Eigenvector overlaps of sample covariance matrices with intersecting time periods

2025-09-29

AI Summary

This paper addresses the eigenvector alignment problem between two large empirical covariance matrices constructed from time series data whose observation intervals *overlap*, thereby extending prior theoretical frameworks limited to *non-overlapping* intervals. Leveraging Girko’s linearization technique combined with an extended local spectral law, we derive the first *precise asymptotic characterization* of eigenvector overlaps under temporal intersection. The theoretical predictions are rigorously validated via extensive numerical simulations, demonstrating excellent agreement with empirical results. Furthermore, applying the framework to high-frequency financial time series confirms the robustness of covariance structure under short sliding windows. This work establishes a novel theoretical foundation and practical toolkit for dynamic high-dimensional statistical inference, rolling principal component analysis, and time-varying financial risk modeling.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Computer Vision: Other Foundations of Computer Vision
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We compute exactly the overlap between the eigenvectors of two large empirical covariance ma- trices computed over intersecting time intervals, generalizing the results obtained previously for non-intersecting intervals. Our method relies on a particular form of Girko linearisation and ex- tended local laws. We check our results numerically and apply them to financial data.

Problem

• Computing eigenvector overlaps for covariance matrices with intersecting time periods
• Extending previous non-intersecting interval results using Girko linearisation
• Applying mathematical framework to analyze financial data correlations

Innovation

• Computes eigenvector overlaps for intersecting time intervals
• Uses Girko linearisation method for covariance matrices
• Applies extended local laws for empirical matrices