Independent component analysis of quaternion Gaussian vectors

This paper addresses the independent component analysis (ICA) of quaternion Gaussian vectors. Firstly, we define the properness profile of a quaternion random variable, which can be seen as the quaternion analogue of the circularity coefficients of complex vectors. The properness profile is a three-dimensional pure quaternion vector, which does not only measure the improperness degree of the quaternion random variable, but also provides the improperness distribution. Secondly, we prove that the quaternion ICA model can be identified up to the trivial scale and permutation ambiguities, and a residual quaternion mixture among the sources with rotationally equivalent properness profiles, i.e., properness profiles related by a quaternion rotation. Finally, the main results of the paper are illustrated by means of some numerical examples.