The motile behaviour of bacteria underlies many important aspects of their actions, including pathogenicity, foraging efficiency, and ability to form biofilms. In this thesis, we apply mathematical modelling and analysis to various aspects of the planktonic motility of flagellated bacteria, guided by experimental observations. We use data obtained by tracking free-swimming Rhodobacter sphaeroides under a microscope, taking advantage of the availability of a large dataset acquired using a recently developed, high-throughput protocol. A novel analysis method using a hidden Markov model for the identification of reorientation phases in the tracks is described. This is assessed and compared with an established method using a computational simulation study, which shows that the new method has a reduced error rate and less systematic bias. We proceed to apply the novel analysis method to experimental tracks, demonstrating that we are able to successfully identify reorientations and record the angle changes of each reorientation phase. The analysis pipeline developed here is an important proof of concept, demonstrating a rapid and cost-effective protocol for the investigation of myriad aspects of the motility of microorganisms. In addition, we use mathematical modelling and computational simulations to investigate the effect that the microscope sampling rate has on the observed tracking data. This is an important, but often overlooked aspect of experimental design, which affects the observed data in a complex manner. Finally, we examine the role of rotational diffusion in bacterial motility, testing various models against the analysed data. This provides strong evidence that R. sphaeroides undergoes some form of active reorientation, in contrast to the mainstream belief that the process is passive.

In this work, we are interested in Growth of Galton-Watson trees under two different models: (1) Galton-Watson (GW) forests with lifetimes and/or immigrants, and (2) Galton-Watson forests with mutation, which we call Galton-Watson-Clone-Mutant forests, or GWCMforests. Under each model, we study certain consistent families (Fλ)λ≥0 of GW/GWCM forests and associated decompositions that include backbone decomposition as studied by many authors. Specifically, consistency here refers to the property that for each μ ≤ λ, the forest Fμ has the same distribution as the subforest of Fλ spanned by the blue leaves in a Bernoulli leaf colouring, where each leaf of Fλ is coloured in blue independently with probability μ/λ. In the first model, the case of exponentially distributed lifetimes and no immigration was studied by Duquesne and Winkel and related to the genealogy of Markovian continuous-state branching processes (CSBP). We characterise here such families in the framework of arbitrary lifetime distributions and immigration according to a renewal process, and show convergence to Sagitov’s (non-Markovian) generalisation of continuous-state branching renewal processes, and related processes with immigration. In the second model, we characterise such families in terms of certain bivariate CSBP with branching mechanisms studied previously by Watanabe and show associated convergence results. This is related to, but more general than Bertoin’s study of GWCM trees, and also ties in with work by Abraham and Delmas, who study directly some of the limiting processes.

Heart disease is the major cause of death in the developed world. Due to its fast, portable, low-cost and harmless way of imaging the heart, echocardiography has become the most frequent tool for diagnosis of cardiac function in clinical routine. However, visual assessment of heart function from echocardiography is challenging, highly operatordependant and is subject to intra- and inter observer errors. Therefore, development of automated methods for echocardiography analysis is important towards accurate assessment of cardiac function. In this thesis we develop new ways to model echocardiography data using Bayesian machine learning methods and concern three problems: (i) wall motion analysis in 2D stress echocardiography, (ii) segmentation of the myocardium in 3D echocardiography, and (iii) standard views extraction from 3D echocardiography. Firstly, we propose and compare four discriminative methods for feature extraction and wall motion classification of 2D stress echocardiography (images of the heart taken at rest and after exercise or pharmalogical stress). The four methods are based on (i) Support Vector Machines, (ii) Relevance Vector Machines, (iii) Lasso algorithm and Regularised Least Squares, (iv) Elastic Net regularisation and Regularised Least Squares. Although all the methods are shown to have superior performance to the state-of-the-art, one conclusion is that good segmentation of the myocardium in echocardiography is key for accurate assessment of cardiac wall motion. We investigate the application of one of the most promising current machine learning techniques, called Decision Random Forests, to segment the myocardium from 3D echocardiograms. We demonstrate that more reliable and ultrasound specific descriptors are needed in order to achieve the best results. Specifically, we introduce two sets of new features to improve the segmentation results: (i) LoCo and GloCo features with a local and a global shape constraint on coupled endoand epicardial boundaries, and (ii) FA features, which use the Feature Asymmetry measure to highlight step-like edges in echocardiographic images. We also reinforce the traditional features such as Haar and Rectangular features by aligning 3D echocardiograms. For that we develop a new registration technique, which is based on aligning centre lines of the left ventricles. We show that with alignment performance is boosted by approximately 15%. Finally, a novel approach to detect planes in 3D images using regression voting is proposed. To the best of our knowledge we are the first to use a one-step regression approach for the task of plane detection in 3D images. We investigate the application to standard views extraction from 3D echocardiography to facilitate efficient clinical inspection of cardiac abnormalities and diseases. We further develop a new method, called the Class- Specific Regression Forest, where class label information is incorporating into the training phase to reinforce the learning from semantically relevant to the problem classes. During testing the votes from irrelevant classes are excluded from voting to maximise the confidence of output predictors. We demonstrate that the Class-Specific Regression Random Forest outperforms the classic Regression Random Forest and produces results comparable to the manual annotations.

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The past decade has seen great progress in research on large scale mapping and perception in static environments. Real world perception requires handling uncertain situations with multiple possible interpretations: e.g. changing appearances, dynamic objects, and varying motion models. These aspects of perception have been largely avoided through the use of heuristics and preprocessing. This thesis is motivated by the challenge of including discrete reasoning directly into the estimation process. We approach the problem by using Conditional Linear Gaussian Networks (CLGNs) as a generalization of least-squares estimation which allows the inclusion of discrete model selection variables. CLGNs are a powerful framework for modeling sparse multi-modal inference problems, but are difficult to solve efficiently. We propose the Iterative Local Model Selection (ILMS) algorithm as a general approximation strategy specifically geared towards the large scale problems encountered in tracking and mapping. Chapter 4 introduces the ILMS algorithm and compares its performance to traditional approximate inference techniques for Switching Linear Dynamical Systems (SLDSs). These evaluations validate the characteristics of the algorithm which make it particularly attractive for applications in robot perception. Chief among these is reliability of convergence, consistent performance, and a reasonable trade off between accuracy and efficiency. In Chapter 5, we show how the data association problem in multi-target tracking can be formulated as an SLDS and effectively solved using ILMS. The SLDS formulation allows the addition of additional discrete variables which model outliers and clutter in the scene. Evaluations on standard pedestrian tracking sequences demonstrates performance competitive with the state of the art. Chapter 6 applies the ILMS algorithm to robust pose graph estimation. A non-linear CLGN is constructed by introducing outlier indicator variables for all loop closures. The standard Gauss-Newton optimization algorithm is modified to use ILMS as an inference algorithm in between linearizations. Experiments demonstrate a large improvement over state-of-the-art robust techniques. The ILMS strategy presented in this thesis is simple and general, but still works surprisingly well. We argue that these properties are encouraging for wider applicability to problems in robot perception.

In mathematical finance, the sensitivities of option prices to various market parameters, also known as the “Greeks”, reflect the exposure to different sources of risk. Computing these is essential to predict the impact of market moves on portfolios and to hedge them adequately. This is commonly done using Monte Carlo simulations. However, obtaining accurate estimates of the Greeks can be computationally costly. Multilevel Monte Carlo offers complexity improvements over standard Monte Carlo techniques. However the idea has never been used for the computation of Greeks. In this work we answer the following questions: can multilevel Monte Carlo be useful in this setting? If so, how can we construct efficient estimators? Finally, what computational savings can we expect from these new estimators? We develop multilevel Monte Carlo estimators for the Greeks of a range of options: European options with Lipschitz payoffs (e.g. call options), European options with discontinuous payoffs (e.g. digital options), Asian options, barrier options and lookback options. Special care is taken to construct efficient estimators for non-smooth and exotic payoffs. We obtain numerical results that demonstrate the computational benefits of our algorithms. We discuss the issues of convergence of pathwise sensitivities estimators. We show rigorously that the differentiation of common discretisation schemes for Ito processes does result in satisfactory estimators of the the exact solutions’ sensitivities. We also prove that pathwise sensitivities estimators can be used under some regularity conditions to compute the Greeks of options whose underlying asset’s price is modelled as an Ito process. We present several important results on the moments of the solutions of stochastic differential equations and their discretisations as well as the principles of the so-called “extreme path analysis”. We use these to develop a rigorous analysis of the complexity of the multilevel Monte Carlo Greeks estimators constructed earlier. The resulting complexity bounds appear to be sharp and prove that our multilevel algorithms are more efficient than those derived from standard Monte Carlo.

In under 6 decades dengue has emerged from South East Asia to become the most widespread arbovirus affecting human populations. Recent dramatic increases in epidemic dengue fever have mainly been attributed to factors such as vector expansion and ongoing ecological, climate and socio-demographic changes. The failure to control the virus in endemic regions and prevent global spread of its mosquito vectors and genetic variants, underlines the urgency to reassess previous research methods, hypotheses and empirical observations. This thesis comprises a set of studies that integrate currently neglected and emerging epidemiological, ecological and evolutionary factors into unified mathematical frameworks, in order to better understand the contemporary population biology of the dengue virus. The observed epidemiological dynamics of dengue are believed to be driven by selective forces emerging from within-host cross-immune reactions during sequential, heterologous infections. However, this hypothesis is mainly supported by modelling approaches that presume all hosts to contribute equally and significantly to the selective effects of cross-immunity both in time and space. In the research presented in this thesis it is shown that the previously proposed effects of cross-immunological reactions are weakened in agent-based modelling approaches, which relax the common deterministic and homogeneous mixing assumptions in host-host and host-pathogen interactions. Crucially, it is shown that within these more detailed models, previously reported universal signatures of dengue's epidemiology and population genetics can be reproduced by demographic and natural stochastic processes alone. While this contrasts with the proposed role of cross-immunity, it presents demographic stochasticity as a parsimonious mechanism that integrates, for the first time, multi-scale features of dengue's population biology. The implications of this research are applicable to many other pathogens, involving challenging new ways of determining the underlying causes of the complex phylodynamics of antigenically diverse pathogens.

We study non-equilibrium and strongly-correlated dynamics in two contexts. We begin by analysing quantum many-body systems out of equilibrium through the lens of cold atomic impurities in Bose gases. Such highly-imbalanced mixtures provide a controlled arena for the study of interactions, dissipation, decoherence and transport in a many-body quantum environment. Specifically we investigate the oscillatory dynamics of a trapped and initially highly-localised impurity interacting with a weakly-interacting trapped quasi low-dimensional Bose gas. This relates to and goes beyond a recent experiment by the Inguscio group in Florence. We witness a delicate interplay between the self-trapping of the impurity and the inhomogeneity of the Bose gas, and describe the dissipation of the energy of the impurity through phononic excitations of the Bose gas. We then study the transport of a driven, periodically-trapped impurity through a quasi one-dimensional Bose gas. We show that placing the weakly-interacting Bose gas in a separate periodic potential leads to a phononic excitation spectrum that closely mimics those in solid state systems. As a result we show that the impurity-Bose gas system exhibits phonon-induced resonances in the impurity current that were predicted to occur in solids decades ago but never clearly observed. Following this, allowing the bosons to interact strongly, we predict the effect of different strongly-correlated phases of the Bose gas on the motion of the impurity. We show that, by observing the impurity, properties of the excitation spectrum of the Bose gas, e.g., gap and bandwidth, may be inferred along with the filling of the bosonic lattice. In other words the impurity acts as a probe of its environment. To describe the dynamics of such a strongly-correlated system we use the powerful and near-exact time-evolving block decimation (TEBD) method, which we describe in detail. The second part of this thesis then analyses, for the first time, the performance of this method when applied to simulate non-equilibrium classical stochastic processes. We study its efficacy for a well-understood model of transport, the totally-asymmetric exclusion process, and find it to be accurate. Next, motivated by the inefficiency of sampling-based numerical methods for high variance observables we adapt and apply TEBD to simulate a path-dependent observable whose variance increases exponentially with system size. Specifically we calculate the expected value of the exponential of the work done by a varying magnetic field on a one-dimensional Ising model undergoing Glauber dynamics. We confirm using Jarzynski's equality that the TEBD method remains accurate and efficient. Therefore TEBD and related methods complement and challenge the usual Monte Carlo-based simulators of non-equilibrium stochastic processes.

This thesis presents a new integrated tool and decision support framework to approach complex problems resulting from the interaction of many multi-criteria issues. The framework is embedded in an integrated tool called OUTDO (Oxford University Tool for Decision Organisation). OUTDO integrates Multi-Criteria Decision Analysis (MCDA), decision rationale management with a modified Issue-Based Information Systems (IBIS) representation, and probabilistic forecasting to effectively capture the essential reasons why decisions are made and to dynamically re-use the rationale. In doing so, it allows exploration of how changes in external parameters affect complicated and uncertain decision making processes in the present and in the future. Once the decision maker constructs his or her own decision process, OUTDO checks if the decision process is consistent and coherent and looks for possible ways to improve it using three new semantic-based decision support approaches. For this reason, two ontologies (the Decision Ontology and the Energy Ontology) were integrated into OUTDO to provide it with these semantic capabilities. The Decision Ontology keeps a record of the decision rationale extracted from OUTDO and the Energy Ontology describes the energy generation domain, focusing on the water requirement in thermoelectric power plants. A case study, with the objective of recommending electricity generation and steam condensation technologies for ten different regions in the UK, is used to verify OUTDO’s features and reach conclusions about the overall work.

Confocal single-molecule fluorescence is a powerful tool for monitoring conformational dynamics, and has provided new insight into the enzymatic activities of complex biological molecules such as DNA and RNA polymerases. Though useful, such studies are typically qualitative in nature, and performed almost exclusively in highly purified, in vitro settings. In this work, I focus on improving the methodology of confocal single-molecule fluorescence in two broad ways: (i) by enabling the quantitative identification of molecular dynamics in proteins and nucleic acids in vitro, and (ii) developing the tools needed to perform these analyses in vivo. Toward the first goal, and together with several colleagues, I have developed three novel methods for the quantitative identification of dynamics in biomolecules: (i) Burst Variance Analysis (BVA), which unambiguously identifies dynamics in single-molecule FRET experiments; (ii) Dynamic Probability Density Analysis (PDA), which hypothesis-tests specific kinetic models against smFRET data and extracts rate information; and (iii) a novel molecular counting method useful for studying single-molecule thermodynamics. We validated these methods against Monte Carlo simulations and experimental DNA controls, and demonstrated their practical application in vitro by analyzing the “fingers-closing” conformational change in E.coli DNA Polymerase I; these studies identified unexpected conformational flexibility which may be important to the fidelity of DNA synthesis. To enable similar studies in the context of a living cell, we generated a nuclease-resistant DNA analogue of the Green Fluorescent Protein, or “Green Fluorescent DNA,” and developed an electroporation method to efficiently transfer it into the cytoplasm of E.coli. We demonstrate in vivo confocal detection of smFRET from this construct, which is both bright and photostable in the cellular milieu. In combination with PDA, BVA and our novel molecular counting method, this Green Fluorescent DNA should enable the characterization of DNA and protein-DNA dynamics in living cells, at the single-molecule level. I conclude by discussing the ways in which these methods may be useful in investigating the dynamics of processes such as transcription, translation and recombination, both in vitro and in vivo.