Advertisement

Chinese Annals of Mathematics, Series B

, Volume 40, Issue 5, pp 811–842 | Cite as

Ergodicity and Accuracy of Optimal Particle Filters for Bayesian Data Assimilation

  • David KellyEmail author
  • Andrew M. StuartEmail author
Article
  • 29 Downloads

Abstract

Data assimilation refers to the methodology of combining dynamical models and observed data with the objective of improving state estimation. Most data assimilation algorithms are viewed as approximations of the Bayesian posterior (filtering distribution) on the signal given the observations. Some of these approximations are controlled, such as particle filters which may be refined to produce the true filtering distribution in the large particle number limit, and some are uncontrolled, such as ensemble Kalman filter methods which do not recover the true filtering distribution in the large ensemble limit. Other data assimilation algorithms, such as cycled 3DVAR methods, may be thought of as controlled estimators of the state, in the small observational noise scenario, but are also uncontrolled in general in relation to the true filtering distribution. For particle filters and ensemble Kalman filters it is of practical importance to understand how and why data assimilation methods can be effective when used with a fixed small number of particles, since for many large-scale applications it is not practical to deploy algorithms close to the large particle limit asymptotic. In this paper, the authors address this question for particle filters and, in particular, study their accuracy (in the small noise limit) and ergodicity (for noisy signal and observation) without appealing to the large particle number limit. The authors first overview the accuracy and minorization properties for the true filtering distribution, working in the setting of conditional Gaussianity for the dynamics-observation model. They then show that these properties are inherited by optimal particle filters for any fixed number of particles, and use the minorization to establish ergodicity of the filters. For completeness we also prove large particle number consistency results for the optimal particle filters, by writing the update equations for the underlying distributions as recursions. In addition to looking at the optimal particle filter with standard resampling, they derive all the above results for (what they term) the Gaussianized optimal particle filter and show that the theoretical properties are favorable for this method, when compared to the standard optimal particle filter.

Keywords

Particle filters Data assimilation Ergodic theory 

2000 MR Subject Classification

60G35 62M20 37H99 60F99 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

DK was supported as an NYU-Courant instructor. The authors are grateful to an anonymous referee for comments on an earlier version of this paper which have led to a correction of the presentation in Subsection 2.2, and pointers to relevant literature that were previously omitted; they are also grateful to Jonathan Mattingly for discussions relating to ergodicity of nonlinear Markov processes, and to Arnaud Doucet and Adam Johansen for helpful pointers to the literature. The authors also highlight, on the occasion of his 70th birthday, the inspirational work of Andy Majda, in applied mathematics in general, and in the area of data assimilation and filtering (the subject of this paper) in particular.

References

  1. [1]
    Akashi, H. and Kumamoto, H., Random sampling approach to state estimation in switching environments, Automatica, 13(4), 1977, 429–434.zbMATHCrossRefGoogle Scholar
  2. [2]
    Atar, R. and Zeitouni, O., Exponential stability for nonlinear filtering, Annales de l’IHP Probabilitiés et statistiques, 33, 1997, 697–725,.MathSciNetzbMATHGoogle Scholar
  3. [3]
    Bauer, P., Thorpe, A. and Brunet, G., The quiet revolution of numerical weather prediction, Nature, 525(7567), 2015, 47–55.CrossRefGoogle Scholar
  4. [4]
    Blömker, D., Law, K., Stuart, A. M. and Zygalakis, K. C., Accuracy and stability of the continuous-time 3DVAR filter for the Navier-Stokes equation, Nonlinearity, 26(8), 2013, 2193.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Budhiraja, A. and Ocone, D., Exponential stability of discrete-time filters for bounded observation noise, Systems & Control Letters, 30(4), 1997, 185–193.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Burgers, G., Leeuwen, P. J. V. and Evensen, G. Analysis scheme in the ensemble Kalman filter, Monthly Weather Review, 126(6), 1998, 1719–1724.CrossRefGoogle Scholar
  7. [7]
    Chorin, A. J. and Tu, X. M., Implicit sampling for particle filters, Proceedings of the National Academy of Sciences, 106(41), 2009, 17249–17254.CrossRefGoogle Scholar
  8. [8]
    Crisan, D. and Heine, K., Stability of the discrete time filter in terms of the tails of noise distributions, Journal of the London Mathematical Society, 78(2), 2008, 441–458.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    Moral, P. D., Kurtzmann, A. and Tugaut, J., On the stability and the uniform propagation of chaos of extended ensemble Kalman-Bucy filters, 2016, arXiv:1606.08256.zbMATHGoogle Scholar
  10. [10]
    Douc, R., Fort, G., Moulines, E. and Priouret, P., Forgetting the initial distribution for hidden Markov models, Stochastic Processes and Their Applications, 119(4), 2009, 1235–1256.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    Douc, R., Moulines, E. and Olsson, J., Optimality of the auxiliary particle filter, Probability and Mathematical Statistics, 29(1), 2009, 1–28.MathSciNetzbMATHGoogle Scholar
  12. [12]
    Doucet, A., Freitas, N. D. and Gordon, N., An introduction to sequential Monte Carlo methods, Sequential Monte Carlo Methods in Practice, Springer-Verlag, New York, 2010.zbMATHGoogle Scholar
  13. [13]
    Doucet, A., Godsill, S. and Andrieu, C., On sequential Monte Carlo sampling methods for Bayesian filtering, Statistics and Computing, 10(3), 2000, 197–208.CrossRefGoogle Scholar
  14. [14]
    Evensen, G., Using the extended Kalman filter with a multilayer quasi-geostrophic ocean model, Journal of Geophysical Research: Oceans, 97(C11), 1992, 17905–17924.CrossRefGoogle Scholar
  15. [15]
    Evensen, G., The ensemble Kalman filter: Theoretical formulation and practical implementation, Ocean Dynamics, 53(4), 2003, 343–367.CrossRefGoogle Scholar
  16. [16]
    Ghil, M. and Malanotte-Rizzoli, P., Data assimilation in meteorology and oceanography, Advances in Geophysics, 33, 1991, 141–266.CrossRefGoogle Scholar
  17. [17]
    Handel, R. V., The stability of conditional Markov processes and Markov chains in random environments, The Annals of Probability, 37(5), 2009, 1876–1925.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    Harlim, J. and Majda, A. J., Catastrophic filter divergence in filtering nonlinear dissipative systems, Communications in Mathematical Sciences, 8(1), 2010, 27–43.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    Houtekamer, P. L. and Mitchell, H. L., Ensemble Kalman filtering, Quarterly Journal of the Royal Meteorological Society, 131(613), 2005, 3269–3289.CrossRefGoogle Scholar
  20. [20]
    Johansen, A. M. and Doucet, A., A note on auxiliary particle filters, Statistics & Probability Letters, 78(12), 2008, 1498–1504.MathSciNetzbMATHGoogle Scholar
  21. [21]
    Kelly, D., Majda, A. J. and Tong, X. T., Concrete ensemble Kalman filters with rigorous catas-trophic filter divergence, Proceedings of the National Academy of Sciences, 112(34), 2015, 10589–10594.CrossRefGoogle Scholar
  22. [22]
    Kelly, D., Vanden-Eijnden, E. and Weare, J., Personal Communications, 2016.Google Scholar
  23. [23]
    Kelly, D. T. B., Law, K. J. H. and Stuart, A. M., Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time, Nonlinearity, 27(10), 2014, 2579.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    Kleptsyna, M. L. and Veretennikov, A. Y., On discrete time ergodic filters with wrong initial data, Probability Theory and Related Fields, 141(3-4), 2008, 411–444.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    Kunita, H., Asymptotic behavior of the nonlinear filtering errors of Markov processes, Journal of Multivariate Analysis, 1(4), 1971, 365–393.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    Law, K., Stuart, A. and Zygalakis, K., Data Assimilation, Springer-Verlag, New York, 2015.zbMATHCrossRefGoogle Scholar
  27. [27]
    Law, K. J. H., Sanz-Alonso, D., Shukla, A. and Stuart, A. M., Filter accuracy for the Lorenz 96 model: fixed versus adaptive observation operators, Physica D: Nonlinear Phenomena, 325, 2016, 1–13.MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    Law, K. J. H., Shukla, A. and Stuart, A. M., Analysis of the 3DVAR filter for the partially observed Lorenz ’63 model, Discrete and Continuous Dynamical Systems A, 34, 2014, 1061–1078.MathSciNetzbMATHCrossRefGoogle Scholar
  29. [29]
    Law, K. J. H., Tembine, H. and Tempone, R., Deterministic mean-field ensemble Kalman filtering, 2014, arXiv:1409.0628.zbMATHGoogle Scholar
  30. [30]
    Leeuwen, P. J. V., Particle filtering in geophysical systems, Monthly Weather Review, 137(12), 2009, 4089–4114.CrossRefGoogle Scholar
  31. [31]
    Leeuwen, P. J. V., Nonlinear data assimilation in geosciences: An extremely efficient particle filter, Quarterly Journal of the Royal Meteorological Society, 136(653), 2010, 1991–1999.CrossRefGoogle Scholar
  32. [32]
    Liu, J. S. and Chen, R., Blind deconvolution via sequential imputations, Journal of the American Statistical Association, 90(430), 1995, 567–576.MathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    Lorenc, A. C., Analysis methods for numerical weather prediction, Quarterly Journal of the Royal Meteorological Society, 112(474), 1986, 1177–1194.CrossRefGoogle Scholar
  34. [34]
    Majda, A. J. and Harlim, J., Filtering Complex Turbulent Systems, Cambridge University Press, Cambridge, 2012.zbMATHCrossRefGoogle Scholar
  35. [35]
    Majda, A. J. and Tong, X. T., Robustness and accuracy of finite ensemble Kalman filters in large dimensions, 2016, arXiv:1606.09321.Google Scholar
  36. [36]
    Moodey, A. J. F., Lawless, A. S., Potthast, R. W. E. and Leeuwen, P. J. V., Nonlinear error dynamics for cycled data assimilation methods, Inverse Problems, 29(2), 2013, 025002.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [37]
    Pitt, M. K. and Shephard, N., Filtering via simulation: Auxiliary particle filters, Journal of the American Statistical Association, 94(446), 1999, 590–599.MathSciNetzbMATHCrossRefGoogle Scholar
  38. [38]
    Poyiadjis, G., Doucet, A. and Singh, S. S., Particle approximations of the score and observed information matrix in state space models with application to parameter estimation, Biometrika, 98(1), 2011, 65–80.MathSciNetzbMATHCrossRefGoogle Scholar
  39. [39]
    Rebeschini, P. and Handel, R. V., Can local particle filters beat the curse of dimensionality? The Annals of Applied Probability, 25(5), 2015, 2809–2866.MathSciNetzbMATHCrossRefGoogle Scholar
  40. [40]
    Sanz-Alonso, D. and Stuart, A. M., Long-time asymptotics of the filtering distribution for partially observed chaotic dynamical systems, SIAM/ASA Journal on Uncertainty Quantification, 3(1), 2015, 1200–1220.MathSciNetzbMATHCrossRefGoogle Scholar
  41. [41]
    Sedigh-Sarvestani, M., Albers, D. J., and Gluckman, B. J., Data assimilation of glucose dynamics for use in the intensive care unit, 2012 Annual International Conference of the IEEE Engineering in Medicine and Biology Society, IEEE, 2012, 5437–5440.Google Scholar
  42. [42]
    Sedigh-Sarvestani, M., Schiff, S. J. and Gluckman, B. J., Reconstructing mammalian sleep dynamics with data assimilation, PLoS Comput Biol, 8(11), 2012, e1002788.MathSciNetCrossRefGoogle Scholar
  43. [43]
    Snyder, C., Particle filters, the “optimal” proposal and high-dimensional systems, Proceedings of the ECMWF Seminar on Data Assimilation for Atmosphere and Ocean, 2011.Google Scholar
  44. [44]
    Snyder, C., Bengtsson, T. and Morzfeld, M., Performance bounds for particle filters using the optimal proposal, Monthly Weather Review, 143(11), 2015, 4750–4761.CrossRefGoogle Scholar
  45. [45]
    Tadic, V. Z. B. and Doucet, A., Asymptotic properties of recursive maximum likelihood estimation in non-linear state-space models, 2018, arXiv:1806.09571.Google Scholar
  46. [46]
    Thrun, S., Particle filters in robotics, Proceedings of the Eighteenth Conference on Uncertainty in artificial Intelligence, Morgan Kaufmann Publishers Inc., San Francisco, CA, 2002, 511–518.Google Scholar
  47. [47]
    Tong, X. T. and Handel, R. V., Ergodicity and stability of the conditional distributions of nondegenerate Markov chains, Annals of Applied Probability, 22(4), 2012, 1495–1540.MathSciNetzbMATHCrossRefGoogle Scholar
  48. [48]
    Tong, X. T., Majda, A. J. and Kelly, D., Nonlinear stability of the ensemble Kalman filter with adaptive covariance inflation, Communications in Mathematical Sciences, 14(5), 2016, 1283–1313.MathSciNetzbMATHCrossRefGoogle Scholar
  49. [49]
    Tong, X. T., Majda, A. J. and Kelly, D., Nonlinear stability and ergodicity of ensemble based Kalman filters, Nonlinearity, 29(2), 2016, 657.MathSciNetzbMATHCrossRefGoogle Scholar
  50. [50]
    Work, D. B., Blandin, S., Tossavainen, O. P. and et al., A traffic model for velocity data assimilation, Applied Mathematics Research Express, 2010(1), 2010, 1–35.MathSciNetzbMATHGoogle Scholar
  51. [51]
    Zaritskii, V. S., Svetnik, V. B. and Šimelevišc, L. I., Monte-Carlo technique in problems of optimal information processing, Avtomatika i Telemekhanika, 36(12), 1975, 95–103.MathSciNetGoogle Scholar

Copyright information

© The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2019

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.The Voleon GroupBerkeleyUSA
  3. 3.Department of Computing and Mathematical SciencesCalifornia Institute of TechnologyPasadenaUSA

Personalised recommendations