Graphical models for multivariate extremes

TL;DR

The fundamental concepts of extremal graphical models and recent advances in the field are provided and different existing perspectives on graphical extremes are presented in a unified way through graphical models for exponent measures.

Abstract

Graphical models in extremes have emerged as a diverse and quickly expanding research area in extremal dependence modeling. They allow for parsimonious statistical methodology and are particularly suited for enforcing sparsity in high-dimensional problems. In this work, we provide the fundamental concepts of extremal graphical models and discuss recent advances in the field. Different existing perspectives on graphical extremes are presented in a unified way through graphical models for exponent measures. We discuss the important cases of nonparametric extremal graphical models on simple graph structures, and the parametric class of Hüsler--Reiss models on arbitrary undirected graphs. In both cases, we describe model properties, methods for statistical inference on known graph structures, and structure learning algorithms when the graph is unknown. We illustrate different methods in an application to flight delay data at US airports.

Figures (9)

• Figure 1: Examples of graph structures on the set of nodes $\mathcal{V} = \{1,\dots, 4\}$.
• Figure 2: All panels show $n$ samples of the random vector $\mathbf{X}$ and illustrate the three main approaches to multivariate extremes: the blue points in the left panel are the random point set that is approximated by the Poisson point process $\Pi_\Lambda$; the solid blue point in the center panel is the componentwise maximum with approximate max-stable distribution; the blue points in the right panel are the threshold exceedances that are approximate samples of a multivariate Pareto distribution.
• Figure 3: Cassiopeia graph
• Figure 4: Example of a block graph and an associated extremal variogram of a Hüsler--Reiss distribution. Edges are labelled with their corresponding variogram entry. Underlined entries, corresponding to non-edges, can be computed using \ref{['tree_metric']}, for example $\Gamma_{16} = 6 + 6 + 10 = 22$.
• Figure 5: Example of a decomposable graph and an associated partial and completed extremal variogram. Underlined entries, corresponding to non-edges, are uniquely recovered from the known variogram entries on edges. Note that unlike in the case of a block graph (\ref{['ch13:fig:GammaCompletionBlock']}), the solution is not simply obtained by summing the entries along paths. For example, $\Gamma_{24} = 15$ is neither equal to $\Gamma_{21} + \Gamma_{14} = 13$, nor to $\Gamma_{23} + \Gamma_{34} = 21$.

Theorems & Definitions (12)

• Remark 2.1
• Example 2.1: Max-linear model
• Example 2.2: Hüsler--Reiss model
• Definition 2.1: Undirected separation
• Example 2.3
• Definition 2.2: Global Markov property and graphical models
• Definition 3.1
• Definition 3.2
• Example 3.1
• Example 3.2