Bifurcating fuzzy sets: Theory and application

Abstract

To overcome the complexities in Type-2 Fuzzy Sets (T2FSs) and to present a general model for chaotic FSs, this paper introduces new FSs called Bifurcating FSs (BFSs). The BFS is inspired from human brain. Mathematically, it is equivalent to the coupled chaotic map in which one of the variables is a fuzzy membership function. Biologically, it is equivalent to the oscillatory neuron in which a fuzzy membership function is an activation function for the input neuron. Graphically, it is represented by an input-as-a-bifurcation-parameter diagram. This diagram demonstrates that a BFS can exhibit a wide range of FSs such as convex or non-convex, T1 or T2 and chaotic or non-chaotic FSs, and also make information processing fast. Using a BFS as an activation function in the neurons of the membership layer of a Fuzzy Neural Network (FNN), a new network called a Bifurcating FNN (BFNN) is presented. The proposed network has a similar design to an Interval T2 FNN (IT2FNN). The BFNN is applied to the problem of forecasting Mackey–Glass chaotic time series under different SNRs and different chaotic degrees. In comparison with T1FNNs and IT2FNNs, results show that BFNNs provide more accurate and robust predictions and handle more uncertainties.