Collaborative Research: Extremes in High Dimensions: Causality, Sparsity, Classification, Clustering, Learning

近年来，透过新闻报道和亲身经历，市民对极端事件有了深刻的认识，特别是极端天气情况，例如热浪延长、极寒期、龙卷风和飓风的次数和强度增加，或创纪录的降水导致前所未有的洪水。就在过去几年中，极端气候事件的保险理赔数额惊人，其中包括2019年4月的密苏里河洪水（108亿美元）、2018年10月的迈克尔飓风（250亿美元）、2017年12月的加州野火（187亿美元）、2012年美国干旱/热浪（339亿美元）和2012年10月的桑迪飓风（734亿美元）。该清单不包括2008年金融危机等非气候极端事件，也不包括当前的covid-19大流行。今天经历的许多与天气、环境、工业、流行病学、经济或社交媒体相关的极端事件正在以更频繁的速度发生，这往往会给我们的社会带来巨大的损失，从经济到人类生活再到我们的生活方式。虽然在稳定状态下极端事件的发生是相当容易理解的，但很明显，极端事件的优势表明稳态假设不再有效。这项研究的主要目的是试图了解各种因素对极端事件频率和强度的因果影响，这些因素来自潜在的大量变量，包括不断变化的环境条件、美国境内的人口流动、不断变化的景观和不断变化的经济条件。从许多变量中，我们希望产生一种方法来提取数据中对描述和预测极端事件有直接影响的重要特征。这项研究可能对需要预测和规划极端事件的政策制定者有用，从而制定合理的策略来减轻极端事件对社会的影响。研究生资助将用于跨学科研究。这个研究项目的主要目标是设计新的工具来分析和模拟极端的无数情况，远远超出了经典的极值理论的界限。这包括检测通常是更小维度的非线性集合，这些集合可以提供对高维极值的充分描述，为此我们希望应用强大的现代学习技术（如基于图的学习方法），使我们能够从数据中确定这种极值支持。一般来说，在描述高维极值的指数测度中检测稀疏性，即定位（通常是许多）具有大多数指数测度支持的低维区域将是本研究的重点。本研究的第二个主要推力集中在小维度和大维度问题的因果关系问题上。在最基本的形式中，如果X的某些变化（有时本身是极端的，但并不总是如此）影响Y的尾部行为，则一组变量X被认为是依赖向量Y的尾部因果关系。这种类型的一个重要设置是极端事件因果关系的潜在结果框架，这将是本项目研究议程的主要焦点。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

In recent years, through news reports and first-hand experience, the general public has become keenly aware of extreme events, in particular, of extreme weather conditions such as extended heat waves, periods of extreme cold, an increase in the number and intensity of tornadoes and hurricanes, or periods of record precipitation resulting in unprecedented floods. Just in the past few years, the insurance claims from extreme climatic events have been staggering, which include the Missouri River flood in April 2019 ($10.8B), Hurricane Michael in October 2018 ($25B), the California wildfires in December 2017 ($18.7B), the US drought/heatwave in 2012 ($33.9B), and Hurricane Sandy in October 2012 ($73.4B). This list does not include non-climatic extreme events such as the financial crisis from 2008 nor the current covid-19 pandemic. Many of the extreme events experienced today that are weather, environmental, industrial, epidemiological, economic, or social media related are occurring at a more frequent rate, which often result in huge losses to our society in a variety of ways from financial to human life to our way of life. While the occurrence of extreme events is reasonably well understood in steady state situations, it has become clear that the preponderance of extremes events suggest that the steady-state assumption is no longer valid. The key objective of this research is to try to understand causal impacts of various factors from a potentially large array of variables including changing environmental conditions, demographic movements within the US, changing landscapes, and changing economic conditions, on the frequency and magnitude of extreme events. From many variables, we hope to produce methodology to extract the important features in the data that have a direct impact on describing and predicting extremes. This research is potentially of use to policymakers who need to anticipate and plan for extreme events leading to sensible strategies for mitigating their impact on society. The graduate student support will be used for interdisciplinary research.The principal goal of this research project is to design new tools for analyzing and modeling extremes in a myriad of situations that go well beyond the boundaries of classical extreme value theory. These include detection of often nonlinear sets of much smaller dimension that can provide an adequate description of extremes in high dimensions, for which we hope to apply the powerful modern learning techniques (such as graph-based learning methods) that allow us to determine this extremal support from the data. In general, detecting sparsity in the exponent measure describing high-dimensional extremes, i.e., locating (often numerous) low-dimensional regions which carry most of the support of exponent measure will be a key focus of this research. A second main thrust of this research centers on the issue of causality in both small and large dimensional problems. In the most basic form, a set of variables X is said to be tail causal to a dependent vector Y if certain changes in X (sometimes themselves extreme but not always so) impact the tail behavior of Y. An important setting of this type is the potential outcomes framework for causality of extreme events, which will be a major focus in this project's research agenda.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.