Numerous attempts have been made to replicate the success of complex-valued algebra in engineering and science to other hypercomplex domains such as quaternions, tessarines, biquaternions, and octonions. Perhaps, none have matched the success of quaternions. The most useful feature of quaternions lies in their ability to model three-dimensional rotations which, in turn, have found various industrial applications such as in aeronautics and computergraphics. Recently, we have witnessed a renaissance of quaternions due to the rise of machine learning. To equip the reader to contribute to this emerging research area, this chapter lays down the foundation for: - augmented statistics for modelling quaternion-valued random processes, - widely linear models to exploit such advanced statistics, - quaternion calculus and algebra for algorithmic derivations, - mean square estimation for practical considerations. For ease of exposure, several examples are offered to facilitate the learning, understanding, and(hopefully) the adoption of this multidimensional domain.

翻译：在工程与科学领域，众多尝试致力于将复数代数的成功经验推广至其他超复数域，如四元数、镶嵌数、双四元数与八元数。其中，四元数取得的成就或许最为显著。四元数最具实用价值的特性在于其能够对三维旋转进行建模，该特性已在航空学与计算机图形学等多个工业领域获得广泛应用。近年来，随着机器学习的兴起，四元数研究迎来了复兴。为帮助读者参与这一新兴研究领域，本章将奠定以下基础：- 面向四元数值随机过程建模的增广统计学方法，- 利用此类先进统计学的广泛线性模型，- 用于算法推导的四元数微积分与代数体系，- 基于均方估计的实际应用考量。为便于理解，本章通过若干实例促进对这一多维域的学习、理解与（期望中的）实际应用。