Three-level Tolerance Multi-granulation Rough Set Modelings And Relevant Three-way Class-specific Reductions

The multi-granulation rough set is a new data method with multi-levels and multi-perspectives.It realizes knowledge discovery under different classification standards.This model has become an important theory for uncertainty information processing.However,the classical multi-granularity rough set model is based on strict equivalence relation.This precise classification makes the model has some limitations when facing the data with noise or missing information.In this paper,three-level tolerance multi-granulation rough set models with strong fault-tolerant ability are constructed by introducing tolerance parameters at relationship level,approximate level,and granulation level,respectively.Based on the three-way regions of the new models,three-way class-specific reductions based on positive preservation,negative preservation and positive-negative preservation are constructed.They are constructed from three stages of granularity preprocessing,attribute set reduction and attribute reduction respectively.The main research findings involve the following three aspects:Firstly,based on the majority inclusion principle,the strict equivalence relation is transformed into probability relation with the majority equality by introducing the tolerance parameter at the relation level.Then,the rough set model based on probability relationship is constructed and its mathematical properties are discussed.It lays a foundation for the subsequent construction of tolerance multi-granulation rough set model.Secondly,three-layer tolerable multi-granularity rough sets are constructed based on the probability relation.The multi-granulation rough set model based on the probability relationship is constructed on the relation level.It includes optimistic multi-granulation rough set and pessimistic multi-granulation rough set,also the mathematical properties of the two models are discussed.Then,variable precision multi-grain rough sets based on probability relations is constructed by introducing the tolerance parameter on the approximate level.It includes variable precision optimistic multi-granulation rough set and variable precision pessimistic multigranulation rough set.Furthermore,the generalized variable-precision multi-granularity rough set model based on probability relationship id constructed by introducing tolerance parameter on granulation level.The three-way regions(positive region,negative region and boundary region)under the three models are defined,and the related properties of approximate operators under the models are discussed.Finally,based on the three-level tolerable multi-granulation rough set models,the threeway class-specific reductions based on region preservations are constructed.Specifically,the three-way regions are defined based on a given class-specific models,and their monotonicity under attribute sets and attribute inclusion chains are discussed respectively.Then,considering the multi-granulation environment,the three-way class-specific reductions are constructed from the three stages based on region preservations,including granulation preprocessing,attribute sets reduction and attribute reduction.Furthermore,the relationships among the three class-specific attribute sets reduction and the three class-specific attribute reductions are analyzed respectively.Concretely,the three-stage reductions are dealt with as follows:(?)The granulation preprocessing is used to delete the attribute subsets with repeated granulation structure.The purpose is to avoid the repeatability of the granulation calculation in the later reduction process and to improve the reduction efficiency;(?)The attribute sets reduction is used to delete the unnecessary attribute subsets in the attribute subset family,in order to keep the region value unchanged and to reduce the search scope of subsequent attribute reduction in the multi-granulation environment;(?)The attribute reduction is to remove unnecessary attributes from the required attribute subsets and achieve final reduction from attributes.In summary,the tolerance multi-granulation rough set models are constructed by introducing different tolerance parameters at relationship level,approximate level,and granulation level,respectively.The models overcome the limitations of the existing multi-granulation rough sets when dealing with noise or missing data.They have stronger fault tolerance and better adaptability.The class-specific reduction methods by three stages fully considers the multigranularity environment which reducing from two aspects of the attribute sets and attribute.It expands and enriches the traditional multi-granularity attribute reduction method,and provides a new idea for attribute reduction in multi-granularity environment.