performance issues in non-gaussian filtering problems
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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
G. HendebyPerformance Issues in Non-Gaussian Filtering Problems
NSSPW ‘06Corpus Christi College, Cambridge
Motivating Problem – Example I
Linear system: non-Gaussian process noise Gaussian measurement noise
Posterior distribution:distinctly non-Gaussian
G. HendebyPerformance Issues in Non-Gaussian Filtering Problems
NSSPW ‘06Corpus Christi College, Cambridge
Motivating Problem – Example II
Estimate target position based on two range measurements Nonlinear measurements but Gaussian noise Posterior distribution: bimodal
G. HendebyPerformance Issues in Non-Gaussian Filtering Problems
NSSPW ‘06Corpus 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)
G. HendebyPerformance Issues in Non-Gaussian Filtering Problems
NSSPW ‘06Corpus Christi College, Cambridge
Filters: EKF
EKF: Linearize the model around the best estimate and apply the Kalman filter (KF) to the resulting system.
G. HendebyPerformance Issues in Non-Gaussian Filtering Problems
NSSPW ‘06Corpus Christi College, Cambridge
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
G. HendebyPerformance Issues in Non-Gaussian Filtering Problems
NSSPW ‘06Corpus Christi College, Cambridge
Filters: PF
Simulate several possible states and compare to the measurements obtained.
G. HendebyPerformance Issues in Non-Gaussian Filtering Problems
NSSPW ‘06Corpus Christi College, Cambridge
Filters: PMF
Grid the state space and propagate the probabilities according to the Bayesian relations
G. HendebyPerformance Issues in Non-Gaussian Filtering Problems
NSSPW ‘06Corpus 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!
G. HendebyPerformance Issues in Non-Gaussian Filtering Problems
NSSPW ‘06Corpus Christi College, Cambridge
Filter Evaluation (2/2)
Kullback divergence Compares the distance between two distributions Captures all moments of the distributions
G. HendebyPerformance Issues in Non-Gaussian Filtering Problems
NSSPW ‘06Corpus 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
G. HendebyPerformance Issues in Non-Gaussian Filtering Problems
NSSPW ‘06Corpus Christi College, Cambridge
Example I
Linear system: non-Gaussian process noise Gaussian measurement noise
Posterior distribution:distinctly non-Gaussian
G. HendebyPerformance Issues in Non-Gaussian Filtering Problems
NSSPW ‘06Corpus 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
G. HendebyPerformance Issues in Non-Gaussian Filtering Problems
NSSPW ‘06Corpus Christi College, Cambridge
Example II
Estimate target position based on two range measurements Nonlinear measurements but Gaussian noise Posterior distribution: bimodal
G. HendebyPerformance Issues in Non-Gaussian Filtering Problems
NSSPW ‘06Corpus 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
G. HendebyPerformance Issues in Non-Gaussian Filtering Problems
NSSPW ‘06Corpus 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
G. HendebyPerformance Issues in Non-Gaussian Filtering Problems
NSSPW ‘06Corpus 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
G. HendebyPerformance Issues in Non-Gaussian Filtering Problems
NSSPW ‘06Corpus Christi College, Cambridge
ThanksThanks forfor listeninglistening
Questions?Questions?