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The eigenstructure of block-structured correlation matrices and its implications for principal component analysis

Author

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  • Jorge Cadima
  • Francisco Lage Calheiros
  • Isabel Preto

Abstract

Block-structured correlation matrices are correlation matrices in which the p variables are subdivided into homogeneous groups, with equal correlations for variables within each group, and equal correlations between any given pair of variables from different groups. Block-structured correlation matrices arise as approximations for certain data sets' true correlation matrices. A block structure in a correlation matrix entails a certain number of properties regarding its eigendecomposition and, therefore, a principal component analysis of the underlying data. This paper explores these properties, both from an algebraic and a geometric perspective, and discusses their robustness. Suggestions are also made regarding the choice of variables to be subjected to a principal component analysis, when in the presence of (approximately) block-structured variables.

Suggested Citation

  • Jorge Cadima & Francisco Lage Calheiros & Isabel Preto, 2010. "The eigenstructure of block-structured correlation matrices and its implications for principal component analysis," Journal of Applied Statistics, Taylor & Francis Journals, vol. 37(4), pages 577-589.
    • Handle: RePEc:taf:japsta:v:37:y:2010:i:4:p:577-589
    DOI: 10.1080/02664760902803263
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    Cited by:

    1. Boyle, Phelim & Jiang, Ruihong, 2023. "A note on portfolios of averages of lognormal variables," Insurance: Mathematics and Economics, Elsevier, vol. 112(C), pages 97-109.
    2. Gong, Tingnan & Zhang, Weiping & Chen, Yu, 2023. "Uncovering block structures in large rectangular matrices," Journal of Multivariate Analysis, Elsevier, vol. 198(C).
    3. Ilya Archakov & Peter Reinhard Hansen, 2020. "A Canonical Representation of Block Matrices with Applications to Covariance and Correlation Matrices," Papers 2012.02698, arXiv.org, revised Nov 2021.

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