A framework of fuzzy variable precision rough sets and its applications

Abstract

The traditional rough sets model was restrictive for many applications since it could only handle the symbolic valued databases without misclassification. Here misclassification means the wrong or missing information in classification. As a generalization, the Variable Precision Rough Set model (VPRS) was introduced to handle databases with misclassification. However it could not effectively handle numerical data (real number data). To handle the databases with non-symbolic values, some generalization models of rough sets, called Fuzzy Rough Sets (FRS), have been introduced by combining fuzzy sets and rough sets. It could handle databases with numerical data. However, FRS was sensitive to misclassification and perturbation. Here perturbation means small change between the observed data and factual data. It is valuable to combine FRS and VPRS so that a powerful tool, which can effectively handle numerical data with misclassification and perturbation, is developed. In this thesis we first study two applications of FRS and find their limitations in real worlds which show the necessity to propose a robust model. Second, we generalize FRS and then propose a robust model named Fuzzy Variable Precision Rough Sets (FVPRS). After compring FVPRS with some flexible generalizations of RS, we find that FVPRS has good prospects for some real applications. Third, we establish two applications with FVPRS: attribute reduction and classifier construction. In these two applications, we propose the concepts of reductions in FVPRS and investigate their structures by using discernibility matrix (or vector). We also develop the algorithms to find the reductions. Thus a unified variable precision framework of FRS is set up. Finally in order to overcome NP-hard problem of finding the optimal solutions, we design some fast heuristic algorithms (the computational complexities of these algorithms increase with the square of the number of attributes) to obtain near-optimal solutions. From the theoretical viewpoints, the main contribution of this thesis is that we propose a robust framework with respect to misclassification and perturbation. From the practical viewpoint, the main contribution of this thesis is that two robust applications of FVPRS are proposed based on the strict mathematical reasoning.