6.891 computer experiments for particle filtering
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6.891 Computer Experiments for Particle Filtering. Yuan Qi MIT Media Lab [email protected] May 7, 2002. Outline. Effect of Resampling in Particle Filtering The Role of Proposal Distribution Transition Prior Proposal EKF Proposal UKF Proposal Effect of Sampling Size Conclusion. - PowerPoint PPT PresentationTRANSCRIPT
6.891 Computer Experiments for Particle Filtering
Yuan Qi
MIT Media Lab
May 7, 2002
Outline
Effect of Resampling in Particle Filtering The Role of Proposal Distribution
– Transition Prior Proposal– EKF Proposal– UKF Proposal
Effect of Sampling Size Conclusion
Tracking Nonlinear and Nonstationary Time Series
known nonlinear process model
known nonlinear and non-stationary observation model
Gamma(3,2) process noise
Zero-mean Gaussian observation noise
The Effect of Resampling
SIS: Sequential Importance-sampling (No Resampling),
200 samples
SIR: Sequential Importance-sampling Resampling,
200 samples
The Effect of Resampling
Without resampling, the variance of the importance weight increases over time. Eventually, one of them comes to one.
Resampling increases the effective sampling size
Problems of Resampling : “Sampling impoverishment”,
Reduction of particle diversity Only resampling when the
effective size is small
Possible Improvements: Increase Number of Samples Regularisation (Parzen
window) MCMC step
The Effect of Proposal Distribution
CONDENSATION: PF with transition prior as the proposal distribution. Only a few particles might survive if the likelihood lies in one of the tails of the prior distribution, or if it is too narrow (low measurement error).
The Effect of Proposal Distribution
PF with Extended Kalman filtering (EKF) proposal PF with Unscented Kalman filtering (UKF) proposal
Unscented Transformation: transform sigma points instead of approximating a nonlinear model
Why UKF? – More accurate variance estimation than EKF. Usually EKF
tends to underestimate the variance.– A heavy-tailed distribution is preferred as proposal
distribution for importance sampling
The Effect of Proposal Distribution
The comparison of PF, PF-EKF, and PF-UKF, 200
samples
Estimated Variances by EKF and UKF for proposal distributions
Particle Histograms of PF, PF-EKF, PF_UKF
Numerical Comparison (1)
Root mean square (RMS) errors
------------------------------------------- PF = 0.60319 PF-MCMC = 0.4572 PF-EKF = 0.50879 PF-EKF-MCMC = 0.5045 PF-UKF = 0.028264 PF-UKF-MCMC = 0.067867
Another Comparison
200 Particles
The Effect of Sampling Size
Estimates by 50 particles Particle Histograms of PF, PF-EKF, PF_UKF
Numerical Comparison (2)
Root mean square (RMS) errors
-------------------------------------------
PF = 0.67369
PF-MCMC = 0.76296
PF-EKF = 0.44347
PF-EKF-MCMC = 0.36801
PF-UKF = 0.16369
PF-UKF-MCMC = 0.11716
Estimates by 10 particles Particle Histograms of PF, PF-EKF, PF_UKF
The Effect of Sampling Size
Numerical Comparison (3)
Root mean square (RMS) errors
-------------------------------------------
PF = 1.223
PF-MCMC = 1.0798
PF-EKF = 0.48827
PF-EKF-MCMC = 0.5141
PF-UKF = 0.54065
PF-UKF-MCMC = 0.48272
Conclusion
Resampling allows a PF relocate particles in important regions.
The quality of proposal distributions greatly affects the performance of a PF.
The performance of a PF degenerates when the sampling size gets smaller.
A MCMC step in a PF often improves the performance.
Future improvement: utilizing heaved tailed distribution, f.g., t distribution, as proposal distribution?
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