logo AFST
Particle Filters for nonlinear data assimilation in high-dimensional systems
Peter Jan van Leeuwen
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 4, p. 1051-1085

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.

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.

Published online : 2017-12-13
DOI : https://doi.org/10.5802/afst.1560
@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},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 26},
     number = {4},
     year = {2017},
     pages = {1051-1085},
     doi = {10.5802/afst.1560},
     language = {en},
     url = {https://afst.centre-mersenne.org/item/AFST_2017_6_26_4_1051_0}
}
van Leeuwen, Peter Jan. Particle Filters for nonlinear data assimilation in high-dimensional systems. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 4, pp. 1051-1085. doi : 10.5802/afst.1560. afst.centre-mersenne.org/item/AFST_2017_6_26_4_1051_0/

[1] M. Ades; Peter Jan van Leeuwen An exploration of the equivalent weights particle filter, Quart. J. Roy. Meteor. Soc., Tome 139 (2013) no. 672, pp. 820-840 | Article

[2] Jeffrey L. Anderson; Stephen L. Anderson A Monte-Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts, Monthly Weather Rev., Tome 127 (1999), pp. 2741-2758 | Article

[3] A. Bain; A. Crisan Fundamentals of Stochastic Filtering, Stochastic Modelling and Applied Probability, Tome 60, Springer, 2009 | Article

[4] Thomas Bengtsson; Chris Snyder; Doug Nychka Toward a nonlinear ensemble filter for high-dimensional systems, J. Geophys. Res., Tome 108 (2003), pp. 8775-8785 | Article

[5] Alexandros Beskos; Dan Crisan; Ajay Jasra On the stability of sequential Monte Carlo methods in high dimensions, Ann. Appl. Probab., Tome 24 (2014) no. 4, pp. 1396-1445 | Article | Zbl 1304.82070

[6] Alexandre J. Chorin; Xuemin Tu Implicit sampling for particle filters, PNAS, Tome 106 (2009) no. 41, pp. 17249-17254 | Article

[7] Pierre Del Moral On the stability of interacting processes with applications to filtering and genetic algorithms, Ann. Inst. Henri Poincaré, Probab. Stat., Tome 37 (2001) no. 2, pp. 155-194 | Article | Zbl 0990.60005

[8] Pierre Del Moral Feynman-Kac Formulae. Genealogical and Interacting Particle Systems with Applications, Probability and Its Applications, Springer, 2004, xviii+555 pages | Zbl 1130.60003

[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 0967.00022

[10] N. J. Gordon; D. J. Salmond; A. F. M. Smith Novel approach to nonlinear/non-Gaussian Bayesian state estimation, IEE Proceedings F, Tome 140 (1993) no. 2, pp. 107-113 | Article

[11] Ramon van Handel When do nonlinear filters achieve maximal accuracy?, SIAM J. Control Optim., Tome 48 (2009) no. 5, pp. 3151-3168 | Article | Zbl 1203.93198

[12] F. Le Gland; V. Monbet; V.-D. Tran Large sample asymptotics for the ensemble Kalman Filter, The Oxford handbook of nonlinear filtering, Oxford University Press, 2011, pp. 598-631 | Zbl 1225.93108

[13] Peter Jan van Leeuwen 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] Peter Jan van Leeuwen Particle Filtering in Geophysical Systems, Monthly Weather Rev., Tome 137 (2009), pp. 4089-4114 | Article

[15] Peter Jan van Leeuwen Nonlinear Data Assimilation in geosciences: an extremely efficient particle filter, Quart. J. Roy. Meteor. Soc., Tome 136 (2010), pp. 1991-1996 | Article

[16] Peter Jan van Leeuwen Efficient non-linear Data Assimilation in Geophysical Fluid Dynamics, Computers & Fluids, Tome 46 (2011) no. 1, pp. 52-58 | Article

[17] Matthias Morzfeld; Xuemin Tu; Ethan Atkins; Alexandre J. Chorin A random map implementation of implicit filters, J. Comput. Phys., Tome 231 (2012) no. 4, pp. 2049-2066 | Article | Zbl 1242.65012

[18] Stephen G. Penny; Takemasa Miyoshi A local particle filter for high dimensional geophysical systems, Nonlin. Processes Geophys., Tome 23 (2016), pp. 391-405 | Article

[19] Michael K. Pitt; Neil Shephard Filtering via simulation: Auxilary particle filters, J. Am. Stat. Ass., Tome 94 (1999) no. 446, pp. 590-599 | Article

[20] Jonathan Poterjoy A localized particle filter for high-dimensional nonlinear systems, Monthly Weather Rev., Tome 144 (2016), pp. 59-76 | Article

[21] Patrick Rebeschini; Ramon van Handel Can local particle filters beat the curse of dimensionality?, Ann. Appl. Probab., Tome 25 (2015) no. 5, pp. 2809-2866 | Article | Zbl 1325.60058

[22] Sebastian Reich; Colin Cotter Probabilistic Forecasting and Bayesian Data Assimilation, Cambridge University Press, 2015, x+297 pages | Zbl 1314.62005

[23] Chris Snyder; Thomas Bengtsson; Peter Bickel; Jeffrey L. Anderson Obstacles to high-dimensional particle filtering, Monthly Weather Rev., Tome 136 (2008), pp. 4629-4640 | Article

[24] Chris Snyder; Thomas Bengtsson; Mathias Morzfeld Performance bounds for particle filters using the optimal proposal, Monthly Weather Rev., Tome 143 (2015), pp. 4750-4761 | Article

[25] Xin Thomson Tong; Ramon van Handel Ergodicity and stability of the conditional distributions of nondegenerate Markov chains, Ann. Appl. Probab., Tome 22 (2012) no. 4, pp. 1495-1540 | Article | Zbl 1252.60069

[26] Mengbin Zhu; Peter Jan van Leeuwen; Javier Amezcua Implicit equal-weights particle filter, Quart. J. Roy. Meteor. Soc., Tome 142 (2016) no. 698, pp. 1904-1919 | Article