In the first part parametric models for which the likelihood is intractable are discussed. A method for fitting such models when simulation from the model is possible is presented, which gives estimates that are linear functions of a possibly large set of candidate features. A combination of simulations based on a fractional design and sets of discriminant analyses is used to find an optimal estimate of the parameter vector and its covariance matrix. The procedure is an alternative to Approximate Bayesian Computation and Indirect Inference methods. A way of assessing goodness of fit is briefly described. In the second part the aim is to give a relationship between the effect of one or more explanatory variables on the response when adjusting for an intermediate variable and when not. This relationship is examined mainly for the cases in which the response depends on the two variables via a logistic regression or a proportional hazards model. Some of the theoretical results are illustrated using a set of data on prostate cancer. Then matched pairs with binary outcomes are discussed, for which two methods of analysis are described and compared.

This thesis is broadly concerned with the dynamics, statistics and phenomenology of the perturbed Universe. By studying the perturbations to cosmological spacetimes, and the subsequent growth of large scale structure, we find that we can link both fundamentally and astrophysically interesting physics to cosmological observables. We use a healthy mix of statistical, analytical and numerical techniques throughout this thesis. In Chapter 2 we introduce and summarise the statistics of random fields, as these are fundamental objects used to model cosmological observables. We introduce the spherical Fourier-Bessel expansion as a tool to perform genuine 3-dimensional studies of cosmological random fields. In Chapter 3 we introduce the theory of inflation and discuss the basic machinery that allows us to calculate the statistical properties of the quantum mechanical flucatuations that seed large scale structure. What we see is that different fundamental physics in the early Universe leads to different statistical properties that we may test. The second half of Chapter 3 introduces the large scale structure of the Universe that describes the clustering of galaxies on cosmological scales. We discuss the growth and evolution of structure under gravitational collapse and the core observables that are predicted, such as the power spectrum, variance and skewness. Chapter 4 introduces the Minkowski functionals. These are a set of topological statistics that probe the morphological properties of random fields. In particular they may be used to quantify deviations from Gaussianity in the large scale structure of galaxies. The deviations from Gaussianity can be generated by two primary mechanisms: 1) The gravitational collapse of perturbations is a non-linear process. Even if we have Gaussian initial conditions, gravitational collapse will induce non-Gaussianity. 2) Different theories for the early Universe will imprint different non- Gaussian features in the primordial perturbations that seed large scale structure, i.e. we have non-Gaussian initial conditions. We can connect the amplitude and momentum dependence of the non-Gaussianity to different fundamental interactions. We introduce a topological statistic based on the Minkowski functionals that retains the momentum dependence giving us greater distinguishing power between different contributions to non-Gaussianity. In Chapter 5 we introduce the Baryon Acoustic Oscillations (BAOs) as described in the spherical Fourier-Bessel formalism. The BAOs are a solid prediction in cosmology and should help us to constrain cosmological parameters. We implement a full 3-dimensional study and study how redshift space distortions, induced by the motion of galaxies, and non-linearities, induced by gravitational collapse, impact the characteristics of these BAOs. Chapter 6 extends the spherical Fourier-Bessel theme by introducing the thermal Sunyaev- Zel’dovich (tSZ) effect and cosmological weak lensing (WL). It is thought that weak lensing will provide an unbiased probe of the dark Universe and that the tSZ effect will probe the thermal history of the Universe. Unfortunately, the tSZ effect loses redshift information as it is a line of sight projection. We study the cross-correlation of the tSZ effect with WL in order to reconstruct the tSZ effect in a full 3-dimensional study in an attmept to recover the lost distance information. We use the halo model, spectroscopic redshift surveys and suvery effects to understand how detailed modelling effects the tSZ-WL cross correlation. Chapter 7 marks a real change in theme and introduces the subject of relativistic cosmology. Inparticular we introduce the 1+3, 1+1+2 and 2+2 formalisms as tools to study cosmological perturbations. We provide rather self-contained introductions and provide some minor corrections to the literature in the 1+1+2 formalism as well as introducing new results. In Chapter 8 we apply the 1+1+2 and 2+2 approaches to the Schwarzschild spacetime. Here we outline the full system of equations in both approaches and how they are related, setting up a correspondence between the two. Our aim is to construct closed, covariant, gauge-invariant and frame-invariant wave equations that govern the gravitational perturbations of the Schwarzschild spacetime. We correct a result in the literature and derive two new equations. The first governs axial gravitational perturbations and is related to the magnetic Weyl scalar. The second is valid for both polar and axial perturbations and is given by a combination of the magnetic and electric Weyl 2-tensors. We discuss their relation to the literature at large. Finally, in Chapter 9 we apply the 1+1+2 and 2+2 approaches the LTB spacetime. This inhomogeneous but spherically symmetric spacetime is the first stepping stone into genuinely inhomogeneous cosmological spacetimes. We seek a closed, covariant master equation for the gravitational perturbations of the LTB spacetime. We present an equation governing axial gravitational perturbations and a preliminary equation, valid for both the polar and axial sectors, that is constructed from the electric and magneticWeyl 2-tensors but is coupled to the energy-momentum content of the LTB spacetime. We discuss how auxilliary equations may be introduced in order to close the master equation for polar and axial perturbations. This last result leads to the identification of H as a master variable for axial perturbations of all vacuum LRS-II spacetimes and the LTB spacetime. It is thought that these results can be extended to non-vacuum LRS-II spacetimes. Likewise, the master variable constructed from Weyl variables constitutes a master variable for all vacuum LRS-II spacetimes and it is thought that this will extend to the non-vacuum case.

Estimating multiple geometric shapes such as tracks or surfaces creates significant mathematical challenges particularly in the presence of unknown data association. In particular, problems of this type have two major challenges. The first is typically the object of interest is infinite dimensional whilst data is finite dimensional. As a result the inverse problem is ill-posed without regularization. The second is the data association makes the likelihood function highly oscillatory. The focus of this thesis is on techniques to validate approaches to estimating problems in geometric statistical inference. We use convergence of the large data limit as an indicator of robustness of the methodology. One particular advantage of our approach is that we can prove convergence under modest conditions on the data generating process. This allows one to apply the theory where very little is known about the data. This indicates a robustness in applications to real world problems. The results of this thesis therefore concern the asymptotics for a selection of statistical inference problems. We construct our estimates as the minimizer of an appropriate functional and look at what happens in the large data limit. In each case we will show our estimates converge to a minimizer of a limiting functional. In certain cases we also add rates of convergence. The emphasis is on problems which contain a data association or classification component. More precisely we study a generalized version of the k-means method which is suitable for estimating multiple trajectories from unlabeled data which combines data association with spline smoothing. Another problem considered is a graphical approach to estimating the labeling of data points. Our approach uses minimizers of the Ginzburg-Landau functional on a suitably defined graph. In order to study these problems we use variational techniques and in particular I-convergence. This is the natural framework to use for studying sequences of minimization problems. A key advantage of this approach is that it allows us to deal with infinite dimensional and highly oscillatory functionals.

Due to the availability of the vast amount of graph-structured data generated in various experiment settings (e.g., biological processes, social connections), the need to rapidly identify network structural differences is becoming increasingly prevalent. In many fields, such as bioinformatics, social network analysis and neuroscience, graphs estimated from the same experimental settings are always defined on a fixed set of objects. We formalize such a problem as a labelled graph comparison problem. The main issue in this area, i.e. measuring the distance between graphs, has been extensively studied over the past few decades. Although a large distance value constitutes evidence of difference between graphs, we are more interested in the issue of inferentially justifying whether a distance value as large or larger than the observed distance could have been obtained simply by chance. However, little work has been done to provide the procedures of statistical inference necessary to formally answer this question. Permutation-based inference has been proposed as a theoretically sound approach and a natural way of tackling such a problem. However, the common permutation procedure is computationally expensive, especially for large graphs. This thesis contributes to the labelled graph comparison problem by addressing three different topics. Firstly, we analyse two labelled graphs by inferentially justifying their independence. A permutation-based testing procedure based on Generalized Hamming Distance (GHD) is proposed. We show rigorously that the permutation distribution is approximately normal for a large network, under three graph models with two different types of edge weights. The statistical significance can be evaluated without the need to resort to computationally expensive permutation procedures. Numerical results suggest the validity of this approximation. With the Topological Overlap edge weight, we suggest that the GHD test is a more powerful test to identify network differences. Secondly, we tackle the problem of comparing two large complex networks in which only localized topological differences are assumed. By applying the normal approximation for the GHD test, we propose an algorithm that can effectively detect localised changes in the network structure from two large complex networks. This algorithm is quickly and easily implemented. Simulations and applications suggest that it is a useful tool to detect subtle differences in complex network structures. Finally, we address the problem of comparing multiple graphs. For this topic, we analyse two different problems that can be interpreted as corresponding to two distinct null hypotheses: (i) a set of graphs are mutually independent; (ii) graphs in one set are independent of graphs in another set. Applications for the multiple graphs problem are commonly found in social network analysis (i) or neuroscience (ii). However, little work has been done to inferentially address the problem of comparing multiple networks. We propose two different statistical testing procedures for (i) and (ii), by again using a normality approximation for GHD. We extend the normality of GHD for the two graphs case to multiple cases, for hypotheses (i) and (ii), with two different permutation strategies. We further build a link between the test of group independence to an existing method, namely the Multivariate Exponential Random Graph Permutation model (MERGP). We show that by applying asymptotic normality, the maximum likelihood estimate of MERGP can be analytically derived. Therefore, the original, computationally expensive, inferential procedure of MERGP can be abandoned.

The widespread use of radio frequencies of wavelengths small compared with the major terrain irregularities has led to the development of theoretical deterministic models for the prediction of field strengths over paths of given profile. The examination of these models is the main objective of the present thesis. Although present radio links are mainly based on empirical developments, theoretical approaches may offer considerable alternative for the design of future wireless communications systems. It is well known that the methods applied are based on multidimensional integral equations, which only in certain and idealised cases reduce to a practical form suitable for realistic utilisation. The present work attempts to reveal the physical processes that characterise the radio channel and how these are approached by certain models for common engineering applications. Since the major mechanism of propagation in radio environments is diffraction, extensive analysis is performed for this physical process. In particular, a new fast implementation of the Vogler multiple knife-edge diffraction algorithm is described with the additional benefit of improved accuracy at path profile configurations where the original solution fails considerably. An entirely new approach to slope-Uniform Theory of Diffraction is introduced and shown to produce essentially identical results to Vogler within much shorter computation times. This is applied to 3D urban propagation and to terrestrial fixed links and is shown to produce accurate results compared with measurements. Finally, new physical-statistical models are introduced in order to overcome the excessive cost of high resolution building databases. Application to both mobilesatellite and to broadband fixed access systems revealed a high degree of statistical accuracy

This thesis studies several topics in the area of weak gravitational lensing and addresses some key statistical problems within this subject. A large part of the thesis concerns the measurement of galaxy shapes for weak gravitational lensing and the systematics they introduce. I focused on studying two key effects, typical for model-fitting shape measurement methods. First is noise bias, which arises due to pixel noise on astronomical images. I measure noise bias as a function of key galaxy and image parameters and found that the results are in good agreement with theoretical predictions. I found that if the statistical power of a survey is to be fully utilised, noise bias effects have to be calibrated. The second effect is called model bias, which stems from using simple models to fit galaxy images, which can have more complicated morphologies. I also investigate the interaction of these two systematics. I found model bias to be small for ground-based surveys, rarely exceeding 1%. Its interaction with noise bias was found to be negligible. These results suggest that for ongoing weak lensing surveys, noise bias is the dominant effect. Chapter 5 describes my search for a weak lensing signal from dark matter filaments in CFHTLenS fields. It presents a novel, model-fitting approach to modelling the mass dis- tribution and combining measurements from multiple filaments. We find that CFHTLenS data does provide very good evidence for dark matter filaments, with detection significance of 3.9σ for the filament density parameter relative to mean halo density of connected halos at their R200. For 19 pairs of the most massive halos, the integrated density contrast of filaments was found on a level of 1 · 1013M⊙/h. The appendices present my contribution to three other papers. They describe practical applications of the calibration of noise bias in the GREAT08 challenge and the Dark Energy Survey. I also present the results of the validation of reconvolution and image rendering using FFTs in the GalSim toolkit.

This thesis considers statistical modelling of natural image data. Obtaining advances in this field can have significant impact for both engineering applications, and for the understanding of the human visual system. Several recent advances in natural image modelling have been obtained with the use of unsupervised feature learning. We consider a class of such models, restricted Boltzmann machines (RBMs), used in many recent state-of-the-art image models. We develop extensions of these stochastic artificial neural networks, and use them as a basis for building more effective image models, and tools for computational vision. We first develop a novel framework for obtaining Boltzmann machines, in which the hidden unit activations co-transform with transformed input stimuli in a stable and predictable way throughout the network. We define such models to be transformation equivariant. Such properties have been shown useful for computer vision systems, and have been motivational for example in the development of steerable filters, a widely used classical feature extraction technique. Translation equivariant feature sharing has been the standard method for scaling image models beyond patch-sized data to large images. In our framework we extend shallow and deep models to account for other kinds of transformations as well, focusing on in-plane rotations. Motivated by the unsatisfactory results of current generative natural image models, we take a step back, and evaluate whether they are able to model a subclass of the data, natural image textures. This is a necessary subcomponent of any credible model for visual scenes. We assess the performance of a state- of-the-art model of natural images for texture generation, using a dataset and evaluation techniques from in prior work. We also perform a dissection of the model architecture, uncovering the properties important for good performance. Building on this, we develop structured extensions for more complicated data comprised of textures from multiple classes, using the single-texture model architecture as a basis. These models are shown to be able to produce state-of-the-art texture synthesis results quantitatively, and are also effective qualitatively. It is demonstrated empirically that the developed multiple-texture framework provides a means to generate images of differently textured regions, more generic globally varying textures, and can also be used for texture interpolation, where the approach is radically dfferent from the others in the area. Finally we consider visual boundary prediction from natural images. The work aims to improve understanding of Boltzmann machines in the generation of image segment boundaries, and to investigate deep neural network architectures for learning the boundary detection problem. The developed networks (which avoid several hand-crafted model and feature designs commonly used for the problem), produce the fastest reported inference times in the literature, combined with state-of-the-art performance.

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This thesis concerns the statistical analysis of open quantum systems subject to an external and non-stationary perturbation. In the first paper, a generalization of the explicit-duration hidden Markov models (EDHMM) which takes into account the presence of sparse data is presented. Introducing a kernel estimator in the estimation procedure increases the accuracy of the estimates, and thus allows one to obtain a more reliable information about the evolution of the unobservable system. A generalization of the Viterbi algorithm to EDHMM is developed. In the second paper, we develop a Markov Chain Monte Carlo (MCMC) procedure for estimating the EDHMM. We improve the flexibility of our formulation by adopting a Bayesian model selection procedure which allows one to avoid a direct specification of the number of states of the hidden chain. Motivated by the presence of sparsity, we make use of a non-parametric estimator to obtain more accurate estimates of the model parameters. The formulation presented turns out to be straightforward to implement, robust against the underflow problem and provides accurate estimates of the parameters. In the third paper, an extension of the Cramér-Rao inequality for quantum discrete parameter models is derived. The latter are models in which the parameter space is restricted to a finite set of points. In some estimation problems indeed, theory provides us with additional information that allow us to restrict the parameter space to a finite set of points. The extension presented sets the ultimate accuracy of an estimator, and determines a discrete counterpart of the quantum Fisher information. This is particularly useful in many experiments in which the parameters can assume only few different values: for example, the direction which the magnetic field points to. We also provide an illustration related to a quantum optics problem.

Chronic liver disease is a worldwide health problem, and hepatic fibrosis (HF) is one of the hallmarks of the disease. Pathology diagnosis of HF is based on textural change in the liver as a lobular collagen network that develops within portal triads. The scale of collagen lobules is characteristically on order of 1mm, which close to the resolution limit of in vivo Gd-enhanced MRI. In this work the methods to collect training and testing images for a Hotelling observer are covered. An observer based on local texture analysis is trained and tested using wet-tissue phantoms. The technique is used to optimize the MRI sequence based on task performance. The final method developed is a two stage model observer to classify fibrotic and healthy tissue in both phantoms and in vivo MRI images. The first stage observer tests for the presence of local texture. Test statistics from the first observer are used to train the second stage observer to globally sample the local observer results. A decision of the disease class is made for an entire MRI image slice using test statistics collected from the second observer. The techniques are tested on wet-tissue phantoms and in vivo clinical patient data.