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VOL. 2 | 2008 Curse-of-dimensionality revisited: Collapse of the particle filter in very large scale systems
Thomas Bengtsson, Peter Bickel, Bo Li

Editor(s) Deborah Nolan, Terry Speed

Abstract

It has been widely realized that Monte Carlo methods (approximation via a sample ensemble) may fail in large scale systems. This work offers some theoretical insight into this phenomenon in the context of the particle filter. We demonstrate that the maximum of the weights associated with the sample ensemble converges to one as both the sample size and the system dimension tends to infinity. Specifically, under fairly weak assumptions, if the ensemble size grows sub-exponentially in the cube root of the system dimension, the convergence holds for a single update step in state-space models with independent and identically distributed kernels. Further, in an important special case, more refined arguments show (and our simulations suggest) that the convergence to unity occurs unless the ensemble grows super-exponentially in the system dimension. The weight singularity is also established in models with more general multivariate likelihoods, e.g. Gaussian and Cauchy. Although presented in the context of atmospheric data assimilation for numerical weather prediction, our results are generally valid for high-dimensional particle filters.

Information

Published: 1 January 2008
First available in Project Euclid: 7 April 2008

zbMATH: 1166.93376
MathSciNet: MR2459957

Digital Object Identifier: 10.1214/193940307000000518

Subjects:
Primary: 60G50 , 62L12 , 86A10 , 86A22 , 86A32 , 93E11

Keywords: ensemble forecast , inverse problem , Monte Carlo , multivariate Cauchy , multivariate likelihood , numerical weather prediction , sample ensemble , state-space model