Quaternion neuro-fuzzy learning algorithm for generation of fuzzy rules.

The conventional neuro-fuzzy learning algorithm with Gaussian membership functions based on the gradient descent method can generate or tune fuzzy rules. However, increasing the number of inputs greatly increases the number of parameters. Thus, representing fuzzy rule tables is difficult, learning time increases, and learning accuracy decreases. To overcome these problems, we have developed a complex-valued neuro-fuzzy learning algorithm that extends the neuro-fuzzy learning algorithm to complex numbers. In the method, the inputs, antecedent membership functions, and consequent singletons are complex numbers, but the outputs are real. For converting complex to real numbers, we proposed two types of activation function. The complex-valued method reduced the parameter numbers and showed better learning accuracy. In this paper, we extend the method to the quaternion domain. In the quaternion neuro-fuzzy learning algorithm, the inputs, antecedent membership functions, and consequent singletons are quaternion, and the outputs are real. For parameter tuning, we derived the quaternion back propagation of quaternion neural networks that outputs real values for quaternion-valued inputs. The quaternion back propagation shows better learning convergence and accuracy than the conventional back propagation, and the tuning process is more complex, although it benefits from the quaternion back propagation. The method assigns a four-dimensional real number to one real and three imaginary parts of a quaternion number, which is used as a single quaternion input. This process greatly reduces the number of tuned parameters, leading to better learning than the conventional method. We compare the proposed and conventional methods using several function identification problems, and show that the proposed method outperforms its counterpart, making it a useful tool for learning in a fuzzy system model. In the best cases, the number of epochs was reduced to one-fortieth and the error to one-thirtieth of those in the conventional method. [ABSTRACT FROM AUTHOR]