Anchored Geodesic Analysis for Multivariate Extremes

2026-07-14

AI Summary

This study addresses the challenges of modeling extremal dependence structures and dimensionality reduction in high-dimensional extreme-value data by proposing Anchored Geodesic Component Analysis (AGCA). The method anchors on directions of complete dependence and approximates the angular measure on the unit sphere via constrained great subspheres, establishing an interpretable dimensionality reduction framework based on a bounded sine-squared geodesic loss. Theoretically, it provides bounds on tail approximation error and establishes statistical consistency. Methodologically, AGCA integrates geodesic principal component analysis, angular measure modeling, and second-moment spectral decomposition in the tangent space. Empirical results on ten-dimensional daily stock portfolio losses show that the first ten AGCA components capture approximately 91% of the anchored variation, yielding an average relative error of only 1.25% for capped excess losses and normalized Value-at-Risk estimates.

Abstract

Extremal dependence is naturally described by the angular law of large multivariate observations. We introduce anchored geodesic component analysis (AGCA), a dimension-reduction method for extremal angular laws on the positive unit sphere. AGCA approximates angular variation by great subspheres constrained to pass through a chosen reference direction, with balanced complete dependence as the default anchor. Under a bounded sine-squared geodesic loss, the population and empirical problems reduce exactly to eigenanalysis of a second-moment matrix of anchored tangent departures. The resulting scores, loadings, residual risks and explained-variation summaries describe departures from the benchmark and remain well defined for face and near-axis extremes. Low-rank AGCA reconstructions also support tail simulation: bounded Lipschitz functionals and homogeneous tail scores, including portfolio capped excesses and value-at-risk, inherit explicit error bounds from the AGCA residual risk. We establish top-\(k\) consistency for oracle and rank-Pareto AGCA summaries and an oracle central limit theorem whose covariance is that of an independent sample from the limiting angular law. In daily equity-portfolio losses, AGCA finds concentrated benchmark-relative tail directions: ten components explain about \(91\%\) of anchored variation and approximate capped-excess and normalized value-at-risk summaries with about \(1.25\%\) average relative error.

Problem

• multivariate extremes
• angular law
• dimension reduction
• tail dependence
• extremal risk

Innovation

• Anchored Geodesic Component Analysis
• extremal angular law
• dimension reduction
• tail risk approximation
• geodesic loss