Efficient Construction of the Characteristic Function of the Cauchy Estimator Using Basis Functions

A newly enhanced, recursive, and robust Bayesian state estimation algorithm for linear and nonlinear systems, referred to as the Multivariate Cauchy Estimator (MCE), is presented. The algorithm enables robust state estimation performance for applications with more volatile system noises than the Gaussian distribution suggests. This is achieved by over-bounding realistic process and measurement noises with additive, heavy-tailed Cauchy random variables. The characteristic function (CF) of the un-normalized conditional probability density function (ucpdf) is propagated as a growing sum of terms in the MCE. Here, the CF is simplified by replacing the original with a representation of linear parameter vectors that operate on bases composed of indicator functions. This insight can lead to eliminating over 99% of terms that previously comprised this CF. Using graphical processing units, the MCE can exploit its parallel mathematical structure and achieve a fast execution rate. A target tracking example shows the robustness of the MCE over the Kalman filter in both heavy-tailed and Gaussian noise.