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.

我们开发了一个强大的四元数循环神经网络（QRNN），用于实时处理具有异常值的三维和四维数据。这通过将实时递归学习（RTRL）算法与最大相关性准则（MCC）作为损失函数相结合来实现。通过基于广义 HR（GHR）微积分推导出这两种算法，GHR 微积分允许对四元数变量的实函数求导，并提供乘积和链式法则，从而实现了优雅而紧凑的推导。通过在肺癌放疗中对胸部内部标记物的运动预测进行模拟实验，包括常规和不规则的呼吸序列，支持了分析的结论。