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.

Zernike radial slope polynomials for wavefront reconstruction and refraction

Ophthalmic wavefront sensors typically measure wavefront slope, from which wavefront phase is reconstructed. We show that ophthalmic prescriptions (in power-vector format) can be obtained directly from slope measurements without wavefront reconstruction. This is achieved by fitting the measurement data with a new set of orthonormal basis functions called Zernike radial slope polynomials. Coefficients of this expansion can be used to specify the ophthalmic power vector using explicit formulas derived by a variety of methods. Zernike coefficients for wavefront error can be recovered from the coefficients of radial slope polynomials, thereby offering an alternative way to perform wavefront reconstruction.

Design of experiments for bivariate binary responses modelled by Copula functions

Optimal design for generalized linear models has primarily focused on univariate data. Often experiments are performed that have multiple dependent responses described by regression type models, and it is of interest and of value to design the experiment for all these responses. This requires a multivariate distribution underlying a pre-chosen model for the data. Here, we consider the design of experiments for bivariate binary data which are dependent. We explore Copula functions which provide a rich and flexible class of structures to derive joint distributions for bivariate binary data. We present methods for deriving optimal experimental designs for dependent bivariate binary data using Copulas, and demonstrate that, by including the dependence between responses in the design process, more efficient parameter estimates are obtained than by the usual practice of simply designing for a single variable only. Further, we investigate the robustness of designs with respect to initial parameter estimates and Copula function, and also show the performance of compound criteria within this bivariate binary setting.

Permutation polynomials of the form (xp −x +δ)s +L(x)

Recently, several classes of permutation polynomials of the form (x2 + x + δ)s + x over F2m have been discovered. They are related to Kloosterman sums. In this paper, the permutation behavior of polynomials of the form (xp − x + δ)s + L(x) over Fpm is investigated, where L(x) is a linearized polynomial with coefficients in Fp. Six classes of permutation polynomials on F2m are derived. Three classes of permutation polynomials over F3m are also presented.

Revisiting Polya's summation techniques using a spreadsheet : from addition tables to Bernoulli polynomials

Recurrence relations in mathematics form a very powerful and compact way of looking at a wide range of relationships. Traditionally, the concept of recurrence has often been a difficult one for the secondary teacher to convey to students. Closely related to the powerful proof technique of mathematical induction, recurrences are able to capture many relationships in formulas much simpler than so-called direct or closed formulas. In computer science, recursive coding often has a similar compactness property, and, perhaps not surprisingly, suffers from similar problems in the classroom as recurrences: the students often find both the basic concepts and practicalities elusive. Using models designed to illuminate the relevant principles for the students, we offer a range of examples which use the modern spreadsheet environment to powerfully illustrate the great expressive and computational power of recurrences.

Bivariate genome-wide association study of genetically correlated neuroimaging phenotypes from DTI and MRI through a seemingly unrelated regression model

Large multisite efforts (e.g., the ENIGMA Consortium), have shown that neuroimaging traits including tract integrity (from DTI fractional anisotropy, FA) and subcortical volumes (from T1-weighted scans) are highly heritable and promising phenotypes for discovering genetic variants associated with brain structure. However, genetic correlations (rg) among measures from these different modalities for mapping the human genome to the brain remain unknown. Discovering these correlations can help map genetic and neuroanatomical pathways implicated in development and inherited risk for disease. We use structural equation models and a twin design to find rg between pairs of phenotypes extracted from DTI and MRI scans. When controlling for intracranial volume, the caudate as well as related measures from the limbic system - hippocampal volume - showed high rg with the cingulum FA. Using an unrelated sample and a Seemingly Unrelated Regression model for bivariate analysis of this connection, we show that a multivariate GWAS approach may be more promising for genetic discovery than a univariate approach applied to each trait separately.

Some Remarks on the Approximation of Transitional Density in Stochastic Differential Equations

Aijt-Sahalia (2002) introduced a method to estimate transitional probability densities of di®usion processes by means of Hermite expansions with coe±cients determined by means of Taylor series. This note describes a numerical procedure to ¯nd these coe±cients based on the calculation of moments. One advantage of this procedure is that it can be used e®ectively when the mathematical operations required to ¯nd closed-form expressions for these coe±cients are otherwise infeasible.

Investigating the links between muscle strength, sun exposure, dietary vitamin D intake and the vitamin D status of ambulatory older adults in South East Queensland

Vitamin D deficiency and insufficiency are now seen as a contemporary health problem in Australia with possible widespread health effects not limited to bone health1. Despite this, the Vitamin D status (measured as serum 25-hydroxyvitamin D (25(OH)D)) of ambulatory adults has been overlooked in this country. Serum 25(OH)D status is especially important among this group as studies have shown a link between Vitamin D and fall risk in older adults2. Limited data also exists on the contributions of sun exposure via ultraviolet radiation and dietary intake to serum 25(OH)D status in this population. The aims of this project were to assess the serum 25(OH)D status of a group of older ambulatory adults in South East Queensland, to assess the association between their serum 25(OH)D status and functional measures as possible indicators of fall risk, obtain data on the sources of Vitamin D in this population and assess whether this intake was related to serum 25(OH)D status and describe sun protection and exposure behaviors in this group and investigate whether a relationship existed between these and serum 25(OH)D status. The collection of this data assists in addressing key gaps identified in the literature with regard to this population group and their Vitamin D status in Australia. A representative convenience sample of participants (N=47) over 55 years of age was recruited for this cross-sectional, exploratory study which was undertaken in December 2007 in south-east Queensland (Brisbane and Sunshine coast). Participants were required to complete a sun exposure questionnaire in addition to a Calcium and Vitamin D food frequency questionnaire. Timed up and go and handgrip dynamometry tests were used to examine functional capacity. Serum 25(OH)D status and blood measures of Calcium, Phosphorus and Albumin were determined through blood tests. The Mean and Median serum 25-Hydroxyvitamin D (25(OH)D) for all participants in this study was 85.8nmol/L (Standard Deviation 29.7nmol/L) and 81.0nmol/L (Range 22-158nmol/L), respectively. Analysis at the bivariate level revealed a statistically significant relationship between serum 25(OH)D status and location, with participants living on the Sunshine Coast having a mean serum 25(OH)D status 21.3nmol/L higher than participants living in Brisbane (p=0.014). While at the descriptive level there was an apparent trend towards higher outdoor exposure and increasing levels of serum 25(OH)D, no statistically significant associations between the sun measures of outdoor exposure, sun protection behaviors and phenotypic characteristics and serum 25(OH)D status were observed. Intake of both Calcium and Vitamin D was low in this sample with sixty-eight (68%) of participants not meeting the Estimated Average Requirements (EAR) for Calcium (Median=771.0mg; Range=218.0-2616.0mg), while eighty-seven (87%) did not meet the Adequate Intake for Vitamin D (Median=4.46ug; Range=0.13-30.0ug). This raises the question of how realistic meeting the new Adequate Intakes for Vitamin D is, when there is such a low level of Vitamin D fortification in this country. However, participants meeting the Adequate Intake (AI) for Vitamin D were observed to have a significantly higher serum 25(OH)D status compared to those not meeting the AI for Vitamin D (p=0.036), showing that meeting the AI for Vitamin D may play a significant role in determining Vitamin D status in this population. By stratifying our data by categories of outdoor exposure time, a trend was observed between increased importance of Vitamin D dietary intake as a possible determinant of serum 25(OH)D status in participants with lower outdoor exposures. While a trend towards higher Timed Up and Go scores in participants with higher 25(OH) D status was seen, this was only significant for females (p=0.014). Handgrip strength showed statistically significant association with serum 25(OH)D status. The high serum 25(OH)D status in our sample almost certainly explains the limited relationship between functional measures and serum 25(OH)D. However, the observation of an association between slower Time Up and Go speeds, and lower serum 25(OH)D levels, even with a small sample size, is significant as slower Timed Up and Go speeds have been associated with increased fall risk in older adults3. Multivariable regression analysis revealed Location as the only significant determinant of serum 25(OH)D status at p=0.014, with trends (p=>0.1) for higher serum 25(OH)D being shown for participants that met the AI for Vitamin D and rated themselves as having a higher health status. The results of this exploratory study show that 93.6% of participants had adequate 25(OH)D status-possibly due to measurement being taken in the summer season and the convenience nature of the sample. However, many participants do not meet their dietary Calcium and Vitamin D requirements, which may indicate inadequate intake of these nutrients in older Australians and a higher risk of osteoporosis. The relationship between serum 25(OH)D and functional measures in this population also requires further study, especially in older adults displaying Vitamin D insufficiency or deficiency.

An investigation into the barriers to the implementation of automation and robotics technologies in the construction industry

The rising problems associated with construction such as decreasing quality and productivity, labour shortages, occupational safety, and inferior working conditions have opened the possibility of more revolutionary solutions within the industry. One prospective option is in the implementation of innovative technologies such as automation and robotics, which has the potential to improve the industry in terms of productivity, safety and quality. The construction work site could, theoretically, be contained in a safer environment, with more efficient execution of the work, greater consistency of the outcome and higher level of control over the production process. By identifying the barriers to construction automation and robotics implementation in construction, and investigating ways in which to overcome them, contributions could be made in terms of better understanding and facilitating, where relevant, greater use of these technologies in the construction industry so as to promote its efficiency. This research aims to ascertain and explain the barriers to construction automation and robotics implementation by exploring and establishing the relationship between characteristics of the construction industry and attributes of existing construction automation and robotics technologies to level of usage and implementation in three selected countries; Japan, Australia and Malaysia. These three countries were chosen as their construction industry characteristics provide contrast in terms of culture, gross domestic product, technology application, organisational structure and labour policies. This research uses a mixed method approach of gathering data, both quantitative and qualitative, by employing a questionnaire survey and an interview schedule; using a wide range of sample from management through to on-site users, working in a range of small (less than AUD0.2million) to large companies (more than AUD500million), and involved in a broad range of business types and construction sectors. Detailed quantitative (statistical) and qualitative (content) data analysis is performed to provide a set of descriptions, relationships, and differences. The statistical tests selected for use include cross-tabulations, bivariate and multivariate analysis for investigating possible relationships between variables; and Kruskal-Wallis and Mann Whitney U test of independent samples for hypothesis testing and inferring the research sample to the construction industry population. Findings and conclusions arising from the research work which include the ranking schemes produced for four key areas of, the construction attributes on level of usage; barrier variables; differing levels of usage between countries; and future trends, have established a number of potential areas that could impact the level of implementation both globally and for individual countries.

What lies beneath? : The pattern and abundance of the subterranean tuber bank of the invasive liana cat’s claw creeper, Macfadyena unguis-cati (Bignoniaceae)

Cat’s claw creeper, Macfadyena unguis-cati (L.) Gentry (Bignoniaceae) is a major environmental weed of riparian areas, rainforest communities and remnant natural vegetation in coastal Queensland and New South Wales, Australia. In densely infested areas, it smothers standing vegetation, including large trees, and causes canopy collapse. Quantitative data on the ecology of this invasive vine are generally lacking. The present study examines the underground tuber traits of M. unguis-cati and explores their links with aboveground parameters at five infested sites spanning both riparian and inland vegetation. Tubers were abundant in terms of density (~1000 per m2), although small in size and low in level of interconnectivity. M. unguis-cati also exhibits multiple stems per plant. Of all traits screened, the link between stand (stem density) and tuber density was the most significant and yielded a promising bivariate relationship for the purposes of estimation, prediction and management of what lies beneath the soil surface of a given M. unguis-cati infestation site. The study also suggests that new recruitment is primarily from seeds, not from vegetative propagation as previously thought. The results highlight the need for future biological-control efforts to focus on introducing specialist seed- and pod-feeding insects to reduce seed-output.

Trends in demographic, health behaviour factors, and self-perceived weight status : influences on obesity in Australia from 1995 to 2005

Overweight and obesity are two of the most important emerging public health issues in our time and regarded by the World Health Organisation [WHO] (1998) as a worldwide epidemic. The prevalence of obesity in the USA is the highest in the world, and Australian obesity rates fall into second place. Currently, about 60% of Australian adults are overweight (BMI „d 25kg/m2). The socio-demographic factors associated with overweight and/or obesity have been well demonstrated, but many of the existing studies only examined these relationships at one point of time, and did not examine whether significant relationships changed over time. Furthermore, only limited previous research has examined the issue of the relationship between perception of weight status and actual weight status, as well as factors that may impact on people¡¦s perception of their body weight status. Aims: The aims of the proposed research are to analyse the discrepancy between perceptions of weight status and actual weight status in Australian adults; to examine if there are trends in perceptions of weight status in adults between 1995 to 2004/5; and to propose a range of health promotion strategies and furth er research that may be useful in managing physical activity, healthy diet, and weight reduction. Hypotheses: Four alternate hypotheses are examined by the research: (1) there are associations between independent variables (e.g. socio -demographic factors, physical activity and dietary habits) and overweight and/or obesity; (2) there are associations between the same independent variables and the perception of overweight; (3) there are associations between the same independent variables and the discrepancy between weight status and perception of weight status; and (4) there are trends in overweight and/or obesity, perception of overweight, and the discrepancy in Australian adults from 1995 to 2004/5. Conceptual Framework and Methods: A conceptual framework is developed that shows the associations identified among socio -demographic factors, physical activity and dietary habits with actual weight status, as well as examining perception of weight status. The three latest National Health Survey data bases (1995 , 2001 and 2004/5) were used as the primary data sources. A total of 74,114 Australian adults aged 20 years and over were recruited from these databases. Descriptive statistics, bivariate analyses (One -Way ANOVA tests, unpaired t-tests and Pearson chi-square tests), and multinomial logistic regression modelling were used to analyse the data. Findings: This research reveals that gender, main language spoken at home, occupation status, household structure, private health insurance status, and exercise are related to the discrepancy between actual weight status and perception of weight status, but only gender and exercise are related to the discrepancy across the three time point s. The current research provides more knowledge about perception of weight status independently. Factors which affect perception of overweight are gender, age, language spoken at home, private health insurance status, and diet ary habits. The study also finds that many factors that impact overweight and/or obesity also have an effect on perception of overweight, such as age, language spoken at home, household structure, and exercise. However, some factors (i.e. private health insurance status and milk consumption) only impact on perception of overweight. Furthermore, factors that are rel ated to people’s overweight are not totally related to people’s underestimation of their body weight status in the study results. Thus, there are unknown factors which can affect people’s underestimation of their body weight status. Conclusions: Health promotion and education activities should provide education about population health education and promotion and education for particular at risk sub -groups. Further research should take the form of a longitudinal study design ed to examine the causal relationship between overweight and/or obesity and underestimation of body weight status, it should also place more attention on the relationships between overweight and/or obesity and dietary habits, with a more comprehensive representation of SES. Moreover, further research that deals with identification of characteristics about perception of weight status, in particular the underestimation of body weight status should be undertaken.