Abstract
This paper studies probability density estimation on the Siegel space. The Siegel space is a generalization of the hyperbolic space. Its Riemannian metric provides an interesting structure to the Toeplitz block Toeplitz matrices that appear in the covariance estimation of radar signals. The main techniques of probability density estimation on Riemannian manifolds are reviewed. For computational reasons, we chose to focus on the kernel density estimation. The main result of the paper is the expression of Pelletier’s kernel density estimator. The computation of the kernels is made possible by the symmetric structure of the Siegel space. The method is applied to density estimation of reflection coefficients from radar observations.
Highlights
Various techniques can be used to estimate the density of probability measure in the Euclidean spaces, such as histograms, kernel methods, or orthogonal series
Regarding the problem of histograms, the case of the Siegel space is similar to the case of the hyperbolic space
In space time adaptative radar processing (STAP), the signal is formed by a succession of matrices
Summary
Various techniques can be used to estimate the density of probability measure in the Euclidean spaces, such as histograms, kernel methods, or orthogonal series. These methods can sometimes be adapted to densities in Riemannian manifolds. Convergence rates of the density estimation using kernels and orthogonal series were gradually generalized to Riemannian manifolds (see [1,2,3]). Toeplitz block Toeplitz matrices can be represented by a symmetric positive definite matrix and a point laying in a product of Siegel disks.
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