Determining the dependence structure of multivariate extremes

Simpson, Emma S.; Wadsworth, Jennifer L.; Tawn, Jonathan A.

doi:10.1093/biomet/asaa018

Cited by 25 publications

(40 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Their error bounds are linear in d and scale as 1/ √ k, where k is the number of order statistics of each component that are considered extreme during the training step. A refinement of the latter framework is proposed in the yet unpublished work of [34]. [6] and [7] aim at identifying subgroups of components for which the probability of a joint excess over a large quantile is not negligible compared to that of an excess by a single component.…”

Section: Dimensionality Reduction For Extreme Values a Brief Overviewmentioning

confidence: 99%

Principal component analysis for multivariate extremes

Drees

Sabourin

2021

Electron. J. Statist.

View full text Add to dashboard Cite

In the probabilistic framework of multivariate regular variation, the first order behavior of heavy-tailed random vectors above large radial thresholds is ruled by a homogeneous limit measure. For a high dimensional vector, a reasonable assumption is that the support of this measure is concentrated on a lower dimensional subspace, meaning that certain linear combinations of the components are much likelier to be large than others. Identifying this subspace and thus reducing the dimension will facilitate a refined statistical analysis. In this work we apply Principal Component Analysis (PCA) to a re-scaled version of radially thresholded observations.Within the statistical learning framework of empirical risk minimization, our main focus is to analyze the squared reconstruction error for the exceedances over large radial thresholds. We prove that the empirical risk converges to the true risk, uniformly over all projection subspaces. As a consequence, the best projection subspace is shown to converge in probability to the optimal one, in terms of the Hausdorff distance between their intersections with the unit sphere. In addition, if the exceedances are rescaled to the unit ball, we obtain finite sample uniform guarantees to the reconstruction error pertaining to the estimated projection subspace. Numerical experiments illustrate the capability of the proposed framework to improve estimators of extreme value parameters.

show abstract

Section: Dimensionality Reduction For Extreme Values a Brief Overviewmentioning

confidence: 99%

Principal component analysis for multivariate extremes

Drees

Sabourin

2021

Electron. J. Statist.

View full text Add to dashboard Cite

show abstract

“…In such a context it is helpful to partition the underlying space, in our case S d−1 + , with understandable subsets (Chautru (2015), Goix et al (2017), Simpson et al (2020)). In this article we consider the subsets…”

Section: Sparsity In Extremesmentioning

confidence: 99%

“…We focus in this section on the subsets C β . A positive value for P(Θ ∈ C β ) entails that the marginals X j for j ∈ β take simultaneously large values while the ones in β c do not (Chautru (2015), Simpson et al (2020), Goix et al (2017)). Our aim is to use Proposition 2 to compare the nullity or not of the probabilities P(Θ ∈ C β ) and P(Z ∈ C β ).…”

Section: Maximal Directionsmentioning

confidence: 99%

See 1 more Smart Citation

Sparse regular variation

Nicolas¹,

Wintenberger²

2021

Adv. Appl. Probab.

View full text Add to dashboard Cite

Regular variation provides a convenient theoretical framework for studying large events. In the multivariate setting, the spectral measure characterizes the dependence structure of the extremes. This measure gathers information on the localization of extreme events and often has sparse support since severe events do not simultaneously occur in all directions. However, it is defined through weak convergence, which does not provide a natural way to capture this sparsity structure. In this paper, we introduce the notion of sparse regular variation, which makes it possible to better learn the dependence structure of extreme events. This concept is based on the Euclidean projection onto the simplex, for which efficient algorithms are known. We prove that under mild assumptions sparse regular variation and regular variation are equivalent notions, and we establish several results for sparsely regularly varying random vectors.

show abstract

“…analyzing the extremes of a high dimensional random vector. Such studies can be divided into the following categories: clustering methods [6,7,39], support identification, [32,33,8,9,53,44], Principal Component Analysis of the angular component of extremes [14,41,20], and graphical models for extremes [36,23,1]; see also [24] and the references therein. Our approach is remotely related to the last category: extremal graphical models.…”

Section: Dimensionality Reduction In Evtmentioning

confidence: 99%

Tail inverse regression for dimension reduction with extreme response

Aghbalou¹,

Portier²,

Sabourin³

et al. 2021

Preprint

View full text Add to dashboard Cite

We consider the problem of dimensionality reduction for prediction of a target Y ∈ R to be explained by a covariate vector X ∈ R p , with a particular focus on extreme values of Y which are of particular concern for risk management. The general purpose is to reduce the dimensionality of the statistical problem through an orthogonal projection on a lower dimensional subspace of the covariate space. Inspired by the sliced inverse regression (SIR) methods, we develop a novel framework (TIREX, Tail Inverse Regression for EXtreme response) relying on an appropriate notion of tail conditional independence in order to estimate an extreme sufficient dimension reduction (SDR) space of potentially smaller dimension than that of a classical SDR space. We prove the weak convergence of tail empirical processes involved in the estimation procedure and we illustrate the relevance of the proposed approach on simulated and real world data.

show abstract

Determining the dependence structure of multivariate extremes

Cited by 25 publications

References 23 publications

Principal component analysis for multivariate extremes

Principal component analysis for multivariate extremes

Sparse regular variation

Tail inverse regression for dimension reduction with extreme response

Contact Info

Product

Resources

About