1. 6.891 Computer Experiments for Particle Filtering
Yuan Qi
MIT Media Lab
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. The Effect of Resampling
SIS: Sequential Importance-sampling (No Resampling),
200 samples
SIR: Sequential Importance-sampling Resampling,
200 samples

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

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

10. Numerical Comparison (1)
Root mean square (RMS) errors
PF =
PF-MCMC =
PF-EKF =
PF-EKF-MCMC =
PF-UKF =
PF-UKF-MCMC =

11. Another Comparison
200 Particles

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

13. Numerical Comparison (2)
Root mean square (RMS) errors
PF =
PF-MCMC =
PF-EKF =
PF-EKF-MCMC =
PF-UKF =
PF-UKF-MCMC =

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 =
PF-EKF =
PF-EKF-MCMC =
PF-UKF =
PF-UKF-MCMC =

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?

17. E N D