Principal Component Analysis for max-stable distributions

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.

主成分分析（Principal Component Analysis，PCA）是统计学中最常用的降维技术之一，尤其适用于多元分布集中于低维子空间附近的场景。然而多元极值分布却为主成分分析的应用带来了挑战：其约束支撑域会阻碍低维结构的检测，且重尾特性可能导致二阶矩不存在，从而无法使用经典的基于方差的主成分分析方法。我们借助回归框架将主成分分析适配至极大稳定分布，并采用极大线性映射将随机向量投影至低维空间，同时保留其极大稳定性。我们还对那些可从低维表示实现完美重构的分布进行了数学刻画。最后，我们演示了如何对最优投影矩阵进行一致估计，并通过模拟实验与基准数据集应用验证了该方法的实际可行性。