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Title: Generalised particle filters
Author: Li, Kai
ISNI:       0000 0004 2732 4101
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2013
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The ability to analyse, interpret and make inferences about evolving dynamical systems is of great importance in different areas of the world we live in today. Various examples include the control of engineering systems, data assimilation in meteorology, volatility estimation in financial markets, computer vision and vehicle tracking. In general, the dynamical systems are not directly observable, quite often only partial information, which is deteriorated by the presence noise, is available. This naturally leads us to the area of stochastic filtering, which is defined as the estimation of dynamical systems whose trajectory is modelled by a stochastic process called the signal, given the information accumulated from its partial observation. A massive scientific and computational effort is dedicated to the development of various tools for approximating the solution of the filtering problem. Classical PDE methods can be successful, particularly if the state space has low dimensions (one to three). In higher dimensions (up to ten), a class of numerical methods called particle filters have proved the most successful methods to-date. These methods produce approximations of the posterior distribution of the current state of the signal by using the empirical distribution of a cloud of particles that explore the signal’s state space. In this thesis, we discuss a more general class of numerical methods which involve generalised particles, that is, particles that evolve through spaces larger than the signal’s state space. Such generalised particles include Gaussian mixtures, wavelets, orthonormal polynomials, and finite elements in addition to the classical particle methods. This thesis contains a rigorous analysis of the approximation of the solution of the filtering problem using Gaussian mixtures. In particular we deduce the L2-convergence rate and obtain the central limit theorem for the approximating system. Finally, the filtering model associated to the Navier-Stokes equation will be discussed as an example.
Supervisor: Crisan, Dan Sponsor: Imperial College London
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral