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

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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.