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

Publication: Research - peer-review › Journal article – Annual report year: 2011

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**Nonlinear tracking in a diffusion process with a Bayesian filter and the finite element method.** / Pedersen, Martin Wæver; Thygesen, Uffe Høgsbro; Madsen, Henrik.

Publication: Research - peer-review › Journal article – Annual report year: 2011

### Harvard

*Computational Statistics & Data Analysis*, vol 55, no. 1, pp. 280-290., 10.1016/j.csda.2010.04.018

### APA

*Computational Statistics & Data Analysis*,

*55*(1), 280-290. 10.1016/j.csda.2010.04.018

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### MLA

*Computational Statistics & Data Analysis*. 2011, 55(1). 280-290. Available: 10.1016/j.csda.2010.04.018

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### Bibtex

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### RIS

TY - JOUR

T1 - Nonlinear tracking in a diffusion process with a Bayesian filter and the finite element method

AU - Pedersen,Martin Wæver

AU - Thygesen,Uffe Høgsbro

AU - Madsen,Henrik

PY - 2011

Y1 - 2011

N2 - 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 offline 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.

AB - 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 offline 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.

U2 - 10.1016/j.csda.2010.04.018

DO - 10.1016/j.csda.2010.04.018

M3 - Journal article

VL - 55

SP - 280

EP - 290

JO - Computational Statistics & Data Analysis

T2 - Computational Statistics & Data Analysis

JF - Computational Statistics & Data Analysis

SN - 0167-9473

IS - 1

ER -