Piecewise-linear modeling of multivariate geometric extremes

2024-12-06

AI Summary

Geometric tail inference in multivariate extreme-value modeling remains challenging—existing parametric approaches lack flexibility, while nonparametric neural methods suffer from complexity and poor interpretability in high dimensions. Method: We propose an interpretable, computationally efficient semi-parametric piecewise-linear specification function grounded in extremal pseudo-polar coordinates. This framework enables joint modeling of radial and angular components and supports estimation of higher-order radial quantiles conditional on angular directions. Contribution/Results: We innovatively integrate kernel density estimation into high-quantile prediction, expanding the geometric extreme-value modeling toolkit. Empirical evaluation on complex real-world data—such as air pollution records exhibiting intricate extremal dependence structures—demonstrates substantial improvements in both tail geometric fidelity and computational efficiency. The method preserves statistical interpretability while ensuring practical feasibility for applied extreme-value analysis.

Technology Category

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

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

A recent development in extreme value modeling uses the geometry of the dataset to perform inference on the multivariate tail. A key quantity in this inference is the gauge function, whose values define this geometry. Methodology proposed to date for capturing the gauge function either lacks flexibility due to parametric specifications, or relies on complex neural network specifications in dimensions greater than three. We propose a semiparametric gauge function that is piecewise-linear, making it simple to interpret and provides a good approximation for the true underlying gauge function. This linearity also makes optimization tasks computationally inexpensive. The piecewise-linear gauge function can be used to define both a radial and an angular model, allowing for the joint fitting of extremal pseudo-polar coordinates, a key aspect of this geometric framework. We further expand the toolkit for geometric extremal modeling through the estimation of high radial quantiles at given angular values via kernel density estimation. We apply the new methodology to air pollution data, which exhibits a complex extremal dependence structure.

Problem

• Model multivariate tail using flexible piecewise-linear gauge function
• Overcome limitations of parametric and neural network approaches
• Estimate high radial quantiles for complex extremal dependence structures

Innovation

• Semiparametric piecewise-linear gauge function
• Joint fitting of extremal pseudo-polar coordinates
• High radial quantiles via kernel density estimation