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6.891 Computer Experiments for Particle Filtering

6.891 Computer Experiments for Particle Filtering. Yuan Qi MIT Media Lab yuanqi@media.mit.edu 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.

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6.891 Computer Experiments for Particle Filtering

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  1. 6.891 Computer Experiments for Particle Filtering Yuan Qi MIT Media Lab yuanqi@media.mit.edu May 7, 2002

  2. Outline • Effect of Resampling in Particle Filtering • The Role of Proposal Distribution • Transition Prior Proposal • EKF Proposal • UKF Proposal • Effect of Sampling Size • Conclusion

  3. 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

  4. SIS: Sequential Importance-sampling (No Resampling), 200 samples SIR: Sequential Importance-sampling Resampling, 200 samples The Effect of Resampling

  5. 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

  6. 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).

  7. 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

  8. The comparison of PF, PF-EKF, and PF-UKF, 200 samples Estimated Variances by EKF and UKF for proposal distributions The Effect of Proposal Distribution

  9. Particle Histograms of PF, PF-EKF, PF_UKF

  10. 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 Numerical Comparison (1)

  11. Another Comparison 200 Particles

  12. Estimates by 50 particles Particle Histograms of PF, PF-EKF, PF_UKF The Effect of Sampling Size

  13. 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

  14. The Effect of Sampling Size Estimates by 10 particles Particle Histograms of PF, PF-EKF, PF_UKF

  15. 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

  16. 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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