Research On Rough Modeling Of Type-2 Fuzzy Sets

During the process of modeling phenomena and laws of motion in the objective world with the help of data,we need to deal with various uncertainties in data.Both fuzzy set theory and rough set theory are useful tools to manipulate uncertainties.Fuzzy set theory can be used to manipulate those vague concepts with indistinct boundary.Whether an object belongs to such a concept cannot be answered by ”yes” or ”no”.An American cybernetician,L.A.Zadeh,proposed fuzzy sets to deal with such fuzzy concepts and a value between 0 and 1 is used to describe the membership degree of an object.Then fuzzy phenomena can be manipulated by mathematical thinking and mathematical methods.The concept of rough set,proposed by a Polish scholar Z.Pawlak,is used to describe the rough concepts that cannot be expressed precisely by the existing knowledge.The rough set theory can manipulate the inconsistency between conditional attributes and decision attributes.For those contradictory samples,whose conditional attributes are same but decision attributes are different,rough set theory admits that they are objective,models them systematically and extracts certain rules and possible rules by defining approximation operators and attributes reduction.The classic rough set theory is based on set theory,and it is limited because only symbolic data can be manipulated.The combination of rough set theory and fuzzy set theory,i.e.,the fuzzy rough set theory,can deal with real-valued data.Type-2 fuzzy set theory was proposed to deal with uncertainties that exist in fuzzy logic systems and it can manipulate uncertainty better.This paper studies the uncertain problems that are depicted by type-2 fuzzy sets based on the idea of rough modeling,and proposes the rough modeling method of type-2 fuzzy sets.They are of great theoretic significance and application value for the modeling of uncertain data.The main content and achievements of this paper can be summed up as follows:(1)For the definition of type-2 fuzzy rough sets,we use the wavy-slice representation of type-2 fuzzy sets to represent the upper approximation and lower approximation of a type-2 fuzzy set as the sum of approximations of its embedded type-2 sets and propose a new definition of type-2 fuzzy rough sets.The new definition builds a bridge between type-2 fuzzy rough sets and fuzzy rough sets and makes it convenient to generalize properties of fuzzy rough sets to type-2 fuzzy rough sets.Also we compare the existing two definitions of type-2 fuzzy rough sets with the new definition and prove that they are equivalent in the ability of approximate expressing.(2)For the granular structures of type-2 fuzzy rough sets,we consider the new definition of type-2 fuzzy rough sets and find type-2 fuzzy points of every embedded type-2 set.Then “equivalence classes” and “complements of equivalence classes” of these points,i.e.,granular type-2 fuzzy sets are constructed.The granular type-2 fuzzy sets are proved to be basic granules and they can be used to represent the upper and lower approximations of type-2 fuzzy sets by the operations of union and intersection.For type-2 fuzzy rough sets over one universe,we generalize the concept of granular fuzzy sets in fuzzy rough set theory and define two granular type-2 fuzzy sets.For type-2 fuzzy rough sets over two universes,we use the definition of type-2 fuzzy rough sets to deduce the granular type-2 fuzzy sets.(3)For the topological properties of type-2 fuzzy approximation space,we prove that “a reflexive and transitive type-2 fuzzy relation determines a type-2 fuzzy topological space,such that the upper and lower approximation operators are the closure operator and interior operator respectively”,and “if the closure operator and the interior operator of a type-2 fuzzy topological space satisfy some conditions,a reflexive and transitive type-2 fuzzy relation can be determined and the upper approximation and lower approximation relative to the type-2 fuzzy relation equal to the closure operator and the interior operator respectively”.The research results clarify the relationship between type-2 fuzzy approximation spaces and type-2 fuzzy topological spaces.(4)For the belief structures of type-2 fuzzy rough sets,we define the probability measure of an interval type-2 fuzzy set at first.Then the probability measure of a type-2 fuzzy set is defined as the mean of probability measures of its ?-planes based on the ?-plane representation of type-2 fuzzy sets.Finally,we study the belief functions and plausibility functions of type-2 fuzzy rough sets.This paper is intended to build a rough model of type-2 fuzzy sets.A new definition of type-2 fuzzy rough sets is proposed based on the wavy-slice representation of type-2 fuzzy sets,a new type-2 fuzzy rough model is built and the granular structures,topological properties and belief structures of type-2 fuzzy rough sets are discussed in this paper.The fuzzy rough set theory is generalized deeply and a new tool to manipulate uncertainties in data modeling is proposed.