A Numerical Solution Using an Adaptively Preconditioned Lanczos Method for a Class of Linear Systems Related with the Fractional Poisson Equation

This study considers the solution of a class of linear systems related with the fractional Poisson equation (FPE) (−∇2)α/2φ=g(x,y) with nonhomogeneous boundary conditions on a bounded domain. A numerical approximation to FPE is derived using a matrix representation of the Laplacian to generate a linear system of equations with its matrix A raised to the fractional power α/2. The solution of the linear system then requires the action of the matrix function f(A)=A−α/2 on a vector b. For large, sparse, and symmetric positive definite matrices, the Lanczos approximation generates f(A)b≈β0Vmf(Tm)e1. This method works well when both the analytic grade of A with respect to b and the residual for the linear system are sufficiently small. Memory constraints often require restarting the Lanczos decomposition; however this is not straightforward in the context of matrix function approximation. In this paper, we use the idea of thick-restart and adaptive preconditioning for solving linear systems to improve convergence of the Lanczos approximation. We give an error bound for the new method and illustrate its role in solving FPE. Numerical results are provided to gauge the performance of the proposed method relative to exact analytic solutions.

Poisson, Poisson-gamma and zero-inflated regression models of motor vehicle crashes: balancing statistical fit and theory

There has been considerable research conducted over the last 20 years focused on predicting motor vehicle crashes on transportation facilities. The range of statistical models commonly applied includes binomial, Poisson, Poisson-gamma (or negative binomial), zero-inflated Poisson and negative binomial models (ZIP and ZINB), and multinomial probability models. Given the range of possible modeling approaches and the host of assumptions with each modeling approach, making an intelligent choice for modeling motor vehicle crash data is difficult. There is little discussion in the literature comparing different statistical modeling approaches, identifying which statistical models are most appropriate for modeling crash data, and providing a strong justification from basic crash principles. In the recent literature, it has been suggested that the motor vehicle crash process can successfully be modeled by assuming a dual-state data-generating process, which implies that entities (e.g., intersections, road segments, pedestrian crossings, etc.) exist in one of two states—perfectly safe and unsafe. As a result, the ZIP and ZINB are two models that have been applied to account for the preponderance of “excess” zeros frequently observed in crash count data. The objective of this study is to provide defensible guidance on how to appropriate model crash data. We first examine the motor vehicle crash process using theoretical principles and a basic understanding of the crash process. It is shown that the fundamental crash process follows a Bernoulli trial with unequal probability of independent events, also known as Poisson trials. We examine the evolution of statistical models as they apply to the motor vehicle crash process, and indicate how well they statistically approximate the crash process. We also present the theory behind dual-state process count models, and note why they have become popular for modeling crash data. A simulation experiment is then conducted to demonstrate how crash data give rise to “excess” zeros frequently observed in crash data. It is shown that the Poisson and other mixed probabilistic structures are approximations assumed for modeling the motor vehicle crash process. Furthermore, it is demonstrated that under certain (fairly common) circumstances excess zeros are observed—and that these circumstances arise from low exposure and/or inappropriate selection of time/space scales and not an underlying dual state process. In conclusion, carefully selecting the time/space scales for analysis, including an improved set of explanatory variables and/or unobserved heterogeneity effects in count regression models, or applying small-area statistical methods (observations with low exposure) represent the most defensible modeling approaches for datasets with a preponderance of zeros

Robust designs for Poisson regression models

We consider the problem of how to construct robust designs for Poisson regression models. An analytical expression is derived for robust designs for first-order Poisson regression models where uncertainty exists in the prior parameter estimates. Given certain constraints in the methodology, it may be necessary to extend the robust designs for implementation in practical experiments. With these extensions, our methodology constructs designs which perform similarly, in terms of estimation, to current techniques, and offers the solution in a more timely manner. We further apply this analytic result to cases where uncertainty exists in the linear predictor. The application of this methodology to practical design problems such as screening experiments is explored. Given the minimal prior knowledge that is usually available when conducting such experiments, it is recommended to derive designs robust across a variety of systems. However, incorporating such uncertainty into the design process can be a computationally intense exercise. Hence, our analytic approach is explored as an alternative.

Robust designs for Poisson regression models

We consider the problem of how to construct robust designs for Poisson regression models. An analytical expression is derived for robust designs for first-order Poisson regression models where uncertainty exists in the prior parameter estimates. Given certain constraints in the methodology, it may be necessary to extend the robust designs for implementation in practical experiments. With these extensions, our methodology constructs designs which perform similarly, in terms of estimation, to current techniques, and offers the solution in a more timely manner. We further apply this analytic result to cases where uncertainty exists in the linear predictor. The application of this methodology to practical design problems such as screening experiments is explored. Given the minimal prior knowledge that is usually available when conducting such experiments, it is recommended to derive designs robust across a variety of systems. However, incorporating such uncertainty into the design process can be a computationally intense exercise. Hence, our analytic approach is explored as an alternative.

Efficient solution of two-sided nonlinear space-fractional diffusion equations using fast Poisson preconditioners

We develop a fast Poisson preconditioner for the efficient numerical solution of a class of two-sided nonlinear space fractional diffusion equations in one and two dimensions using the method of lines. Using the shifted Gr¨unwald finite difference formulas to approximate the two-sided(i.e. the left and right Riemann-Liouville) fractional derivatives, the resulting semi-discrete nonlinear systems have dense Jacobian matrices owing to the non-local property of fractional derivatives. We employ a modern initial value problem solver utilising backward differentiation formulas and Jacobian-free Newton-Krylov methods to solve these systems. For efficient performance of the Jacobianfree Newton-Krylov method it is essential to apply an effective preconditioner to accelerate the convergence of the linear iterative solver. The key contribution of our work is to generalise the fast Poisson preconditioner, widely used for integer-order diffusion equations, so that it applies to the two-sided space fractional diffusion equation. A number of numerical experiments are presented to demonstrate the effectiveness of the preconditioner and the overall solution strategy.

Space-time variation of malaria incidence in Yunnan province, China

Abstract Background Understanding spatio-temporal variation in malaria incidence provides a basis for effective disease control planning and monitoring. Methods Monthly surveillance data between 1991 and 2006 for Plasmodium vivax and Plasmodium falciparum malaria across 128 counties were assembled for Yunnan, a province of China with one of the highest burdens of malaria. County-level Bayesian Poisson regression models of incidence were constructed, with effects for rainfall, maximum temperature and temporal trend. The model also allowed for spatial variation in county-level incidence and temporal trend, and dependence between incidence in June–September and the preceding January–February. Results Models revealed strong associations between malaria incidence and both rainfall and maximum temperature. There was a significant association between incidence in June–September and the preceding January–February. Raw standardised morbidity ratios showed a high incidence in some counties bordering Myanmar, Laos and Vietnam, and counties in the Red River valley. Clusters of counties in south-western and northern Yunnan were identified that had high incidence not explained by climate. The overall trend in incidence decreased, but there was significant variation between counties. Conclusion Dependence between incidence in summer and the preceding January–February suggests a role of intrinsic host-pathogen dynamics. Incidence during the summer peak might be predictable based on incidence in January–February, facilitating malaria control planning, scaled months in advance to the magnitude of the summer malaria burden. Heterogeneities in county-level temporal trends suggest that reductions in the burden of malaria have been unevenly distributed throughout the province.

Determinants of pre-hospital care non-usage for patients with emergency care needs [including commentary by Snooks H.]

Objective-To establish the demographic, health status and insurance determinants of pre-hospital ambulance non-usage for patients with emergency medical needs. Methods-Triage category, date of birth, sex, marital status, country of origin, method and time of arrival, ambulance insurance status, diagnosis, and disposal were collected for all patients who presented over a four month period (n=10 229) to the emergency department of a major provincial hospital. Data for patients with urgent (n=678) or critical care needs (n=332) who did not use pre-hospital care were analysed using Poisson regression. Results-Only a small percentage (6.6%) of the total sample were triaged as having urgent medical needs or critical care needs (3.2%). Predictors of usage for those with urgent care needs included age greater than 65 years (prevalence ratio (PR)=0.54; 95% confidence interval (CI)= 0.35 to 0.83), being admitted to intensive care or transferred to another hospital (PR=0.62; 95% CI=0.44 to 0.89) or ward (PR=0.72; 95% CI=0.56 to 0.93) and ambulance insurance status (PR=0.67; 95% CI=052 to 0.86). Sex, marital status, time of day and country of origin were not predictive of usage and non-usage. Predictors of usage for those with critical care needs included age 65 years or greater (PR=0.45; 95% CI=0.25 to 0.81) and a diagnosis of trauma (PR=0.49; 95% CI=0.26 to 0.92). A non-English speaking background was predictive of non-usage (PR=1.98; 95% CI=1.06 to 3.70). Sex, marital status, time of day, triage and ambulance insurance status were not predictive of non-usage. Conclusions-Socioeconomic and medical factors variously influence ambulance usage depending on the severity or urgency of the medical condition. Ambulance insurance status was less of an influence as severity of condition increased suggesting that, at a critical level of urgency, patients without insurance are willing to pay for a pre-hospital ambulance service.

Vision, functional and cognitive determinants of motor vehicle incidents in older drivers

Background: The proportion of older individuals in the driving population is predicted to increase in the next 50 years. This has important implications for driving safety as abilities which are important for safe driving, such as vision (which accounts for the majority of the sensory input required for driving), processing ability and cognition have been shown to decline with age. The current methods employed for screening older drivers upon re-licensure are also vision based. This study, which investigated social, behavioural and professional aspects involved with older drivers, aimed to determine: (i) if the current visual standards in place for testing upon re-licensure are effective in reducing the older driver fatality rate in Australia; (ii) if the recommended visual standards are actually implemented as part of the testing procedures by Australian optometrists; and (iii) if there are other non-standardised tests which may be better at predicting the on-road incident-risk (including near misses and minor incidents) in older drivers than those tests recommended in the standards. Methods: For the first phase of the study, state-based age- and gender-stratified numbers of older driver fatalities for 2000-2003 were obtained from the Australian Transportation Safety Bureau database. Poisson regression analyses of fatality rates were considered by renewal frequency and jurisdiction (as separate models), adjusting for possible confounding variables of age, gender and year. For the second phase, all practising optometrists in Australia were surveyed on the vision tests they conduct in consultations relating to driving and their knowledge of vision requirements for older drivers. Finally, for the third phase of the study to investigate determinants of on-road incident risk, a stratified random sample of 600 Brisbane residents aged 60 years and were selected and invited to participate using an introductory letter explaining the project requirements. In order to capture the number and type of road incidents which occurred for each participant over 12 months (including near misses and minor incidents), an important component of the prospective research study was the development and validation of a driving diary. The diary was a tool in which incidents that occurred could be logged at that time (or very close in time to which they occurred) and thus, in comparison with relying on participant memory over time, recall bias of incident occurrence was minimised. Association between all visual tests, cognition and scores obtained for non-standard functional tests with retrospective and prospective incident occurrence was investigated. Results: In the first phase,rivers aged 60-69 years had a 33% lower fatality risk (Rate Ratio [RR] = 0.75, 95% CI 0.32-1.77) in states with vision testing upon re-licensure compared with states with no vision testing upon re-licensure, however, because the CIs are wide, crossing 1.00, this result should be regarded with caution. However, overall fatality rates and fatality rates for those aged 70 years and older (RR=1.17, CI 0.64-2.13) did not differ between states with and without license renewal procedures, indicating no apparent benefit in vision testing legislation. For the second phase of the study, nearly all optometrists measured visual acuity (VA) as part of a vision assessment for re-licensing, however, 20% of optometrists did not perform any visual field (VF) testing and only 20% routinely performed automated VF on older drivers, despite the standards for licensing advocating automated VF as part of the vision standard. This demonstrates the need for more effective communication between the policy makers and those responsible for carrying out the standards. It may also indicate that the overall higher driver fatality rate in jurisdictions with vision testing requirements is resultant as the tests recommended by the standards are only partially being conducted by optometrists. Hence a standardised protocol for the screening of older drivers for re-licensure across the nation must be established. The opinions of Australian optometrists with regard to the responsibility of reporting older drivers who fail to meet the licensing standards highlighted the conflict between maintaining patient confidentiality or upholding public safety. Mandatory reporting requirements of those drivers who fail to reach the standards necessary for driving would minimise potential conflict between the patient and their practitioner, and help maintain patient trust and goodwill. The final phase of the PhD program investigated the efficacy of vision, functional and cognitive tests to discriminate between at-risk and safe older drivers. Nearly 80% of the participants experienced an incident of some form over the prospective 12 months, with the total incident rate being 4.65/10 000 km. Sixty-three percent reported having a near miss and 28% had a minor incident. The results from the prospective diary study indicate that the current vision screening tests (VA and VF) used for re-licensure do not accurately predict older drivers who are at increased odds of having an on-road incident. However, the variation in visual measurements of the cohort was narrow, also affecting the results seen with the visual functon questionnaires. Hence a larger cohort with greater variability should be considered for a future study. A slightly lower cognitive level (as measured with the Mini-Mental State Examination [MMSE]) did show an association with incident involvement as did slower reaction time (RT), however the Useful-Field-of-View (UFOV) provided the most compelling results of the study. Cut-off values of UFOV processing (>23.3ms), divided attention (>113ms), selective attention (>258ms) and overall score (moderate/ high/ very high risk) were effective in determining older drivers at increased odds of having any on-road incident and the occurrence of minor incidents. Discussion: The results have shown that for the 60-69 year age-group, there is a potential benefit in testing vision upon licence renewal. However, overall fatality rates and fatality rates for those aged 70 years and older indicated no benefit in vision testing legislation and suggests a need for inclusion of screening tests which better predict on-road incidents. Although VA is routinely performed by Australian optometrists on older drivers renewing their licence, VF is not. Therefore there is a need for a protocol to be developed and administered which would result in standardised methods conducted throughout the nation for the screening of older drivers upon re-licensure. Communication between the community, policy makers and those conducting the protocol should be maximised. By implementing a standardised screening protocol which incorporates a level of mandatory reporting by the practitioner, the ethical dilemma of breaching patient confidentiality would also be resolved. The tests which should be included in this screening protocol, however, cannot solely be ones which have been implemented in the past. In this investigation, RT, MMSE and UFOV were shown to be better determinants of on-road incidents in older drivers than VA and VF, however, as previously mentioned, there was a lack of variability in visual status within the cohort. Nevertheless, it is the recommendation from this investigation, that subject to appropriate sensitivity and specificity being demonstrated in the future using a cohort with wider variation in vision, functional performance and cognition, these tests of cognition and information processing should be added to the current protocol for the screening of older drivers which may be conducted at licensing centres across the nation.

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.