Currently, the quaternion-valued feedforward neural network (QFNN) has been proposed for image compression and has a more superior performance than the real-valued feedforward neural network (FNN). However, the used quaternion activation function is a split quaternion function, thus it may not preserve the cross-information within the components of the data and for time series forecast, the established model is a strictly linear model which may not be appropriate for noncircular quaternion-valued signal processing.
In this paper, a fully quaternion activation function is employed to design the QFNN and an augmented QFNN (AQFNN) is proposed. They are derived by using recent studies in the augmented quaternion statistics and the HR-calculus. With the augmented quaternion statistics, the AQFNN can process quaternion-valued noncircular signals, effectively. Simulations on both benchmark circular and noncircular quaternion-valued signals, and real-world quaternion-valued signals support the analysis.