Higher Type, Higher Order Fuzzy Sets and Hybrid Fuzzy Sets

Abstract

In this chapter, we provide an introduction to more advanced, hierarchically structured information granules such as those of higher type and higher order. In general, when talking about information granules of higher type, say type-2, we mean information granules whose elements are characterized by membership grades, which themselves are information granules (instead of being plain numeric values). For instance, in type-2 fuzzy sets, membership grades are quantified as fuzzy sets in [0,1] or intervals in the unit interval. Of course, one could envision a plethora of the constructs along this line; for instance, the membership grades could be rough sets or probability density functions (as this is the case in probabilistic sets). When talking about higher order information granules, we mean constructs for which the universe of discourse comprises a family of information granules instead of single elements. Hybridization, on the other hand, is about bringing several formalisms of information granules and using them in an orthogonal setting. This situation is visible in fuzzy probabilities.