nach oben

Erschienen in:

2015 | OriginalPaper | Buchkapitel

4. Discrete Time: Filtering Algorithms

verfasst von : Kody Law, Andrew Stuart, Konstantinos Zygalakis

Erschienen in: Data Assimilation

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

In this chapter, we describe various algorithms for the filtering problem. Recall from Section 2.4 that filtering refers to the sequential update of the probability distribution on the state given the data, as data is acquired, and that \(Y _{j} =\{ y_{\ell}\}_{\ell=1}^{j}\) denotes the data accumulated up to time j. The filtering update from time j to time j + 1 may be broken into two steps: prediction , which is based on the equation for the state evolution, using the Markov kernel for the stochastic or deterministic dynamical system that maps \(\mathbb{P}(v_{j}\vert Y _{j})\) into \(\mathbb{P}(v_{j+1}\vert Y _{j})\); and analysis , which incorporates data via Bayes’s formula and maps \(\mathbb{P}(v_{j+1}\vert Y _{j})\) into \(\mathbb{P}(v_{j+1}\vert Y _{j+1})\). All but one of the algorithms we study (the optimal proposal version of the particle filter) will also reflect these two steps.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Vorheriges Kapitel Discrete Time: Smoothing Algorithms

Nächstes Kapitel Discrete Time: MATLAB Programs

Nur mit Berechtigung zugänglich

We do not need to match the constant terms (with respect to v), since the normalization constant in Bayes’s theorem deals with matching these.

J. Anderson. A method for producing and evaluating probabilistic forecasts from ensemble model integrations. Journal of Climate, 9(7):1518–1530, 1996.CrossRef

J. Anderson. An ensemble adjustment Kalman filter for data assimilation. Mon. Wea. Rev., 129(12):2884–2903, 2001.CrossRef

J. Anderson and S. Anderson. A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts. Mon. Wea. Rev., 127(12):2741–2758, 1999.CrossRef

A. Andrews. A square root formulation of the Kalman covariance equations. AIAA Journal, 6(6):1165–1166, 1968.CrossRefMATH

20.

C. Bishop, B. Etherton, and S. Majumdar. Adaptive sampling with the ensemble transform Kalman filter. Part I: Theoretical aspects. Mon. Wea. Rev., 129(3):420–436, 2001.

24.

C. E. A. Brett, K. F. Lam, K. J. H. Law, D. S. McCormick, M. R. Scott, and A. M. Stuart. Accuracy and stability of filters for dissipative pdes. Physica D: Nonlinear Phenomena, 245(1):34–45, 2013.MathSciNetCrossRefMATH

27.

A. Carrassi, M. Ghil, A. Trevisan, and F. Uboldi. Data assimilation as a nonlinear dynamical systems problem: Stability and convergence of the prediction–assimilation system. Chaos: An Interdisciplinary Journal of Nonlinear Science, 18:023112, 2008.MathSciNetCrossRefMATH

28.

A. Chorin, M. Morzfeld, and X. Tu. Implicit particle filters for data assimilation. Comm. Appl. Math. Comp. Sc., 5:221–240, 2010.MathSciNetCrossRefMATH

30.

A. Chorin and X. Tu. Implicit sampling for particle filters. Proc. Nat. Acad. Sc., 106:17249–17254, 2009.CrossRef

36.

P. Courtier, E. Andersson, W. Heckley, D. Vasiljevic, M. Hamrud, A. Hollingsworth, F. Rabier, M. Fisher, and J. Pailleux. The ECMWF implementation of three-dimensional variational assimilation (3d-Var). I: Formulation. Quart. J. R. Met. Soc., 124(550):1783–1807, 1998.

39.

D. Crisan and A. Doucet. A survey of convergence results on particle filtering methods for practitioners. Signal Processing, IEEE Transactions on, 50(3):736–746, 2002.MathSciNetCrossRef

46.

P. Del Moral and A. Guionnet. On the stability of interacting processes with applications to filtering and genetic algorithms. In Annales de l’Institut Henri Poincaré (B) Probability and Statistics, volume 37, pages 155–194. Elsevier, 2001.

47.

A. Doucet, S. Godsill, and C. Andrieu. On sequential Monte Carlo sampling methods for Bayesian filtering. Statistics and computing, 10(3):197–208, 2000.CrossRef

50.

G. Evensen. Data Assimilation: The Ensemble Kalman Filter. Springer, 2006.

52.

M. Fisher, Mand Leutbecher and G. Kelly. On the equivalence between Kalman smoothing and weak-constraint four-dimensional variational data assimilation. Q. J. Roy. Met. Soc., 131(613):3235–3246, 2005.

58.

C. González-Tokman and B. R. Hunt. Ensemble data assimilation for hyperbolic systems. Physica D, 243:128–142, 2013.CrossRefMATH

59.

G. A. Gottwald and A. Majda. A mechanism for catastrophic filter divergence in data assimilation for sparse observation networks. Nonlinear Processes in Geophysics, 20(5):705–712, 2013.CrossRef

65.

A. Harvey. Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge Univ Pr, 1991.

67.

K. Hayden, E. Olson, and E. Titi. Discrete data assimilation in the Lorenz and Dd Navier–Stokes equations. Physica D: Nonlinear Phenomena, 2011.

70.

I. Hoteit, X. Luo, and D.-T. Pham. Particle Kalman filtering: A nonlinear Bayesian framework for ensemble Kalman filters. Mon. Wea. Rev., 140(2), 2012.

71.

I. Hoteit, D.-T. Pham, G. Triantafyllou, and G. Korres. A new approximate solution of the optimal nonlinear filter for data assimilation in meteorology and oceanography. Mon. Wea. Rev., 136(1), 2008.

72.

P. Houtekamer and H. Mitchell. A sequential ensemble Kalman filter for atmospheric data assimilation. Mon. Wea. Rev., 129(1):123–137, 2001.CrossRef

77.

A. Jazwinski. Stochastic Processes and Filtering Theory, volume 63. Academic Pr, 1970.

79.

R. Kalman. A new approach to linear filtering and prediction problems. Journal of Basic Engineering, 82(1):35–45, 1960.CrossRef

81.

E. Kalnay. Atmospheric Modeling, Data Assimilation and Predictability. Cambridge, 2003.

82.

D. T. B. Kelly, K. J. H. Law, and A. M. Stuart. Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time. Nonlinearity, 27(10):2579, 2014.

86.

P. Lancaster and L. Rodman. Algebraic Riccati Equations. Oxford University Press, 1995.

87.

K. J. H. Law, D. Sanz-Alonso, A. Shukla, and A. M. Stuart. Filter accuracy for chaotic dynamical systems: fixed versus adaptive observation operators. http://arxiv.org/abs/1411.3113.

88.

K. J. H. Law, A. Shukla, and A. M. Stuart. Analysis of the 3DVAR filter for the partially observed Lorenz ’63 model. Discrete and Continuous Dynamical Systems A, 34:1061–1078, 2014.MathSciNetCrossRefMATH

89.

K. J. H. Law and A. M. Stuart. Evaluating data assimilation algorithms. Mon. Wea. Rev., 140:3757–3782, 2012.CrossRef

90.

F. Le Gland, V. Monbet, and V.-D. Tran. Large sample asymptotics for the ensemble Kalman filter. 2009.MATH

91.

J. Li and D. Xiu. On numerical properties of the ensemble Kalman filter for data assimilation. Computer Methods in Applied Mechanics and Engineering, 197(43):3574–3583, 2008.MathSciNetCrossRefMATH

93.

A. C. Lorenc. Analysis methods for numerical weather prediction. Quart. J. R. Met. Soc., 112(474):1177–1194, 2000.CrossRef

94.

A. C. Lorenc, S. P. Ballard, R. S. Bell, N. B. Ingleby, P. L. F. Andrews, D. M. Barker, J. R. Bray, A. M. Clayton, T. Dalby, D. Li, T. J. Payne, and F. W. Saunders. The Met. Office global three-dimensional variational data assimilation scheme. Quart. J. R. Met. Soc., 126(570):2991–3012, 2000.

98.

D. G. Luenberger. Optimization by Vector Space Methods. John Wiley & Sons, 1968.

100.

A. Majda and J. Harlim. Filtering Complex Turbulent Systems. Cambridge University Press, 2012.

101.

A. Majda, J. Harlim, and B. Gershgorin. Mathematical strategies for filtering turbulent dynamical systems. Disc. Cont. Dyn. Sys., 2010.

103.

J. Mandel, L. Cobb, and J. D. Beezley. On the convergence of the ensemble Kalman filter. Applications of Mathematics, 56(6):533–541, 2011.MathSciNetCrossRefMATH

109.

A. Moodey, A. Lawless, R. Potthast, and P. van Leeuwen. Nonlinear error dynamics for cycled data assimilation methods. Inverse Problems, 29(2):025002, 2013.

110.

L. Nerger, T. Janjic, J. Schröter, and W. Hiller. A unification of ensemble square root Kalman filters. Mon. Wea. Rev., 140(7):2335–2345, 2012.CrossRef

115.

D. Oliver, A. Reynolds, and N. Liu. Inverse Theory for Petroleum Reservoir Characterization and History Matching. Cambridge Univ Pr, 2008.

117.

E. Ott, B. Hunt, I. Szunyogh, A. Zimin, E. Kostelich, M. Corazza, E. Kalnay, D. Patil, and J. Yorke. A local ensemble Kalman filter for atmospheric data assimilation. Tellus A, 56:273–277, 2004.CrossRef

119.

D. F. Parrish and J. C. Derber. The national meteorological centers spectral statistical-interpolation analysis system. Mon. Wea. Rev., 120(8):1747–1763, 1992.CrossRef

121.

D. Pham, J. Verron, and L. Gourdeau. Singular evolutive Kalman filters for data assimilation in oceanography. Oceanographic Literature Review, 45(8), 1998.

122.

P. Rebeschini and R. van Handel. Can local particle filters beat the curse of dimensionality? Ann. Appl. Probab., 25, 2809–2866 (2015).MathSciNetCrossRefMATH

123.

S. Reich. A Gaussian-mixture ensemble transform filter. Q.J.Roy.Met.Soc., 138(662):222–233, 2012.

126.

D. Sanz-Alonso and A. M. Stuart. Long-time asymptotics of the filtering distribution for partially observed chaotic dynamical systems. arXiv:1406.1936, 2014 (to Appear in SIAM J. Uncertainty Quantification)

128.

L. Slivinski, E. Spiller, A. Apte, and B. Sandstede. A hybrid particle-ensemble Kalman filter for Lagrangian data assimilation. Mon. Wea. Rev., 145:195–211, 2015.CrossRef

129.

T. Snyder, T. Bengtsson, P. Bickel, and J. Anderson. Obstacles to high-dimensional particle filtering. Mon. Wea. Rev.., 136:4629–4640, 2008.CrossRef

136.

A. Tarantola. Inverse Problem Theory. SIAM, 2005.

138.

M. Tippett, J. Anderson, C. Bishop, T. Hamill, and J. Whitaker. Ensemble square root filters. Mon. Wea. Rev., 131(7):1485–1490, 2003.CrossRef

142.

P. Van Leeuwen. Particle filtering in geophysical systems. Mon. Wea. Rev., 137:4089–4114, 2009.CrossRef

143.

P. van Leeuwen. Nonlinear data assimilation in geosciences: an extremely efficient particle filter. Q.J.Roy.Met.Soc., 136(653):1991–1999, 2010.

144.

P. Van Leeuwen and G. Evensen. Data assimilation and inverse methods in terms of a probabilistic formulation. Mon. Wea. Rev., 124:2892–2913, 1996.

146.

J. Whitaker and T. Hamill. Ensemble data assimilation without perturbed observations. Mon. Wea. Rev., 130(7):1913–1924, 2002.CrossRef

Titel: Discrete Time: Filtering Algorithms
verfasst von: Kody Law
Andrew Stuart
Konstantinos Zygalakis
Verlag: Springer International Publishing
Buch: Data Assimilation
Print ISBN: 978-3-319-20324-9

Electronic ISBN: 978-3-319-20325-6

Copyright-Jahr: 2015
DOI: https://doi.org/10.1007/978-3-319-20325-6_4

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"