School of Mathematics and Statistics - Theses
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ItemNo Preview AvailableThe mathematics of individual-based modelling: developing a realistic model of simple epithelial tissue( 2023-05)Simple epithelia are the functioning components of many tissues found throughout the body. However, they are susceptible to disruption, which can lead to diseases such as cancer, asthma, cardiac disease, and viral infections. Before we can understand how these diseases occur, we must first understand how these tissues are normally maintained. Individual-based modelling is one such way to study simple epithelia. This thesis aims to gain a better understanding of the mathematics and mechanisms that underpin realistic individual-based models. We use these findings to develop a realistic model of simple epithelia. This research consists of two key parts. The first part focuses on understanding the fundamental mathematical constructions of individual-based models. In this research, we investigate three individual-based models of tissue dynamics: Overlapping Spheres, Voronoi Tessellations and Vertex Models. We investigate how particular modelling assumptions made at the tissue and cell boundaries affects both tissue growth and tissue collision. We find that all models are sensitive to their boundary description, with Overlapping Spheres models being highly sensitive to evolutionary time-scale, tissue structures of Voronoi Tessellation models being highly sensitive to their tissue shape, and Vertex Models being the lest sensitive description. This research emphasises the importance of thorough mathematical understanding to undertake model selection for specific problems, as to ensure macroscopic tissue behaviours are not artefacts of model selection. Upon understanding the importance of model selection, we then consider the sensitivity of the Centre-based models of Overlapping Spheres and Voronoi Tessellation models. By investigating the models’ parameters, we demonstrate how they contain two independent time scales of tissue evolution. We also provide a guide for numerically solving the equations of motion and demonstrate how naive parameter choices can result in unstable behaviour. Finally, to ensure biologically realistic dynamics in the model, a degree of Brownian motion should be incorporated, unless a tissue maintains high cell renewal. After understanding the fundamental mathematics of individual-based models, the second part of this research introduces a novel three-dimensional model of simple epithelia. Our description of the simple epithelia is deformable and consists of multiple layers. The movement of cells within the tissue is regulated by minimising a bending potential, cell-cell adhesion, and cell viscosity. We demonstrate that this model is robust to tissue relaxation and dynamic homoeostasis while undergoing renewal. Lastly, we also show how the description is capable of maintaining the structure at dynamic homoeostasis under regeneration via cell migration and removal, and we show the model is comparable to that of a fixed geometry, without the need for the unrealistic limitations. Finally, we show how our novel model can describe tissues with curved surfaces, applying the model to describe spherical organoids under regimes of relaxation and renewal, showing that dynamic homoeostasis is maintained. We propose a novel extension that is capable of maintaining actively deforming structures, in specified regions within the tissue, to describe highly generalised tissue structures. We demonstrate that this extended model exhibits robustness under tissue relaxation and renewal, while undergoing active tissue deformations. Finally, we show our description of general simple epithelial can describe tissue regeneration via cell migration and removal, while undergoing active tissue deformations. The results and findings of this research will prove valuable to better understanding the mechanisms that contribute to simple epithelial tissue maintenance and homoeostasis within the human body.
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ItemEconomic and Social Aspects of Heterogeneous Assembly Lines( 2023-10)Assembly lines with heterogeneous resources are widely present in manufacturing industries. The procedures to build a broad range of products employ various skilled workers or robots equipped with diverse tools. In order to remain competitive, it is crucial for a company to efficiently meet the market demand and reduce expenses at the same time. Since resources such as facilities, workforce wages, machines, and tools are quite costly, this requirement gives rise to the need to design economically viable lines. Conversely, we must take a myriad of technological and physical factors into account as well. For instance, assembly lines might appear in straight or U-shaped layouts, have continuous/(a)synchronous paces, operate with two-sided or multi-manned stations, manufacture multiple products in a mixed-model fashion, and employ a specialised workforce with different capabilities. Furthermore, as demonstrated by the COVID-19 pandemic scenario, possessing some resiliency for quick adaptations is a desirable feature. In addition, taking good care of employees fosters job satisfaction and reduces ergonomic risks, avoiding unnecessary turnover. Finally, it is also manageable to positively contribute towards societal issues, such as integrating workers with disabilities among ``conventional'' or robotic ones. Therefore, this thesis proposes three problems to investigate the complexities of economic and social aspects found in assembly lines with resource and workforce heterogeneity: the Resource-Constrained Assembly Line Balancing Problem (RCALBP), the Assembly Line Worker Assignment and Rebalancing Problem (ALWARP), and the Multi-manned Assembly Line Worker Integration and Balancing Problem (MALWIBP) are herein studied and discussed. We develop Mixed-Integer Linear Programming (MILP) formulations for all three problems. They either aim at minimising the cycle time given limited resources or wages and facilities costs at the desired production rate. Moreover, a state-of-the-art survey on Benders Decomposition (BD) approaches applied to the Assembly Line Balancing Problem (ALBP) is also presented. In the RCALBP and ALWARP, we explore resource and workforce heterogeneity appearing in real-world industrial applications. The former minimises cycle time given a limited number of stations and resources, whilst the latter aims at preserving jobs while minimising labour costs. We consider dedicated and alternative resource types for tasks, take scenarios with falling demands into account, and impose regularity metrics on workload reductions. These problems are mathematically modelled within a MILP framework, and benchmark instances are solved in commercial solvers along with case studies. As both RCALBP and ALWARP instances have very restrictive constraints, we can optimally solve large cases with commercial solvers by making problem-specific adjustments in the parameter tuning, as well as incorporating strong valid inequalities, variable reduction, and lower bounding techniques into the formulation. However, when the problem grows too complex, we may have to resort to heuristic approaches to obtain near-optimal solutions in a reduced computational time. The MALWIBP examines the balancing of assembly lines with multi-manned stations running on a heterogeneous workforce. This union creates a highly combinatorial problem: we must further link the already coupled decisions on assigning tasks to heterogeneous workers and workers to stations with task scheduling assessments. Thus, two heuristic solution procedures are developed, which tackle the problem with a hierarchical decomposition approach, showing that multi-operated stations can reduce the assembly line's length even in the presence of a heterogeneous workforce. Lastly, inspired by the decomposition methods realm, a well-established exact strategy for solving large optimisation problems is surveyed. More specifically, a comprehensive literature review on applying classical and logic-based BD approaches to ALBP variations is inspected. As several literature contributions have recently employed BD algorithms to tackle ALBPs with practical extensions, this survey attempts to consolidate the current body of knowledge by providing a detailed literature review on each application's particular aspects and ideas. We summarise existing gaps to offer insights into the BD efficiency for combinatorial problems such as ALBPs from a managerial perspective and indicate a shift in the research trend. In concluding remarks, the contributions of the developed works are summarised and future research avenues are suggested for all problems.
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ItemParisian Ruin with Random Delays for Spectrally Negative Lévy Processes( 2023-12)This thesis is devoted to studying Parisian ruin problems for spectrally negative Levy processes. The thesis consists of six chapters. Chapter 1 is a general introduction to spectrally negative Levy processes. We give definitions, examples and general properties of such processes. We also review the basics of stochastic calculus. Our exposition in this chapter often presents sketches of arguments rather than rigorous proofs. To compensate for the lack of rigour, we included references to more specialised texts where the proofs of the cited/used results can be found. Chapter 2 is devoted to the scale functions of spectrally negative Levy processes. We give basic definitions, examples and general properties. We also discuss some applications of scale functions. Chapter 3 presents previously known results of Parisian ruin theory and some related topics. We give the definitions and well-known facts in these areas. Chapter 4 introduces a new interesting and natural extension to the Parisian ruin problem under the assumption that the risk reserve dynamics are given by a spectrally negative Levy process with trajectories of locally bounded variation. The novel feature of this extension is that the distributions of the lengths of the random implementation delays can depend on the deficit at the epochs when the risk reserve process turns negative, starting a new negative excursion. Moreover, this extension allows for the possibility of an immediate ruin when the deficit hits a certain subset. In this setting, we derive a closed-form expression for the Parisian ruin probability and the joint Laplace transform of the Parisian ruin time and the deficit at that time of ruin. Chapter 5 is devoted to extending the results obtained in Chapter 4 to the case of spectrally negative Levy processes with trajectories of unbounded variation. Chapter 6 deals with the Parisian ruin time with arbitrary delays being independent of the deficit for the compound Poisson risk model. We show that the absolute distance between two Parisian ruin probabilities with different delays is bounded from above by the Levy-Prokhorov distance between the distributions of the two delays multiplied by a function of the initial reserve value. This means that the Parisian ruin probability with an arbitrary delay distribution can be approximated by the Parisian ruin probability with delay windows following a (finite) mixture of Erlang distributions, and the latter probability admits a closed-form expression.
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ItemMathematical approaches to pattern formation in dermatology(University of Melbourne, 2005)
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ItemAspects of mixed longitudinal growth analysis(University of Melbourne, 2010)This thesis presents practical approaches to the analysis of mixed longitudinal growth data. Longitudinal studies of the human population are specifically designed to investigate changes over a limited age range in a characteristic which is measured repeatedly for each study participant. This type of data poses several methodological challenges. First, models for the analysis of longitudinal data must recognize the relationship between the observations taken from each study participant. The mixed nature of the data calls for the use of random effects and variance and correlation structures for the within group errors. Secondly, the models must be flexible enough so that they can be easily differentiated for the timing of the population growth spurts. And thirdly, longitudinal growth data of human subjects is more often than not affected by the missing data problem. In practice, the missing data mechanism needs to be understood and taken into consideration when fitting the models. These aspects of mixed longitudinal growth analysis are covered in detail in this thesis using a comprehensive data set of repeated measures of human height of hundreds of Melbourne school children ranging form the ages of 5 to 18 years.
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ItemPointwise axiomatic spectral theory in Banach algebras(University of Melbourne, 2008)
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ItemOn sample size determination for discrete data(University of Melbourne, 1993)
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ItemNo Preview AvailableModelling high-dimensional spatial-temporal data: a focus on nonstationary and nonlinear phenomena for a future-focused landslide early-warning system( 2023-11)Rainfall-induced landslides are seeing an increase as the by-products of climate change and present significant damage to the environment, society, and human lives. These landslides are nonstationary and nonlinear phenomena often recorded as high-dimensional vector time series manifesting spatiotemporal dependence. Modelling and forecasting landslides are difficult, with the challenges coming from the complexity of the underlying time series and the remote-sensing techniques used in obtaining these monitoring data. Also, these time series may be of irregular frequency and contain missing values. We tackle these challenges by developing statistical forecasting tools for a future-focused landslide early-warning system (LEWS). These forecasting tools include our developed statistical models characterising complex time series, dimension reduction for efficient dynamic data representation, and three complementary risk assessment prongs to turn the derived forecasts into early-warning predictions of slope failure. Our proposed models are based on a novel spatial dimension reduction technique called empirical dynamic quantiles (EDQ). The idea behind this technique is to use a small number of representative EDQ series from the observed time series to surmise the whole dataset. We then perform various statistical analyses based on these representative EDQ series which will be computationally feasible. The general form of our time series model combined two advanced econometric methods error-correction cointegration (ECC), vector autogregression (VAR) and a nonlinear function $\boldsymbol{c}(t)$ with the EDQ method named ECC-VAR-$\boldsymbol{c}(t)$-EDQ model. We use this model to deal with these high-dimensional, spatial-temporal dependent vector time series with nonstationary and nonlinear phenomena. For different purposes in practice, we provide two methods to estimate the nonlinear function $\boldsymbol{c}(t)$ and further improve the forecasting accuracy. One is the \emph{empirical function-based method} and the other is \emph{physical-based method}. Once the form of $\boldsymbol{c}(t)$ has been determined, we can use the generalised least square (GLS) to estimate the unknown parameters involved in this ECC-VAR-$\boldsymbol{c}(t)$-EDQ model after performing our developed nonlinear cointegration test. The above-mentioned model is in a situation where there are no missing values and for high-frequency data. To apply the general time series analysis for low-frequency data with some missing values, we develop a model that combines the stochastic differential equations (SDE) and Markov chain Monte Carlo (MCMC) approach with the EDQ technique named SDE-MCMC-EDQ model. The basic idea behind this model is that we can convert these low-frequency time series to high-frequency by introducing some additional data between every two consecutive observations implies the estimation of these unknown data in addition to the SDE model parameters, where both these imputed data and the parameters in SDE are treated as random variables. The reproducibility and robustness of all these developed models are assessed by the application of different real-world ground motion data or simulation studies. Results found well fitted these different slope data with the goodness of fit statistic ($0.94\leq R^2\leq0.99$) close to 1. In addition to the forecast values derived from our developed models, we use three risk assessments in parallel to predict where, when, and risk of failure for supporting a complete future-focused LEWS which more accessible to the public.
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ItemA class of smooth, possibly data-adaptive, nonparametric copula estimators and corresponding resampling techniques with applications( 2023-08)Copulas are mathematical tools for modeling the dependence between the components of a random vector. They are frequentely used in fields such as finance, economics, and risk management. Chapter 1 and 2 of this thesis provide a review of the main results in the study of copulas including their basic properties, estimation methods, the empirical copula processes and appropriate resampling schemes for the latter. Chapter 3 proposes a broad class of smooth, possibly data-adaptive nonparametric copula estimators that contains empirical Bernstein copulas introduced by Sancetta and Satchell (2004) (and thus the empirical beta copula proposed by Segers et al. (2017)). A specific subclass that performs uniformly better than the empirical beta copula in Monte Carlo experiments is identified. Furtheremore, conditions under which related sequential empirical copula processes converge weakly are provided. Chapter 4 proposes two resampling techniques for the class of estimators considered in Chapter 3. One technique builds up on the work of Kiriliouk et al. (2021) and can be used to bootstrap related empirical copula processes in the i.i.d. case. The other technique is a smooth extension of the dependent multiplier bootstrap proposed in Bucher and Kojadinovic (2016) and can be used to bootstrap related empirical copula processes in the sequential time series case. In addition, two classes of smooth estimators of the first-order partial derivatives of the copula are also theoretically and empirically studied. The last chapter discusses potential future research directions, such as applying the studied estimators and corresponding resampling techniques for change-point detection and inference in factor copula models.