Les filtres particulaires sont des méthodes de Monte-Carlo pour l’inférence bayésienne. Cette dernière s’appuie sur le théorème de Bayes qui exprime de quelle manière la connaissance a priori d’un système, représentée par une fonction de densité de probabilité, doit être modifiée lorsque de nouvelles informations provenant d’observations de ce système deviennent disponibles. Ce procédé est appelé assimilation de données dans les sciences de la Terre. Ces notes introduisent les filtres particulaires et se concentrent sur les problèmes spécifiques à leur utilisation dans les sciences de la Terre, où les problèmes d’assimilation sont généralement posés en très grande dimension. Un exemple est le problème de la prévision météorologique, dont la taille de l’espace d’état peut dépasser le milliard. Nous discutons ensuite les récents progrès et outils développés en vue de gérer ce fameux « fléau de la dimension », tels que la localisation ou la méthode des « proposal densities », dans laquelle on modifie légèrement le modèle étudié en vue d’améliorer la densité de probabilité a posteriori. Toutes ces considérations amènent à une nouvelle classe de filtres particulaires qui est effectivement capable d’estimer les densités de probabilité a posteriori. Notre exposition privilégie la présentation des idées principales de cette direction de recherche en pleine expansion, parfois au détriment de la rigueur mathématique.
Particle Filters are Monte-Carlo methods used for Bayesian Inference. Bayesian Inference is based on Bayes Theorem that states how prior information about a system, encoded in a probability density function, is updated when new information in the form of observations of that system become available. This process is called data assimilation in the geosciences. This contribution discusses what particle filters are and what the main issue is when trying to use them in the geosciences, in which the data-assimilation problem is typically very high dimensional. An example is numerical weather forecasting, with a state-space size of a billion or more. Then it discusses recent progress made in trying to beat the so-called “curse of dimensionality”, such as localisation and clever ways to slightly change the model equations to obtain better approximations to the posterior probability density via so-called proposal densities. This culminates in a new class of particle filters that is indeed able to provide estimates of the posterior probability density. The emphasis is not on mathematical rigour but on conveying the main new ideas in this rapidly growing field.
Peter Jan van Leeuwen 1
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@article{AFST_2017_6_26_4_1051_0,
author = {Peter Jan van Leeuwen},
title = {Particle {Filters} for nonlinear data assimilation in high-dimensional systems},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {1051--1085},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 26},
number = {4},
year = {2017},
doi = {10.5802/afst.1560},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1560/}
}
TY - JOUR AU - Peter Jan van Leeuwen TI - Particle Filters for nonlinear data assimilation in high-dimensional systems JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2017 SP - 1051 EP - 1085 VL - 26 IS - 4 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1560/ DO - 10.5802/afst.1560 LA - en ID - AFST_2017_6_26_4_1051_0 ER -
%0 Journal Article %A Peter Jan van Leeuwen %T Particle Filters for nonlinear data assimilation in high-dimensional systems %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2017 %P 1051-1085 %V 26 %N 4 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1560/ %R 10.5802/afst.1560 %G en %F AFST_2017_6_26_4_1051_0
Peter Jan van Leeuwen. Particle Filters for nonlinear data assimilation in high-dimensional systems. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 4, pp. 1051-1085. doi : 10.5802/afst.1560. https://afst.centre-mersenne.org/articles/10.5802/afst.1560/
[1] An exploration of the equivalent weights particle filter, Quart. J. Roy. Meteor. Soc., Volume 139 (2013) no. 672, pp. 820-840 | DOI
[2] A Monte-Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts, Monthly Weather Rev., Volume 127 (1999), pp. 2741-2758 | DOI
[3] Fundamentals of Stochastic Filtering, Stochastic Modelling and Applied Probability, 60, Springer, 2009 | DOI
[4] Toward a nonlinear ensemble filter for high-dimensional systems, J. Geophys. Res., Volume 108 (2003), pp. 8775-8785 | DOI
[5] On the stability of sequential Monte Carlo methods in high dimensions, Ann. Appl. Probab., Volume 24 (2014) no. 4, pp. 1396-1445 | DOI | Zbl
[6] Implicit sampling for particle filters, PNAS, Volume 106 (2009) no. 41, pp. 17249-17254 | DOI
[7] On the stability of interacting processes with applications to filtering and genetic algorithms, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 37 (2001) no. 2, pp. 155-194 | DOI | Zbl
[8] Feynman-Kac Formulae. Genealogical and Interacting Particle Systems with Applications, Probability and Its Applications, Springer, 2004, xviii+555 pages | Zbl
[9] Sequential Monte-Carlo methods in practice (Arnaud Doucet; Nando De Freitas; Neil Gordon, eds.), Statistics for Engineering and Information Science, Springer, 2001, xiv+581 pages | Zbl
[10] Novel approach to nonlinear/non-Gaussian Bayesian state estimation, IEE Proceedings F, Volume 140 (1993) no. 2, pp. 107-113 | DOI
[11] When do nonlinear filters achieve maximal accuracy?, SIAM J. Control Optim., Volume 48 (2009) no. 5, pp. 3151-3168 | DOI | Zbl
[12] Large sample asymptotics for the ensemble Kalman Filter, The Oxford handbook of nonlinear filtering, Oxford University Press, 2011, pp. 598-631 | Zbl
[13] Nonlinear ensemble data assimilation for the ocean, Recent developments in data assimilation for atmosphere and ocean, 8-12 September 2003 (2003), pp. 265-286
[14] Particle Filtering in Geophysical Systems, Monthly Weather Rev., Volume 137 (2009), pp. 4089-4114 | DOI
[15] Nonlinear Data Assimilation in geosciences: an extremely efficient particle filter, Quart. J. Roy. Meteor. Soc., Volume 136 (2010), pp. 1991-1996 | DOI
[16] Efficient non-linear Data Assimilation in Geophysical Fluid Dynamics, Computers & Fluids, Volume 46 (2011) no. 1, pp. 52-58 | DOI
[17] A random map implementation of implicit filters, J. Comput. Phys., Volume 231 (2012) no. 4, pp. 2049-2066 | DOI | Zbl
[18] A local particle filter for high dimensional geophysical systems, Nonlin. Processes Geophys., Volume 23 (2016), pp. 391-405 | DOI
[19] Filtering via simulation: Auxilary particle filters, J. Am. Stat. Ass., Volume 94 (1999) no. 446, pp. 590-599 | DOI
[20] A localized particle filter for high-dimensional nonlinear systems, Monthly Weather Rev., Volume 144 (2016), pp. 59-76 | DOI
[21] Can local particle filters beat the curse of dimensionality?, Ann. Appl. Probab., Volume 25 (2015) no. 5, pp. 2809-2866 | DOI | Zbl
[22] Probabilistic Forecasting and Bayesian Data Assimilation, Cambridge University Press, 2015, x+297 pages | Zbl
[23] Obstacles to high-dimensional particle filtering, Monthly Weather Rev., Volume 136 (2008), pp. 4629-4640 | DOI
[24] Performance bounds for particle filters using the optimal proposal, Monthly Weather Rev., Volume 143 (2015), pp. 4750-4761 | DOI
[25] Ergodicity and stability of the conditional distributions of nondegenerate Markov chains, Ann. Appl. Probab., Volume 22 (2012) no. 4, pp. 1495-1540 | DOI | Zbl
[26] Implicit equal-weights particle filter, Quart. J. Roy. Meteor. Soc., Volume 142 (2016) no. 698, pp. 1904-1919 | DOI
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