The main aim of this thesis is to develop new spatial clustering approaches which can simultaneously identify different areal clusters and guarantee their geographical contiguity. The second aim is to adjust the finite mixture model in order to cope with the issues caused by outliers or singletons (clusters with only one object). In addition, the thesis also aims to extend the applications of these newly proposed spatial clustering techniques from univariate to multivariate space. In Chapter 1, I will review some available clustering techniques in grouping spatial data and will also introduce different types of clustering data and the Glasgow housing market data which will be used in the thesis’s application. At the end of this chapter, I will outline the structure of this thesis. In Chapter 2, I will give the general statistical theory and inference methodologies used across this thesis, including frequentist and Bayesian statistical inferences, multidimensional scaling and the Procrustes transformation. In Chapter 3, I will introduce techniques that could be used in transforming between two types of clustering data introduced in Chapter 1. Chapter 4 will define some cluster and graph terminology and will also introduce different clustering techniques, such as hierarchical clustering, Chameleon hierarchical clustering and model-based clustering. In this chapter, I will also cover some techniques used in cluster comparisons, methods for number of clusters decisions and number of dimensions decisions. Chapter 6 will introduce more detail about spatial hierarchical clustering. The simulation results from spatial hierarchical clustering will be used as the reference results for comparison with the results from the proposed novel spatial clustering techniques in later chapters. The newly proposed clustering techniques, Chameleon spatial hierarchical clustering, spatially constrained finite mixture model with noise component or with priors and spatially constrained Bayesian model-based clustering with dissimilarities, in clustering areal data will be introduced in Chapters 7, 8 and 9 respectively. Also, the simulations and the application in Glasgow housing market will be given at the end of each of these three chapters. Chameleon spatial hierarchical clustering combined the spatial contiguity with Chameleon hierarchical clustering, so areas grouped together are spatially contiguous. Spatially constrained finite mixture models incorporate the spatial prior distribution into the classical finite mixture model to deal with the spatial contiguity issue. Also, I will make the spatially constrained finite mixture model more robust by incorporating a uniform distribution to model the noise points or adding prior distributions to the model. In Chapter 9, I will add a spatial prior to the model-based clustering with dissimilarities model and then will use a Bayesian approach to obtain a spatial contiguous clustering. Chapter 10 will be conclusions and discussion about the newly proposed clustering methods.

The thesis deals with 1D and 2D scalar equations governing dynamic behaviour of strongly inhomogeneous layered structures. Harmonic vibrations of a composite rod and antiplane shear motions of a cylindrical body consisting of several components are studied paying particular attention to the lowest frequencies. The main focus is on a strong contrast between the parameters characterising structure components, including their sizes, material stiffness, and densities. We start with a multi-parametric analysis of the near-rigid body motions of rods and cylindrical bodies with piecewise uniform properties. The listed problems allow exact analytical solutions demonstrating that the values of all lowest eigenfrequencies tend to zero at large/small ratios of material and geometric parameters. The low-frequency behaviour is considered for so-called global and local regimes, and simple explicit conditions on the problem parameters, underlying each of the regimes, are derived. Further, we present a perturbation procedure for a more general setup based on the evaluation of the almost rigid body motions of “stronger” components assuming a high contrast of material parameters. The proposed approach is extended to structures of arbitrary shape, with variable material parameters, as well as to multi-component structures. We obtain asymptotic formulae for the lowest natural frequencies and also present illustrative examples for each of the studied problems. Many of asymptotic estimations are compared with exact solutions. The results of the thesis are applicable to a mathematical justification of shear deformation theories for multi-layered plates and shells with a strong transverse inhomogeneity.

The interest in the clinical use of stem cell therapies is increasing rapidly, with a need for more control over cell populations cultured/expanded in vitro. This is particularly relevant for the treatment of neurological disorders such as Parkinson's disease where positive outcome measures of clinical trials will be limited by the number of derived neurons and their specific sub-types. The aim is to generate enhanced neural cell populations from stem cells through the design of the cell-material interface. The niche micro-environment is complex, being responsible for cell attachment, proliferation and differentiation. Material engineering approaches to better control cell responses have looked towards surface chemical, topographical and mechanical cues. The many permutations of these factors pose a major challenge in the optimisation of biomaterial design. Machine learning techniques will be used to assess the impact of surface properties on the biological micro-environment. Cell interaction/response provides computational outputs, with input variables being derived from material properties such as surface chemical characteristics (logP, charge, density, wettability, etc.) and topography (nano- and micro-scale, aspect ratio, etc). The aim is to unravel the relationship between cells and biomaterial surface of in vitro cell culture. In vitro experiments and in silico modelling will continually inform each other towards the optimisation of neural cell characteristic responses.

The work examines response of the upper ocean to time-varying winds. In the Ekman paradigm the effect of wind is considered as time-varying horizontally uniform tangential wind stress applied to the ocean surface and the turbulent diffusion of momentum is described employing the Boussinesq closure hypothesis via a single scalar eddy viscosity. In contrast to all previous studies we take into account both its depth and time dependence and examine effects of density strati�cation. We found exact general solution to the full Navier-Stokes equations which describes dynamics of the Ekman boundary layer in terms of the Green's function. Several cases of varying eddy viscosity have been examined: (a) According to the Zikanov et al. [2003] parameterization (justified by LES) the eddy viscosity in non-strati�ed fluid increases linearly with depth in the upper part of the fluid, reaching the maximum value at some depth specifi�ed by the wind speed, and then decreases linearly with depth in the lower layer. For this model the explicit analytic solution describing Ekman response to arbitrary wind has been obtained and thoroughly compared with the available models employing more simple eddy viscosity pro�les lacking the LES validation. The range of situations where much simpler models can be used with acceptable accuracy has been identi�fied. (b) We considered the simplest model of the upper ocean with mixed layer at the top and strati�fied fluid below, which in terms of the Ekman model reduces to a two-layer model: the top (turbulent) layer is characterized by a high constant value of eddy viscosity, while the bottom layer has a much smaller viscosity also assumed to be constant. Basic scenarios such as sharp increase of wind and switch o� of the wind have been analysed from the viewpoint of fi�nding how and when the vertical profi�le of strati�cation affects the surface current caused by wind varying in time. It has been found under what conditions the surface velocity vector is noticeably affected by the presence of strati�cation. The parameter controlling whether the presence of strati�cation will manifest itself on the surface is shown to be the non-dimensional depth of the pycnocline: the surface velocity �field is quite sensitive to the depth of the mixed layer, but is much less sensitive to the strength of strati�cation. From the perspective of remote sensing of the characteristics of strati�fication the using HF radars, it has been concluded that these fi�ndings open new possibilities. (c) When the eddy viscosity is assumed to be both time and depth dependent, three basic scenarios have been thoroughly examined: (i) An increase of wind ending up with a plateau; (ii) Switch-off of the wind; (iii) Periodic wind. Their analysis shows that accounting for time dependence of eddy viscosity substantially changes the response, compared to the predictions of the models with constant in time viscosity. We also report a severe limitation of the Ekman type models employed in modelling of the oceanic surface boundary layer. The Ekman current caused by a growing wind quickly becomes unstable with respect to inviscid inflectional instability. These instabilities are fast, which suggests spikes of dramatically enhanced mixing in the corresponding parts of the water column. The instabilities also break down a fundamental element of the Ekman-type models the assumed spatial uniformity. The results require a radical revision of the existing paradigm.

It is widely known that positive and conditionally negative definite functions take finite values at the origin. Nevertheless, there exist functions with a singularity at zero, arising naturally e.g.\ in potential theory or the study of (continuous) extremal measures, which still exhibit the general characteristics of positive or conditional negative definiteness. Taking a framework set up by Lionel Cooper as a motivation, we study the general properties of functions which are positive definite in an extended sense. We prove a Bochner-type theorem and, as a consequence, show how unbounded positive definite functions arise as limits of classical positive definite functions, as well as that their space is closed under convolution. Moreover, we provide criteria for a function to be positive definite in the extended sense, showing in particular that complete monotonicity in conjunction with local absolute integrability is sufficient. The celebrated Schoenberg theorem establishes a relation between positive definite and conditionally negative definite functions. By introducing a notion of conditional negative definiteness which accounts for the classical, non-singular conditionally negative definite functions, as well as functions which are unbounded at the origin, we extend this result to real-valued functions with a singularity at zero. Moreover, we demonstrate how singular conditionally negative definite functions arise as limits of classical conditionally negative definite functions and provide several examples of functions which are unbounded at the origin and conditionally negative definite in an extended sense. Finally, we study the convexity and minimisation of the energy associated with various singular, completely monotone functions, which have not previously been considered in the field of potential theory or experimental design and solve the corresponding energy problems by means of numerically computing approximations to the (optimal) minimising measures.

The linear stability properties of the boundary layer generated above a disk of infinite extent which rotates around its azimuth are explored for a novel configuration. The rotation rate is taken to be temporally periodic, motivated by findings from Thomas et. al. (Proc. Royal Soc. A, 2011) that the addition of an oscillatory component to an otherwise steady flow has stabilising effects. The vorticity-based methods that were first adopted by Davies and Carpenter (J. Comput. Phys., 2001) are utilised in a novel way for the solution of steady and temporally periodic eigenvalue dispersion relations. Validation of this method is provided by archetypal flow configurations such as the steady Blasius boundary layer and the temporally periodic Stokes layer, where Floquet theory is incorporated. Floquet stability theory is applied to the periodically modulated rotating disk for fixed wavenumber and fixed frequency disturbances, where it is shown that the addition of a modulated rotation rate has a stabilising effect on the boundary layer across a range of modulation frequencies. Confirmation is provided by frozen profile analyses and direct numerical simulations of the subsequent flow development. An energy analysis of the perturbation quantities is conducted to provide insights into the physical mechanisms for the stabilisation. The flow response to impulsive excitations in the periodically modulated rotating disk boundary layer is explored. Direct numerical simulations of radially homogeneous and inhomogeneous configurations are conducted and global stability behaviour is investigated.

No description available.

Motivated by the needs of Aneurin Bevan University Health Board, this thesis ex- plores three themes: the phenomenon of deadlock in queueing systems, the develop- ment of discrete event simulation software, and applying modelling to the evaluation of the effects of a new healthcare intervention, Stay Well Plans, for older people in Gwent. When customers in a restricted queueing network become mutually blocked, and all possible movement ceases, that system becomes deadlocked. This thesis novelly investigates deadlock. A graph theoretical method of detecting deadlock in discrete event simulations is given, analytical models of deadlocking systems are built, and these are used to investigate the effect of system parameters on the expected time until reaching deadlock. Furthermore a deadlock resolution procedure is proposed. An open source discrete event simulation software, Ciw, is developed. This software is designed and developed using best practice principles. Furthermore it permits the use of best practice, such as reproducibility, in simulation modelling. Ciw is used for the modelling of a healthcare system, in order to evaluate the effect of Stay Well Plans. During the development of these models, a number of techniques are employed to overcome the difficulties of lack of data. Insightful results from these models are obtained, indicating a shift in demand from residential care services to community care services.

No description available.

Lakes are vital components of the global biosphere, supporting complex ecosystems and playing important roles in the global biogeochemical cycle. However, they are vulnerable to the threat from climate change and their responses to climate forcing, eutrophication and other pressures, and their possibly confounding interactions, are not yet well understood. Monitoring lake health is therefore essential, in order to understand the changing patterns over space and time. Traditionally, in-situ data, which are collected directly from within lakes and analysed in laboratories, have been available for analysis. However, although these data are assumed to be accurate within measurement error, they are expensive to collect, so that few, if any, in-situ sampling locations are available for each lake, often with infrequent sampling at each location. On the other hand, remotely-sensed data, which are derived from reflectance measurements of the Earth's surface, obtained from satellites, have recently become widely available. These data have good spatial coverage of up to 300 metre resolution, covering entire lakes, often with a monthly-average time-scale, but they must firstly be calibrated with the in-situ data to ensure accuracy, before inferences are made. The data for this research were provided by the GloboLakes project (www.globolakes.ac.uk), which is a consortium research project that is investigating the state of lakes and their responses to environmental drivers on a global scale. The research primarily focusses on log(chlorophyll-a) data for Lake Balaton, in Hungary, and for the Great Lakes of North America. The key question of interest for this research is: ``How can data fusion be performed for in-situ and remotely-sensed lake water quality data, accounting for the spatiotemporal change of support between the point-location, point-time in-situ data and the grid-cell-scale, monthly-averaged remotely-sensed data, producing a fused dataset that takes accuracy from the in-situ data and spatial and temporal information from the remotely-sensed data?" In order to answer this question, this thesis presents the following work: An initial analysis of the data for Lake Balaton motivates the following work, by demonstrating the spatial and temporal patterns in the data, using mixed-effects models, generalised additive models, kriging and principal components analysis. Following the identification of statistical downscaling as an appropriate method for fusion of the data, statistical downscaling models are developed, specifically in the framework of Bayesian hierarchical models with spatially-varying coefficients, for the novel application to data for log(chlorophyll-a), producing fully calibrated maps of fused data across lake surfaces, with associated comprehensive uncertainty measures. Bivariate and multiple-lakes statistical downscaling models are developed and applied, motivated by the assumption that sharing information between variables and between lakes can improve the accuracy of model predictions. The statistically novel method of nonparametric statistical downscaling is developed, to account for both the spatial and temporal aspects of the change of support between the in-situ and remotely-sensed data. Using methodology from both functional data analysis and statistical downscaling, the model treats in-situ and remotely-sensed data at each location as observations of smooth functions over time, estimated using bases, with the basis coefficients related via a spatially-varying coefficient regression. This is computed within a Bayesian hierarchical model, enabling the calculation of comprehensive uncertainties. This thesis presents the background, motivation, model development and application of the novel method of nonparametric statistical downscaling, filling the gap in the literature of accounting for changing temporal support in statistical downscaling modelling. Results are presented throughout this thesis, to demonstrate the utility of the method for real lake water quality data.