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Nonlinear and Non-Gaussian Estimation with A Focus on Particle Filters Prasanth Jeevan Mary Knox May 12, 2006

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Background Optimal linear filters Wiener Stationary Kalman Gaussian Posterior, p(x|y) Filters for nonlinear systems Extended Kalman Particle

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Extended Kalman Filter (EKF) Locally linearize the non-linear functions Assume p(x k |y 1,…,k ) is Gaussian

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Particle Filter (PF) Weighted point mass or “particle” representation of possibly intractable posterior probability density functions, p(x|y) Estimates recursively in time allowing for online calculations Attempts to place particles in important regions of the posterior pdf O(N) complexity on number of particles

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Particle Filter Background [Ristic et. al. 2004] Monte Carlo Estimation Pick N>>1 “particles” with distribution p(x) Assumption: x i is independent

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Importance Sampling Cannot sample directly from p(x) Instead sample from known importance density, q(x), where: Estimate I from samples and importance weights where

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Sequential Importance Sampling (SIS) Iteratively represent posterior density function by random samples with associated weights Assumptions: x k Hidden Markov process, y k conditionally independent given x k

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Degeneracy Variance of sample weights increases with time if importance density not optimal [Doucet 2000] In a few cycles all but one particle will have negligible weights PF will updating particles that contribute little in approximating the posterior N eff, estimate of effective sample size [Kong et. al. 1994] :

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Optimal Importance Density [Doucet et. al. 2000] Minimizes variance of importance weights to prevent degeneracy Rarely possible to obtain, instead often use

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Resampling Generate new set of samples from: Weights are equal after i.i.d. sampling O(N) complexity Coupled with SIS, these are the two key components of a PF

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Sample Impoverishment Set of particles with low diversity Particles with high weights are selected more often

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Sampling Importance Resampling (SIR) [Gordon et. al. 1993] Importance density is the transitional prior Resampling at every time step

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SIR Pros and Cons Pro: importance density and weight updates are easy to evaluate Con: Observations not used when transitioning state to next time step

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A Cycle of SIR

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Auxiliary SIR - Motivation [Pitt and Shephard 1999] Want to use observation when exploring the state space ( ’s) To have particles in regions of high likelihood Incorporate into resampling at time k-1 Looking one step ahead to choose particles

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ASIR - from SIR From SIR we had If we move the likelihood inside we get: We don’t have though Use, a characterization of given such as

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ASIR continued So then we get: And the new importance weight becomes:

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ASIR Pros & Cons Pro Can be less sensitive to peaked likelihoods and outliers by using observation Outliers - Model-improbable states that can result in a dramatic loss of high-weight particles Cons Added computation per cycle If is a bad characterization of (ie. large process noise), then resampling suffers, and performance can degrade

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Simulation Linear System Equations: where v ~ N(0,6) and w ~ N(0,5)

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Simulation Linear 10 Samples

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Simulation Linear 50 Samples

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Simulation Linear Table 1: Mean Squared Error Per Time Step Number of Particles Filter10501001000 KF0.03490.0351 0.0350 0.0352 ASIR0.77920.08860.04170.0350 SIR0.90530.09770.04960.0354

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Simulation Nonlinear System Equations: where v ~ N(0,6) and w ~ N(0,5)

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Simulation Nonlinear 10 Samples

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Simulation Nonlinear 50 Samples

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Simulation Nonlinear 100 Samples

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Simulation Nonlinear 1000 Samples

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Simulation Nonlinear Table 2: Mean Squared Error Per Time Step Number of Particles Filter10501001000 EKF 812.08826.20827.94838.75 ASIR 30.14 20.1518.8117.86 SIR37.9722.62 21.49 19.78

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Conclusion PF approaches KF optimal estimates as N PF better than EKF for nonlinear systems ASIR generates ‘better particles’ in certain conditions by incorporating the observation PF is applicable to a broad class of system dynamics Simulation approaches have their own limitations Degeneracy and sample impoverishment

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Conclusion (2) Particle filters composed of SIS and resampling Many variations to improve efficiency (both computationally and for getting ‘better’ particles) Other PFs: Regularized PF, (EKF/UKF)+PF, etc.

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