A Study On The Theory Of Fuzzy Rough Sets

Rough set theory, a new mathematical tool dealing with vagueness and uncertainty, was introduced by Zdzislaw Pawlak in 1982. It has been widely used in the area of AI, data mining, pattern recognition, fault diagnostics, etc.In the past several years, numerous papers have been published that deal with the relationships between the theories of fuzzy sets and rough sets, and with the possibility of combining them. One of the conclusions of most of those researches is that both theories are complementary, as they represent different aspects of uncertainty. Indeed, fuzzy sets theory is an optimal tool in knowledge engineering to deal with the vagueness implicit in human language. Rough sets theory, on the other hand, is mainly concerned with the coarseness with which a set or concept can be defined.In practice, the extraction of knowledge from a set of measured data will face both problems: the attributes of this data can be presented in linguistic terms (which will usually be subjective and vague), and a rough description of the data elements may be a cause for indiscernibility among some of them. For this reason, several extensions of the original and deterministic rough sets theory have been proposed that can handle data that may have a fuzzy nature. This has resulted in the so called Fuzzy Rough Sets concept, which can be conceived under several forms and from which an increasing number of applications has been derived.The concept of fuzzy rough sets was initially proposed by Dubois and Prade (1990). By extending the crisp set X to the fuzzy set F, Dubois and Prade defined the notion of rough fuzzy sets (a special case of fuzzy rough sets), and fuzzy rough sets with respect to the fuzzy relation R. Bodjanova (1997) presented a new model named as modified fuzzy rough sets which was based on degrees of inclusion of fuzzy sets. Moris and Yakout (1998) studied fuzzy T-rough sets defined by fuzzy T-similarity relations, and originally brought in fuzzy logical operator R-implication.Up to now, the most distinguish studies were made by Radzikowska and Kerre (2002). They defined a broad family of fuzzy rough sets, each one of which, called an (T, J)-fuzzy rough set, is determined by an implicator T and a triangular norm T.Based on this, we replace the fuzzy similarity relation by fuzzy proximity relation (not necessarily transitive), to investigate a more general model of fuzzy rough sets built on the triangular conorm S and the triangular norm T, rather than on the implicator I and the triangular norm T. The new model satisfies the two important identities (—|R)F (?) F (?) (R|—)F and (—|R)F =~ (R|—)(~ F), as well as some other interesting properties.This paper is a study on the theory of fuzzy rough sets. In the first chapter, we introduces the basic ideas of rough sets theory. A survey on the fuzzy rough sets theory is proposed in the second chapter. The aim of the third chapter is to generalize the Radzikowska-Kerre definition and present a more general model of fuzzy rough sets, called (S, T)-fuzzy rough sets , based on fuzzy similarity relations as well as fuzzy logical operators. The forth chapter is to try to introduce the variable precision methods into the fuzzy information processing. The last chapter is the conclusion part.