Parametric Bayesian Filters for Nonlinear Stochastic Dynamical Systems: A Survey

Nonlinear stochastic dynamical systems are commonly used to model physical processes. For linear and Gaussian systems, the Kalman filter is optimal in minimum mean squared error sense. However, for nonlinear or non-Gaussian systems, the estimation of states or parameters is a challenging problem. Furthermore, it is often required to process data online. Therefore, apart from being accurate, the feasible estimation algorithm also needs to be fast. In this paper, we review Bayesian filters that possess the aforementioned properties. Each filter is presented in an easy way to implement algorithmic form. We focus on parametric methods, among which we distinguish three types of filters: filters based on analytical approximations (extended Kalman filter, iterated extended Kalman filter), filters based on statistical approximations (unscented Kalman filter, central difference filter, Gauss-Hermite filter), and filters based on the Gaussian sum approximation (Gaussian sum filter). We discuss each of these filters, and compare them with illustrative examples.

[1]  H. Cramér.  Mathematical methods of statistics , 1947 .

[2]  C. Striebel,et al.  On the maximum likelihood estimates for linear dynamic systems , 1965 .

[3]  Y. Ho,et al.  A Bayesian approach to problems in stochastic estimation and control , 1964 .

[4]  Harry L. Van Trees,et al.  Detection, Estimation, and Modulation Theory, Part I , 1968 .

[5]  Michael Athans,et al.  A comparison of three non-linear filters , 1969, Autom..

[6]  R. Mehra.  On the identification of variances and adaptive Kalman filtering , 1970 .

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

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

[9]  D. Alspach.  Comments on "On the identification of variances and adaptive Kalman filtering" , 1972 .

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

[11]  V. Benes.  Exact finite-dimensional filters for certain diffusions with nonlinear drift , 1981 .

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

[13]  Chaw-Bing Chang,et al.  Application of state estimation to target tracking , 1984 .

[14]  F. Daum.  Exact finite dimensional nonlinear filters , 1985, 1985 24th IEEE Conference on Decision and Control.

[15]  S. Haykin,et al.  Adaptive Filter Theory , 1986 .

[16]  K. Spingarn.  Passive Position Location Estimation Using the Extended Kalman Filter , 1987, IEEE Transactions on Aerospace and Electronic Systems.

[17]  B. Øksendal.  Stochastic differential equations : an introduction with applications , 1987 .

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

[19]  Brian D. Ripley,et al.  Stochastic Simulation , 2005 .

[20]  Harold W. Sorenson,et al.  Recursive Bayesian estimation using piece-wise constant approximations , 1988, Autom..

[21]  T. Kerr.  Status of CR-like lower bounds for nonlinear filtering , 1989 .

[22]  B. Øksendal.  The Filtering Problem , 1989 .

[23]  S. Sangsuk-Iam,et al.  Analysis of discrete-time Kalman filtering under incorrect noise covariances , 1990 .

[24]  R. G. Reynolds.  Robust estimation of covariance matrices , 1990 .

[25]  Paul Zarchan,et al.  Tactical and strategic missile guidance , 1990 .

[26]  Lennart Ljung,et al.  Adaptation and tracking in system identification - A survey , 1990, Autom..

[27]  C. R. Rao,et al.  Information and the Accuracy Attainable in the Estimation of Statistical Parameters , 1992 .

[28]  Hisashi Tanizaki,et al.  Nonlinear Filters: Estimation and Applications , 1993 .

[29]  F. W. Cathey,et al.  The iterated Kalman filter update as a Gauss-Newton method , 1993, IEEE Trans. Autom. Control..

[30]  Greg Welch,et al.  Welch & Bishop , An Introduction to the Kalman Filter 2 1 The Discrete Kalman Filter In 1960 , 1994 .

[31]  Jón Dańıelsson.  Stochastic volatility in asset prices estimation with simulated maximum likelihood , 1994 .

[32]  J. Quadrat.  Numerical methods for stochastic control problems in continuous time , 1994 .

[33]  J. Mendel.  Lessons in Estimation Theory for Signal Processing, Communications, and Control , 1995 .

[34]  Denis Talay,et al.  Simulation of stochastic differential systems , 1995 .

[35]  S. Julier,et al.  A General Method for Approximating Nonlinear Transformations of Probability Distributions , 1996 .

[36]  Hisashi Tanizaki,et al.  Nonlinear filters based on taylor series expansions , 1996 .

[37]  Hisashi Tanizaki.  Application of Nonlinear Filters , 1996 .

[38]  Dimitrios Hatzinakos,et al.  An adaptive Gaussian sum algorithm for radar tracking , 1997, Proceedings of ICC'97 - International Conference on Communications.

[39]  Carlos H. Muravchik,et al.  Posterior Cramer-Rao bounds for discrete-time nonlinear filtering , 1998, IEEE Trans. Signal Process..

[40]  G. Evensen,et al.  Analysis Scheme in the Ensemble Kalman Filter , 1998 .

[41]  T. Schei.  A finite-difference method for linearization in nonlinear estimation algorithms , 1998 .

[42]  X. Rong Li,et al.  Survey of maneuvering target tracking: dynamic models , 2000, SPIE Defense + Commercial Sensing.

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

[44]  Hugh F. Durrant-Whyte,et al.  A new method for the nonlinear transformation of means and covariances in filters and estimators , 2000, IEEE Trans. Autom. Control..

[45]  Kazufumi Ito,et al.  Gaussian filters for nonlinear filtering problems , 2000, IEEE Trans. Autom. Control..

[46]  Niels Kjølstad Poulsen,et al.  New developments in state estimation for nonlinear systems , 2000, Autom..

[47]  Rudolph van der Merwe,et al.  The unscented Kalman filter for nonlinear estimation , 2000, Proceedings of the IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium (Cat. No.00EX373).

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

[49]  S. Roweis,et al.  Learning Nonlinear Dynamical Systems Using the Expectation–Maximization Algorithm , 2001 .

[50]  M. Verlaan,et al.  Nonlinearity in Data Assimilation Applications: A Practical Method for Analysis , 2001 .

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

[52]  Petr Tichavský,et al.  Filtering, predictive, and smoothing Cramér-Rao bounds for discrete-time nonlinear dynamic systems , 2001, Autom..

[53]  Arnaud Doucet,et al.  A survey of convergence results on particle filtering methods for practitioners , 2002, IEEE Trans. Signal Process..

[54]  Simon J. Julier,et al.  The scaled unscented transformation , 2002, Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301).

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

[56]  Torsten Söderström,et al.  Anticipative grid design in point-mass approach to nonlinear state estimation , 2002, IEEE Trans. Autom. Control..

[57]  Silverio Bolognani,et al.  Extended Kalman filter tuning in sensorless PMSM drives , 2002, Proceedings of the Power Conversion Conference-Osaka 2002 (Cat. No.02TH8579).

[58]  Rudolph van der Merwe,et al.  The Unscented Kalman Filter , 2002 .

[59]  Jeffrey K. Uhlmann,et al.  Reduced sigma point filters for the propagation of means and covariances through nonlinear transformations , 2002, Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301).

[60]  V. Pugachev,et al.  Stochastic Systems: Theory and Applications , 2002 .

[61]  Joris De Schutter,et al.  Kalman filters for nonlinear systems , 2002 .

[62]  A. Farina,et al.  Tracking a ballistic target: comparison of several nonlinear filters , 2002 .

[63]  T. Singh,et al.  The higher order unscented filter , 2003, Proceedings of the 2003 American Control Conference, 2003..

[64]  X. R. Li,et al.  Survey of maneuvering target tracking. Part I. Dynamic models , 2003 .

[65]  Jason L. Williams.  Gaussian Mixture Reduction for Tracking Multiple Maneuvering Targets in Clutter , 2003 .

[66]  Geir Evensen,et al.  The Ensemble Kalman Filter: theoretical formulation and practical implementation , 2003 .

[67]  Larry H. Matthies,et al.  Kalman filter-based algorithms for estimating depth from image sequences , 1989, International Journal of Computer Vision.

[68]  Matthew A. Wilson,et al.  Dynamic Analyses of Information Encoding in Neural Ensembles , 2004, Neural Computation.

[69]  Jeffrey K. Uhlmann,et al.  Unscented filtering and nonlinear estimation , 2004, Proceedings of the IEEE.

[70]  Branko Ristic,et al.  A comparison of two Crame/spl acute/r-Rao bounds for nonlinear filtering with P/sub d/<1 , 2004, IEEE Transactions on Signal Processing.

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

[72]  Yuanxin Wu,et al.  An Improvement to Unscented Transformation , 2004, Australian Conference on Artificial Intelligence.

[73]  A. Honkela.  Approximating nonlinear transformations of probability distributions for nonlinear independent component analysis , 2004, 2004 IEEE International Joint Conference on Neural Networks (IEEE Cat. No.04CH37541).

[74]  Branko Ristic,et al.  Beyond the Kalman Filter: Particle Filters for Tracking Applications , 2004 .

[75]  Jeffrey K. Uhlmann,et al.  Corrections to "Unscented Filtering and Nonlinear Estimation" , 2004, Proc. IEEE.

[76]  F. Daum.  Nonlinear filters: beyond the Kalman filter , 2005, IEEE Aerospace and Electronic Systems Magazine.

[77]  Nikos A. Vlassis,et al.  Accelerated EM-based clustering of large data sets , 2006, Data Mining and Knowledge Discovery.

[78]  Eric Moulines,et al.  Inference in hidden Markov models , 2010, Springer series in statistics.

[79]  Xin Zhang,et al.  Dynamic Cramer-Rao bound for target tracking in clutter , 2005, IEEE Transactions on Aerospace and Electronic Systems.

[80]  Liu Si-hua.  Suboptimal Nonlinear Filters for Tracking a Ballistic Target , 2005 .

[81]  Jesús Rodríguez-Millán,et al.  Three Mathematica supported proposals for the discretization of nonlinear dynamical control systems , 2005 .

[82]  J. Vermaak,et al.  Online sensor registration , 2005, 2005 IEEE Aerospace Conference.

[83]  D. Simon.  Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches , 2006 .

[84]  Torsten Söderström,et al.  Advanced point-mass method for nonlinear state estimation , 2006, Autom..

[85]  I. Bilik,et al.  Target tracking in glint noise environment using nonlinear non-Gaussian Kalman filter , 2006, 2006 IEEE Conference on Radar.

[86]  Sten Bay Jørgensen,et al.  A Generalized Autocovariance Least-Squares Method for Kalman Filter Tuning , 2008 .

[87]  Haikady N. Nagaraja,et al.  Inference in Hidden Markov Models , 2006, Technometrics.

[88]  G. Evensen.  Data Assimilation: The Ensemble Kalman Filter , 2006 .

[89]  Krzysztof Pomorski.  Gauss-Hermite approximation formula , 2006, Comput. Phys. Commun..

[90]  C. Manohar,et al.  New forms of extended Kalman filter via transversal linearization and applications to structural system identification , 2007 .

[91]  Gene H. Golub,et al.  Calculation of Gauss quadrature rules , 1967, Milestones in Matrix Computation.

[92]  Robert J. Elliott,et al.  Discrete-Time Nonlinear Filtering Algorithms Using Gauss–Hermite Quadrature , 2007, Proceedings of the IEEE.

[93]  Kristine L. Bell,et al.  Matrix CRLB Scaling Due to Measurements of Uncertain Origin , 2007 .

[94]  Gamini Dissanayake,et al.  Mobile Robot Localization and Mapping using a Gaussian Sum Filter , 2007 .

[95]  Peter D. Scott,et al.  A novel Gaussian Sum Filter Method for accurate solution to the nonlinear filtering problem , 2008, 2008 11th International Conference on Information Fusion.

[96]  Simo Ali-Loytty,et al.  On the convergence of the Gaussian mixture filter , 2008 .

[97]  Bayesian Filtering,et al.  Efficient Gaussian Mixture Filter for Hybrid Positioning , 2008 .

[98]  K. Xiong,et al.  Adaptive robust extended Kalman filter for nonlinear stochastic systems , 2008 .

[99]  Sundeep Rangan,et al.  Asymptotic Analysis of MAP Estimation via the Replica Method and Compressed Sensing , 2009, NIPS.

[100]  S. Särkkä,et al.  BAYESIAN ESTIMATION OF TIME-VARYING SYSTEMS: Discrete-Time Systems , 2009 .

[101]  Jonathan Weare,et al.  Particle filtering with path sampling and an application to a bimodal ocean current model , 2009, J. Comput. Phys..

[102]  Nilanjan Saha,et al.  Extended Kalman filters using explicit and derivative-free local linearizations , 2009 .

[103]  James B. Rawlings,et al.  Estimation of the disturbance structure from data using semidefinite programming and optimal weighting , 2009, Autom..

[104]  F. Gland,et al.  Large sample asymptotics for the ensemble Kalman filter , 2009 .

[105]  Jindrich Duník,et al.  Derivative-free estimation methods: New results and performance analysis , 2009, Autom..

[106]  S. Haykin,et al.  Cubature Kalman Filters , 2009, IEEE Transactions on Automatic Control.

[107]  O. Cappé,et al.  On‐line expectation–maximization algorithm for latent data models , 2009 .

[108]  Ondrej Straka,et al.  Adaptive choice of scaling parameter in derivative-free local filters , 2010, 2010 13th International Conference on Information Fusion.

[109]  Biao Huang,et al.  Estimation and control of solid oxide fuel cell system , 2010, Comput. Chem. Eng..

[110]  Simo Särkkä,et al.  On Gaussian Optimal Smoothing of Non-Linear State Space Models , 2010, IEEE Transactions on Automatic Control.

[111]  Simon Haykin,et al.  Cubature Kalman Filtering for Continuous-Discrete Systems: Theory and Simulations , 2010, IEEE Transactions on Signal Processing.

[112]  Ramon van Handel,et al.  Nonlinear Filtering and Systems Theory , 2010 .

[113]  Henk A. P. Blom,et al.  Decomposed particle filtering and track swap estimation in tracking two closely spaced targets , 2011, 14th International Conference on Information Fusion.

[114]  B. Rozovskii,et al.  The Oxford Handbook of Nonlinear Filtering , 2011 .

[115]  Yong Qi,et al.  Online Estimation of the Approximate Posterior Cramer-Rao Lower Bound for Discrete-Time Nonlinear Filtering , 2011, IEEE Transactions on Aerospace and Electronic Systems.

[116]  Arunabha Bagchi,et al.  A theoretical analysis of Bayes-optimal multi-target tracking and labelling , 2011 .

[117]  Sean P. Meyn,et al.  A mean-field control-oriented approach to particle filtering , 2011, Proceedings of the 2011 American Control Conference.

[118]  Shengli Zhou,et al.  Posterior Cramér-Rao bounds for Doppler biased multistatic range-only tracking , 2011, 14th International Conference on Information Fusion.

[119]  Aubrey B. Poore,et al.  Adaptive Gaussian Sum Filters for Space Surveillance , 2011, IEEE Transactions on Automatic Control.

[120]  Pramod K. Varshney,et al.  Conditional Posterior Cramér–Rao Lower Bounds for Nonlinear Sequential Bayesian Estimation , 2012, IEEE Transactions on Signal Processing.

[121]  Henk A. P. Blom,et al.  Optimal decomposed particle filtering of two closely spaced Gaussian targets , 2011, IEEE Conference on Decision and Control and European Control Conference.

[122]  Thomas R. Bewley,et al.  Efficient grid-based Bayesian estimation of nonlinear low-dimensional systems with sparse non-Gaussian PDFs , 2012, Autom..