Type-Reduced Set structure and the truncated type-2 fuzzy set

In this paper, the Type-Reduced Set (TRS) of the continuous type-2 fuzzy set is considered as an object in its own right. The structures of the TRSs of both the interval and generalised forms of the type-2 fuzzy set are investigated. In each case the respective TRS structure is approached by first examining the TRS of the discretised set. The TRS of a continuous interval type-2 fuzzy set is demonstrated to be a continuous horizontal straight line, and that of a generalised type-2 fuzzy set, a continuous, convex curve. This analysis leads on to the concept of truncation, and the definition of the truncation grade. The truncated type-2 fuzzy set is then defined, whose TRS (and hence defuzzified value) is identical to that of the non-truncated type-2 fuzzy set. This result is termed the Type-2 Truncation Theorem, an immediate corollary of which is the Type-2 Equivalence Theorem which states that the defuzzified values of type-2 fuzzy sets that are equivalent under truncation are equal. Experimental corroboration of the equivalence of the non-truncated and truncated generalised type-2 fuzzy set is provided. The implications of these theorems for uncertainty quantification are explored. The theorem's repercussions for type-2 defuzzification employing the α-Planes Representation are examined; it is shown that the known inaccuracies of the α-Planes Method are deeply entrenched.