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Curse-of-dimensionality revisited: Collapse of the particle filter in very large scale systems

Thomas Bengtsson, Peter Bickel, and Bo Li

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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.

Chapter information

Source
Deborah Nolan and Terry Speed, eds., Probability and Statistics: Essays in Honor of David A. Freedman (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2008), 316-334

Dates
First available in Project Euclid: 7 April 2008

Permanent link to this document
https://projecteuclid.org/euclid.imsc/1207580091

Digital Object Identifier
doi:10.1214/193940307000000518

Subjects
Primary: 93E11: Filtering [See also 60G35] 62L12: Sequential estimation 86A22: Inverse problems [See also 35R30] 60G50: Sums of independent random variables; random walks 86A32: Geostatistics 86A10: Meteorology and atmospheric physics [See also 76Bxx, 76E20, 76N15, 76Q05, 76Rxx, 76U05]

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

Rights
Copyright © 2008, Institute of Mathematical Statistics

Citation

Bengtsson, Thomas; Bickel, Peter; Li, Bo. Curse-of-dimensionality revisited: Collapse of the particle filter in very large scale systems. Probability and Statistics: Essays in Honor of David A. Freedman, 316--334, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2008. doi:10.1214/193940307000000518. https://projecteuclid.org/euclid.imsc/1207580091


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