No description available.

This thesis consists of three chapters. The first chapter focuses on adaptive smoothing splines for fitting functions with varying roughness. In the first part of the first chapter, we study an asymptotically optimal procedure to choose the value of a discretized version of the variable smoothing parameter in adaptive smoothing splines. With the choice given by the multivariate version of the generalized cross validation, the resulting adaptive smoothing spline estimator is shown to be consistent and asymptotically optimal under some general conditions. In the second part, we derive the asymptotically optimal local penalty function, which is subsequently used for the derivation of the locally optimal smoothing spline estimator. In the second chapter, we propose a Lipschitz regularity based statistical model, and apply it to coordinate measuring machine (CMM) data to estimate the form error of a manufactured product and to determine the optimal sampling positions of CMM measurements. Our proposed wavelet-based model takes advantage of the fact that the Lipschitz regularity holds for the CMM data. The third chapter focuses on the classification of functional data which are known to be well separable within a particular interval. We propose an interval based classifier. We first estimate a baseline of each class via convex optimization, and then identify an optimal interval that maximizes the difference among the baselines. Our interval based classifier is constructed based on the identified optimal interval. The derived classifier can be implemented via a low-order-of-complexity algorithm.

People at risk for blood clots are often treated with anticoagulants, warfarin is such an anticoagulant. The dose's effect is measured by comparing the time for blood to clot to a control time called an INR value. Previous anticoagulant studies have addressed agreement between fingerstick (POC) devices and the standard laboratory, however these studies rely on mathematical formulas as criteria for clinical evaluations, i.e. clinical evaluation vs. precision and bias. Fourteen such criteria were found in the literature. There exists little consistency among these criteria for assessing clinical agreement, furthermore whether these methods of assessing agreement are reasonable estimates of clinical decision-making is unknown and has yet to be validated. One previous study compared actual clinical agreement by having two physicians indicate a dosing decision based on patient history and INR values. This analysis attempts to justify previously used mathematical criteria for clinical agreement. Generalized additive models with smoothing spline estimates were calculated for each of the 14 criteria and compared to the smoothing spline estimate for the method using actual physician decisions (considered the "gold standard"). The area between the criteria method spline and the gold standard method spline served as the comparison, using bootstrapping for statistical inference. Although some of the criteria methods performed better than others, none of them matched the gold standard. This stresses the need for clinical assessment of devices.

Spline smoothers as implemented in common mixed model software provide a familiar framework for estimating semi-parametric and non-parametric models. Following a review of literature on splines and mixed models, details for implementing mixed model splines are presented. The examples use an experiment in the health sciences to demonstrate how to use mixed models to generate the smoothers. The first example takes a simple one-group case, while the second example fits an expanded model using three groups simultaneously. The second example also demonstrates how to fit confidence bands to the three-group model. The examples use mixed model software as implemented in lme() in R. Following the examples a discussion of the method is presented.

In the first part of the thesis, we propose a new class of designs called multi-layer sliced Latin hypercube design (DSLHD) for running computer experiments. A general recursive strategy for constructing MLSLHD has been developed. Ordinary Latin hypercube designs and sliced Latin hypercube designs are special cases of MLSLHD with zero and one layer respectively. A special case of MLSLHD with two layers, doubly sliced Latin hypercube design, is studied in detail. The doubly sliced structure of DSLHD allows more flexible batch size than SLHD for collective evaluation of different computer models or batch sequential evaluation of a single computer model. Both finite-sample and asymptotical sampling properties of DSLHD are examined. Numerical experiments are provided to show the advantage of DSLHD over SLHD for both sequential evaluating a single computer model and collective evaluation of different computer models. Other applications of DSLHD include design for Gaussian process modeling with quantitative and qualitative factors, cross-validation, etc. Moreover, we also show the sliced structure, possibly combining with other criteria such as distance-based criteria, can be utilized to sequentially sample from a large spatial data set when we cannot include all the data points for modeling. A data center example is presented to illustrate the idea. The enhanced stochastic evolutionary algorithm is deployed to search for optimal design. In the second part of the thesis, we propose a new smoothing technique called completely-data-driven smoothing, intended for smoothing over irregular regions. The idea is to replace the penalty term in the smoothing splines by its estimate based on local least squares technique. A close form solution for our approach is derived. The implementation is very easy and computationally efficient. With some regularity assumptions on the input region and analytical assumptions on the true function, it can be shown that our estimator achieves the optimal convergence rate in general nonparametric regression. The algorithmic parameter that governs the trade-off between the fidelity to the data and the smoothness of the estimated function is chosen by generalized cross validation (GCV). The asymptotic optimality of GCV for choosing the algorithm parameter in our estimator is proved. Numerical experiments show that our method works well for both regular and irregular region smoothing. The third part of the thesis deals with uncertainty quantification in building energy assessment. In current practice, building simulation is routinely performed with best guesses of input parameters whose true value cannot be known exactly. These guesses affect the accuracy and reliability of the outcomes. There is an increasing need to perform uncertain analysis of those input parameters that are known to have a significant impact on the final outcome. In this part of the thesis, we focus on uncertainty quantification of two microclimate parameters: the local wind speed and the wind pressure coefficient. The idea is to compare the outcome of the standard model with that of a higher fidelity model. Statistical analysis is then conducted to build a connection between these two. The explicit form of statistical models can facilitate the improvement of the corresponding modules in the standard model.

The goal of this thesis is to analyze the mathematical apparatus of the most widespread methods used for the yield curves construction. It aims to introduce not only the various of construction models but also to describe the whole process of creation, while discussing the advantages and disadvantage of individual methods. The first chapter focus on the general theory and the use of the term structure of interest rates in practice. The second part deals with the construction process itself and describes the most frequently used methods. The last chapter then shows the real application of selected methods on given data set and the use of the constructed yield curves for interest rate derivative valuation too.

MSc (Statistics) / Department of Statistics / The baseline parameter vector in a discrete-time survival model is determined by the number of time points. The larger the number of the time points, the higher the dimension of the baseline parameter vector which often leads to biased maximum likelihood estimates. One of the ways to overcome this problem is to use a simpler parametrization that contains fewer parameters. A simulation approach was used to compare the accuracy of three variants of penalised regression spline methods in smoothing the baseline hazard function. Root mean squared error (RMSE) analysis suggests that generally all the smoothing methods performed better than the model with a discrete baseline hazard function. No single smoothing method outperformed the other smoothing methods. These methods were also applied to data on age at rst alcohol intake in Thohoyandou. The results from real data application suggest that there were no signi cant di erences amongst the estimated models. Consumption of other drugs, having a parent who drinks, being a male and having been abused in life are associated with high chances of drinking alcohol very early in life. / NRF

Διάφορες κλιματολογικές, υδρολογικές και περιβαλλοντικές μελέτες απαιτούν ακριβή γνώση της χωρικής κατανομής των σταθερών ισοτόπων του υδρογόνου και του οξυγόνου στον υετό. Δεδομένου ότι ο αριθμός των σταθμών συλλογής δειγμάτων υετού για ισοτοπική ανάλυση είναι μικρός και όχι ομογενώς κατανεμημένος σε πλανητικό επίπεδο, η πλανητκή κατανομή των σταθερών ισοτόπων μπορεί να υπολογισθεί μέσω της δημιουργίας πλεγματικών ισοτοπικών δεδομένων, για τη δημιουργία των οποίων έχουν προταθεί διάφορες μέθοδοι. Ορισμένες χρησιμοποιούν εμπειρικές σχέσεις και γεωστατιστικές μεθόδους ώστε να ελαχιστοποιήσουν τα σφάλματα λόγω παρεμβολής. Στην εργασία αυτή γίνεται μια προσπάθεια να δημιουργηθούν βάσεις πλεγματικών δεδομένων της ισοτοπικής σύστασης του υετού με ανάλυση 10΄ × 10΄ για την περιοχή της Κεντρικής και Ανατολικής Μεσογείου. Προσδιορίζονται στατιστικά πρότυπα λαμβάνοντας υπ’ όψιν γεωγραφικές και μετεωρολογικές παραμέτρους, ως ανεξάρτητες μεταβλητές. Η αρχική μεθοδολογία χρησιμοποιεί μόνο το υψόμετρο της περιοχής και το γεωγραφικό της πλάτος ως ανεξάρτητες μεταβλητές. Επειδή η ισοτοπική σύσταση εξαρτάται και από το γεωγραφικό μήκος προστέθηκαν στα υφιστάμενα πρότυπα, εκτός των γεωγραφικών μεταβλητών και μετεωρολογικές. Προτείνεται σειρά προτύπων τα οποία περιλαμβάνουν είτε ορισμένες είτε συνδυασμό αυτών των παραμέτρων. Η αξιολόγηση των προτύπων γίνεται με εφαρμογή των μεθόδων Thin Plate Splines (TPSS) και Ordinary Kriging (ΟΚ). / Several climatic, hydrological and environmental studies require the accurate knowledge of the spatial distribution of stable isotopes in precipitation. Since the number of rain sampling stations for isotope analysis is small and not evenly distributed around the globe, the global distribution of stable isotopes can be calculated via the production of gridded isotopic data sets. Several methods have been proposed for this purpose. Some of them use empirical equations and geostatistical methods in order to minimize eventual errors due to interpolation. In this work a methodology is proposed for the development of 10΄ × 10΄ gridded isotopic data of precipitation in Central and Eastern Mediterranean. Statistical models are developed taking into account geographical and meteorological parameters as independent variables. The initial methodology takes into account only the altitude and latitude of an area. Since however the isotopic composition of precipitation depends also on longitude, the existing models have been modified by adding meteorological parameters as independent variables also. A series of models is proposed taking into account some or a combination of the above mentioned variables. The models are validated using the Thin Plate Smoothing Splines (TPSS) and the Ordinary Kriging (OK) methods.

Thesis (MSc (Geography and Environmental Studies))--University of Stellenbosch, 2007. / Due to the relative scarcity of weather stations, the climate conditions of large areas are not adequately represented by a weather station. This is especially true for regions with complex topographies or low population densities. Various interpolation techniques and software packages are available with which the climate of such areas can be calculated from surrounding weather stations’ data. This study investigates the possibility of using the software package ANUSPLIN to create accurate climate maps for the Western Cape, South Africa. ANUSPLIN makes use of thin plate smoothing splines and a digital elevation model to convert point data into grid format to represent an area’s climatic conditions. This software has been used successfully throughout the world, therefore a large body of literature is available on the topic, highlighting the limitations and successes of this interpolation method. Various factors have an effect on a region’s climate, the most influential being location (distance from the poles or equator), topography (height above sea level), distance from large water bodies, and other topographical factors such as slope and aspect. Until now latitude, longitude and the elevation of a weather station have most often been used as input variables to create climate grids, but the new version of ANUSPLIN (4.3) makes provision for additional variables. This study investigates the possibility of incorporating the effect of the surrounding oceans and topography (slope and aspect) in the interpolation process in order to create climate grids with a resolution of 90m x 90m. This is done for monthly mean daily maximum and minimum temperature and the mean monthly rainfall for the study area for each month of the year. Not many projects where additional variables have been incorporated in the interpolation process using ANUSPLIN are to be found in the literature, thus further investigation into the correct transformation and the units of these variables had to be done before they could be successfully incorporated. It was found that distance to oceans influences a region’s maximum and minimum temperatures, and to a lesser extent rainfall, while aspect and slope has an influence on a region’s rainfall. In order to assess the accuracy of the interpolation process, two methods were employed, namely statistical values produced during the spline function calculations by ANUSPLIN, and the removal of a selected number of stations in order to compare the interpolated values with the actual measured values. The analysis showed that more accurate maps were obtained when additional variables were incorporated into the interpolation process. Once the best transformations and units were identified for the additional variables, climate maps were produced in order to compare them with existing climate grids available for the study area. In general the temperatures were higher than those of the existing grids. For the rainfall grids ANUSPLIN’s produced higher rainfall values throughout the study region compared to the existing grids, except for the Southwestern Cape where the rainfall values were lower on north-facing slopes and high-lying area

For multivariate data, when testing homogeneity of covariance matrices arising from two or more groups, Bartlett's (1937) modified likelihood ratio test statistic is appropriate to use under the null hypothesis of equal covariance matrices where the null distribution of the test statistic is based on the restrictive assumption of normality. Zhang and Boos (1992) provide a pooled bootstrap approach when the data cannot be assumed to be normally distributed. We give three alternative bootstrap techniques to testing homogeneity of covariance matrices when it is both inappropriate to pool the data into one single population as in the pooled bootstrap procedure and when the data are not normally distributed. We further show that our alternative bootstrap methodology can be extended to testing Flury's (1988) hierarchy of covariance structure models. Where deviations from normality exist, we show, by simulation, that the normal theory log-likelihood ratio test statistic is less viable compared with our bootstrap methodology. For functional data, Ramsay and Silverman (2005) and Lee et al (2002) together provide four computational techniques for functional principal component analysis (PCA) followed by covariance structure estimation. When the smoothing method for smoothing individual profiles is based on using least squares cubic B-splines or regression splines, we find that the ensuing covariance matrix estimate suffers from loss of dimensionality. We show that ridge regression can be used to resolve this problem, but only for the discretisation and numerical quadrature approaches to estimation, and that choice of a suitable ridge parameter is not arbitrary. We further show the unsuitability of regression splines when deciding on the optimal degree of smoothing to apply to individual profiles. To gain insight into smoothing parameter choice for functional data, we compare kernel and spline approaches to smoothing individual profiles in a nonparametric regression context. Our simulation results justify a kernel approach using a new criterion based on predicted squared error. We also show by simulation that, when taking account of correlation, a kernel approach using a generalized cross validatory type criterion performs well. These data-based methods for selecting the smoothing parameter are illustrated prior to a functional PCA on a real data set.