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Elastic scattering spectroscopy (ESS), is a non-invasive and real-time in vivo optical diagnosis technique sensitive to changes in the physical properties of human tissue, and thus able to detect early cancer and precancerous changes. This thesis focuses on the statistical issue on how to eliminate irrelevant variations in the high-dimensional ESS spectra and extract the most useful information to enable the classification of tissue as normal or abnormal. Multivariate statistical methods have been used to tackle the problems, among which principal component discriminant analysis and partial least squares discriminant analysis are the most explored throughout the thesis as general tools for supervised dimension reduction and classification. Customized multivariate methods are proposed in the specific context of ESS. When ESS spectra are measured in vivo by a hand-held optical probe, differences in the angle and pressure of the probe are a major source of variability between the spectra from replicate measurements. A customized spectral pre-treatment called error removal by orthogonal subtraction (EROS) is designed to ameliorate the effect of this variability. This pre-treatment reduces the complexity and increases both the accuracy and interpretability of the subsequent classification models when applied to early detection of cancer risk in Barrett’s oesophagus. For the application of ESS to diagnosis of sentinel lymph node metastases in breast cancer, an automated ESS scanner was developed to take measurements from a larger area of tissue to produce ESS images for cancer diagnosis. Problems arise due to the existence of background area in the image with considerable between-node variation and no training data available. A partially supervised Bayesian multivariate finite mixture classification model with a Markov random field spatial prior in a reduced dimensional space is proposed to recognise the background area automatically at the same time as distinguishing normal from metastatic tissue.

One of the most important goals of unsupervised learning is to discover meaningful clusters in data. Clustering algorithms strive to discover groups, or clusters, of data points which belong together because they are in some way similar. The research presented in this thesis focuses on using Bayesian statistical techniques to cluster data. We take a model-based Bayesian approach to defining a cluster, and evaluate cluster membership in this paradigm. Due to the fact that large data sets are increasingly common in practice, our aim is for the methods in this thesis to be efficient while still retaining the desirable properties which result from a Bayesian paradigm. We develop a Bayesian Hierarchical Clustering (BHC) algorithm which efficiently addresses many of the drawbacks of traditional hierarchical clustering algorithms. The goal of BHC is to construct a hierarchical representation of the data, incorporating both finer to coarser grained clusters, in such a way that we can also make predictions about new data points, compare different hierarchies in a principled manner, and automatically discover interesting levels of the hierarchy to examine. BHC can also be viewed as a fast way of performing approximate inference in a Dirichlet Process Mixture model (DPM), one of the cornerstones of nonparametric Bayesian Statistics. We create a new framework for retrieving desired information from large data collections, Bayesian Sets, using Bayesian clustering techniques. Unlike current retrieval methods, Bayesian Sets provides a principled framework which leverages the rich and subtle information provided by queries in the form of a set of examples. Whereas most clustering algorithms are completely unsupervised, here the query provides supervised hints or constraints as to the membership of a particular cluster. We call this "clustering on demand", since it involves forming a cluster once some elements of that cluster have been revealed. We use Bayesian Sets to develop a content-based image retrieval system. We also extend Bayesian Sets to a discriminative setting and use this to perform automated analogical reasoning. Lastly, we develop extensions of clustering in order to model data with more complex structure than that for which traditional clustering is intended. Clustering models traditionally assume that each data point belongs to one and only one cluster, and although they have proven to be a very powerful class of models, this basic assumption is somewhat limiting. For example, there may be overlapping regions where data points actually belong to multiple clusters, like movies which can each belong to multiple genres. We extend traditional mixture models to create a statistical model for overlapping clustering, the Infinite Overlapping Mixture Model (IOMM), in a non-parametric Bayesian setting, using the Indian Buffet Process (IBP). We also develop a Bayesian Partial Membership model (BPM), which allows data points to have partial membership in multiple clusters via a continuous relaxation of a finite mixture model.

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I present two new methods for the calculation of surface properties. Firstly, a method of thermodynamic integration to calculate surface free energies. A strain is applied to a unit cell of the bulk material, that opens up a vacuum gap and creates two surfaces. A parameter s describes this process, from s = 0 (the bulk material) to s = si (large vacuum gap). The difference in free energy between these two systems is then calculated by the integration of the stress on the unit cell over s. I use this general theory to find the surface free energy of the titanium dioxide (110) surface using density functional theory. The second part of the thesis gives a general transition state theory method for the calculation of the desorption rate of a molecule from a surface, at any coverage and temperature. This approach depends on the density of molecules as a function of the distance from the surface, and I show that this can be found from the potential of mean force. This is especially useful at low temperatures, where experiments are conducted but brute force simulation is computationally unfeasible. I use this theory to calculate the desorption rate of water from the (001) surface of magnesium oxide at 100 1200K and 0 2/3 coverage, with classical potentials. An important outcome of these calculations is that the frequency prefactor (from the Polanyi-Wigner equation) is dependent on temperature.

The focus of this thesis is biological systems whose dynamics present an interesting feature: only some dimensions drive the whole system. In our examples, the dynamics is expressed as ODEs, such that the ith equation depends on all the variables xi = f (x1,...,xi,xi+1,...), so that they cannot be solved by classical methods. The authors in the literature found that one could express the variable of order bigger than N as a function of the first N variables, thus closing the differential equations; the approximations obtained were exponentially close to the nonapproximated result. In Nonlinear Dynamics, such functions are called Inertial Manifolds. They are defined as manifolds that are invariant under the flow of the dynamical system, and attract all trajectories exponentially. The first example gives rise to a generalisation of a theorem which, in the literature, is proved for the PDE u= -Au + V(u). We prove existence for the most general case u= -A(u)u + V(u) and consider the validity of the results for the biological parameters. We also present a theoretical discussion, by providing examples. The second example arises from Statistics applied to population biology. The infinite number of differential equations for the moments are approximated using a Moment Closure technique, that is expressing moments of order higher than N as a function of the first moments, generally using the function valid for the normal distribution. The example shows exceptional approximation. Though this technique is often used, there is no complete mathematical justification. We examine the relation between the Moment Closure technique and Inertial Manifolds. We prove that the approximated system can be seen as a perturbation of the original system, that it admits an Inertial Manifold, which is close to the original one for \epsilon \rightarrow 0 and t \rightarrow \infty.

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This thesis presents new solutions for two acceptance decisions problems. First, we present methods for basic acceptance sampling for attributes, based on the nonparametric predictive inferential approach for Bernoulli data, which is extended for this application. We consider acceptance sampling based on destructive tests and on non-destructive tests. Attention is mostly restricted to single stage sampling, but extension to two-stage sampling is also considered and discussed. Secondly, sequential acceptance decision problems are considered with the aim to select one or more candidates from a group, with the candidates observed sequentially, either per individual or in subgroups, and with the ordering of an individual compared to previous candidates and those in the same subgroup available. While, for given total group size, this problem can in principle be solved by dynamic programming, the computational effort required makes this not feasible for problems once the number of candidates to be selected, and the total group size are not small. We present a new heuristic approach to such problems, based on the principles of nonparametric predictive inference, and we study its performance via simulations. The approach is very flexible and computationally straightforward, and has advantages over alternative heuristic rules that have been suggested in the literature.

In this thesis, we study the vibrational modes of some Skyrmion solutions within the rational map approximation. When the Skyrme field is perturbed around a static solution, the radial and angular vibrations of the field are decoupled; this gives us two sets of eigenvalue equations that provide us the vibrational modes of the solutions when solved. Using the symmetry properties of the solutions of the equation of motion of the model, the conjugate relations among these solutions can be identified. With a physical contraint on the dispersion relation between the mass and the vibration frequency, we obtain an upper bound for the critical pion mass. Finally, we compare between our results with some numerical results obtained by Houghton et al.