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Graphical lasso for extremes

Phyllis Wan, Chen Zhou

TL;DR

This paper proposes the extreme graphical lasso procedure to estimate the sparsity in the tail dependence, similar to the Gaussian graphicalLasso method in high dimensional statistics, and proves its consistency in identifying the graph structure and estimating model parameters.

Abstract

In this paper, we estimate the sparse dependence structure in the tail region of a multivariate random vector, potentially of high dimension. The tail dependence is modeled via a graphical model for extremes embedded in the Hüsler-Reiss distribution. We propose the extreme graphical lasso procedure to estimate the sparsity in the tail dependence, similar to the Gaussian graphical lasso in high dimensional statistics. We prove its consistency in identifying the graph structure and estimating model parameters. The efficiency and accuracy of the proposed method are illustrated by simulations and real data examples.

Graphical lasso for extremes

TL;DR

This paper proposes the extreme graphical lasso procedure to estimate the sparsity in the tail dependence, similar to the Gaussian graphicalLasso method in high dimensional statistics, and proves its consistency in identifying the graph structure and estimating model parameters.

Abstract

In this paper, we estimate the sparse dependence structure in the tail region of a multivariate random vector, potentially of high dimension. The tail dependence is modeled via a graphical model for extremes embedded in the Hüsler-Reiss distribution. We propose the extreme graphical lasso procedure to estimate the sparsity in the tail dependence, similar to the Gaussian graphical lasso in high dimensional statistics. We prove its consistency in identifying the graph structure and estimating model parameters. The efficiency and accuracy of the proposed method are illustrated by simulations and real data examples.
Paper Structure (35 sections, 12 theorems, 198 equations, 7 figures)

This paper contains 35 sections, 12 theorems, 198 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Hüsler-Reiss graphical models
  4. Graphical models for extremes
  5. Hüsler-Reiss graphical models
  6. Charaterization using variagoram matrix $\Gamma$ and sub-covariance matrices $\Sigma^{(k)}$'s
  7. Characterization using precision matrix $\Theta$
  8. Characterization using covariance matrix $\Sigma$
  9. The extreme graphical lasso
  10. Graphical lasso estimator of $\Theta$
  11. Solving the extreme graphical lasso
  12. Theoretical results
  13. Applying the extreme graphical lasso
  14. Estimators for $\Sigma^{(k)}$
  15. Convergence of $S$ to $\Sigma$
  16. ...and 20 more sections

Key Result

Proposition 2.1

Given a multivariate random vector $\mathbf{W}=(W_1,\ldots,W_d)$, define its variogram matrix to be $\Gamma = (\Gamma_{ij}) \in \mathbb{R}^{d\times d}$ such that The parameter space $\mathcal{D}_0$ is the collection of variogram matrices $\Gamma$ for all random vectors $\mathbf{W}$ with positive definite covariance matrices.

Figures (7)

  • Figure 1: (a) Star graph; (b) Diamond graph.
  • Figure 2: The curves of $|||\Omega_{E^cE}(\Omega_{EE})^{-1}|||_{\infty}$ versus $M$ for the star graph and the diamond graph, each with parameter values $x=0.8$ and $x=1.2$. The two dash vertical lines in each graph indicates the values of $\lambda_2$ and $\lambda_d$ for the corresponding $\Sigma$.
  • Figure 3: The average F1 score and the fraction of times being optimal versus the penalizing parameter $\gamma_n$ (in log scale) for the $BA(20,1)$ (upper panel) and $BA(20,2)$ (lower panel) graphs. The solid line provides the F1 score for the estimated graphs (left scale). The dash line provides the fraction of times (right scale) when each value is chosen as the optimal penalizing parameter. The plots are over 100 simulated samples.
  • Figure 4: The aggregated graph structure based on 100 simulated samples. The true graph follows the $BA(20,1)$ (upper panel) and $BA(20,2)$ (lower panel) model.
  • Figure 5: The boxplots for the F1 scores versus different $k_n/d$ ratios, based on 100 simulated samples. The upper (lower) panel shows the result for $d=20$ ($d=100$). The left (right) graph is based on the $BA(20,1)$ ($BA(20,2)$) model.
  • ...and 2 more figures

Theorems & Definitions (24)

  • Proposition 2.1
  • Proposition 2.2: Lemma 1, eng2018a
  • Proposition 2.3
  • Proposition 2.4: Relationship between $\Sigma$ and $\tilde{\Sigma}^{(k)}$
  • Remark 2.5
  • Proposition 2.6: Relationship between $\Sigma$ and $\Theta$
  • Remark 2.7
  • Corollary 2.8: Parameter space of $\Sigma$ and $\Theta$
  • Proposition 3.1
  • Remark 3.2
  • ...and 14 more