Quaternion Neuro-fuzzy Learning Algorithm for Fuzzy Rule Generation

In order to generate or tune fuzzy rules, Neuro-Fuzzy learning algorithms with Gaussian type membership functions based on gradient-descent method are well known. In this paper, we propose a new learning approach, the Quaternion Neuro-Fuzzy learning algorithm. This method is an extension of the conventional method to four-dimensional space by using a quaternion neural network that maps quaternion to real values. Input, antecedent membership functions and consequent singletons are quaternion, and output is real. Four-dimensional input can be better represented by quaternion than by real values. We compared it with the conventional method by several function identification problems, and revealed that the proposed method outperformed the counterpart: The number of rules was reduced to 5 from 625, the number of epochs by one fortieth, and error by one tenth in the best cases.