Model selection and error estimation

We study model selection strategies based on penalized empirical loss minimization. We point out a tight relationship between error estimation and data-based complexity penalization: any good error estimate may be converted into a data-based penalty function and the performance of the estimate is governed by the quality of the error estimate. We consider several penalty functions, involving error estimates on independent test data, empirical VC dimension, empirical VC entropy, and margin-based quantities. We also consider the maximal difference between the error on the first half of the training data and the second half, and the expected maximal discrepancy, a closely related capacity estimate that can be calculated by Monte Carlo integration. Maximal discrepancy penalty functions are appealing for pattern classification problems, since their computation is equivalent to empirical risk minimization over the training data with some labels flipped.

Fast rates for estimation error and oracle inequalities for model section

We consider complexity penalization methods for model selection. These methods aim to choose a model to optimally trade off estimation and approximation errors by minimizing the sum of an empirical risk term and a complexity penalty. It is well known that if we use a bound on the maximal deviation between empirical and true risks as a complexity penalty, then the risk of our choice is no more than the approximation error plus twice the complexity penalty. There are many cases, however, where complexity penalties like this give loose upper bounds on the estimation error. In particular, if we choose a function from a suitably simple convex function class with a strictly convex loss function, then the estimation error (the difference between the risk of the empirical risk minimizer and the minimal risk in the class) approaches zero at a faster rate than the maximal deviation between empirical and true risks. In this paper, we address the question of whether it is possible to design a complexity penalized model selection method for these situations. We show that, provided the sequence of models is ordered by inclusion, in these cases we can use tight upper bounds on estimation error as a complexity penalty. Surprisingly, this is the case even in situations when the difference between the empirical risk and true risk (and indeed the error of any estimate of the approximation error) decreases much more slowly than the complexity penalty. We give an oracle inequality showing that the resulting model selection method chooses a function with risk no more than the approximation error plus a constant times the complexity penalty.

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.

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.

Extracellular matrix of the human cyclic corpus luteum

Extracellular matrix regulates many cellular processes likely to be important for development and regression of corpora lutea. Therefore, we identified the types and components of the extracellular matrix of the human corpus luteum at different stages of the menstrual cycle. Two different types of extracellular matrix were identified by electron microscopy; subendothelial basal laminas and an interstitial matrix located as aggregates at irregular intervals between the non-vascular cells. No basal laminas were associated with luteal cells. At all stages, collagen type IV α1 and laminins α5, β2 and γ1 were localized by immunohistochemistry to subendothelial basal laminas, and collagen type IV α1 and laminins α2, α5, β1 and β2 localized in the interstitial matrix. Laminin α4 and β1 chains occurred in the subendothelial basal lamina from mid-luteal stage to regression; at earlier stages, a punctate pattern of staining was observed. Therefore, human luteal subendothelial basal laminas potentially contain laminin 11 during early luteal development and, additionally, laminins 8, 9 and 10 at the mid-luteal phase. Laminin α1 and α3 chains were not detected in corpora lutea. Versican localized to the connective tissue extremities of the corpus luteum. Thus, during the formation of the human corpus luteum, remodelling of extracellular matrix does not result in basal laminas as present in the adrenal cortex or ovarian follicle. Instead, novel aggregates of interstitial matrix of collagen and laminin are deposited within the luteal parenchyma, and it remains to be seen whether this matrix is important for maintaining the luteal cell phenotype.

Endogenous mortality, human capital and economic growth

We consider growth and welfare effects of lifetime-uncertainty in an economy with human capital-led endogenous growth. We argue that lifetime uncertainty reduces private incentives to invest in both physical and human capital. Using an overlapping generations framework with finite-lived households we analyze the relevance of government expenditure on health and education to counter such growth-reducing forces. We focus on three different models that differ with respect to the mode of financing of education: (i) both private and public spending, (ii) only public spending, and (iii) only private spending. Results show that models (i) and (iii) outperform model (ii) with respect to long-term growth rates of per capita income, welfare levels and other important macroeconomic indicators. Theoretical predictions of model rankings for these macroeconomic indicators are also supported by observed stylized facts.