The objective of this thesis is to study the e ect of under-reporting in epidemics. In particular, there are two broad questions we investigate: In the situation of under-reporting in epidemics, what would happen if the data were treated as if no under-reporting were occurring? Such assumption leads to an under-estimation of the contact rate, implying an under-estimation of the reproduction number. By allowing for the fact that under-reporting is occurring, how and how well can we estimate the reporting rate and other parameters of the model? We explore the above questions by considering the stochastic Markovian SIR epidemic in which various reporting processes are incorporated. We consider cases of constant reporting probability and move on to more realistic assumptions such as the reporting probability depending on time, the number of reported cases and the dependence on the source of infection for each infected individual. We develop various methodologies, based on temporal data, to account for underreporting in the Bayesian framework using MCMC to sample from the posterior distributions of the model parameters. An introduction to the spatial aspect is also considered with the SIR model with reporting process on Z.

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This thesis determines some of the implications of non-universal and emergent universal statistics on arithmetic correlations and fluctuations of arithmetic functions, in particular correlations amongst prime numbers and the variance of the expected number of prime numbers over short intervals are generalised by associating these concepts to L-functions arising from number theoretic objects. Inspired by work in quantum chaology, which shares the property of displaying emergent universality, in chapter 2 a heuristic is given to determine the behaviour of a correlation function associated to functions in the Selberg class from the universal form of the 2-point correlation statistic conjectured for this class. Also in this chapter, the Riemann zeta function is taken as an example of an L-function from which the correlations between pairs of prime numbers arise from a non-universal form of the 2-point correlation statistic for its zeros. Chapter 3 explores the implications of the 2-point correlation statistic on an arithmetic variance associated to functions in the Selberg class, generalising the variance of primes in short intervals. Many of the ideas in this thesis are based on the preprint, [BKS15], joint with Hung Bui and Jon Keating.

In this thesis we study some critical problems in the area of Data Envelopment Analysis (DEA) within the unifying framework of polyhedral characteristics of the production possibility sets and efficiency frontiers of important DEA models. Recent developments in DEA have made it possible to identify the efficient frontier explicitly. This thesis builds on these developments to make the following contributions. We establish theoretical results on the efficiency classifications of surfaces of the boundaries of the production possibility sets. These systematise existing research in the field and fill in many gaps. Our main results provide necessary and sufficient conditions for characterising fully-dimensional efficient surfaces. In addition, the new theoretical framework leads us to discover and address inconsistencies in the related literature. Next we study the sensitivity of efficiency classifications of Decision Making Units (DMUs) to data perturbations. In contrast to existing approaches, we study the effects of arbitrary data perturbations on the efficiency classifications of all DMUs. Theoretical constructs based on the polyhedral nature of production possibility sets lead to identifying a Conditional Stability Region for each DMU within which its data can be perturbed without affecting the efficiency classification of any other DMU. Finally, we develop a new methodology for cross-evaluation in DEA which replaces the traditional approach of peer evaluation by evaluating DMUs across all possible weights obtained from our explicit identification of the DEA production possibility set. The new approach eliminates some major flaws and weaknesses of the traditional approach and produces more meaningful results. Moreover, a set of extensions to the new approach lead to tools that allow identification of DMUs with unrealistic efficiency scores as well as the identification of under-achieving DMUs, a concept that is introduced here.

Discovering clusters in data is a common goal of statistical data analysis. Two kinds of clustering, hierarchical clustering and community discovery, are considered here, as well as their composition-discovering hierarchies of communities. Hierarchical clustering discovers hierarchies of clusters of data, represented as a tree whose leaves are the data. Community discovery finds clusters of people, most commonly from the adjacency matrix of a graph of the relationships between the people. We shall leverage Bayesian statistics to construct several models and corresponding efficient learning algorithms for discovering hierarchies, communities and hierarchies of communities. This thesis has three main contributions, each being a model and a learning algorithm for tackling one of these clustering problems. First we develop an efficient model-based hierarchical clustering algorithm using greedy model selection. Unlike many other hierarchical clustering algorithms, our model is not necessarily a binary tree, but can be any tree where each internal node may have any number of children. This can lead to simpler hierarchies that we find are just as predictive of the data, but are more interpretable as the hierarchies are less visually cluttered and the underlying model has fewer parameters than a binary tree-based model. We then adapt this hierarchical clustering model and algorithm to discovering communities in social networks based upon their adjacency matrix, where the leaves of the discovered tree correspond to people. This adaptation is not straightforward as a naive adaptation leads to an inefficient learning algorithm. We develop a dynamic programming scheme and number of approximations that yield several fast algorithms. We then show empirically that these approximations are faster than the Infinite Relational Model, producing similar or better predictions in less time for the task of predicting unobserved edges in a graph. Finally we tackle the problem of discovering communities directly from interactions among individuals, rather than from the adjacency matrix of a graph. We develop a model that uses a statistical notion of reciprocity to discover communities from time-series interaction data. We then develop a Markov Chain Monte Carlo method for inference and show empirically that this model is much better at predicting future interactions among individuals than several alternate models.

Abnormal change detection techniques can be used to solve a range of real world problems but many of the available methods have been developed to address specific application problems, such as change detection of land disturbances, typhoon image analysis and forest fire prediction. The designing of general, scalable and statistically relevant abnormal change detection methods is very impOltant. Computational intelligence and statistical methods provide an effective way of detecting abnormal change in sequential time series data. In this thesis, the aim is to develop methods for detecting and categorizing abnormal changes in sequential data. We propose three new methods to detect changes in data streams: a Geometric Moving Average Martingale (GMAM) method for change detection based on the Martingale theory, two feature extraction methods Piecewise Linear Representation Morphological Feature Points and Piecewise Linear Representation Important Points, and an anomaly detection method in sequential data based on subsequence identification and the weighted local outlier factor method. We also extend the GMAM method and apply it for detecting seismic anomalies in outgoing long-wave radiation data. There are some findings in this thesis. Firstly, there are two components underpinning the GMAM method. One is the exponential weighting of observations which has the capability of reducing false changes. Another is the use of the GMAM value for hypothesis testing. Secondly, the proposed piecewise representation method based on morphological feature points and important points can extract the features of time series data and help the weighted local outlier factor method to find the anomalies of time series. Thirdly, the weighted local outlier factor method can obtain higher accuracy when applied to 17 data sets than the LOF method and Hot SAX methods. Finally, an extension of the GMAM method to the Average GMAM method (AG) has been applied to analyse seismic anomalies within Outgoing Long-wave Radiation (OLR) data observed by Satellites from 2006 to 2015 for the two recent Wenchuan and Lushan earthquakes and four comparative study areas: Wenchuan, Puer, Beijing and Northeastern areas. The Yushu earthquake and Hetian earthquake have also been examined. The experimental results show that the proposed AG method can effectively discover abnormal changes within OLR data and that there are large AG values in the pre- and post-occurrence of the earthquakes in these areas, which could be viewed as seismic anomalies.

In this thesis we explore prototype theory and random set theory as a foundation for antecedent labels in rule-based systems. In particular, we introduce three learning algorithms for regression problems and we also investigate the use of genetic algorithm as a generic parallleter optimization method for each of these approaches. Two of the learning algorithms generate Takagi-Sugeno rules whilst the third corresponds to a form of regression tree. Within the rules fuzzy labels are represented either through a combination of random set and prototype theory or through a novel extension of prototype theory to allow for fuzzy prototypes. Through out the thesis the algorithms are assessed on a number of benchmark data sets and compared with several state-of-the-art regression methods.

A marginal structural model parameterises the distribution of an outcome given a treatment intervention, where such a distribution is the fundamental probabilistic representation of the causal effect of treatment on the outcome. Causal inference methods are designed to consistently estimate aspects of these causal distributions, in the presence of interference from non-causal associations which typically occur in observational data. One such method, which involves the application of inverse probability of treatment weights, directly targets the parameters of marginal structural models. The asymptotic properties and practical applicability of this method are well established, but little attention has been paid to its finite-sample performance. This is because simulating data from known distributions which are entirely suitable for such investigations generally presents a significant challenge, especially in scenarios where the outcome is survival time. We illuminate these issues, and propose and implement certain solutions, considering separately the cases of static (pre-determined) and dynamic (tailored) treatment interventions. In so doing, we explore both theoretical and practical aspects of marginal structural models for survival outcomes, and the associated inference method.

In this thesis, we firstly introduce the application of the Generalized Gaussian Markov Random Field (GGMRF) to the problem of video super-resolution. The GGMRF prior is employed to perform a maximum a posteriori (MAP) estimation of the desired high-resolution image. Compared with traditional prior models, the GGMRF can describe the distribution of the high-resolution image much better and can also preserve better the discontinuities (edges) of the original image. Previous work had used GGMRF for image restoration in which the temporal dependencies among video frames are not considered. Since the corresponding energy function is convex, gradient descent optimisation techniques are used to solve the MAP estimation. Results show the super-resolved images using the GGMRF prior not only offers a good visual quality enhancement, but also contain a significantly smaller amount of noise. We then propose a Bayesian-based super resolution algorithm that uses approximations of symmetric alpha-stable (SaS) Markov Random Fields (MRF) as prior. The approximated SaS prior is employed to perform MAP estimation for the high-resolution (RR) image reconstruction process. Compared with other state-of-the-art prior models, the proposed prior can better capture the heavy tails of the distribution of the HR image. Thus, the edges of the reconstructed HR image are preserved better in our method. Since the corresponding energy function is non-convex, the graduated nonconvexity (GNC) method is used to solve the MAP estimation. Experiments confirm the better fit achieved by the proposed model to the actual data distribution and the consequent improvement in terms of visual quality over previously proposed super resolution algorithms . . A joint video fusion and super-resolution algorithm is also proposed in this thesis. The method addresses the problem of generating a high-resolution HR image from infrared (IR) and visible (VI) low-resolution (LR) images, in a Bayesian framework. In order to preserve better the discontinuities, a Generalized Gaussian Markov Random Field (MRF) is used to formulate the prior. Experimental results demonstrate that information from both visible and infrared bands is recovered from the LR frames in an effective way. Finally, a novel video super-resolution image reconstruction algorithm that based on low rank matrix completion algorithm is presented. The proposed algorithm addresses the problem of generating a HR image from several LR images, based on sparse representation and low-rank matrix completion. The approach represents observed LR frames in the form of sparse matrices and rearranges those frames into low dimensional constructions. Experimental results demonstrate that, high-frequency details in the super resolved images are recovered from the LR frames .

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