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

• Published in: Journal of applied statistics Vol. 37; no. 4; pp. 577 - 589
• Main Authors: Cadima, Jorge, Calheiros, Francisco Lage, Preto, Isabel P.
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 01-04-2010; Taylor and Francis Journals; Taylor & Francis Ltd
• Series: Journal of Applied Statistics
• Subjects: between-group eigenpairs; block-structured correlation matrices; eigendecomposition; Mathematical analysis; Mathematical models; principal component analysis; Principal components analysis; Statistics; Variables; within-group eigenpairs
• ISSN: 0266-4763; 1360-0532
• DOI: 10.1080/02664760902803263

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.