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The chapter develops quaternion calculus and algebra needed for algorithmic derivations and introduces mean square estimation for practical considerations. 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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 computer graphics. 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  \n• augmented statistics for modelling quaternion-valued random processes,• widely linear models to exploit such advanced statistics,  \n• quaternion calculus and algebra for algorithmic derivations,  \n• mean square estimation for practical considerations.  \nFor ease of exposure, several examples are offered to facilitate the learning, understanding, and (hopefully) the adoption of this multidimensional domain.  \nKeywords: Quaternion Algebra; Augmented Quaternion Variable; Augmented Quaternion Statistics; HR-Calculus; Quaternion-Valued Learning Techniques.  \nMATHEMATICAL NOMENCLATURE  \nNomenclature: Scalars, column vectors, and matrices are denoted by lowercase, bold lowercase, and bold uppercase letters respectively. The remainder of nomenclature is summarised as follows:  \nN set of natural numbers  \nR set of real-valued numbers  \n2  \nR+ set of positive real-valued numbers  \nC set of complex-valued numbers  \nH set of quaternion-valued numbers  \nℜ{·} operator returning the real component ℑ{·} operator returning the imaginary component  \n{􀁹, 􀁺, 􀁞} imaginary units spanning the imaginary subspace of H  \n􀁛 a generic imaginary unit, which can be one of {􀁹, 􀁺, 􀁞}  \nℑ􀁪 (·) operator returning the imaginary component alongside 􀁪 ∈ {􀁹, 􀁺, 􀁞}∥ · ∥ second-order norm  \nI identity matrix of appropriate size  \n(·) ∗ conjugate operator  \n(·)T transpose operator (·)H Hermitian operator  \n􀀴 Euler’s number  \nE {·} statistical expectation operator  \n􀁭 partial derivative operator  \nd total differential operators  \n∇􀁪 gradient operator with respect to 􀁪  \ndet (·) determinant operator ⟨·, ·⟩ inner product × cross product  \n6.1 INTRODUCTION  \nMultidimensional signal processing is typically associated with the real domain, R, and the vector space. Perhaps, a possible explanation for this assumption lies in the maturity of these domains over the hypercomplex domain of quaternions. Indeed, quaternions remain a relatively unexplored area in multidimensional signal processing. This presents opportunities for further research. To exploit such opportunities, this chapter aims to present the fundamentals of this hypercomplex domain based on the so-called widely linear processing model introduced in Section 6.4. This model implies that considering a quaternion variable in isolation is not sufficient to describe or capture the complete information of thequaternion. The aim of this chapter is to address this shortcoming. We first start with the history of quaternions to put in context the recent advances in quaternion-valued signal processing.  \nThe need for a number system beyond R became pressing in the 16th century with the quest to derive a general solution to find roots of polynomials of arbitrary degree. Most notable in this area is the work of Niccolo Tartagila and Girolamo Cardano, who contributed to closed-form solutions for finding roots of polynomials of third and fourth degree Mandic and Goh (2009) . Rafael Bombelli, in his work on","cbCainSFyaSxaT0x","https://ap.wps.com/l/cbCainSFyaSxaT0x","pdf",1700309,39,"English","# Chapter 6: Hypercomplex Widely Linear Processing Fundamentals for Quaternion Machine Learning\n## Abstract and keywords\n## Mathematical nomenclature\n## 6.1 Introduction","[{\"question\":\"Why can’t quaternion variables be treated in isolation for learning and signal processing?\",\"answer\":\"Because a quaternion’s complete information cannot be fully captured by considering only the isolated quaternion variable. The chapter motivates a widely linear processing model to address this shortcoming.\"},{\"question\":\"What is the role of augmented statistics in the proposed framework?\",\"answer\":\"Augmented statistics are used to model quaternion-valued random processes and to enable widely linear models to exploit the richer second-order information.\"},{\"question\":\"Which mathematical tools does the chapter emphasize for derivations and implementation?\",\"answer\":\"The chapter emphasizes quaternion calculus and algebra for algorithmic derivations, and mean square estimation for practical considerations.\"}]","Hypercomplex Widely Linear Processing Fundamentals for Quaternion Machine Learning - Chapter 6 - Fundamentals | PDF",1789642647,98]