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.

中文翻译：

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四元数随机变量的增强统计：四元数学习机的关键[超复杂信号和图像处理]


矢量传感器数据的学习机自然是在四元数域中开发的，并以四元数统计为基础。为此，我们重新审视离散四元数随机变量（RV）的“增强”表示基础${\bf{q}}^{a}[n]$ ， IE， ${[}{\bf{q}}{[}{n}{]}\;{\bf{q}}^{\imath}{[}{n}{]}\;{\bf{q }}^{\jmath}{[}{n}{]}{\bf{q}}^{\kappa}{[}{n}{]]}$ ，并证明其在处理四元数 RV 的通用性方面的关键作用。这是通过严格考虑增强四元数 RV 并通过除了传统协方差之外还涉及额外的二阶统计量来实现的$E\{{\bf{q}}\mathbf{[}{n}\mathbf{]}{\bf{q}}^{{*}}\mathbf{[}{n}\mathbf{]} \}$ [1] 。为了阐明四元数的有用性，我们考虑其最著名的应用 - 3D 方向 - 并提供纯虚数（纯）四元数的增强统计说明。这里提出的四元数统计可以用于分析现有的和开发新颖的统计机器学习方法，因此可以作为四元数学习机的关键。