The eigenstructure of block-structured correlation matrices and its implications for principal component analysis |
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Authors: | Jorge Cadima Francisco Lage Calheiros Isabel P Preto |
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Institution: | 1. Departamento de Matemática, Instituto Superior de Agronomia , Universidade Técnica de Lisboa , Tapada da Ajuda, 1349-017 , Lisboa , Portugal;2. Departamento de Engenharia Civil, Faculdade de Engenharia , Universidade do Porto , Portugal |
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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. |
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Keywords: | block-structured correlation matrices eigendecomposition principal component analysis within-group eigenpairs between-group eigenpairs |
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