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Dimension Reduction in Multivariate Extremes via Latent Linear Factor Models

Alexis Boulin, Axel Bücher

Abstract

We propose a new and interpretable class of high-dimensional tail dependence models based on latent linear factor structures. Specifically, extremal dependence of an observable vector is assumed to be driven by a lower-dimensional latent $K$-factor model, where $K \ll d$, thereby inducing an explicit form of dimension reduction. Geometrically, this is reflected in the support of the associated spectral dependence measure, whose intrinsic dimension is at most $K-1$. The loading structure may additionally exhibit sparsity, meaning that each component is influenced by only a small number of latent factors, which further enhances interpretability and scalability. Under mild structural assumptions, we establish identifiability of the model parameters and provide a constructive recovery procedure based on a margin-free tail pairwise dependence matrix, which also yields practical rank-based estimation methods. The framework combines naturally with marginal tail models and is particularly well suited to high-dimensional settings. We illustrate its applicability in a spatial wind energy application, where the latent factor structure enables tractable estimation of the risk that a large proportion of turbines simultaneously fall below their cut-in wind speed thresholds.

Dimension Reduction in Multivariate Extremes via Latent Linear Factor Models

Abstract

We propose a new and interpretable class of high-dimensional tail dependence models based on latent linear factor structures. Specifically, extremal dependence of an observable vector is assumed to be driven by a lower-dimensional latent -factor model, where , thereby inducing an explicit form of dimension reduction. Geometrically, this is reflected in the support of the associated spectral dependence measure, whose intrinsic dimension is at most . The loading structure may additionally exhibit sparsity, meaning that each component is influenced by only a small number of latent factors, which further enhances interpretability and scalability. Under mild structural assumptions, we establish identifiability of the model parameters and provide a constructive recovery procedure based on a margin-free tail pairwise dependence matrix, which also yields practical rank-based estimation methods. The framework combines naturally with marginal tail models and is particularly well suited to high-dimensional settings. We illustrate its applicability in a spatial wind energy application, where the latent factor structure enables tractable estimation of the risk that a large proportion of turbines simultaneously fall below their cut-in wind speed thresholds.
Paper Structure (19 sections, 15 theorems, 121 equations, 10 figures, 2 algorithms)

This paper contains 19 sections, 15 theorems, 121 equations, 10 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries on tail dependence and multivariate regular variation
  3. Latent linear factor models for tail dependence
  4. Estimation of latent linear factor models for tail dependence
  5. Estimation of the number of factors and the pure variables
  6. Estimating the loading matrix
  7. Estimation of the latent spectral measure
  8. Estimation of derived tail parameters
  9. Hyperparameter selection and model validation
  10. Case study
  11. Conclusion and Outlook
  12. Latent linear factor modeling for regularly varying random vectors
  13. Proofs for Section \ref{['sec:latent-linear-factor-models']}
  14. Proofs for Section \ref{['sec:estimating-latent-linear-factor-models']}
  15. Proofs for Section \ref{['sec:latent-linear-factor-models-regular-variation']}
  16. ...and 4 more sections

Key Result

Proposition 3.1

Let $K \in [d]$, $A \in \mathcal{A}_s(d,K)$ and $\psi\in \mathcal{S}_{\mathrm d}(\mathbb S_{+}^{K-1}, {\|\cdot\|_1})$. Then the random vector where $R$ is standard Pareto and $\bm \Lambda \sim \psi$ is independent of $R$, has STDF $L=L_{K, A, \psi}$ as in eq:stdf-factor-introduction, that is, Further, the spectral dependence measure of $\bm Z$ with respect to the 1-norm is given by $\Psi_{\bm Z

Figures (10)

  • Figure 1: Illustration of the support of a spectral dependence measure with respect to the Euclidean norm (dashed black) when $A \in \mathbb{R}^{3\times 2}$ has columns $(1, 1/3, 0)$ and $(0,2/3,1)$.
  • Figure 2: (a) Detailed turbine locations in Schleswig-Holstein (black dots) with January 2023 mean wind speeds from HOSTRADA (grid cells of size 1km x 1km). Pixels outside HOSTRADA resolution data are shown in grey. (b) A typical wind turbine power curve. Turbines produce no power below the cut-in speed (3 m/s), operate at rated capacity between 14 and 25 m/s, and shut down above the cut-out speed (25 m/s).
  • Figure : (a) $k'=135$
  • Figure : (a) $\alpha = 0.5$
  • Figure : (a) $k'=135$
  • ...and 5 more figures

Theorems & Definitions (35)

  • Proposition 3.1
  • Definition 3.2: Latent linear factor tail dependence
  • Lemma 3.3: Latent linear factor tail dependence on the level of spectral dependence measures
  • Definition 3.4: Tail pairwise dependence matrix
  • Proposition 3.5: TPDM for latent linear factor tail dependence
  • Theorem 3.6
  • Remark 3.7: Reconstructing $(K,A,\psi)$ from $L$
  • Proposition 3.8
  • Lemma 4.1
  • Proposition A.1
  • ...and 25 more