Flexible Modeling of Nonstationary Extremal Dependence using Spatially-Fused LASSO and Ridge Penalties

Statistical modeling of a nonstationary spatial extremal dependence structure is challenging. Max-stable processes are common choices for modeling spatially-indexed block maxima, where an assumption of stationarity is usual to make inference feasible. However, this assumption is often unrealistic for data observed over a large or complex domain. We propose a computationally-efficient method for estimating extremal dependence using a globally nonstationary, but locally-stationary, max-stable process by exploiting nonstationary kernel convolutions. We divide the spatial domain into a fine grid of subregions, assign each of them its own dependence parameters, and use LASSO ($L_1$) or ridge ($L_2$) penalties to obtain spatially-smooth parameter estimates. We then develop a novel data-driven algorithm to merge homogeneous neighboring subregions. The algorithm facilitates model parsimony and interpretability. To make our model suitable for high-dimensional data, we exploit a pairwise likelihood to draw inferences and discuss computational and statistical efficiency. An extensive simulation study demonstrates the superior performance of our proposed model and the subregion-merging algorithm over the approaches that either do not model nonstationarity or do not update the domain partition. We apply our proposed method to model monthly maximum temperatures at over 1400 sites in Nepal and the surrounding Himalayan and sub-Himalayan regions; we again observe significant improvements in model fit compared to a stationary process and a nonstationary process without subregion-merging. Furthermore, we demonstrate that the estimated merged partition is interpretable from a geographic perspective and leads to better model diagnostics by adequately reducing the number of subregion-specific parameters.