Elements of information theory
Elements of information theory
Principal component neural networks: theory and applications
Principal component neural networks: theory and applications
On testing for impropriety of complex-valued Gaussian vectors
IEEE Transactions on Signal Processing
The quaternion LMS algorithm for adaptive filtering of hypercomplex processes
IEEE Transactions on Signal Processing
Second-order statistics of complex signals
IEEE Transactions on Signal Processing
Canonical coordinates and the geometry of inference, rate, andcapacity
IEEE Transactions on Signal Processing
Deterministic CCA-Based Algorithms for Blind Equalization of FIR-MIMO Channels
IEEE Transactions on Signal Processing - Part II
Second-order analysis of improper complex random vectors and processes
IEEE Transactions on Signal Processing
Widely linear estimation with complex data
IEEE Transactions on Signal Processing
The Theory of Quaternion Orthogonal Designs
IEEE Transactions on Signal Processing
Second-order complex random vectors and normal distributions
IEEE Transactions on Signal Processing
A Unifying Discussion of Correlation Analysis for Complex Random Vectors
IEEE Transactions on Signal Processing
Quaternion-MUSIC for vector-sensor array processing
IEEE Transactions on Signal Processing
Detection and estimation of improper complex random signals
IEEE Transactions on Information Theory
Complex random vectors and ICA models: identifiability, uniqueness, and separability
IEEE Transactions on Information Theory
The multivariate complex normal distribution-a generalization
IEEE Transactions on Information Theory
Augmented second-order statistics of quaternion random signals
Signal Processing
A class of quaternion valued affine projection algorithms
Signal Processing
| Hi-index | 754.84 |
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 Cη-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.