Foster, Neil; (2025) Sequential Monte Carlo Methods for High-Dimensional State Space Models. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

This thesis focuses on developing methodologies to approximate the filtering distribution in partially observed state space models, with particular attention to addressing particle degeneracy issues that arise in high-dimensional state spaces and when dealing with informative observations. Chapter 1 introduces the challenge of sequential inference within state space models, along with an overview of the Sequential Monte Carlo (SMC) methodology. The chapter also presents the bootstrap filter as a tool for approximating the filtering distribution, discussing its limitations and providing background information. Chapter 2 reviews three popular existing methodologies for filtering discretely observed stochastic differential equations (SDEs). Chapters 3 and 4 propose a new methodology for filtering partially observed SDEs. This approach combines likelihood-informed proposals, tempering steps targeting intermediate distributions between prior and posterior densities, and mutations through Markov Chain Monte Carlo (MCMC) steps to enhance particle diversity. The methodology’s effectiveness is evaluated through a small-scale experiment involving a double-well potential process. Chapter 5 applies filtering techniques to a tsunami wave field model, treated as a discrete-time state space model. The optimal filter can be derived under a linear observation scheme with Gaussian noise processes for both observations and process noise. We present an efficient approach for computing the optimal filter using the Fast Fourier Transform, which enables efficient sampling of conditional Gaussian random fields and rapid matrix-vector multiplication. Numerical experiments demonstrate the accuracy and efficiency of the proposed method. Chapter 6 extends the methodology from Chapter 3 to the filtering of a tsunami wave field model represented as a stochastic partial differential equation. This complex setting is tested through experiments that examine the algorithm’s robustness with respect to both spatial resolution and the informativeness of observations.

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