inclusion relation ⊆.

Algebras of Definable and Rough Sets in Quasi Order-based Approximation Spaces

A pair of approximation operators, based on the notion of granules in generalized ap- proximation spaces, was studied in an earlier work by the authors. In this article, we investigate algebraic structures formed by the definable sets and also by the rough sets determined by this pair of approximation operators. The definable sets are open sets of an Alexandrov topological space, and form a completely distributive lattice in which the set of completely join irreducible elements is join dense. The collection of rough sets also forms a similar structure. Representation results for such classes of completely distributive lattices as well as Heyting algebras in terms of definable and rough sets are obtained. Further, two unary operators on rough sets are considered, making the latter constitute a structure that is named a 'rough lattice'. Repr esentation results for rough lattices are proved.

Classification of coverings in the finite approximation spaces

Based on the notions of neighborhood and complementary neighborhood, we consider the classification of coverings in the covering rough set theory. We present a classification rule under which the coverings on a universe U are classified such that for any given pair of neighborhood (or complementary neighborhood)-based lower and upper approximation operators on U, all the different coverings in the same class generate the same pair of lower and upper approximations of each X⊂U. We show that there is a one to one correspondence between the equivalence classes of coverings and the topologies of the universe. Thus the number of the equivalence classes of coverings is equal to the number of topologies of the universe, and therefore, this number is much smaller than that of the coverings. We also give an illustrative example to show how we can classify some given coverings by calculating the topologies.Moreover, based on the relationship between the neighborhood and the complementary neighborhood, we find that each class of coverings has a dual class, and each pair of approximation operators has a dual pair of approximation operators. For a finite universe, considering one pair of approximation operators under one equivalence class of coverings is equivalent to considering its dual approximation operator pair under the dual class of coverings. Finally, we also present a sufficient condition under which, for any given pair of neighborhood-based lower and upper approximation operators, two coverings of the universe U generate the same pair of approximations of each X⊂U if and only if they belong to the same equivalence class.

Duality, conjugacy and adjointness of approximation operators in covering-based rough sets

Many different proposals exist for the definition of lower and upper approximation operators in covering-based rough sets. In this paper, we establish relationships between the most commonly used operators, using especially concepts of duality, conjugacy and adjointness (also referred to as Galois connection). We highlight the importance of the adjointness condition as a way to provide a meaningful link, aside from duality, between a pair of approximation operators. Moreover, we show that a pair of a lower and an upper approximation operator can be dual and adjoint at the same time if and only if the upper approximation is self-conjugate, and we relate this result to a similar characterization obtained for the generalized rough set model based on a binary relation.

Matroidal structure of covering-based rough sets through the upper approximation number

Covering-based rough set theory is a generalisation of rough set theory. Matroids are based on linear algebra and graph theory, and have a variety of applications in many fields. In this paper, we introduce matroid theory to covering-based rough sets, and explore the matroidal structure and properties of covering-based rough sets. Specifically, we define the upper approximation number to establish the matroidal structure of covering-based rough sets. So many important concepts and methods in matroid theory can be employed to investigate covering-based rough sets. The rank plays a very important role in a matrix, so we use the rank function of the matroid induced by a covering to measure the covering. With the rank function, a pair of approximation operators, namely, matroid approximation operators, are constructed. This type of approximation operators not only inherits the properties of those traditional ones which are defined from the perspective of set theory, but also presents some new properties. Finally, the matroid upper approximations are compared with the second upper approximations in covering-based rough sets.

Rough function model and rough membership function

Two pairs of approximation operators, which are the scale lower and upper approximations as well as the real line lower and upper approximations, are defined. Their properties and antithesis characteristics are analyzed. The rough function model is generalized based on rough set theory, and the scheme of rough function theory is made more distinct and complete. Therefore, the transformation of the real function analysis from real line to scale is achieved. A series of basic concepts in rough function model including rough numbers, rough intervals, and rough membership functions are defined in the new scheme of the rough function model. Operating properties of rough intervals similar to rough sets are obtained. The relationship of rough inclusion and rough equality of rough intervals is defined by two kinds of tools, known as the lower (upper) approximation operator in real numbers domain and rough membership functions. Their relative properties are analyzed and proved strictly, which provides necessary theoretical foundation and technical support for the further discussion of properties and practical application of the rough function model.