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Principal component models for correlation matrices

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  • Robert J. Boik

Abstract

Distributional theory regarding principal components is less well developed for correlation matrices than it is for covariance matrices. The intent of this paper is to reduce this disparity. Methods are proposed that enable investigators to fit and to make inferences about flexible principal components models for correlation matrices. The models allow arbitrary eigenvalue multiplicities and allow the distinct eigenvalues to be modelled parametrically or nonparametrically. Local parameterisations and implicit functions are used to construct full-rank unconstrained parameterisations. First-order asymptotic distributions are obtained directly from the theory of estimating functions. Second-order accurate distributions for making inferences under normality are obtained directly from likelihood theory. Simulation studies show that the Bartlett correction is effective in controlling the size of the tests and that first-order approximations to nonnull distributions are reasonably accurate. The methods are illustrated on a dataset. Copyright Biometrika Trust 2003, Oxford University Press.

Suggested Citation

  • Robert J. Boik, 2003. "Principal component models for correlation matrices," Biometrika, Biometrika Trust, vol. 90(3), pages 679-701, September.
    • Handle: RePEc:oup:biomet:v:90:y:2003:i:3:p:679-701
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    Cited by:

    1. Zongliang Hu & Zhishui Hu & Kai Dong & Tiejun Tong & Yuedong Wang, 2021. "A shrinkage approach to joint estimation of multiple covariance matrices," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(3), pages 339-374, April.
    2. Pourahmadi, Mohsen & Daniels, Michael J. & Park, Trevor, 2007. "Simultaneous modelling of the Cholesky decomposition of several covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 98(3), pages 568-587, March.
    3. Ryan Browne & Paul McNicholas, 2014. "Estimating common principal components in high dimensions," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 8(2), pages 217-226, June.
    4. Boik, Robert J., 2013. "Model-based principal components of correlation matrices," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 310-331.
    5. Changqing Ye & Linfeng Yao & Yuan Meng & Yu Zhang & Guoqing He, 2022. "Post-Occupancy Evaluation of Green Technologies for a High-Rise Building Based on User Experience," Sustainability, MDPI, vol. 14(15), pages 1-17, August.
    6. 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.
    7. Robert Boik, 2008. "Newton Algorithms for Analytic Rotation: an Implicit Function Approach," Psychometrika, Springer;The Psychometric Society, vol. 73(2), pages 231-259, June.

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