An Explicit Link between Extreme Value Theory and Compositional Data Analysis

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.