From their inception, quaternions and their division algebra have proven to be advantageous in modelling rotation/orientation in three-dimensional spaces and have seen use from the initial formulation of electromagnetic filed theory through to forming the basis of quantum filed theory. Despite their impressive versatility in modelling real-world phenomena, adaptive information processing techniques specifically designed for quaternion-valued signals have only recently come to the attention of the machine learning, signal processing, and control communities. The most important development in this direction is introduction of the HR-calculus, which provides the required mathematical foundation for deriving adaptive information processing techniques directly in the quaternion domain. In this article, the foundations of the HR-calculus are revised and the required tools for deriving adaptive learning techniques suitable for dealing with quaternion-valued signals, such as the gradient operator, chain and product derivative rules, and Taylor series expansion are presented. This serves to establish the most important applications of adaptive information processing in the quaternion domain for both single-node and multi-node formulations. The article is supported by Supplementary Material, which will be referred to as SM.

翻译：从提出之初，四元数及其除代数就在三维空间旋转/取向建模中展现出优势，其应用范围从电磁场理论的初始形式化直至量子场论的基础构建。尽管四元数在现实世界现象建模中具有卓越的通用性，但专为四元数信号设计的自适应信息处理技术直至近年才引起机器学习、信号处理与控制领域的关注。该方向最重要的进展是HR-微积分的提出，它为直接在四元数域推导自适应信息处理技术提供了必要的数学基础。本文重新审视了HR-微积分的基本原理，并给出了推导适用于四元数信号的自适应学习技术所需的核心工具，包括梯度算子、链式法则与乘积法则、以及泰勒级数展开。这为建立四元数域自适应信息处理在单节点与多节点公式中的最重要应用奠定了基础。本文附有补充材料（以SM指代）。