performance issues in non-gaussian filtering problems

G. HendebyPerformance Issues in Non-Gaussian Filtering Problems

NSSPW ‘06Corpus 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



Motivating Problem – Example I

Linear system: non-Gaussian process noise Gaussian measurement noise

Posterior distribution:distinctly non-Gaussian



Motivating Problem – Example II

Estimate target position based on two range measurements Nonlinear measurements but Gaussian noise Posterior distribution: bimodal



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)



Filters: EKF

EKF: Linearize the model around the best estimate and apply the Kalman filter (KF) to the resulting system.



Filters: MMF

Run several EKF in parallel, and combine the results based on measurements and switching probabilities

Filter 1Filter 1

Filter 2

Filter M

Filter 1Filter 1

Filter 2

Filter M

Mix



Filters: PF

Simulate several possible states and compare to the measurements obtained.



Filters: PMF

Grid the state space and propagate the probabilities according to the Bayesian relations



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!




Kullback divergence Compares the distance between two distributions Captures all moments of the distributions




Kullback divergence – Gaussian example Let

The result depends on the normalized difference in mean and the relative difference in variance



Example I

Linear system: non-Gaussian process noise Gaussian measurement noise

Posterior distribution:distinctly non-Gaussian



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



Example II

Estimate target position based on two range measurements Nonlinear measurements but Gaussian noise Posterior distribution: bimodal



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



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



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



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performance issues in non-gaussian filtering problems

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