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978-3-8439-0869-6, Reihe Mathematik
A Modified Particle Filter with Adaptive Stepsize for Continuous-Time Models with Measurement Time Uncertainties
121 Seiten, Dissertation Technische Universität Kaiserslautern (2012), Softcover, A5
In this thesis, a modification of the particle filter algorithm is introduced, which works on continuous-time stochastic state space systems. The newly developed MTU-PF algorithm extends the scope of application of particle filter (PF) methods to systems with high variation in the measurement times by allowing the direct inclusion of measurement time uncertainties (MTU) into the underlying model. While in the standard PF measurement time uncertainties are lumped with the measurement value error, the MTU-PF introduces additional random variables modelling the measurement time uncertainties. In this way complex correlations between time and value can be modelled more accurately leading to a significant improvement of the estimation results. The modifications additionally allow the use of adaptive time-stepping strategies for improving the performance of the algorithm.
After presenting the standard PF for discrete-time stochastic state space models, the theoretical foundations for the new MTU-PF algorithm are established by generalizing the standard PF to the context of time-continuous states with time-discrete measurements and by introducing a new formulation of the filter model for the system states. Then the idea of introducing measurement time uncertainties into the model is illustrated, together with a motivating example. Afterwards, a new full model and new filter models, which include random variables modelling the measurement times, are developed. Moreover, it is shown how to effectively compute the filter distributions by introducing a new concept of partial weights which are modelled as stochastic processes. Furthermore, adaptations for the resampling procedure and for the computation of the data likelihood are derived. It is shown how the MTU-PF gives the opportunity of an adaptively chosen stepsize and a stepsize determination method is developed. The application of the standard PF and the MTU-PF to the motivating example clearly demonstrates the advantages of the MTU-PF. Furthermore, an example of a PK/PD study is considered. The underlying leucine kinetics ODE model is extended to an SDE model. Simulation runs are performed for both methods, with data taken from a clinical study involving diabetes patients. In addition, the appendix provides a detailed review on different discretization techniques for stochastic differential equations, which can be used for the sampling and update steps of the MTU-PF.