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.

Impact of simulated visual impairment on the cognitive test performance of young adults

Aims: This study investigated the effect of simulated visual impairment on the speed and accuracy of performance on a series of commonly used cognitive tests. ----- Methods: Cognitive performance was assessed for 30 young, visually normal subjects (M=22.0yrs ± 3.1 yrs) using the Digit Symbol Substitution Test (DSST), Trail Making Test (TMT) A and B and the Stroop Colour Word Test under three visual conditions: normal vision and two levels of visually degrading filters (VistechTM) administered in a random order. Distance visual acuity and contrast sensitivity were also assessed for each filter condition. ----- Results: The visual filters, which degraded contrast sensitivity to a greater extent than visual acuity, significantly increased the time to complete (p<0.05), but not the number of errors made, on the DSST and the TMT A and B and affected only some components of the Stroop test.----- Conclusions: Reduced contrast sensitivity had a marked effect on the speed but not the accuracy of performance on commonly used cognitive tests, even in young individuals; the implications of these findings are discussed.

Effect of amblyopia on the developmental eye movement test in children

Purpose. To investigate the functional impact of amblyopia in children, the performance of amblyopic and age-matched control children on a clinical test of eye movements was compared. The influence of visual factors on test outcome measures was explored. Methods. Eye movements were assessed with the Developmental Eye Movement (DEM) test, in a group of children with amblyopia (n = 39; age, 9.1 ± 0.9 years) of different causes (infantile esotropia, n = 7; acquired strabismus, n = 10; anisometropia, n = 8; mixed, n = 8; deprivation, n = 6) and in an age-matched control group (n = 42; age, 9.3 ± 0.4 years). LogMAR visual acuity (VA), stereoacuity, and refractive error were also recorded in both groups. Results. No significant difference was found between the amblyopic and age-matched control group for any of the outcome measures of the DEM (vertical time, horizontal time, number of errors and ratio(horizontal time/vertical time)). The DEM measures were not significantly related to VA in either eye, level of binocular function (stereoacuity), history of strabismus, or refractive error. Conclusions. The performance of amblyopic children on the DEM, a commonly used clinical measure of eye movements, has not previously been reported. Under habitual binocular viewing conditions, amblyopia has no effect on DEM outcome scores despite significant impairment of binocular vision and decreased VA in both the better and worse eye.

Type Test Elimination using Typeflow Analysis

Programs written in languages of the Oberon family usually contain runtime tests on the dynamic type of variables. In some cases it may be desirable to reduce the number of such tests. Typeflow analysis is a static method of determining bounds on the types that objects may possess at runtime. We show that this analysis is able to reduce the number of tests in certain plausible circumstances. Furthermore, the same analysis is able to detect certain program errors at compile time, which would normally only be detected at program execution. This paper introduces the concepts of typeflow analysis and details its use in the reduction of runtime overhead in Oberon-2.

Australian student reactions to U.S. tourism advertising : a test of advertising as public diplomacy

A study among Australian college students gauged their reactions to a television commercial produced for the U.S. Commerce Department to bolster sagging tourism numbers among international visitors. In addition to using traditional measures applied to tourism advertisements, the student also concluded items to measure attitudes toward the U.S. government and its people Pre- and post-viewing results indicated that while the Hollywood-movie-themed commercial was not well received by the Australian students as a tourism message, it did result in more favorable attitudes toward the U.S. government, though not the U.S. people. The findings lend partial support for the potential of tourism advertising efforts to exert a "bleed-over effect" in terms of their contributions to overall attitudes toward a country, regardless of whether viewers plan to visit the country whose travel advertisements they see.

Australian student reactions to US tourism advertising : a test of advertising as public diplomacy

A study among Australian college students gauged their reactions to a television commercial produced for the US Commerce Department to bolster sagging tourism numbers among international visitors. In additional to using traditional measures applied to tourism advertisements, the study also included items to measure attitudes toward the US government and its people. Pre- and post-viewing results indicated that although the Hollywood-movie-themes commercial was not well received by the Australian students as a tourism message, it did result in more favourable attitudes toward the US government, although not the US people. The findings lend partial support for the potential of tourism advertising efforts to exert a 'bleed-over effect' in terms of their contribution to overall attitudes toward a country, regardless of whether viewers plan to visit the country whose travel advertisements of which they see.

Simulation and enhancement of a cardiovascular device test rig

Cardiovascular assist devices are tested in mock circulation loops (MCLs) prior to animal and clinical testing. These MCLs rely on characteristics such as pneumatic parameters to create pressure and flow, and pipe dimensions to replicate the resistance, compliance and fluid inertia of the natural cardiovascular system. A mathematical simulation was developed in SIMULINK to simulate an existing MCL. Model validation was achieved by applying the physical MCL characteristics to the simulation and comparing the resulting pressure traces. These characteristics were subsequently altered to improve and thus predict the performance of a more accurate physical system. The simulation was successful in simulating the physical mock circulation loop, and proved to be a useful tool in the development of improved cardiovascular device test rigs.

Implementing change within a school science department: progressive and dissonant voices

The purpose of this study was to describe the teaching and leadership experiences of a science teacher who, as head of department, was preparing to introduce changes in the science department of an independent school in response to the requirements of the new junior science syllabus in Queensland, Australia. This teacher consented to classroom observations and interviews with the researchers where his beliefs about teaching practice and change were explored. Other science teachers at the school also were interviewed about their reactions to the planned changes. Interpretive analysis of the data provides an account of the complex interactions, negotiations, compromises, concessions, and trade-offs faced by the teacher during a period of education reform. Perceived barriers existing within the school that impeded proposed change are identified

Adiabatic compression testing - part II : background and approach to estimating severity of test methodology

Adiabatic compression testing of components in gaseous oxygen is a test method that is utilized worldwide and is commonly required to qualify a component for ignition tolerance under its intended service. This testing is required by many industry standards organizations and government agencies; however, a thorough evaluation of the test parameters and test system influences on the thermal energy produced during the test has not yet been performed. This paper presents a background for adiabatic compression testing and discusses an approach to estimating potential differences in the thermal profiles produced by different test laboratories. A “Thermal Profile Test Fixture” (TPTF) is described that is capable of measuring and characterizing the thermal energy for a typical pressure shock by any test system. The test systems at Wendell Hull & Associates, Inc. (WHA) in the USA and at the BAM Federal Institute for Materials Research and Testing in Germany are compared in this manner and some of the data obtained is presented. The paper also introduces a new way of comparing the test method to idealized processes to perform system-by-system comparisons. Thus, the paper introduces an “Idealized Severity Index” (ISI) of the thermal energy to characterize a rapid pressure surge. From the TPTF data a “Test Severity Index” (TSI) can also be calculated so that the thermal energies developed by different test systems can be compared to each other and to the ISI for the equivalent isentropic process. Finally, a “Service Severity Index” (SSI) is introduced to characterizing the thermal energy of actual service conditions. This paper is the second in a series of publications planned on the subject of adiabatic compression testing.

Improving automatic speaker verification using SVM techniques

Automatic recognition of people is an active field of research with important forensic and security applications. In these applications, it is not always possible for the subject to be in close proximity to the system. Voice represents a human behavioural trait which can be used to recognise people in such situations. Automatic Speaker Verification (ASV) is the process of verifying a persons identity through the analysis of their speech and enables recognition of a subject at a distance over a telephone channel { wired or wireless. A significant amount of research has focussed on the application of Gaussian mixture model (GMM) techniques to speaker verification systems providing state-of-the-art performance. GMM's are a type of generative classifier trained to model the probability distribution of the features used to represent a speaker. Recently introduced to the field of ASV research is the support vector machine (SVM). An SVM is a discriminative classifier requiring examples from both positive and negative classes to train a speaker model. The SVM is based on margin maximisation whereby a hyperplane attempts to separate classes in a high dimensional space. SVMs applied to the task of speaker verification have shown high potential, particularly when used to complement current GMM-based techniques in hybrid systems. This work aims to improve the performance of ASV systems using novel and innovative SVM-based techniques. Research was divided into three main themes: session variability compensation for SVMs; unsupervised model adaptation; and impostor dataset selection. The first theme investigated the differences between the GMM and SVM domains for the modelling of session variability | an aspect crucial for robust speaker verification. Techniques developed to improve the robustness of GMMbased classification were shown to bring about similar benefits to discriminative SVM classification through their integration in the hybrid GMM mean supervector SVM classifier. Further, the domains for the modelling of session variation were contrasted to find a number of common factors, however, the SVM-domain consistently provided marginally better session variation compensation. Minimal complementary information was found between the techniques due to the similarities in how they achieved their objectives. The second theme saw the proposal of a novel model for the purpose of session variation compensation in ASV systems. Continuous progressive model adaptation attempts to improve speaker models by retraining them after exploiting all encountered test utterances during normal use of the system. The introduction of the weight-based factor analysis model provided significant performance improvements of over 60% in an unsupervised scenario. SVM-based classification was then integrated into the progressive system providing further benefits in performance over the GMM counterpart. Analysis demonstrated that SVMs also hold several beneficial characteristics to the task of unsupervised model adaptation prompting further research in the area. In pursuing the final theme, an innovative background dataset selection technique was developed. This technique selects the most appropriate subset of examples from a large and diverse set of candidate impostor observations for use as the SVM background by exploiting the SVM training process. This selection was performed on a per-observation basis so as to overcome the shortcoming of the traditional heuristic-based approach to dataset selection. Results demonstrate the approach to provide performance improvements over both the use of the complete candidate dataset and the best heuristically-selected dataset whilst being only a fraction of the size. The refined dataset was also shown to generalise well to unseen corpora and be highly applicable to the selection of impostor cohorts required in alternate techniques for speaker verification.

The Federal circuit in Bilski : the machine-or-transformation test

In a much anticipated judgment, the Federal Circuit has sought to clarify the standards applicable in determining whether a claimed method constitutes patent-eligible subject matter. In Bilski, the Federal Circuit identified a test to determine whether a patentee has made claims that pre-empt the use of a fundamental principle or an abstract idea or whether those claims cover only a particular application of a fundamental principle or abstract idea. It held that the sole test for determining subject matter eligibility for a claimed process under § 101 is that: (1) it is tied to a particular machine or apparatus, or (2) it transforms a particular article into a different state or thing. The court termed this the “machine-or-transformation test.” In so doing it overruled its earlier State Street decision to the extent that it deemed its “useful, tangible and concrete result” test as inadequate to determine whether an alleged invention recites patent-eligible subject matter.

Learning approach and perception of learning context in economics education : a causality test

In the study of student learning literature, the traditional view holds that when students are faced with heavy workload, poor teaching, and content that they cannot relate to – important aspects of the learning context, they will more likely utilise the surface approach to learning due to stresses, lack of understanding and lack of perceived relevance of the content (Kreber, 2003; Lizzio, Wilson, & Simons, 2002; Ramdsen, 1989; Ramsden, 1992; Trigwell & Prosser, 1991; Vermunt, 2005). For example, in studies involving health and medical sciences students, courses that utilised student-centred, problem-based approaches to teaching and learning were found to elicit a deeper approach to learning than the teacher-centred, transmissive approach (Patel, Groen, & Norman, 1991; Sadlo & Richardson, 2003). It is generally accepted that the line of causation runs from the learning context (or rather students’ self reported data on the learning context) to students’ learning approaches. That is, it is the learning context as revealed by students’ self-reported data that elicit the associated learning behaviour. However, other research studies also found that the same teaching and learning environment can be perceived differently by different students. In a study of students’ perceptions of assessment requirements, Sambell and McDowell (1998) found that students “are active in the reconstruction of the messages and meanings of assessment” (p. 391), and their interpretations are greatly influenced by their past experiences and motivations. In a qualitative study of Hong Kong tertiary students, Kember (2004) found that students using the surface learning approach reported heavier workload than students using the deep learning approach. According to Kember if students learn by extracting meanings from the content and making connections, they will more likely see the higher order intentions embodied in the content and the high cognitive abilities being assessed. On the other hand, if they rote-learn for the graded task, they fail to see the hierarchical relationship in the content and to connect the information. These rote-learners will tend to see the assessment as requiring memorising and regurgitation of a large amount of unconnected knowledge, which explains why they experience a high workload. Kember (2004) thus postulate that it is the learning approach that influences how students perceive workload. Campbell and her colleagues made a similar observation in their interview study of secondary students’ perceptions of teaching in the same classroom (Campbell et al., 2001). The above discussions suggest that students’ learning approaches can influence their perceptions of assessment demands and other aspects of the learning context such as relevance of content and teaching effectiveness. In other words, perceptions of elements in the teaching and learning context are endogenously determined. This study attempted to investigate the causal relationships at the individual level between learning approaches and perceptions of the learning context in economics education. In this study, students’ learning approaches and their perceptions of the learning context were measured. The elements of the learning context investigated include: teaching effectiveness, workload and content. The authors are aware of existence of other elements of the learning context, such as generic skills, goal clarity and career preparation. These aspects, however, were not within the scope of this present study and were therefore not investigated.

Are organ donation communication decisions reasoned or reactive? A test of the utility of an augmented Theory of Planned Behaviour with the Prototype/Willingness Model

Objectives: To explore whether people's organ donation consent decisions occur via a reasoned and/or social reaction pathway. --------- Design: We examined prospectively students' and community members' decisions to register consent on a donor register and discuss organ donation wishes with family. --------- Method: Participants completed items assessing theory of planned behaviour (TPB; attitude, subjective norm, perceived behavioural control (PBC)), prototype/willingness model (PWM; donor prototype favourability/similarity, past behaviour), and proposed additional influences (moral norm, self-identity, recipient prototypes) for registering (N=339) and discussing (N=315) intentions/willingness. Participants self-reported their registering (N=177) and discussing (N=166) behaviour 1 month later. The utility of the (1) TPB, (2) PWM, (3) augmented TPB with PWM, and (4) augmented TPB with PWM and extensions was tested using structural equation modelling for registering and discussing intentions/willingness, and logistic regression for behaviour. --------- Results: While the TPB proved a more parsimonious model, fit indices suggested that the other proposed models offered viable options, explaining greater variance in communication intentions/willingness. The TPB, augmented TPB with PWM, and extended augmented TPB with PWM best explained registering and discussing decisions. The proposed and revised PWM also proved an adequate fit for discussing decisions. Respondents with stronger intentions (and PBC for registering) had a higher likelihood of registering and discussing. --------- Conclusions: People's decisions to communicate donation wishes may be better explained via a reasoned pathway (especially for registering); however, discussing involves more reactive elements. The role of moral norm, self-identity, and prototypes as influences predicting communication decisions were highlighted also.