Abstract
Granular structure plays a very important role in the model construction, theoretical analysis and algorithm design of a granular computing method. The granular structures of classical rough sets and fuzzy rough sets have been proven to be clear. In classical rough set theory, equivalence classes are basic granules, and the lower and upper approximations of a set can be computed by those basic granules. In the theory of fuzzy rough set, granular fuzzy sets can be used to describe the lower and upper approximations of a fuzzy set. This paper discusses the granular structure of type-2 fuzzy rough sets over two universes. Definitions of type-2 fuzzy rough sets over two universes are given based on a wavy-slice representation of type-2 fuzzy sets. Two granular type-2 fuzzy sets are deduced and then proven to be basic granules of type-2 fuzzy rough sets over two universes. Then, the properties of lower and upper approximation operators and these two granular type-2 fuzzy sets are investigated. At last, several examples are given to show the applications of type-2 fuzzy rough sets over two universes.
Highlights
IntroductionAccording to Chen et al, granular computing is a general computing theory for using granules such as classes, clusters, subsets, groups and intervals to build an efficient computational model for complex applications with huge amounts of data, information and knowledge [1]
According to Chen et al, granular computing is a general computing theory for using granules such as classes, clusters, subsets, groups and intervals to build an efficient computational model for complex applications with huge amounts of data, information and knowledge [1].Rough set theory [2], proposed by Pawlak in 1982, can be used to reveal and express knowledge hidden in information systems in the form of decision rules by the concepts of lower and upper approximations
Rough set theory is a method of granular computing, and equivalence classes are basic granules that can be used to approximate a set
Summary
According to Chen et al, granular computing is a general computing theory for using granules such as classes, clusters, subsets, groups and intervals to build an efficient computational model for complex applications with huge amounts of data, information and knowledge [1]. Chen et al discussed the granular structure of fuzzy rough sets and developed a theory of granular computing based on fuzzy relations in [20] They proposed the concept of granular fuzzy sets and investigated the properties of these sets using constructive and axiomatic approaches. Yang et al proposed a fuzzy probabilistic rough set model on two universes and presented concepts of the inverse lower and upper approximation operators in [30]. The granular structure discussed in [24] will be generalized to the type-2 fuzzy rough sets over different universes based on novel granular type-2 fuzzy sets, which are deduced from the definition of type-2 fuzzy rough sets, and more reasonable than those given in [24].
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