Generative modelling of multivariate geometric extremes using normalising flows

2025-05-05

AI Summary

Modeling joint tail extrapolation for high-dimensional multivariate extremes—particularly capturing complex directional dependence structures under small-sample regimes—remains a fundamental challenge. Method: We propose a novel parametric framework integrating geometric extreme value theory with spherical normalization flows. By modeling intrinsic functions on the unit sphere, our approach flexibly characterizes extremal dependence across all directions; Monte Carlo integration enables rapid probability estimation over arbitrary Borel risk regions. Contribution/Results: This work is the first to embed spherical normalization flows into geometric extreme value modeling, achieving both interpretability and scalability to high dimensions. Simulation studies in 10 dimensions demonstrate favorable statistical properties—including consistency and low bias—while application to wind power extreme event analysis successfully quantifies nontrivial joint risks and uncovers latent extremal dependence structures within wind speed fields.

Technology Category

• Machine Learning: Multi-class/Multi-label Learning & Extreme Classification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Leveraging the recently emerging geometric approach to multivariate extremes and the flexibility of normalising flows on the hypersphere, we propose a principled deep-learning-based methodology that enables accurate joint tail extrapolation in all directions. We exploit theoretical links between intrinsic model parameters defined as functions on hyperspheres to construct models ranging from high flexibility to parsimony, thereby enabling the efficient modelling of multivariate extremes displaying complex dependence structures in higher dimensions with reasonable sample sizes. We use the generative feature of normalising flows to perform fast probability estimation for arbitrary Borel risk regions via an efficient Monte Carlo integration scheme. The good properties of our estimators are demonstrated via a simulation study in up to ten dimensions. We apply our methodology to the analysis of low and high extremes of wind speeds. In particular, we find that our methodology enables probability estimation for non-trivial extreme events in relation to electricity production via wind turbines and reveals interesting structure in the underlying data.

Problem

• Accurate joint tail extrapolation in all directions
• Efficient modelling of complex multivariate extremes
• Fast probability estimation for extreme events

Innovation

• Normalising flows model multivariate geometric extremes
• Flexible hypersphere-based deep learning methodology
• Efficient Monte Carlo integration for risk estimation