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Wasserstein-Aitchison GAN for angular measures of multivariate extremes

Stéphane Lhaut, Holger Rootzén, Johan Segers

TL;DR

The paper tackles the challenge of estimating probabilities of multivariate extreme events in high dimensions by integrating marginal GP tail modeling with a nonparametric GAN-based model of angular dependence on the Aitchison simplex. WA-GAN learns the angular measure Φ in the L1-norm framework, via a Wasserstein GAN applied to coordinates in the Aitchison space, and then uses multivariate GP theory to map simulated angles back to the original data scale. The approach is validated on simulated data with logistic, Hüsler–Reiss, and Gaussian dependencies and on a 30-dimensional financial dataset, showing competitive or superior performance in tail dependence and extremal value generation relative to HTGAN and GPGAN. Limitations appear under asymptotic independence when the angular measure concentrates on the simplex boundary, motivating future enhancements such as heavier-tailed latent distributions or time-varying conditioning. Overall, WA-GAN provides a scalable, theory-grounded framework for simulating extremes in high dimensions with practical applicability to risk assessment.

Abstract

Economically responsible mitigation of multivariate extreme risks -- extreme rainfall in a large area, huge variations of many stock prices, widespread breakdowns in transportation systems -- requires estimates of the probabilities that such risks will materialize in the future. This paper develops a new method, Wasserstein--Aitchison Generative Adversarial Networks (WA-GAN) to, which provides simulated values of $d$-dimensional multivariate extreme events and which can hence be used to give estimates of such probabilities. The main hypothesis is that, after transforming the observations to the unit-Pareto scale, their distribution is regularly varying in the sense that the distributions of their radial and angular components (with respect to the $L_1$-norm) converge and become asymptotically independent as the radius gets large. The method is a combination of standard extreme value analysis modeling of the tails of the marginal distributions with nonparametric GAN modeling of the angular distribution. For the latter, the angular values are transformed to Aitchison coordinates in a full $(d-1)$-dimensional linear space, and a Wasserstein GAN is trained on these coordinates and used to generate new values. A reverse transformation is then applied to these values and gives simulated values on the original data scale. Our method is applied to simulated data and to a financial data set from the Kenneth French Data Library. The method shows good performance compared to other existing methods in the literature, both in terms of capturing the dependence structure of the extremes in the data and in generating accurate new extremes.

Wasserstein-Aitchison GAN for angular measures of multivariate extremes

TL;DR

The paper tackles the challenge of estimating probabilities of multivariate extreme events in high dimensions by integrating marginal GP tail modeling with a nonparametric GAN-based model of angular dependence on the Aitchison simplex. WA-GAN learns the angular measure Φ in the L1-norm framework, via a Wasserstein GAN applied to coordinates in the Aitchison space, and then uses multivariate GP theory to map simulated angles back to the original data scale. The approach is validated on simulated data with logistic, Hüsler–Reiss, and Gaussian dependencies and on a 30-dimensional financial dataset, showing competitive or superior performance in tail dependence and extremal value generation relative to HTGAN and GPGAN. Limitations appear under asymptotic independence when the angular measure concentrates on the simplex boundary, motivating future enhancements such as heavier-tailed latent distributions or time-varying conditioning. Overall, WA-GAN provides a scalable, theory-grounded framework for simulating extremes in high dimensions with practical applicability to risk assessment.

Abstract

Economically responsible mitigation of multivariate extreme risks -- extreme rainfall in a large area, huge variations of many stock prices, widespread breakdowns in transportation systems -- requires estimates of the probabilities that such risks will materialize in the future. This paper develops a new method, Wasserstein--Aitchison Generative Adversarial Networks (WA-GAN) to, which provides simulated values of -dimensional multivariate extreme events and which can hence be used to give estimates of such probabilities. The main hypothesis is that, after transforming the observations to the unit-Pareto scale, their distribution is regularly varying in the sense that the distributions of their radial and angular components (with respect to the -norm) converge and become asymptotically independent as the radius gets large. The method is a combination of standard extreme value analysis modeling of the tails of the marginal distributions with nonparametric GAN modeling of the angular distribution. For the latter, the angular values are transformed to Aitchison coordinates in a full -dimensional linear space, and a Wasserstein GAN is trained on these coordinates and used to generate new values. A reverse transformation is then applied to these values and gives simulated values on the original data scale. Our method is applied to simulated data and to a financial data set from the Kenneth French Data Library. The method shows good performance compared to other existing methods in the literature, both in terms of capturing the dependence structure of the extremes in the data and in generating accurate new extremes.
Paper Structure (33 sections, 2 theorems, 61 equations, 12 figures, 4 tables, 2 algorithms)

This paper contains 33 sections, 2 theorems, 61 equations, 12 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. A brief literature survey
  4. Regular variation and extreme value analysis
  5. Multivariate regular variation and angular measure.
  6. Univariate generalized Pareto distributions.
  7. Multivariate generalized Pareto distributions.
  8. Tail modeling and rare event probabilities.
  9. Wasserstein Generative Adversial Networks
  10. Aitchison coordinates
  11. Combining GAN, Aitchison coordinates, and Extremes
  12. Numerical experiments
  13. Performance and computation
  14. Performance measures.
  15. Architectures.
  16. ...and 18 more sections

Key Result

Proposition 1

Under Assumptions ass:multRV--ass:margins_rv, we have where $\bm{Y}$ is a multivariate Pareto random vector, with distribution with $Y$ a unit-Pareto random variable independent of $\bm{\Theta} \sim \Phi$, and where $\boldsymbol{\xi} = (\xi_j)_{j=1}^d$ is the vector of marginal coefficients as in Assumption ass:margins_rv; $\bm{a}(\cdot) = (a_j(\cdot))_{j=1}^d$ and $\bm{b}(\cdot) = (b_j(\cdot))_

Figures (12)

  • Figure 1: Illustration of a Wasserstein GAN with gradient-based optimization
  • Figure 2: Relation between thresholds in the $\bm{V}$ space. The gray zone is where we can safely sample using Algorithm \ref{['alg:MGP']}.
  • Figure 3: Logistic dependence structure. Left: $d=10$; right: $d=50$. -- Extremal coefficients of order $k=2$ (blue) and $k=3$ (red). Solid black line is the diagonal. Dashed blue and red lines are the true values of the coefficients for $k=2$ and $k=3$ respectively.
  • Figure 4: Logistic dependence structure. Left: $d=10$; right: $d=50$. -- First two margins of extremes in the test set and the different generative methods.
  • Figure 5: Hüsler--Reiss dependence structure. Left: $d=10$; right: $d=50$. -- Extremal coefficients of order $k=2$ (blue) and $k=3$ (red).
  • ...and 7 more figures

Theorems & Definitions (6)

  • Remark 1: Equivalent formulations of Assumption \ref{['ass:margins_rv']}
  • Proposition 1: Weak convergence of excesses to the MGP distribution
  • Remark 2: Standard MGP random vectors
  • Proposition 2: An orthonormal basis for the Aitchison simplex
  • Remark 3: Standardization of the margins to unit-Pareto
  • Remark 4: Relation between thresholds in both algorithms