A Deterministic Filter for non-Gaussian Bayesian Estimation

We present a fully deterministic method to compute sequential updates for stochastic state estimates of dynamic models from noisy measurements. It does not need any assumptions about the type of distribution for either data or measurement — in particular it does not have to assume any of them as Gaussian. It is based on a polynomial chaos expansion (PCE) of the stochastic variables of the model. We use a minimum variance estimator that combines an a priori state estimate and noisy measurements in a Bayesian way. For computational purposes, the update equation is projected onto a finite-dimensional PCE-subspace. The resulting Kalman-type update formula for the PCE coefficients can be efficiently computed solely within the PCE. As it does not rely on sampling, the method is deterministic, robust, and fast. In this paper we discuss the theory and practical implementation of the method. The original Kalman filter is shown to be a low-order special case. In a first experiment, we perform a bi-modal identification using noisy measurements. Additionally, we provide numerical experiments by applying it to the well known Lorenz-84 model and compare it to a related method, the ensemble Kalman filter.