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This paper addresses the independent component analysis (ICA) of quaternion random vectors. In particular, we focus on the Gaussian case and therefore only consider the quaternion second-order statistics (SOS), which are given by the covariance matrix and three complementary covariance matrices. First, we derive the necessary and sufficient conditions for the identifiability of the quaternion ICA model, which are based on the definition of the properness profile of a quaternion random variable and more specifically on the concept of rotationally equivalent properness profiles. Second, we show that the maximum-likelihood (ML) approach to the quaternion ICA problem reduces to the approximated joint diagonalization (AJD) of the sample-mean estimates of the covariance and complementary covariance matrices. Unlike the complex case, these four matrices cannot be simultaneously diagonalized in general, and we have to resort to a particular AJD algorithm. The proposed technique, which can be seen as a quasi-Newton method, is based on the local approximation of the nonconvex ML-ICA cost function (a measure of the entropy loss due to the residual correlation among the estimated quaternion sources), and it provides a satisfactory solution of the quaternion ICA model. The performance of the proposed quaternion ML-ICA algorithm, as well as its relationship to the identifiability conditions, are illustrated by means of several numerical examples.

In a recent paper, the second-order statistical analysis of quaternion random vectors has shown that there exist two different kinds of quaternion widely linear processing, which are associated with the two main types of quaternion properness. In this paper, we consider the problem of determining, from a finite number of independent vector observations, whether a quaternion Gaussian vector is proper or not. Specifically, we derive three generalized likelihood ratio tests (GLRTs) for testing the two main kinds of quaternion properness and show that the GLRTs reduce to the estimation of three previously proposed quaternion improperness measures. Interestingly, the three GLRT statistics (improperness measures) can be interpreted as an estimate of the entropy loss due to the quaternion improperness. Additionally, we analyze the case in which the orthogonal basis for the representation of the quaternion vector is unknown, which results in the problem of estimating the principal C-properness direction, i.e., the pure unit quaternion minimizing the C-improperness measure. Although this estimation problem is not convex, we propose a technique based on successive convex approximations, which can be solved in closed form. Finally, some simulation examples illustrate the performance and practical application of the proposed tests.

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. © 2010 IEEE.