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Classical multivariate analysis techniques such as principal components analysis (PCA), canonical correlation analysis (CCA) and discriminant analysis (DA) can be badly affected when extreme outliers are present. The purpose of this thesis is to present new robust versions of these methods. Our approach is based on the following observation: the classical approaches to PCA, CCA and DA can all be interpreted as operations on a Gaussian likelihood function. Consequently, PCA, CCA and DA can be robustified by replacing the Gaussian likelihood with a Cauchy likelihood. The performance of the Cauchy version of each of these procedures is studied in detail both theoretically, through calculation of the relevant influence function, and numerically, through numerous examples involving real and simulated data. Our results demonstrate that the new procedures have good robustness properties which are certainly far superior to these of the classical versions.

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The human placenta is characterised by a unique circulatory arrangement, with numerous villous trees containing fetal vessels immersed in maternal blood. Placental tissue therefore manifests a multiscale structure balancing microscopic delivery of nutrients and macroscopic flow. The aims of this study are to examine the interaction between these scales and to understand the influence of placental organisation on the effectiveness of nutrient uptake, which can be compromised in pathologies like pre-eclampsia and diabetes. We first systematically analyse solute transport by a unidirectional flow past an array of microscopic sinks, taking up a dissolved nutrient or gas, for both regular and random sink distributions. We classify distinct asymptotic transport regimes, each characterised by the dominance of advective, diffusive or uptake effects at the macroscale, and analyse a set of simplified model problems to assess the accuracy of homogenization approximations as a function of governing parameters (Peclet and Damkohler numbers) and the statistical properties of the sink distribution. The difference between the leading-order homogenization approximation and the exact solute distribution is determined by large spatial gradients at the scale of individual villi (depending on transport parameter values) and substantial fluctuations that can be correlated over lcngthscales comparable to the whole domain. In addition, we consider the nonlinear advective effects of solute-carriers, such as red blood cells carrying oxygen. Homogenization of the solute-carrier-facilitated transport introduces an effective Peclet number that depends on the slowly varying leading-order concentration, so that an asymptotic transport regime can be changed within the domain. At large Peclet and Damkohler numbers (typical for oxygen transport in the human placenta), nonlinear advection due to solute-carriers leads to a more uniform solute distribution than for a linear carrier-free transport, suggesting a "homogenizing" effect of red blood cells on placental oxygen transport. We then use image analysis and homogenization concepts to extract the effective transport properties (diffusivity and hydraulic resistance) from the microscopic images of histological sections of the normal human placenta. The resulting two-dimensional tensor quantities allow us to assess the anisotropy of placental tissue for solute transport. We also show how the pattern of villous centres of mass can be characterised using an integral correlation measure, and identify the minimum spatial scale over which the distribution of villous branches appears statistically homogeneous. Finally, we propose a mathematical model for maternal blood flow in a placental functional unit (a placentone), describing flow of maternal blood via Darcy's law and steady advective transport of a dissolved nutrient. An analytical method of images and computational integration along streamlines are employed to find flow and solute concentration distributions, which are illustrated for a range of governing system parameters. Predictions of the model agree with experimental radioangiographic studies of tracer dynamics in the intervillous space. The model supports the hypothesis that basal veins are located on the periphery of the placentone in order to optimise delivery of nutrients. We also explain the importance of dilatation of maternal spiral arteries and suggest the existence of an optimal volume fraction of villous tissue, which can both be involved in the placental dysfunction. Theoretical studies of this thesis thus constitute a step towards modelling-based diagnostics and treatment of placental disorders.

This thesis is devoted to Ruin Theory which sometimes referred to the collective ruin theory. In Actuarial Science, one of the most important problems is to determine the finite time or infinite time ruin probability of the risk process in an insurance company. To treat a realistic economic situation, the random interest factor should be taken into account. We first define the model with the interest rate and approximate the ruin probability for the model by the Brownian motion and develop several numerical methods to evaluate the ruin probability. Then we construct several models which incorporate possible investment strategies. We estimate the parameters from the simulated data. Then we find the optimal investment strategy with a given upper bound on the ruin probability. Finally we study the ruin probability for our class of models with the Heavy- Tailed claim size distribution.

Healthcare-associated infections (HCAIs) remain a problem worldwide, and can cause severe illness and death. The increasing level of antibiotic resistance among bacteria that cause HCAIs limits infection treatment options, and is a major concern. Statistical modelling is a vital tool in developing an understanding of HCAI transmission dynamics. In this thesis, stochastic epidemic models are developed and used with the aim of investigating methicillin-resistant Staphylococcus aureus (MRSA) transmission and intervention measures in hospital wards. A detailed analysis of MRSA transmission and the effectiveness of patient isolation was performed, using data collected from several general medical wards in London. A Markov chain Monte Carlo (MCMC) algorithm was used to derive parameter estimates, accounting for unobserved transmission dynamics. A clear reduction in transmission associated with the use of patient isolation was estimated. A Bayesian framework offers considerable benefits and flexibility when dealing with missing data; however, model comparison is difficult, and existing methods are far from universally accepted. Two commonly used Bayesian model selection tools, reversible jump MCMC and the deviance information criterion (DIC), were thoroughly investigated in a transmission model setting, using both simulated and real data. The collection of whole genome sequence (WGS) data is becoming easier, faster and cheaper than ever before. With WGS data likely to become abundant in the near future, the development of sophisticated analytical tools and models to exploit such genetic information is of great importance. New methods were developed to model MRSA transmission, using both genetic and epidemiological data, allowing for the reconstruction of transmission networks and simultaneous estimation of key transmission parameters. This approach was tested with simulated data and employed on WGS data collected from two Thai intensive care units. This work offers much scope for future investigations into genetic diversity and more complex transmission models, once suitable data become available.

This thesis considers the interplay between the continuous and discrete properties of random stochastic processes. It is shown that the special cases of the one-sided Lévy-stable distributions can be connected to the class of discrete-stable distributions through a doubly-stochastic Poisson transform. This facilitates the creation of a one-sided stable process for which the N-fold statistics can be factorised explicitly. The evolution of the probability density functions is found through a Fokker-Planck style equation which is of the integro-differential type and contains non-local effects which are different for those postulated for a symmetric-stable process, or indeed the Gaussian process. Using the same Poisson transform interrelationship, an exact method for generating discrete-stable variates is found. It has already been shown that discrete-stable distributions occur in the crossing statistics of continuous processes whose autocorrelation exhibits fractal properties. The statistical properties of a nonlinear filter analogue of a phase-screen model are calculated, and the level crossings of the intensity analysed. It is found that rather than being Poisson, the distribution of the number of crossings over a long integration time is either binomial or negative binomial, depending solely on the Fano factor. The asymptotic properties of the inter-event density of the process are found to be accurately approximated by a function of the Fano factor and the mean of the crossings alone.

Over the last ten years questions related to the safety of nanoparticles and their possible toxic effects have become well-established. The government's Health and Safety Laboratories (HSL) at Buxton are currently attempting to determine their possible toxicity in the workplace. It is their responsibility to establish what levels are exposure can be considered safe in the workplace. This project is a CASE studentship with HSL and aims to start developing mathematical models relating to nanotoxicology. After reviewing the available literature, three key mechanisms which are involved in the possible toxicity of nanoparticles emerge. One mechanism is the oxidative stress they cause once they enter individual cells. The second mechanism is the damage done to the surface of the lung if they are not successfully phagocytosed on inhalation. Finally, the third mechanism is their propensity to aggregate both when dispersed in the air or when they are found inside the body. These three topics are dealt with in Parts I, II and III respectively. There has been much concern over how carbon nanotubes (CNTs) may cause oxidative stress. Oxidative stress occurs when there is an overload negatively charged species in the cell. These are collectively known as Reactive Oxidative Species (ROS). ROS are always present in a cell as they are the natural product of the metabolic pathways. By their reactivity, they readily cause damage to other molecules in the cell, so every cell produces anti-oxidants in order to control the concentration of ROS. However, when the concentration of ROS becomes too high the concentration of anti-oxidants becomes depleted and the cell can become too damaged to function. In this case it dies by necrosis. When a cell dies by necrosis is can cause irritation and further damage to surrounding cells. Oxidative stress can also trigger the immune response so that the cell dies by self-programmed apoptotic cell death which limits this damage to surrounding cells. It is best to avoid unnecessary cell death, however, not undergoing apoptosis risks a more damaging necrotic death. Part I introduces develops models of Tumor Necrosis Factor-alpha (TNF-alpha) activated pathways. This model consists of three signalling cascades. One pathway triggers apoptosis while a second inhibits apoptosis. These two models are based on pre-existing models. This work introduces a third pathway which activates Activator Protein-1 (AP-1). This pathway includes two well-known ROS sensitive elements. These are the ROS-sensitive activation of the Mitogen-Activated Protein Kinase (MAPK) cascade and the ROS-sensitive deactivation of the MAPK phosphatases. These three pathways are regulated by three sets of inhibiting reactions and inhibitors to these inhibitors. The effect of these inhibitors is to introduce a time-lag between the initial TNF-alpha extracellular signal and the death of the cell by apoptosis. This time-lag is regulated by the concentration of intracellular ROS and the concentration of anti-oxidants. Different combinations of inhibitors can be switched on or off before running the model. The effectiveness of the oxidative stress sensitive elements in regulating apoptosis can therefore be optimised while different sets of inhibitors are active. Two qualitatively different types of solutions are found. The cell can be either only transiently active, over a shorter period of time, or persistently active, over a longer period of time. This could provide some guidance to biologists investigation TNF-alpha activation of the immune system. On inhalation, CNTs have been found to reach the alveoli, where air exchange occurs in the lung. The only mechanism available to remove debris in these delicate regions of the lung are lung macrophages. Macrophages work by enclosing unwanted matter in an organelle called a lysosome and then moving this debris away to where it can be cleared by cilia. Non-organic material does not trigger a macrophage response as strongly as organic material, which also triggers the immune system. The shape of fibrous material makes it more difficult for a macrophage to successfully form a lysosome and to move the material away once it has been engulfed. Frustrated phagocytosis releases harmful acids and enzymes which can damage the alveoli causing oxidative stress. If debris cannot be removed, then dead cells may form around the debris to protect the surrounding tissue, forming a granuloma. Both scarring from frustrated phagocytosis and granuloma formation will impair the function of the lung. In Part II, insight is gained on how a cell membrane can engulf an object with a high aspect ratio. The mechanisms of phagocytosis are complex in terms of both cell signalling cascades and the polymerisation and de-polymerisation of the actin network. In order to find a model which takes into account the geometry of a cell as a whole, this picture has been simplified. An energy minimisation approach is used where the surface of a cell is taken to be a surface of rotation around an axis, which is taken to be the axis of a fibre. In Chapter 4, the free energy is taken to be of a liquid drop, resting on a solid surface, in vapour where only the surface and volume energies are considered. The surface tension is taken to account for the tension in the lipid bilayer on the surface of the macrophage. In Chapter 5, the free energy is extended to also include a Helfrich or bending energy which specifically takes into account the energy taken to bend a lipid bilayer. It is assumed that, in order to conserve the limited resources of a macrophage, the shape of a lipid membrane which has successfully engulfed a particle will be energetically stable with regards to these surface, volume and bending energies as a macrophage reaches the final stages of phagocytosis. This does not take into account the energy required to remodel the cytoskeleton for the cell to reach this shape. However, the bending energy associated with cell membranes of increasing length can be used to suggest the amount of energy required in this dynamical process. It is found that in Chapter 4, when no Helfrich energy is included in the energy minimisation, the only limiting shape possible in the limit of increasing length to radius ratio of the fibre is a sphere. When the Helfrich energy is included, three different boundary conditions are imposed. The first boundary condition sets the forces associated with the bending energy to zero at the edge of the membrane. At the point of contact between the membrane and the fibre, the forces reduce to that of a classical solid/liquid/vapour interface. The second boundary condition is imposes the length of the droplet. This length can be incrementally increased to find solutions of increasing length. Finally, a third boundary condition is imposed which sets the contact angle of the membrane at the surface of the fibre to zero. By imposing these three boundary conditions, a variety of membrane shapes were obtained. These results are expected to be a useful guide to experimentalists observing different shapes of macrophages under different conditions. Part III in Section 7.1 pin-points frameworks of models which use concepts from polymer physics to possibly predict the volume of an aggregate of CNTs and also to understand how nanoparticles interact with chain-like protein. However, no new results are presented in Part III.

Nosocomial pathogens are usually organisms such as fungi and bacteria that are associated with infections caused in a hospital environment. Examples include Clostridium difficile, Pseudomonas aeruginosa, Vancomycin-resistant enterococcus and Methicillin-resistant Staphylococcus aureus (MRSA). MRSA, like most of the nosocomial pathogens, is resistant to antibiotics and is one of the most serious causes of infections. In this thesis we assess the effects of antibiotics and antiseptics on carriage and transmission of MRSA. We use highly detailed patient level data taken from two Intensive Care Unit (ICU) wards in St. Guys and Thomas’s hospital in London, where patients were receiving daily antimicrobial treatment and a decolonisation protocol was used. We work in discrete time and employ three different patient-level stochastic models in a Bayesian framework to explore the effectiveness of antimicrobial treatment on MRSA in discrete time. We also develop suitable methods of model assessment. The first two models assume that there is no transmission between patients in the ICU wards. Initially a Markov model is used, assuming perfect swab test specificity and sensitivity, to describe the colonisation status of an individual on a daily basis. Results are obtained using Gaussian random walk Metropolis- Hastings algorithms. We find some evidence that decolonisation treatment and Oxazolidinone have a positive effect in clearing MRSA carriage. The second model is a hidden Markov model and assumes perfect swab test specificity but imperfect sensitivity. We obtain the results using data- augmented Markov Chain Monte Carlo (MCMC) algorithms to make inference for the unobserved patient colonisation states. We find evidence that the Antiseptic treatment used during the decolonisation period is effective in the clearance of MRSA carriage. In the third case we assume that there is MRSA transmission between the patients in the ICUs. We use three different stochastic transmission models which overcome many of the unrealistic assumptions of other models. A data- augmented MCMC algorithm is employed in order to estimate the transmission rates of MRSA between the patients assuming imperfect swab test sensitivity. We found no or limited evidence that antibiotic use affects the transmission process, whereas antiseptic treatment was found to have an effect.

This thesis is concerned with developing statistical methods for evaluating and comparing molecular shapes. Techniques from statistical shape analysis serve as a basis for our methods. However, as molecules are fuzzy objects of electron clouds which constantly undergo vibrational motions and conformational changes, these techniques should be modified to be more suitable for the distinctive features of molecular shape. The first part of this thesis is concerned with the continuous nature of molecules. Based on molecular properties which have been measured at the atom positions, a continuous field--based representation of a molecule is obtained using methods from spatial statistics. Within the framework of reproducing kernel Hilbert spaces, a similarity index for two molecular shapes is proposed which can then be used for the pairwise alignment of molecules. The alignment is carried out using Markov chain Monte Carlo methods and posterior inference. In the Bayesian setting, it is also possible to introduce additional parameters (mask vectors) which allow for the fact that only part of the molecules may be similar. We apply our methods to a dataset of 31 steroid molecules which fall into three activity classes with respect to the binding activity to a common receptor protein. To investigate which molecular features distinguish the activity classes, we also propose a generalisation of the pairwise method to the simultaneous alignment of several molecules. The second part of this thesis is concerned with the dynamic aspect of molecular shapes. Here, we consider a dataset containing time series of DNA configurations which have been obtained using molecular dynamic simulations. For each considered DNA duplex, both a damaged and an undamaged version are available, and the objective is to investigate whether or not the damage induces a significant difference to the the mean shape of the molecule. To do so, we consider bootstrap hypothesis tests for the equality of mean shapes. In particular, we investigate the use of a computationally inexpensive algorithm which is based on the Procrustes tangent space. Two versions of this algorithm are proposed. The first version is designed for independent configuration matrices while the second version is specifically designed to accommodate temporal dependence of the configurations within each group and is hence more suitable for the DNA data.