Search

Foundational basis for optimal climate change detection from energy-balance and cointegration models This thesis has critically examined the validity of optimal fingerprinting methods for the detection and attribution (D&A) of climate change trends. The validity is called into question because optimal fingerprinting involves a linear regression of non-stationary time series. Such non-stationary regressions are in general statistically inconsistent, meaning they are liable to produce spurious results. This thesis has investigated, using an idealized linear-response-model framework motivated by energy-balance considerations, whether the standard assumptions of optimal fingerprinting are sufficient to guarantee consistency, and hence whether detected climate trends are likely to be genuine or artefacts of spurious correlation. The principal reasoning tool in the thesis is the linear impulse-response model, familiar to many climatologists when parameterized as an energy-balance model (EBM), a simplified representation of global climate. A rigorous and efficient maximum likelihood method has been developed for estimating parameters of EBMs with any k > 0 number of boxes from CO2-quadrupling general circulation model (GCM) experiments and the method implemented as a free software package. It has been found that a three-box ocean is optimal for emulating the global mean surface temperature (GMST) impulse responses of GCMs in the Coupled Model Intercomparison Project Phase 5 (CMIP5). A new linear-filtering method has also been developed for estimating historical effective radiative forcing (ERF) from time series of GMST. It has been shown that the response of any k-box EBM can be represented as an ARMA(k, k-1) autoregressive moving-average filter and that, by inverting the ARMA filter, time series of surface temperature may be converted into radiative forcing. A comparison with an established method ("ERF_trans"), using historical simulations from HadGEM3-GC31-LL, found that the new method gives an ERF time series that closely matches published results (correlation of 0.83). Applying the new method to historical temperature observations, in combination with HadGEM3, produces evidence of a significant increase in ERF over the historical period with an estimated forcing in 2018 of 1.45 +- 0.504 Watts per square metre. It has been proved, using an idealized linear-response-model framework where forcing is represented as an integrated process, that if standard assumptions hold then the optimal fingerprinting estimator is consistent, and hence robust against spurious regression. Hypothesis tests, conducted using historical GMST observations and simulation output from 13 GCMs of the CMIP6 generation, have produced no evidence that these assumptions are violated in practice. The historical trends in GMST which are detected and attributed using these GCMs are therefore very likely not spurious. Consistency of the fingerprinting estimator was found to depend on "cointegration" between historical observations and GCM output. Detection of such a cointegration for the GMST variable indicates that the least-squares estimator is "superconsistent", with better convergence properties than might previously have been assumed. Furthermore, a new method has been developed for quantifying D&A uncertainty, which exploits the connection between cointegration and error-correction time series models to eliminate the need for pre-industrial control simulations.

We consider a family of stochastic interacting particle systems introduced by Kac as a model for a spatially homogeneous gas undergoing elastic collisions, corresponding to the spatially homogeneous Boltzmann equation. We consider Kac's problem of showing propagation of chaos - that if the velocities of the particles are initially approximately independent, then the same is true at later times - which is equivalent to the convergence of the empirical measures, and which derives the spatially homogeneous Boltzmann equation from the underlying molecular dynamics. The first two results concern the propagation of chaos for different Kac models. In the first case, we consider the hard spheres kernel, which is appropriate for modelling interactions arising from localised interactions. For this model, we build on previous analyses of the same problem to show that the expected deviation between the Kac process and the Boltzmann equation, measured in expected Wasserstein distance, is of the order N -α. Particular care is paid to the time-dependence of the estimates, using the stability properties of the Boltzmann equation: we will show that the expected deviation at a single fixed time is typically of the order N -α, uniformly in time, while for the largest deviation on a time interval [0,tfin], there is a prefactor (1 + tfin)ß, where ß > 0 can be chosen as close to 0 as desired. We also show that similar estimates hold, possibly for smaller a, as soon as the initial data are only assumed to have 2 + ε moments. We next consider the case of non-cutoff hard potentials, which arise from modelling a family of long-range interactions. In this case, the collision kernel is doubly-unbounded, both unbounded as the relative velocity increases, and with a non-integrable angular singularity, so that every particle undergoes infinitely many collisions on any nontrivial time-interval. In this context, we introduce a Tanaka-style coupling for a well-chosen distance function on Rd, which allows us to exploit a negative Povzner-type term. This leads to a proof of propagation of chaos and a new uniqueness and stability result for the corresponding Boltzmann equation, assuming only that the initial measure is a probability measure with finitely many moments, whereas previously established results have required either some additional regularity or an exponential moment. In the case of the corresponding hard-potential Landau equation, the same argument can be further refined to show uniqueness and stability for probability measures with 2 + ε moments. We also show that energy-conserving solutions to the Landau equation exist, assuming only that the initial data has finite energy, and we use the new uniqueness result to show that all solutions to the Landau equation immediately admit analytic densities, aside from the degenerate cases where the initial data is a point mass. By contrast, previous results have only shown that such regular solutions exist. We further study the dynamical large deviations of the N-particle system, either for the case of a cutoff Maxwell molecules kernel, or a caricature of the hard spheres kernel, for a range of initial conditions including equilibrium. We seek large deviation estimates jointly with an auxiliary flux measure which records the collision history. We prove a large deviation upper bound, and a lower bound restricted to classes of sufficiently regular paths, with a rate function analagous to those found elsewhere in the literature for similar problems. However, we show by exhibiting a family of counterexamples that this rate function does not capture all possible large deviation behaviours; although the particle system almost surely conserves energy, possible large deviation behaviour includes energy nonconserving solutions to the Boltzmann equation, as found by Lu and Wennberg, but these occur strictly more rarely than predicted by the rate function. The final section concerns Smoluchowski and Flory coagulation equations with a particular bilinear form, which arise when studying the interaction structure of the Kac process by forming clusters of particles joined by chains of collisions, corresponding to the cumulant expansion. Exploiting the bilinear structure and a coupling to random graphs, we are able to give a detailed analysis of the coagulation particle system and limiting Flory equation, including showing the emergence of a unique macroscopic cluster at a finite time tg ∈ (0,∞), which we characterise exactly in the case of the particle system.

This journal format thesis presents interdisciplinary research on characterising the biological process of intracellular transport using random walk models. The first two chapters outline the aims of the thesis, summarise the main chapters, and introduce intracellular transport, anomalous diffusion and various random walk models. This includes uncoupled continuous time random walks both in discrete and continuous space with extension to Levy walks, the fractional diffusion equation using the structured density approach, velocity random walks and fractional Brownian motion. The third chapter outlines experimental observations for power-law distributed running times in active intracellular transport. In addition, a mesoscopic Levy walk model is introduced making connections to a previously established microscopic framework involving multiple detachment and attachment rates of motor proteins. This work shows that the Levy walk model is capable of accurately describing intracellular transport and is compatible with the microscopic mechanisms of previously proposed theories. Furthermore, a new persistent random walk model with finite velocities that describe the phenomena of self-reinforcing directionality in the transport of endosomes is presented. Through this model, superdiffusion is exhibited despite exponential running times. Exact analytical expressions for the first and second moments are derived and validated using Monte Carlo simulations. The fourth chapter models the passive intracellular transport of lysosomes using the variable-order fractional diffusion equation. In doing so, the asymptotic representation of the solution of the variable-order fractional diffusion equation is found for an anomalous exponent that monotonically increases with position away from the origin. Furthermore, the asymptotic representation is validated using Monte Carlo simulations and experimental evidence for variable-order fractional diffusion in lysosomes is presented. The final chapter presents a neural network, trained on fractional Brownian motion, which estimates the Hurst exponent of a trajectory using as few as seven data points. This neural network is then applied using a windowed approach to trajectories of GFP-Rab5 tagged endosomes, GFP-SNX1 tagged endosomes and lysosomes. The Hurst exponents estimated locally in time for single experimental trajectories form distinct probability density functions for each different intracellular vesicle and various statistics on the vesicle dynamics are extracted. In doing so, the potential of this new technique is demonstrated.

Finite Markov chains are probabilistic network models that are commonly used as representations of dynamical processes in the physical sciences, biological sciences, economics, and elsewhere. Markov chains that appear in realistic modelling tasks are frequently observed to be nearly reducible, incorporating a mixture of fast and slow processes that leads to ill-conditioning of the underlying matrix of probabilities for transitions between states. Hence, the wealth of established theoretical results that makes Markov chains attractive and convenient models often cannot be used straightforwardly in practice, owing to numerical instability associated with the standard computational procedures to evaluate the expressions. This work is concerned with the development of theory, algorithms, and simulation methods for the efficient and numerically stable analysis of finite Markov chains, with a primary focus on exact approaches that are robust and therefore applicable to nearly reducible networks. New methodologies are presented to determine representative paths, identify the dominant transition mechanisms for a particular process of interest, and analyze the local states that have a strong influence on the characteristics of the global dynamics. The novel approaches yield new insights into the behaviour of Markovian networks, addressing and overcoming numerical challenges. The methodology is applied to example models that are relevant to current problems in chemical physics, including Markov chains representing a protein folding transition, and a configurational transition in an atomic cluster. Relevant classical theory of finite Markov chains and a description of existing robust algorithms for their numerical analysis is given in Chapter 1. The remainder of this thesis considers the problem of investigating a transition from an initial set of states in a Markovian network to an absorbing (target) macrostate. A formal approach to determine a finite set of representative transition paths is proposed in Chapter 2, based on exact pathwise decomposition of the total productive flux. This analysis allows for the importance of competing dynamical processes to be rigorously quantified. A robust state reduction algorithm to compute the expectation of any path property for a transition between two endpoint states is also described in Chapter 2. Chapter 3 reports further numerically stable state reduction algorithms to compute quantities that characterize the features of a transition at a statewise level of detail, allowing for identification of the local states that play a key role in modulating the slow dynamics. An expression is derived for the probability that a state is visited on a path that proceeds directly to the absorbing state without revisiting the initial state, which characterizes the dynamical relevance of an individual state to the overall transition process. In Chapter 4, an unsupervised strategy is proposed to utilize a highly efficient simulation algorithm for sampling paths on a Markov chain. The framework employs a scalable community detection algorithm to obtain an initial clustering of the network into metastable sets of states, which is subsequently refined by a variational optimization procedure. The optimized clustering is then used as the basis for simulating trajectory segments that necessarily escape from the metastable macrostates. The thesis is concluded with an overview of recent related advances that are beyond the scope of the current work (Chapter 5), and a discussion of potential applications where the novel methodology reported herein may be applied to perform insightful analyses that were previously intractable.

Though ubiquitous in applications, global optimisation problems are generally the most computationally intense due to their solution time growing exponentially with linear increase in their dimensions (this is the well known/so called 'curse of dimensionality'). In this thesis, we show that this scalability - and sometimes even tractability - challenges can be overcome in the global optimization of functions with low effective dimensionality, that are constant along an (unknown) linear subspace and only vary over the effective (complement) subspace. Such functions can often be found in applications, for example, in hyper-parameter optimization for neural networks, heuristic algorithms for combinatorial optimization problems and complex engineering simulations. We consider using random embeddings to reduce the problem dimension and search space, while attempting to preserve optimal values. In particular, we investigate two randomly reduced (sub)problem formulations that aim to solve the corresponding unconstrained and bound-constrained cases of the global optimization problem. In our REGO formulation for unconstrained global optimization, a Gaussian random, low-dimensional problem with bound constraints is formulated and solved in a reduced space by means of a(ny) global optimization algorithm. We prove novel probabilistic bounds for the success of REGO in solving the original, low effective-dimensionality unconstrained problem, which show its independence of the (potentially large) ambient dimension and its precise dependence on the dimensions of the effective and randomly embedding subspaces. These results significantly improve existing theoretical analyses by providing the exact distribution of a reduced minimizer and its Euclidean norm and by the general assumptions required on the problem. We validate our theoretical findings by extensive numerical testing of REGO with three types of global optimization solvers, illustrating the improved scalability of REGO compared to the full-dimensional application of the respective solvers. For the bound-constrained global optimization problem with special structure, a reduced subproblem formulation is investigated that solves the original problem over a Gaussian random low-dimensional subspace subject to affine constraints, so as to preserve feasibility with respect to the given domain. Under reasonable assumptions, we show that the probability that the reduced problem is successful in solving the original, full-dimensional problem is positive. Furthermore, in the case when the objective's effective subspace is aligned with the coordinate axes, we provide an asymptotic bound on this success probability that captures its algebraic dependence on the effective and, surprisingly, ambient dimensions. We then propose X-REGO, a generic algorithmic framework that uses multiple random embeddings, solving the above reduced problem repeatedly, approximately and possibly, adaptively. Using the success probability of the reduced subproblems, we prove that X-REGO converges globally, with probability one, and linearly in the number of embeddings, to a neighbourhood of a constrained global minimizer. Our numerical experiments on special structure functions illustrate our theoretical findings and the improved scalability of X-REGO variants when coupled with state-of-the-art global - and even local - optimization solvers for the subproblems. In an attempt to improve the scalability of generic global optimization problems, that may not possess low effective dimensionality, we propose to extend the use of the random embeddings framework above to this context. For Lipschitz continuous objectives, we develop a novel analysis that estimates the probability of success of the feasible randomly reduced subproblems based on connections to the field of conic integral geometry. To evaluate the quality of our bound, we compare it to the success of uniform sampling, in the asymptotic regime. Finally, again using our success probability bound, we establish that the X- REGO algorithmic framework applied to the generic bound-constrained global optimization problem is convergent with probability one, and linearly in the number of embeddings, to a neighbourhood of a constrained global minimizer.

Many biological systems are comprised of active particles, that is entities which move by consuming energy from their environment such as flocks of fish, cell tissues or bacterial swarms. The interaction of active particles with each other or complex environments, such as porous media and chemical fields, leads to behaviour unlike that of classical equilibrium systems. The present thesis is concerned with the theoretical and experimental investigation of the transport behaviour of active particles to elucidate how it results from their environmental interactions. To investigate the impact of different boundary conditions, I studied the diffusive transport of Active Brownian Particles (ABP) in an obstacle lattice using agent-based simulations. These show that boundary conditions which preserve the particle’s orientation can result in high diffusivities even at high obstacle densities, unlike classical specular reflection. The dynamics are well described by a model based on Run-and-Tumble particles (RTP) with microscopically derived parameters. The study was then extended to investigate the transport of RTP in structured environments in presence of chemical gradients, which bias the active particles in a process termed chemotaxis. Results of this model show how the reduction of chemotaxis in obstacle lattices depends on the boundary condition. To complement theoretical analysis, I investigated bacterial scattering experimentally by tracking wild-type and smooth-swimming mutants of the model bacterium Escherichia coli swimming in microfluidic channels with lattices of obstacles. Based on the microscopic analysis of scattering events, the diffusive transport was modelled and compared to the experimental measurements. In a final investigation, I considered how heterogeneities in the environment affect active particle transport. As non-classical surface interactions reduce the effective speed at obstacles, introducing a non-homogenous distribution of obstacles can introduce a spatial dependence of the swimming speed. Combining simulations and preliminary experiments, I show that this introduces a drift towards denser obstacle regions. A spatially varying speed can also be introduced directly by bacteria via chemokinesis, which is a change in swimming speed according to the absolute level of a chemical. I extend a Keller-Segel type model to include chemokinesis and apply it to predict the dynamics of bacterial populations in experimentally inspired chemical fields. I find that chemokinesis can not only enhance the chemotactic population response, but also modifies it qualitatively.

Developed to give meaning to differential equations driven by rough signals, rough path theory has opened in recent years a new approach to tackle certain problems in other fields such as mathematical finance and machine learning. This is due to certain algebraic and analytical properties of an object called the rough path signature. This thesis aims to make a contribution in this direction by introducing signature-based methods to study various problems arising in finance and machine learning. In the context of finance, we consider two problems: the pricing and hedging of financial derivatives, and the optimal execution of financial securities. We propose numerical methods based on signatures to solve both problems. Finally, in the context of machine learning we begin by introducing a new neural network architecture called the deep signature model, based on the deep signature transform. The thesis concludes with a practical application of the signature transform on a real-world problem, where we aim to make predictions from psychiatric data using signatures.

Noise has come to be accepted as a quintessential part of social and biological processes. It has transcended the misconception of being an obstacle, which hinders our understanding of `true mechanisms' hiding behind the randomness, and is now recognised as the cause of many important phenomena. Different sources of noise exist, and their combined effect is not trivial to understand. In this thesis, we contribute by studying models which combine intrinsic and extrinsic noise. We consider systems with discrete interacting components; as a consequence, they are subject to intrinsic noise. At the same time, we explore how two sources of extrinsic noise modify the time evolution of these models: motion, and agent-to-agent heterogeneity. We investigate motion of individuals in a two dimensional setting. Members of the population take positions in space and are moved by an external flow. The position of agents defines an interaction graph, so the population is structured. The interaction network is modified as the flow advects individuals in space. We choose an evolutionary dynamics setting, and study how the changing population structure alters the probability with which a mutant invades a population of wild-type individuals. We find that seemingly subtle changes in the mechanics of evolution, which implement birth and death events, can lead to significant changes in the mutant's chances of success. Therefore, we propose these differences can be used to identify the underlying mechanism in a given experimental setting. Furthermore, we debate that the commonly used term to describe the invasion process in unstructured populations, `well-mixed', is a misnomer, which must be used with care. To study agent-to-agent heterogeneity we use models of epidemics and opinion dynamics. For the latter we explore how the achievement, maintenance or alternation of consensus are affected by the presence of more than two co-evolving opinions, with potentially different conviction strengths. In the model of disease spread, we study how heterogeneity in the susceptibility and infectiousness of individuals influences the frequency and amplitude of outbreaks. In both models heterogeneity is represented by an arbitrary number of compartments that describe the `type' of the agents. To be able to simplify the mathematical description of these systems, we approach them by aggregating these compartments into bigger groups. In both cases, we find that this marginalised description provides a good approximation of the model dynamics. We are able to characterise the recurrence and severity of outbreaks using the aggregated components; similarly, we provide an analytical description of the simplified opinion dynamics, which is an approximation when conviction strengths differ, and exact when they are homogeneous.

Many processes in physics and biology can be understand through the framework of escape from a metastable state, including (but not limited to) the rates of chemical reactions, the unfolding of proteins, the nucleation of bubbles, and the condensation of gases. To understand the kinetics of these processes, we have to be able to calculate the rate of escape. In this thesis, I solve several of such of escape problems, each addressing a specific physical or biological system. I first show how the forced unfolding of heteropolymers could be a process with non-exponential kinetics, developing ideas about the importance of unfolding pathways in determining kinetics of unfolding. Then, I consider forced unfolding when a molecule is attached to a yielding (viscoelastic) substrate, and a constant force is applied. I show that the rates of unfolding depend on both the elastic and viscous response of the substrate. This problem is related to the biological process of mechanosensing, when the unfolding `sensor' protein exposes catalytic residues and generates a chemical signal to the cell. Related to this is the analysis of population-dynamics study of cells adhesion on substrates, which allows me to extract key characteristics and parameters of the adhesome complex. Then, I apply the ideas of escape from a metastable state to ask about the rates of a ligand at the end of a tethered polymer binding to a surface receptor, using a mean field approach to reduce the problem to one dimension. I show that there is a trade-off between the entropic cost of reaching to a receptor vs the volumetric cost of expanding the tether length. I then show that for a Gaussian chain with multiple ligands along its length, there exists a finite, non-zero optimal number of ligands to minimise the time taken for the end of the chain to bind to the surface. Finally, I consider the problem of microswimmers in an obstacle lattice, calculating their transport properties, and showing how we can use lattices to examine the underlying stochastic dynamics.

This thesis studies physical and stochastic aspects of microorganisms. From the point of view of $\textit{physics}$, the studies in this thesis are motivated by the goal of gaining biological insight using the machinery of physics and mathematics. From the point of view of $\textit{biology}$, the studies in this thesis focus primarily on choanoflagellates, eukaryotes that are the closest living unicellular relatives of animals. This choice of model organism was motivated by the important biological question of the origin of multicellularity. Why was it that single-celled organisms evolved to become multicellular? In particular, we study closely the species $\textit{Salpingoeca rosetta}$, which has the ability to form colonies that resemble true multicellular organisms. A large part of this thesis deals with the random walks of microorganisms. We study these active random walks both for single cells and those composed of individual organisms adhered together. The latter colonial random walkers are typified by choanoflagellates. We develop quantitative theories and use these to extract physical parameters. The increasing ocean oxygen levels in the Precambrian era are thought to be an important factor in the emergence of complex multicellular, animal life. As a first step, we address this situation by studying the response of $\textit{S. rosetta}$ to oxygen gradients. We find that $\textit{S. rosetta}$ displays positive aerotaxis. Analysis of the spatial population distributions provides evidence for logarithmic sensing of oxygen, which enhances sensing in low oxygen neighbourhoods. Analysis of search strategy models on the experimental colony trajectories finds that choanoflagellate aerotaxis is consistent with stochastic navigation, the statistics of which are captured using an effective continuous version of classical run-and-tumble chemotaxis. We compare this continuous run-to-tumble with the run-and-tumble seen in bacteria by formulating a general model for persistent run-and-tumble. We find that although an optimal persistence does exist for a given tumble frequency, in the full parameter space there is a continuum of optimal solutions. We develop this model further by introducing finite tumble times. Efficient uptake of prey and nutrients from the environment is an important component in the fitness of all microorganisms, and its dependence on size may reveal clues to the origins of evolutionary transitions to multicellularity. We examine these issues in depth for choanoflagellates, finding that in the absence of other requirements and in a homogeneously nutritious environment, the optimal strategy to maximise filter feeding is to swim fast which favours swimming unicells. In contrast, in large external flows, a sessile form becomes advantageous. Effects of prey diffusion are discussed and are also found to be advantageous for the swimming unicell. Finally, we consider the switching between synchronous and anti-synchronous beating of flagella in the green alga $\textit{Chlamydomonas}$, a phenomenon that results in run-and-tumble behaviour in eukaryotes. We develop a theoretical model to describe this beating and use it to argue that the synchrony itself is obtained intracellularly, whereas the flagella shapes are most likely strongly influenced by hydrodynamic interactions.

Two optimisation problems were considered, in which market exposure is indirectly controlled. The first one models the capital of a company and an independent portfolio of new businesses, each one represented by a Cram\'{e}r-Lundberg process. The company can choose the proportion of new business it wants to take on and can alter this proportion over time. Here the objective is to find a strategy that maximises the survival probability. We use a point processes framework to deal with the impact of an adapted strategy in the intensity of the new business. We prove that when Cram\'{e}r-Lundberg processes with exponentially distributed claims, it is optimal to choose a threshold type strategy, where the company switches between owning all new businesses or none depending on the capital level. For this type of processes that change both drift and jump measure when crossing the constant threshold, we solve the one and two-sided exit problems. This optimisation problem is also solved when the capital of the company and the new business are modelled by spectrally positive L\'{e}vy processes of bounded variation. Here the one-sided exit problem is solved and we prove optimality of the same type of threshold strategy for any jump distribution. The second problem is a stochastic variation of the work done by Taylor about underwriting in a competitive market. Taylor maximised discounted future cash flows over a finite time horizon in a discrete time setting when the change of exposure from one period to the next has a multiplicative form involving the company's premium and the market average premium. The control is the company's premium strategy over a the mentioned finite time horizon. Taylor's work opened a rich line of research, and we discuss some of it. In contrast with Taylor's model, we consider the market average premium to be a Markov chain instead of a deterministic vector. This allows to model uncertainty in future conditions of the market. We also consider an infinite time horizon instead of finite. This solves the time dependency in Taylor's optimal strategies that were giving unrealistic results. Our main result is a formula to calculate explicitly the value function of a specific class of pricing strategies. Further we explore concrete examples numerically. We find a mix of optimal strategies where in some examples the company should follow the market while in other cases should go against it.

Point processes are mechanisms that beget point patterns. Realisations of point processes are observed in many contexts, for example, locations of stars in the sky, or locations of trees in a forest. Inferring the mechanisms that drive point processes relies on the development of models that appropriately account for the dependencies inherent in the data. Fitting models that adequately capture the complex dependency structures in either space, time, or both is often problematic. This is commonly due to—but not restricted to—the intractability of the likelihood function, or computational burden of the required numerical operations. This thesis primarily focuses on developing point process models with some hierarchical structure, and specifically where this is a latent structure that may be considered as one of the following: (i) some unobserved construct assumed to be generating the observed structure, or (ii) some stochastic process describing the structure of the point pattern. Model fitting procedures utilised in this thesis include either (i) approximate-likelihood techniques to circumvent intractable likelihoods, (ii) stochastic partial differential equations to model continuous spatial latent structures, or (iii) improving computational speed in numerical approximations by exploiting automatic differentiation. Moreover, this thesis extends classic point process models by considering multivariate dependencies. This is achieved through considering a general class of joint point process model, which utilise shared stochastic structures. These structures account for the dependencies inherent in multivariate point process data. These models are applied to data originating from various scientific fields; in particular, applications are considered in ecology, medicine, and geology. In addition, point process models that account for the second order behaviour of these assumed stochastic structures are also considered.

This thesis explores the stochastic features of models of ecological systems in discrete and in continuous time. Our interest lies in models formulated at the microscale, from which a mesoscopic description can be derived. The stochasticity present in the models, constructed in this way, is intrinsic to the systems under consideration and stems from their finite size. We start by exploring a susceptible-infectious-recovered model for epidemic spread on a network. We are interested in the case where the connectivity, or degree, of the individuals is characterised by a very broad, or heterogeneous, distribution, and in the effects of stochasticity on the dynamics, which may depart wildly from that of a homogeneous population. The model at the mesoscale corresponds to a system of stochastic differential equations with a very large number of degrees of freedom which can be reduced to a two-dimensional model in its deterministic limit. We show how this reduction can be carried over to the stochastic case by exploiting a time-scale separation in the deterministic system and carrying out a fast-variable elimination. We use simulations to show that the temporal behaviour of the epidemic obtained from the reduced stochastic model yields reasonably good agreement with the microscopic model under the condition that the maximum allowed degree that individuals can have is not too close to the population size. This is illustrated using time series, phase diagrams and the distribution of epidemic sizes. The general mesoscopic theory used in continuous-time models has only very recently been developed for discrete-time systems in one variable. Here, we explore this one-dimensional theory and find that, in contrast to the continuous-time case, large jumps can occur between successive iterates of the process, and this translates at the mesoscale into the need for specifying `boundary' conditions everywhere outside of the system. We discuss these and how to implement them in the stochastic difference equation in order to obtain results which are consistent with the microscopic model. We then extend the theoretical framework to make it applicable to systems containing an arbitrary number of degrees of freedom. In addition, we extend a number of analytical results from the one-dimensional stochastic difference equation to arbitrary dimension, for the distribution of fluctuations around fixed points, cycles and quasi-periodic attractors of the corresponding deterministic map. We also derive new expressions, describing the autocorrelation functions of the fluctuations, as well as their power spectrum. From the latter, we characterise the appearance of noise-induced oscillations in systems of dimension greater than one, which have been previously observed in continuous-time systems and are known as quasi-cycles. Finally, we explore the ability of intrinsic noise to induce chaotic behaviour in the system for parameter values for which the deterministic map presents a non-chaotic attractor; we find that this is possible for periodic, but not for quasi-periodic, states.

Recent experiments have shown the fundamental role that random fluctuations play in many chemical systems in living cells, such as gene regulatory networks. Mathematical models are thus indispensable to describe such systems and to extract relevant biological information from experimental data. Recent decades have seen a considerable amount of modelling effort devoted to this task. However, current methodologies still present outstanding mathematical and computational hurdles. In particular, models which retain the discrete nature of particle numbers incur necessarily severe computational overheads, greatly complicating the tasks of characterising statistically the noise in cells and inferring parameters from data. In this thesis we study analytical approximations and inference methods for stochastic reaction dynamics. The chemical master equation is the accepted description of stochastic chemical reaction networks whenever spatial effects can be ignored. Unfortunately, for most systems no analytic solutions are known and stochastic simulations are computationally expensive, making analytic approximations appealing alternatives. In the case where spatial effects cannot be ignored, such systems are typically modelled by means of stochastic reaction-diffusion processes. As in the non-spatial case an analytic treatment is rarely possible and simulations quickly become infeasible. In particular, the calibration of models to data constitutes a fundamental unsolved problem. In the first part of this thesis we study two approximation methods of the chemical master equation; the chemical Langevin equation and moment closure approximations. The chemical Langevin equation approximates the discrete-valued process described by the chemical master equation by a continuous diffusion process. Despite being frequently used in the literature, it remains unclear how the boundary conditions behave under this transition from discrete to continuous variables. We show that this boundary problem results in the chemical Langevin equation being mathematically ill-defined if defined in real space due to the occurrence of square roots of negative expressions. We show that this problem can be avoided by extending the state space from real to complex variables. We prove that this approach gives rise to real-valued moments and thus admits a probabilistic interpretation. Numerical examples demonstrate better accuracy of the developed complex chemical Langevin equation than various real-valued implementations proposed in the literature. Moment closure approximations aim at directly approximating the moments of a process, rather then its distribution. The chemical master equation gives rise to an infinite system of ordinary differential equations for the moments of a process. Moment closure approximations close this infinite hierarchy of equations by expressing moments above a certain order in terms of lower order moments. This is an ad hoc approximation without any systematic justification, and the question arises if the resulting equations always lead to physically meaningful results. We find that this is indeed not always the case. Rather, moment closure approximations may give rise to diverging time trajectories or otherwise unphysical behaviour, such as negative mean values or unphysical oscillations. They thus fail to admit a probabilistic interpretation in these cases, and care is needed when using them to not draw wrong conclusions. In the second part of this work we consider systems where spatial effects have to be taken into account. In general, such stochastic reaction-diffusion processes are only defined in an algorithmic sense without any analytic description, and it is hence not even conceptually clear how to define likelihoods for experimental data for such processes. Calibration of such models to experimental data thus constitutes a highly non-trivial task. We derive here a novel inference method by establishing a basic relationship between stochastic reaction-diffusion processes and spatio-temporal Cox processes, two classes of models that were considered to be distinct to each other to this date. This novel connection naturally allows to compute approximate likelihoods and thus to perform inference tasks for stochastic reaction-diffusion processes. The accuracy and efficiency of this approach is demonstrated by means of several examples. Overall, this thesis advances the state of the art of modelling methods for stochastic reaction systems. It advances the understanding of several existing methods by elucidating fundamental limitations of these methods, and several novel approximation and inference methods are developed.

Individual-based models have been applied to study a broad spectrum of problems across multiple disciplines, such as the spread of epidemics or the outcome of social dilemma. They are used to investigate the macroscopic effects that arise from the microscopic dynamics of interacting individuals. Fixation describes the taking over of the population by a single type of individual or species. It is a prominent feature in the field of population genetics, which interprets many biological scenarios of evolution. Equilibration describes the process of reaching a heterogeneous steady state. In this thesis we analyse these macroscopic features through techniques derived from statistical physics and the theory of stochastic processes. Birth-death processes are used to describe the interaction of two types of individual in a population, such as competing strains of bacteria. These interactions are often specified using the framework of evolutionary game theory. The environment in which the population evolves can have a crucial impact on selection. In systems where the environment switches between multiple states we develop a general theory to calculate the fixation time statistics of a mutant individual in a population of wild-types, as well as the stationary distributions when mutations are present in the dynamics. In some birth--death processes, and in particular those described by evolutionary game theory, the mean fixation time contains only limited information. By diagonalising the master equation that describes the process, we are able to obtain closed-form expressions for the complete fixation time distributions. Individual-based models can also be used to describe the accumulation of mutations in a cell. This has important consequences for the initiation and progression of cancer. We find that such systems exhibit metastable states in the dynamics, and we can exploit the separation of timescales between relaxing to the quasi-stationary state and reaching fixation to characterise these phenomena. In this scenario we employ the WKB method to describe the population-level dynamics. Although this method has been used to describe numerous stochastic processes, a clear and coherent description is lacking in the literature. Through the use of multiple examples, including the aforementioned cancer initiation model, we carefully explain the multitude of constructs and equations that result from the application of this methodThe analytical characterisation of the evolutionary dynamics that are observed in these stochastic processes has resulted in a greater understanding of fixation and equilibration. This thesis promotes the benefits of analytical, or even semi-analytical methods, and on a more general level contributes toward a more complete understanding of evolutionary processes.

All real-world populations are composed of a finite number of individuals. Due to the intrinsically random nature of interactions between individuals, the dynamics of finite-sized populations are stochastic processes. Additionally, for many types of interaction not all effects occur instantaneously. Instead there are delays before effects are felt. The centrepiece of this thesis is a method of analytically studying stochastic population dynamics with delay reactions. Dynamics with delay reactions are non-Markovian, meaning many of the widely used techniques to study stochastic processes break down. It is not always possible to formulate the master equation, which is a common starting point for analysis of stochastic effects in population dynamics. We follow an alternative method, and derive an exact functional integral approach which is capable of capturing the effects of both stochasticity and delay in the same modelling framework. Our work builds on previous techniques developed in statistical physics, in particular the Martin-Siggia-Rose-Janssen-de Dominicis functional integral. The functional integral approach does not rely on an particular constraints on the population dynamics, for example the choice of delay distribution. Functional integrals can not in general be solved exactly. We show how the functional integral can be used to derive the deterministic, chemical Langevin, and linear-noise approximations for stochastic dynamics with delay. In the later chapters we extend Gillespie’s approximate method of studying stochastic dynamics with delay reactions, which can be used to derive the chemical Langevin equation, by-pass the functional integral. We also derive an extension to the functional integral approach so that it also covers systems with interruptible delay reactions. To demonstrate the applicability of our results we consider various models of population dynamics, arising from ecology, epidemiology, developmental biology, and chemistry. Our analytical calculations are found to provide excellent agreement with exact numerical simulations.

We develop and study the high-density limit of various new models in mathematical pop- ulation genetics. These models extend the Λ-Fleming–Viot process when there are two genetic types at the locus of study. Given a finite sample from a population undergoing these dynamics, a key tool for understanding the corresponding genealogy is the method of duality. We introduce the reproduction-linked mutation mechanism and consider how this affects the process of relative allelic frequencies and the genealogy. The second generalization incorporates two forms of natural selection – differential killing and differential birth. We contrast the structure of their genealogies. Several properties of the block size spectra of the Kingman and Beta coalescents are also investigated, including their behaviour as they come down from infinity.

Sequence alignment is one of the most intensely studied problems in bioinformatics, and is an important step in a wide range of analyses. An issue that has gained much attention in recent years is the fact that downstream analyses are often highly sensitive to the specific choice of alignment. One way to address this is to jointly sample alignments along with other parameters of interest. In order to extend the range of applicability of this approach, the first chapter of this thesis introduces a probabilistic evolutionary model for protein structures on a phylogenetic tree; since protein structures typically diverge much more slowly than sequences, this allows for more reliable detection of remote homologies, improving the accuracy of the resulting alignments and trees, and reducing sensitivity of the results to the choice of dataset. In order to carry out inference under such a model, a number of new Markov chain Monte Carlo approaches are developed, allowing for more efficient convergence and mixing on the high-dimensional parameter space. The second part of the thesis presents a directed acyclic graph (DAG)-based approach for representing a collection of sampled alignments. This DAG representation allows the initial collection of samples to be used to generate a larger set of alignments under the same approximate distribution, enabling posterior alignment probabilities to be estimated reliably from a reasonable number of samples. If desired, summary alignments can then be generated as maximum-weight paths through the DAG, under various types of loss or scoring functions. The acyclic nature of the graph also permits various other types of algorithms to be easily adapted to operate on the entire set of alignments in the DAG. In the final part of this work, methodology is introduced for alignment-DAG-based sequence annotation using hidden Markov models, and RNA secondary structure prediction using stochastic context-free grammars. Results on test datasets indicate that the additional information contained within the DAG allows for improved predictions, resulting in substantial gains over simply analysing a set of alignments one by one.

In this thesis we study three different modelling frameworks for biological systems of dispersal and combinations thereof. The three frameworks involved are individual-based models, group-level models in the form of partial differential equations (PDEs) and robot swarms. In the first two chapters of the thesis, we present ways of coupling individual based models with PDEs in so-called hybrid models, with the aim of achieving improved performance of simulations. Two classes of such hybrid models are discussed that allow an efficient simulation of multi-species systems of dispersal with reactions, but involve individual resolution for certain species and in certain parts of a computational domain if desired. We generally consider two types of example systems: bacterial chemotaxis and reaction-diffusion systems, and present results in the respective application area as well as general methods. The third chapter of this thesis introduces swarm robotic experiments as an additional tool to study systems of dispersal. In general, those experiments can be used to mimic animal behaviour and to study the impact of local interactions on the group-level dynamics. We concentrate on a target finding problem for groups of robots. We present how PDE descriptions can be adjusted to incorporate the finite turning times observed in the robotic system and that the adjusted models match well with experimental data. In the fourth and last chapter, we consider interactions between robots in the form of hard-sphere collisions and again derive adjusted PDE descriptions. We show that collisions have a significant impact on the speed with which the group spreads across a domain. Throughout these two chapters, we apply a combination of experiments, individual-based simulations and PDE descriptions to improve our understanding of interactions in systems of dispersal.

Operating and interacting in an environment requires the ability to manage uncertainty and to choose definite courses of action. In this thesis we look to Bayesian probability theory as the means to achieve the former, and find that through rigorous application of the rules it prescribes we can, in theory, solve problems of decision making under uncertainty. Unfortunately such methodology is intractable in realworld problems, and thus approximation of one form or another is inevitable. Many techniques make use of heuristic procedures for managing uncertainty. We note that such methods suffer unreliable performance and rely on the specification of ad-hoc variables. Performance is often judged according to long-term asymptotic performance measures which we also believe ignores the most complex and relevant parts of the problem domain. We therefore look to develop principled approximate methods that preserve the meaning of Bayesian theory but operate with the scalability of heuristics. We start doing this by looking at function approximation in continuous state and action spaces using Gaussian Processes. We develop a novel family of covariance functions which allow tractable inference methods to accommodate some of the uncertainty lost by not following full Bayesian inference. We also investigate the exploration versus exploitation tradeoff in the context of the Multi-Armed Bandit, and demonstrate that principled approximations behave close to optimal behaviour and perform significantly better than heuristics on a range of experimental test beds.

The genome is a treasure trove of information about the history of an individual, his population, and his species. For as long as genomic data have been available, methods have been developed to retrieve this information and learn about population history. Over the last decade, large international genomic projects (e.g. the HapMap Project and the 1000 Genomes Project) have offered access to high quality data collected from thousands of individuals from a vast number of populations. Freely available to all, these databases offer the possibility to develop new methods to uncover the history of the peopling of the world by modern humans. Due to the complexity of the problem and the large amount of available data, all developed methods either simplify the model with strong assumptions or use an approximation; they also dramatically down-sample their data by either using fewer individuals or only portions of the genome. In this thesis, we present a novel method to jointly estimate the time of divergence of a pair of populations and their variable sizes, a previously unsolved problem. The method uses multiple regions of the genome with low recombination rate. For each region, we use an importance sampler to build a large number of possible genealogies, and from those we estimate the likelihood function of parameters of interest. By modelling the population sizes as piecewise constant within fixed time intervals, we aim to capture population size variation through time. We show via simulation studies that the method performs well in many situations, even when the model assumptions are not totally met. We apply the method to five populations from the 1000 Genomes Project, obtaining estimates of split times between European groups and among Europe, Africa and Asia. We also infer shared and non-shared bottlenecks in out-of- Africa groups, expansions following population separations, and the sizes of ancestral populations further back in time.

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.

This thesis focus on the effect that selection has on the ancestry of a spatially structured population. In the absence of selection, the ancestry of a sample from the population behaves as a system of random walks that coalesce upon meeting. Backwards in time, each ancestral lineage jumps, at the time of its birth, to the location of its parent, and whenever two ancestral lineages have the same parent they jump to the same location and coalesce. Introducing selective forces to the evolution of a population translates into branching when we follow ancestral lineages, a by-product of biased sampling forwards in time. We study populations that evolve according to the Spatial Lambda-Fleming-Viot process with selection. In order to assess whether the picture under selection differs from the neutral case we must consider the timescale dictated by the neutral mutation rate Theta. Thus we look at the rescaled dual process with n=1/Theta. Our goal is to find a non-trivial rescaling limit for the system of branching and coalescing random walks that describe the ancestral process of a population. We show that the strength of selection (relative to the mutation rate) required to do so depends on the dimension; in one and two dimensions selection needs to be stronger in order to leave a detectable trace in the population. The main results in this thesis can be summarised as follows. In dimensions three and higher we take the selection coefficient to be proportional to 1/n, in dimension two we take it to be proportional to log(n)/n and finally, in dimension one we take the selection coefficient to be proportional to 1/sqrt(n). We then proceed to prove that in two and higher dimensions the ancestral process of a sample of the population converges to branching Brownian motion. In one dimension, provided we do not allow ancestral lineages to jump over each other, the ancestral process converges to a subset of the Brownian net. We also provide numerical results that show that the non-crossing restriction in one dimension cannot be lifted without a qualitative change in the behaviour of the process. Finally, through simulations, we study the rate of convergence in the two-dimensional case.

The discovery of genetic regulatory networks was an important advancement in science. Not only do they help understand how organisms behave but the development of synthetic genetic networks has aided in other fields of science and industry. Many genetic networks have been modelled deterministically by using differential equations to provide an insight into the network's behaviour. However, within a biological environment, a certain degree of intrinsic noise should be expected and the robustness of these networks should be tested. Creating and analysing a genetic network in a biological environment can be a time consuming task so applying stochastic methods, such as the Gillespie Algorithm, to a computer model will provide an important, initial insight into the behaviour of the system. One interesting genetic network is the coupled repressilator due to its relatively simplistic design and the broad, multistable dynamics it offers. The inhomogeneous solutions that it can yield are particularly interesting as they may help explain certain biological phenomena, and may be used as a tool to assist with further research into genetic networks. In this thesis, the Gillespie Algorithm will be applied to the coupled repressilator so that its robustness can be tested. Biologically feasible modifications will be made to the system to produce much more stable and predictable dynamics so that the broad range of solutions can exist within a noisy environment. The methods developed will take into account previously made assumptions and potential errors in biological data so that they can be applied to other genetic system. One further objective in this thesis is to explore computational limitations that may occur when modelling large, stochastic networks. Issues such as rounding errors and dealing with very small and very large numbers were encountered and methods to circumvent these without sacrificing computational run-time will be developed and applied.

Real-world data often exhibits sequential dependence, across diverse domains such as human behavior, medicine, finance, and climate modeling. Probabilistic methods capture the inherent uncertainty associated with prediction in these contexts, with autoregressive models being especially prominent. This dissertation focuses on using autoregressive models to answer complex probabilistic queries that go beyond single-step prediction, such as the timing of future events or the likelihood of a specific event occurring before another. In particular, we develop a broad class of novel and efficient approximation techniques for marginalization in sequential models that are model-agnostic. These techniques rely solely on access to and sampling from next-step conditional distributions of a pre-trained autoregressive model, including both traditional parametric models as well as more recent neural autoregressive models. Specific approaches are presented for discrete sequential models, for marked temporal point processes, and for stochastic jump processes, each tailored to a well-defined class of informative, long-range probabilistic queries.

In the last decades, the explosion of data from quantitative techniques has revolutionised our understanding of biological processes. In this scenario, advanced statistical methods and algorithms are becoming fundamental to decipher the dynamics of biochemical mechanisms such those involved in the regulation of gene expression. Here we develop mechanistic models and approximate inference techniques to reverse engineer the dynamics of gene regulation, from mRNA and/or protein time series data. We start from an existent variational framework for statistical inference in transcriptional networks. The framework is based on a continuous-time description of the mRNA dynamics in terms of stochastic differential equations, which are governed by latent switching variables representing the on/off activity of regulating transcription factors. The main contributions of this work are the following. We speeded-up the variational inference algorithm by developing a method to compute a posterior approximate distribution over the latent variables using a constrained optimisation algorithm. In addition to computational benefits, this method enabled the extension to statistical inference in networks with a combinatorial model of regulation. A limitation of this framework is the fact that inference is possible only in transcriptional networks with a single-layer architecture (where a single or couples of transcription factors regulate directly an arbitrary number of target genes). The second main contribution in this work is the extension of the inference framework to hierarchical structures, such as feed-forward loop. In the last contribution we define a general structure for transcription-translation networks. This work is important since it provides a general statistical framework to model complex dynamics in gene regulatory networks. The framework is modular and scalable to realistically large systems with general architecture, thus representing a valuable alternative to traditional differential equation models. All models are embedded in a Bayesian framework; inference is performed using a variational approach and compared to exact inference where possible. We apply the models to the study of different biological systems, from the metabolism in E. coli to the circadian clock in the picoalga O. tauri.

This thesis considers two areas of research in non-equilibrium soft matter at the mesoscale. In the first part we introduce active colloids in the context of active matter and focus on the particular case of phoretic colloids. The general theory of phoresis is presented along with an expression for the phoretic velocity of a colloid and its rotational diffusion in two and three dimensions. We introduce a model for thermally active colloids that absorb light and emit heat and propel through thermophoresis. Using this model we develop the equations of motion for their collective dynamics and consider excluded volume through a lattice gas formalism. Solutions to the thermoattractive collective dynamics are studied in one dimension analytically and numerically. A few numerical results are presented for the collective dynamics in two dimensions. We simulate an unconfined system of thermally active colloids under directed illumination with simple projection based geometric optics. This system self-organises into a comet-like swarm and exhibits a wide range of non- equilibrium phenomena. In the second part we review the background of polymer translocation, including key experiments, theoretical progress and simulation studies. We present, discuss and use a common model to investigate the potential of patterned nanopores for stochastic sensing and identification of polynucleotides and other heteropolymers. Three pore patterns are characterised in terms of the response of a homopolymer with varying attractive affinity. This is extended to simple periodic block co-polymer heterostructures and a model device is proposed and demonstrated with two stochastic sensing algorithms. We find that mul- tiple sequential measurements of the translocation time is sufficient for identification with high accuracy. Motivated by fluctuating biological channels and the prospect of frequency based selectivity we investigate the response of a homopolymer through a pore that has a time dependent geometry. We show that a time dependent mobility can capture many features of the frequency response.

Every year in the UK, tens of thousands of hospital patients suffer adverse events, such as un-planned transfers to Intensive Therapy Units or unexpected cardiac arrests. Studies have shown that in a large majority of cases, significant physiological abnormalities can be observed within the 24-hour period preceding such events. Such warning signs may go unnoticed, if they occur between observations by the nursing staff, or are simply not identified as such. Timely detection of these warning signs and appropriate escalation schemes have been shown to improve both patient outcomes and the use of hospital resources, most notably by reducing patients’ length of stay. Automated real-time early-warning systems appear to be cost-efficient answers to the need for continuous vital-sign monitoring. Traditionally, a limitation of such systems has been their sensitivity to noisy and artefactual measurements, resulting in false-alert rates that made them unusable in practice, or earned them the mistrust of clinical staff. Tarassenko et al. (2005) and Hann (2008) proposed a novelty detection approach to the problem of continuous vital-sign monitoring, which, in a clinical trial, was shown to yield clinically acceptable false alert rates. In this approach, an observation is compared to a data fusion model, and its “normality” assessed by comparing a chosen statistic to a pre-set threshold. The method, while informed by large amounts of training data, has a number of heuristic aspects. This thesis proposes a principled approach to multivariate novelty detection in stochastic time- series, where novelty scores have a probabilistic interpretation, and are explicitly linked to the starting assumptions made. Our approach stems from the observation that novelty detection using complex multivariate, multimodal generative models is generally an ill-defined problem when attempted in the data space. In situations where “novel” is equivalent to “improbable with respect to a probability distribution ”, formulating the problem in a univariate probability space allows us to use classical results of univariate statistical theory. Specifically, we propose a multivariate extension to extreme value theory and, more generally, order statistics, suitable for performing novelty detection in time-series generated from a multivariate, possibly multimodal model. All the methods introduced in this thesis are applied to a vital-sign monitoring problem and compared to the existing method of choice. We show that it is possible to outperform the existing method while retaining a probabilistic interpretation. In addition to their application to novelty detection for vital-sign monitoring, contributions in this thesis to existing extreme value theory and order statistics are also valid in the broader context of data-modelling, and may be useful for analysing data from other complex systems.

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 doctoral thesis is concerned with some theoretical and practical questions related to backward stochastic differential equations (BSDEs) and more specifically their connection with some parabolic partial differential equations (PDEs). The thesis is made of three parts. In the first part, we study the probabilistic representation for a class of multidimensional PDEs with quadratic nonlinearities of a special form. We obtain a representation formula for the PDE solution in terms of the solutions to a Lipschitz BSDE. We then use this representation to obtain an estimate on the gradient of the PDE solutions by probabilistic means. In the course of our analysis, we are led to prove some results for the associated multidimensional quadratic BSDEs, namely an existence result and a partial uniqueness result. In the second part, we study the well-posedness of a very general quadratic reflected BSDE driven by a continuous martingale. We obtain the comparison theorem, the special comparison theorem for reflected BSDEs (which allows to compare the increasing processes of two solutions), the uniqueness and existence of solutions, as well as a stability result. The comparison theorem (from which uniqueness follows) and the special comparison theorem are obtained through natural techniques and minimal assumptions. The existence is based on a perturbative procedure, and holds for a driver whis is Lipschitz, or slightly-superlinear, or monotone with arbitrary growth in y. Finally, we obtain a stability result, which gives in particular a local Lipschitz estimate in BMO for the martingale part of the solution. In the third and last part, we study the time-discretization of BSDEs having nonlinearities that are monotone but with polynomial growth in the primary variable. We show that in that case, the explicit Euler scheme is likely to diverge, while the implicit scheme converges. In fact, by studying the family of θ-schemes, which are mixed explicit-implicit, θ characterizing the degree of implicitness, we find that the scheme converges when the implicit component is dominant (θ ≥ 1/2 ). We then propose a tamed explicit scheme, which converges. We show that the implicit-dominant schemes with θ > 1/2 and our tamed explicit scheme converge with order 1/2 , while the trapezoidal scheme (θ = 1/2) converges with order 7/4.

A recent application of point processes has emerged from the electronic trading of financial assets. Many securities are now traded on purely electronic exchanges where demand and supply are aggregated in limit order books. Buy and sell trades in the asset as well as quote additions and cancellations can then be interpreted as events that not only determine the shape of the order book, but also define point processes that exhibit a rich internal structure. A large class of such point processes are those driven by a diffusive intensity process. A flexible choice with favourable analytic properties is a Cox-Ingersoll-Ross (CIR) diffusion. We adopt a Bayesian perspective on the statistical inference for these doubly stochastic processes, and focus on filtering the latent intensity process. We derive analytic results for the moment generating function of its posterior distribution. This is achieved by solving a partial differential equation for a linearised version of the filtering equation. We also establish an efficient and simple numerical evaluation of the posterior mean and variance of the intensity process. This relies on extending an equivalence result between a point process with CIR-intensity and a partially observed population process. We apply these results to empirical datasets from foreign exchange trading. One objective is to assess whether a CIR-driven point process is a satisfactory model for the variations in trading activity. This is answered in the negative, as sudden bursts of activity impair the fit of any diffusive intensity model. Controlling for such spikes, we conclude with a discussion of the stochastic control of a market making strategy when the only information available are the times of buy and sell trades.

Among all stochastic processes with independent increments, essentially only Brownian motion and Poisson process have a chaotic representation property. In the case of a Levy process, several approaches have been proposed in order to construct an orthogonal decomposition of the corresponding L2-space. In this dissertation, we deal with orthogonal (chaotic) decompositions for generalized processes with independent values. We do not suppose stationarity of the process, as a result the Levy measure of the process depends on points of the space. We first construct, in Chapter 3, a unitary isomorphism between a certain symmetric Fock space and the space L2 (D',mu). Here D' is a co-nuclear space of generalized functions (distributions), and mu is a generalized stochastic process with independent values. This isomorphism is constructed by employing the projection spectral theorem for an (uncountable) family of commuting self-adjoint operators. We next derive, in Chapter 4, a counterpart of the Nualart Schoutens decomposition for generalized stochastic process with independent values. Our results here extend those in the papers of Nualart Schoutens and Lytvynov. In Chapter 5, we construct an orthogonal decomposition of the space L2 (D',mu) in terms of orthogonal polynomials on D'. We observe a deep relation between this decomposition and the results of the two previous chapters. Finally, in Chapter 6, we briefly discuss the situation of the generalized stochastic processes of Meixner's type.

In this thesis I explore three new topics in Dynamical Systems. In Chapters 2 and 3 I investigate the dynamics of a family of asynchronous linear systems. These systems are of interest as models for asynchronous processes in economics and computer science and as novel ways to solve linear equations. I find a tight sandwich of bounds relating the Lyapunov exponents of these asynchronous systems to the eigenvalue of their synchronous counterparts. Using ideas from the theory of IFSs I show how the random behavior of these systems can be quickly sampled and go some way to characterizing the associated probability measures. In Chapter 4 I consider another family of random linear dynamical system but this time over the Max-plus semi-ring. These models provide a linear way to model essentially non-linear queueing systems. I show how the topology of the queue network impacts on the dynamics, in particular I relate an eigenvalue of the adjacency matrix to the throughput of the queue. In Chapter 5 I consider non-smooth systems which can be used to model a wide variety of physical systems in engineering as well as systems in control and computer science. I introduce the Moving Average Transformation which allows us to systematically 'smooth' these systems enabling us to apply standard techniques that rely on some smoothness, for example computing Lyapunov exponents from time series data.

In this thesis we discuss the synchronization of stochastic population oscillators in ecology and neuroscience. Traditionally, the synchronization of oscillators has been studied in deterministic systems with various modes of synchrony induced by coupling between the oscillators. However, recent developments have shown that an ensemble of uncoupled oscillators can be synchronized by a common noise source alone. By considering the effects of noise-induced synchronization on biological oscillators, we are able to explain various biological phenomena in ecological and neurobiological contexts - most importantly, the long-observed Moran effect. Our formulation of the systems as limit cycle oscillators arising from populations of individuals, each with a random element to its behaviour, also allows us to examine the interaction between an external noise source and this intrinsic stochasticity. This provides possible explanations as to why in ecological systems large-amplitude cycles may not be observed in the wild. In neural population oscillators, we were able to observe not just synchronization, but also clustering in some pa- rameter regimes. Finally, we are also able to extend our methods to include coupling in our models. In particular, we examine the competing effects of dispersal and extrinsic noise on the synchronization of a pair of Rosenzweig-Macarthur predator-prey systems. We discover that common environmental noise will ultimately synchronize the oscillators, but that the approach to synchrony depends on whether or not dispersal in the absence of noise supports any stable asynchronous states. We also show how the combination of correlated (shared) and uncorrelated (unshared) noise with dispersal can lead to a multistable steady-state probability density. Similar analysis on a coupled system of neural oscillators would be an interesting project for future work, which, among other future directions of research, is discussed in the concluding section of this thesis.

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.

The study of social network dynamics has become an increasingly important component of many disciplines in the social sciences. In the past decade, statistical models and methods have been proposed which permit researchers to draw statistical inference on these dynamics. This thesis builds on one such family of models, the stochastic actor oriented model (SAOM) proposed by Snijders [2001]. Goodness of fit for SAOMs is an area that is only just beginning to be filled in with appropriate methods. This thesis proposes a Mahalanobis distance based, Monte Carlo goodness of fit test that can depend on arbitrary features of the observed network data and covariates. As remediating poor fit can be a difficult process, a modified model distance (MMD) estimator is devised that can help researchers to choose among a set of model elaborations. In practice, panel data is typically used to draw SAOM-based inference. This thesis also proposes a score-type test for time heterogeneity between the waves in the panel that is computationally cheap and fits into a convenient, forward model selecting workflow. Next, this thesis proposes a rigorous method for aggregating so-called relational event data (e.g. emails and phone calls) by extending the SAOM family to a family of hidden Markov models that suppose a latent social network is driving the observed relational events. Finally, this thesis proposes a measurement model for SAOMs inspired by error-in-variables (EiV) models employed in an array of disciplines. Like the relational event aggregation model, the measurement model is a hidden Markov model extension to the SAOM family. These models allow the researcher to specify the form of the mesurement error and buffer against potential attenuating biases and other problems that can arise if the errors are ignored.

This thesis studies three models: Multi-type TASEP in discrete time, long-range lastpassage percolation on the line and convoy formation in a travelling servers model. All three models are relatively easy to state but they show a very rich and interesting behaviour. The TASEP is a basic model for a one-dimensional interacting particle system with non-reversible dynamics. We study some aspects of the TASEP in discrete time and compare the results to recently obtained results for the TASEP in continuous time. In particular we focus on stationary distributions for multi-type models, speeds of secondclass particles, collision probabilities and the speed process. We consider various natural update rules. The second model we study is directed last-passage percolation on the random graph G = (V,E) where V = ℤ and each edge (i,j), for i <: j ∈ , is present in E independently with some probability p ∈ (0,1]. To every (i,j) ∈ E we attach i.i.d. random weights vi,j > 0. We are interested in the behaviour of w0,n, which is the maximum weight of all directed paths from 0 to n, as n tends to infinity. We see two very different types of behaviour, depending on whether 𝔼[vi,j2] is finite or infinite. In the case where 𝔼[vi,j2] is finite we show that the process has a certain regenerative structure, and prove a strong law of large numbers and, under an extra assumption, a functional central limit theorem. In the situation where 𝔼[vi,j2] is infinite we obtain scaling laws and asymptotic distributions expressed in terms of a continuous last-passage percolation model on [0,1]. In the last model customers arrive on the non-negative half-line as a Poisson process of rate λ ∈ (0,∞) and n servers start at the origin at time 0. After completing some initial holding times, each server jumps to the first free customer it sees to its right and serves this customer. All service times are i.i.d. exponentials with parameter ν. After completion of a service the customer leaves the system and the server jumps to the next customer to the right, ignoring customers that are currently being served. We study the formation of convoys, groups of servers that travel together at the same speed, and the asymptotic behaviour of Xt(j), see position of server j at time t.

An active particle converts energy to motion. An active suspension is a population of active particles, typically microscale, that are immersed in a viscous and/or elastic medium. This thesis is about the statistics of active suspensions. Unlike a suspension at thermodynamic equilibrium, we show that an active suspension inherently has non-Gaussian fluctuations due to an interplay between self-driven constituents and microscopic physics. Consequently, the diffusion of a tracer in an active suspension is not Gaussian. Our results explain some experiments with active suspensions that contain either swimming microorganisms or molecular motors. We provide different models for the fluctuations in dilute active suspensions, ranging from phenomenological to exact. The fundamental ingredient of such non-Gaussian fluctuations is an ultraslow convergence to the central limit theorem caused by truncated power-laws. Without any truncation, there is an intimate relation to the generalized central limit theorem. We suggest similar effects occur in many other systems. These may be associated with probability distributions that appear to be exponential.

The past five years has seen the discovery of a wealth of genetics variants associated with an incredible range of diseases and traits that have been identified in genome- wide association studies (GWAS). These GWAS have typically been performed in in- dividuals of European descent, prompting a call for such studies to be conducted over a more diverse range of populations. These include groups such as African Ameri- cans and Latinos as they are recognised as bearing a disproportionately large burden of disease in the U.S. population. The variation in ancestry among such groups must be correctly accounted for in association studies to avoid spurious hits arising due to differences in ancestry between cases and controls. Such ancestral variation is not all problematic as it may also be exploited to uncover loci associated with disease in an approach known as admixture mapping, or to estimate recombination rates in admixed individuals. Many models have been proposed to infer genetic ancestry and they differ in their accuracy, the type of data they employ, their computational efficiency, and whether or not they can handle multi-way admixture. Despite the number of existing models, there is an unfulfilled requirement for a model that performs well even when the ancestral populations are closely related, is extendible to multi-way admixture scenarios, and can handle whole- genome data while remaining computationally efficient. In this thesis we present a novel method of ancestry estimation named MULTIMIX that satisfies these criteria. The underlying model we propose uses a multivariate nor- mal to approximate the distribution of a haplotype at a window of contiguous SNPs given the ancestral origin of that part of the genome. The observed allele types and the ancestry states that we aim to infer are incorporated in to a hidden Markov model to capture the correlations in ancestry that we expect to exist between neighbouring sites. We show via simulation studies that its performance on two-way and three-way admixture is competitive with state-of-the-art methods, and apply it to several real admixed samples of the International HapMap Project and the 1000 Genomes Project.

The subject of this thesis is the study of certain stochastic models arising in Population Genetics. The study of biological evolution naturally motivates the construction and use of sometimes sophisticated mathematical models. We contribute to the study of the so-called Lambda models. Our work is divided into two parts. In Part I, we study non-spatial models, introduced in 1999. Although there is a very rich literature concerning the description of genetic diversity thanks to the genealogies arising in these models, we obtain new results by considering the dynamics of the full population. We also contribute by presenting the first Bayesian method that allows us to reconstruct the genealogies generated by these models from data. In Part II, we study a recent extension of these models to the spatial setting. In particular, we prove a non trivial result concerning the geographical dispersal of a new mutant under this model.

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.

A statistical meta-analysis combines the results of several studies that address a set of related research hypotheses, thus increasing the power and reliability of the inference. Meta-analytic methods are over 50 years old and play an important role in science; pooling evidence from many trials to provide answers that any one trial would have insufficient samples to address. On the other hand, the number of neuroimaging studies is growing dramatically, with many of these publications containing conflicting results, or being based on only a small number of subjects. Hence there has been increasing interest in using meta-analysis methods to find consistent results for a specific functional task, or for predicting the results of a study that has not been performed directly. Current state of neuroimaging meta-analysis is limited to coordinate-based meta-analysis (CBMA), i.e., using only the coordinates of activation peaks that are reported by a group of studies, in order to "localize" the brain regions that respond to a certain type of stimulus. This class of meta-analysis suffers from a series of problems and hence cannot result in as accurate results as desired. In this research, we describe the problems that existing CBMA methods are suffering from and introduce a hierarchical mixed-effects image-based metaanalysis (IBMA) solution that incorporates the sufficient statistics (i.e., voxel-wise effect size and its associated uncertainty) from each study. In order to improve the statistical-inference stage of our proposed IBMA method, we introduce a nonparametric technique that is capable of adjusting such an inference for spatial nonstationarity. Given that in common practice, neuroimaging studies rarely provide the full image data, in an attempt to improve the existing CBMA techniques we introduce a fully automatic model-based approach that employs Gaussian-process regression (GPR) for estimating the meta-analytic statistic image from its corresponding sparse and noisy observations (i.e., the collected foci). To conclude, we introduce a new way to approach neuroimaging meta-analysis that enables the analysis to result in information such as “functional connectivity” and networks of the brain regions’ interactions, rather than just localizing the functions.

This dissertation is about some of the most important philosophical aspects of chaos research, a famous recent mathematical area of research about deterministic yet unpredictable and irregular, or even random behaviour. It consists of three parts. First, as a basis for the dissertation, I examine notions of unpredictability in ergodic theory, and I ask what they tell us about the justification and formulation of mathematical definitions. The main account of the actual practice of justifying mathematical definitions is Lakatos's account on proof-generated definitions. By investigating notions of unpredictability in ergodic theory, I present two previously unidentified but common ways of justifying definitions. Furthermore, I criticise Lakatos's account as being limited: it does not acknowledge the interrelationships between the different kinds of justification, and it ignores the fact that various kinds of justification - not only proof-generation - are important. Second, unpredictability is a central theme in chaos research, and it is widely claimed that chaotic systems exhibit a kind of unpredictability which is specific to chaos. However, I argue that the existing answers to the question "What is the unpredictability specific to chaos?" are wrong. I then go on to propose a novel answer, viz. the unpredictability specific to chaos is that for predicting any event all sufficiently past events are approximately probabilistically irrelevant. Third, given that chaotic systems are strongly unpredictable, one is led to ask: are deterministic and indeterministic descriptions observationally equivalent, i.e., do they give the same predictions? I treat this question for measure-theoretic deterministic systems and stochastic processes, both of which are ubiquitous in science. I discuss and formalise the notion of observational equivalence. By proving results in ergodic theory, I first show that for many measure-preserving deterministic descriptions there is an observationally equivalent indeterministic description, and that for all indeterministic descriptions there is an observationally equivalent deterministic description. I go on to show that strongly chaotic systems are even observationally equivalent to some of the most random stochastic processes encountered in science. For instance, strongly chaotic systems give the same predictions at every observation level as Markov processes or semi-Markov processes. All this illustrates that even kinds of deterministic and indeterministic descriptions which, intuitively, seem to give very different predictions are observationally equivalent. Finally, I criticise the claims in the previous philosophical literature on observational equivalence.

Nicotinic acid adenine dinucleotide phosphate (NAADP) is a potent Ca²⁺ mobilising messenger in mammalian and non-mammalian cells. Studies on a variety of cell types suggest that NAADP evokes Ca²⁺ release from a lysosome-related store and via activation of a receptor distinct from either ryanodine receptors (RyR) or inositol 1,4,5-trisphosphate (IP₃) receptors (IP₃R). However, the identity of the NAADP receptor has, until now, remained elusive. In this thesis I have shown that NAADP-evoked Ca²⁺ release from lysosomes is underpinned by two-pore channels (TPCs), of which there are 3 subtypes, TPC1, TPC2 and TPC3. When stably over-expressed in HEK293 cells, TPC2 was found to be specifically targeted to lysosomes, while TPC1 and TPC3 were targeted to endosomes. Initial Ca²⁺ signals via TPC2, but not those via TPC1, were amplified into global Ca²⁺ waves by Ca²⁺-induced Ca²⁺ release (CICR) from the endoplasmic reticulum (ER) via IP₃Rs. I have shown that, consistent with a role for TPCs in NAADP-mediated Ca²⁺ release, TPC2 is expressed in pulmonary arterial smooth muscle cells (PASMCs), is likely targeted to lysosomal membranes, and that TPCs also underpin NAADP-evoked Ca²⁺ signalling in this cell type. However, and in contrast to HEK293 cells, in PASMCs NAADP evokes spatially restricted Ca²⁺ bursts that are amplified into global Ca²⁺ waves by CICR from the sarcoplasmic reticulum (SR) via a subpopulation of RyRs, but not via IP₃Rs. I have demonstrated that lysosomes preferentially co-localise with RyR subtype 3 (RyR3) in the perinuclear region of PASMCs to comprise a “trigger zone” for Ca²⁺ signalling by NAADP, away from which a propagating Ca²⁺ wave may be carried by subsequent recruitment of RyR2. The identification of TPCs as a family of NAADP receptors may further our understanding of the mechanisms that confer the versatility of Ca²⁺ signalling which is required to regulate such diverse cellular functions as gene expression, fertilization, cell growth, and ultimately cell death.

In this thesis we establish a relationship between the Potential Approach to interest rates and the Market Models. This relationship allows us to derive the dynamics of forward LIBOR rates and forward swap rates by modelling the state price density. It means that we are able to secure the arbitrage-free condition and positive interest rate feature when we model the volatility drifts of those dynamics. On the other hand, we develop the Potential Approach, particularly the Hughston-Rafailidis Chaotic Interest Rate Model. The early argument enables us to infer that the Chaos Models belong to the Stochastic Volatility Market Models. In particular, we propose One-variable Chaos Models with the application of exponential polynomials. This maintains the generality of the Chaos Models and performs well for yield curves comparing with the Nelson-Siegel Form and the Svensson Form. Moreover, we calibrate the One-variable Chaos Model to European Caplets and European Swaptions. We show that the One-variable Chaos Models can reproduce the humped shape of the term structure of caplet volatility and also the volatility smile/skew curve. The calibration errors are small compared with the Lognormal Forward LIBOR Model, the SABR Model, traditional Short Rate Models, and other models under the Potential Approach. After the calibration, we introduce some new interest rate models under the Potential Approach. In particular, we suggest a new framework where the volatility drifts can be indirectly modelled from the short rate via the state price density.

We first review recent work on stable and multistable random processes and their localisability. Then most of the thesis concerns a new approach to these topics based on characteristic functions. Our aim is to construct processes on R, which are α(x)-multistable, where the stability index α(x) varies with x. To do this we first use characteristic functions to define α(x)-multistable random integrals and measures and examine their properties. We show that an α(x)-multistable random measure may be obtained as the limit of a sequence of measures made up of α-stable random measures restricted to small intervals with α constant on each interval. We then use the multistable random integrals to define multistable random processes on R and study the localisability of these processes. Thus we find conditions that ensure that a process locally ‘looks like’ a given stochastic process under enlargement and appropriate scaling. We give many examples of multistable random processes and examine their local forms. Finally, we examine the dimensions of graphs of α-stable random functions defined by series with α-stable random variables as coefficients.

Mergers are the combining of two or more firms to create synergies. These synergies may come from various sources such as operational synergies come from economies of scale or financial synergies come from increased value of securities of the firm. There are vast amount of studies analysing operational synergies of mergers. This study analyses the financial ones. This way the dynamics of purely financial synergies can be revealed. Purely financial synergies can be transformed into financial instruments such as securitization. While analysing financial synergies the puzzle of distribution of financial synergies between claimholders is investigated. Previous literature on mergers showed that bondholders may gain more than existing shareholders of the merging firms. This may become rather controversial. A merger may be synergistic but it does not necessarily mean that shareholders' wealth will increase. Managers and/or shareholders are the parties making the merger decision. If managers are acting to the best interest of shareholders then they would try to increase shareholders' wealth. To solve this problem first the dynamics of mergers were analysed and then new strategies developed and demonstrated to transfer the financial synergies to the shareholders.

From a clinical perspective retinal vascular segmentation and analysis are important tasks in aiding quantification of vascular disease progression for such prevalent pathologies as diabetic retinopathy, arteriolosclerosis and hypertension. Combined with the emergence of inexpensive digital imaging, retinal fundus images are becoming increasingly available through public databases fuelling interest in retinal vessel research. Vessel segmentation is a challenging task which needs to fulfil many requirements: the accurate segmentation of both normal and pathological vessels; the extraction of vessels of different sizes from large high contrast to small low contrast; minimal user interaction; low computational requirements; and the potential for application among different imaging modalities. We demonstrate a novel and significant improvement on an emerging stochastic vessel segmentation technique, particle filtering, in terms of improved performance at vascular bifurcations and extensibility. An alternative deterministic approach is also presented in the form of a framework utilising morphological Tramline filtering and non-parametric windows pdf estimation. Results of the deterministic algorithm on retinal images match those of state-of-art unsupervised methods in terms of pixel accuracy. In analysing retinal vascular networks, an important initial step is to distinguish between arteries and veins in order to proceed with pathological metrics such as branching angle, diameter, length and arteriole to venule diameter ratio. Practical difficulties include the lack of intensity and textural differences between arteries and veins in all but the largest vessels and the obstruction of vessels and connectivity by low contrast or other vessels. To this end, an innovative Markov Chain Monte Carlo Metropolis-Hastings framework is formulated for the separation of vessel trees. It is subsequently applied to both synthetic and retinal image data with promising results.

Accurate estimation of environmental dose rate is essential for high-resolution trapped-charge dating. Beta and gamma emissions from simulated sediments containing radioactive uranium, thorium and potassium are modelled in contexts that are spatially heterogeneous. Dose rate was modelled using Monte Carlo radiation transport codes MCNP and PENELOPE. A number of key issues that affect dose rate evaluation are examined and updated corrections are calculated. Granular structures used for geometrical input into the models were simulated using randomly packed ellipsoids. The pair correlation function and chord length distributions were derived. The effects of water content on dose rate were modelled and compared with cavity theory. Apart from activity dilution, the variation of grain size or water content was shown to be significant for gamma radiations due to the transition from charged particle equilibrium. The standard correction for beta dose rate due to grain size was found to be satisfactory although sensitivity to grain shape and material should be taken into account. Dose rate modeling was applied to three dating studies of early human fossils: Skhul V, Israel skull; Hofmeyr, South Africa skull and the Forbes’ Quarry, Gibraltar skull. The spatial modelling was implemented using computerised tomographic (CT) images and dose rate found to be modified significantly by the presence of the skull in the sediment. Time evolution of the dose rate was examined for the latter two skulls and dates of 36±3ka (Hofmeyr) and 55-95ka (Forbes’ Quarry) were calculated.