Analyzing Dependent Extremes via Joint Quantile Regression

飓风哈维和经济大衰退有什么共同之处？两者都是普通和松散连接的进程以同步方式达到非凡水平的例子。准确预测同步极端后果的规模和频率对于健全的风险管理至关重要。但对于数据科学来说，这项任务仍然是一个艰巨的挑战，因为数据科学主要是围绕平均行为和独立的概念建立的。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关系，也可以对单个结果从普通水平到特殊水平的转变进行建模。但没有什么令人满意的东西可以同时处理这两个问题。本研究利用基于分位数驱动回归分析的新数据分析工具填补了这一空白。这些新工具专门用于分析跨越空间和时间或在网络集群内记录的数据。它们的准确性和可靠性都要经过详细的数学检验。应用于金融和环境科学，评估新工具在科学调查和政策制定中的有用性。该项目将统计研究与软件开发和研究生教育相结合。分位数只是用从0到1的变化水平表示的百分位数，而不是一个百分点。变量的分位数直接反映了它在普通水平和特殊水平之间的平稳过渡。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，这种估计容易进行；预测-结果关系不需要详细的概率模型。但是，具有已知依赖结构的数据所面临的挑战要复杂得多；准确的估计需要调整近距离单元之间的固有噪声相关性。由于普通分位数回归方法的无模型性质以及它们孤立地关注单个分位数，因此这种任务仍然超出了普通分位数回归方法的范围。相比之下，联合分位数回归通过结合结果的完整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。该研究探讨了在联合分位数回归中使用copula建模来解决噪声相关性的问题，重点是为每种数据类型定制适当的copula公式。对所得工具的统计保证进行了严格的渐近分析。将开发用于模型评估和选择的定量和可视化诊断工具。所有的工具都将被整合到R包“qrjoint”中，可以通过综合R档案网络获得。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。

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.