Krylov subspace methods for approximating functions of symmetric positive definite matrices with applications to applied statistics and anomalous diffusion

Matrix function approximation is a current focus of worldwide interest and finds application in a variety of areas of applied mathematics and statistics. In this thesis we focus on the approximation of A^(-α/2)b, where A ∈ ℝ^(n×n) is a large, sparse symmetric positive definite matrix and b ∈ ℝ^n is a vector. In particular, we will focus on matrix function techniques for sampling from Gaussian Markov random fields in applied statistics and the solution of fractional-in-space partial differential equations. Gaussian Markov random fields (GMRFs) are multivariate normal random variables characterised by a sparse precision (inverse covariance) matrix. GMRFs are popular models in computational spatial statistics as the sparse structure can be exploited, typically through the use of the sparse Cholesky decomposition, to construct fast sampling methods. It is well known, however, that for sufficiently large problems, iterative methods for solving linear systems outperform direct methods. Fractional-in-space partial differential equations arise in models of processes undergoing anomalous diffusion. Unfortunately, as the fractional Laplacian is a non-local operator, numerical methods based on the direct discretisation of these equations typically requires the solution of dense linear systems, which is impractical for fine discretisations. In this thesis, novel applications of Krylov subspace approximations to matrix functions for both of these problems are investigated. Matrix functions arise when sampling from a GMRF by noting that the Cholesky decomposition A = LL^T is, essentially, a `square root' of the precision matrix A. Therefore, we can replace the usual sampling method, which forms x = L^(-T)z, with x = A^(-1/2)z, where z is a vector of independent and identically distributed standard normal random variables. Similarly, the matrix transfer technique can be used to build solutions to the fractional Poisson equation of the form ϕn = A^(-α/2)b, where A is the finite difference approximation to the Laplacian. Hence both applications require the approximation of f(A)b, where f(t) = t^(-α/2) and A is sparse. In this thesis we will compare the Lanczos approximation, the shift-and-invert Lanczos approximation, the extended Krylov subspace method, rational approximations and the restarted Lanczos approximation for approximating matrix functions of this form. A number of new and novel results are presented in this thesis. Firstly, we prove the convergence of the matrix transfer technique for the solution of the fractional Poisson equation and we give conditions by which the finite difference discretisation can be replaced by other methods for discretising the Laplacian. We then investigate a number of methods for approximating matrix functions of the form A^(-α/2)b and investigate stopping criteria for these methods. In particular, we derive a new method for restarting the Lanczos approximation to f(A)b. We then apply these techniques to the problem of sampling from a GMRF and construct a full suite of methods for sampling conditioned on linear constraints and approximating the likelihood. Finally, we consider the problem of sampling from a generalised Matern random field, which combines our techniques for solving fractional-in-space partial differential equations with our method for sampling from GMRFs.

Spatial analysis of notified cryptosporidiosis infections in Brisbane, Australia

Purpose: This study explored the spatial distribution of notified cryptosporidiosis cases and identified major socioeconomic factors associated with the transmission of cryptosporidiosis in Brisbane, Australia. Methods: We obtained the computerized data sets on the notified cryptosporidiosis cases and their key socioeconomic factors by statistical local area (SLA) in Brisbane for the period of 1996 to 2004 from the Queensland Department of Health and Australian Bureau of Statistics, respectively. We used spatial empirical Bayes rates smoothing to estimate the spatial distribution of cryptosporidiosis cases. A spatial classification and regression tree (CART) model was developed to explore the relationship between socioeconomic factors and the incidence rates of cryptosporidiosis. Results: Spatial empirical Bayes analysis reveals that the cryptosporidiosis infections were primarily concentrated in the northwest and southeast of Brisbane. A spatial CART model shows that the relative risk for cryptosporidiosis transmission was 2.4 when the value of the social economic index for areas (SEIFA) was over 1028 and the proportion of residents with low educational attainment in an SLA exceeded 8.8%. Conclusions: There was remarkable variation in spatial distribution of cryptosporidiosis infections in Brisbane. Spatial pattern of cryptosporidiosis seems to be associated with SEIFA and the proportion of residents with low education attainment.

Temperature variation and emergency hospital admissions for stroke in Brisbane, Australia, 1996-2005

Stroke is a leading cause of disability and death. This study evaluated the association between temperature variation and emergency admissions for stroke in Brisbane, Australia. Daily emergency admissions for stroke, meteorologic and air pollution data were obtained for the period of January 1996 to December 2005. The relative risk of emergency admissions for stroke was estimated with a generalized estimating equations (GEE) model. For primary intracerebral hemorrhage (PIH) emergency admissions, the average daily PIH for the group aged < 65 increased by 15% (95% Confidence Interval (CI): 5, 26%) and 12% (95% CI: 2, 22%) for a 1°C increase in daily maximum temperature and minimum temperature in summer, respectively, after controlling for potential confounding effects of humidity and air pollutants. For ischemic stroke (IS) emergency admissions, the average daily IS for the group aged ≥ 65 decreased by 3% (95% CI: -6, 0%) for a 1°C increase in daily maximum temperature in winter after adjustment for confounding factors. Temperature variation was significantly associated with emergency admissions for stroke, and its impact varied with different type of stroke. Health authorities should pay greater attention to possible increasing emergency care for strokes when temperature changes, in both summer and winter.

Diurnal variation of corneal shape and thickness

Purpose: To investigate associations between the diurnal variation in a range of corneal parameters, including anterior and posterior corneal topography, and regional corneal thickness. ----- Methods: Fifteen subjects had their corneas measured using a rotating Scheimpflug camera (Pentacam) every 3-7 hours over a 24-hour period. Anterior and posterior corneal axial curvature, pachymetry and anterior chamber depth were analysed. The best fitting corneal sphero-cylinder from the axial curvature, and the average corneal thickness for a series of different corneal regions were calculated. Intraocular pressure and axial length were also measured at each measurement session. Repeated measures ANOVA were used to investigate diurnal change in these parameters. Analysis of covariance was used to examine associations between the measured ocular parameters. ----- Results: Significant diurnal variation was found to occur in both the anterior and posterior corneal curvature and in the regional corneal thickness. Flattening of the anterior corneal best sphere was observed at the early morning measurement (p < 0.0001). The posterior cornea also underwent a significant steepening (p < 0.0001) and change in astigmatism 90/180° at this time. A significant swelling of the cornea (p < 0.0001) was also found to occur immediately after waking. Highly significant associations were found between the diurnal variation in corneal thickness and the changes in corneal curvature. ----- Conclusions: Significant diurnal variation occurs in the regional thickness and the shape of the anterior and posterior cornea. The largest changes in the cornea were typically evident upon waking. The observed non-uniform regional corneal thickness changes resulted in a steepening of the posterior cornea, and a flattening of the anterior cornea to occur at this time.

The malnutrition screening tool is a useful tool for identifying malnutrition risk in residential aged care

Background: The Malnutrition Screening Tool (MST) is a valid nutrition screening tool in the acute hospital setting but has not been assessed in residential aged care facilities. The aim of this secondary analysis was to determine whether the MST could be a useful nutrition screening tool when compared with a full nutrition assessment by Subjective Global Assessment (SGA) in the residential aged care setting. ----- Methods: Two hundred and eighty-five residents (29% male; mean age: 84 ± 9 years) from eight residential aged care facilities in Australia participated. A secondary analysis of data collected during a nutrition intervention study was conducted. The MST consists of two questions related to recent weight loss and appetite. While the MST was not specifically applied, weight loss and appetite information was available and an estimated MST score (0-5) calculated. Nutritional status was assessed by a research assistant trained in using SGA. ----- Results: Malnutrition prevalence was 42.8% (122 malnourished out of 285 residents). Compared to the SGA, the MST was an effective predictor of nutritional risk (sensitivity = 83.6%, specificity = 65.6%, positive predictive value = 0.65, negative predictive value =0.84). ----- Conclusions: The components of the MST have acceptable sensitivity and specificity suggesting it can play a valuable role in quickly identifying malnutrition risk in the residential aged care setting. Further prospective research using the MST tool against a broader array of objective and subjective nutritional parameters is required to confirm its validity as a screening tool in aged care settings.

The statistics of refractive error maps : managing wavefront aberration analysis without Zernike polynomials

The refractive error of a human eye varies across the pupil and therefore may be treated as a random variable. The probability distribution of this random variable provides a means for assessing the main refractive properties of the eye without the necessity of traditional functional representation of wavefront aberrations. To demonstrate this approach, the statistical properties of refractive error maps are investigated. Closed-form expressions are derived for the probability density function (PDF) and its statistical moments for the general case of rotationally-symmetric aberrations. A closed-form expression for a PDF for a general non-rotationally symmetric wavefront aberration is difficult to derive. However, for specific cases, such as astigmatism, a closed-form expression of the PDF can be obtained. Further, interpretation of the distribution of the refractive error map as well as its moments is provided for a range of wavefront aberrations measured in real eyes. These are evaluated using a kernel density and sample moments estimators. It is concluded that the refractive error domain allows non-functional analysis of wavefront aberrations based on simple statistics in the form of its sample moments. Clinicians may find this approach to wavefront analysis easier to interpret due to the clinical familiarity and intuitive appeal of refractive error maps.

Variation of the Stiles-Crawford effect with accommodation and myopia

Background: Mechanical forces either due to accommodation or myopia may stretch the retina and/or cause shear between the retina and choroid. This can be investigated by making use of the Stiles-Crawford effect (SCE), which is the phenomenon of light changing in apparent brightness as it enters through different positions in the pupil. The SCE can be measured by psychophysical and objective techniques, with the SCE parameters being directionality (rate of change across the pupil), and orientation (the location of peak sensitivity in the pupil). Aims: 1. To study the changes in foveal SCE with accommodation in emmetropes and myopes using a subjective (psychophysical) technique. 2. To develop and evaluate a quick objective technique of measuring the SCE using the multifocal electroretinogram. Methods: The SCE was measured in 6 young emmetropes and 6 young myopes for up to 8 D accommodation stimulus with a psychophysical technique and its variants. An objective technique using the multifocal electroretinogram was developed and evaluated with 5 emmetropes. Results: Using the psychophysical technique, the SCE directionality increased by similar amounts in both emmetropes and myopes as accommodation increased, with an increase of 15-20% with 6 D of accommodation. However, there were no significant orientation changes. Additional measurements showed that most of the change in the directionality was probably an artefact of optical factors such as higher-order aberrations and accommodative lag rather a true effect of accommodation. The multifocal technique demonstrated the presence of the SCE, but results were noisy and too variable to detect any changes in SCE directionality or orientation with accommodation. Conclusion: There is little true change in the SCE with accommodation responses up to 6 D in either emmetropes or myopes, although it is possible that substantial changes might occur at very high accommodation levels. The objective technique using the multifocal electroretinogram was quicker and less demanding for the subjects than the psychophysical technique, but as implemented in this thesis, it is not a reliable method of measuring the SCE.

Latitude variation in pancreatic cancer mortality in Australia

Objectives: Ecological studies support the hypothesis that there is an association between vitamin D and pancreatic cancer (PaCa) mortality, but observational studies are somewhat conflicting. We sought to contribute further data to this issue by analyzing the differences in PaCa mortality across the eastern states of Australia and investigating if there is a role of vitamin D-effective ultraviolet radiation (DUVR), which is related to latitude. ---------- Methods: Mortality data from 1968 to 2005 were sourced from the Australian General Record of Incidence and Mortality books. Negative binomial models were fitted to calculate the association between state and PaCa mortality. Clear sky monthly DUVR in each capital city was also modeled. ---------- Results: Mortality from PaCa was 10% higher in southern states than in Queensland, with those in Victoria recording the highest mortality risk (relative risk, 1.13; 95% confidence interval, 1.09-1.17). We found a highly significant association between DUVR and PaCa mortality, with an estimated 1.5% decrease in the risk per 10-kJ/m2 increase in yearly DUVR. ---------- Conclusions: These data show an association between latitude, DUVR, and PaCa mortality. Although this study cannot be used to infer causality, it supports the need for further investigations of a possible role of vitamin D in PaCa etiology.

Building the toolkit of quality monitoring clinical outcomes as a means of assessing quality in residential care

Residential aged care in Australia does not have a system of quality assessment related to clinical outcomes, creating a significant gap in quality monitoring. Clinical outcomes represent the results of all inputs into care, thus providing an indication of the success of those inputs. To fill this gap, an assessment tool based on resident outcomes (the ResCareQA) was developed and evaluated in collaboration with residential care providers. A useful output of the ResCareQA is a profile of resident clinical status, and this paper will use such outputs to present a snapshot of nine residential facilities. Such comprehensive data has not yet been available within Australia, so this will provide an important insight. ResCareQA data was collected from all residents (N=498) of nine aged care facilities from two major aged care providers. For each facility, numerator–denominator data were calculated to assess the degree of potential clinical problems. Results varied across clinical areas and across facilities, and rank-ordered facility results for selected clinical areas are reviewed and discussed. Use of the ResCareQA to generate clinical outcome data provides a concrete means of monitoring care quality within residential facilities; regular use of the ResCareQA could thus contribute to improved care outcomes within residential aged care.