1. G. Hendeby Performance Issues in Non-Gaussian Filtering Problems NSSPW ‘06 Corpus Christi College, Cambridge Performance Issues in Non-Gaussian Filtering Problems G. Hendeby, LiU, Sweden R. Karlsson, LiU, Sweden F. Gustafsson, LiU, Sweden N. Gordon, DSTO, Australia

2. G. Hendeby Performance Issues in Non-Gaussian Filtering Problems NSSPW ‘06 Corpus Christi College, Cambridge Motivating Problem – Example I  Linear system:  non-Gaussian process noise  Gaussian measurement noise  Posterior distribution: distinctly non-Gaussian

3. G. Hendeby Performance Issues in Non-Gaussian Filtering Problems NSSPW ‘06 Corpus Christi College, Cambridge Motivating Problem – Example II  Estimate target position based on two range measurements  Nonlinear measurements but Gaussian noise  Posterior distribution: bimodal

4. G. Hendeby Performance Issues in Non-Gaussian Filtering Problems NSSPW ‘06 Corpus Christi College, Cambridge Filters The following filters have been evaluated and compared  Local approximation:  Extended Kalman Filter ( EKF )  Multiple Model Filter ( MMF )  Global approximation:  Particle Filter ( PF )  Point Mass Filter ( PMF, representing truth)

5. G. Hendeby Performance Issues in Non-Gaussian Filtering Problems NSSPW ‘06 Corpus Christi College, Cambridge Filters: EKF EKF: Linearize the model around the best estimate and apply the Kalman filter ( KF ) to the resulting system.

6. G. Hendeby Performance Issues in Non-Gaussian Filtering Problems NSSPW ‘06 Corpus Christi College, Cambridge Filters: MMF  Run several EKF in parallel, and combine the results based on measurements and switching probabilities Filter 1 Filter 2 Filter M Filter 1 Filter 2 Filter M Mix

7. G. Hendeby Performance Issues in Non-Gaussian Filtering Problems NSSPW ‘06 Corpus Christi College, Cambridge Filters: PF Simulate several possible states and compare to the measurements obtained.

8. G. Hendeby Performance Issues in Non-Gaussian Filtering Problems NSSPW ‘06 Corpus Christi College, Cambridge Filters: PMF  Grid the state space and propagate the probabilities according to the Bayesian relations

9. G. Hendeby Performance Issues in Non-Gaussian Filtering Problems NSSPW ‘06 Corpus Christi College, Cambridge Filter Evaluation (1/2) Mean square error ( MSE )  Standard performance measure  Approximates the estimate covariance  Bounded by the Cramér-Rao Lower Bound ( CRLB )  Ignores higher-order moments!

10. G. Hendeby Performance Issues in Non-Gaussian Filtering Problems NSSPW ‘06 Corpus Christi College, Cambridge Filter Evaluation (2/2) Kullback divergence  Compares the distance between two distributions  Captures all moments of the distributions

11. G. Hendeby Performance Issues in Non-Gaussian Filtering Problems NSSPW ‘06 Corpus Christi College, Cambridge Filter Evaluation (2/2) Kullback divergence – Gaussian example  Let  The result depends on the normalized difference in mean and the relative difference in variance

12. G. Hendeby Performance Issues in Non-Gaussian Filtering Problems NSSPW ‘06 Corpus Christi College, Cambridge Example I  Linear system:  non-Gaussian process noise  Gaussian measurement noise  Posterior distribution: distinctly non-Gaussian

13. G. Hendeby Performance Issues in Non-Gaussian Filtering Problems NSSPW ‘06 Corpus Christi College, Cambridge Simulation results – Example I  MSE similar for both KF and PF!  KL is better for PF, which is accounted for by multimodal target distribution which is closer to the truth

14. G. Hendeby Performance Issues in Non-Gaussian Filtering Problems NSSPW ‘06 Corpus Christi College, Cambridge Example II  Estimate target position based on two range measurements  Nonlinear measurements but Gaussian noise  Posterior distribution: bimodal

15. G. Hendeby Performance Issues in Non-Gaussian Filtering Problems NSSPW ‘06 Corpus Christi College, Cambridge Simulation results – Example II (1/2)  MSE differs only slightly for EKF and PF  KD differs more, again since PF handles the non-Gaussian posterior distribution better

16. G. Hendeby Performance Issues in Non-Gaussian Filtering Problems NSSPW ‘06 Corpus Christi College, Cambridge Simulation results – Example II (2/2)  Using the estimated position to determine the likelihood to be in the indicated region  The EKF based estimate differs substantially from the truth

17. G. Hendeby Performance Issues in Non-Gaussian Filtering Problems NSSPW ‘06 Corpus Christi College, Cambridge Conclusions  MSE and Kullback divergence evaluated as performance measures  Important information is missed by the MSE, as shown in two examples  The Kullback divergence can be used as a complement to traditional MSE evaluation

18. G. Hendeby Performance Issues in Non-Gaussian Filtering Problems NSSPW ‘06 Corpus Christi College, Cambridge Thanks for listening Questions?