Extremal Behavior of Aggregated Data with an Application to Downscaling

TL;DR

This paper introduces the $ ext{l}$-extremal coefficient to analyze how aggregation affects the tail behavior of spatial data, providing a theoretical basis for statistical downscaling of extreme temperature events.

Contribution

It develops a novel extremal dependence framework linking aggregated data distribution to the underlying process, with explicit formulas and a practical downscaling application.

Findings

Derived explicit formulas for $ ext{l}$-extremal coefficients.

Established the joint extremal dependence for multiple functionals.

Successfully downscaled temperature maxima during a heatwave.

Abstract

The distribution of spatially aggregated data from a stochastic process $X$ may exhibit a different tail behavior than its marginal distributions. For a large class of aggregating functionals $\ell$ we introduce the $\ell$-extremal coefficient that quantifies this difference as a function of the extremal spatial dependence in $X$. We also obtain the joint extremal dependence for multiple aggregation functionals applied to the same process. Explicit formulas for the $\ell$-extremal coefficients and multivariate dependence structures are derived in important special cases. The results provide a theoretical link between the extremal distribution of the aggregated data and the corresponding underlying process, which we exploit to develop a method for statistical downscaling. We apply our framework to downscale daily temperature maxima in the south of France from a gridded data set and use our model to generate high resolution maps of the warmest day during the 2003 heatwave.