Hierarchical models for 2D presence/absence data having ambiguous zeroes: With a biogeographical case study on dingo behaviour

This dissertation is primarily an applied statistical modelling investigation, motivated by a case study comprising real data and real questions. Theoretical questions on modelling and computation of normalization constants arose from pursuit of these data analytic questions. The essence of the thesis can be described as follows. Consider binary data observed on a two-dimensional lattice. A common problem with such data is the ambiguity of zeroes recorded. These may represent zero response given some threshold (presence) or that the threshold has not been triggered (absence). Suppose that the researcher wishes to estimate the effects of covariates on the binary responses, whilst taking into account underlying spatial variation, which is itself of some interest. This situation arises in many contexts and the dingo, cypress and toad case studies described in the motivation chapter are examples of this. Two main approaches to modelling and inference are investigated in this thesis. The first is frequentist and based on generalized linear models, with spatial variation modelled by using a block structure or by smoothing the residuals spatially. The EM algorithm can be used to obtain point estimates, coupled with bootstrapping or asymptotic MLE estimates for standard errors. The second approach is Bayesian and based on a three- or four-tier hierarchical model, comprising a logistic regression with covariates for the data layer, a binary Markov Random field (MRF) for the underlying spatial process, and suitable priors for parameters in these main models. The three-parameter autologistic model is a particular MRF of interest. Markov chain Monte Carlo (MCMC) methods comprising hybrid Metropolis/Gibbs samplers is suitable for computation in this situation. Model performance can be gauged by MCMC diagnostics. Model choice can be assessed by incorporating another tier in the modelling hierarchy. This requires evaluation of a normalization constant, a notoriously difficult problem. Difficulty with estimating the normalization constant for the MRF can be overcome by using a path integral approach, although this is a highly computationally intensive method. Different methods of estimating ratios of normalization constants (N Cs) are investigated, including importance sampling Monte Carlo (ISMC), dependent Monte Carlo based on MCMC simulations (MCMC), and reverse logistic regression (RLR). I develop an idea present though not fully developed in the literature, and propose the Integrated mean canonical statistic (IMCS) method for estimating log NC ratios for binary MRFs. The IMCS method falls within the framework of the newly identified path sampling methods of Gelman & Meng (1998) and outperforms ISMC, MCMC and RLR. It also does not rely on simplifying assumptions, such as ignoring spatio-temporal dependence in the process. A thorough investigation is made of the application of IMCS to the three-parameter Autologistic model. This work introduces background computations required for the full implementation of the four-tier model in Chapter 7. Two different extensions of the three-tier model to a four-tier version are investigated. The first extension incorporates temporal dependence in the underlying spatio-temporal process. The second extensions allows the successes and failures in the data layer to depend on time. The MCMC computational method is extended to incorporate the extra layer. A major contribution of the thesis is the development of a fully Bayesian approach to inference for these hierarchical models for the first time. Note: The author of this thesis has agreed to make it open access but invites people downloading the thesis to send her an email via the 'Contact Author' function.

Nonlinear diffusion and exclusion processes with contact interactions

Exclusion processes on a regular lattice are used to model many biological and physical systems at a discrete level. The average properties of an exclusion process may be described by a continuum model given by a partial differential equation. We combine a general class of contact interactions with an exclusion process. We determine that many different types of contact interactions at the agent-level always give rise to a nonlinear diffusion equation, with a vast variety of diffusion functions D(C). We find that these functions may be dependent on the chosen lattice and the defined neighborhood of the contact interactions. Mild to moderate contact interaction strength generally results in good agreement between discrete and continuum models, while strong interactions often show discrepancies between the two, particularly when D(C) takes on negative values. We present a measure to predict the goodness of fit between the discrete and continuous model, and thus the validity of the continuum description of a motile, contact-interacting population of agents. This work has implications for modeling cell motility and interpreting cell motility assays, giving the ability to incorporate biologically realistic cell-cell interactions and develop global measures of discrete microscopic data.

The Galerkin finite element approximation of the fractional cable equation

The cable equation is one of the most fundamental equations for modeling neuronal dynamics. Cable equations with a fractional order temporal derivative have been introduced to model electrotonic properties of spiny neuronal dendrites. In this paper, the fractional cable equation involving two integro-differential operators is considered. The Galerkin finite element approximations of the fractional cable equation are proposed. The main contribution of this work is outlined as follow: • A semi-discrete finite difference approximation in time is proposed. We prove that the scheme is unconditionally stable, and the numerical solution converges to the exact solution with order O(Δt). • A semi-discrete difference scheme for improving the order of convergence for solving the fractional cable equation is proposed, and the numerical solution converges to the exact solution with order O((Δt)2). • Based on the above semi-discrete difference approximations, Galerkin finite element approximations in space for a full discretization are also investigated. • Finally, some numerical results are given to demonstrate the theoretical analysis.

A preconditioned Lanczos method for space-fractional reaction-diffusion equations on two-dimensional unstructured meshes

We consider a two-dimensional space-fractional reaction diffusion equation with a fractional Laplacian operator and homogeneous Neumann boundary conditions. The finite volume method is used with the matrix transfer technique of Ilić et al. (2006) to discretise in space, yielding a system of equations that requires the action of a matrix function to solve at each timestep. Rather than form this matrix function explicitly, we use Krylov subspace techniques to approximate the action of this matrix function. Specifically, we apply the Lanczos method, after a suitable transformation of the problem to recover symmetry. To improve the convergence of this method, we utilise a preconditioner that deflates the smallest eigenvalues from the spectrum. We demonstrate the efficiency of our approach for a fractional Fisher’s equation on the unit disk.

Non-contact ACL injuries : association with clegg soil impact test values for sports fields, soil moisture and prevailing weather

Introduction: There is a recognised relationship between dry weather conditions and increased risk of anterior cruciate ligament (ACL) injury. Previous studies have identified 28 day evaporation as an important weather-based predictor of non-contact ACL injuries in professional Australian Football League matches. The mechanism of non-contact injury to the ACL is believed to increased traction and impact forces between footwear and playing surface. Ground hardness and the amount and quality of grass are factors that would most likely influence this and are inturn, related to the soil moisture content and prevailing weather conditions. This paper explores the relationship between soil moisture content, preceding weather conditions and the Clegg Soil Impact Test (CSIT) which is an internationally recognised standard measure of ground hardness for sports fields. Methodology: The 2.25 kg Clegg Soil Impact Test and a pair of 12 cm soil moisture probes were used to measure ground hardness and percentage moisture content. Five football fields were surveyed at 13 prescribed sites just before seven football matches from October 2008 to January 2009 (an FC Women’s WLeague team). Weather conditions recorded at the nearest weather station were obtained from the Bureau of Meteorology website and total rainfall less evaporation was calculated for 7 and 28 days prior to each match. All non-contact injuries occurring during match play and their location on the field were recorded. Results/conclusions: Ground hardness varied between CSIT 5 and 17 (x10G) (8 is considered a good value for sports fields). Variations within fields were typically greatest in the centre and goal areas. Soil moisture ranged from 3 to 40% with some fields requiring twice the moisture content of others to maintain similar CSIT values. There was a non-linear, negative relationship for ground hardness versus moisture content and a linear relationship with weather (R2, of 0.30 and 0.34, respectively). Three non-contact ACL injuries occurred during the season. Two of these were associated with hard and variable ground conditions.

Count-min sketches for estimating password frequency within Hamming distance two

The count-min sketch is a useful data structure for recording and estimating the frequency of string occurrences, such as passwords, in sub-linear space with high accuracy. However, it cannot be used to draw conclusions on groups of strings that are similar, for example close in Hamming distance. This paper introduces a variant of the count-min sketch which allows for estimating counts within a specified Hamming distance of the queried string. This variant can be used to prevent users from choosing popular passwords, like the original sketch, but it also allows for a more efficient method of analysing password statistics.

Simulating the effect of acoustic treatment types for residential balconies with road traffic noise

Development of design guides to estimate the difference in speech interference level due to road traffic noise between a reference position and balcony position or façade position is explored. A previously established and validated theoretical model incorporating direct, specular and diffuse reflection paths is used to create a database of results across a large number of scenarios. Nine balcony types with variable acoustic treatments are assessed to provide acoustic design guidance on optimised selection of balcony acoustic treatments based on location and street type. In total, the results database contains 9720 scenarios on which multivariate linear regression is conducted in order to derive an appropriate design guide equation. The best fit regression derived is a multivariable linear equation including modified exponential equations on each of nine deciding variables, (1) diffraction path difference, (2) ratio of total specular energy to direct energy, (3) distance loss between reference position and receiver position, (4) distance from source to balcony façade, (5) height of balcony floor above street, (6) balcony depth, (7) height of opposite buildings, (8) diffusion coefficient of buildings, and; (9) balcony average absorption. Overall, the regression correlation coefficient, R2, is 0.89 with 95% confidence standard error of ±3.4 dB.

Two-scale computational modelling of unsaturated flow in soils exhibiting small scale heterogeneities

Unsaturated water flow in soil is commonly modelled using Richards’ equation, which requires the hydraulic properties of the soil (e.g., porosity, hydraulic conductivity, etc.) to be characterised. Naturally occurring soils, however, are heterogeneous in nature, that is, they are composed of a number of interwoven homogeneous soils each with their own set of hydraulic properties. When the length scale of these soil heterogeneities is small, numerical solution of Richards’ equation is computationally impractical due to the immense effort and refinement required to mesh the actual heterogeneous geometry. A classic way forward is to use a macroscopic model, where the heterogeneous medium is replaced with a fictitious homogeneous medium, which attempts to give the average flow behaviour at the macroscopic scale (i.e., at a scale much larger than the scale of the heterogeneities). Using the homogenisation theory, a macroscopic equation can be derived that takes the form of Richards’ equation with effective parameters. A disadvantage of the macroscopic approach, however, is that it fails in cases when the assumption of local equilibrium does not hold. This limitation has seen the introduction of two-scale models that include at each point in the macroscopic domain an additional flow equation at the scale of the heterogeneities (microscopic scale). This report outlines a well-known two-scale model and contributes to the literature a number of important advances in its numerical implementation. These include the use of an unstructured control volume finite element method and image-based meshing techniques, that allow for irregular micro-scale geometries to be treated, and the use of an exponential time integration scheme that permits both scales to be resolved simultaneously in a completely coupled manner. Numerical comparisons against a classical macroscopic model confirm that only the two-scale model correctly captures the important features of the flow for a range of parameter values.

Investigating the sensitivity of OSLD responses for varying field sizes across matched linear accelerators

When radiation therapy centres are equipped with two or more linear accelerators from the same vendor, they are usually beam-matched. This work tested the sensitivity of optically stimulated luminescence dosimeters (OSLDs) across matched linear accelerators. The responses were compared with an unshielded diode detector for varying field sizes. Clinical studies are currently done with thermoluminescent dosimeters (TLD), which absorb radiation then emit some levels of light determined by the radiation absorption when heated.

Association between single-nucleotide polymorphisms in growth factor genes and quality of life in men with prostate cancer and the general population

Purpose Improved survival for men with prostate cancer has led to increased attention to factors influencing quality of life (QOL). As protein levels of vascular endothelial growth factor (VEGF) and insulin-like growth factor 1 (IGF-1) have been reported to be associated with QOL in people with cancer, we sought to identify whether single-nucleotide polymorphisms (SNPs) of these genes were associated with QOL in men with prostate cancer. Methods Multiple linear regression of two data sets (including approximately 750 men newly diagnosed with prostate cancer and 550 men from the general population) was used to investigate SNPs of VEGF and IGF-1 (10 SNPs in total) for associations with QOL (measured by the SF-36v2 health survey). Results Men with prostate cancer who carried the minor ‘T’ allele for IGF-1 SNP rs35767 had higher mean Role-Physical scale scores (≥0.3 SD) compared to non-carriers (p < 0.05). While this association was not identified in men from the general population, one IGF-1 SNP rs7965399 was associated with higher mean Bodily Pain scale scores in men from the general population that was not found in men with prostate cancer. Men from the general population who carried the rare ‘C’ allele had higher mean Bodily Pain scale scores (≥0.3 SD) than non-carriers (p < 0.05). Conclusions Through identifying SNPs that are associated with QOL in men with prostate cancer and men from the general population, this study adds to the mapping of complex interrelationships that influence QOL and suggests a role for IGF-I in physical QOL outcomes. Future research may identify biomarkers associated with increased risk of poor QOL that could assist in the provision of pre-emptive support for those identified at risk.

Fatigue Laws Via Functional Equations

In routine industrial design, fatigue life estimation is largely based on S-N curves and ad hoc cycle counting algorithms used with Miner's rule for predicting life under complex loading. However, there are well known deficiencies of the conventional approach. Of the many cumulative damage rules that have been proposed, Manson's Double Linear Damage Rule (DLDR) has been the most successful. Here we follow up, through comparisons with experimental data from many sources, on a new approach to empirical fatigue life estimation (A Constructive Empirical Theory for Metal Fatigue Under Block Cyclic Loading', Proceedings of the Royal Society A, in press). The basic modeling approach is first described: it depends on enforcing mathematical consistency between predictions of simple empirical models that include indeterminate functional forms, and published fatigue data from handbooks. This consistency is enforced through setting up and (with luck) solving a functional equation with three independent variables and six unknown functions. The model, after eliminating or identifying various parameters, retains three fitted parameters; for the experimental data available, one of these may be set to zero. On comparison against data from several different sources, with two fitted parameters, we find that our model works about as well as the DLDR and much better than Miner's rule. We finally discuss some ways in which the model might be used, beyond the scope of the DLDR.

Stability of a Random Diffusion with Linear Drift

We consider a Linear system with Markovian switching which is perturbed by Gaussian type noise, If the linear system is mean square stable then we show that under certain conditions the perturbed system is also stable, We also shaw that under certain conditions the linear system with Markovian switching can be stabilized by such noisy perturbation.

Unsteady Incompressible Two-Dimensional and Axisymmetric Turbulent Boundary Layer Flows

The unsteady turbulent incompressible boundary-layer flow over two-dimensional and axisymmetric bodies with pressure gradient has been studied. An eddy-viscosity model has been used to model the Reynolds shear stress. The unsteadiness is due to variations in the free stream velocity with time. The nonlinear partial differential equation with three independent variables governing the flow has been solved using Keller's Box method. The results indicate that the free stram velocity distribution exerts strong influence on the boundary-layer characteristics. The point of zero skin friction is found to move upstream as time increases.

Statistically steady turbulence in thin films: direct numerical simulations with Ekman friction

We present a detailed direct numerical simulation (DNS) of the two-dimensional Navier-Stokes equation with the incompressibility constraint and air-drag-induced Ekman friction; our DNS has been designed to investigate the combined effects of walls and such a friction on turbulence in forced thin films. We concentrate on the forward-cascade regime and show how to extract the isotropic parts of velocity and vorticity structure functions and hence the ratios of multiscaling exponents. We find that velocity structure functions display simple scaling, whereas their vorticity counterparts show multiscaling, and the probability distribution function of the Weiss parameter 3, which distinguishes between regions with centers and saddles, is in quantitative agreement with experiments.

Equilibrium studies of ammonium exchange with Australian natural zeolites

This paper is concerned with the study of the equilibrium exchange of ammonium ions with two natural zeolite samples sourced in Australia from Castle Mountain Zeolites and Zeolite Australia. A range of sorption models including Langmuir Vageler, Competitive Langmuir, Freundlich, Temkin, Dubinin Astakhov and Brouers–Sotolongo were applied in order to gain an insight as to the exchange process. In contrast to most previous studies, non-linear regression was used in all instances to determine the best fit of the experimental data. Castle Mountain natural zeolite was found to exhibit higher ammonium capacity than Zeolite Australia material when in the freshly received state, and this behavior was related to the greater amount of sodium ions present relative to calcium ions on the zeolite exchange sites. The zeolite capacity for ammonium ions was also found to be dependent on the solution normality, with 35–60% increase inuptake noted when increasing the ammonium concentration from 250 to 1000 mg/L. The optimal fit ofthe equilibrium data was achieved by the Freundlich expression as confirmed by use of Akaikes Information Criteria. It was emphasized that the bottle-point method chosen influenced the isotherm profile in several ways, and could lead to misleading interpretation of experiments, especially if the constant zeolite mass approach was followed. Pre-treatment of natural zeolite with acid and subsequently sodium hydroxide promoted the uptake of ammonium species by at least 90%. This paper highlighted the factors which should be taken into account when investigating ammonium ion exchange with natural zeolites.