Properness and Widely Linear Processing of Quaternion Random Vectors

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 Q-proper or C^η-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.