ASSIGNMENT

Assignment

Assignment

Introduction

In fields as diverse as finance, biology and the physical sciences, dynamical systems are naturally modeled using continuous-time stochastic processes. Such equations are a mainstay of stock market prediction, neural modelling, environmental monitoring and other applications. More recently, they have arisen in the hemodynamics underpinning Functional Magnetic Resonance Imaging (fMRI) , which particularly motivates this work.

Continuous time modelling can provide an expressiveness that discrete time cannot. The list of notable behaviours more amicable to continuous-time includes jumps, phase transitions and bistabilities. All of these behaviours are actuated only by the continual injection of stochasticity into a system. The introduction of noise at only preset discrete times rarely suffices to capture the onset of these behaviours effectively.

Stochasticity not only drives these intrinsic behaviours of the phenomena under study, but can also be used to account for uncertainty in the model itself. Stochasticity introduced to poorly understood components reflects inconfidence in the model, while limited stochasticity implies firmer knowledge. This is particularly important in the fMRI application of this work, where an understanding of the coupling between neural and hemodynamic activity in the brain is limited. Almost surely the same can be said of other application areas, particularly young fields where computational models are yet to mature.

Given observations and a parameterised model of a continuous-time dynamical system, we are interested in estimating the underlying state and parameters of the model. To do so, some of our most powerful machinery is that of Bayesian filtering, fundamentally the Kalman filter, and in more recent years Monte Carlo techniques such as the particle filter. Largely developed for discrete-time systems, and at most the special case of linear-Gaussian systems in continuous time, the application of such methods to general non-linear, non-Gaussian continuous-time models is far from straightforward. Their applicability is important, however, as such methods can effectively combine a physical model describing system dynamics with actual observations. In the case of fMRI, purely statistical methods such as Structural Equation Modelling (SEM) oppose deterministic methods such as Dynamic Causal Modelling (DCM). Bayesian filtering represents an intermediate between the extremes of these two paradigms.

Continuous Time modeling

In the context of financial modelling, where SDEs are prevalent, presents the idea of introducing m - 1 latent points between each pair of observations of a partially observed stochastic process. This corresponds precisely to an Euler Maruyama discretisation, with m tuned to control error appropriately. Batch Markov Chain Monte Carlo (MCMC) may then be performed over the system.

We find such fixed time-step Euler-Maruyama discretisation computationally expensive, even untenable in the worst cases, and discuss this at length in §4-5. Our own methods facilitate an adaptive time-step in the first instance, and higher order discretisation than Euler-Maruyama in the second. This potentially delivers substantially leaner runtimes, which we demonstrate through empirical results. The batch MCMC approach may also suffer from poor mixing, particularly in high dimensional systems where we might expect many modes in the posterior. While the importance sampling techniques of this work also suffer under dimensionality for other reasons, ...