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

Functions for modeling nonstationary extremal dependence using locally-stationary max-stable processes with LASSO and ridge penalization. The provided code is in support of Shao, X., Hazra, A., Richards, J., and Huser, R. (2023+). Flexible modeling of non-stationary extremal dependence using spatially-fused LASSO and ridge penalties. ArXiv.

The two main R scripts are:

1. `Modeling.R` - Fits the extremal dependence model with a provided dataset. Can be run for either the simulation study or the application.
2. `Summary.R` - Provides a summary of the fitted model with related plots and tables.

They should be run sequentially. By changing the code to `dat = "Simulated"` or `dat = "NepalExtended"` at the top of each script will run model fitting and summary for a simple simulated data on the true partition P1 (square partition) or the Nepal temperature data used in the application, respectively. Note that the former takes only a few minutes to compile, whilst the latter will take a few hours! Compiling Summary.R with `dat = "NepalExtended"` will provide Figures 4-6 and Table 3 in the paper.

Auxillary scripts include:

1. `Simulate_data.R` - A simple simulation of Brown-Resnick processes with nonstationary extremal dependence;
2. `Algorithm1.R` - Function for Algorithm 1 described in the paper;
3. `Merge_subr.R` - Function for the subregion merging process described in the paper;
4. `Lambda_tuning.R` - Function for the $\mathbf{\lambda}$-tuning described in the paper;
5. `Fit.R` - Some fitting function for convenience, using r-optim;
6. `Objectives.R` - Pairwise likelihood functions for the Brown-Resnick process (and inverted counterpart);
7. `Utils.R` - Various other utility functions.

Included in the repo are two Rdata files:

1. `NepalExtended.Rdata` - The gridded data of monthly maximum temperature dataset from Nepal and its surrounding Himalayan and sub-Himalayan regions used in the data application of the paper. This file includes the marginal parameter estiamtes (GEV parameters) derived using the Max-and-Smooth method: postman.mu (location), postman.sigma (scale), postman.xi (shape).
2. `Simulated.Rdata` - Simulated data from `Simulate_data.R`, including the coordinate and true (dependence) parameter information. The true partition is partition P1 mentioned in the paper.

Some further remarks:

• The current program only works for gridded data, but an extension to general lattice data is available.
• Nonstationarity is assumed for the input data.
• Input data must be renormalized to unit Fréchet margins to fit the max-stable processes.
• Some difficulties in range estimation may emerge.