Particle Gaussian Mixture Filters for Nonlinear Non-Gaussian Bayesian Estimation
Abstract
Nonlinear filtering is the problem of estimating the state of a stochastic nonlinear
dynamical system using noisy observations. It is well known that the posterior state
estimates in nonlinear problems may assume non-Gaussian multimodal probability
densities. We present an unscented Kalman-particle hybrid filtering framework
for tracking the three dimensional motion of a space object. The hybrid filtering
scheme is designed to provide accurate and consistent estimates when measurements
are sparse without incurring a large computational cost. It employs an unscented
Kalman filter (UKF) for estimation when measurements are available. When the
target is outside the field of view (FOV) of the sensor, it updates the state probability
density function (PDF) via a sequential Monte Carlo method. The hybrid
filter addresses the problem of particle depletion through a suitably designed filter
transition scheme. The performance of the hybrid filtering approach is assessed by
simulating two test cases of space objects that are assumed to undergo full three
dimensional orbital motion.
Having established its performance in the space object tracking problem, we extend
the hybrid approach to the general multimodal estimation problem. We propose
a particle Gaussian mixture-I (PGM-I) filter for nonlinear estimation that is free of
the particle depletion problem inherent to most particle filters. The PGM-I filter
employs an ensemble of randomly sampled states for the propagation of state probability
density. A Gaussian mixture model (GMM) of the propagated PDF is then
recovered by clustering the ensemble. The posterior density is obtained subsequently
through a Kalman measurement update of the mixture modes. We prove the convergence
in probability of the resultant density to the true filter density assuming
exponential forgetting of initial conditions by the true filter. The PGM-I filter is
capable of handling the non-Gaussianity of the state PDF arising from dynamics,
initial conditions or process noise. A more general estimation scheme titled PGM-II
filter that can also handle non-Gaussianity related to measurement update is considered
next. The PGM-II filter employs a parallel Markov chain Monte Carlo (MCMC)
method to sample from the posterior PDF. The PGM-II filter update is asymptotically
exact and does not enforce any assumptions on the number of Gaussian modes.
We test the performance of the PGM filters on a number of benchmark filtering
problems chosen from recent literature. The PGM filtering performance is compared
with that of other general purpose nonlinear filters such as the feedback particle filter
and the log homotopy based particle flow filters. The results also indicate that the
PGM filters can perform at par with or better than other general purpose nonlinear
filters such as the feedback particle filter (FPF) and the log homotopy based particle
flow filters. Based on the results, we derive important guidelines on the choice between
the PGM-I and PGM-II filters. Furthermore, we conceive an extension of the
PGM-I filter, namely the augmented PGM-I filter, for handling the nonlinear/non-
Gaussian measurement update without incurring a large computational penalty. A
preliminary design for a decentralized PGM-I filter for the distributed estimation
problem is also obtained. Finally we conduct a more detailed study on the performance
of the parallel MCMC algorithm. It is found that running several parallel
Markov chains can lead to significant computational savings in sampling problems
that involve multi modal target densities. We also show that the parallel MCMC
method can be used to solve global optimization problems.
Subject
Nonlinear FilteringGaussian Mixture Models
Kalman Filters
Particle Filtering
Markov Chain Monte Carlo
Optimization
Space Object Tracking.
Citation
Akkam Veettil, Dilshad Raihan (2019). Particle Gaussian Mixture Filters for Nonlinear Non-Gaussian Bayesian Estimation. Doctoral dissertation, Texas A & M University. Available electronically from https : / /hdl .handle .net /1969 .1 /184950.