School of Mathematics and Statistics - Theses
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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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ItemRisk Analysis and Probabilistic Decision Making for Censored Failure Data( 2019)Operation and maintenance of a fleet always require a high level of readiness, reduced cost, and improved safety. In order to achieve these goals, it is essential to develop and determine an appropriate maintenance programme for the components in use. A failure analysis involving failure model selection, robust parameter estimation, probabilistic decision making, and assessing the cost-effectiveness of the decisions are the key to the selection of a proper maintenance programme. Two significant challenges faced in failure analysis studies are, minimizing the uncertainty associated with model selection and making strategic decisions based on few observed failures. In this thesis, we try to resolve some of these problems and evaluate the cost-effectiveness of the selections. We focus on choosing the best model from a model space and robust estimation of quantiles leading to the selection of optimal repair and replacement time of units. We first explore the repair and replacement cost of a unit in a system. We design a simulation study to assess the performance of the parameter estimation methods, maximum likelihood estimation (MLE), and median rank regression method (MRR) in estimating quantiles of the Weibull distribution. Then, we compare the models; Weibull, gamma, log-normal, log-logistic, and inverse-Gaussian in failure analysis. With an example, we show that the Weibull and the gamma distributions provide competing fits to the failure data. Next, we demonstrate the use of Bayesian model averaging in accounting for that model uncertainty. We derive an average model for the failure observations with respective posterior model probabilities. Then, we illustrate the cost-effectiveness of the selected model by comparing the distribution of the total replacement and repair cost. In the second part of the thesis, we discuss the prior information. Initially, we assume, the parameters of the Weibull distribution are dependent by a function of the form rho = sigma/mu and re-parameterize the Weibull distribution. Then we propose a new Jeffreys’ prior for the parameters mu and rho. Finally, we designed a simulation study to assess the performance of the new Jeffreys’ prior compared to the MLE.
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ItemBiorthogonal Polynomial Sequences and the Asymmetric Simple Exclusion Process( 2019)The diffusion algebra equations of the stationary state of the three parameter Asymmetric Simple Exclusion Process are represented as a linear functional, acting on a tensor algebra. From the linear functional, a pair of sequences (P and Q) of monic polynomials are constructed which are bi-orthogonal, that is, they are orthogonal with respect to each other and not necessarily themselves. The uniqueness and existence of the pair of sequences arises from the determinant of the bi-moment matrix whose elements satisfy a pair of q-recurrence relations. The determinant is evaluated using an LDU-decomposition. If the action of the linear functional is represented as an inner product, then the action of the polynomials Q on a boundary vector V, generates a basis whose orthogonal dual vectors are given by the action of P on the dual boundary vector W}. This basis gives the representation of the algebra which is associated with the Al-Salam-Chihara polynomials obtained by Sasamoto. Several theorems associated with the three parameter asymmetric simple exclusion process are proven combinatorially. The theorems involve the linear functional which, for the three parameter case, is a substitution morphism on a q-Weyl algebra. The two polynomial sequences, P and Q, are represented in terms of q-binomial lattice paths. A combinatorial representation for the value of the linear functional defining the matrix elements of a bi-moment matrix is established in terms of the value of a q-rook polynomial and utilised to provide combinatorial proofs for results pertaining to the linear functional. Combinatorial proofs are provided for theorems in terms of the p,q-binomial coefficients, which are closely related to the combinatorics of the three parameter ASEP. The results for the three parameter diffusion algebra of the Asymmetric Simple Exclusion Process are extended to five parameters. A pair of basis changes are derived from the LDU decomposition of the bi-moment matrix. In order to derive the LDU decomposition a recurrence relation satisfied by the lower triangular matrix elements is conjectured. Associated with this pair of bases are three sequences of orthogonal polynomials. The first pair of orthogonal polynomials generate the new basis vectors (the boundary basis) by their action on the boundary vectors (written is the standard basis), whilst the third orthogonal polynomials are essentially the Askey-Wilson polynomials. All theses results are ultimately related to the LDU decomposition of a matrix.
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ItemStress testing mixed integer programming solvers through new test instance generation methods( 2019)Optimisation algorithms require careful tuning and analysis to perform well in practice. Their performance is strongly affected by algorithm parameter choices, software, and hardware and must be analysed empirically. To conduct such analysis, researchers and developers require high-quality libraries of test instances. Improving the diversity of these test sets is essential to driving the development of well-tested algorithms. This thesis is focused on producing synthetic test sets for Mixed Integer Programming (MIP) solvers. Synthetic data should be carefully designed to be unbiased, diverse with respect to measurable features of instances, have tunable properties to replicate real-world problems, and challenge the vast array of algorithms available. This thesis outlines a framework, methods and algorithms developed to ensure these requirements can be met with synthetically generated data for a given problem. Over many years of development, MIP solvers have become increasingly complex. Their overall performance depends on the interactions of many different components. To cope with this complexity, we propose several extensions over existing approaches to generating optimisation test cases. First, we develop alternative encodings for problem instances which restrict consideration to relevant instances. This approach provides more control over instance features and reduces the computational effort required when we have to resort to search-based generation approaches. Second, we consider more detailed performance metrics for MIP solvers in order to produce test cases which are not only challenging but from which useful insights can be gained. This work makes several key contributions: 1. Performance metrics are identified which are relevant to component algorithms in MIP solvers. This helps to define a more comprehensive performance metric space which looks beyond benchmarking statistics such as CPU time required to solve a problem. Using these more detailed performance metrics we aim to produce explainable and insightful predictions of algorithm performance in terms of instance features. 2. A framework is developed for encoding problem instances to support the design of new instance generators. The concepts of completeness and correctness defined in this framework guide the design process and ensure all problem instances of potential interest are captured in the scheme. Instance encodings can be generalised to develop search algorithms in problem space with the same guarantees as the generator. 3. Using this framework new generators are defined for LP and MIP instances which control feasibility and boundedness of the LP relaxation, and integer feasibility of the resulting MIP. Key features of the LP relaxation solution, which are directly controlled by the generator, are shown to affect problem difficulty in our analysis of the results. The encodings used to control these properties are extended into problem space search operators to generate further instances which discriminate between solver configurations. This work represents the early stages of an iterative methodology required to generate diverse test sets which continue to challenge the state of the art. The framework, algorithms and codes developed in this thesis are intended to support continuing development in this area.
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ItemIntelligent Management of Elective Surgery Patient Flow( 2019)Rapidly growing demand and soaring costs for healthcare services in Australia and across the world are jeopardising the sustainability of government-funded healthcare systems. We need to be innovative and more efficient in delivering healthcare services in order to keep the system sustainable. In this thesis, we utilise a number of scientific tools to improve the patient flow in a surgical suite of a hospital and subsequently develop a structured approach for intelligent patient flow management. First, we analyse and understand the patient flow process in a surgical suite. Then we obtain data from the partner hospital and extract valuable information from a large database. Next, we use machine learning techniques, such as classification and regression tree analysis, random forest, and k-nearest neighbour regression, to classify patients into lower variability resource user groups and fit discrete phase-type distributions to the clustered length of stay data. We use length of stay scenarios sampled from the fitted distributions in our sequential stochastic mixed-integer programming model for tactical master surgery scheduling. Our mixed-integer programming model has the particularity that the scenarios are utilised in a chronologically sequential manner, not in parallel. Moreover, we exploit the randomness in the sample path to reduce the requirement of optimising the process for many scenarios which helps us obtain high-quality schedules while keeping the problem algorithmically tractable. Last, we model the patient flow process in a healthcare facility as a stochastic process and develop a model to predict the probability of the healthcare facility exceeding capacity the next day as a function of the number of inpatients and the next day scheduled patients, their resource user groups, and their elapsed length of stay. We evaluate the model's performance using the receiver operating characteristic curve and illustrate the computation of the optimal threshold probability by using cost-benefit analysis that helps the hospital management make decisions.
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ItemCopula-based spatio-temporal modelling for count data( 2019)Modelling of spatio-temporal count data has received considerable attention in recent statistical research. However, the presence of massive correlation between locations, time points and variables imposes a great computational challenge. In existing literature, latent models under the Bayesian framework are predominately used. Despite numerous theoretical and practical advantages, likelihood analysis of spatio-temporal modelling on count data is less wide spread, due to the difficulty in identifying the general class of multivariate distributions for discrete responses. In this thesis, we propose a Gaussian copula regression model (copSTM) for the analysis of multivariate spatio-temporal data on lattice. Temporal effects are modelled through the conditional marginal expectations of the response variables using an observation-driven time series model, while spatial and cross-variable correlations are captured in a block dependence structure, allowing for both positive and negative correlations. The proposed copSTM model is flexible and sufficiently generalizable to many situations. We provide pairwise composite likelihood inference tools. Numerical examples suggest that the proposed composite likelihood estimator produces satisfactory estimation performance. While variable selection of generalized linear models is a well developed topic, model subsetting in applications of Gaussian copula models remains a relatively open research area. The main reason is the computational burden that is already quite heavy for simply fitting the model. It is therefore not computationally affordable to evaluate many candidate sub-models. This makes penalized likelihood approaches extremely inefficient because they need to search through different levels of penalty strength, apart from the fact suggested by our numerical experience that optimization of penalized composite likelihoods with many popular penalty terms (e.g LASSO and SCAD) usually does not converge in copula models. Thus, we propose to use a criterion-based selection approach that borrows strength from the Gibbs sampling technique.The methodology guarantees to converge to the model with the lowest criterion value, yet without searching through all possible models exhaustively. Finally, we present an R package implementing the estimation and selection of the copSTM model in C++. We show examples comparing our package to many available R packages (on some special cases of the copSTM), confirming the correctness and efficiency of the package functions. The package copSTM provides a competitive toolkit option for the analysis spatio-temporal count data on lattice in terms of both model flexibility and computational efficiency.
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ItemSingular vectors for the WN algebras and the BRST cohomology for relaxed highest-weight Lk(sl(2)) modules( 2019)This thesis presents the computation of singular vectors of the W_n algebras and the BRST cohomology of modules of the simple vertex operator algebra L_k(sl2) associated to the affine Lie algebra of sl2 in the relaxed category We will first recall some general theory on vertex operator algebras. We will then introduce the module categories that are relevant for conformal field theory. They are the category O of highest-weight modules and the relaxed category which contains O as well as the relaxed highest-weight modules with spectral flow and non-split extensions. We will then introduce the W_n algebras and the simple vertex operator algebra L_k(sl2). Properties of the Heisenberg algebra, the bosonic and the fermionic ghosts will be discussed as they are required in the free field realisations of W_n and L_k(sl2) as well as the construction of the BRST complex. We will then compute explicitly the singular vectors of W_n algebras in their Fock representations. In particular, singular vectors can be realised as the image of screening operators of the W_n algebras. One can then realise screening operators in terms of Jack functions when acting on a highest-weight state, thereby obtaining explicit formulae of the singular vectors in terms of symmetric functions. We will then discuss the BRST construction and the BRST cohomology for modules in category O. Lastly we compute the BRST cohomology for L_k(sl2) modules in the relaxed category. In particular, we compute the BRST cohomology for the highest-weight modules with positive spectral flow for all degrees and the BRST cohomology for the highest-weight modules with negative spectral flow for one degree.