Expected likelihood approach for covariance matrix estimation: Complex angular central Gaussian case

We consider the problem of covariance matrix estimation for strongly non-homogeneous data (clutter), modelled as spherically invariant complex random vectors. When normalized, such data are described by a complex angular Gaussian distribution. For a number of independent identically distributed (i.i.d.) samples with this distribution, we introduce the likelihood ratio and demonstrate that for the true covariance matrix, this LR has a p.d.f. which does not depend on the true matrix itself, but rather just its dimension and sample support. This “scenario-invariance” is used to select regularizing parameters that reduce the LR of the MLE estimate to a level statistically equal to the likelihood of the true covariance matrix. We apply this approach to select the loading factor in fixed point loaded covariance matrix estimates, and the order in the fixed point time-varying auto regressive (TVAR) covariance matrix estimate.

I. Background

In most studies (see [1]–[4] for example), inhomogeneous sets of training data are modelled as a set of spherically invariant random samples. Each -variant single sample is modelled as a product of a positive valued random number and a traditional -variate (complex) Gaussian vector

$${\bf x}(t)=\nu_{t}{\bf y}(t),\ t=1, \ldots, T,\ \nu_{t}>0.\eqno{\hbox{(1)}}$$ Here is a positive definite covariance matrix. For presentation (1), is distributed with a p.d.f. that may be expressed as $$\omega({\bf x})=\gamma({\bf x}^{{\rm H}}{\bf R}_{0}^{-1}{\bf x}), \eqno{\hbox{(2)}}$$ where is a function related to the p.d.f. of . For covariance matrix estimation, the spherically invariant random vector (SIRV) is usually transformed into the normalized vector : $${\bf z}(t)={{\bf x}(t)\over \Vert {\bf x}(t)\Vert _{2}}\equiv {{\bf y}(t)\over \Vert {\bf y}(t)\Vert _{2}};\Vert {\bf x}\Vert _{2}=\sqrt{\sum_{i=1}^{M}\vert {\bf x}(t)_{i}\vert ^{2}}.\eqno{\hbox{(3)}}$$