Current neighborhood rough sets have been usually used to solve complete information system, not incomplete system. In order to solve this problem, an extended tolerance relation was proposed to deal with the incomplete mixed information system, and associative definitions were provided. The degree of complete tolerance and neighborhood threshold were used as the constraint conditions to find the extended tolerance neighborhood. The attribute importance of the system was got by the decision positive region within the neiborhood, and the attribute reduction algorithm based on the extended tolerance relation was proposed, which was given by the importance as the heuristic factor. Seven different types of data sets on UCI database was used for simulation, and the proposed method was compared with Extension Neighborhood relation (EN), Tolerance Neighborhood Entropy (TRE) and Neighborhood Rough set (NR) respectively. The experimental results show that, the proposed algorithm can ensure accuracy of classification, select less attributes by reduction. Finally, the influence of neighborhood threshold in extended tolerance relation on classification accuracy was discussed.

Since the algorithm of attribute reduction based on positive region is based on the thought of lower approximation, it just considers the right distinguished samples. Using the thought of upper approximation and the concept of neighborhood information granule, the distinguished object set with its basic characteristics was designed and analyzed, then the new attribute importance measurement based on distinguished object set and heuristic attribute reduction algorithm was proposed. The proposed algorithm considered both the relative positive region of information decision table and the influence on boundary samples when growing condition attributes. The feasibility of the algorithm was discussed by instance analysis, and the comparative experiments on UCI data set with attribute reduction algorithm based on positive region were carried out. The experimental results show that the proposed attribute reduction algorithm can get better reduction, and the classification precision of sample set can remain the same or has certain improvement.

Focusing on the underdeveloped robustness when the existing extended rough set model encounters the noise for the incomplete information system, the necessity of adjusting the size of basic knowledge granule as well as introducing the relative degree of misclassification was analyzed. Then the Variable Precision Rough Set model based on Variable-Precision Tolerance Relation (VPRS-VPTR) was established on the basis of the object connection weight matrix, which was proposed according to the lack probability of system attribute value. Moreover, the properties of the VPRS-VPTR model were discussed, the classification accuracy under the basic knowledge granule size and the relative degree of misclassification was analyzed, the corresponding algorithm was depicted and the time complexity analysis was given afterwards. The experimental results show that the VPRS-VPTR model has higher classification accuracy compared with some other research about the expanded rough set, and the change trend of the classification accuracy is similar for the train set and the test set of several groups of incomplete data sets in UCI database. It proves that the proposed model is more precise and flexible, and the algorithm is feasible and effective.