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Model-based principal components of correlation matrices

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  • Boik, Robert J.

Abstract

A model for principal components of correlation matrices is proposed. The model satisfies the correlation constraint (i.e., unit valued diagonal elements) as well as optional constraints on eigenvalues and/or eigenvectors. The model yields simplified principal components that retain both orthogonality and variance maximization properties. Inference procedures for eigenvalues, eigenvectors, and loadings on rotated or raw principal components are given. Multivariate normality is not required. A major issue in the modeling process is that the eigen-structure of the population correlation matrix can induce rank deficiencies in the submatrix of the constraint Jacobian matrix that is associated with the correlation constraint. This rank deficiency is a property of the population constraint Jacobian matrix; it is not necessarily a property of the sample Jacobian matrix evaluated at the solution to the estimating equation. Furthermore, if degenerate constraints are eliminated, then the fitted correlation matrix need not satisfy the correlation constraint. Procedures are proposed for detecting rank deficiencies, eliminating degenerate constraints, and constructing auxiliary constraints that ensure that the correlation constraint is satisfied. The procedures are illustrated on two real data sets.

Suggested Citation

  • Boik, Robert J., 2013. "Model-based principal components of correlation matrices," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 310-331.
    • Handle: RePEc:eee:jmvana:v:116:y:2013:i:c:p:310-331
    DOI: 10.1016/j.jmva.2012.11.017
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    References listed on IDEAS

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    1. Schott, James R., 1998. "Estimating correlation matrices that have common eigenvectors," Computational Statistics & Data Analysis, Elsevier, vol. 27(4), pages 445-459, June.
    2. Robert Boik, 2008. "Newton Algorithms for Analytic Rotation: an Implicit Function Approach," Psychometrika, Springer;The Psychometric Society, vol. 73(2), pages 231-259, June.
    3. Krishnaiah, P. R. & Lee, Jack C., 1979. "On the asymptotic joint distributions of certain functions of the eigenvalues of four random matrices," Journal of Multivariate Analysis, Elsevier, vol. 9(2), pages 248-258, June.
    4. Kollo, T. & Neudecker, H., 1993. "Asymptotics of Eigenvalues and Unit-Length Eigenvectors of Sample Variance and Correlation Matrices," Journal of Multivariate Analysis, Elsevier, vol. 47(2), pages 283-300, November.
    5. Boik, Robert J., 2005. "Second-order accurate inference on eigenvalues of covariance and correlation matrices," Journal of Multivariate Analysis, Elsevier, vol. 96(1), pages 136-171, September.
    6. Ke-Hai Yuan & Peter Bentler, 2000. "On equivariance and invariance of standard errors in three exploratory factor models," Psychometrika, Springer;The Psychometric Society, vol. 65(1), pages 121-133, March.
    7. Peter Bentler & Ke-Hai Yuan, 1998. "Tests for linear trend in the smallest eigenvalues of the correlation matrix," Psychometrika, Springer;The Psychometric Society, vol. 63(2), pages 131-144, June.
    8. Robert J. Boik, 2002. "Spectral models for covariance matrices," Biometrika, Biometrika Trust, vol. 89(1), pages 159-182, March.
    9. Haruhiko Ogasawara, 2002. "Concise formulas for the standard errors of component loading estimates," Psychometrika, Springer;The Psychometric Society, vol. 67(2), pages 289-297, June.
    10. Lee, Sik-Yum, 1985. "Analysis of covariance and correlation structures," Computational Statistics & Data Analysis, Elsevier, vol. 2(4), pages 279-295, February.
    11. Robert J. Boik, 2003. "Principal component models for correlation matrices," Biometrika, Biometrika Trust, vol. 90(3), pages 679-701, September.
    12. Boik, Robert J., 1998. "A Local Parameterization of Orthogonal and Semi-Orthogonal Matrices with Applications," Journal of Multivariate Analysis, Elsevier, vol. 67(2), pages 244-276, November.
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