This work proposes a method of model-based speech enhancement that uses a network of HMMs to first decode noisy speech and to then synthesise a set of features that enables a speech production model to reconstruct clean speech. The motivation is to remove the distortion and residual and musical noises that are associated with conventional filteringbased methods of speech enhancement. STRAIGHT forms the speech production model for speech reconstruction and requires a time-frequency spectral surface, aperiodicity and a fundamental frequency contour. The technique of HMM-based synthesis is used to create the estimate of the timefrequency surface, and aperiodicity after the model and state sequence is obtained from HMM decoding of the input noisy speech. Fundamental frequency were found to be best estimated using the PEFAC method rather than synthesis from the HMMs. For the robust HMM decoding in noisy conditions it is necessary for the HMMs to model noisy speech and consequently noise adaptation is investigated to achieve this and its resulting effect on the reconstructed speech measured. Even with such noise adaptation to match the HMMs to the noisy conditions, decoding errors arise, both in terms of incorrect decoding and time alignment errors. Confidence measures are developed to identify such errors and then compensation methods developed to conceal these errors in the enhanced speech signal. Speech quality and intelligibility analysis is first applied in terms of PESQ and NCM showing the superiority of the proposed method against conventional methods at low SNRs. Three way subjective MOS listening test then discovers the performance of the proposed method overwhelmingly surpass the conventional methods over all noise conditions and then a subjective word recognition test shows an advantage of the proposed method over speech intelligibility to the conventional methods at low SNRs.

The study of random orders and of rankings models has attracted much work in the combinatorics and statistics literature. However, there has been little focus on random partial order models for statistical modelling. A partial order on a set P corresponds to a transitively closed, directed acyclic graph h(P) with vertices in P. Such orders generalize orders defined by partitioning the elements of P and ranking the elements of the partition. Observed orders are modelled as random linear extensions of suborders of an unobserved partial order h(P) evolving according to a stochastic process, and inference on the partial order is of interest. Chapter 1 reviews some static random partial order models. Of particular interest for modelling is a latent variables model based on the random k-dimensional orders and a parameter controlling the mean depth of a partial order. In Chapter 2, it is extended to a stochastic process on latent variables to describe a partial order evolving continuously in time. As a Hidden Markov model, the process is observed by taking random linear extensions from suborders of the partial order at a sequence of sampling times. The posterior distribution for the unobserved process is doubly-intractable. The basis for a numerical inference algorithm in Chapter 3 is Particle MCMC with an efficient particle filtering transition distribution on the latent variables. Sampling latent variables relates to the well studied problem of estimating multivariate normal orthant probabilities, for which Chapter 4 gives a new importance sampler. It is competitive with existing samplers under some conditions on the covariance. Inference on the partial order process is computational, and Chapter 5 gives some numerical algorithms to reduce the complexity of some common latent variables computations. Lastly, Chapter 6 applies Chapter 2 and 3 to dynamic ranking problems in the areas of historical research and sport tournaments.

In this thesis we consider stochastic resonance for a diffusion with drift given by a potential, which has two metastable states and two pathways between them. Depending on the direction of the forcing the height of the two barriers, one for each path, will either oscillate alternating or in synchronisation. We consider a simplified model given by discrete and continuous time Markov Chains with two states. This was done for alternating and synchronised wells. The invariant measures are derived for both cases and shown to be constant for the synchronised case. A PDF for the escape time from an oscillatory potential is reviewed. Methods of detecting stochastic resonance are presented, which are linear response, signal-to-noise ratio, energy, out-of-phase measures, relative entropy and entropy. A new statistical test called the conditional Kolmogorov-Smirnov test is developed, which can be used to analyse stochastic resonance. An explicit two dimensional potential is introduced, the critical point structure derived and the dynamics, the invariant state and escape time studied numerically. The six measures are unable to detect the stochastic resonance in the case of synchronised saddles. The distribution of escape times however not only shows a clear sign of stochastic resonance, but changing the direction of the forcing from alternating to synchronised saddles an additional resonance at double the forcing frequency starts to appear. The conditional KS test reliably detects the stochastic resonance even for forcing quick enough and for data so sparse that the stochastic resonance is not obvious directly from the histogram of escape times.

An epidemic of an infectious disease can be modelled by using either a deterministic model or a stochastic model. In this thesis, we consider the effect that different types of noise has on the dynamical behaviour of deterministic SIS models and SIR/SIRS models as well as an HIV model. We start off with a literature review giving previous work and the mathematical background to the area. Next, we introduce demographic stochasticity into the well-established deterministic SIS model with births and deaths and derive a stochastic differential equation (SDE). We assume that an infected individual or a susceptible individual who dies is immediately replaced by a susceptible individual and thus the population size is kept constant. In order for our model to make sense, we then prove that the SDE has a strong unique nonnegative solution which is bounded above and establish the conditions needed for the disease to become extinct. Based on the idea of the Feller test, we also calculate the respective probabilities of the solution first hitting zero or the upper limit. Numerical simulations are then produced using the Milstein method with both theoretical and realistic parameter values to confirm our theoretical results. Motivated by the model discussed in the first topic, we then continue our study on the effect of demographic stochasticity on the deterministic SIS model by now assuming that the births and deaths of individuals are independent of each other and thus the population size can vary with respect to time. In this case, the per capita disease contact rate may be dependent on the population size and we have shown that this model allows us to consider the cases when the population size tends to a large number and when the population size tends to a small number. First we look at the SDE model for the total population size and show that there exists a strong unique nonnegative solution. Then we look at the two-dimensional SDE SIS model and show that there also exists a strong unique nonnegative solution which is bounded above given the total population size. We then obtain the conditions needed in order for the disease to become extinct in finite time almost surely. Numerical simulations with both theoretical and realistic parameter values are also produced to confirm our theoretical results. Next we look at a different type of noise, namely the telegraph noise, which is an example of an environmental noise. Telegraph noise could be modelled as changing between two or more regimes of environment which differ by factors such as rainfalls or nutrition. This form of switching can be modelled using a finite-state Markov Chain. We incorporate the telegraph noise into the SIRS epidemic model. First we start with a two-state Markov Chain and show that there exists a unique nonnegative solution and establish the conditions for extinction and persistence for the stochastic SIRS model. We then explain how the results can be generalised to a finite-state Markov Chain. Furthermore we also show that the results for the SIR model with Markov switching are a special case of the SIRS model. Numerical simulations are produced using theoretical and realistic parameter values to confirm our theoretical results. Lastly we look at the modified Kaplan HIV model amongst injecting drug users. We introduce environmental stochasticity into the deterministic HIV model by the well-known standard technique of parameter perturbation. We then prove that the resulting SDE has a unique global nonnegative solution. As well as constructing the conditions required for extinction and persistence we also show that there exists a stationary distribution for the persistence case. Simulations using the Euler-Maruyama method with realistic parameter values are then constructed to illustrate and support our theoretical results. A brief discussion and summary section is given at the end to conclude the thesis.

Electrical impedance tomography (EIT) is a non-intrusive and portable imaging technique which has been used widely in many medical, geological and industrial applications for imaging the interior electrical conductivity distribution within a region from the knowledge of the injected currents through attached electrodes and resulting voltages, or boundary potential and current flux. If the quantities involved are all real then EIT is called electrical resistance tomography (ERT). The work in this thesis focuses on solving inverse geometric problems in ERT where we seek detecting the size, the shape and the location of inner objects within a given bounded domain. These ERT problems are governed by Laplace’s equation subject either to the most practical and general boundary conditions, forming the socalled complete-electrode model (CEM), in two dimensions or to the more idealised boundary conditions in three-dimensions called the continuous model. Firstly, the method of the fundamental solutions (MFS) is applied to solve the forward problem of the two-dimensional complete-electrode model of ERT in simplyconnected and multiple-connected domains (rigid inclusion, cavity and composite bimaterial), as well as providing the corresponding MFS solutions for the three-dimensional continuous model. Secondly, a Bayesian approach and the Markov Chain Monte Carlo (MCMC) estimation technique are employed in combinations with the numerical MFS direct solver in order to obtain the inverse solution. The MCMC algorithm is not only used for reconstruction, but it also deals with uncertainty assessment issues. The reliability and accuracy of a fitted object is investigated through some meaningful statistical aspects such as the object boundary histogram and object boundary credible intervals.

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This thesis firstly considers a modelling framework for multivariate volatility in financial time series. As most financial returns exhibit heavy tails and skewness, we are considering a model for the returns based on the skew-t distribution, while the volatility is assumed to follow a Wishart autoregressive process. We define a new type of Wishart autoregressive process and highlight some of its properties and some of its advantages. Particle filter based inference for this model is discussed and a novel approach of estimating static parameters is provided. Furthermore, an alternative methodology for estimating higher dimension data is developed. Secondly, inspired from the idea of Ulig's Wishart process, a new Wishart-Newton model is developed. The approach combines conjugate Bayesian inference while the hyper parameters are estimated by a Newton-Raphson method and here an online volatility estimate algorithm is proposed. The two proposed models are compared with the benchmarking GO-GARCH model in both function execution time and cumulative returns of different dimensional datasets.

This thesis focuses on theoretical research of optimal and robust control theory for a class of nonlinear stochastic systems. The nonlinearities that appear in the diffusion terms are of a square-root type. Under such systems the following problems are investigated: optimal stochastic control in both finite and infinite horizon; robust stabilization and robust H∞ control; H₂/H∞ control in both finite and infinite horizon; and risk-sensitive control. The importance of this work is that explicit optimal linear controls are obtained, which is a very rare case in the nonlinear system. This is regarded as an advantage because with explicit solutions, our work becomes easier to be applied into the real problems. Apart from the mathematical results obtained, we have also introduced some applications to finance.

This thesis develops computational methods that can provide insights into the behaviour of biomolecular processes. The methods extract a simplified representation/model from samples characterising the profiles of different biomolecular functional units. The simplified representation helps us gain a better understanding of the relations between the functional units or between the samples. The proposed computational methods integrate graph theoretical algorithms and probabilistic models. Firstly, we were interested in finding proteins that have a similar role in the transcription cycle. We performed a clustering analysis on an experimental dataset using a graph partitioning algorithm. We found groups of proteins associated with different stages of the transcription cycle. Furthermore, we estimated a network model describing the relations between the clusters and identified proteins that are representative for a cluster or for the relation between two clusters. Secondly, we proposed a computational framework that unravels the structure of a biological process from high-dimensional samples characterising different stages of the process. The framework integrates a feature selection procedure and a feature extraction algorithm in order to extract a low-dimensional projection of the high-dimensional samples. We analysed two microarray datasets characterising different cell types part of the blood system and found that the extracted representations capture the structure of the hematopoietic stem cell differentiation process. Furthermore, we showed that the low-dimensional projections can be used as a basis for analysis of gene expression patterns. Finally, we introduced the geometric hidden Markov model (GHMM), a probabilistic model for multivariate time series data. The GHMM assumes that the time series lie on a noisy low-dimensional manifold and infers a dynamical model that reflects the low-dimensional geometry. We analysed multivariate time series data generated with a stochastic model of a biomolecular circuit and showed that the estimated GHMM captures the oscillatory behaviour of the circuit.