Bibliography: p. 176-187. / This thesis addresses a number of special topics in spatial interpolation and smoothing. The motivation for the thesis comes from two projects, one being to extend the availability of a daily rainfall model for southern Africa to sites at which little or no rainfall data is available, using data from nearby sites, and the other arising from a need to improve the species abundance estimates used to produce maps for the Southern African Bird Atlas Project in areas where the original presence/absence data is sparse. Although problems of spatial interpolation and smoothing have been the subject of much research in recent years, leading to the development of the specialised discipline of geostatistics, these two problems have features which render the available methodology inappropriate in certain respects. The semi-variogram plays a central role in geostatistical work. In both of the applications considered here, the raw semi-variogram is 'contaminated' by error, but the error variance varies widely between data points, so that the spatial autocorrelation structure of the underlying error-free variable is blurred. An adjusted semi-variogram, which removes the effect of the measurement error, is defined and incorporated into the kriging equations. A number of measures have been proposed for kriging in the presence of trend, ranging from explicit modelling of a deterministic trend function to 'moving window' kriging, which assumes local stationarity as an approximation. The former approach is often inappropriate over large non-homogenous regions, while the latter approach tends to underestimate the kriging variance. As an alternative strategy it is proposed here that the trend function be considered as another random variable, with a long-range spatial autocorrelation. This approach is simple to implement, and can also be used as a basis for filtering the data to separate trend from local or high-frequency variation. The daily rainfall model is based on a Fourier series representation giving rise to amplitude and phase parameters; the latter are circular in nature, and not amenable to analysis by standard techniques. This thesis describes a method of interpolation and smoothing, analogous to kriging, which is appropriate for unit vector data available at a number of spatial locations. The cumulated values of species counts in the SABAP are essentially binomially distributed and thus again specialised techniques are required for interpolation. New geostatistical methods which cater for both binomial and Poisson data are presented. Another problem arises from the need to improve interpolated values of the rainfall model parameters by incorporating information on altitude. Although a number of approaches are possible, for example, using co-kriging or kriging with external drift, difficulties are caused by the complexity of the relationship between the rainfall at a point and the surrounding topography. This problem is overcome by the use of orthogonal functions of altitude to model the patterns of topography.

Includes bibliographical references.

The objective of this research is to develop a framework of multi-objective optimization (MOO) models that are better capable of providing decision support on future long-term electricity generation planning (EGP), in the context of insufficient electricity capacity and to apply it to the electricity system for a developing country. The problem that motivated this study was a lack of EGP models in developing countries to keep pace with the countries' socio-economic and demographic dynamics. This research focused on two approaches: mathematical programming (MP) and system dynamics (SD). Detailed model descriptions, formulations, and implementation results are presented in the thesis along with the observations and insights obtained during the course of this research.

Markowitz's modem portfolio theory has played a vital role in investment portfolio management, which is constantly pushing the development on volatility models. Particularly, the stochastic volatility model which reveals the dynamics of conditional volatility. Financial time series and volatility models has become one of the hot spots in operations research. In this thesis, one of the areas we explore is the theoretical formulation of the optimal portfolio selection problem under Ito calculus framework. Particularly, a stochastic variation calculus problem, i.e., seeking the optimal stochastic volatility diffusion family for facilitating the best portfolio selection identified under the continuous-time stochastic optimal control theoretical settings. One of the properties this study examines is the left-shifting role of the GARCH(1, 1) (General Autoregressive Conditional Heteroskedastic) model's efficient frontier. This study considers many instances where the left shifting superior behaviour of the GARCH(1, 1) is observed. One such instance is when GARCH(1, 1) is compared within the volatility modelling extensions of the GARCH environ in a single index framework. This study will demonstrate the persistence of the superiority of the G ARCH ( 1, 1) frontier within a multiple and single index context of modem portfolio theory. Many portfolio optimization models are investigated, particularly the Markowitz model and the Sharpe Multiple and Single index models. Includes bibliographical references (p. 313-323).

Includes bibliographical references (p. 177-190).

Includes bibliographical references. / Patterns of renewal of feathers are poorly known in African birds. Moult is energetically costly and is thus an important part of a bird's annual cycle; moult needs to be fitted in with breeding activities, and in some species, migration. Ringers in southern Africa have been submitting primary moult data to SAFRING, the South African Bird Ringing Unit, since 1998, providing a large amount of data that the author has been curating and checking on an on-going basis as Ringing Coordinator. May main interest is in the weaverbirds and I have ringed several thousand Southern Masked Weavers Ploceus velatus and many other species. Les Underhill and Walter Zucchini developed a statistical model to analyse timing and duration of primary moult in a standardized way in 1988. Primary moult was analysed in the southern African weaverbird family using the Underhill-Zucchini method throughout. In some species, this method was applied to individual feathers as well as the whole wing.

This thesis's research focus is on computational statistics along with DEAR (abbreviation of differential equation associated regression) model direction, and that in mind, the journal papers are written as contributions to statistical machine learning algorithm literature.

Bibliography: leaves 181-206. / This thesis focuses on the empirical analysis of the fuzzy feature of the Johannesburg Stock Exchange Overall Index using fuzzy logic techniques. The data for the periods 1985 - 2001 is used in the analysis. Description of the fuzzy feature is crucial to the proper understanding of the movement of the JSE Overall Index and the South African economy. The fuzzy feature of the Johannesburg Security Exchange Overall Index if understood would impact on the financial and economical decisions. A preliminary Fractal analysis is carried out before the fuzzy analysis to investigate the nature of the Johannesburg Security Exchange Overall Index. The Johannesburg Security Exchange Overall Index experiences the Hurst phenomena of long memory for periods of 100 days (approximately three months). Outside the long memory periods, the Johannesburg Security Exchange Overall Index is found to exhibit antipersistent or short-range dependency characteristics. The fuzzy feature of the Johannesburg Security Exchange Overall Index is described by many aspects of fuzzy logic. The analysis of the fuzzy feature is carried out according to time periods of approximately 4 years each of the Johannesburg Security Exchange Overall Index. The index in each time period is partitioned in three fuzzy states: "low", "middle" and "high". The fuzzy states are important in assessing the fuzzy nature of the Johannesburg Security Exchange Overall Index. The partitioning reveals that the fuzzy states of the Johannesburg Security Exchange Overall Index do not possess sharp boundaries. The sizes of the fuzzy states are found to change with time. This reflects changes in the forces behind the dynamics of the index.

The first major contribution of the thesis is the development of sensitivity analysis strategy for dealing with incomplete longitudinal data. The second important contribution is setting up of simulation experiment to evaluate the performance of some of the sensitivity analysis approaches. The third contribution is that the thesis offers recommendations on which sensitivity analysis strategy to use and in what circumstance. It is recommended that when drawing statistical inferences in the presence of missing data, methods of analysis based on plausible scientific assumptions should be used. One major issue is that such assumptions cannot be verified using the data at hand. In order to verify these assumptions, sensitivity analysis should be performed to investigate the robustness of statistical inferences to plausible alternative assumptions about the missing data. The thesis implemented various sensitivity analysis strategies to incomplete longitudinal CD4 count data in order to investigate the effect of tuberculosis pericarditis (TBP) treatment on CD4 count changes over time. The thesis achieved the first contribution by formulating primary analysis (which assume that the data are missing at random) and then conducting sensitivity analyses to assess whether statistical inferences under the primary analysis model are sensitive to models that assume that the data are not missing at random. The second contribution was achieved via simulation experiment involving formulating hypotheses on how sensitivity analysis strategies would performed under varying rate of missing values and model mis-specification (when the model is mis-specified). The third contribution was achieved based on our experience from the development and application of the sensitivity analysis strategies as well as the simulation experiment. Using the CD4 count data, we observed that statistical inferences under the primary analysis formulation are robust to the sensitivity analyses formulations, suggesting that the mechanism that generated the missing CD4 count measurements is likely to be missing at random. The results also revealed that TBP does not interact with the HIV/AIDS treatment and that TBP treatment had no significant effect on CD4 count changes over time. We have observed in our simulation results that the sensitivity analysis strategies produced unbiased statistical inferences except when a strategy is inappropriately applied in a given trial setting and also, when a strategy is mis-specified. Although the methods considered were applied to data in the IMPI trial setting, these methods can also be applied to clinical trials with similar settings. A sensitivity analysis strategy may not necessarily give bias results because it has been mis-specified, but it may also be that it has been applied in a wrongly defined trial setting. We therefore strongly encourage analysts to carefully study these sensitivity analysis frameworks together with a clearly and precise definition of the trial objective in order to decide on which sensitivity analysis strategy to use.

Includes bibliographical references. / The Southern African Bird Atlas Project (Harrison at el. 1997a, b) and the Mozambique Bird Atlas Project (Parker 1999) generated data on reporting rates for birds that takes into account the likelihood of species detection in a given area. Our main objective in this thesis is to explore methods for analysing and summarising reporting rate data. The observed reporting rate data are subject to bias due to differential sampling effort and observer errors. We use a logistic regression model suitable for binomial type data to replace the observed reporting rates with smoothed probabilities of detection. To base our prediction on data from the surrounding neighbourhood, we choose as explanatory variables the north-south and west-east coordinates relative to the target grid cell for which the prediction is being made. We explore some variants of our general smoothing approach that relates to the presentation of the smoothed distributions. The smoothed distributions of detection probabilities are presented as multicoloured maps. We consider two alternative ways of subdividing the range of detection probabilities into sub-intervals. One approach is species-specific, while the other imposes an absolute subdivision on all species distributions. For species with highly fragmented distributions, we introduce the possibility of using a weighted average between observed reporting rates and smoothed detection probabilities as the final value to be plotted. The weights are based on the extent of coverage and the underlying degree of fragmentation. The identification of patterns of distributions for species is an important part of biogeography and plays a major role in the identification of areas where conservation efforts should be targeted. Interest centres around identifying areas of species richness, centres for narrow eudemism and zones of transition in species composition. We explore the benefits of using a range of detection probabilities in comparison to the use of presence-absence data to identify areas rich in species and rich in narrow-endemic species. We transform existing measures for species richness and species endemism by replacing presence-absence data with detection probability deciles that reflect the relative likelihood of detecting a species in a given grid cell. The resulting measures give more weight to the areas where species have the core of their distributions and down-weight the peripheral edges of the species distributions, where detection probabilities may be too small to guarantee continued survival. The use of a mathematical model to generate smoothed distributions of detection probabilities enables us to calculate gradients for the detection probability surfaces for species. We can define the concept of individual species gradients that reflect the relative degree of change among detection probabilities within the overall range of occurrence for the species. We combine the gradients for all species in several different ways. Large values for the overall sums of the gradients indicate areas of large fluctuation in species composition. On the other hand, small values for the overall gradient sums indicate areas of relative stability. We also sum the gradients in one of 16 directions. These directional gradient sums distinguish between areas where the changes in species detection probability distributions occur in isolated directions, thus indicating ecological transition zones, and areas of random fluctuation, indicative of species fragmentation. In this thesis we do not derive new statistical methods. We adapt existing techniques to deal with the abundance component of the data generated by the bird atlas projects. We show how the measures based on reporting rate data, rather than presence-absence data, add substantial insight into patterns of distribution of bird species in southern Africa.