Thesis (M.Sc. (Statistics)) -- University of Limpopo, 2020 / Several studies indicated a growing trend in terms of frequency and severity of extreme events. Extreme rainfall could cause disasters that lead to loss of property and life. The aim of the study was to model the monthly rainfall variability in selected provinces of South Africa using extreme value distributions. This study investigated the best-fit probability distributions in the five provinces of South Africa. Five probability distributions: gamma, Gumbel, log-normal, Pareto and Weibull, were fitted and the best was selected from the five distributions for each province. Parameters of these distributions were estimated by the method of maximum likelihood estimators. Based on the Akaike information criteria (AIC) and Bayesian information criteria (BIC), the Weibull distribution was found to be the best-fit probability distribution for Eastern Cape, KwaZulu-Natal, Limpopo and Mpumalanga, while in Gauteng the best-fit probability distribution was found to be the gamma distribution. Monthly rainfall trends detected using the Mann–Kendall test revealed significant monotonic decreasing long-term trend for Eastern Cape, Gauteng and KwaZulu-Natal, and insignificant monotonic decreasing longterm trends for Limpopo and Mpumalanga. Non-stationary generalised extreme value distribution (GEVD) and non-stationary generalized Pareto distribution (GPD) were applied to model monthly rainfall data. The deviance statistic and likelihood ratio test (LRT) were used to select the most appropriate model. Model fitting supported stationary GEVD model for Eastern Cape, Gauteng and KwaZulu-Natal. On the other hand, model fitting supported non-stationary GEVD models for maximum monthly rainfall with nonlinear quadratic trend in the location parameter and a linear trend in the scale parameter for Limpopo, while in Mpumalanga the non-stationary GEVD model, which has a nonlinear quadratic trend in the scale parameter and no variation in the location parameter fitted well to the maximum monthly rainfall data. Results from the non-stationary GPD models showed that inclusion of the time covariate in our models was not significant for Eastern Cape, hence the bestfit model was the stationary GPD model. Furthermore, the non-stationary GPD model with a linear trend in the scale parameter provided the best-fit for KwaZulu-Natal and Mpumalanga, while in Gauteng and Limpopo the nonstationary GPD model with a nonlinear quadratic trend in the scale parameter fitted well to the monthly rainfall data. Lastly, GPD with time-varying thresholds was applied to model monthly rainfall excesses, where a penalised regression cubic smoothing spline was used as a time-varying threshold and the GPD model was fitted to cluster maxima. The estimate of the shape parameter showed that the Weibull family of distributions is appropriate in modelling the upper tail of the distribution for Limpopo and Mpumalanga, while for Eastern Cape, Gauteng and KwaZulu-Natal, the exponential family of distributions was found to be appropriate in modelling the upper tail of the distribution. The dissertation contributes positively to the body of knowledge in extreme value theory application to rainfall data and makes recommendations to the government agencies on the long-term rainfall variability and their negative impact on the economy.

Thesis (M. Sc. (Statistics)) -- University of Limpopo, 2020 / The main purpose of modelling rare events such as heavy rainfall, heat waves, wind speed, interest rate and many other rare events is to try and mitigate the risk that might arise from these events. Heavy rainfall and floods are still troubling many countries. Almost every incident of heavy rainfall or floods might result in loss of lives, damages to infrastructure and roads, and also financial losses. In this dissertation, the interest was in modelling average monthly rainfall for South Africa using extreme value theory (EVT). EVT is made up mainly of two approaches: the block maxima and peaks-over thresh old (POT). This leads to the generalised extreme value and the generalised Pareto distributions, respectively. The unknown parameters of these distri butions were estimated using the method of maximum likelihood estimators in this dissertation. According to goodness-of-fit test, the distribution in the Weibull domain of attraction, Gumbel domain and generalised Pareto distri butions were appropriate distributions to model the average monthly rainfall for South Africa. When modelling using the POT approach, the point process model suggested that some areas within South Africa might experience high rainfall in the coming years, whereas the GPD model suggested otherwise. The block maxima approach using the GEVD and GEVD for r-largest order statistics also revealed similar findings to that of the GPD. The study recommend that for future research on average monthly rainfall for South Africa the findings might be improved if we can invite the Bayesian approach and multivariate extremes. Furthermore, on the POT approach, time-varying covariates and thresholds are also recommended. / National Research Foundation (NRF) and South African Weather Service (SAWS)

A project report submitted to the Faculty of Engineering, University of the Witwatersrand, Johannesburg, in partial fulfilment of the requirements for the degree Of Master Of Science in Engineering. / The temporal distribution of rainfall can have a significant effect on peak runoff, especially so in the small catchments that are typical of the Witwatersrand. This report investigates the shape of the natural hyetoraph and its use in the analysis of peak runoff. It describes the climatology of the sub-continent and rain-producing systems. Then more specifically, aspects of rainfall over Johannesburg are discussed. Some Of the more commonly-used temporal distributions of rainfall are reviewed and the relationship between intensity-time distributions and mass curves is illustrated. Mass curves are derived using data from a rain gauge in Norwood, Johannesburg. The data is analysed with the assistance of a computer program and classified into quartiles. The quartiles are further analysed in an attempt to define their characteristics in greater detail. The mass curves are used wIth a hydrological model to generate hydrographs. The values of runoff peaks are found to be comparable with those obtained using currently accepted temporal rainfall distributions. / AC 2018