A Hierarchical Max-Infinitely Divisible Spatial Model for Extreme Precipitation

Abstract

Understanding the spatial extent of extreme precipitation is necessary for determining flood risk and adequately designing infrastructure (e.g., stormwater pipes) to withstand such hazards. While environmental phenomena typically exhibit weakening spatial dependence at increasingly extreme levels, limiting max-stable process models for block maxima have a rigid dependence structure that does not capture this type of behavior. We propose a flexible Bayesian model from a broader family of (conditionally) max-infinitely divisible processes that allows for weakening spatial dependence at increasingly extreme levels, and due to a hierarchical representation of the likelihood in terms of random effects, our inference approach scales to large datasets. The proposed model is constructed using flexible random basis functions that are estimated from the data, allowing for straightforward inspection of the predominant spatial patterns of extremes. In addition, the described process possesses (conditional) max-stability as a special case, making inference on the tail dependence class possible. We apply our model to extreme precipitation in eastern North America, and show that the proposed model adequately captures the extremal behavior of the data.

Max‐infinitely divisible models and inference for spatial extremes

“…Alternatively, we could assume that the prior for 1/ c (or ) is a mixture between a continuous distribution on (0, ∞ ) and a point mass at zero. In the Bayesian context, the recent work of Bopp et al (2020) proposes a hierarchical max‐id model extending the max‐stable Reich and Shaby (2012) model.…”

“…Therefore, we always need to choose a finite, and often relatively small, block size, which casts doubts on the validity of the max‐stability assumption in practice. A fast growing body of empirical studies on environmental extremes in the literature has indeed revealed that the max‐stability assumption arising asymptotically is often violated at finite levels (Bopp, Shaby, & Huser, 2020), and that the spatial dependence strength is often weakening as events become more extreme (see, e.g., Bacro, Gaetan, Opitz, & Toulemonde, 2020; Castro‐Camilo & Huser, 2020; Castro‐Camilo, Mhalla, & Opitz, 2020; Davison, Huser, & Thibaud, 2013; Huser, Opitz, & Thibaud, 2017; Huser & Wadsworth, 2020; Tawn, Shooter, Towe, & Lamb, 2018). In particular, under asymptotic independence , maxima become ultimately independent at the highest levels, requiring specialized models capturing the decay rate towards independence.…”

“…Padoan (2013) proposed a max‐id model based on Gaussian process ratios, whose dependence strength varies with the intensity of the extreme event. Moreover, a recent implementation of a specific spatial max‐id model using simulation‐based Bayesian inference (Bopp et al, 2020) bears witness of the versatility of such models. Compared to the models proposed by Padoan (2013) and Bopp et al (2020), our proposed max‐id models can control the joint tail decay rate more flexibly, and unlike Padoan (2013), they can be arbitrarily close to popular max‐stable models.…”

New Exploratory Tools for Extremal Dependence: $$\chi $$ Networks and Annual Extremal Networks

“…First, the χ statistic is a measure of pairwise tail dependence and hence not able to describe the higher-order extremal dependence. Second, χ is an appropriate extremal dependence metric in the case of asymptotic dependence, it is less clear about the usefulness of this measure if the environmental process of interest exhibits weakening spatial dependence as events become more extreme (Huser et al 2017;Wadsworth et al 2017;Huser and Wadsworth 2018;Bopp et al 2018).…”

Estimating high-resolution Red Sea surface temperature hotspots, using a low-rank semiparametric spatial model

“…Max-stable and Pareto processes are asymptotically justified models for spatial extremes, but likelihood computations are usually challenging, even for low or moderate spatial dimensions (see, e.g., Castruccio, Huser and Genton (2016) and ). An exception is the max-stable model of Reich and Shaby (2012) which is computationally tractable for higher spatial dimensions; see also Bopp, Shaby and Huser (2021). However, this max-stable model has been criticized for its lack of flexibility in a variety of applications.…”