Closed-Form Solutions to Forward–Backward Smoothing

We propose a closed-form Gaussian sum smoother and, more importantly, closed-form smoothing solutions for increasingly complex problems arising from practice, including tracking in clutter, joint detection and tracking (in clutter), and multiple target tracking (in clutter) via the probability hypothesis density. The solutions are based on the corresponding forward-backward smoothing recursions that involve forward propagation of the filtering densities, followed by backward propagation of the smoothed densities. The key to the exact solutions is the use of alternative forms of the backward propagations, together with standard Gaussian identities. Simulations are also presented to verify the proposed solutions.

[1]  Masahito Yamada,et al.  Structural Time Series Models and the Kalman Filter , 1989 .

[2]  Thia Kirubarajan,et al.  Multitarget Tracking using Probability Hypothesis Density Smoothing , 2011, IEEE Transactions on Aerospace and Electronic Systems.

[3]  R. Mahler,et al.  PHD filters of higher order in target number , 2006, IEEE Transactions on Aerospace and Electronic Systems.

[4]  Y. Bresler.  Two-filter formulae for discrete-time non-linear bayesian smoothing , 1986 .

[5]  Ning Ma,et al.  Multi-Sensor Joint Detection and Tracking with the Bernoulli Filter , 2012, IEEE Transactions on Aerospace and Electronic Systems.

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

[7]  J. Meditch.  A survey of data smoothing for linear and nonlinear dynamic systems , 1973 .

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

[9]  Andrew Harvey,et al.  Forecasting, Structural Time Series Models and the Kalman Filter , 1990 .

[10]  Ba-Ngu Vo,et al.  On performance evaluation of multi-object filters , 2008, 2008 11th International Conference on Information Fusion.

[11]  Ba-Ngu Vo,et al.  A Consistent Metric for Performance Evaluation of Multi-Object Filters , 2008, IEEE Transactions on Signal Processing.

[12]  Ba Tuong Vo,et al.  Random finite sets in Multi-object filtering , 2008 .

[13]  P. Fearnhead,et al.  A sequential smoothing algorithm with linear computational cost. , 2010 .

[14]  Ba-Ngu Vo,et al.  A closed form solution to the Probability Hypothesis Density Smoother , 2010, 2010 13th International Conference on Information Fusion.

[15]  H. Sorenson,et al.  Recursive bayesian estimation using gaussian sums , 1971 .

[16]  A. Doucet,et al.  Sequential Monte Carlo methods for multitarget filtering with random finite sets , 2005, IEEE Transactions on Aerospace and Electronic Systems.

[17]  Thiagalingam Kirubarajan,et al.  Estimation with Applications to Tracking and Navigation , 2001 .

[18]  A. Doucet,et al.  Smoothing algorithms for state–space models , 2010 .

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

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

[21]  Ba-Ngu Vo,et al.  Bayesian Filtering With Random Finite Set Observations , 2008, IEEE Transactions on Signal Processing.

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

[23]  Simon J. Godsill,et al.  Monte Carlo smoothing with application to audio signal enhancement , 2002, IEEE Trans. Signal Process..

[24]  Ba-Ngu Vo,et al.  Bernoulli Forward-Backward Smoothing for Joint Target Detection and Tracking , 2011, IEEE Transactions on Signal Processing.

[25]  James Ting-Ho Lo,et al.  Finite-dimensional sensor orbits and optimal nonlinear filtering , 1972, IEEE Trans. Inf. Theory.

[26]  R. Mahler.  Multitarget Bayes filtering via first-order multitarget moments , 2003 .

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

[28]  Michael A. West,et al.  Bayesian Forecasting and Dynamic Models (Springer Series in Statistics) , 2013 .

[29]  G. Kitagawa.  Non-Gaussian State—Space Modeling of Nonstationary Time Series , 1987 .

[30]  M. West,et al.  Bayesian forecasting and dynamic models , 1989 .

[31]  Ronald P. S. Mahler,et al.  Statistical Multisource-Multitarget Information Fusion , 2007 .

[32]  A. Doucet,et al.  Monte Carlo Smoothing for Nonlinear Time Series , 2004, Journal of the American Statistical Association.

[33]  Ba-Ngu Vo,et al.  Analytic Implementations of the Cardinalized Probability Hypothesis Density Filter , 2007, IEEE Transactions on Signal Processing.

[34]  B. Anderson,et al.  Optimal Filtering , 1979, IEEE Transactions on Systems, Man, and Cybernetics.

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

[36]  Ba-Ngu Vo,et al.  The Gaussian Mixture Probability Hypothesis Density Filter , 2006, IEEE Transactions on Signal Processing.

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

[38]  G. Kitagawa.  Smoothness priors analysis of time series , 1996 .