State Estimation in Unknown Non-Gaussian Measurement Noise using Variational Bayesian Technique

The problem of state space estimation of linear systems in an unknown non-Gaussian noise field is considered. A finite Gaussian mixture model (GMM) is used to model the non-Gaussian measurement noise with unknown statistics. A variational Bayesian expectation maximization (VBEM) algorithm is proposed to estimate the system states as well as the unknown parameters. In the variational Bayesian expectation (VBE) step, approximate inference is established to estimate the system state. The Gaussian mixture parameters are then updated in the variational Bayesian maximization (VBM) step. We also derive the true marginal posteriors to verify the performance of the proposed VBEM method. Computer simulations show that the proposed method has an improved estimation performance compared with other conventional approaches.

[1]  Y. Bar-Shalom,et al.  An interacting multiple model approach for target tracking with glint noise , 1994, Proceedings of 1994 IEEE National Radar Conference.

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

[3]  N. Nasios,et al.  Variational learning for Gaussian mixture models , 2006, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[4]  Adrian Corduneanu,et al.  Variational Bayesian Model Selection for Mixture Distributions , 2001 .

[5]  Nando de Freitas,et al.  An Introduction to Sequential Monte Carlo Methods , 2001, Sequential Monte Carlo Methods in Practice.

[6]  Teng Joon Lim,et al.  The Variational Inference Approach to Joint Data Detection and Phase Noise Estimation in OFDM , 2007, IEEE Transactions on Signal Processing.

[7]  Matthew J. Beal.  Variational algorithms for approximate Bayesian inference , 2003 .

[8]  Michael I. Jordan,et al.  An Introduction to Variational Methods for Graphical Models , 1999, Machine Learning.

[9]  Y. Bar-Shalom,et al.  IMM estimator versus optimal estimator for hybrid systems , 2005, IEEE Transactions on Aerospace and Electronic Systems.

[10]  I. Bilik,et al.  Maneuvering Target Tracking in the Presence of Glint using the Nonlinear Gaussian Mixture Kalman Filter , 2010, IEEE Transactions on Aerospace and Electronic Systems.

[11]  Jun S. Liu,et al.  Mixture Kalman filters , 2000 .

[12]  R. Shumway,et al.  AN APPROACH TO TIME SERIES SMOOTHING AND FORECASTING USING THE EM ALGORITHM , 1982 .

[13]  Terrence J. Sejnowski,et al.  Variational Learning for Switching State-Space Models , 2001 .

[14]  Xin Li,et al.  Patch-Based Video Processing: A Variational Bayesian Approach , 2009, IEEE Transactions on Circuits and Systems for Video Technology.

[15]  Thia Kirubarajan,et al.  Estimation with Applications to Tracking and Navigation: Theory, Algorithms and Software , 2001 .

[16]  G. Hewer,et al.  Robust Preprocessing for Kalman Filtering of Glint Noise , 1987, IEEE Transactions on Aerospace and Electronic Systems.

[17]  W. Wu,et al.  Target racking with glint noise , 1993 .

[18]  Fredrik Gustafsson,et al.  Adaptive filtering and change detection , 2000 .

[19]  Naonori Ueda,et al.  Variational bayesian estimation and clustering for speech recognition , 2004, IEEE Transactions on Speech and Audio Processing.

[20]  Karl J. Friston,et al.  Variational Bayesian identification and prediction of stochastic nonlinear dynamic causal models , 2009, Physica D. Nonlinear phenomena.

[21]  Simo Särkkä,et al.  Recursive Noise Adaptive Kalman Filtering by Variational Bayesian Approximations , 2009, IEEE Transactions on Automatic Control.

[22]  WU WENG-RONG,et al.  Target Tracking With Glint Noise , 2010 .

[23]  Radford M. Neal.  Pattern Recognition and Machine Learning , 2007, Technometrics.

[24]  David Barber,et al.  Expectation Correction for Smoothed Inference in Switching Linear Dynamical Systems , 2006, J. Mach. Learn. Res..

[25]  William D. Penny,et al.  Bayesian nonstationary autoregressive models for biomedical signal analysis , 2002, IEEE Transactions on Biomedical Engineering.

[26]  Petar M. Djuric,et al.  Gaussian sum particle filtering , 2003, IEEE Trans. Signal Process..

[27]  Neil J. Gordon,et al.  A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking , 2002, IEEE Trans. Signal Process..

[28]  Nikolas P. Galatsanos,et al.  Variational Bayesian Sparse Kernel-Based Blind Image Deconvolution With Student's-t Priors , 2009, IEEE Transactions on Image Processing.

[29]  H. Sorenson,et al.  Nonlinear Bayesian estimation using Gaussian sum approximations , 1972 .

[30]  D. Magill.  Optimal adaptive estimation of sampled stochastic processes , 1965 .

[31]  J. Tabrikian,et al.  MMSE-Based Filtering in Presence of Non-Gaussian System and Measurement Noise , 2010, IEEE Transactions on Aerospace and Electronic Systems.

[32]  William D. Penny,et al.  Variational Bayes for generalized autoregressive models , 2002, IEEE Trans. Signal Process..

[33]  Brendan J. Frey,et al.  A comparison of algorithms for inference and learning in probabilistic graphical models , 2005, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[34]  H. Leung,et al.  Maximum Likelihood State Estimation of Semi-Markovian Switching System in Non-Gaussian Measurement Noise , 2010, IEEE Transactions on Aerospace and Electronic Systems.

[35]  Václav Smídl,et al.  Variational Bayesian Filtering , 2008, IEEE Transactions on Signal Processing.

[36]  Aristidis Likas,et al.  Bayesian feature and model selection for Gaussian mixture models , 2006, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[37]  C. Masreliez.  Approximate non-Gaussian filtering with linear state and observation relations , 1975 .