Since the mid 1970s and increasingly over the last decade, causal inference has generated interest and controversy in statistics. Mathematical frame works have been developed to make causal inference in fields ranging from epidemiology to social science. However, most frameworks rely on the existence of counterfactuals, and the assumptions that underpin them are not always made explicit. This thesis analyses such assumptions and proposes an alternative model. This is then used to tackle problems that have been formulated in counterfactual terms. The proposed framework is based on decision theory. Causes are seen in terms of interventions which in turn are seen as decisions. Decisions are thus explicitly included as intervention variables, in both algebraic expressions for causal effects and the in DAGs which represent the probabilistic structure between the variables. The non-counterfactual framework introduces a novel way of determining whether causal quantities are identifiable. Two such quantities are considered and conditions for their identification are presented. These are the direct effect of treatment on response in the presence of a mediating variable, and the effect of treatment on the treated. To determine whether these are identifiable, intervention nodes are introduced on the variables that are thought to be causal in the problem. By manipulating the conditional independences between the observed variables and the intervention nodes it is possible to determine whether the quantities of interest can be expressed in terms of the a) specific settings and/or b) the idle setting of the intervention nodes, corresponding to experimental regimes and the observational regimes of the causal variables. This method can be easily tailored to any specific context, as it relies only on the understanding of conditional independences.

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Many problems of image understanding can be formulated as semantic segmentation, or the assignment of a 'class' label to every pixel in the image. Until recently, for reasons of efficiency, the problem of generating a good labelling of an image has been formulated as the minimisation of a pairwise Markov random field. However, these pairwise fields are unable to capture the higher-order statistics of natural images which can be used to enforce the coherence of regions in the image or to encourage particular regions to belong to a certain class. Despite these limitations, the use of pairwise Markov models is prevalent in vision. This can largely be attributed to the pragmatism of computer vision researchers; although such models do not fully capture image statistics, they service as an effective discriminative model that prevents individual pixels from being mislabelled. Moreover, unlike many optimisation approaches for higher-order models, approximation algorithms exist for pairwise models, that are guaranteed to find a solution whose cost must lie within a fixed bound of the cost of global optima. In this thesis, we show that the optimisation of many higher-order models can also be performed by approximate algorithms which have the same guarantees and effectiveness as those used for the optimisation of pairwise algorithms. We first consider the optimisation of the higher-order Associative Hierarchical Networks, and by transforming them into pairwise models, propose new approximate algorithms for efficiently minimising them. This work is the first to prove approximation bounds, independent of the size of cliques, for the widespread P" and robust P" models. We consider the problem of optimising the set of labels present in an image and the labelling of the image concurrently, and show how they can be optimised simultaneously using a variety of techniques. In the final chapter,

This PhD is concerned with the causal Bayesian framework account of probabilistic judgement (Krynski & Tenenbaum, 2007). which posits that accurate Bayesian reasoning is contingent upon the reasoner being able to intuitively represent evidence in terms of a sufficient causal model. According to this account, base rate neglect whereby reasoners fail to consider the prior probability of the hypothesis, irrespective of the evidence - can be overcome by explicitly clarifying the causal basis of all of the given evidence. Chapter 1 reviews the literature on base rate neglect; details the main accounts of base rate neglect in Bayesian reasoning; considers some issues with the causal Bayesian framework; and finally outlines the main aims of the PhD. Chapter 2 presents two dual-task experiments which aimed to test the intuitiveness of causal facilitation. Whilst secondary load was not found to have an overall effect on reasoning, subsequent non-load experiments in the chapter found similar levels of causal facilitation to the dual-task experiments. leading to the overall conclusion that facilitation occurs outside of working memory considerations. Chapter 3 indicated that the causal facilitation effect was present only • in the absence of an additional intervention designed to highlight nested set relations in the data, indicating that reasoners may not employ a strictly causal model, but instead represent different causes as interrelated sets of data. Chapter 4 demonstrated that the causal facilitation effect was limited to reasoners of sufficiently high numeracy, which explained the consistently lower levels of Bayesian responding reported throughout the rest of the PhD. • Overall, the thesis furthers our understanding of how mental representations can positively influence judgements over the classical, purely statistical approach. Clarifying the causal basis of the given evidence can help reasoners of good numerical ability to intuitively recognise the set relations between data, leading to Significant improvements in performance.

Multi-parametric programming is a mathematical theory to address optimization problems involving varying parameters. This thesis is concerned with the development of model-based controllers via parametric programming and their application to the design, operation and control of process systems. In part I of this thesis two new algorithms are presented for solving parametric optimization problem of linear state space models via dynamic programming. The first algorithm solves the nominal case, while the second introduce the case of uncertainty in the system matrices. Moreover, an algorithm for robust Explicit model based controller for box-constrained linear systems. These algorithms constitute the basis for the development of model based controllers in the rest of the thesis. In part II of this thesis, dynamic mathematical models for the cases of metal hydride tank reactor, Proton Exchange Membrane (PEM) fuel cell unit and tunnel kiln process are presented. These mathematical models are used to derive reduced order linear models in order to design explicit/multi-parametric Model Predictive Controllers. Moreover, the extensive design of an experimental PEM fuel cell unit which includes the process and instrumentation diagram (PID), the complete list of materials and the three dimension design of the unit, is presented. Based on the experimental results provided by the manufacturer, a validated dynamic mathematical model of the PEM fuel cell unit is presented. Finally, mathematical modelling, dynamic optimization and design of PI controller for the firing process of a tunnel kiln is presented.

Statistical shape analysis is a field for which there is growing demand. One of the major drivers for this growth is the number of practical applications which can use statistical shape analysis to provide useful insight. An example of one of these practical applications is investigating and comparing facial shapes. An ever improving suite of digital imaging technology can capture data on the three-dimensional shape of facial features from standard images. A field for which this offers a large amount of potential analytical benefit is the reconstruction of the facial surface of children born with a cleft lip or a cleft lip and palate. This thesis will present two potential methods for analysing data on the facial shape of children who were born with a cleft lip and/or palate using data from two separate studies. One form of analysis will compare the facial shape of one year old children born with a cleft lip and/or palate with the facial shape of control children. The second form of analysis will look for relationships between facial shape and psychological score for ten year old children born with a cleft lip and/or palate. While many of the techniques in this thesis could be extended to different applications much of the work is carried out with the express intention of producing meaningful analysis of the cleft children studies. Shape data can be defined as the information remaining to describe the shape of an object after removing the effects of location, rotation and scale. There are numerous techniques in the literature to remove the effects of location, rotation and scale and thereby define and compare the shapes of objects. A method which does not require the removal of the effects of location and rotation is to define the shape according to the bending of important shape curves. This method can naturally provide a technique for investigating facial shape. When considering a child's face there are a number of curves which outline the important features of the face. Describing these feature curves gives a large amount of information on the shape of the face. This thesis looks to define the shape of children's faces using functions of bending, called curvature functions, of important feature curves. These curvature functions are not only of use to define an object, they are apt for use in the comparison of two or more objects. Methods to produce curvature functions which provide an accurate description of the bending of face curves will be introduced in this thesis. Furthermore, methods to compare the facial shape of groups of children will be discussed. These methods will be used to compare the facial shape of children with a cleft lip and/or palate with control children. There is much recent literature in the area of functional regression where a scalar response can be related to a functional predictor. A novel approach for relating shape to a scalar response using functional regression, with curvature functions as predictors, is discussed and illustrated by a study into the psychological state of ten year old children who were born with a cleft lip or a cleft lip and palate. The aim of this example is to investigate whether any relationship exists between the bending of facial features and the psychological score of the children, and where relationships exist to describe their nature. The thesis consists of four parts. Chapters 1 and 2 introduce the data and give some background to the statistical techniques. Specifically, Chapter 1 briefly introduces the idea of shape and how the shape of objects can be defined using curvature. Furthermore, the two studies into facial shape are introduced which form the basis of the work in this thesis. Chapter 2 gives a broad overview of some standard shape analysis techniques, including Procrustes methods for alignment of objects, and gives further details of methods based on curvature. Functional data analysis techniques which are of use throughout the thesis are also discussed. Part 2 consists of Chapters 3 to 5 which describe methods to find curvature functions that define the shape of important curves on the face and compare these functions to investigate differences between control children and children born with a cleft lip and/or palate. Chapter 3 considers the issues with finding and further analysing the curvature functions of a plane curve whilst Chapter 4 extends the methods to space curves. A method which projects a space curve onto two perpendicular planes and then uses the techniques of Chapter 3 to calculate curvature is introduced to facilitate anatomical interpretation. Whilst the midline profile of a control child is used to illustrate the methods in Chapters 3 and 4, Chapter 5 uses curvature functions to investigate differences between control children and children born with a cleft lip and/or palate in terms of the bending of their upper lips. Part 3 consists of Chapters 6 and 7 which introduce functional regression techniques and use these to investigate potential relationships between the psychological score and facial shape, defined by curvature functions, of cleft children. Methods to both display graphically and formally analyse the regression procedure are discussed in Chapter 6 whilst Chapter 7 uses these methods to provide a systematic analysis of any relationship between psychological score and facial shape. The final part of the thesis presents conclusions discussing both the effectiveness of the methods and some brief anatomical/psychological findings. There are also suggestions of potential future work in the area.

The aim of this thesis is to propose bootstrap and empirical likelihood confidence regions and hypothesis tests for use in statistical shape analysis. Bootstrap and empirical likelihood methods have some advantages when compared to conventional methods. In particular, they are nonparametric methods and so it is not necessary to choose a family of distribution for building confidence regions or testing hypotheses. There has been very little work on bootstrap and empirical likelihood methods in statistical shape analysis. Only one paper (Bhattacharya and Patrangenaru, 2003) has considered bootstrap methods in statistical shape analysis, but just for constructing confidence regions. There are no published papers on the use of empirical likelihood methods in statistical shape analysis. Existing methods for building confidence regions and testing hypotheses in shape analysis have some limitations. The Hotelling and Goodall confidence regions and hypothesis tests are not appropriate for data sets with low concentration. The main reason is that these methods are designed for data with high concentration, and if this hypothesis is violated, the methods do not perform well. On the other hand, simulation results have showed that bootstrap and empirical likelihood methods developed in this thesis are appropriate to the statistical shape analysis of low concentrated data sets. For highly concentrated data sets all the methods show similar performance. Theoretical aspects of bootstrap and empirical likelihood methods are also considered. Both methods are based on asymptotic results and those results are explained in this thesis. It is proved that the bootstrap methods proposed in this thesis are asymptotically pivotal. Computational aspects are discussed. All the bootstrap algorithms are implemented in “R”. An algorithm for computing empirical likelihood tests for several populations is also implemented in “R”.