Recursive Bayesian inference on stochastic differential equations

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Doctoral thesis (monograph)
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Date
2006-04-24
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Language
en
Pages
228
Series
Helsinki University of Technology Laboratory of Computational Engineering publications. Report B, 54
Abstract
This thesis is concerned with recursive Bayesian estimation of non-linear dynamical systems, which can be modeled as discretely observed stochastic differential equations. The recursive real-time estimation algorithms for these continuous-discrete filtering problems are traditionally called optimal filters and the algorithms for recursively computing the estimates based on batches of observations are called optimal smoothers. In this thesis, new practical algorithms for approximate and asymptotically optimal continuous-discrete filtering and smoothing are presented. The mathematical approach of this thesis is probabilistic and the estimation algorithms are formulated in terms of Bayesian inference. This means that the unknown parameters, the unknown functions and the physical noise processes are treated as random processes in the same joint probability space. The Bayesian approach provides a consistent way of computing the optimal filtering and smoothing estimates, which are optimal given the model assumptions and a consistent way of analyzing their uncertainties. The formal equations of the optimal Bayesian continuous-discrete filtering and smoothing solutions are well known, but the exact analytical solutions are available only for linear Gaussian models and for a few other restricted special cases. The main contributions of this thesis are to show how the recently developed discrete-time unscented Kalman filter, particle filter, and the corresponding smoothers can be applied in the continuous-discrete setting. The equations for the continuous-time unscented Kalman-Bucy filter are also derived. The estimation performance of the new filters and smoothers is tested using simulated data. Continuous-discrete filtering based solutions are also presented to the problems of tracking an unknown number of targets, estimating the spread of an infectious disease and to prediction of an unknown time series.
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Keywords
Bayesian inference, continuous-discrete filtering, unscented Kalman filter, particle filter
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https://urn.fi/urn:nbn:fi:tkk-006778