Extremes Models and Methods from Transformed Linear Operations

在许多科学、工程和商业应用中，与极端量级和罕见事件相关的风险的量化和评估是重要的。单变量极值方法已经发展得很好，但是需要易于实现的统计方法来描述和建模高维、时间序列和空间背景下的极值依赖。 线性统计方法包括传统的多元分析、时间序列和空间统计，在非极值统计中普遍存在。 最近，PI和合著者通过利用变换线性运算将线性统计方法和极值分析的看似不同的领域联系起来。这种方法的扩展将用于开发用于分析高维极值依赖，建模极值依赖时间和建模空间极值的方法。由于所提出的工作是在非极端设置现有的线性模型和方法的启发，模型和方法将相对简单和熟悉。这个项目将产生通过应用变换线性运算来描述和建模极值依赖的统计方法。在最近提交的一篇论文中，PI和合著者通过变换的线性运算将线性代数与正则变分联系起来，正则变分是极值依赖的一个广泛使用和理论上合理的框架。PI和合著者获得了一个合理的向量空间的极端产生的基础的概念，提出了一个极端的模拟协方差矩阵，并执行此矩阵的特征分解有助于理解高维尾部依赖。 本项目旨在为极端空间数据开发一个简单的线性模型，类似于非极端空间统计学中的空间自回归模型，以及受熟悉的阿尔马模型启发的变换线性时间序列模型，但适用于极端时间序列数据。本项目还将为空间和时间序列模型开发推理程序。 此外，该项目还将进一步开发用于理解高维极值依赖的线性方法，并探索极值的条件依赖和独立的想法。该奖项反映了NSF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而被认为值得支持。

Quantification and assessment of risk associated with extreme-in-magnitude and rare events are important in many science, engineering, and business applications. Univariate extremes methods are well-developed, but there is a need for easily implementable statistical methods to describe and model extremal dependence in high dimensions, and in the time series and spatial contexts. Linear statistical methods including traditional multivariate analysis, time series, and spatial statistics are ubiquitous in non-extreme statistics. Recently, the PI and coauthor connected the seemingly disparate areas of linear statistical methods and extreme value analyses by utilizing transformed-linear operations. An extension of this approach will be used to develop methods for analyzing high-dimensional extremal dependence, modeling extremal dependence in time, and modeling spatial extremes. Because the proposed work is inspired by existing linear models and methods in the non-extreme setting, the models and methods will be relatively simple and familiar. This project will produce statistical methods for describing and modeling extremal dependence via applying transformed-linear operations. In a recently submitted paper, the PI and coauthor link linear algebra to regular variation, a widely-used and theoretically-justified framework for extremal dependence, via transformed linear operations. The PI and coauthor obtain a sensible vector space for extremes yielding the notion of basis, propose an extremes analog to the covariance matrix, and perform an eigendecomposition of this matrix useful for understanding high-dimensional tail dependence. This project aims to develop a simple linear model for extreme spatial data, analogous to the spatial autoregressive model in non-extreme spatial statistics, and transformed-linear time series models inspired by familiar ARMA models, but appropriate for extreme time series data. This project will also develop inference procedures for both the spatial and time series models. Furthermore, this project will further develop linear methods for understanding extremal dependence in high dimensions and explore the idea of conditional dependence and independence for extremes.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.