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首页> 外文期刊>Information Theory, IEEE Transactions on >Properness and Widely Linear Processing of Quaternion Random Vectors
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Properness and Widely Linear Processing of Quaternion Random Vectors

机译：四元数随机向量的性质和广义线性处理

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摘要

In this paper, the second-order circularity of quaternion random vectors is analyzed. Unlike the case of complex vectors, there exist three different kinds of quaternion properness, which are based on the vanishing of three different complementary covariance matrices. The different kinds of properness have direct implications on the Cayley–Dickson representation of the quaternion vector, and also on several well-known multivariate statistical analysis methods. In particular, the quaternion extensions of the partial least squares (PLS), multiple linear regression (MLR) and canonical correlation analysis (CCA) techniques are analyzed, showing that, in general, the optimal linear processing is full-widely linear. However, in the case of jointly $BBQ $-proper or $BBC ^{eta} $-proper vectors, the optimal processing reduces, respectively, to the conventional or semi-widely linear processing. Finally, a measure for the degree of improperness of a quaternion random vector is proposed, which is based on the Kullback–Leibler divergence between two zero-mean Gaussian distributions, one of them with the actual augmented covariance matrix, and the other with its closest proper version. This measure quantifies the entropy loss due to the improperness of the quaternion vector, and it admits an intuitive geometrical interpretation based on Kullback–Leibler projections onto sets of proper augmented covariance matrices.
机译：本文分析了四元数随机向量的二阶圆度。与复矢量的情况不同，存在三种不同的四元数性质，它们基于三种不同的互补协方差矩阵的消失。不同种类的性质对四元数向量的Cayley-Dickson表示以及几种众所周知的多元统计分析方法都有直接的影响。特别地，分析了偏最小二乘（PLS），多元线性回归（MLR）和规范相关分析（CCA）技术的四元数扩展，表明通常，最佳线性处理是全线性的。但是，在联合$ BBQ $ -proper或$ BBC ^ {eta} $ -proper向量的情况下，最佳处理分别减少到常规或半宽线性处理。最后，基于四项零均值高斯分布之间的Kullback-Leibler散度，提出了一种四元数随机向量的不恰当程度的度量，其中两个零均值高斯分布中的一个具有实际的增强协方差矩阵，而另一个具有其最接近的分布正确的版本。这种度量量化了由于四元数向量的不当性而引起的熵损失，并且它接受了基于Kullback-Leibler投影的直观几何解释，并将其应用于适当的增强协方差矩阵集。

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