Sequential Bayesian Filtering via Minimum Distortion Quantization

  • Graham C. Goodwin
  • Arie Feuer
  • Claus Müller


Bayes Rule provides a conceptually simple, closed form, solution to the sequential Bayesian nonlinear filtering problem. The solution, in general, depends upon the evaluation of high dimensional multivariable integrals and is thus computationally intractable save in a small number of special cases. Hence some form of approximation is inevitably required. An approximation in common use is based upon the use of Monte Carlo sampling techniques. This general class of methods is referred to as Particle Filtering. In this paper we advocate an alternative deterministic approach based on the use of minimum distortion quantization. Accordingly we use the term Minimum Distortion Nonlinear Filtering (MDNF) for this alternative class of algorithms. Here we review the theoretical support for MDNF and illustrate its performance via simulation studies.


Posterior Distribution Vector Quantization Distortion Measure Quantization Point Posterior Probability Density Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Arulampalam, S., Maskell, S., Gordon, N., Clapp, T.: A Tutorial on Particle Filters for On-line Non-linear/Non-Gaussian Bayesian Tracking. IEEE Trans. On Signal Proces. 50(2), 174–188 (2002)CrossRefGoogle Scholar
  2. 2.
    Chen, Z.: Bayesian Filtering: From Kalman Filters to Particle Filters, and Beyond. Available at:
  3. 3.
    Gersho, A., Gray, R.M.: Vector Quantization and Signal Compression. Kluwer Academic Publishers, London (1992)zbMATHCrossRefGoogle Scholar
  4. 4.
    Goodwin, G.C., Feuer, A., Müller, C.: Gradient interpretation of the Lloyd algorithm in vector quantization. Available at:
  5. 5.
    Goodwin, G.C., Sin, K.S.: Adaptive Filtering Prediction and Control. Prentice-Hall, Englewood Cliffs (1984)zbMATHGoogle Scholar
  6. 6.
    Goodwin, G.C., Østergaard, J., Quevedo, D.E., Feuer, A.: A vector quantization approach to scenario generation for stochastic NMPC. In: Int. Workshop on Assessment and Future Directions of NMPC, Pavia, Italy (September 2008)Google Scholar
  7. 7.
    Gordon, N., Salmond, D., Smith, A.F.M.: Novel Approach to Non-linear and Non-Gaussian Baysian State Estimation. IEE Proceedings-F 140, 107–113 (1993)Google Scholar
  8. 8.
    Graf, S., Luschgy, H.: Foundation of Quantization for Probability Distribution. Lecture Notes in Mathematics, vol. 1730. Springer, Berlin (2000)CrossRefGoogle Scholar
  9. 9.
    Hammersley, J.M., Hanscomb, D.C.: Monte Carlo Methods. Chapman&Hall, London (1964)zbMATHCrossRefGoogle Scholar
  10. 10.
    Handschin, J.E., Mayne, D.Q.: Monte Carlo techniques to estimate the conditional expectation in multistage nonlinear filtering. International Journal of Control 9(5), 547–559 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Ho, Y.C., Lee, R.C.K.: A Baysian approach to problems in stochastic estimation and control. IEEE Trans. Automatic Control 9, 333–339 (1964)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kulhavy, R.: Quo vadis, Baysian identification? Int. J. Adaptive Control and Signal Processing 13, 469–485 (1999)zbMATHCrossRefGoogle Scholar
  13. 13.
    Lloyd, S.P.: Least squares quantization in PCM. IEEE Trans. Inform. Theory 28, 127–135 (1982)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Müller, C., Goodwin, G.C., Feuer, A.: The gradient and Hessian of distortion measures in vector quantization. Available at:
  15. 15.
    Pages, G.: A space vector quantization method for numerical integration. Journal of Computational and Applied Mathematics, 89, 19997, 1–38Google Scholar
  16. 16.
    Pages, G., Pham, H.: Optimal quantization methods for nonlinear filtering with discrete-time observations. Bernoulli 5, 893–932 (2005)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Rojas, C.R., Goodwin, G.C., Seron, M.M.: Open-cut mine planning via closed loop receding horizon optimal control. In: Sánchez-Pen̈o, R., Quevedo, J., Puig Cayuela, V. (eds.) Identification and Control: The gap between theroy and practice, Springer, Heidelberg (2007)Google Scholar
  18. 18.
    Sellami, A.: Comparative survey on nonlinear filtering methods: the quantization and the particle filtering approaches. J. Statist. Comp. and Simul. 78(2), 93–113 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Sorenson, H.W.: On the development of practical nonlinear filters. Inform. Sci. 7, 253–270 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Sorenson, H.W., Alspach, D.L.: Recursive Bayesian estimation using Gaussian sums. Automatica 7, 465–479 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Wan, E., van der Merwe, R.: The unscented Kalman filter. In: Haykin, S. (ed.) Kalman Filtering and Neural Networks, Wiley, New York (2001)Google Scholar
  22. 22.
    Wang, A.H., Klein, R.L.: Optimal quadrature formula nonlinear estimators. Inform. Sci. 16, 169–184 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    West, M., Harrison, J.: Bayesian Forecasting and Dynamic Models, 2nd edn. Springer, New York (1997)zbMATHGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2010

Authors and Affiliations

  • Graham C. Goodwin
    • 1
  • Arie Feuer
    • 2
  • Claus Müller
    • 1
  1. 1.School of Electrical Engineering and Computer ScienceThe University of NewcastleAustralia
  2. 2.Electrical Engineering DepartmentTechnion-Israel Institute of TechnologyIsrael

Personalised recommendations