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.

翻译：主成分分析（PCA）是统计学中最常用的降维技术之一，当多元分布集中于较低维子空间附近时尤其有效。多元极值分布对PCA的应用提出了挑战，因为其约束支撑阻碍了低维结构的检测，且重尾特性可能导致二阶矩不存在，从而无法应用基于经典方差的PCA技术。本文通过回归框架将PCA适配于极大稳定分布，并采用极大线性映射将随机向量投影至低维空间，同时保持极大稳定性。我们还刻画了能够从低维表示实现完美重构的分布特征。最后，我们论证了最优投影矩阵如何被一致估计，并通过模拟研究和基准数据集的应用验证了该方法的实际可行性。