Bayesian Quadrature Variance in Sigma-Point Filtering

  • Jakub Prüher
  • Miroslav Šimandl
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 383)


Sigma-point filters are algorithms for recursive state estimation of the stochastic dynamic systems from noisy measurements, which rely on moment integral approximations by means of various numerical quadrature rules. In practice, however, it is hardly guaranteed that the system dynamics or measurement functions will meet the restrictive requirements of the classical quadratures, which inevitably results in approximation errors that are not accounted for in the current state-of-the-art sigma-point filters. We propose a method for incorporating information about the integral approximation error into the filtering algorithm by exploiting features of a Bayesian quadrature—an alternative to classical numerical integration. This is enabled by the fact that the Bayesian quadrature treats numerical integration as a statistical estimation problem, where the posterior distribution over the values of the integral serves as a model of numerical error. We demonstrate superior performance of the proposed filters on a simple univariate benchmarking example.


Nonlinear filtering Sigma-point filter Gaussian filter Integral variance Bayesian quadrature Gaussian process 



This work was supported by the Czech Science Foundation, project no. GACR P103-13-07058J.


  1. 1.
    Arasaratnam, I., Haykin, S.: Cubature kalman filters. IEEE Trans. Autom. Control 54(6), 1254–1269 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bhar, R.: Stochastic filtering with applications in finance. World Scientific (2010)Google Scholar
  3. 3.
    Deisenroth, M. P., Huber, M. F., Hanebeck, U. D.: Analytic moment-based Gaussian process filtering. In: Proceedings of the 26th Annual International Conference on Machine Learning—ICML ’09, pp. 1–8. ACM Press (2009)Google Scholar
  4. 4.
    Deisenroth, M. P., Ohlsson, H.: A general perspective on gaussian filtering and smoothing: explaining current and deriving new algorithms. In: IEEE (June 2011) American Control Conference (ACC), pp. 1807–1812 (2011)Google Scholar
  5. 5.
    Deisenroth, M.P., Turner, R.D., Huber, M.F., Hanebeck, U.D., Rasmussen, C.E.: Robust filtering and smoothing with gaussian processes. IEEE Trans. Autom. Control 57(7), 1865–1871 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Duník, J., Straka, O., Šimandl, M.: Stochastic integration filter. IEEE Trans. Autom. Control 58(6), 1561–1566 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gelb, A.: Applied Optimal Estimation. The MIT Press (1974)Google Scholar
  8. 8.
    Gelman, A.: Bayesian Data Analysis. Chapman and Hall/CRC, 3rd edn (2013)Google Scholar
  9. 9.
    Gillijns, S., Mendoza, O., Chandrasekar, J., De Moor, B., Bernstein, D., Ridley, A.: What is the ensemble kalman filter and how well does it work? In: American Control Conference, 2006. p. 6 (2006)Google Scholar
  10. 10.
    Girard, A., Rasmussen, C. E., Quiñonero Candela, J., Murray-Smith, R.: Gaussian process priors with uncertain inputs application to multiple-step ahead time series forecasting. In: Becker, S., Thrun, S., Obermayer, K. (eds.) Advances in Neural Information Processing Systems 15, pp. 545–552. MIT Press (2003)Google Scholar
  11. 11.
    Gordon, N.J., Salmond, D.J., Smith, A.F.M.: Novel approach to nonlinear/non-gaussian bayesian state estimation. IEE Proceedings F (Radar and Signal Processing) 140(2), 107–113 (1993)CrossRefGoogle Scholar
  12. 12.
    Grewal, M. S., Weill, L. R., Andrews, A. P.: Global Positioning Systems, Inertial Navigation, and Integration. Wiley (2007)Google Scholar
  13. 13.
    Ito, K., Xiong, K.: Gaussian filters for nonlinear filtering problems. IEEE Trans. Autom. Control 45(5), 910–927 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Jiang, T., Sidiropoulos, N., Giannakis, G.: Kalman filtering for power estimation in mobile communications. IEEE Trans. Wireless Commun. 2(1), 151–161 (2003)CrossRefGoogle Scholar
  15. 15.
    Julier, S.J., Uhlmann, J.K., Durrant-Whyte, H.F.: A new method for the nonlinear transformation of means and covariances in filters and estimators. IEEE Trans. Autom. Control 45(3), 477–482 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kalman, R.E.: A new approach to linear filtering and prediction problems. J. Basic Eng. 82(1), 35–45 (1960)CrossRefGoogle Scholar
  17. 17.
    Li, X. R., Zhao, Z.: Measuring estimator’s credibility: noncredibility index. In: 2006 9th International Conference on Information Fusion, pp. 1–8 (2006)Google Scholar
  18. 18.
    Maybeck, P. S.: Stochastic Models, Estimation and Control: Volume 2. Academic Press (1982)Google Scholar
  19. 19.
    Minka, T.P.: Deriving Quadrature Rules from Gaussian Processes. Tech. rep., Statistics Department, Carnegie Mellon University, Tech. Rep (2000)Google Scholar
  20. 20.
    Nørgaard, M., Poulsen, N.K., Ravn, O.: New developments in state estimation for nonlinear systems. Automatica 36, 1627–1638 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    O’Hagan, A.: Bayes-Hermite quadrature. J. Stat. Plann. Infer. 29(3), 245–260 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Osborne, M. A., Rasmussen, C. E., Duvenaud, D. K., Garnett, R., Roberts, S. J.: Active learning of model evidence using bayesian quadrature. In: Advances in Neural Information Processing Systems (NIPS), pp. 46–54 (2012)Google Scholar
  23. 23.
    Rasmussen, C. E., Williams, C. K.: Gaussian Processes for Machine Learning. The MIT Press (2006)Google Scholar
  24. 24.
    Rasmussen, C. E., Ghahramani, Z.: Bayesian monte carlo. In: Becker, S.T., Obermayer, K. (eds.) Advances in Neural Information Processing Systems 15, pp. 489–496. MIT Press, Cambridge, MA (2003)Google Scholar
  25. 25.
    Särkkä, S.: Bayesian Filtering and Smoothing. Cambridge University Press, New York (2013)CrossRefzbMATHGoogle Scholar
  26. 26.
    Särkkä, S., Hartikainen, J., Svensson, L., Sandblom, F.: Gaussian process quadratures in nonlinear sigma-point filtering and smoothing. In: 2014 17th International Conference on Information Fusion (FUSION), pp. 1–8 (2014)Google Scholar
  27. 27.
    Sarmavuori, J., Särkkä, S.: Fourier-Hermite Kalman filter. IEEE Trans. Autom. Control 57(6), 1511–1515 (2012)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Šimandl, M., Duník, J.: Derivative-free estimation methods: new results and performance analysis. Automatica 45(7), 1749–1757 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Smith, G. L., Schmidt, S. F., McGee, L. A.: Application of statistical filter theory to the optimal estimation of position and velocity on board a circumlunar vehicle. Tech. rep., NASA Tech. Rep. (1962)Google Scholar
  30. 30.
    Wasserman, L.: All of Nonparametric Statistics. Springer (2007)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.European Center of Excellence - New Technologies for the Information SocietyFaculty of Applied Sciences, University of West BohemiaPilsenCzech Republic

Personalised recommendations