Online sequential extreme learning machine (OS-ELM) is one of the most popular real-time learning strategy for feedforward neural networks with single hidden layer due to its fast learning speed and excellent generalization ability. When dealing with quaternion signals, traditional real-valued learning models usually provide only suboptimal solutions compared with their quaternion-valued counterparts. However, online sequential quaternion extreme learning machine (OS-QELM) model is still lacking in literature. To fill this gap, this paper aims to establish a framework for the derivation and the design of OS-QELM. Specifically, we first derive a standard OS-QELM, and then propose two augmented OS-QELM models which can capture the complete second-order statistics of noncircular quaternion signals. The corresponding regularized models and two approaches to reducing the computational complexity are also derived and discussed respectively. Benefiting from the quaternion algebra and the augmented structure, the proposed models exhibit superiority over OS-ELM in simulation results on several benchmark quaternion regression problems and colour face recognition problems.

在线顺序极限学习机（OS-ELM）由于具有快速的学习速度和出色的泛化能力，是具有单个隐藏层的前馈神经网络最流行的实时学习策略之一。当处理四元数信号时，与四元数值对应的模型相比，传统的实值学习模型通常仅提供次优的解决方案。但是，文献中仍然缺少在线顺序四元数极限学习机（OS-QELM）模型。为了填补这一空白，本文旨在为OS-QELM的推导和设计建立一个框架。具体来说，我们首先导出标准OS-QELM，然后提出两个增强的OS-QELM模型，它们可以捕获非圆形四元数信号的完整二阶统计量。分别推导并讨论了相应的正则化模型和两种降低计算复杂度的方法。受益于四元数代数和增强结构，在几个基准四元数回归问题和彩色人脸识别问题的仿真结果中，所提出的模型在OS-ELM方面表现出优势。