FRG: Collaborative Research:Extreme Value Theory for Spatially Indexed Functional Data

该项目的重点是开发统计工具，以模拟环境和气候极端事件的空间和时间结构。大多数气候和环境数据集可被视为曲线的集合，每年一条曲线，可在一个区域内的几个地点获得。例如，特定位置的温度具有年度模式。这种年曲线的形状每年都在变化，从一个地方到另一个地方。在相当大的区域内，极端偏离典型模式会影响农业生产和公共卫生。 经济影响是相当大的，特别是如果它们发生在意想不到的时间和地点。 热浪在大面积地区出现的不寻常时间可能会因作物歉收和森林火灾造成重大经济损失，也会影响公共卫生服务的准备水平。 同样，长时间的寒冷，无风暴的冬季天气往往导致人口稠密的山谷中颗粒污染水平的增加。 公职人员必须充分了解此类极端事件的可能范围和影响。 该项目将有助于严格和客观地了解所涉及的风险，并为农业、公共卫生、精算学、气候学和生态学领域的研究人员和决策者提供定量工具，该项目力求建立一个统计框架，对一个区域可能出现的偏离通常年度模式的极端情况进行定量评估，即，在历史记录中没有观察到的形式的偏离，但可以以正概率发生。该项目的主要重点是建立一个数学框架，并通过开发统计软件加以实施。建立在功能数据分析，极值理论和时空统计的最新进展，建模的极端分布曲线在空间位置将开发的方法。 极值曲线将由定义在曲线所在的函数空间上的泛函确定。 这项工作将通过对几个历史数据集、衍生数据集和计算机数据集的分析来指导和验证。 探索性分析将揭示极端形状的最突出的特性。随后将建立模型和发展渐近理论，以评估以前未观察到的事件的概率。这些模型将揭示探索性分析中不明显的空间索引功能数据的极值特征。 将获得建立置信区的程序，在置信区中，可能以规定的概率发生极端偏离。还将推导出用于评估极端形状及其发生区域的趋势的探索和推理工具。

This project focuses on the development of statistical tools to model the spatial and temporal structure of environmental and climate extreme events. Most climate and environmental data sets can be viewed as collections of curves, one curve per year, available at several locations within a region. For example, temperature at a specific location has an annual pattern. The shapes of such annual curves change from year to year, and from location to location. Extreme departures from a typical pattern over a sizeable region can impact agricultural production and public health. The economic impacts are considerable, particularly, if they occur at unexpected times and locations. An unusual timing of a heat wave over a large area may cause significant economic damage due to crop failure and forest fires, and also affect the level of preparedness of public health services. Similarly, long spells of cold, storm-free winter time weather often lead to an increase in particulate pollution levels in densely populated mountain valleys. It is important that public officials are well-informed about the possible range and impact of such extreme events. This project will contribute toward a rigorous and objective understanding of the risks involved, and provide quantitative tools for researchers and decision makers in the fields of agriculture, public health, actuarial science, climatology and ecology.The project seeks to develop a statistical framework for a quantitative assessment of possible extremal departures from the usual annual pattern over a region, i.e. departures of the form that have not been observed in historical records, but can occur with a positive probability. The primary focus of the project is the creation of a mathematical framework, and implementation through the development of statistical software. Building on recent advances in functional data analysis, extreme value theory and spatio-temporal statistics, methodology for modeling the extremal distributions of curves observed at spatial locations will be developed. Extreme curves will be determined by functionals defined on a function space in which the curves live. The work will be guided and validated by the analysis of several historical, derived, and computer data sets. Exploratory analysis will reveal the most prominent properties of extremal shapes. This will be followed by model building and the development of asymptotic theory needed to evaluate probabilities of events not previously observed. The models will reveal extremal features of the spatially indexed functional data that are not apparent from the exploratory analysis. Procedures for the construction of confidence regions, where extremal departures may occur with prescribed probability, will be obtained. Exploratory and inferential tools for the assessment of trends in the extremal shapes and regions over which they occur will also be derived.