Analyzing Dependent Extremes via Joint Quantile Regression

飓风哈维和大衰退之间有什么共同点？两者都是普通和松散联系的过程以同步的方式达到非凡水平的例子。准确预测同步极端结果的规模和频率对于稳健的风险管理至关重要。但这项任务对数据科学来说仍然是一个艰巨的挑战，因为数据科学主要是围绕着平均行为和独立性的概念建立的。一个恰当的例子是回归分析，其中研究一组结果如何同时受到一组共同预测因子的影响。目前的统计方法既可以解决多个结果之间的相互依赖性，也可以模拟单一结果从普通水平到特殊水平的过渡。但没有什么令人满意的东西可以同时处理这两个问题。这项研究填补了这一空白，新的数据分析工具的基础上分位数驱动的回归分析。新工具专门用于分析在空间和时间上或在网络集群内记录的数据。他们受到详细的数学审查的准确性和可靠性。在金融和环境科学方面的应用是为了评估新工具在科学调查和决策方面的效用。该项目将统计研究与软件开发和研究生教育相结合。分位数是用从零到一的水平表示的简单的分位数，而不是一个百分点。变量的分位数可以直接访问其在普通水平和非常水平之间的平滑过渡。在标准分位数回归中，人们估计预测因子在结果的任何给定分位数水平上的影响。当观测单元相互独立时，这种估计很容易进行;不需要预测结果关系的详细概率模型。但是，具有已知依赖结构的数据提出了一个更为复杂的挑战;准确的估计需要调整邻近单元之间的固有噪声相关性。这样的任务仍然超出了普通分位数回归方法的范围，因为它们的无模型性质和它们孤立地关注单个分位数。相比之下，联合分位数回归通过为结果引入完整的概率模型并同时在所有分位数水平上进行联合估计，使这项任务变得可行。本研究探讨使用Copula建模来解决联合分位数回归中的噪声相关性，重点是为每种数据类型适当定制Copula公式。对所得工具的统计保证进行了严格的渐近分析。将为模型评估和选择开发定量和可视化诊断工具。所有的工具都将被整合到R软件包“qrjoint”中，可通过The Comprehensive R Archive Network获得。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，被认为值得支持。

What is common between Hurricane Harvey and the Great Recession? Both are examples of ordinary and loosely connected processes attaining extraordinary levels in a synchronized manner. Accurately predicting the size and frequency of synchronized extreme outcomes is crucial to robust risk management. But this task remains a difficult challenge to data science that has been mostly built around the notions of average behavior and independence. A case in point is regression analyses where one studies how a group of outcomes are simultaneously influenced by a common set of predictors. Current statistical methods can either address inter-dependency between multiple outcomes, or model transitions from ordinary to extraordinary levels of a single outcome. But nothing satisfactory exists to handle both. This research fills this gap with new data analysis tools based on quantile driven regression analysis. The new tools are specialized for analyzing data recorded over space and time or within network clusters. They are subjected to detailed mathematical scrutiny for accuracy and reliability. Applications to finance and environmental sciences are carried out to assess the usefulness of the new tools in scientific investigation and policy making. The project integrates statistical research with software development and graduate education. Quantiles are simply percentiles expressed in terms of a level varying from zero through one, as opposed to a percentage point. The quantiles of a variable give direct access to its smooth transitions between ordinary and extraordinary levels. In standard quantile regression, one estimates the effects of predictors at any given quantile level of an outcome. Such estimation is easy to carry out when observation units are mutually independent; there is no need of a detailed probabilistic model for the predictor-outcome relation. But data with known dependency structures present a far more complex challenge; accurate estimation requires adjusting for intrinsic noise correlation between units in close proximity. Such a task remains beyond the scope of ordinary quantile regression methods due to their model-free nature and their focus on single quantiles in isolation. In contrast, joint quantile regression makes this task feasible by incorporating a full probabilistic model for the outcome and enabling a joint estimation at all quantile levels at once. The research investigates the use of copula modeling to address noise correlation in joint quantile regression, focusing greatly on appropriate customization of the copula formulation for each data type. A rigorous asymptotic analysis is carried out toward statistical guarantees of the resulting tools. Quantitative and visualization-based diagnostic tools will be developed for model assessment and selection. All tools will be incorporated in the R package “qrjoint” available through The Comprehensive R Archive Network.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.