Discrete phylogeographic models enable the inference of the geographic history of biological organisms along phylogenetic trees. Frequently applied in the context of epidemiological modelling, phylogeographic generalised linear models were developed to allow for the evaluation of multiple predictors of spatial diffusion. The standard phylogeographic generalised linear model formulation, however, assumes that rates of spatial diffusion are a noiseless deterministic function of the set of covariates, admitting no other unobserved sources of variation. Under a variety of simulation scenarios, we demonstrate that the lack of a term modelling stochastic noise results in high false positive rates for predictors of spatial diffusion. We further show that the false positive rate can be controlled by including a random effect term, thus allowing unobserved sources of rate variation. Finally, we apply this random effects model to three recently published datasets and contrast the results of analysing these datasets with those obtained using the standard model. Our study demonstrates the prevalence of false positive results for predictors under the standard phylogeographic model in multiple simulation scenarios and, using empirical data from the literature, highlights the importance of a model accounting for random variation.

Commercial sugarcane farming requires large quantities of water to be delivered to the fields. Ideal irrigation schedules are produced indicating how much water to be supplied to fields considering multiple objectives in the farming process. Software packages do not fully account for the fact that the ideal irrigation schedule may not be met due to limitations in the water distribution network. This dissertation proposes the use of mathematical modelling to better understand water supply and demand management on a commercial sugarcane farm. Due to the complex nature of water stress on sugarcane, non-linearities occur in the model. A piecewise linear approximation is used to handle the non-linearity in the water allocation model and is solved in a commercial optimisation software package. A test data set is first used to exercise and evaluate the model performance, then to illustrate the practical applicability of the model, a commercial sized data set is used and analysed.

The e-retail sector in South Africa has a significant opportunity to capture a large portion of the country's retail industry. Central to seizing this opportunity is leveraging the advantages that the online setting affords. In particular, the e-retailer can offer an extremely large catalogue of products; far beyond what a traditional retailer is capable of supporting. However, as the catalogue grows, it becomes increasingly difficult for a customer to efficiently discover desirable products. As a consequence, it is important for the e-retailer to develop tools that automatically explore the catalogue for the customer. In this dissertation, we develop a recommender system (RS), whose purpose is to provide suggestions for products that are most likely of interest to a particular customer. There are two primary contributions of this dissertation. First, we describe a set of six characteristics that all effective RS's should possess, namely; accuracy, responsiveness, durability, scalability, model management, and extensibility. Second, we develop an RS that is capable of serving recommendations in an actual e-retail environment. The design of the RS is an attempt to embody the characteristics mentioned above. In addition, to show how the RS supports model selection, we present a proof-of-concept experiment comparing two popular methods for generating recommendations that we implement for this dissertation, namely, implicit matrix factorisation (IMF) and Bayesian personalised ranking (BPR).

Since the establishment of Markowitz's theory, numerous studies have been carried out over the past six decades or so that cover the benefits, limitations, modifications and enhancements of Mean Variance (MV) optimisation. This study endeavours to extend on this, by means of adding factors to the minimum variance framework, which would increase the likelihood of outperforming both the market and the minimum variance portfolio (MVP). An analysis of the impact of these factor tilts on the MVP is carried out in the South African environment, represented by the FTSE-JSE Shareholder weighted Index as the benchmark portfolio. The main objective is to examine if the systematic and robust methods employed, which involve the incorporation of factor tilts into the multicriteria problem, together with covariance shrinkage – improve the performance of the MVP. The factor tilts examined include Active Distance, Concentration and Volume. Additionally, the constant correlation model is employed in the estimation of the shrinkage intensity, structured covariance target and shrinkage estimator. The results of this study showed that with specific levels of factor tilting, one can generally improve both absolute and risk-adjusted performance and lower concentration levels in comparison to both the MVP and benchmark. Additionally, lower turnover levels were observed across all tilted portfolios, relative to the MVP. Furthermore, covariance shrinkage enhanced all portfolio statistics examined, but significant improvement was noted on drawdown levels, capture ratios and risk. This is in contrast to the results obtained when the standard sample covariance matrix was employed.