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

该项目的重点是开发统计工具，对环境和气候极端事件的时空结构进行建模。大多数气候和环境数据集可以看作是一组曲线，每年一条，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。这种年度曲线的形状因年而异，因地而异。在相当大的区域内，极端偏离典型模式可能会影响农业生产和公众健康。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。由于农作物歉收和森林火灾，热浪在大片地区出现的不同寻常的时机可能会造成重大经济损失，并影响公共卫生服务的防备水平。同样，长期寒冷、无风暴的冬季天气往往会导致人口稠密的山谷中颗粒物污染水平的增加。重要的是，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、气候学和生态学领域的研究人员和决策者提供量化工具。该项目旨在建立一个统计框架，对一个区域可能出现的与通常年度模式的极端偏离进行量化评估，即历史记录中没有观察到的形式偏离，但可能以肯定的概率发生。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。在函数数据分析、极值理论和时空统计学最新进展的基础上，将制定对在空间位置观测到的曲线的极值分布进行建模的方法。极端曲线将由定义在曲线所在的函数空间上的泛函来确定。这项工作将通过对几个历史、派生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最显著的特性。在此之后，将建立模型和发展所需的渐近理论，以评估以前未观察到的事件的概率。这些模型将揭示空间索引功能数据的极端特征，这些特征在探索性分析中并不明显。将获得构建置信域的程序，在该置信域中，可能会以规定的概率发生极端偏离。还将获得用于评估极端形状及其发生区域趋势的探索性和推论工具。

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.