A Tutorial on Bayesian Estimation and
Tracking Techniques Applicable to
Nonlinear and Non-Gaussian Processes
February 2005
A.J. Haug, The MITRE Corporation
ABSTRACT
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.
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