Data-based analysis of extreme events: inference, numerics and applications

The concept of extreme events describes the above average behavior of a process, for instance, heat waves in climate or weather research, earthquakes in geology and financial crashes in economics. It is significant to study the behavior of extremes, in order to reduce their negative impacts. Key objectives include the identification of the appropriate mathematical/statistical model, description of the underlying dependence structure in the multivariate or the spatial case, and the investigation of the most relevant external factors. Extreme value analysis (EVA), based on Extreme Value Theory, provides the necessary statistical tools. Assuming that all relevant covariates are known and observed, EVA often deploys statistical regression analysis to study the changes in the model parameters. Modeling of the dependence structure implies a priori assumptions such as Gaussian, locally stationary or isotropic behavior. Based on EVA and advanced time-series analysis methodology, this thesis introduces a semiparametric, nonstationary and non- homogenous framework for statistical regression analysis of spatio-temporal extremes. The involved regression analysis accounts explicitly for systematically missing covariates; their influence was reduced to an additive nonstationary offset. The nonstationarity was resolved by the Finite Element Time Series Analysis Methodology (FEM). FEM approximates the underlying nonstationarity by a set of locally stationary models and a nonstationary hidden switching process with bounded variation (BV). The resulting FEM-BV- EVA approach goes beyond a priori assumptions of standard methods based, for instance, on Bayesian statistics, Hidden Markov Models or Local Kernel Smoothing. The multivariate/spatial extension of FEM-BV-EVA describes the underlying spatial variability by the model parameters, referring to hierarchical modeling. The spatio-temporal behavior of the model parameters was approximated by locally stationary models and a spatial nonstationary switching process. Further, it was shown that the resulting spatial FEM-BV-EVA formulation is consistent with the max-stability postulate and describes the underlying dependence structure in a nonparametric way. The proposed FEM-BV-EVA methodology was integrated into the existent FEM MATLAB toolbox. The FEM-BV-EVA framework is computationally efficient as it deploys gradient free MCMC based optimization methods and numerical solvers for constrained, large, structured quadratic and linear problems. In order to demonstrate its performance, FEM-BV-EVA was applied to various test-cases and real-data and compared to standard methods. It was shown that parametric approaches lead to biased results if significant covariates are unresolved. Comparison to nonparametric methods based on smoothing regression revealed their weakness, the locality property and the inability to resolve discontinuous functions. Spatial FEM-BV-EVA was applied to study the dynamics of extreme precipitation over Switzerland. The analysis identified among others three major spatially dependent regions