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Nonlinear tracking in a diffusion process with a Bayesian filter and the finite element method

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  • Pedersen, M.W.
  • Thygesen, U.H.
  • Madsen, H.

Abstract

A new approach to nonlinear state estimation and object tracking from indirect observations of a continuous time process is examined. Stochastic differential equations (SDEs) are employed to model the dynamics of the unobservable state. Tracking problems in the plane subject to boundaries on the state-space do not in general provide analytical solutions. A widely used numerical approach is the sequential Monte Carlo (SMC) method which relies on stochastic simulations to approximate state densities. For off-line analysis, however, accurate smoothed state density and parameter estimation can become complicated using SMC because Monte Carlo randomness is introduced. The finite element (FE) method solves the Kolmogorov equations of the SDE numerically on a triangular unstructured mesh for which boundary conditions to the state-space are simple to incorporate. The FE approach to nonlinear state estimation is suited for off-line data analysis because the computed smoothed state densities, maximum a posteriori parameter estimates and state sequence are deterministic conditional on the finite element mesh and the observations. The proposed method is conceptually similar to existing point-mass filtering methods, but is computationally more advanced and generally applicable. The performance of the FE estimators in relation to SMC and to the resolution of the spatial discretization is examined empirically through simulation. A real-data case study involving fish tracking is also analysed.

Suggested Citation

  • Pedersen, M.W. & Thygesen, U.H. & Madsen, H., 2011. "Nonlinear tracking in a diffusion process with a Bayesian filter and the finite element method," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 280-290, January.
    • Handle: RePEc:eee:csdana:v:55:y:2011:i:1:p:280-290
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    References listed on IDEAS

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    1. Arnaud Doucet & Vladislav Tadić, 2003. "Parameter estimation in general state-space models using particle methods," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 55(2), pages 409-422, June.
    2. Creal, Drew D., 2008. "Analysis of filtering and smoothing algorithms for Lévy-driven stochastic volatility models," Computational Statistics & Data Analysis, Elsevier, vol. 52(6), pages 2863-2876, February.
    3. Golightly, A. & Wilkinson, D.J., 2008. "Bayesian inference for nonlinear multivariate diffusion models observed with error," Computational Statistics & Data Analysis, Elsevier, vol. 52(3), pages 1674-1693, January.
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    1. Li, Junye, 2013. "An unscented Kalman smoother for volatility extraction: Evidence from stock prices and options," Computational Statistics & Data Analysis, Elsevier, vol. 58(C), pages 15-26.
    2. Woillez, Mathieu & Fablet, Ronan & Ngo, Tran-Thanh & Lalire, Maxime & Lazure, Pascal & de Pontual, Hélène, 2016. "A HMM-based model to geolocate pelagic fish from high-resolution individual temperature and depth histories: European sea bass as a case study," Ecological Modelling, Elsevier, vol. 321(C), pages 10-22.

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