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.

翻译：我们提出了一种鲁棒的四元数递归神经网络（QRNN），用于对含有异常值的3D和4D数据进行实时处理。该方法通过结合实时递归学习（RTRL）算法与作为损失函数的最大相关熵准则（MCC）实现。尽管均方误差和最大相关熵准则均为可行的代价函数，但研究表明非二次型最大相关熵损失函数对异常值具有更低敏感性，因此更适用于包含多维噪声或不确定性数据的应用场景。两种算法均基于新型广义HR（GHR）微积分推导，该微积分框架支持对四元数变量实函数进行微分运算，并具备乘积法则与链式法则，从而实现了简洁紧凑的公式推导。针对肺癌放射治疗中胸内标记点运动预测的仿真实验（包含规则与不规则呼吸序列）验证了理论分析的有效性。