Statistical Inference for H uslerReiss Graphical Models Through Matrix Completions

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Statistical Inference for H¨usler–Reiss

Graphical Models Through Matrix

Completions

Manuel Hentschel

Research Center for Statistics, University of Geneva

and

Sebastian Engelke

Research Center for Statistics, University of Geneva

and

Johan Segers

ISBA/LIDAM, UCLouvain

October 16, 2023

Abstract

The severity of multivariate extreme events is driven by the dependence between the

largest marginal observations. The H¨usler–Reiss distribution is a versatile model for this

extremal dependence, and it is usually parameterized by a variogram matrix. In order

to represent conditional independence relations and obtain sparse parameterizations,

we introduce the novel H¨usler–Reiss precision matrix. Similarly to the Gaussian case,

this matrix appears naturally in density representations of the H¨usler–Reiss Pareto

distribution and encodes the extremal graphical structure through its zero pattern.

For a given, arbitrary graph we prove the existence and uniqueness of the completion

of a partially speciﬁed H¨usler–Reiss variogram matrix so that its precision matrix

has zeros on non-edges in the graph. Using suitable estimators for the parameters

on the edges, our theory provides the ﬁrst consistent estimator of graph structured

H¨usler–Reiss distributions. If the graph is unknown, our method can be combined with

recent structure learning algorithms to jointly infer the graph and the corresponding

parameter matrix. Based on our methodology, we propose new tools for statistical

inference of sparse H¨usler–Reiss models and illustrate them on large ﬂight delay data

in the U.S., as well as Danube river ﬂow data.

Keywords: extreme value analysis; multivariate generalized Pareto distribution; sparsity;

variogram

arXiv:2210.14292v2 [stat.ME] 13 Oct 2023

1 Introduction

In statistical modelling, conditional independence and graphical models are well-

established concepts for analyzing structural relationships in data (Lauritzen, 1996;

Wainwright and Jordan, 2008). Particularly important are Gaussian graphical models,

also known as Gaussian Markov random ﬁelds (Rue and Held, 2005). The graphical

structure of a multivariate normal distribution with positive deﬁnite covariance matrix

Σ can be read oﬀ from the zeros of its precision matrix Σ−1.

For risk assessment in ﬁelds such as climate science, hydrology, or ﬁnance, the

primary interest is in extreme observations, with attention to both the marginal tails

and the dependence between multiple risk factors. In view of the growing complexity

and dimensionality of modern data sets, sparsity and graphical models are becoming

crucial notions for the analysis of extremes (e.g., Engelke and Ivanovs, 2021).

There are two diﬀerent ways of deﬁning graphical models for extreme value

distributions. The ﬁrst is based on max-linear models (Gissibl and Kl¨uppelberg, 2018)

and the second one studies multivariate Pareto distributions (Engelke and Hitz, 2020).

We follow the second approach since their new notions of conditional independence

and extremal graphical models link naturally to the well-known Hammersley–Cliﬀord

theorem for density factorizations. Moreover, in the case of tree graphs, Segers (2020)

shows that the extremes of regularly varying Markov trees converge to these extremal

tree models. Lee and Joe (2018) propose parsimonious models for extreme value

copulas; the link with extremal graphical models is made in Asenova et al. (2021).

The class of extremal graphical models with H¨usler–Reiss Pareto distributions is

of particular interest. In

dimensions, the parameter of this family is a variogram

matrix Γ

∈Rd×d

. Because of their ﬂexibility and stochastic properties, H¨usler–Reiss

distributions can be seen as the counterpart of the Gaussian family for multivariate

extremes. In combination with extremal graphical models, the H¨usler–Reiss family

constitutes a powerful tool for sparse extreme value modelling, with many open

questions still to explore.

For a connected, undirected graph

= (

V, E

) with nodes

{

, . . . , d}

and

edges

, Engelke and Hitz (2020) show that such a distribution’s graphical structure

can be read oﬀ from a set of (

d−

(

d−

1) precision matrices Θ

(k)

, for

k∈V

While zeros in Θ

(k)

correspond to extremal conditional independence of nodes

i, j ̸

the information on edges involving the

th node is encoded only indirectly through

the row sums of this matrix. A natural question, appearing also in the discussion of

Engelke and Hitz (2020), is if there exists a symmetric approach involving a single

d×dprecision matrix.

Statistical inference for H¨usler–Reiss graphical models is limited so far to the

simple structures of trees and block graphs (Engelke and Volgushev, 2020; Asenova

and Segers, 2023). The parameter matrix Γ is then additive on the graph and the

maximum likelihood estimator is an explicit combination of the bivariate estimators

on the edges. Since block graphs lack ﬂexibility for general applications, several

discussion contributions of Engelke and Hitz (2020) have emphasized the need for

estimators suitable for more general graphs.

In this paper, we obtain new theoretical results on H¨usler–Reiss distributions that

answer the two open questions above and enable statistical inference on extremal

graphical models on decomposable and non-decomposable graphs. Our main contri-

butions are threefold. Firstly, in Section 3, we introduce the H¨usler–Reiss precision

matrix Θ

∈Rd×d

as Θ

= Θ

(k)

for some

k̸

i, j

, a deﬁnition which—surprisingly—is

independent of the particular choice of

k∈V

. This positive semi-deﬁnite matrix

indeed reﬂects the sparsity of the extremal graph by zero oﬀ-diagonal entries. We give

several characterizations of this matrix, one of them as the Moore–Penrose inverse of

a projection of the parameter matrix Γ.

Secondly, we study how a H¨usler–Reiss distribution on a given, general graph

= (

V, E

) and ﬁxed marginal distributions on the edges of the graph can be

constructed. Thanks to the new H¨usler–Reiss precision matrix, this task can be

framed as a matrix completion problem, which aims to ﬁnd a conditionally negative

deﬁnite variogram matrix Γ that has speciﬁed values Γ

in the entries corresponding

to the edges (

i, j

)

∈E

of graph

and whose precision matrix Θ has zeros in the

remaining entries, i.e., Θ

= 0 for (

i, j

)

/∈E

. A concrete example in dimension

= 4

for the graph in Figure 1b is

Γ = 





0 3 ? 1

3 0 10 2

? 10 0 ?

1 2 ? 0





,Θ = 





? ? 0 ?

? ? ? ?

0 ? ? 0

? ? 0 ?





,(1.1)

where the completion problem corresponds to ﬁnding the entries with

“?”

. In Section 4,

we show that such a completion exists and is unique. Our results can be seen as

a semi-deﬁnite extension of matrix completion problems for the covariance matrix

of Gaussian distributions studied in Speed and Kiiveri (1986) and Bakonyi and

Woerdeman (2011).

Thirdly, we leverage our theoretical results to provide eﬀective statistical tools for

extremal graphical models. The H¨usler–Reiss precision matrix allows us to represent

the maximum (surrogate-)likelihood estimate of Γ on the graph

as the maximizer

of the constrained optimization problem

log |Θ|++1

2tr(b

ΓΘ),s.t. Θij = 0 if (i, j)/∈E,

where

is the empirical variogram (Engelke and Volgushev, 2020) and

|·|+

denotes

the pseudo-determinant. In Section 5, we prove that the solution to this optimization

problem is given by the solution of the above matrix completion problem. We further

combine recent structure learning methods (Engelke et al., 2022c) with our completion

to yield the ﬁrst estimator that is jointly sparsistent for the extremal graph and

consistent for the model parameters. Our methodology enables new model assessment

plots that allow for model interpretation and comparison of models with diﬀerent

degrees of sparsity. This is illustrated in a case study of large delays in the domestic

U.S. air travel network in Section 6. The supplementary material contains another

case study, together with mathematical details and proofs.

The H¨usler–Reiss precision matrix and the properties we derive in this paper

have already been used for multiple purposes, including the parameterization of

sparse statistical models (R¨ottger and Schmitz, 2023; R¨ottger et al., 2023a), structure

estimation (Engelke et al., 2022c; Wan and Zhou, 2023), eﬃcient statistical inference

(R¨ottger et al., 2023b; Lederer and Oesting, 2023), and a characterization in terms of

pairwise interaction models (Lalancette, 2023).

2 Extremal graphical models

2.1 Multivariate generalized Pareto distributions

Multivariate extreme value theory studies the tail behavior of a random vector

(X1, . . . , Xd)

. A ﬁrst summary of the extremal dependence structure of the

bivariate margins for i, j ∈V:= {1, . . . , d}is the extremal correlation

χij := lim

p→0χij(p) := lim

p→0P(Fi(Xi)>1−p|Fj(Xj)>1−p)∈[0,1],(2.1)

deﬁned whenever the limit exists and where

is the distribution function of

When

χij >

0 we say that

and

are asymptotically dependent, and when

χij

= 0

we speak of asymptotic independence. In the former case, there are two diﬀerent, but

closely related approaches for modelling extremal dependence: through component-

wise maxima of independent copies of

leading to max-stable distributions (de Haan

and Resnick, 1977); and through threshold exceedances of

resulting in multivariate

generalized Pareto distributions (Rootz´en and Tajvidi, 2006). Here, we concentrate

on the threshold exceedance approach since it is well-suited for graphical modelling

(Engelke and Hitz, 2020; Segers, 2020). For statistical models for asymptotic indepen-

dence, we refer to Heﬀernan and Tawn (2004), for instance, and to Papastathopoulos

et al. (2017) in the context of extremes of Markov chains as well as Casey and

Papasthatopoulos (2023) for extremes in decomposable graphical models.

To make abstraction of the univariate marginal distributions and concentrate on

the extremal dependence, it is usually assumed that all variables

follow a given

continuous distribution. Throughout, we use standard exponential margins, that is,

(

Xi≤x

) = 1

−exp

(

−x

) for

x≥

0 and

i∈V

. Let 0,1, and

∞

denote vectors

of adequate size with all elements equal to 0, 1, and

∞

respectively. A random

vector

= (

Y1, . . . , Yd

) is said to follow a multivariate generalized Pareto distribution

(Rootz´en and Tajvidi, 2006) if for any z∈ L =x∈Rd:x̸≤ 0, we have

P(Y≤z) := lim

u→∞ P(X−u1≤z|X̸≤ u1) = Λc(z∧0)−Λc(z)

Λc(0),(2.2)

for some random vector

, which is then said to be in the domain of attraction of

;

the simple normalization by subtracting

1comes from the assumption of exponential

margins of

. Multivariate generalized Pareto distributions are threshold-stable

in the sense that for a vector

a≥

0, the conditional random vector

Y−a

given

Y≰a

is again multivariate generalized Pareto. This also implies useful stochastic

representations; see Rootz´en et al. (2018) for details. The so-called exponent measure

Λ is a measure on [

−∞,∞

)

d\ {−∞}

that is ﬁnite on sets bounded away from

−∞

, and we write Λ

(

) := Λ

[−∞,∞)d\[−∞, z]

. Multivariate generalized Pareto

distributions are the only ones that can arise as limits of threshold exceedances as

in (2.2).

We assume Λ to be absolutely continuous with respect to the

-dimensional

Lebesgue measure and let λdenote its Radon–Nikodym derivative. The set of valid

exponent measure densities λis characterized by the following two properties:

λ(y+t1) = exp(−t)λ(y),∀t∈R, y ∈Rd,

Zyi>0

λ(y) dy= 1,∀i∈V.

Since the distribution of

is proportional to the restriction of Λ to

, its density

then

also exists and is proportional to the exponent measure density

f(y)

λ(y)/

c(0)

for all y∈ L, since Λc(0) = Λ(L).

When considering marginal distributions of multivariate generalized Pareto dis-

tributions, it is often more useful to work with the limit distributions that arise in

(2.2) by replacing

with the sub-vector

, also in the conditioning event, for some

non-empty

I⊂ {1, . . . , d}

. The resulting “generalized Pareto marginal distribution”,

here denoted

Y(I)

, is supported on

x∈R|I|:x̸≤ 0

, and its corresponding

exponent measure and density are the actual marginals Λ

and

λI

. Note that

Y(I)

is equal in distribution to

conditioned on the event

YI̸≤

0. See Rootz´en and

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StatisticalInferenceforH¨usler–ReissGraphicalModelsThroughMatrixCompletionsManuelHentschelResearchCenterforStatistics,UniversityofGenevaandSebastianEngelkeResearchCenterforStatistics,UniversityofGenevaandJohanSegersISBA/LIDAM,UCLouvainOctober16,2023AbstractTheseverityofmultivariateextremeeventsisdri...

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