A 3uFunction representation for Gaussian type-2 fuzzy sets

The computational complexities in Gaussian type-2 fuzzy sets (GT2FSs) are main obstacles to apply these sets to modeling high order uncertainties. To model a Gaussian uncertainty this paper introduces a new representation named 3uFunction for GT2FSs. This representation is a development of 2uFunction representation. In fact, it is based on a Hierarchical Fuzzy Convolution (HFC) and consists of two steps, first step is to apply Fuzzy Convolution (FC) on two Interval Type-2 Fuzzy Sets (IT2FSs), as a result a Trapezoidal Type-2 Fuzzy Set (TT2FS) is obtained, and last step is to apply FC on the obtained TT2FS and a third IT2FS. Results show that applying HFC on three IT2FSs can represent a GT2FS or a Bell Type-2 Fuzzy Sets (BT2FS).