Augmented Statistics of Quaternion Random Variables: A lynchpin of quaternion learning machines [Hypercomplex Signal and Image Processing]

Learning machines for vector sensor data are naturally developed in the quaternion domain and are underpinned by quaternion statistics. To this end, we revisit the “augmented” representation basis for discrete quaternion random variables (RVs)
${\bf{q}}^{a}[n]$
, i.e.,
${[}{\bf{q}}{[}{n}{]}\;{\bf{q}}^{\imath}{[}{n}{]}\;{\bf{q}}^{\jmath}{[}{n}{]}{\bf{q}}^{\kappa}{[}{n}{]]}$
, and demonstrate its pivotal role in the treatment of the generality of quaternion RVs. This is achieved by a rigorous consideration of the augmented quaternion RV and by involving the additional second-order statistics, besides the traditional covariance
$E\{{\bf{q}}\mathbf{[}{n}\mathbf{]}{\bf{q}}^{{*}}\mathbf{[}{n}\mathbf{]}\}$

[1]
. To illuminate the usefulness of quaternions, we consider their most well-known application—3D orientation—and offer an account of augmented statistics for purely imaginary (pure) quaternions. The quaternion statistics presented here can be exploited in the analysis of existing and the development of novel statistical machine learning methods, hence acting as a lynchpin for quaternion learning machines.