  We develop a robust quaternion recurrent neural network (QRNN) for real-time
processing of 3D and 4D data with outliers. This is achieved by combining the
real-time recurrent learning (RTRL) algorithm and the maximum correntropy
criterion (MCC) as a loss function. While both the mean square error and
maximum correntropy criterion are viable cost functions, it is shown that the
non-quadratic maximum correntropy loss function is less sensitive to outliers,
making it suitable for applications with multidimensional noisy or uncertain
data. Both algorithms are derived based on the novel generalised HR (GHR)
calculus, which allows for the differentiation of real functions of quaternion
variables and offers the product and chain rules, thus enabling elegant and
compact derivations. Simulation results in the context of motion prediction of
chest internal markers for lung cancer radiotherapy, which includes regular and
irregular breathing sequences, support the analysis.
