26,139 research outputs found

    Particle Gaussian Mixture Filters for Nonlinear Non-Gaussian Bayesian Estimation

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    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

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    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

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    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

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    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

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    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

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    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 LpL_p 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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