Bayesian Inference for Peaks Over Threshold Models for Multivariate and Spatial Extremes

极值理论是概率统计的一个分支，主要研究罕见事件。这种方法在科学和技术的许多领域都有应用。例如，对精算风险的量化、对金融市场大幅波动的估计以及对最大水量的估计。与我们社会特别相关的是对极端气候事件的研究。气候相关变量的历史记录提供了极端天气加剧的证据。气候预测表明，具有灾难性可能性的事件的频率和强度将进一步增加。这项研究的重点是开发统计方法，使之能够仔细评估与极端事件有关的不确定性。建议的方法将侧重于联合观察几个变量的模型，并适用于在大空间领域收集的观测数据。对罕见事件发生的不确定性的概率评估将通过贝叶斯方法实现。这将为理性决策和决策提供有力的工具。在本项目中，提出了一种新的极值分布统计分析方法。这些方法是基于对感兴趣变量使用超过固定阈值的量，或超过阈值的峰值(POT)。将开发和实施对(A)多变量观测、(B)空间索引场和(C)空间多变量观测场进行贝叶斯推断的POT方法。在极值理论中，重点是外推，因为极值观测是用来描述分布尾部的行为的。一元极值推理的理论和方法已经建立并得到了充分的发展。对于多变量问题，建立不同变量的联合尾部相关性模型是关键。从这个意义上讲，这个理论是很好理解的，但推理方法并不像单变量情况下那样简单。对于锅法来说，情况尤其如此。在处理地理参考数据时引入了更高级别的复杂性。事实上，在空间环境中，不可能为在任意数量的地点收集的观测数据编写现实POT模型的全部可能性。这项研究的重点是开发以下方法：(A)在概念上清楚地使用作为贝叶斯分层模型核心的简单因式分解来具体说明；(B)允许完全集成的贝叶斯推理，其解释所有估计不确定性并以概率方式量化它；(C)具有理论上合理的渐近性质；(D)提供广泛的尾部相关性的灵活表征；以及(E)在计算上对于大空间域是可行的。

Extreme value theory is a branch of probability and statistics that focuses on the study of rare events. There are many areas of science and technology where such methods find applications. Examples include the quantification of actuarial risk, estimation of large fluctuations in financial markets, and the estimation of maximum water flow. Of particular relevance for our society is the study of extreme climate events. Historical records of climate related variables provide evidence that there is an intensification of extreme weather. Climate projections indicate that the frequency and intensity of events with catastrophic potential will increase even further. This research focuses on the development of statistical methods that will enable careful assessment of the uncertainties related to extreme events. The proposed methods will focus on models that look jointly at several variables and apply to observations collected in large spatial domains. Probabilistic assessment of the uncertainties in the occurrence of rare events will be made possible by a Bayesian approach. This will provide a powerful tool for rational decision and policy making.In this project, novel methodology for the statistical analysis of the distributions of extreme values is proposed. The methods are based on using the amounts in excess of a fixed threshold for the variables of interest, or peaks over thresholds (POT). POT methods to perform Bayesian inference for (a) multivariate observations, (b) spatially indexed fields, and (c) fields of multivariate observations in space will be developed and implemented. In extreme value theory, the focus is on extrapolation as scarce extreme observations are used to describe the behavior of the tails of the distribution. The theory and the methods for inference on univariate extreme values are firmly established and fully developed. For multivariate problems, it is key to model the joint tail dependence of the different variables. In this sense, the theory is well understood, but inferential methods are not as straightforward as in the univariate case. This is especially true for POT methods. A further level of complication is introduced when dealing with georeferenced data. In fact, in the spatial setting, it is impossible to write the full likelihood of realistic POT models for observations collected at an arbitrary number of locations. This research focuses on the development of methods that (a) are conceptually clear to specify using a simple factorization that is at the core of Bayesian hierarchical models, (b) allow for fully integrated Bayesian inference that accounts for all estimation uncertainty and quantifies it probabilistically, (c) have theoretically sound asymptotic properties, (d) provide flexible characterizations of a wide range of tail dependence, and (e) are computationally feasible for large spatial domains.