A Point Process Characterisation of Extreme Temperatures: an Application to South African Data

The point process (PP) modelling approach is considered a more elegant alternative of extreme value analysis. This is because of its capability in modelling both the frequency and intensity rates of the occurrence of extremes. In this paper, we demonstrate the use of the PP modelling approach in which stationary and non-stationary models are used in modelling average maximum daily temperature (AMDT) in South Africa. The data constitutes average daily temperature observations that are collected by the South African Weather Services over the period 1 January 2000 to 30 August 2010. This study is interested on the occurrence of extreme high temperature and because of that the data for non-winter season (1 September to 30 April) of each year is used. A penalised regression cubic smoothing spline function is used for non-linear detrending of the data and determining a fixed threshold above which excesses are extracted and used. An extremal mixture model is then fitted to determine a threshold in which a boundary corrected kernel density is fitted to the bulk model and a generalised Pareto distribution (GPD) fitted to the tail of the distribution. The data exhibits properties of short-range dependence and strong seasonality, leading to declustering. An interval estimator method is used to decluster data for the purpose of fitting PP models to cluster maxima. The models that are used in this paper are nested and, as a result, likelihood ratio tests are conducted using the deviance statistic. The tests support the fit of the stationary PP model. We further fitted the stationary GPD and used the formal tests which are the Cramér-von Mises test and the Anderson-Darling test to diagnose fit. These tests and the diagnostic plots support fit of the stationary GPD to cluster maxima. Uncertainty of the estimates of GPD parameters is assessed in this paper using bootstrap re-sampling approach. The stationary PP model was used with the reparameterisation approach to determine frequency of the occurrence of extremely hot days, which are found to be 15 times per year. The modelling framework and results of this study are important to power utility companies in scheduling and dispatching electricity to customers during a hot spell.