Zugl.: Clausthal, Techn. Univ., Diss., 2008

Techn. Universiẗat, Diss., 2002--Berlin.

Universiẗat, Diss., 2004--Münster (Westfalen).

Techn. Hochsch., Diss., 2005--Aachen.

Statistical modeling of a nonstationary spatial extremal dependence structure is a challenging problem. In practice, parametric max-stable processes are commonly used for modeling spatially-indexed block maxima data, where the stationarity assumption is often made to make inference easier. However, this assumption is unreliable for data observed over a large or complex domain. In this work, we develop a computationally-efficient method to estimate nonstationary extremal dependence using max-stable processes, which builds upon and extends an approach recently proposed in the classical geostatistical literature. More precisely, we divide the spatial domain into a fine grid of subregions, each having its own set of dependence-related parameters, and then impose LASSO ($L_1$) or Ridge ($L_2$) penalties to obtain spatially-smooth estimates. We then also subsequently merge the subregions sequentially together with a new algorithm to enhance the model's performance. Here we focus on the popular Brown-Resnick process, although extensions to other classes of max-stable processes are also possible. We discuss practical strategies for adequately defining the subregions and merging them back together. To make our method suitable for high-dimensional datasets, we exploit a pairwise likelihood approach and discuss the choice of pairs to achieve reasonable computational and statistical efficiency. We apply our proposed method to a dataset of annual maximum temperature in Nepal and show that our approach fits reasonably and realistically captures the complex non-stationarity in the extremal dependence.

La hausse des températures et les modifications du régime des précipitations affectent les écosystèmes particulièrement fragiles des régions arctiques et subarctiques et ont des conséquences socio-économiques pour les populations locales. La perception et l'évaluation des opportunités et des risques qui y sont associés, dépendent des moyens de subsistance, des normes, des valeurs et des représentations du monde des individus qui y sont confrontés.La perception du climat est culturellement et socialement construite. Cette perception varie dans le temps et dans l'espace et, parfois même, diffère parmi différents groupes sociaux en fonction des valeurs et des modes de représentation du monde. Pour toutes ces raisons, nous avons non seulement étudié certains phénomènes météorologiques mais aussi intégré le cadre culturel, politique et historique dans lesquels ils s'inscrivent. Nous avons analysé l'environnement et le climat – et par extension, les événements météorologiques à forts impacts – comme des objets socio-culturels afin de mieux comprendre à la fois leurs impacts mais aussi leurs perceptions par les habitants. Ce travail se situe à la croisée de ces chemins, en confrontant l'observation, et donc les changements, et la perception qu'en ont différents acteurs, en considérant deux régions distinctes, les mers nordiques et la République de Touva.Dans une première partie, nous avons analysés l'impact des Polar Lows, d'intenses cyclones de méso-échelle qui se développent sur les mers libres de glace de l'Arctique pendant l'hiver, sur les régions côtières du nord de la Norvège. Ces systèmes sont associés à des vents de surface forts avec bien souvent des rafales qui peuvent être très violentes (Heinemann et Claud, 1997). Les conditions en mer lors du passage d'un PL peuvent s'avérer dangereuses avec des fortes vagues, des précipitations neigeuses brutales et du blizzard. Ces événements météorologiques extrêmes représentent un véritable risque pour les activités maritimes et côtières de la région, notamment pour le transport maritime, la pêche et les plateformes pétrolières et gazières offshore. En effet, les nouvelles zones libres de glace offrent de multiples opportunités économiques dans ces régions, en particulier en mer de Barents. Or dans le même temps, le recul de la banquise élargit mécaniquement les régions de formation des PLs.Dans une seconde partie, nous avons regardé les impacts du changement climatique et des événements météorologiques sévères en République de Touva. La République de Touva se localise entre 49°5 et 53°5 N en latitude, 88°5 et le 99°E en longitude, c'est par conséquent une région subarctique très méridionale. Le climat y est extrêmement continental et les précipitations ont tendance à être faibles en raison de la faible teneur en humidité dans l'air froid. La plupart du territoire est caractérisé par une végétation forestière de taïga ou de steppe semi-aride. Au cours du XXème siècle, la région a connu des changements socio-économiques majeurs, parfois brutaux, avec notamment le passage d'une société communiste à l'économie planifiée à une économie de marché au début des années 1990. Malgré ces récents bouleversements, les pasteurs nomades (chevaux, vaches, yaks, moutons, chameaux) en Touva occidentale et les chasseurs-cueilleurs éleveurs de rennes en Touva orientale, vivent toujours en étroite relation avec l’environnement naturel. Les populations autochtones de Touva, confrontées à un changement rapide de la société et à des changements globaux causés par certaines politiques régionales et nationales contemporaines, avec notamment l’expansion de l'industrie minière et par le développement de mégaprojets (complexe hydroéléctrique, construction d'une voie chemin de fer), offrent des points de vue variés, en fonction de leurs modes de vie, sur les changements environnementaux qu'ils observent et leurs impacts sur leurs activités quotidiennes. / The perception of the climate is culturally and socially constructed. For this reason, we have studied some weather events integrating the cultural, political and historical contexts in which they occur.In a first part, we analyzed the impact of Polar Lows, intense mesocyclones that develop over ice-free Arctic seas during winter time, on coastal regions of Norway. The passage of PL can provoke dangerous sea conditions with strong waves, sudden snowfall and blizzard. This phenomenon may represent a risk to maritime and coastal activities in the region, particularly for shipping, fishing and oil and gas offshore platforms.In a second part we studied the impacts of climate change and severe weather events in the Republic of Tuva. Tuva is a very southern subarctic region. Its climate is extremely continental and precipitation tend to be low due to the low moisture content in the cold air. During the twentieth century, the region has experienced major socio-economic changes, sometimes brutal, including the transition from a communist and planned economy to a market economy in the early 1990. Despite these recent changes, pastoralists in western Tuva (horses, cows, yaks, sheep, camels) and reindeer herders in eastern Tuva, still live in close contact with the natural environment. Indigenous peoples of Tuva are facing global changes caused by certain contemporary regional and national policies, including the expansion of the mining industry and the development of mega projects. They offer different points of view, describing environmental changes and their impact on their daily activities.

The classical Extreme Value Theory deals with independent random variables. If random variables are dependent, large values tend to cluster (that is, one large value is followed by a series of large values). It is of interest to describe probabilistically the clustering and estimate the relevant cluster functionals. We consider disjoint blocks, sliding blocks and runs estimators of cluster indices. Using a modern theory of multivariate, regularly varying time series, we obtain consistency results and central limit theorems under conditions that can be easily verified for a large class of short-range dependent models. In particular, we show that in the Peak-over-Threshold framework, all the estimators have the same limiting variances. This solves a longstanding open problem and is in contrast to the Block Maxima method. Our findings are illustrated by simulation experiments.

Cette thèse présente des contributions à la modélisation multivariée et spatiale des valeurs extrêmes. Au travers d'une extension de la représentation par coordonnées pseudo-polaires, représentation très utilisée en théorie des valeurs extrêmes, une approche unifiée et générale pour la modélisation en valeurs extrêmes est proposée. La variable radiale de ces coordonnées est donnée par une fonction non négative et homogène dite fonction d'agrégation permettant d'agréger un vecteur dans un scalaire. La loi de la variable d'angle est caractérisée par une mesure dite angulaire ou spectrale. Nous définissons les lois radiales de Pareto et une version inversée de ces lois, toutes deux motivées dans le cadre de la variation régulière multivariée. Cette classe de modèles est assez souple et permet de modéliser les valeurs extrêmes de vecteurs aléatoires dont la variable agrégée est à décroissance de type Pareto ou Pareto inversé. Dans le cadre spatial, nous mettons l'accent sur les lois bivariées à l'instar des méthodes couramment utilisées. Des approches inférentielles originales sont développées, fondées sur un nouvel outil de représentation appelé spectrogramme. Le spectrogramme est constitué des mesures spectrales caractérisant le comportement extrémalbivarié. Enfin, la construction dite spectrale du processus limite max-stable des processus elliptiques, à savoir le processus t-extrémal, est présentée. Par ailleurs, nous énonçons des méthodesd'inférence et explorons des méthodes de simulation des processus de type max-stable et de type Pareto. L'intérêt pratique des modèles et méthodes proposés est illustré au travers d'applications à des données environnementales et financières. / This PhD thesis presents contributions to the modelling of multivariate andspatial extreme values. Using an extension of commonly used pseudo-polar representations inextreme value theory, we propose a general unifying approachto modelling of extreme value dependence. The radial variable of such coordinates is obtained from applying a nonnegative and homogeneous function, called aggregation function, allowing us to aggregate a vector into a scalar value. The distribution of the angle component is characterized by a so-called angular or spectral measure. We define radial Pareto distribution and an inverted version of thesedistributions, both motivated within the framework of multivariateregular variation. This flexible class of models allows for modelling of extreme valuesin random vectors whose aggregated variable shows tail decay of thePareto or inverted Pareto type. For the purpose of spatial extreme value analysis, we follow standard methodology in geostatistics of extremes and put the focus on bivariatedistributions. Inferentialapproaches are developed based on the notion of a spectrogram,a tool composed of thespectral measures characterizing bivariate extreme value behavior. Finally, the so-called spectral construction of the max-stable limit processobtained from elliptical processes, known as extremal-t process, ispresented. We discuss inference and explore simulation methods for the max-stableprocess and the corresponding Pareto process. The utility of the proposed models and methods is illustrated throughapplications to environmental and financial data.

Urbanisation alters the hydrologic response of a catchment, resulting in increased runoff rates and volumes, and loss of infiltration and base flow. Quantification of uncertainties is important in hydrologic designs of urban infrastructure. Major sources of uncertainty in the Intensity Duration Frequency (IDF) relationships are due to insufficient quantity and quality of data leading to parameter uncertainty and, in the case of projections of future IDF relationships under climate change, uncertainty arising from use of multiple General Circulation Models (GCMs) and scenarios. The work presented in the thesis presents methodologies to quantify the uncertainties arising from parameters of the distribution fitted to data and the multiple GCMs using a Bayesian approach. High uncertainties in GEV parameters and return levels are observed at shorter durations for Bangalore City. Twenty six GCMs from the CMIP5 datasets, along with four RCP scenarios are considered for studying the effects of climate change. It is observed that the uncertainty in short duration rainfall return levels is high when compared to the longer durations. Further it is observed that parameter uncertainty is large compared to the model uncertainty. Disaggregation of precipitation extremes from larger time scales to smaller time scales when the extremes are modeled with the GPD is burdened with difficulties arising from varying thresholds for different durations. In this study, the scale invariance theory is used to develop a disaggregation model for precipitation extremes exceeding specified thresholds. A scaling relationship is developed for a range of thresholds obtained from a set of quantiles of non-zero precipitation of different durations. The disaggregation model is applied to precipitation datasets of Berlin City, Germany and Bangalore City, India. From both the applications, it is observed that the uncertainty in the scaling exponent has a considerable effect on uncertainty in scaled parameters and return levels of shorter durations. A Bayesian hierarchical model is used to obtain spatial distribution of return levels of precipitation extremes in urban areas and quantify the associated uncertainty. Applicability of the methodology is demonstrated with data from 19 telemetric rain gauge stations in Bangalore City, India. For this case study, it is inferred that the elevation and mean monsoon precipitation are the predominant covariates for annual maximum precipitation. For the monsoon maximum precipitation, it is observed that the geographic covariates dominate while for the summer maximum precipitation, elevation and mean summer precipitation are the predominant covariates. In this work, variation in the dependence structure of extreme precipitation within an urban area and its surrounding non-urban areas at various durations is studied. The Berlin City, Germany, with surrounding non-urban area is considered to demonstrate the methodology. For this case study, the hourly precipitation shows independence within the city even at small distances, whereas the daily precipitation shows a high degree of dependence. This dependence structure of the daily precipitation gets masked as more and more surrounding non-urban areas are included in the analysis. The geographical covariates are seen to be predominant within the city and the climatological covariates prevail when non-urban areas are added. These results suggest the importance of quantification of dependence structure of spatial precipitation at the sub-daily timescales, as well as the need to more precisely model spatial extremes within the urban areas. The work presented in this thesis thus contributes to quantification of uncertainty in precipitation extremes through developing methodologies for generating probabilistic future IDF relationships under climate change, spatial mapping of probabilistic return levels and modeling dependence structure of extreme precipitation in urban areas at fine resolutions.

The global optimization problem -- i.e., the problem of finding global extreme points of given function on restricted domain of values -- often appears in many real-world applications. Improving efficiency of this task can reduce the latency of the application or provide more precise result since the task is usually solved by an approximative algorithm. This thesis focuses on the practical aspects of global optimization algorithms, especially in the domain of algorithmic trading data analysis. Successful implementations of the global optimization solver already exist for CPUs, but they are quite time demanding. The main objective of this thesis is to design a GO solver that utilizes the raw computational power of the GPU devices. Despite the fact that the GPUs have significantly more computational cores than the CPUs, the parallelization of a known serial algorithm is often quite challenging due to the specific execution model and the memory architecture constraints of the existing GPU architectures. Therefore, the thesis will explore multiple approaches to the problem and present their experimental results.