26,139 research outputs found
Particle Gaussian Mixture Filters for Nonlinear Non-Gaussian Bayesian Estimation
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
Parallelized Particle and Gaussian Sum Particle Filters for Large Scale Freeway Traffic Systems
Large scale traffic systems require techniques able to: 1) deal with high amounts of data and heterogenous data coming from different types of sensors, 2) provide robustness in the presence of sparse sensor data, 3) incorporate different models that can deal with various traffic regimes, 4) cope with multimodal conditional probability density functions for the states. Often centralized architectures face challenges due to high communication demands. This paper develops new estimation techniques able to cope with these problems of large traffic network systems. These are Parallelized Particle Filters (PPFs) and a Parallelized Gaussian Sum Particle Filter (PGSPF) that are suitable for on-line traffic management. We show how complex probability density functions of the high dimensional trafc state can be decomposed into functions with simpler forms and the whole estimation problem solved in an efcient way. The proposed approach is general, with limited interactions which reduces the computational time and provides high estimation accuracy. The efciency of the PPFs and PGSPFs is evaluated in terms of accuracy, complexity and communication demands and compared with the case where all processing is centralized
A New Reduction Scheme for Gaussian Sum Filters
In many signal processing applications it is required to estimate the
unobservable state of a dynamic system from its noisy measurements. For linear
dynamic systems with Gaussian Mixture (GM) noise distributions, Gaussian Sum
Filters (GSF) provide the MMSE state estimate by tracking the GM posterior.
However, since the number of the clusters of the GM posterior grows
exponentially over time, suitable reduction schemes need to be used to maintain
the size of the bank in GSF. In this work we propose a low computational
complexity reduction scheme which uses an initial state estimation to find the
active noise clusters and removes all the others. Since the performance of our
proposed method relies on the accuracy of the initial state estimation, we also
propose five methods for finding this estimation. We provide simulation results
showing that with suitable choice of the initial state estimation (based on the
shape of the noise models), our proposed reduction scheme provides better state
estimations both in terms of accuracy and precision when compared with other
reduction methods
An Introduction to Twisted Particle Filters and Parameter Estimation in Non-linear State-space Models
Twisted particle filters are a class of sequential Monte Carlo methods
recently introduced by Whiteley and Lee to improve the efficiency of marginal
likelihood estimation in state-space models. The purpose of this article is to
extend the twisted particle filtering methodology, establish accessible
theoretical results which convey its rationale, and provide a demonstration of
its practical performance within particle Markov chain Monte Carlo for
estimating static model parameters. We derive twisted particle filters that
incorporate systematic or multinomial resampling and information from
historical particle states, and a transparent proof which identifies the
optimal algorithm for marginal likelihood estimation. We demonstrate how to
approximate the optimal algorithm for nonlinear state-space models with
Gaussian noise and we apply such approximations to two examples: a range and
bearing tracking problem and an indoor positioning problem with Bluetooth
signal strength measurements. We demonstrate improvements over standard
algorithms in terms of variance of marginal likelihood estimates and Markov
chain autocorrelation for given CPU time, and improved tracking performance
using estimated parameters.Comment: This work has been submitted to the IEEE for possible publication.
Copyright may be transferred without notice, after which this version may no
longer be accessibl
Particle Learning and Smoothing
Particle learning (PL) provides state filtering, sequential parameter
learning and smoothing in a general class of state space models. Our approach
extends existing particle methods by incorporating the estimation of static
parameters via a fully-adapted filter that utilizes conditional sufficient
statistics for parameters and/or states as particles. State smoothing in the
presence of parameter uncertainty is also solved as a by-product of PL. In a
number of examples, we show that PL outperforms existing particle filtering
alternatives and proves to be a competitor to MCMC.Comment: Published in at http://dx.doi.org/10.1214/10-STS325 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Analysis of error propagation in particle filters with approximation
This paper examines the impact of approximation steps that become necessary
when particle filters are implemented on resource-constrained platforms. We
consider particle filters that perform intermittent approximation, either by
subsampling the particles or by generating a parametric approximation. For such
algorithms, we derive time-uniform bounds on the weak-sense error and
present associated exponential inequalities. We motivate the theoretical
analysis by considering the leader node particle filter and present numerical
experiments exploring its performance and the relationship to the error bounds.Comment: Published in at http://dx.doi.org/10.1214/11-AAP760 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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