Bayesian Overbounding Filter Using Gaussian-Pareto Distributions

We introduce a recursive estimator designed to maintain bounded tail probabilities in Positioning, Navigation and Timing (PNT) problems. This estimator leverages Gaussian-pareto distributions and the Dvoretzky–Kiefer–Wolfowitz inequality to provide conservative upper bounds for sensor inputs constructed from limited experimental data. To facilitate the propagation of Gaussian-pareto distributions through PNT algorithms, we develop a method that employs Gaussian Mixture Models (GMM) to conservatively approximate these distributions. This approach enables efficient evaluation of convolution integrals necessary for density propagation in PNT algorithms. Given that convolutions with Gaussian mixtures can result in an exponential growth in the number of terms required to represent the Gaussian-pareto distribution accurately, we devise a culling algorithm that periodically prunes the Gaussian mixture while ensuring conservative overbounding. We assess the recursive algorithm’s performance using a simplistic, one-dimensional PNT problem and compare it with the traditional Gaussian overbounding methods. Our results consistently demonstrate that the protection levels computed using the proposed algorithm are smaller than those obtained through the traditional Gaussian overbounding method. Given that this recursive estimator exhibited superior performance on this simplistic and linear problem, simulation results suggest that employing it on more complex PNT algorithms may lead to increased availability of PNT systems in practice.