Remarks on Quaternion Multi–Layer Neural Network Based on the Generalised HR Calculus

This study investigates a training method of a quaternion multi–layer neural network based on a gradient– descent method extended to quaternion numbers. The gradient of the cost function is calculated using the generalised ${\mathbb{H}}{\mathbb{R}}$ calculus to derive the training rule for the network parameters. Computational experiments for identifying and controlling a discrete–time nonlinear plant were conducted to evaluate the proposed method. The results confirmed the feasibility of using the G ${\mathbb{H}}{\mathbb{R}}$ calculus in the quaternion neural network and showed the capability of using the quaternion neural network for a control system application.