Widely Linear Quaternion Unscented Kalman Filter for Quaternion-Valued Feedforward Neural Network

Recently, the quaternion-valued feedforward neural network (QFNN) has been developed to process three dimensional (3-D) and 4-D signals in quaternion domain, and the weight matrices and bias vectors of the QFNN were obtained based on the quaternion backward propagation (QBP) method. However, it should be noted that the QBP is a first-order quaternion gradient descent algorithm. The convergence speed of the QBP is usually slow and may not be very suitable to process nonstationary quaternion-valued signals. To address this problem, a widely linear quaternion unscented Kalman filter (WLQUKF) algorithm is proposed to train the QFNN. This is derived by utilizing some recent studies in the augmented quaternion statistics and the $\mathbb {HR}$ -calculus. With the augmented quaternion statistics, the WLQUKF is able to process general quaternion-valued noncircular, nonlinear, and nonstationary signals, effectively. Simulations on both benchmark circular and noncircular quaternion-valued signals, and on real-world quaternion-valued signals support the analysis.