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(2022) Zhang, Jun ZeThesisThis thesis investigates some properties of complex structures on Lie algebras. In particular, we focus on nilpotent complex structures that are characterized by a suitable Jinvariant 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 Jinvariant stratification on a step 2 nilpotent Lie algebra with a complex structure.

(2022) Biswas, Raaj KishoreThesisRearend crashes are a major part of road injury burden, accounting for onethird of all vehicletovehicle crashes in New South Wales, Australia. Close following or driving with short headways is a key cause, yet the role of driver behaviour in rearend crash risk is not well researched. The primary aim of this research was to develop a better understanding of rearend crashes by assessing headways on Australian roads and investigating driver behaviour and performance associated with close following in crash and noncrash 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 rearend casualty and multiple vehicle crashes. The modelling of these associated factors were confirmed as contributing factors in rearend 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 rearend crashes than other crash types. Rearend 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 rearend 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. Overthespeedlimit driving was observed in all headway scenarios, but especially in higher speed zones. The findings challenge the notion that rearend 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 rearend crash risk.

(2022) Han, BruceThesisThe Hawkes process was first proposed by Alan G. Hawkes in which the arrival of events exhibits a selfexciting 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 Bspline functions. We also derive the Rosenblatt residuals of the modulated renewal Hawkes process, which can serve as a basis for assessing the goodnessoffit 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 Bspline trend function is applied to model extreme intraday negative returns on several cryptocurrencies. The estimated trend function suggests an inverse Ushaped 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 goodnessoffit check and by the Akaike Information Criterion.

(2022) Yang, YuThesisResearch 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 aggregatedlevel 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 pseudomarginal 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 pseudomarginal 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 nonparametric 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 regularisationinduced confounding issue which is often observed in the direct application of BART in causal inference.

(2022) Rock, ChristopherThesisThis thesis investigates links between the eigenvalues and eigenfunctions of the LaplaceBeltrami 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 LaplaceBeltrami eigenvalues in terms of the corresponding higher Cheeger constant. The level sets of LaplaceBeltrami eigenfunctions sometimes reveal sets with small Cheeger ratio, representing wellseparated 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 softthresholding 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 LaplaceBeltrami 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 softthresholded 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 softthresholded linear combinations of the inputted eigenfunctions. We develop a new Weylinspired eigengap heuristic and heuristics based on the sparse basis vectors, suggesting how many eigenvectors to pass to our method.

(2022) Shahriari, SiroosThesisTime 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. Featurerich timeseries 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 decisionmaking, 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 timeseries 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 nonparametric method) for shortterm 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 ARMAGARCH 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 COVID19 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.

(2022) Akhymbek, MeiramThesisThe TrotterKato product formula is a mathematical clarification of path integration in quantum theory [62]. It gives a precise meaning to Feynman’s path integral representation of the solutions to Schrodinger equations with timedependent potentials. In this thesis, we consider the TrotterKato product formula in arbitrary symmetrically Fnormed ideal closed with respect to the logarithmic submajorization. An abstract nonautonomous evolution equation is widely used in various fields of mathematics and quantum mechanics. For example, Schrodinger equation and linear partial differential equations of parabolic or hyperbolic type [53, 70]. The second problem we consider is the existence of the propagator for such an equation and its approximation formula in an arbitrary symmetric Banach ideal. The approximation formula in the autonomous case corresponds to the Trotter product formula.

(2022) Ross, JamesThesisGraphs are combinatorial objects commonly used to model relationships between pairs of entities. Hypergraphs are a generalization of graphs in which edges connect an arbitrary number of vertices. We consider hypergraphs in which each edge has size k, each vertex has a degree specified by a degree sequence d, and all edges are unique. These are known as simple kuniform hypergraphs with degree sequence d. We focus on algorithms for sampling these hypergraphs, particularly when the degree sequence is approximately regular and sufficiently sparse. The goal is an algorithm which produces approximately uniform output with expected running time that is polynomial in the number of vertices. We first discuss an algorithm for this problem which used a rejection sampling approach and a blackbox bipartite graph sampler. This algorithm was presented in a paper by myself and coauthors: my specific contributions to the publication are described. As a new contribution (not contained in the paper), the rejection sampling approach is extended to give an algorithm for sampling linear hypergraphs, which are hypergraphs in which no two distinct edges share more than one common vertex. We also define and analyse an algorithm for sampling simple kuniform hypergraphs with degree sequence d. Our algorithm uses a blackbox sampler A for producing (possibly nonsimple) hypergraphs and a ‘switchings’ process to remove any repeated edges from the hypergraph. This analysis additionally produces explicit tail bounds for the number and multiplicity of repeated edges in uniformly distributed random hypergraphs, under certain conditions for d and k. We show that our algorithm is asymptotically approximately uniform and has an expected running time that is polynomial in the number of vertices for a large range of degree sequences d, provided d is nearregular. This extends the range of degree sequences for which efficient sampling schemes are known.

(2022) Robertson, GavinThesisIn this thesis we study certain roundness inequalities in metric spaces. The properties roundness and generalised roundness of metric spaces were originally introduced by Enflo where they were used to act as obstructions to uniform embeddings. Since then the relation of these properties to other embeddings such as isometric embeddings and coarse embeddings of metric spaces has been the subject of much study. A major result in this area is that generalised roundness, and the equivalent property of $p$negative type, can act as a sufficient condition for certain isometric embeddings into Euclidean space. In particular a finite metric space $(X,d_{X})$ embeds isometrically into some Euclidean space if and only if it has $2$negative type. We start by studying roundness in the setting of Banach spaces. Here we are able to unify and expand upon results of Enflo that aid in the calculation of the maximal roundness of many classical Banach spaces. We then consider the problem of computing the maximal roundness of more general Banach spaces whose unit spheres are easy to visualise but whose norm is more complicated to write down. In doing so we are led naturally to a slightly broader class of inequalities than that of the usual roundness. This new class of inequalities is then shown to have a close connection to the geometric concepts of smoothness and convexity. In the next section of this thesis we investigate the possibility of an analogous class of inequalities in the setting of generalised roundness, or equivalently $p$negative type. Our starting point is a theorem of Linial, London and Rabinovich which characterises those finite metric spaces that admit a bilipschitz embedding into some Euclidean space with a given amount of distortion. Using this we are able to define the new concept of distorted $p$negative type which we then show is a generalisation of the usual $p$negative type. Due to the theorem of Linial, London and Rabinovich the concept of distorted $p$negative type acts a sufficient condition for certain bilipschitz embeddings into Euclidean space. In particular we show that a finite metric space $(X,d_{X})$ admits a bilipschitz embedding into some Euclidean space with distortion at most $C$ if and only if it has $2$negative type with distortion $C$. We are also able to prove an analogous result for infinite metric spaces. We then proceed to generalise the properties of the usual $p$negative type such as strictness and polygonal equalities to this distorted setting. Explicit examples of finite metric spaces with distorted $p$negative type and examples of their distorted polygonal equalities are then given. Finally, we prove a certain linearisation of the theorem of Linial, London and Rabinovich which provides a complete characterisation of those Banach spaces that are linearly isomorphic to a Hilbert space with a given level of distortion.

(2022) Denes, MichaelThesisThe ocean is dominated by kinematic features, such as gyres, fronts, and mesoscale eddies, that persist for much longer than typical dynamical timescales. Due to their capacity to transport heat, salt, carbon, and other biogeochemical tracers over long distances, these coherent structures play an important role in climate, biology, and smallscale mixing. However, because of their Lagrangian (or flowfollowing) nature, identifying and tracking these features, and ultimately quantifying their contribution to transport processes, is challenging. In this thesis, we study transport and mixing in the ocean by coherent structures through the framework of finitetime coherent sets. This approach is motivated by a dynamic isoperimetric problem whose approximate solution is derived from the associated dynamic Laplace operator. Coherent sets describe regions of phase space that minimise mixing along their boundaries over a finite time window. They identify barriers to transport and provide the skeleton around which more complex or turbulent dynamics occurs. Chapter 2 introduces the formalism of the dynamic isoperimetric problem and associated dynamic Laplace operator and develops the necessary extensions for oceanographic applications. Chapter 3 examines the persistence and material coherence of a mesoscale ocean eddy in the South Atlantic Ocean and applies the framework to quantify, for the first time, transport by the outer ring of the eddy. Chapter 4 extends the framework to mixed boundary conditions, which we use to investigate material transport across circumpolar Southern Ocean fronts. We reveal a previously unobserved global pattern of alternating poleward and equatorward transport across Southern Ocean fronts, resulting from frontal meandering influenced by prominent seafloor obstacles. Chapter 5 adapts the framework to global and regional domains containing multiple coherent features and studies two applications: basinscale gyres and a train of ocean eddies. The results of this thesis demonstrate that the finitetime coherent set framework, used in conjunction with theoretical and numerical extensions developed here, is an effective method for examining the quasicoherent nature of ocean features embedded in incoherent, complex, and turbulent ocean flows. By extending and applying a rigorous mathematical framework, we shed light on previously unexplored transport processes in the global ocean. Our results motivate further study of longlived dynamical features from the Lagrangian perspective.