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We consider estimating the parameters of a Gaussian mixture density with a given number of components best representing a given set of weighted samples. We adopt a density interpretation of the samples by viewing them as a discrete Dirac mixture density over a continuous domain with weighted components. Hence, Gaussian mixture fitting is viewed as density re-approximation. In order to speed up computation, an expectation-maximization method is proposed that properly considers not only the sample locations, but also the corresponding weights. It is shown that methods from literature do not treat the weights correctly, resulting in wrong estimates. This is demonstrated with simple counterexamples. The proposed method works in any number of dimensions with the same computational load as standard Gaussian mixture estimators for unweighted samples.
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Covariance matrix estimation concerns the problem of estimating the covariance matrix from a collection of samples, which is of extreme importance in many applications. Classical results have shown that $O(n)$ samples are sufficient to accurately estimate the covariance matrix from $n$-dimensional independent Gaussian samples. However, in many practical applications, the received signal samples might be correlated, which makes the classical analysis inapplicable. In this paper, we develop a non-asymptotic analysis for the covariance matrix estimation from correlated Gaussian samples. Our theoretical results show that the error bounds are determined by the signal dimension $n$, the sample size $m$, and the shape parameter of the distribution of the correlated sample covariance matrix. Particularly, when the shape parameter is a class of Toeplitz matrices (which is of great practical interest), $O(n)$ samples are also sufficient to faithfully estimate the covariance matrix from correlated samples. Simulations are provided to verify the correctness of the theoretical results.
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This work examines the problem of using finite Gaussian mixtures (GM) probability density functions in recursive Bayesian peer-to-peer decentralized data fusion (DDF). It is shown that algorithms for both exact and approximate GM DDF lead to the same problem of finding a suitable GM approximation to a posterior fusion pdf resulting from the division of a `naive Bayes fusion GM (representing direct combination of possibly dependent information sources) by another non-Gaussian pdf (representing removal of either the actual or estimated `common information between the information sources). The resulting quotient pdf for general GM fusion is naturally a mixture pdf, although the fused mixands are non-Gaussian and are not analytically tractable for recursive Bayesian updates. Parallelizable importance sampling algorithms for both direct local approximation and indirect global approximation of the quotient mixture are developed to find tractable GM approximations to the non-Gaussian `sum of quotients mixtures. Practical application examples for multi-platform static target search and maneuverable range-based target tracking demonstrate the higher fidelity of the resulting approximations compared to existing GM DDF techniques, as well as their favorable computational features.
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