Kalman Filters for Nonlinear Systems and Heavy-Tailed Noise

This thesis is on filtering in state space models. First, we examine approximate Kalman filters for nonlinear systems, where the optimal Bayesian filtering recursions cannot be solved exactly. These algorithms rely on the computation of certain expected values. Second, the problem of filtering in linear systems that are subject to heavy-tailed process and measurement noise is addressed. Expected values of nonlinearly transformed random vectors are an essential ingredient in any Kalman filter for nonlinear systems, because of the required joint mean vector and joint covariance of the predicted state and measurement. The problem of computing expected values, however, goes beyond the filtering context. Insights into the underlying integrals and useful simplification schemes are given for elliptically contoured distributions, which include the Gaussian and Student’s t distribution. Furthermore, a number of computation schemes are discussed. The focus is on methods that allow for simple implementation and that have an assessable computational cost. Covered are basic Monte Carlo integration, deterministic integration rules and the unscented transformation, and schemes that rely on approximation of involved nonlinearities via Taylor polynomials or interpolation. All methods come with realistic accuracy statements, and are compared on two instructive examples. Heavy-tailed process and measurement noise in state space models can be accounted for by utilizing Student’s t distribution. Based on the expressions for conditioning and marginalization of t random variables, a compact filtering algorithm for linear systems is derived. The algorithm exhibits some similarities with the Kalman filter, but involves nonlinear processing of the measurements in form of a squared residual in one update equation. The derived filter is compared to state-of-the-art filtering algorithms on a challenging target tracking example, and outperforms all but one optimal filter that knows the exact instances at which outliers occur. The presented material is embedded into a coherent thesis, with a concise introduction to the Bayesian filtering and state estimation problems; an extensive survey of available filtering algorithms that includes the Kalman filter, Kalman filters for nonlinear systems, and the particle filter; and an appendix that provides the required probability theory basis.