Model selection for extremal dependence structures using deep learning: Application to environmental data.

This repository contains all the code, data, and resources used in our study:
“Model Selection for Extremal Dependence Structures Using Deep Learning: Application to Environmental Data.”

The goal of this research is to better understand and model the spatial dependence structure of extreme 2 m air temperatures across Iraq. We focus on selecting the most appropriate max-stable dependence structure using a deep learning approach.

Our approach uses convolutional neural networks (CNNs) to learn spatial dependence patterns from datasets simulating the max-stable models fitted to 2m air temperature phenomena. The idea is to train the networks to recognize which model and covariance structure best fit the data.

We propose two selection strategies:
• Scheme 1: A single CNN (CNN-C) that predicts both the max-stable model and its covariance function at once.
• Scheme 2: A two-stage approach, where one CNN (CNN-M) predicts the model family, and then a second CNN, selected from CNN-S (for Schlather), CNN-G (for Geometric), or CNN-E (for Extremal-t), determines the specific covariance function.

To evaluate performance, we compare these CNN-based results to a classical model selection method: the Composite Likelihood Information Criterion (CLIC). We also validate our findings using a parametric bootstrap approach based on extremal coefficients.

What the codes and dataset supports:
• Simulating spatial dependence structures under different max-stable processes
• Fitting models using composite likelihood and comparing them using CLIC
• Training CNNs to classify spatial dependence structures
• Validating model selection using extremal coefficient diagnostics
• Comparing deep learning–based selection with traditional statistical methods, e.g., CLIC

本仓库包含我们开展题为《基于深度学习的极值依赖结构模型选择：环境数据应用》（Model Selection for Extremal Dependence Structures Using Deep Learning: Application to Environmental Data）的研究时所用的全部代码、数据与相关资源。

本研究旨在更好地理解并建模伊拉克境内2米高度极端气温的空间依赖结构，重点采用深度学习方法选取最适配的max-stable（极值稳定）依赖结构。

我们的方法采用卷积神经网络（Convolutional Neural Networks, CNNs），从针对2米气温现象拟合的极值稳定模型生成的模拟数据集中学习空间依赖模式，其核心思路是训练神经网络以识别最适配给定数据的模型与协方差结构。

我们提出两种模型选择策略：
• 方案1：单卷积神经网络（CNN-C），可同时预测极值稳定模型及其协方差函数。
• 方案2：两阶段方法，首先通过一个卷积神经网络（CNN-M）预测模型族，随后从对应Schlather模型的CNN-S、对应Geometric模型的CNN-G以及对应Extremal-t模型的CNN-E中选取第二阶段卷积神经网络，以确定具体的协方差函数。

为验证模型性能，我们将基于卷积神经网络的模型选择结果与经典统计方法——复合似然信息准则（Composite Likelihood Information Criterion, CLIC）进行对比。此外，我们还采用基于极值系数的参数自助法（parametric bootstrap）对研究结果进行验证。

本仓库的代码与数据集可支持以下功能：
• 在不同极值稳定过程下模拟空间依赖结构
• 采用复合似然法拟合模型，并通过复合似然信息准则（CLIC）完成模型对比
• 训练卷积神经网络以实现空间依赖结构分类
• 基于极值系数诊断法完成模型选择验证
• 将基于深度学习的模型选择方法与传统统计方法（如CLIC）进行对比