Abstract
The nonlinear filtering problem occurs in many scientific areas. Sequential Monte Carlo solutions with the correct asymptotic behavior such as particle filters exist, but they are computationally too expensive when working with high-dimensional systems. The ensemble Kalman filter (EnKF) is a more robust method that has shown promising results with a small sample size, but the samples are not guaranteed to come from the true posterior distribution. By approximating the model error with a Gaussian distribution, one may represent the posterior distribution as a sum of Gaussian kernels. The resulting Gaussian mixture filter has the advantage of both a local Kalman type correction and the weighting/resampling step of a particle filter. The Gaussian mixture approximation relies on a bandwidth parameter which often has to be kept quite large in order to avoid a weight collapse in high dimensions. As a result, the Kalman correction is too large to capture highly non-Gaussian posterior distributions. In this paper, we have extended the Gaussian mixture filter (Hoteit et al., Mon Weather Rev 136:317–334, 2008) and also made the connection to particle filters more transparent. In particular, we introduce a tuning parameter for the importance weights. In the last part of the paper, we have performed a simulation experiment with the Lorenz40 model where our method has been compared to the EnKF and a full implementation of a particle filter. The results clearly indicate that the new method has advantages compared to the standard EnKF.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Anderson, B.D., Moore, J.B.: Optimal Filtering. Dover, New York (2005)
Apte, A., Hairer, M., Stuart, A.M., Voss, J.: Sampling the posterior: an approach to non-Gaussian data assimilation. Physica D 230, 50–64 (2007)
Bengtsson, T., Bickel, P., Li, B.: Curse-of-dimensionality revisited: collapse of particle filter in very large scale systems. IMS Collect. 2, 316–334 (2008)
Bengtsson, T., Snyder, C., Nychka, D.: Toward a nonlinear ensemble filter for high-dimensional systems. J. Geophys. Res. 108, 35–45 (2003)
Chen, R., Liu, J.S.: Mixture Kalman filters. J. R. Stat. Soc., Ser. B Stat. Methodol. 60, 493–508 (2000)
Doucet, A., de Freitas, N., Gordon, N.: Sequential Monte Carlo Methods in Practice. Springer, New York (2001)
Doucet, A., Godsill, S., Andrieu, C.: On sequential Monte Carlo sampling methods for Bayesian filtering. Stat. Comput. 10, 197–208 (2000)
Evensen, G.: Data Assimilation: The Ensemble Kalman Filter. Springer, New York (2007)
Gneiting, T., Raftery, A., Westveld III, A., Goldman, T.: Calibrated probabilistic forecasting using ensemble model output statistics and minimum CRPS estimation. Mon. Weather Rev. 133(5), 1098–1118 (2005)
Gordon, N.: Bayesian Methods for Tracking. Ph.D. thesis, University of London (1993)
Hoteit, I., Pham, D.-T., Korres, G., Triantafyllou, G.: Particle Kalman filtering for data assimilation in meteorology and oceanography. In: 3rd WCRP International Conference on Reanalysis, p. 6. Tokyo, Japan (2008)
Hoteit, I., Pham, D.-T., Triantafyllou, G., Korres, G.: A new approximative solution of the optimal nonlinear filter for data assimilation in meteorology and oceanography. Mon. Weather Rev. 136, 317–334 (2008)
Kong, A., Liu, J., Wong, W.: Sequential imputations and Bayesian missing data problems. J. Am. Stat. Assoc. 89(425), 278–288 (1994)
Kotecha, J.H., Djurić, P.M.: Gaussian sum particle filtering. IEEE 51(10), 2602–2612 (2003)
Liu, J., West, M.: Sequential Monte Carlo Methods in Practice, pp. 197–223. Springer, New York (2001)
Lorenz, E.N., Emanuel, K.A.: Optimal sites for supplementary weather observations: simulations with a small model. J. Atmos. Sci. 55, 399–414 (1998)
Mandel, J., Beezley, J.D.: An ensemble Kalman-particle predictor–corrector filter for non-Gaussian data assimilation. In: ICCS 2009: Proceedings of the 9th International Conference on Computational Science, pp. 470–478. Springer, Berlin (2009). ISBN 978-3-642-01972-2
Robert, C.P., Casella, G.: Monte Carlo Statistical Methods. Springer, New York (2004)
Sakov, P., Oke, P.R.: Implications of the form of the ensemble transformation in the ensemble square root filters. Mon. Weather Rev. 136, 1042–1053 (2008)
Silverman, B.W.: Density Estimation for Statistics and Data Analysis. Chapman and Hall, New York (1986)
Snyder, C., Bengtsson, T., Bickel, P., Anderson, J.: Obstacles to high-dimensional particle filtering. Mon. Weather Rev. 136, 4629–4640 (2008)
Stordal, A.: Sequential Monte Carlo Methods for Nonlinear Bayesian Filtering. M.Sc. thesis, Department of Mathematics, University of Bergen (2008)
Titterington, D.: Statistical Analysis for Finite Mixture Distributions. Wiley, Chichester (1985)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Stordal, A.S., Karlsen, H.A., Nævdal, G. et al. Bridging the ensemble Kalman filter and particle filters: the adaptive Gaussian mixture filter. Comput Geosci 15, 293–305 (2011). https://doi.org/10.1007/s10596-010-9207-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10596-010-9207-1