TAIL DEPENDENCE MODELING FOR HIGH DIMENSIONAL SPATIAL EXTREMES

Abstract

This dissertation develops flexible statistical models for tail dependence in high-dimensional spatial extremes, with an emphasis on environmental applications in which rare events can exhibit substantial spatial heterogeneity and different behaviors across spatial lags. Existing spatial extremes models often impose a single extremal dependence class over an entire domain and become computationally prohibitive for high-dimensional threshold exceedance data. To address these limitations, this dissertation contributes methodology for flexible dependence modeling, scalable inference, and extensions toward multivariate tail dependence and causal structure.To accurately characterize dependence in extreme events, we propose a mixture model that achieves flexible dependence properties and allows high-dimensional inference for extremes of spatial processes. We modify the popular random scale construction that multiplies a Gaussian random field by a single radial variable; we allow the radial variable to vary smoothly across space and add non-stationarity to the Gaussian process. As the level of extremeness increases, this single model exhibits both asymptotic independence at long ranges and either asymptotic dependence or independence at short ranges. We make joint inference on the dependence model and a marginal model using a copula approach within a Bayesian hierarchical model. Three different simulation scenarios show close to nominal frequentist coverage rates. And we apply the model to a dataset of extreme summertime precipitation over the central United States. We find that the joint tail of precipitation exhibits non-stationary dependence structure that cannot be captured by limiting extreme value models or current state-of-the-art sub-asymptotic models. Flexible spatial extreme models based on Gaussian random scale mixtures provide a framework for capturing a broad range of extremal dependence structures. However, likelihood-based inference under the peaks-over-threshold setting is often computationally infeasible, due to the censored likelihood requiring repeated evaluation of high-dimensional Gaussian distribution functions. We propose a multiplicative log-Laplace nugget that yields conditional independence in the censored likelihood, resulting in a joint likelihood function that is the product of univariate densities which are available in closed form. This eliminates multivariate Gaussian distribution function evaluations and thereby enables inference for threshold exceedances in high dimensions, which represents a major shift for spatial extremes modeling as the total computational cost is now primarily driven by standard spatial statistics operations. We show that under mild conditions, the spatial process that includes the proposed nugget retains the extremal dependence structure of the underlying smooth process. The approach applies broadly to several modern transformed Gaussian scale mixture models for spatial extremes, and is illustrated through simulation studies and an application to precipitation extremes. The dissertation concludes with a future research direction on multivariate extremes. This direction aims to develop models that can represent both strong and weak forms of tail dependence among subsets of variables, use a latent indicator matrix interpretable as a directed graph to encode directional tail dependence structure, and apply these methods to critical risk analysis problems involving river discharge and multiple-pollutant health outcomes.