On the optimality of sample-based estimates of the expectation of the empirical minimizer

We study sample-based estimates of the expectation of the function produced by the empirical minimization algorithm. We investigate the extent to which one can estimate the rate of convergence of the empirical minimizer in a data dependent manner. We establish three main results. First, we provide an algorithm that upper bounds the expectation of the empirical minimizer in a completely data-dependent manner. This bound is based on a structural result due to Bartlett and Mendelson, which relates expectations to sample averages. Second, we show that these structural upper bounds can be loose, compared to previous bounds. In particular, we demonstrate a class for which the expectation of the empirical minimizer decreases as O(1/n) for sample size n, although the upper bound based on structural properties is Ω(1). Third, we show that this looseness of the bound is inevitable: we present an example that shows that a sharp bound cannot be universally recovered from empirical data.

Krylov subspace approximations for the exponential Euler method: Error estimates and the harmonic Ritz approximant

We study Krylov subspace methods for approximating the matrix-function vector product φ(tA)b where φ(z) = [exp(z) - 1]/z. This product arises in the numerical integration of large stiff systems of differential equations by the Exponential Euler Method, where A is the Jacobian matrix of the system. Recently, this method has found application in the simulation of transport phenomena in porous media within mathematical models of wood drying and groundwater flow. We develop an a posteriori upper bound on the Krylov subspace approximation error and provide a new interpretation of a previously published error estimate. This leads to an alternative Krylov approximation to φ(tA)b, the so-called Harmonic Ritz approximant, which we find does not exhibit oscillatory behaviour of the residual error.

Explanations of police racism

This paper describes the development of a substantive theory about police racism as the most significant factor, from among a number of competing societal based explanations, in accounting for Aboriginal over-representation in police arrest rates.

Measurement of total body water (TBW) and total energy expenditure (TEE) using stable isotopes

Understanding the relationship between diet, physical activity and health in humans requires accurate measurement of body composition and daily energy expenditure. Stable isotopes provide a means of measuring total body water and daily energy expenditure under free-living conditions. While the use of isotope ratio mass spectrometry (IRMS) for the analysis of 2H (Deuterium) and 18O (Oxygen-18) is well established in the field of human energy metabolism research, numerous questions remain regarding the factors which influence analytical and measurement error using this methodology. This thesis was comprised of four studies with the following emphases. The aim of Study 1 was to determine the analytical and measurement error of the IRMS with regard to sample handling under certain conditions. Study 2 involved the comparison of TEE (Total daily energy expenditure) using two commonly employed equations. Further, saliva and urine samples, collected at different times, were used to determine if clinically significant differences would occur. Study 3 was undertaken to determine the appropriate collection times for TBW estimates and derived body composition values. Finally, Study 4, a single case study to investigate if TEE measures are affected when the human condition changes due to altered exercise and water intake. The aim of Study 1 was to validate laboratory approaches to measure isotopic enrichment to ensure accurate (to international standards), precise (reproducibility of three replicate samples) and linear (isotope ratio was constant over the expected concentration range) results. This established the machine variability for the IRMS equipment in use at Queensland University for both TBW and TEE. Using either 0.4mL or 0.5mL sample volumes for both oxygen-18 and deuterium were statistically acceptable (p>0.05) and showed a within analytical variance of 5.8 Delta VSOW units for deuterium, 0.41 Delta VSOW units for oxygen-18. This variance was used as “within analytical noise” to determine sample deviations. It was also found that there was no influence of equilibration time on oxygen-18 or deuterium values when comparing the minimum (oxygen-18: 24hr; deuterium: 3 days) and maximum (oxygen-18: and deuterium: 14 days) equilibration times. With regard to preparation using the vacuum line, any order of preparation is suitable as the TEE values fall within 8% of each other regardless of preparation order. An 8% variation is acceptable for the TEE values due to biological and technical errors (Schoeller, 1988). However, for the automated line, deuterium must be assessed first followed by oxygen-18 as the automated machine line does not evacuate tubes but merely refills them with an injection of gas for a predetermined time. Any fractionation (which may occur for both isotopes), would cause a slight elevation in the values and hence a lower TEE. The purpose of the second and third study was to investigate the use of IRMS to measure the TEE and TBW of and to validate the current IRMS practices in use with regard to sample collection times of urine and saliva, the use of two TEE equations from different research centers and the body composition values derived from these TEE and TBW values. Following the collection of a fasting baseline urine and saliva sample, 10 people (8 women, 2 men) were dosed with a doubly labeled water does comprised of 1.25g 10% oxygen-18 and 0.1 g 100% deuterium/kg body weight. The samples were collected hourly for 12 hrs on the first day and then morning, midday, and evening samples were collected for the next 14 days. The samples were analyzed using an isotope ratio mass spectrometer. For the TBW, time to equilibration was determined using three commonly employed data analysis approaches. Isotopic equilibration was reached in 90% of the sample by hour 6, and in 100% of the sample by hour 7. With regard to the TBW estimations, the optimal time for urine collection was found to be between hours 4 and 10 as to where there was no significant difference between values. In contrast, statistically significant differences in TBW estimations were found between hours 1-3 and from 11-12 when compared with hours 4-10. Most of the individuals in this study were in equilibrium after 7 hours. The TEE equations of Prof Dale Scholler (Chicago, USA, IAEA) and Prof K.Westerterp were compared with that of Prof. Andrew Coward (Dunn Nutrition Centre). When comparing values derived from samples collected in the morning and evening there was no effect of time or equation on resulting TEE values. The fourth study was a pilot study (n=1) to test the variability in TEE as a result of manipulations in fluid consumption and level of physical activity; the magnitude of change which may be expected in a sedentary adult. Physical activity levels were manipulated by increasing the number of steps per day to mimic the increases that may result when a sedentary individual commences an activity program. The study was comprised of three sub-studies completed on the same individual over a period of 8 months. There were no significant changes in TBW across all studies, even though the elimination rates changed with the supplemented water intake and additional physical activity. The extra activity may not have sufficiently strenuous enough and the water intake high enough to cause a significant change in the TBW and hence the CO2 production and TEE values. The TEE values measured show good agreement based on the estimated values calculated on an RMR of 1455 kcal/day, a DIT of 10% of TEE and activity based on measured steps. The covariance values tracked when plotting the residuals were found to be representative of “well-behaved” data and are indicative of the analytical accuracy. The ratio and product plots were found to reflect the water turnover and CO2 production and thus could, with further investigation, be employed to identify the changes in physical activity.

Improved image quality for x-ray CT imaging of gel dosimeters

Purpose: This study provides a simple method for improving precision of x-ray computed tomography (CT) scans of irradiated polymer gel dosimetry. The noise affecting CT scans of irradiated gels has been an impediment to the use of clinical CT scanners for gel dosimetry studies. Method: In this study, it is shown that multiple scans of a single PAGAT gel dosimeter can be used to extrapolate a ‘zero-scan’ image which displays a similar level of precision to an image obtained by averaging multiple CT images, without the compromised dose measurement resulting from the exposure of the gel to radiation from the CT scanner. Results: When extrapolating the zero-scan image, it is shown that exponential and simple linear fits to the relationship between Hounsfield unit and scan number, for each pixel in the image, provides an accurate indication of gel density. Conclusions: It is expected that this work will be utilised in the analysis of three-dimensional gel volumes irradiated using complex radiotherapy treatments.

A note on trader Sharpe Ratios

Traders in the financial world are assessed by the amount of money they make and, increasingly, by the amount of money they make per unit of risk taken, a measure known as the Sharpe Ratio. Little is known about the average Sharpe Ratio among traders, but the Efficient Market Hypothesis suggests that traders, like asset managers, should not outperform the broad market. Here we report the findings of a study conducted in the City of London which shows that a population of experienced traders attain Sharpe Ratios significantly higher than the broad market. To explain this anomaly we examine a surrogate marker of prenatal androgen exposure, the second-to-fourth finger length ratio (2D:4D), which has previously been identified as predicting a trader's long term profitability. We find that it predicts the amount of risk taken by traders but not their Sharpe Ratios. We do, however, find that the traders' Sharpe Ratios increase markedly with the number of years they have traded, a result suggesting that learning plays a role in increasing the returns of traders. Our findings present anomalous data for the Efficient Markets Hypothesis.