Principal component analysis for max-stable distributions

TL;DR

This paper adapts principal component analysis to max-stable distributions by using max-linear maps and regression, enabling dimension reduction and reconstruction despite the challenges posed by heavy tails and support constraints.

Contribution

It introduces a novel PCA framework for max-stable distributions using max-linear maps and provides methods for optimal projection estimation and distribution reconstruction.

Findings

Successful adaptation of PCA to max-stable distributions.

Consistent estimation of the optimal projection matrix.

Demonstrated effectiveness through simulation and real data application.

Abstract

Principal component analysis (PCA) is one of the most popular dimension reduction techniques in statistics and is especially powerful when a multivariate distribution is concentrated near a lower-dimensional subspace. Multivariate extreme value distributions have turned out to provide challenges for the application of PCA since their constraint support impedes the detection of lower-dimensional structures and heavy-tails can imply that second moments do not exist, thereby preventing the application of classical variance-based techniques for PCA. We adapt PCA to max-stable distributions using a regression setting and employ max-linear maps to project the random vector to a lower-dimensional space while preserving max-stability. We also provide a characterization of those distributions which allow for a perfect reconstruction from the lower-dimensional representation. Finally, we demonstrate how an optimal projection matrix can be consistently estimated and show viability in practice with a simulation study and application to a benchmark dataset.