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

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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