Asymptotics of eigenstructure of sample correlation matrices for high-dimensional spiked models

Area of Science:

• Statistics
• Data Analysis
• High-Dimensional Data

Background:

• Sample correlation matrices are crucial for analyzing relationships in data.
• Limited understanding exists for spectral properties of correlation matrices in high-dimensional settings, especially beyond null models.
• Spiked models offer a framework to study deviations from independence in data.

Purpose of the Study:

• To investigate the asymptotic spectral properties of sample correlation matrices in high-dimensional data.
• To compare the behavior of eigenvalues and eigenvectors of correlation matrices with those of covariance matrices.
• To provide theoretical results for leading eigenvalues and eigenvectors under spiked models.

Main Methods:

• Application of random matrix theory to analyze sample correlation matrices.
• Focus on high-dimensional regime where the ratio of variables (p) to sample size (n) converges to a constant.
• Derivation of asymptotic first-order and distributional results for eigenvalues and eigenvectors.

Main Results:

• First-order spectral properties of sample correlation matrices align with those of sample covariance matrices.
• Asymptotic distributions of eigenvalues and eigenvectors can differ significantly between correlation and covariance matrices.
• Fluctuations in eigenvalues and eigenvectors derived from correlation matrices are notably smaller than those from covariance matrices.

Conclusions:

• The spectral properties of sample correlation matrices exhibit distinct distributional behaviors compared to covariance matrices in high dimensions.
• Understanding these differences is vital for accurate statistical inference with high-dimensional correlation data.
• The findings contribute to the theoretical foundation of random matrix theory applied to correlation structures.