Extremes Models and Methods from Transformed Linear Operations

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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.