  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.
