Characteristic matrices of compound operations of coverings and their relationships with rough sets

Abstract

As the matrix can compactly represent numeric data, simplify problem formulation and reduce time complexity, it has many applications in most of the scientific fields. For this purpose, some types of generalized rough sets have been connected with matrices. However, covering-based rough sets which play an important role in data mining and machine learning are seldom connected with matrices. In this paper, we define three composition operations of coverings and study their characteristic matrices; Moreover, the relationships between the characteristic matrices and covering approximation operators are investigated. First, for a covering, an existing matrix representation of indiscernible neighborhoods called the type-1 characteristic matrix of the covering is recalled and a new matrix representation of neighborhoods called the type-2 characteristic matrix of the covering is proposed. Second, considering the importance of knowledge fusion and decomposition, we define three types of composition operations of coverings. Specifically, their type-1 and type-2 characteristic matrices are studied. Finally, we also explore the representable properties of covering approximation operators with respect to any covering generated by each composition operation. It is interesting to find that three types of approximation operators, which are induced by each type of composition operation of coverings, can be expressed as the Boolean product of a coefficient matrix and a characteristic vector. These interesting results suggest the potential for studying covering-based rough sets by matrix approaches.