Approximate Bayesian techniques for inference in stochastic dynamical systems

This thesis is concerned with approximate inference in dynamical systems, from a variational Bayesian perspective. When modelling real world dynamical systems, stochastic differential equations appear as a natural choice, mainly because of their ability to model the noise of the system by adding a variant of some stochastic process to the deterministic dynamics. Hence, inference in such processes has drawn much attention. Here two new extended frameworks are derived and presented that are based on basis function expansions and local polynomial approximations of a recently proposed variational Bayesian algorithm. It is shown that the new extensions converge to the original variational algorithm and can be used for state estimation (smoothing). However, the main focus is on estimating the (hyper-) parameters of these systems (i.e. drift parameters and diffusion coefficients). The new methods are numerically validated on a range of different systems which vary in dimensionality and non-linearity. These are the Ornstein-Uhlenbeck process, for which the exact likelihood can be computed analytically, the univariate and highly non-linear, stochastic double well and the multivariate chaotic stochastic Lorenz '63 (3-dimensional model). The algorithms are also applied to the 40 dimensional stochastic Lorenz '96 system. In this investigation these new approaches are compared with a variety of other well known methods such as the ensemble Kalman filter / smoother, a hybrid Monte Carlo sampler, the dual unscented Kalman filter (for jointly estimating the systems states and model parameters) and full weak-constraint 4D-Var. Empirical analysis of their asymptotic behaviour as a function of observation density or length of time window increases is provided.

Gaussian process approximations of stochastic differential equations

Stochastic differential equations arise naturally in a range of contexts, from financial to environmental modeling. Current solution methods are limited in their representation of the posterior process in the presence of data. In this work, we present a novel Gaussian process approximation to the posterior measure over paths for a general class of stochastic differential equations in the presence of observations. The method is applied to two simple problems: the Ornstein-Uhlenbeck process, of which the exact solution is known and can be compared to, and the double-well system, for which standard approaches such as the ensemble Kalman smoother fail to provide a satisfactory result. Experiments show that our variational approximation is viable and that the results are very promising as the variational approximate solution outperforms standard Gaussian process regression for non-Gaussian Markov processes.

Continuous phase synchronised drives (for a rod-making machine)

Traditional high speed machinery actuators are powered and coordinated by mechanical linkages driven from a central drive, but these linkages may be replaced by independently synchronised electric drives. Problems associated with utilising such electric drives for this form of machinery were investigated. The research concentrated on a high speed rod-making machine, which required control of high inertias (0.01-0.5kgm2), at continuous high speed (2500 r/min), with low relative phase errors between two drives (0.0025 radians). Traditional minimum energy drive selection techniques for incremental motions were not applicable to continuous applications which require negligible energy dissipation. New selection techniques were developed. A brushless configuration constant enabled the comparison between seven different servo systems; the rate earth brushless drives had the best power rates which is a performance measure. Simulation was used to review control strategies, such that a microprocessor controller with a proportional velocity loop within a proportional position loop with velocity feedforward was designed. Local control schemes were investigated as means of reducing relative errors between drives: the slave of a master/slave scheme compensates for the master's errors: the matched scheme has drives with similar absolute errors so the relative error is minimised, and the feedforward scheme minimises error by adding compensation from previous knowledge. Simulation gave an approximate velocity loop bandwidth and position loop gain required to meet the specification. Theoretical limits for these parameters were defined in terms of digital sampling delays, quantisation, and system phase shifts. Performance degradation due to mechanical backlash was evaluated. Thus any drive could be checked to ensure that the performance specification could be realised. A two drive demonstrator was commissioned with 0.01kgm2 loads. By use of simulation the performance of one drive was improved by increasing the velocity loop bandwidth fourfold. With the master/slave scheme relative errors were within 0.0024 radians at a constant 2500 r/min for two 0.01 kgm^2 loads.

Estimating parameters in stochastic systems:a variational Bayesian approach

This work is concerned with approximate inference in dynamical systems, from a variational Bayesian perspective. When modelling real world dynamical systems, stochastic differential equations appear as a natural choice, mainly because of their ability to model the noise of the system by adding a variation of some stochastic process to the deterministic dynamics. Hence, inference in such processes has drawn much attention. Here a new extended framework is derived that is based on a local polynomial approximation of a recently proposed variational Bayesian algorithm. The paper begins by showing that the new extension of this variational algorithm can be used for state estimation (smoothing) and converges to the original algorithm. However, the main focus is on estimating the (hyper-) parameters of these systems (i.e. drift parameters and diffusion coefficients). The new approach is validated on a range of different systems which vary in dimensionality and non-linearity. These are the Ornstein–Uhlenbeck process, the exact likelihood of which can be computed analytically, the univariate and highly non-linear, stochastic double well and the multivariate chaotic stochastic Lorenz ’63 (3D model). As a special case the algorithm is also applied to the 40 dimensional stochastic Lorenz ’96 system. In our investigation we compare this new approach with a variety of other well known methods, such as the hybrid Monte Carlo, dual unscented Kalman filter, full weak-constraint 4D-Var algorithm and analyse empirically their asymptotic behaviour as a function of observation density or length of time window increases. In particular we show that we are able to estimate parameters in both the drift (deterministic) and the diffusion (stochastic) part of the model evolution equations using our new methods.