🤖 AI Summary
Conventional PCA fails for max-stable distributions due to their restricted support and heavy tails, which violate PCA’s reliance on finite second moments and linear variance structure. Method: We propose the first PCA paradigm tailored to extreme-value statistics, based on a max-linear regression framework that preserves max-stability in low-dimensional projections. Our approach integrates extremal regression, nonlinear projection optimization, and stability-constrained inference. Contribution/Results: Theoretically, we establish necessary and sufficient conditions for perfect reconstruction and prove consistent estimability of the optimal projection matrix. Empirically, simulations and real-world extreme-value data demonstrate substantial improvements in low-dimensional representation fidelity and reconstruction accuracy for heavy-tailed extremes. The method provides an interpretable, computationally tractable dimensionality reduction foundation for high-dimensional extreme-value modeling.
📝 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.