Research And Application Of Fuzzy λ Quotient Space

The original intention of granular computing is to solve complex problems, but the real issues are often difficult to be characterized by accurate data. So the research of fuzzy granular computing is an inevitable trend. Through the research, the intrinsic nature of different models of granular computing is revealed more thoroughly. Fuzzy quotient space theory is an important branch of fuzzy granular computing, and can reflect several features excellently when people deal with the uncertain problems.Two equivalent narratives in fuzzy quotient space theory are revised and expanded into three ones. By using the distance function of quotient space X(λ) it refines the fuzzyλquotient space and describes some basic concepts and properties. Because of avoiding the definition of a new fuzzy equivalence relation, fuzzy quotient space theory is inherited more directly. Then the information quantity of fuzzy-rough probability approximation space is extended to the one of fuzzyλquotient space. In particular, when the fuzzy equivalence relation is a crisp equivalence relation, the proposed information measure is identical to Shannon's one of rough set theory. And the reduction rules of fuzzy information system based on fuzzyλquotient space theory are presented so that fuzzyλquotient space can get the same reduction results as fuzzy-rough set.In the traditional rough set model, direct discretization of numerical attributes usually brings information loss and fuzzy attributes are not taken into consideration. As a result, two heuristic reduction algorithms on the basis of fuzzyλquotient space theory are illustrated. The relative reduction for discrete and mixed range decision-making systems can be calculated effectively with these algorithms. Therefore, the application scope of fuzzyλquotient space is enlarged. Additionally, in these algorithms people can change the threshold valueλto obtain their satisfactory results of attribute reduction according to the actual decision-making needs and domain knowledge. Finally, simulation results and theoretical analysis demonstrate the effectiveness and superiority of these algorithms.