The fabrication of Bit Patterned Media has become highly developed, with samples fabricated of over 1.5 Tb/in2. However, writing BPM presents significant challenges and for a system to be developed studies must be made into writing. This work has investigated a number of effects on the writing of Bit Pattterned Media (BPM). Magnetostatic interactions between islands have been used to investigate the effect of patterns of magnetisation on the write-window of a BPM system. A method of acquiring a distribution of patterns was determined and used to vary the probability of a target island switching. This showed that magnetostatic interactions between islands could be modelled as a variation in the anisotropy field. The relationship between island parameter distributions, the write-window and error rates was also explored. The effect of non-Gaussian distributions on the error in a BPM system was studied. It was concluded that tails of island parameter distributions have a significant effect on errors occurring in the write process of a BPM system. Therefore an accurate distribution of island parameters must be known and the necessary accuracy of such a distribution was established. Furthermore a model of BPM writing where the shape of the head field is approximated from the value at the maximum head field gradient will not account for switching in the tail of a real head field. This led onto a study of the ideal write point in BPM. In conventional recording theory the medium is designed to switch when the applied head field is at the position of its maximum gradient, which produces sharp transitions between magnetisation regions. A natural assumption in BPM is that the system could be optimised by setting the island switching field in a similar manner. This strategy of optimisation was investigated to see what gives the minimum error, or maximum write-window. It was concluded that optimisation could not be solely based on the maximum head field gradient, furthermore assuming the shape of the head field from this point will not produce an accurate estimation of the error in a BPM system.

This report studies when and why two Hidden Markov Models (HMMs) may represent the same stochastic process. HMMs are characterized in terms of equivalence classes whose elements represent identical stochastic processes. This characterization yields polynomial time algorithms to detect equivalent HMMs. We also find fast algorithms to reduce HMMs to essentially unique and minimal canonical representations. The reduction to a canonical form leads to the definition of 'Generalized Markov Models' which are essentially HMMs without the positivity constraint on their parameters. We discuss how this generalization can yield more parsimonious representations of stochastic processes at the cost of the probabilistic interpretation of the model parameters.

Statistical modelling of data from the embryo transfer process of In-Vitro Fertilization (IVF) treatments is motivated by the need to perform statistical inference for potential factors and to develop predictive models for these treatments. The biggest issue arising when modelling these treatments is that a number of embryos are transferred but unless all of the embryos get implanted or fail to implant then it is not possible to identify which of the embryos implanted. Little work has been done to address this partial observability of the outcome as it arises in this context. We adopt an Embryo-Uterus (EU) framework where a patient response has distinct uterine and embryo components. This framework is used to construct statistical models, expand them to allow for clustering effects and develop a package that will enable the fitting and prediction of these models in STATA. The capabilities of this package are demonstrated in two real datasets, aimed in investigating the effect of a new embryo prognostic variable and the effect of patient clustering in these treatments. In a simulation study EU models are shown to be capable of identifying a patient covariate either as a predictor of uterine receptivity or embryo viability. However a simulation case study shows that a considerable amount of information about the embryo covariate is lost due to the partial observability of the outcome. Further simulation work evaluating the performance of a number of proposed alternatives to the EU model shows that these alternatives are either biased or conservative. The partially observed cycles are finally considered as a missing data problem and two novel modelling approaches are developed which are able to handle the structure of these treatments. These novel models, based on multiple imputation and probability weighting, are compared to the EU model using simulation in terms of predictive accuracy and are found to have similar predictive accuracy to the EU model.

Deploying virtual network functions (VNFs) such as WAN accelerators, network address translators (NATs) and 5G functions at the network edge (NE) can significantly reduce the experienced latency of delay-ultrasensitive applications (e.g., autonomous vehicles and Internet of things). Nonetheless, a major challenge to their anticipated large-scale deployment is the ability to efficiently allocate and manage the scarce NE resources hosting these functions. In this thesis, we describe a novel containerized infrastructure manager (cIM) that extends current managers, such as Kubernetes, with the necessary building blocks to provide an accurate yet elastic resource allocation service to containerized VNFs at scale. The proposed cIM treats the main modules of the VNFs, i.e., the containerized VNF components (cNFCs), as atomic special-purpose functions that can be rapidly deployed to form complex network services. The main component of the proposed cIM, the resource reservation manager (RRM), employs concepts of risk pooling in the insurance industry to accurately reserve the needed resources for the hosting containers. More precisely, to meet anticipated cNFCs demand fluctuation, the RRM accurately reserves a quota of additional resources that are shared by the containerized functions collected together in clusters. The reserved quota of resources ensures the desired availability level of the cNFCs without over-provisioning the scarce resources of the NE. The RRM considers three different situations namely that of a cNFC instance, a cluster of cNFCs or multiple cNFC clusters sharing the reserved resources. Different allocation approaches are then presented for each of these three situations. Simulation experiments are conducted to evaluate the performance of our reservation schemes from different aspects. The corresponding experimental results demonstrate that our proposed cIM can significantly improve the performance of the cNFCs and guarantee their desired availability with minimal resource reservation. Optimal allocation solutions of the resource pools are further proposed considering the desired availability level and the limit of resource pools. The evaluation results demonstrate that our optimization models and solutions obtain the best performance of relevant testing parameters, e.g., availability.

We develop the necessary theory in algebraic geometry to place Bayesian networks into the realm of algebraic statistics. This allows us to create an algebraic geometry--statistics dictionary. In particular, we study the algebraic varieties defined by the conditional independence statements of Bayesian networks. A complete algebraic classification, in terms of primary decomposition of polynomial ideals, is given for Bayesian networks on at most five random variables. Hidden variables are related to the geometry of higher secant varieties. Moreover, a complete algebraic classification, in terms of generating sets of polynomial ideals, is given for Bayesian networks on at most three random variables and one hidden variable. The relevance of these results for model selection is discussed. / Ph. D.

Prognostic models have clinical appeal to aid therapeutic decision making. In the UK, the Nottingham Prognostic Index (NPI) has been used, for over two decades, to inform patient management. However, it has been commented that NPI is not capable of identifying a subgroup of patients with a prognosis so good that adjuvant therapy with potential harmful side effects can be withheld safely. Tissue Microarray Analysis (TMA) now makes possible measurement of biological tissue microarray features of frozen biopsies from breast cancer tumours. These give an insight to the biology of tumour and hence could have the potential to enhance prognostic modelling. I therefore wished to investigate whether biomarkers can add value to clinical predictors to provide improved prognostic stratification in terms of Recurrence Free Survival (RFS). However, there are very many biomarkers that could be measured, they usually exhibit skewed distribution and missing values are common. The statistical issues raised are thus number of variables being tested, form of the association, imputation of missing data, and assessment of the stability and internal validity of the model. Therefore the specific aim of this study was to develop and to demonstrate performance of statistical modelling techniques that will be useful in circumstances where there is a surfeit of explanatory variables and missing data; in particular to achieve useful and parsimonious models while guarding against instability and overfitting. I also sought to identify a subgroup of patients with a prognosis so good that a decision can be made to avoid adjuvant therapy. I aimed to provide statistically robust answers to a set of clinical question and develop strategies to be used in such data sets that would be useful and acceptable to clinicians. A unique data set of 401 Estrogen Receptor positive (ER+) tamoxifen treated breast cancer patients with measurement for a large panel of biomarkers (72 in total) was available. Taking a statistical approach, I applied a multi-faceted screening process to select a limited set of potentially informative variables and to detect the appropriate form of the association, followed by multiple imputations of missing data and bootstrapping. In comparison with the NPI, the final joint model derived assigned patients into more appropriate risk groups (14% of recurred and 4% of non-recurred cases). The actuarial 7-year RFS rate for patients in the lowest risk quartile was 95% (95% C.I.: 89%, 100%). To evaluate an alternative approach, biological knowledge was incorporated into the process of model development. Model building began with the use of biological expertise to divide the variables into substantive biomarker sets on the basis of presumed role in the pathway to cancer progression. For each biomarker family, an informative and parsimonious index was generated by combining family variables, to be offered to the final model as intermediate predictor. In comparison with NPI, patients into more appropriate risk groups (21% of recurred and 11% of non-recurred patients). This model identified a low-risk group with 7-year RFS rate at 98% (95% C.I.: 96%, 100%).

The representation of physical processes by a climate model depends on its structure, numerical schemes, physical parameterizations and resolution, with initial conditions and future emission scenarios further affecting the output. The extent to which climate models agree is therefore of great interest, often with greater confidence in robust results across models. This has led to climate model output being analysed as ensembles rather than in isolation, and quantifying the sources of variation across these ensembles is the aim of many recent studies. Statistical attempts to do this include the use of variants of the mixed-effects analysis of variance or covariance (mixed-effects ANOVA/ANCOVA). This work usually focuses on identifying variation in a variable of interest that is due to differences in model structure, carbon emissions scenario, etc. Quantifying such variation is important in determining where models agree or disagree, but further statistical approaches can be used to diagnose the reasons behind the agreements and disagreements by representing the physical processes within the climate models. A process-based approach is presented that uses simulation with statistical models to perform a global sensitivity analysis and quantify the sources of variation in multi-model ensembles. This approach is a general framework that can be used with any generalised linear mixed model (GLMM), which makes it applicable to use with statistical models designed to represent (sometimes complex) physical relationships within different climate models. The method decomposes the variation in the response variable into variation due to 1) temporal variation in the driving variables, 2) variation across ensemble members in the distributions of the driving variables, 3) variation across ensemble members in the relationship between the response and the driving variables, and 4) variation unexplained by the driving variables. The method is used to quantify the extent to which, and diagnose why, precipitation varies across and within the members of two different climate model ensembles on various different spatial and temporal scales. Change in temperature in response to increased CO2 is related to change in global-mean annual-mean precipitation in a multi-model ensemble of general circulation models (GCMs). A total of 46% of the variation in the change in precipitation in the ensemble is found to be due to the differences between the GCMs, largely because the distribution of the changes in temperature varies greatly across different GCMs. The total variation in the annual-mean change in precipitation that is due to the differences between the GCMs depends on the area over which the precipitation is averaged, and can be as high as 63%. The second climate model ensemble is a perturbed physics ensemble using a regional climate model (RCM). This ensemble is used for three different applications. Firstly, by using lapse rate, saturation specific humidity and relative humidity as drivers of daily-total summer convective precipitation at the grid-point level over southern Britain, up to 8% of the variation in the convective precipitation is found to be due to the uncertainty in RCM parameters. This is largely because given atmospheric conditions lead to different rates of precipitation in different ensemble members. This could not be detected by analysing only the variation across the ensemble members in mean precipitation rate (precipitation bias). Secondly, summer-total precipitation at the grid-point level over the British Isles is used to show how the values of the RCM parameters can be incorporated into a GLMM to quantify the variation in precipitation due to perturbing each individual RCM parameter. Substantial spatial variation is found in the effect on precipitation of perturbing different RCM parameters. Thirdly, the method is extended to focus on extreme events, and the simulation of extreme winter pentad (five-day mean) precipitation events averaged over the British Isles is found to be robust to the uncertainty in RCM parameters.

Background: Transferring cost-effectiveness information between geographic domains offers the potential for more efficient use of analytical resources. However, it is difficult for decision-makers to know when they can rely on costeffectiveness evidence produced for another context. Objectives: This thesis explores the transferability of economic evaluation results produced for one geographic area to another location of interest, and develops an approach to identify factors to predict when this is appropriate. Methods: Multilevel statistical models were developed for the integration of published international costeffectiveness data to assess the impact of contextual effects on country-level; whilst controlling for baseline characteristics within, and across, a set of economic evaluation studies. Explanatory variables were derived from a list of factors suggested in the literature as possible constraints on the transferability of costeffectiveness evidence. The approach was illustrated using published estimates of the cost-effectiveness of statins for the primary and secondary prevention of cardiovascular disease from 67 studies and related to 23 geographic domains, together with covariates on data, study and country-level. Results: The proportion of variation at the country-level observed depends on the appropriate multilevel model structure and never exceeds 15% for incremental effects and 21% for incremental cost. Key sources of variability are patient and disease characteristics, intervention cost and a number of methodological characteristics defined on the data-level. There were fewer significant covariates on the study and country-levels. Conclusions: Analysis suggests that variability in cost-effectiveness data is primarily due to differences between studies, not countries. Further, comparing different models suggests that data from multinational studies severely underestimates country-level variability. Additional research is needed to test the robustness of these conclusions on other sets of cost-effectiveness data, to further explore the appropriate set of covariates, and to foster the development of multilevel statistical modelling for economic evaluation data in health.

Engine manufacturers are constantly seeking avenues to build cleaner and more ef cient engines to meet ever increasing stringent emission legislations. This requires a closer under- standing of the in-cylinder physical and chemical processes, which can be obtained either through experiments or simulations. The advent of computational hardware, methodologies and modelling approaches in recent times make computational uid dynamics (CFD) an important and cost-effective tool for gathering required insights on the in-cylinder ow, combustion and their interactions. Traditional Reynolds-Averaged Navier-Stokes (RANS) methods and emerging Large Eddy Simulation (LES) techniques are being used as a reli- able mathematical framework tools for the prediction of turbulent ow in such conditions. Nonetheless, the combustion submodels commonly used in combustion calculations are developed using insights and results obtained for atmospheric conditions. However, The combustion characteristics and its interaction with turbulence at Internal combustion (IC) engine conditions with, high pressure and temperatures can be quite different from those in conventional conditions and are yet to be investigated in detail. The objective here is to apply FlaRe (Flamelets revised for physical consistencies) model for IC engines conditions and assess its performance. This model was developed in earlier studies for continuous combustion systems. It is well accepted that the laminar burning velocity, SL, is an essential parameter to determine the fuel burn rate and consequently the power output and ef ciency of IC engines. Also, it is involved in almost all of the sophisticated turbulent combustion models for premixed and partially premixed charges. The burning velocities of these mixtures at temperatures of 850 ≤ T ≤ 950 decrease with pressure up to about 3 MPa as it is well known, but it starts to increase beyond this pressure. This contrasting behaviour observed for the rst time is explained and it is related to the role of pressure dependent reaction for iso-octane and involving OH and the in uence of this radical on the fuel consumption rate. The results iv seem to suggest that the overall order of the combustion reaction for iso-octane and gasoline mixture with air is larger than 2 at pressures higher than 3 MPa. The FlaRe combustion is used to simulate premixed combustion inside a spark-ignition engine. The predictive capabilities of the proposed approach and sensitivity of the model to various parameters have been studied. FlaRe approach includes a parameter βc representing the effects of ame curvature on the burning rate. Since the reactant temperature and pressure inside the cylinder are continually varying with time, the mutual in uence of ame curvature and thermo-chemical activities may be stronger in IC engines and thus this parameter is less likely to be constant. The sensitivity of engine simulation results to this parameter is investigated for a range of engine speed and load conditions. The results indicate some sensitivity and so a careful calibration of this parameter is required for URANS calculation which can be avoided using dynamic evaluations for LES. The predicted pressure variations show fair agreement with those obtained using the level-set approach. DNS data of a hydrogen air turbulent premixed ame in a rectangular constant volume vessel has been analysed to see the effect of higher pressure and temperature on the curvature parameter βc. Since the reactant temperature and pressure inside the cylinder are continually varying with time, the mutual in uence of ame curvature and thermo-chemical activities are expected to be stronger in IC engines and thus the parameter βc may not be constant. To shed more light on this, two time steps from the DNS data has been analysed using dynamic βc procedure. The results show that the effect of higher pressure and temperature need to be considered and taken into account while evaluating βc. When combustion takes place inside a closed vessel as in an IC engine the compression of the un-burnt gases by the propagating ame causes the pressure to rise. In the nal part of this thesis, the FlaRe combustion model is implemented in a commercial computational uid dynamics (CFD) code, STAR-CD, in the LES framework to study swirling combustion inside a closed vessel. Different values of βc has been tested and the need for dynamic evaluation is observed.

Intratumoural heterogeneity complicates the molecular interpretation of biopsies, as multiple distinct tumour genomes are sampled and analysed at once. Ignoring the presence of these populations can lead to erroneous conclusions, and so a correct analysis must account for the clonal structure of the sample. Several methods to reconstruct tumour clonality from sequencing data have been proposed, spanning methods that either do not consider phylogenetic constraints or posit a perfect phylogeny. Models of the first type are typically latent feature models that can describe the observed data flexibly, but whose results may not be reconcilable with a phylogeny. The second type, instead, generally comprises non-parametric mixture models, with strict assumptions on the tumour’s evolutionary process. The focus of this dissertation is on the development of a phylogenetic latent feature model that can bridge the advantages of these two approaches, allowing deviations from a perfect phylogeny. The work is recounted by three statistical models of increasing complexity. First, I present a non-parametric model based on the Indian Buffet Process prior, and highlight the need for phylogenetic constraints. Second, I develop a finite, phylogenetic extension of the previous model, and show that it can outperform competing methods. Third, I generalise the phylogenetic model to arbitrary copy-number states. Markov chain Monte Carlo algorithms are presented to perform inference. The models are tested on datasets that include synthetic data, controlled biological data, and clinical data. In particular, the copy-number generalisation is applied to longitudinal circulating tumour DNA samples. Liquid biopsies that leverage circulating tumour DNA require sensitive techniques in order to detect mutations at low allele fractions. One method that allows sensitive mutation calling is the amplicon sequencing strategy TAm-Seq. I present bioinformatic tools to improve both the development of TAm-Seq amplicon panels and the analysis of its sequencing data. Finally, an enhancement of this method is presented and shown to detect mutations de novo and in a multiplexed manner at allele fractions less than 0.1%.