Mixture Kalman Filters by Rong Chen & Jun Liu Presented by Yusong Miao Dec. 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:

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.

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