The statistical modeling of spatial extreme events, augmented by graphical models, provides a comprehensive framework for the development of techniques and models to describe natural phenomena in a variety of environmental, geoscience, and climate science applications. In a changing climate, the impact of natural hazards, such as wildfires, is believed to have evolved in frequency, size, and spatial extent, although regional responses may vary. The aforementioned impacts are of great significance due to their association with air pollution, irreversible harm to the environment and atmosphere, and the fact that they put human lives at risk. The prediction of wildfires holds significant importance within the realm of wildfire management due to its influence on the allocation of resources, the mitigation of detrimental consequences, and the subsequent recovery endeavors. Therefore, the development of robust statistical methodologies that can accurately forecast extreme wildfire occurrences across spatial and temporal dimensions is of great significance. In this thesis, we develop new spatial statistical models, combined with popular machine learning techniques, as well as novel extreme-value methods to enhance the prediction of wildfire risk using graphical models. First, in order to jointly efficiently model high-dimensional wildfire counts and burnt areas over the whole continguous United States, we propose a four-stage zero-inflated bivariate spatiotemporal model combining low-rank spatial models and random forests. Second, to model high values of the McArthur Forest Fire Danger Index over Australia, we develop a novel spatial extreme-value model based on mixtures of tree-based multivariate Pareto distributions. Our new methodology combines theoretically justified spatial extreme models with a computationally convenient graphical model framework to spatial problems in high dimensions efficiently. Third, we exploit recent advancements in deep learning and build a parametric regression model using graphic convolutional neural networks and the extended Generalized Pareto distribution, allow us to jointly model moderate and extreme wildfires observed on irregular spatial grid. We work with a novel dataset of Australian wildfires from 1999 to 2019, and analyse monthly spread over areas correspond to Statistical Area Level 1 regions. We highlight the efficacy of our newly proposed model and perform risk assessment for Australia and dense communities.

Thesis (Ph. D. (Statistics)) -- University of Limpopo, 2016. / Statistics of extremes has seen much growth both in theory and application since its early theoretical developments almost a century ago in the 1920s and its first major applications to real-life problems pioneered by Emil Gumbel in the early 1940s. Although the theory and applications of extreme value theory (EVT) have been extensively advanced and utilised in most developed countries,intermsofapplicationslittlehasbeendoneinmanydevelopingcountries in Africa despite the abundance of areas of applications and raw data in some ofthesecountries. Inhydrology,thechoiceoffloodfrequencyprobabilitydistributions for a particular site or region remains the subject of ongoing research. The work contained in this thesis is a contribution towards this area and it addresses this problem in one of the developing and economically challenged countries in Africa, Mozambique, in the lower Limpopo River basin (LLRB). The LLRB is a basin characterised by extreme natural hazards, alternating between extreme floods and severe droughts. ThisthesisisbasedonanextensiveapplicationofEVTtoextremefloodheights data in the LLRB of Mozambique at three sites: Chokwe, Combomune and Sicacate hydrometric stations. Two fundamental approaches of EVT, block maxima and peaks-over-threshold (POT), are used in this thesis. Recent theoretical results by Ferreira and de Haan (2015) have shown that despite its inefficiency due to data lost as a result of blocking, the block maxima approach is more efficient in a number of situations than the POT approach, and the two approaches are quite comparable for large sample sizes. A number of ii candidate distributions are investigated for their goodness-of-fit to the annual daily maximum flood heights in a block maxima realisation at each site. The findings reveal that the GEV distribution is the most appropriate distribution to apply in the LLRB and the distribution can be recommended as the likelihood function for regional and spatial extremes flood frequency analysis in the basin. The thesis addresses the issue of cumulative effects on daily flood heights through a comparative analysis of six annual maxima moving sums. The findings demonstrate that the six annual maxima time series models are notsignificantlydifferentbasedonthecharacteristicsconsideredinthisthesis. In an attempt to reduce uncertainties in the estimates, a Bayesian Markov chain Monte Carlo (MCMC) approach with a conjugate prior and a GEV likelihood function is used to model the tails of the extreme flood heights in the basin. The findings reveal that the addition of prior information in Bayesian MCMC substantially reduces uncertainties in the estimates and improves precision in the predicted extreme floods. The r largest order statistics models developed in this thesis are generally promising and the standard errors of the estimates of the parameters are substantially reduced. In order to account for climate change impact, nonstationary models are considered with the longterm trend and seasonal oscillation index (SOI) (a meteorological variable indicator) as covariates of the parameters of the GEV distribution and the generalised Pareto distribution (GPD). Among the major contributions of this thesis is a proposed procedure for the determination of the 8 days window period used in extracting independent r largest order values within the same year for the r largest order statistics approach. A summary of the key findings and contributions of this thesis are given in Chapter 9. Moreover, contributions by the study topic in each chapter are given at the end of each chapter. / DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoEMaSS) of South Africa

A análise de freqüência de cheias através da distribuição de probabilidade generalizada de valores extremos-GEV tem crescido nos últimos anos. A estimação de altos quantis de cheias é comumente praticada extrapolando o ajuste, representado por uma das 3 formas inversas de distribuição GEV, para períodos de retorno bem superiores ao período dos dados observados. Eventos hidrológicos ocorrem na natureza com valores finitos, tal que, seus valores máximos seguem a forma assintótica da GEV limitada. Neste trabalho estuda-se a estimabilidade da distribuição GEV através de momentos LH, usando séries de cheias anuais com diferentes características e comprimentos, obtidas de séries de vazões diária gerada de diversas formas. Primeiramente, sequências estocásticas de vazões diárias foram obtidas da distribuição limitada como subjacente da distribuição GEV limitada. Os resultados da estimação dos parâmetros via momentos-LH, mostram que o ajuste da distribuição GEV as amostras de cheias anuais com menos de 100 valores, pode indicar qualquer forma de distribuição de valores extremos e não somente a forma limitada como seria esperado. Também, houve grande incerteza na estimação dos parâmetros obtidos de 50 séries geradas de uma mesma distribuição. Ajustes da distribuição GEV às séries de vazões anuais, obtidas séries de fluxo diários gerados com 4 modelos estocásticos disponíveis na literatura e calibrados aos dados dos rio Paraná e dos Patos, resultaram na forma de Gumbel. Propõe-se um modelo de geração diária que simula picos de vazões usando a distribuição limitada. O ajuste do novo modelo às vazões diárias do rio Paraná reproduziu as estatísticas diárias, mensais, anuais, assim como os valores extremos da série histórica. Além disso, a série das cheias anuais com longa duração, foi adequadamente descrita pela forma da distribuição GEV limitada. / Frequency analysis of floods by Generalized Extreme Value probability distribution has multiplied in the last few years. The estimations of high quantile floods is commonly practiced extrapolating the adjustment represented by one of the three forms of inverse GEV distribution for the return periods much greater than the period of observation. The hydrologic events occur in nature with finite values such that their maximum values follow the asymptotic form of limited GEV distribution. This work studies the identifiability of GEV distribution by LH-moments using annual flood series of different characteristics and lengths, obtained from daily flow series generated by various methods. Firstly, stochastic sequences of daily flows were obtained from the limited distribution underlying the GEV limited distribution. The results from the LH-moment estimation of parameters show that fitting GEV distribution to annual flood samples of less than 100 values may indicate any form of extreme value distribution and not just the limited form as one would expect. Also, there was great uncertainty noticed in the estimated parameters obtained for 50 series generated from the some distribution. Fitting GEV distribution to annual flood series, obtained from daily flow series generated by 4 stochastic model available in literature calibrated for the data from Paraná and dos Patos rivers, indicated Gumbel distribution. A daily flow generator is proposed which simulated the high flow pulses by limited distribution. It successfully reproduced the statistics related to daily, monthly and annual values as well as the extreme values of historic data. Further, annual flood series of long duration are shown to follow the form of asymptotic limited GEV distribution.

A análise de freqüência de cheias através da distribuição de probabilidade generalizada de valores extremos-GEV tem crescido nos últimos anos. A estimação de altos quantis de cheias é comumente praticada extrapolando o ajuste, representado por uma das 3 formas inversas de distribuição GEV, para períodos de retorno bem superiores ao período dos dados observados. Eventos hidrológicos ocorrem na natureza com valores finitos, tal que, seus valores máximos seguem a forma assintótica da GEV limitada. Neste trabalho estuda-se a estimabilidade da distribuição GEV através de momentos LH, usando séries de cheias anuais com diferentes características e comprimentos, obtidas de séries de vazões diária gerada de diversas formas. Primeiramente, sequências estocásticas de vazões diárias foram obtidas da distribuição limitada como subjacente da distribuição GEV limitada. Os resultados da estimação dos parâmetros via momentos-LH, mostram que o ajuste da distribuição GEV as amostras de cheias anuais com menos de 100 valores, pode indicar qualquer forma de distribuição de valores extremos e não somente a forma limitada como seria esperado. Também, houve grande incerteza na estimação dos parâmetros obtidos de 50 séries geradas de uma mesma distribuição. Ajustes da distribuição GEV às séries de vazões anuais, obtidas séries de fluxo diários gerados com 4 modelos estocásticos disponíveis na literatura e calibrados aos dados dos rio Paraná e dos Patos, resultaram na forma de Gumbel. Propõe-se um modelo de geração diária que simula picos de vazões usando a distribuição limitada. O ajuste do novo modelo às vazões diárias do rio Paraná reproduziu as estatísticas diárias, mensais, anuais, assim como os valores extremos da série histórica. Além disso, a série das cheias anuais com longa duração, foi adequadamente descrita pela forma da distribuição GEV limitada. / Frequency analysis of floods by Generalized Extreme Value probability distribution has multiplied in the last few years. The estimations of high quantile floods is commonly practiced extrapolating the adjustment represented by one of the three forms of inverse GEV distribution for the return periods much greater than the period of observation. The hydrologic events occur in nature with finite values such that their maximum values follow the asymptotic form of limited GEV distribution. This work studies the identifiability of GEV distribution by LH-moments using annual flood series of different characteristics and lengths, obtained from daily flow series generated by various methods. Firstly, stochastic sequences of daily flows were obtained from the limited distribution underlying the GEV limited distribution. The results from the LH-moment estimation of parameters show that fitting GEV distribution to annual flood samples of less than 100 values may indicate any form of extreme value distribution and not just the limited form as one would expect. Also, there was great uncertainty noticed in the estimated parameters obtained for 50 series generated from the some distribution. Fitting GEV distribution to annual flood series, obtained from daily flow series generated by 4 stochastic model available in literature calibrated for the data from Paraná and dos Patos rivers, indicated Gumbel distribution. A daily flow generator is proposed which simulated the high flow pulses by limited distribution. It successfully reproduced the statistics related to daily, monthly and annual values as well as the extreme values of historic data. Further, annual flood series of long duration are shown to follow the form of asymptotic limited GEV distribution.