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

近年来，通过新闻报道和亲身体验，广大公众对极端事件，特别是极端天气条件，如延长的热浪、极端寒冷的时期、龙卷风和飓风的数量和强度增加，或导致史无前例的洪水的创纪录降水时期，都有了敏锐的认识。就在过去几年，极端气候事件的保险索赔一直令人震惊，包括2019年4月的密苏里河洪水(108亿美元)、2018年10月的飓风迈克尔(250亿美元)、2017年12月的加州野火(187亿美元)、2012年美国的干旱/热浪(339亿美元)和2012年10月的飓风桑迪(734亿美元)。这份清单不包括非气候极端事件，如2008年的金融危机，也不包括当前的新冠肺炎疫情。今天经历的许多与天气、环境、工业、流行病、经济或社交媒体有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造成从经济到人类生活到我们的生活方式的各种方式的巨大损失。虽然在稳态情况下对极端事件的发生有相当好的理解，但很明显，极端事件的优势表明稳态假设不再有效。这项研究的关键目标是试图从一系列潜在的变量中了解各种因素对极端事件的频率和规模的因果影响，这些变量包括不断变化的环境条件、美国境内的人口变动、不断变化的地形和不断变化的经济状况。我们希望从许多变量中提取数据中对描述和预测极端情况有直接影响的重要特征。这项研究可能对需要预测和计划极端事件的政策制定者有用，他们需要制定合理的战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各种情况下的极端情况，这些情况远远超出了经典极值理论的界限。其中包括检测往往是小得多的非线性集合，这些集合可以对高维中的极值提供足够的描述，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，检测描述高维极值的指数度量中的稀疏性，即定位(通常是大量的)具有指数度量最大支持的低维区域将是本研究的重点。这项研究的第二个主旨集中在小维度和大维度问题中的因果关系问题上。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被称为对依赖向量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NSF的法定使命，并通过使用基金会的智力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situations, it has become clear that the preponderance of extremes events suggest that the steady-state assumption is no longer valid.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.