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Sharp failure rates for the bootstrap particle filter in high dimensions
We prove that the maximum of the sample importance weights in a high-dimensional Gaussian particle filter converges to unity unless the ensemble size grows exponentially in the system dimension. Our work is motivated by and parallels the derivations of Bengtsson, Bickel and Li (2007); however, we weaken their assumptions on the eigenvalues of the covariance matrix of the prior distribution and establish rigorously their strong conjecture on when weight collapse occurs. Specifically, we remove the assumption that the nonzero eigenvalues are bounded away from zero, which, although the dimension of the involved vectors grow to infinity, essentially permits the effective system dimension to be bounded. Moreover, with some restrictions on the rate of growth of the maximum eigenvalue, we relax their assumption that the eigenvalues are bounded from above, allowing the system to be dominated by a single mode.
First available in Project Euclid: 28 April 2008
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Mathematical Reviews number (MathSciNet)
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]
Bayesian filter curse of dimensionality ensemble forecast ensemble methods importance sampling large deviations Monte Carlo numerical weather prediction sample size requirements state-space model
Copyright © 2008, Institute of Mathematical Statistics
Bickel, Peter; Li, Bo; Bengtsson, Thomas. Sharp failure rates for the bootstrap particle filter in high dimensions. Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh, 318--329, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2008. doi:10.1214/074921708000000228. https://projecteuclid.org/euclid.imsc/1209398477
-  Anderson, J. and Anderson, S. (1999). A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts. Monthly Weather Review 127 2741–2758.
-  Arulampalam, M., Maskell, S., Gordon, N. and Clapp, T. (2002). A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking. IEEE Transactions of Signal Processing 50 174–188.
-  Bengtsson, T., Bickel, P. and Li, B. (2007). Probability and Statistics: Essays in Honor of David A. Freedman. IMS Monograph Series 337–356.
-  Del Moral, P., Doucet, A. and Jasra, A. (2006). Sequential Monte Carlo samplers. J. Roy. Statist. Soc. Ser. B 68 411–436.
-  Deltuviene, D. and Saulis, L. (2003). Asymptotic expansion of the distribution density function for the sum of random varaibles in the series scheme in large deviations zones. Acta Appl. Math. 78 87–97.
-  Doucet, A., de Freitas, N. and Gordon, N., eds. (2001). Sequential Monte Carlo Methods in Practice. Springer, New York.
-  Furrer, R. and Bengtsson, T. (2007). Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants. J. Multivariate Anal. 98 227–255.
-  Gordon, N., Salmon, D. and Smith, A. (1993). Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proceedings-F 140 107–113.
-  Liu, J. (2001). Monte Carlo Strategies in Scientific Computing. Springer, New York.
-  Liu, J. and Chen, R. (1998). Sequential Monte Carlo methods for dynamic systems. J. Amer. Statist. Assoc. 93 1032–1044.
-  Pitt, M. and Shepard, N. (1999). Filtering via simulation: Auxilliary particle filters. J. Amer. Statist. Assoc. 94 590–599.
-  Saulis, L. and Statulevicius, V. (2000). Limit Theorems of Probability Theory. Springer, New York.
-  van Leeuwen, P. (2003). A variance minimizing filter for large-scale applications. Monthly Weather Review 131 2071–2084.