Hybridization of Fuzzy and Rough Sets: Present and Future

Though fuzzy set theory has been a very popular technique to represent vagueness between sets and their elements, the approximation of a subset in a universe that contains finite objects was still not resolved until the Pawlak’s rough set theory was introduced. The concept of rough sets was introduced by Pawlak in 1982 as a formal tool for modeling and processing incomplete information in information systems. Rough sets describe the approximation of an arbitrary subset of a universe by two definable or observable subsets called lower and upper approximations. Even though Pawlak’s rough set theory has been widely applied to solve many real world problems, the problem of being not able to deal with real attribute values had been spotted and found. This problem is originated in the crispness of upper and lower approximation sets in traditional rough set theory (TRS). Under the TRS philosophy, two nearly identical real attribute values are unreasonably treated as two different values. TRS theory deals with this problem by discretizing the original dataset, which may result in unacceptable information loss for a large amount of applications. To solve the above problem, a natural way of combining fuzzy sets and rough sets has been proposed. Since 1990’s, researchers had put a lot of efforts on this area and two fuzzy rough set techniques that hybridize fuzzy and rough sets had been proposed to extend the capabilities of both fuzzy sets and rough sets. This chapter does not intend to cover all fuzzy rough set theories. Rather, it firstly gives a brief introduction of the state of the art in this research area and then goes into details to discuss two kinds of well developed hybridization approaches, i.e., constructive and axiomatic approaches. The generalization for equivalence relationships, the definitions of lower and upper approximation sets and the attribute reduction techniques based on these two hybridization frameworks are introduced in different sections. After that, to help readers apply the fuzzy rough set techniques, this chapter also introduces some applications that have successfully applied fuzzy rough set techniques. The final section of this chapter gives some remarks on the merits and problems of each fuzzy rough hybridization technique and the possible research directions in the future.