Error analysis for numerical formulation of particle filter

Xiaoying Han; Jinglai Li; Dongbin Xiu

doi:10.3934/dcdsb.2015.20.1337

Article Contents

2015, Volume 20, Issue 5: 1337-1354. Doi: 10.3934/dcdsb.2015.20.1337

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Error analysis for numerical formulation of particle filter

1.
221 Parker Hall, Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849
2.
Institute of Natural sciences, Department of Mathematics, MOE Key Lab of Scientic and Engineering Computing, Shanghai JiaoTong University, 800 Dongchuan Rd, Minhang 200240, Shanghai
3.
Department of Mathematics, Scientific Computing and Imagining Institute, The University of Utah, Salt Lake City, UT 84112

Received: December 2013

Revised: January 2015

Published: July 2015

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Abstract

As an approximation of the optimal stochastic filter, particle filter is a widely used tool for numerical prediction of complex systems when observation data are available. In this paper, we conduct an error analysis from a numerical analysis perspective. That is, we investigate the numerical error, which is defined as the difference between the numerical implementation of particle filter and its continuous counterpart, and demonstrate that the error consists of discretization errors for solving the dynamic equations numerically and sampling errors for generating the random particles. We then establish convergence of the numerical particle filter to the continuous optimal filter and provide bounds for the convergence rate. Remarkably, our analysis suggests that more frequent data assimilation may lead to larger numerical errors of the particle filter. Numerical examples are provided to verify the theoretical findings.

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Mathematics Subject Classification: Primary: 65C35; Secondary: 65C05.

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