Estimating Directional Statistics Using Wavefield Modeling and Mixtures of von-Mises Distributions

This letter considers the problem of estimating the directional probability distribution of wavefields observed by sensor arrays. In particular, the angular distributions of wavefields are assumed to be mixtures of von-Mises distributions. Mixture models facilitate estimating multimodal and skewed angular distributions. The von-Mises distribution is fully defined with two parameters, namely the mean direction (circular mean) and the concentration parameter. The widely-employed Gaussian distribution is not appropriate in directional statistics since its support is the entire real-line instead of the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$[- \pi,\pi)$</tex> </formula> angular domain. A covariance-matching based estimator is proposed for the parameters of a mixture of von-Mises distributions and the corresponding Cramér-Rao lower bound is derived. A closed-form expression for the covariance matrix of the array response due to scattering is also derived based on the wavefield modeling principle. These results remain valid even for real-world conformal arrays with nonidealities including mutual coupling, mounting platform reflections, and array elements with individual directional beampatterns.