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(2022)ThesisThis thesis investigates some properties of complex structures on Lie algebras. In particular, we focus on nilpotent complex structures that are characterized by a suitable J-invariant ascending or descending central series dj and dj respectively. In this thesis, we introduce a new descending series pj and use it to give proof of a new characterization of nilpotent complex structures. We examine also whether nilpotent complex structures on stratified Lie algebras preserve the strata. We find that there exists a J-invariant stratification on a step 2 nilpotent Lie algebra with a complex structure.
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(2020)ThesisThis thesis contains results about the distribution of integers with prescribed arithmetic structure and an application. These include a counting problem in Diophantine approximation, an asymptotic formula for the number of solutions to congruence's with certain arithmetic conditions, lower bounds on the number of smooth square-free integers in arithmetic progression, an estimate on the smallest square-full number in almost all residue classes modulo a prime, a relaxation of Goldbach's conjecture from the point of view of Ramare's local model, and lastly a refinement of the classical Burgess bound.
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(2020)ThesisIn this thesis we provide new results in additive combinatorics which in turn lead us to new bounds of certain exponential sums. We also use known bounds on exponential and character sums to give new results in additive combinatorics. Specifically we will see how bounds on some quantities from additive combinatorics appear naturally when trying to bound multilinear exponential sums. We then find applications to bounds of exponential sums of sparse polynomials. We also give new bounds for an analogue of the energy variant of the sum-product problem over arbitrary finite fields.
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(2020)ThesisThe linear response of a dynamical system refers to changes to properties of the system under small external perturbations. We consider two applications of linear response theory to dynamical systems. In the first application (covering two settings) we consider the optimal perturbation that (i) maximises the linear response of the equilibrium distribution of the system, (ii) maximises the linear response of the expectation of a specified observable and (iii) maximises the linear response of the rate of convergence of the system to the equilibrium distribution. We also consider problems (i) and (ii) in the time-dependent situation where the governing dynamics is not stationary. We initially solve these problems for finite-state Markov chains. We numerically apply the theory developed in the finite-state setting to stochastically perturbed dynamical systems, where the Markov chain is replaced by a matrix representation of an approximate annealed transfer operator for the random dynamical system. In the second setting, we consider problems (ii) and (iii) for Hilbert-Schmidt integral operators with stochastic kernels. By representing a deterministic dynamical system with additive noise as an integral operator, we develop theory to compute optimal map perturbations that address problems (ii) and (iii); we also provide numerical examples in this setting. The second application of linear response is to finite-time coherent sets. Finite-time coherent sets represent minimally mixing objects in general nonlinear dynamics and are spatially mobile features that are the most predictable in the medium term. Under a small parameter change to the dynamical system, one can ask about the rate of change of the location and shape of the coherent sets, and one can also ask about the mixing properties (how much more or less mixing) with respect to the parameter change. We answer these questions by developing linear response theory for the eigenfunctions of the dynamic Laplace operator, from which one readily obtains the linear response of the corresponding coherent sets. We construct efficient numerical methods based on a recent finite-element approach and provide numerical examples.
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(2021)ThesisThe central aim of this thesis is to construct a fuller and firmer mathematical foundation for the solutions to various classes of nonlinear differential equations than is currently available in the literature. This includes boundary value problems (BVPs) that involve ordinary differential equations, and initial value problems (IVPs) for fractional differential equations. In particular, we establish new conditions that guarantee the existence, uniqueness and approximation of solutions to second-order BVPs, third-order BVPs, and fourth-order BVPs for ordinary differential equations. The results enable us, in turn, to shed new light on problems from applied mathematics, engineering and physics, such as: the Emden and Thomas-Fermi equations; the bending of elastic beams through an application of our general theories; and laminar flow in channels with porous walls. We also ensure the existence, uniqueness and approximation of solutions to some IVPs for fractional differential equations. An understanding of the existence, uniqueness and approximation of solutions to these problems is fundamental from both pure and applied points of view. Our methods involve an analysis of nonlinear operators through fixed-point theory in new and interesting ways. Part of the novelty involves generating new conditions under which these operators are contractive, invariant and/or establishing new a priori bounds on potential solutions. As such, we draw on: Banach fixed- point theorem, Schauder fixed-point theorem, Rus's contraction mapping theorem, and a continuation theorem due to A. Granas and its constructive version known as continuation method for contractive maps. The ideas in this thesis break new ground at the intersection of pure and applied mathematics. Thus, this work will be of interest to those who are researching the theoretical aspects of differential equations, and those who are interested in better understanding their applications.
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(2022)ThesisRear-end crashes are a major part of road injury burden, accounting for one-third of all vehicle-to-vehicle crashes in New South Wales, Australia. Close following or driving with short headways is a key cause, yet the role of driver behaviour in rear-end crash risk is not well researched. The primary aim of this research was to develop a better understanding of rear-end crashes by assessing headways on Australian roads and investigating driver behaviour and performance associated with close following in crash and non-crash scenarios. Two systematic reviews of headway were conducted. First, a review of research on headway identified the need for a consistent and accurate definition of headway, so the thesis puts forward an improved definition. The second review identified the range of external factors that increase the risk of short headway and increase crash risk including speed, task engagement, lead vehicle type, traffic conditions, road characteristics, weather/visibility, drug use, driving fatigue, innovative lane markings, and various warning systems. These factors were then explored in New South Wales data on rear-end casualty and multiple vehicle crashes. The modelling of these associated factors were confirmed as contributing factors in rear-end crashes, congruent with the review of headway. Higher speed, free flowing traffic, volitional task engagement, low cue environments, and collision warning lead to longer headway. Despite lower fatalities, higher odds of injury were observed for rear-end crashes than other crash types. Rear-end crashes were more likely to lead to multiple vehicle crashes, which had a higher chance of fatality than other types of crashes. Finally, naturalistic driving study data was used to investigate headway during normal driving, exploring close following at different speeds and classifying potential risky driving at various headways. In 64 hrs accumulated across 2101 trips, short headways of under 1 s occurred in around 15% of driving. Common manoeuvres to avoid rear-end crashes when close following were changing lanes, or braking, almost always by the following driver. Headway was associated with both driver speed and posted speed limits, decreasing as posted speed limits increased. Over-the-speed-limit driving was observed in all headway scenarios, but especially in higher speed zones. The findings challenge the notion that rear-end crashes are less severe with low injuries. Road users should be made aware of how frequently safe headways are violated and severity of injury outcomes. Driver education, community engagement, application of driver assistance technology consistent with driver behaviour and safety campaigns need to focus on safer speed and headway management to reduce rear-end crash risk.
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(2022)ThesisThe Hawkes process was first proposed by Alan G. Hawkes in which the arrival of events exhibits a self-exciting behaviour. One extension of the classical Hawkes process is the renewal Hawkes process, which allows the underlying process for background events to be a renewal process, rather than the homogeneous Poisson process in the classical Hawkes process. The renewal Hawkes process is stationary in nature, so it is not suitable in situations where there are systematic trends in event occurrence rate. Therefore, in this thesis, we propose a renewal Hawkes process in which a trend function is employed to account for the systematic patterns in the event occurrence rate. We term the process the modulated renewal Hawkes process. Due to the lack of an explicit expression for the intensity process, likelihood evaluation for the modulated renewal Hawkes process model is not trivial. However, by modifying the likelihood evaluation algorithm for renewal Hawkes process in Chen & Stindl (2018), we are able to propose an algorithm to evaluate the exact likelihood of the modulated renewal Hawkes process model. The evaluated likelihood can then be maximised to obtain the maximum likelihood estimator (MLE) of the model parameters. We also propose a method to obtain fast and accurate approximations to the likelihood. In the case where a suitable parametric form of the trend function is not available, we approximate the trend function using B-spline functions. We also derive the Rosenblatt residuals of the modulated renewal Hawkes process, which can serve as a basis for assessing the goodness-of-fit of the model. Simulation experiments were conducted to assess the performance of the MLE of the modulated renewal Hawkes process with either exact or approximate likelihood evaluation, both in the parametric model and in the semiparametric model with an unspecified trend function. We also present an application of the modulated renewal Hawkes process model to the analysis of cryptocurrency data. The modulated renewal Hawkes process model with a B-spline trend function is applied to model extreme intraday negative returns on several cryptocurrencies. The estimated trend function suggests an inverse U-shaped trend in the intraday occurrence times of extreme negative returns on cryptocurrencies. We also compared the model fitting results with several simpler models, such as the nonstationary Hawkes process and the renewal Hawkes process. On most of the cryptocurrency data sets considered in this work, the modulated renewal Hawkes process was found to provide the best fit both by the Rosenblatt residuals based goodness-of-fit check and by the Akaike Information Criterion.
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(2022)ThesisResearch in computational statistics develops numerically efficient methods to estimate statistical models, with Monte Carlo algorithms a subset of such methods. This thesis develops novel Monte Carlo methods to solve three important problems in Bayesian statistics. For many complex models, it is prohibitively expensive to run simulation methods such as Markov chain Monte Carlo (MCMC) on the model directly when the likelihood function includes an intractable term or is computationally challenging in some other way. The first two topics investigate models having such likelihoods. The third topic proposes a novel model to solve a popular question in causal inference, which requires solving a computationally challenging problem. The first application is to symbolic data analysis, where classical data are summarised and represented as symbolic objects. The likelihood function of such aggregated-level data is often intractable as it usually includes a high dimensional integral with large exponents. Bayesian inference on symbolic data is carried out in the thesis by using a pseudo-marginal method, which replaces the likelihood function with its unbiased estimate. The second application is to doubly intractable models, where the likelihood includes an intractable normalising constant. The pseudo-marginal method is combined with the introduction of an auxiliary variable to obtain simulation consistent inference. The proposed algorithm offers a generic solution to a wider range of problems, where the existing methods are often impractical as the assumptions required for their application do not hold. The last application is to causal inference using Bayesian additive regression trees (BART), a non-parametric Bayesian regression technique. The likelihood function is complex as it is based on a sum of trees whose structures change dynamically with the MCMC iterates. An extension to BART is developed to estimate the heterogeneous treatment effect, aiming to overcome the regularisation-induced confounding issue which is often observed in the direct application of BART in causal inference.
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(2022)ThesisThis thesis investigates links between the eigenvalues and eigenfunctions of the Laplace-Beltrami operator, and the higher Cheeger constants of smooth Riemannian manifolds, possibly with boundary. The higher Cheeger constants give a loose description of the major geometric features of a manifold. We obtain a new lower bound on the negative Laplace-Beltrami eigenvalues in terms of the corresponding higher Cheeger constant. The level sets of Laplace-Beltrami eigenfunctions sometimes reveal sets with small Cheeger ratio, representing well-separated features of the manifold. Some manifolds have their major features entwined across several eigenfunctions, and no single eigenfunction contains all the major features. In this case, there may exist carefully chosen linear combinations of the eigenfunctions, each with large values on a single feature, and small values elsewhere. We can then apply a soft-thresholding operator to these linear combinations to obtain new functions, each supported on a single feature. We show that the Cheeger ratios of the level sets of these functions also give an upper bound on the Laplace-Beltrami eigenvalues. We extend these level set results to nonautonomous dynamical systems, and show that the dynamic Laplacian eigenfunctions reveal sets with small dynamic Cheeger ratios. In a later chapter, we propose a numerical method for identifying features represented in eigenvectors arising from spectral clustering methods when those features are not cleanly represented in a single eigenvector. This method provides explicit candidates for the soft-thresholded linear combinations of eigenfunctions mentioned above. Many data clustering techniques produce collections of orthogonal vectors (e.g. eigenvectors) which contain connectivity information about the dataset. This connectivity information must be disentangled by some secondary procedure. We propose a method for finding an approximate sparse basis for the space spanned by the leading eigenvectors, by applying thresholding to linear combinations of eigenvectors. Our procedure is natural, robust and efficient, and it provides soft-thresholded linear combinations of the inputted eigenfunctions. We develop a new Weyl-inspired eigengap heuristic and heuristics based on the sparse basis vectors, suggesting how many eigenvectors to pass to our method.
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(2022)ThesisTime series models are used to model, simulate, and forecast the behaviour of a phenomenon over time based on data recorded over consistent intervals. The digital era has resulted in data being captured and archived in unprecedented amounts, such that vast amounts of information are available for analysis. Feature-rich time-series datasets are one of the data sets that have become available due to the expanding trend of data collection technologies worldwide. With the application of time series analysis to support financial and managerial decision-making, the development and advancement of time series models in the transportation domain are unavoidable. As a result, this thesis redefines time series models for transportation planning use with the following three aims: (1) To combine parametric and bootstrapping techniques within time series models; (2) to develop a time series model capable of modelling both temporal and spatial dependencies in time-series data; and (3) to leverage the hierarchical Bayesian modelling paradigm to accommodate flexible representations of heterogeneity in data. The first main chapter introduces an ensemble of ARIMA models. It compares its performance against conventional ARIMA (a parametric method) and LSTM models (a non-parametric method) for short-term traffic volume prediction. The second main chapter introduces a copula time series model that describes correlations between variables through time and space. Temporal correlations are modelled by an ARMA-GARCH model which enables a modeller to describe heteroscedastic data. The copula model has a flexible correlation structure and is used to model spatial correlations with the ability to model nonlinear, tailed and asymmetric correlations. The third main chapter provides a Bayesian modelling framework to raise awareness about using hierarchical Bayesian approaches for transport time series data. In addition, this chapter presents a Bayesian copula model. The combination of the two models provides a fully Bayesian approach to modelling both temporal and spatial correlations. Compared with frequentist models, the proposed modelling structures can incorporate prior knowledge. In the fourth main chapter, the fully Bayesian model is used to investigate mobility patterns before, during and after the COVID-19 pandemic using social media data. A more focused analysis is conducted on the mobility patterns of Twitter users from different zones and land use types.