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Mixture Kalman Filtersby Rong Chen & Jun Liu
Presented by Yusong MiaoDec. 10, 2003
Structure
Introduce the originals of the problem Mixture Kalman Filters (MKF) model
setup and method Two related extended models w/
examples Applications to show the advantages of
MKF Conclusions
Original of the problem Interest in on-line estimation and prediction of the
dynamic changing system. (Hidden pattern along observations)
Kalman filter (1960) technique can OK Gaussian linear system.
How about non-linear & non-Gaussian system? ------Sequential Monte Carlo approach including: Bootstrip filter / practical filter; Sequential imputation; From on now, call it as “Monte Carlo filters” ------Mixed kalman filter, the role of this paper. Will see the comparisons.
Original of the problem
Before we start MKF, recall the task:
MKF model setup Conditional dynamic linear model (CDLM):
Given trajectory of an indicator variable, the system is Gaussian & linear-- can derive a MC filter focusing on attention on the space of indicator variables, named Mixed Kalman Filter.
MKF model setup Example 1: A special CDLM (Linear system with non-
Gaussian errors) as the follow:
In the CDLM system MKF is more sophisticated, outperform other methods (i.e.. bootstrap filter).
Use a mixture of Gaussian distribution to estimate the target posterior distribution.
MKF model setup The method of MKF
Use a weighted sample of the indicators:
To represent the distribution p(Λt|yt) a random mixture of Gaussian distribution:
Can approximate the target distribution p(xt|yt).
MKF model setup Algorithm (updating weights)---If you are interested in:
Extended MKF with examples
Beside the situation of CDLM, there is a extended one called partial conditional dynamic linear models (PCDLM).
PCDLM are interested in non-linear component of state variables.
No absolute distinction between CDLM and PCDLM.
Extended MKF with examples
Example: Fading channel modeling system (mobile communication channel can be modeled as Rayleigh flat fading
channels)
Extended MKF with examples
Example: Blind deconvolution digital communication system
Where St is a discrete process taking vales on a known set S. It is to be estimated from the observed signals {y1,…,yt}, without knowing the channel coefficients θi.
There two examples can be solved by extended MKF called EMKF.
Extended MKF with examples Why EMKF? It can deal with as many linear and
Gaussian components from systems as possible. P(xt1,xt2|yt)=P(xt1|xt2,y)*P(xt2|yt), Monte Carlo
approximation of P(xt2|yt) and an Gaussian conditional distribution p(xt1|xt2,y).
Need to generate discrete samples in the joint space of the indicators and the non-linear state components.
Extended MKF with examples Algorithm (updating
weights)
Application of MKF-Target tracking
Situation setup:
The tracking errors (differences between the estimated and true target location) are generated and compared with other methods.
Application of MKF-Target tracking
The result proves the advantage of MKF:
Application of MKF-2 D target’s position
A 2-D target’s position is sampled every T=10s. We know the movement and velocities on both x and y directions. Use MKF to simulate the results and compare them with the actual data.
Model setup:
10
01,
10
01
50
05
,
10
01
50
05
,0010
0001,
1000
0100
10010
01001
22vw VWFGH
Application of MKF-2 D target’s position
The result proves the advantage of MKF:
Better than the tradition way done by Bar-Shalom & Fortmann
Other applications of MKF
There are still several other applications with brief introduction can be found on the paper.
Random (non-Gaussian) accelerated target (no clutter).
Digital signal extraction in fading channels. They are both improved under MKF approach
comparing with traditional Monte Carlo approach.
Conclusions MKF can perform real time estimation and prediction
in CDLM situation, which outperform Sequential Monte Carlo approaches.
Similar to EMKF in PCDLM situation. MKF can combine with other Monte Carlo techniques
(Markov chain Monte Carlo updates, delayed estimation, fixed lag filter, etc.) to improve effectiveness.
Sequential Monte Carlo method can be a platform for designing efficient non-linear filtering algorithms.
Questions & Answers