[Paper Review] A Hierarchical Max-infinitely Divisible Process for Extreme Areal Precipitation Over Watersheds

Gregory P. Bopp, Benjamin A. Shaby+1 more|arXiv (Cornell University)|May 16, 2018

Hydrology and Drought Analysis3 citations

TL;DR

This paper proposes a hierarchical max-infinitely divisible process that models extreme areal precipitation with weakening spatial dependence at higher extremes—addressing a key limitation of traditional max-stable models. Using flexible random basis functions and a hierarchical likelihood structure, the model enables scalable Bayesian inference and captures spatial patterns of extremes while retaining max-stability as a special case.

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 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 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.

Motivation & Objective

• To address the limitation of max-stable processes in capturing weakening spatial dependence at extreme precipitation levels.
• To develop a flexible Bayesian model that adapts to data-driven spatial patterns of extremes.
• To enable scalable inference on large spatial datasets through a hierarchical representation with random effects.
• To allow for inspection of dominant spatial patterns via estimated random basis functions.
• To maintain max-stability as a special case, supporting inference on tail dependence classes.

Proposed method

• The model is constructed within the family of max-infinitely divisible processes, allowing for flexible extremal dependence structures.
• A hierarchical likelihood formulation uses random effects to enable scalable Bayesian inference on large datasets.
• Flexible random basis functions are estimated from data to represent spatial patterns of extreme precipitation.
• The model's dependence structure weakens at higher extremes, reflecting observed environmental behavior.
• The process includes max-stability as a special case, enabling statistical inference on tail dependence.
• Inference is performed using a Markov chain Monte Carlo approach tailored to the hierarchical structure.

Experimental results

Research questions

• RQ1 Can a max-infinitely divisible process model weakening spatial dependence at extreme precipitation levels?
• RQ2 How can a flexible, scalable Bayesian model be constructed to capture spatial patterns of extremes in large datasets?
• RQ3 To what extent does the model's hierarchical structure improve computational efficiency compared to standard max-stable models?
• RQ4 Can the model identify dominant spatial patterns of extreme precipitation through estimated basis functions?
• RQ5 Does the model maintain max-stability as a special case while allowing for more flexible dependence structures?

Key findings

• The proposed model successfully captures weakening spatial dependence at increasingly extreme precipitation levels, which standard max-stable models fail to represent.
• The hierarchical structure enables efficient inference on large spatial datasets, overcoming computational limitations of existing methods.
• Estimated random basis functions reveal clear spatial patterns of extreme precipitation, providing interpretable insights into dominant extremal structures.
• The model includes max-stability as a special case, allowing for formal inference on tail dependence classes.
• Empirical application to eastern North America demonstrates adequate fit to observed extreme precipitation data.
• The model's flexibility allows for better representation of real-world extremal behavior compared to rigid max-stable alternatives.