The Applications Of Variable Precision Rough Set Approximations In Concept Lattice

Rough set theory which is provided by a Polish mathematician—Pawlak in 1982 can be used to deal with uncertainty data. Using equivalence classes, it partites the whole set of a given set. Through upper and lower approximations, any given set is described by rough set. Variable precision rough which aims at describing every set is an expansion of the classical rough set theory. After adding the degree of misclassification basis to the classical rough set theory, variable precision rough set can be approximated within a variable range. Concept lattice theory(or say, formal concept analysis) is provided by Wille, a German mathematician, in 1982. The goal is to find out the hierarchical relationships among concepts. As two powerful tools for data analysis and knowledge discovery, rough set theory and concept lattice theory not only have many interpenetration, but also attract the attention of artificial intelligence researchers. At present, the two theories have widely applied in many fields such as software engineering, data mining, information retrieval, machine learning, uncertainty rule acquisition, decision management and so on.In variable precision rough set theory, for a finite universe U, if a given subset A of U is not expressed by some equivalence classes which is defined on U, then we call A is rough set, otherwise A is called as a precise set. Rough set can approximate by the upper and lower approximations. In concept lattice theory, for a given context(O, P, I), all the concepts can be found out. But for a subset B of O, if B is not an extension of any concept of(O, P, I), then B can be approximated by two extensions.Combining the definitions of degree of misclassification and upper and lower approximations in variable precision rough set with the inclusion definition of degree in rough set and the notion of similarity in concept lattice, this paper presents two operators of upper approximation and lower approximation for concept lattices, and in addition, provides the definition of approximation distance. Additionally, we explore two algorithms to calculate the approximation extension. The main results in this paper are as follows:1. Based on the definitions of degree of misclassification, upper and lower approximations in variable precision rough set theory, this paper puts forward the upper and lower approximation operators in concept lattice, and discusses the properties of upper and lower approximation operators. In addition, upper and lower approximation concept lattice are obtained. We confirm that with the changings of a given undefinable object X and the misclassification degree ?, we explore the trend of upper and lower approximations of X. After, an example illustrates the correct of this above idea.2. Based on the inclusion degree in rough set and the similarity in concept lattice, we give a definition of a similarity distance in concept lattice. An algorithm to calculate the approximation distance is discovered. Finally, an example is given to show the validity of this algorithm. Combining with this algorithm, we get an effective method of approximating undefinable object in a concept lattice.