Iterative algorithms for state estimation of jump Markov linear systems

Jump Markov linear systems (JMLSs) are linear systems whose parameters evolve with time according to a finite state Markov chain. Given a set of observations, our aim is to estimate the states of the finite state Markov chain and the continuous (in space) states of the linear system. In this paper, we present original deterministic and stochastic iterative algorithms for optimal state estimation of JMLSs. The first stochastic algorithm yields minimum mean square error (MMSE) estimates of the finite state space Markov chain and of the continuous state of the JMLS. A deterministic and a stochastic algorithm are given to obtain the marginal maximum a posteriori (MMAP) sequence estimate of the finite state Markov chain. Finally, a deterministic and a stochastic algorithm are derived to obtain the MMAP sequence estimate of the continuous state of the JMLS. Computer simulations are carried out to evaluate the performance of the proposed algorithms. The problem of deconvolution of Bernoulli-Gaussian (BG) processes and the problem of tracking a maneuvering target are addressed.

[1]  David Q. Mayne,et al.  A solution of the smoothing problem for linear dynamic systems , 1966, Autom..

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

[3]  Jitendra Tugnait,et al.  Adaptive estimation and identification for discrete systems with Markov jump parameters , 1981, 1981 20th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[4]  Jerry M. Mendel,et al.  Maximum likelihood detection and estimation of Bernoulli - Gaussian processes , 1982, IEEE Trans. Inf. Theory.

[5]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[6]  A. F. Smith,et al.  Statistical analysis of finite mixture distributions , 1986 .

[7]  Stuart German,et al.  Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images , 1988 .

[8]  Lawrence R. Rabiner,et al.  A tutorial on hidden Markov models and selected applications in speech recognition , 1989, Proc. IEEE.

[9]  J. Mendel,et al.  Maximum-Likelihood Deconvolution: A Journey into Model-Based Signal Processing , 1990 .

[10]  Marc Lavielle,et al.  Bayesian deconvolution of Bernoulli-Gaussian processes , 1993, Signal Process..

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

[12]  C. Robert,et al.  Estimation of Finite Mixture Distributions Through Bayesian Sampling , 1994 .

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

[14]  G. Kitagawa.  The two-filter formula for smoothing and an implementation of the Gaussian-sum smoother , 1994 .

[15]  N. Shephard,et al.  The simulation smoother for time series models , 1995 .

[16]  Yakov Bar-Shalom,et al.  Multitarget-Multisensor Tracking: Principles and Techniques , 1995 .

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

[18]  William Dale Blair,et al.  Fixed-interval smoothing for Markovian switching systems , 1995, IEEE Trans. Inf. Theory.

[19]  Yves Goussard,et al.  Unsupervised deconvolution of sparse spike trains using stochastic approximation , 1996, IEEE Trans. Signal Process..

[20]  Antonio Artés-Rodríguez,et al.  Sparse deconvolution using adaptive mixed-Gaussian models , 1996, Signal Process..

[21]  Robin J. Evans,et al.  Probabilistic data association for systems with multiple simultaneous measurements , 1996, Autom..

[22]  G. McLachlan,et al.  The EM algorithm and extensions , 1996 .

[23]  R. Kohn,et al.  Markov chain Monte Carlo in conditionally Gaussian state space models , 1996 .

[24]  Kjetil F. Kaaresen,et al.  Deconvolution of sparse spike trains by iterated window maximization , 1997, IEEE Trans. Signal Process..

[25]  Michael A. West,et al.  Bayesian forecasting and dynamic models (2nd ed.) , 1997 .

[26]  Patrick Duvaut,et al.  Bayesian estimation of state-space models applied to deconvolution of Bernoulli - Gaussian processes , 1997, Signal Process..

[27]  Eric Moulines,et al.  Simulation-based methods for blind maximum-likelihood filter identification , 1999, Signal Process..

[28]  Vikram Krishnamurthy,et al.  Expectation maximization algorithms for MAP estimation of jump Markov linear systems , 1999, IEEE Trans. Signal Process..

[29]  Vikram Krishnamurthy,et al.  Adaptive nonlinear filters for narrow-band interference suppression in spread-spectrum CDMA systems , 1999, IEEE Trans. Commun..

[30]  Hoon Kim,et al.  Monte Carlo Statistical Methods , 2000, Technometrics.

[31]  Arnaud Doucet,et al.  Stochastic sampling algorithms for state estimation of jump Markov linear systems , 2000, IEEE Trans. Autom. Control..

[32]  Neil J. Gordon,et al.  Editors: Sequential Monte Carlo Methods in Practice , 2001 .

[33]  Nando de Freitas,et al.  Sequential Monte Carlo Methods in Practice , 2001, Statistics for Engineering and Information Science.

[34]  Arnaud Doucet,et al.  Particle filters for state estimation of jump Markov linear systems , 2001, IEEE Trans. Signal Process..

[35]  Timothy J. Robinson,et al.  Sequential Monte Carlo Methods in Practice , 2003 .