| dc.description.abstract | Temporal phenomena in a range of disciplines are more naturally modelled in
continuous-time than coerced into a discrete-time formulation. Differential systems
form the mainstay of such modelling, in fields from physics to economics,
geoscience to neuroscience. While powerful, these are fundamentally limited by
their determinism. For the purposes of probabilistic inference, their extension
to stochastic differential equations permits a continuous injection of noise and
uncertainty into the system, the model, and its observation.
This thesis considers Bayesian filtering for state and parameter estimation in general
non-linear, non-Gaussian systems using these stochastic differential models.
It identifies a number of challenges in this setting over and above those of discrete
time, most notably the absence of a closed form transition density. These are addressed
via a synergy of diverse work in numerical integration, particle filtering
and high performance distributed computing, engineering novel solutions for this
class of model.
In an area where the default solution is linear discretisation, the first major
contribution is the introduction of higher-order numerical schemes, particularly
stochastic Runge-Kutta, for more efficient simulation of the system dynamics.
Improved runtime performance is demonstrated on a number of problems, and
compatibility of these integrators with conventional particle filtering and smoothing
schemes discussed.
Finding compatibility for the smoothing problem most lacking, the major theoretical
contribution of the work is the introduction of two novel particle methods, the
kernel forward-backward and kernel two-filter smoothers. By harnessing kernel
density approximations in an importance sampling framework, these attain cancellation
of the intractable transition density, ensuring applicability in continuous
time. The use of kernel estimators is particularly amenable to parallelisation, and
provides broader support for smooth densities than a sample-based representation
alone, helping alleviate the well known issue of degeneracy in particle smoothers.
Implementation of the methods for large-scale problems on high performance
computing architectures is provided. Achieving improved temporal and spatial
complexity, highly favourable runtime comparisons against conventional techniques are presented.
Finally, attention turns to real world problems in the domain of Functional
Magnetic Resonance Imaging (fMRI), first constructing a biologically motivated
stochastic differential model of the neural and hemodynamic activity underlying
the observed signal in fMRI. This model and the methodological advances of
the work culminate in application to the deconvolution and effective connectivity
problems in this domain. | en |
