An Explicit Link between Extreme Value Theory and Compositional Data Analysis

2026-07-10

AI Summary

Extreme value theory and compositional data analysis have developed distinct frameworks for modeling dependence structures of relative information, yet their interconnection remains unclear. This work systematically establishes the equivalence between these approaches in terms of covariance, variogram, and precision matrix representations through algebraic transformations—specifically, oblique projection, the Hüsler–Reiss inverse transformation, and variogram mapping. Building on this equivalence, we introduce the intrinsic log-normal graphical model and enable cross-domain methodological transfer: the Hüsler–Reiss graphical model is adapted to compositional data analysis, while dimensionality reduction techniques from compositional data are extended to multivariate extreme value analysis. This dual integration substantially enriches the modeling toolkit available in both fields.

Abstract

Extreme value theory and compositional data analysis both study settings where relative information plays a central role. In multivariate extreme value theory, threshold exceedance limits satisfy homogeneity properties that separate the radial size of an extreme event from its relative profile. In compositional data analysis, positive vectors are analysed up to multiplicative scale, and inference is based on ratios or log-ratios between components. Consequently, both fields have developed several covariance and dependence representations of the underlying relative structure. In the Hüsler-Reiss model for extremes, these include variogram, covariance, and precision parametrizations. In compositional data analysis, analogous representations arise from pairwise log-ratios, centred log-ratios, and additive log-ratios. We establish an explicit link between the two fields that relates these different representations by a small set of simple transformations, including oblique projections, Hüsler-Reiss inverses, and the variogram map. From a methodological perspective, leveraging this algebraic connection enables the transfer of statistical approaches from one field to the other. For instance, we introduce intrinsic logistic-normal graphical models for compositional data, which are based on Hüsler-Reiss graphical models for extremes. Conversely, we explore how dimensionality reduction methods from compositional data analysis can be applied to the analysis of multivariate extremes.

Problem

• Extreme Value Theory
• Compositional Data Analysis
• Relative Structure
• Hüsler-Reiss Model
• Log-ratio Transformations

Innovation

• Extreme Value Theory
• Compositional Data Analysis
• Hüsler-Reiss Model
• Log-ratio Transformations
• Graphical Models