Multigranulation Fuzzy Interval-set Rough Sets Based On The Relation Matrix

Rough set theory is an important tool for studying and dealing with uncertain problems.The main idea of it is to use a pair of definable concepts(the lower and upper approximations)to approximate and describe the undefinable subset of the universe.At present,the method of the relation matrix is widely used in the research of rough set theory,such as the rough approximations based on a relation matrix,the fuzzy rough approximations based on a relation matrix,and the multi-granularity rough approximations based on a relation matrix.All these give the corresponding lower and upper approximations from the perspective of a relation matrix.This paper presents the vector interval representation of an interval set,which is used to describe an unknown concept.According to the study of the equivalent relation matrix,vector interval,interval-set rough sets,fuzzy interval-set rough sets and multi-granularity rough sets,we discuss the interval-set rough sets,multi-granularity interval-set rough sets and multi-granularity interval-set fuzzy rough sets,respectively,based on the relation matrix.The main research contents are shown as follows:Firstly,applying the vector interval representation of an interval set and the equivalence relation matrix,a quantity multiplication operation and a Boolean operation are defined between the relation matrix and the vector interval.An interval-set rough sets based on a relation matrix is then proposed.The corresponding interval-set rough sets based on the quantity multiplication and the Boolean operation,respectively,are studied.The related algorithms of these two kinds of rough sets are also constructed.Furthermore,these two matrix-based algorithms are experimentally tested by using the actual UCI data set.Compared with the algorithm of the interval-set rough sets obtained by using the definition,the results show the feasibility of the algorithms for the interval-set rough sets based on the relation matrix.Moreover,the experiments also display that the algorithm of the interval-set rough sets based on the Boolean operation is better than that of the interval-set rough sets based on the quantity multiplication operation.Secondly,for the multi-granularity interval-set rough sets,we propose the approach to obtain them by using the relation matrix.The optimistic and pessimistic multi-granularity interval-set rough sets are discussed based on the Boolean operation between the relation matrix and interval vector.Then the vector representations of the multi-granularity interval-set rough lower and upper approximations are also investigated.Lastly,we extend the vector interval to the fuzzy vector interval,define a new representation of a fuzzy interval set,and propose the multi-granularity fuzzy interval-set rough sets.Applying the operations of a relation matrix and a fuzzy vector interval,the fuzzy interval-set rough lower and upper approximations can be denoted by using the vector representations.Furthermore,applying the idea,we extend the single granular space to multi-granular one,and introduce the multi-granularity fuzzy interval-set rough sets based on a relation matrix.The validity of the multi-granularity fuzzy interval-set lower and upper approximations based on the matrix method is illustrated by an example.