A Unifying Discussion of Correlation Analysis for Complex Random Vectors

The assessment of multivariate association between two complex random vectors is considered. A number of correlation coefficients based on three popular correlation analysis techniques, namely canonical correlation analysis, multivariate linear regression, and partial least squares, are reviewed and connected to performance measures in signal processing and communications, such as mean-squared estimation error, mutual information, and signal-to-noise ratio (SNR). For complex data, there are three types of correlation coefficients, which account for rotational, reflectional, and total (i.e., rotational and reflectional) dependencies between two random vectors. These three types are defined and analyzed for different correlation coefficients, and a numerical example is given. It is often required to compare two complex random vectors in a lower-dimensional subspace. For the large class of increasing, Schur-convex correlation coefficients, it is shown that the low-rank approximations of two random vectors maximizing a particular correlation coefficient are determined only by the constraints imposed on the correlation analysis technique. In this context, the correlation spread is defined as a normalized measure of how much of the overall correlation is contained in a low-dimensional subspace.