Mixture Kalman filters

In treating dynamic systems, sequential Monte Carlo methods use discrete samples to represent a complicated probability distribution and use rejection sampling, importance sampling and weighted resampling to complete the on‐line ‘filtering’ task. We propose a special sequential Monte Carlo method, the mixture Kalman filter, which uses a random mixture of the Gaussian distributions to approximate a target distribution. It is designed for on‐line estimation and prediction of conditional and partial conditional dynamic linear models, which are themselves a class of widely used non‐linear systems and also serve to approximate many others. Compared with a few available filtering methods including Monte Carlo methods, the gain in efficiency that is provided by the mixture Kalman filter can be very substantial. Another contribution of the paper is the formulation of many non‐linear systems into conditional or partial conditional linear form, to which the mixture Kalman filter can be applied. Examples in target tracking and digital communications are given to demonstrate the procedures proposed.

[1]  R. E. Kalman,et al.  A New Approach to Linear Filtering and Prediction Problems , 2002 .

[2]  K. Ito,et al.  On State Estimation in Switching Environments , 1970 .

[3]  A. Jazwinski.  Stochastic Processes and Filtering Theory , 1970 .

[4]  A.H. Haddad,et al.  Applied optimal estimation , 1976, Proceedings of the IEEE.

[5]  Hiromitsu Kumamoto,et al.  Random sampling approach to state estimation in switching environments , 1977, Autom..

[6]  M. Smith,et al.  On the detection of target trajectories in a multi target environment , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[7]  Reuven Y. Rubinstein,et al.  Simulation and the Monte Carlo method , 1981, Wiley series in probability and mathematical statistics.

[8]  Jitendra K. Tugnait,et al.  Detection and estimation for abruptly changing systems , 1981, 1981 20th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[9]  D. Rubin,et al.  The calculation of posterior distributions by data augmentation , 1987 .

[10]  Donald B. Rubin,et al.  Comment : A noniterative sampling/importance resampling alternative to the data augmentation algorithm for creating a few imputations when fractions of missing information are modest : The SIR Algorithm , 1987 .

[11]  Y. Bar-Shalom.  Tracking and data association , 1988 .

[12]  Michael A. West.  Mixture Models, Monte Carlo, Bayesian Updating and Dynamic Models , 1992 .

[13]  Nicholas G. Polson,et al.  A Monte Carlo Approach to Nonnormal and Nonlinear State-Space Modeling , 1992 .

[14]  N. Gordon,et al.  Novel approach to nonlinear/non-Gaussian Bayesian state estimation , 1993 .

[15]  Jun S. Liu,et al.  Sequential Imputations and Bayesian Missing Data Problems , 1994 .

[16]  N. Shephard.  Partial non-Gaussian state space , 1994 .

[17]  Jun S. Liu,et al.  Covariance structure of the Gibbs sampler with applications to the comparisons of estimators and augmentation schemes , 1994 .

[18]  R. Kohn,et al.  On Gibbs sampling for state space models , 1994 .

[19]  D. Avitzour.  Stochastic simulation Bayesian approach to multitarget tracking , 1995 .

[20]  Petros G. Voulgaris,et al.  On optimal ℓ∞ to ℓ∞ filtering , 1995, Autom..

[21]  Jun S. Liu,et al.  Blind Deconvolution via Sequential Imputations , 1995 .

[22]  G. Kitagawa.  Monte Carlo Filter and Smoother for Non-Gaussian Nonlinear State Space Models , 1996 .

[23]  Jun S. Liu,et al.  Sequential Monte Carlo methods for dynamic systems , 1997 .

[24]  N. G. Best,et al.  Dynamic conditional independence models and Markov chain Monte Carlo methods , 1997 .

[25]  Simon J. Godsill,et al.  On sequential simulation-based methods for Bayesian filtering , 1998 .

[26]  Markus Hürzeler,et al.  Monte Carlo Approximations for General State-Space Models , 1998 .

[27]  G. Peters,et al.  Monte Carlo Approximations for General State-Space Models , 1998 .

[28]  Jun S. Liu,et al.  Rejection Control and Sequential Importance Sampling , 1998 .

[29]  Jun S. Liu,et al.  Sequential importance sampling for nonparametric Bayes models: The next generation , 1999 .

[30]  M. Pitt,et al.  Filtering via Simulation: Auxiliary Particle Filters , 1999 .

[31]  P. Fearnhead,et al.  An improved particle filter for non-linear problems , 1999 .

[32]  Simon J. Godsill,et al.  On sequential Monte Carlo sampling methods for Bayesian filtering , 2000, Stat. Comput..

[33]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .