A Tutorial on Bayesian Estimation and Tracking Techniques Applicable to Nonlinear and Non-Gaussian Processes

Nonlinear filtering is the process of estimating and tracking the state of a nonlinear stochastic system from non-Gaussian noisy observation data. In this technical memorandum, we present an overview of techniques for nonlinear filtering for a wide variety of conditions on the nonlinearities and on the noise. We begin with the development of a general Bayesian approach to filtering which is applicable to all linear or nonlinear stochastic systems. We show how Bayesian filtering requires integration over probability density functions that cannot be accomplished in closed form for the general nonlinear, non-Gaussian multivariate system, so approximations are required. Next, we address the special case where both the dynamic and observation models are nonlinear but the noises are additive and Gaussian. The extended Kalman filter (EKF) has been the standard technique usually applied here. But, for severe nonlinearities, the EKF can be very unstable and performs poorly. We show how to use the analytical expression for Gaussian densities to generate integral expressions for the mean and covariance matrices needed for the Kalman filter which include the nonlinearities directly inside the integrals. Several numerical techniques are presented that give approximate solutions for these integrals, including Gauss-Hermite quadrature, unscented filter, and Monte Carlo approximations. We then show how these numerically generated integral solutions can be used in a Kalman filter so as to avoid the direct evaluation of the Jacobian matrix associated with the extended Kalman filter. For all filters, step-by-step block diagrams are used to illustrate the recursive implementation of each filter. To solve the fully nonlinear case, when the noise may be non-additive or non-Gaussian, we present several versions of particle filters that use importance sampling. Particle filters can be subdivided into two categories: those that re-use particles and require resampling to prevent divergence, and those that do not re-use particles and therefore require no resampling. For the first category, we show how the use of importance sampling, combined with particle re-use at each iteration, leads to the sequential importance sampling (SIS) particle filter and its special case, the bootstrap particle filter. The requirement for resampling is outlined and an efficient resampling scheme is presented. For the second class, we discuss a generic importance sampling particle filter and then add specific implementations, including the Gaussian particle filter and combination particle filters that bring together the Gaussian particle filter, and either the Gauss-Hermite, unscented, or Monte Carlo Kalman filters developed above to specify a Gaussian importance density. When either the dynamic or observation models are linear, we show how the Rao-Blackwell simplifications can be applied to any of the filters presented to reduce computational costs. We then present results for two nonlinear tracking examples, one with additive Gaussian noise and one with non-Gaussian embedded noise. For each example, we apply the appropriate nonlinear filters and compare performance results.