Analyzing Dependent Extremes via Joint Quantile Regression

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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.