A comparative study of approximations for perturbation analysis of principal components
Author(s) / Creator(s)
Bénasséni, Jacques
Mom, Alain
Abstract / Description
Principal component analysis is a well known method for dimension reduction based on the covariance matrix associated to a multivariate data table. Therefore, a large amount of work has been devoted to analyzing the sensitivity of the eigenstructure of this matrix to influential observations. In order to evaluate the effect of deleting one or a small subset of observations, several approximations for the perturbed eigenelements have been proposed. This paper provides a theoretical and numerical comparison of the main approximations. A special emphasis is given to those based on Rayleigh quotients since they are under-utilized given their excellent performance. A general approach, using refined inequalities, is proposed in order to get a precise evaluation of their accuracy without having to recompute the exact perturbed eigenvalues and eigenvectors. This approach is of specific interest from a computational standpoint. Theoretical developments are illustrated with a numerical study which emphasizes the accuracy of approximations based on Rayleigh quotients.
Keyword(s)
approximation eigenvalue and eigenvector covariance matrix principal component analysis perturbation Rayleigh quotient.Persistent Identifier
Date of first publication
2025-03-31
Journal title
Methodology
Volume
21
Issue
1
Page numbers
27–45
Publisher
PsychOpen GOLD
Publication status
publishedVersion
Review status
peerReviewed
Is version of
Citation
Bénasséni, J. & Mom, A. (2025). A comparative study of approximations for perturbation analysis of principal components. Methodology, 21(1), 27-45. https://doi.org/10.5964/meth.15357
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Author(s) / Creator(s)Bénasséni, Jacques
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Author(s) / Creator(s)Mom, Alain
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PsychArchives acquisition timestamp2025-04-25T11:32:59Z
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Made available on2025-04-25T11:32:59Z
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Date of first publication2025-03-31
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Abstract / DescriptionPrincipal component analysis is a well known method for dimension reduction based on the covariance matrix associated to a multivariate data table. Therefore, a large amount of work has been devoted to analyzing the sensitivity of the eigenstructure of this matrix to influential observations. In order to evaluate the effect of deleting one or a small subset of observations, several approximations for the perturbed eigenelements have been proposed. This paper provides a theoretical and numerical comparison of the main approximations. A special emphasis is given to those based on Rayleigh quotients since they are under-utilized given their excellent performance. A general approach, using refined inequalities, is proposed in order to get a precise evaluation of their accuracy without having to recompute the exact perturbed eigenvalues and eigenvectors. This approach is of specific interest from a computational standpoint. Theoretical developments are illustrated with a numerical study which emphasizes the accuracy of approximations based on Rayleigh quotients.en_US
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Publication statuspublishedVersion
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Review statuspeerReviewed
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CitationBénasséni, J. & Mom, A. (2025). A comparative study of approximations for perturbation analysis of principal components. Methodology, 21(1), 27-45. https://doi.org/10.5964/meth.15357
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ISSN1614-2241
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Persistent Identifierhttps://hdl.handle.net/20.500.12034/11700
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Persistent Identifierhttps://doi.org/10.23668/psycharchives.16288
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Language of contenteng
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PublisherPsychOpen GOLD
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Is version ofhttps://doi.org/10.5964/meth.15357
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Keyword(s)approximationen_US
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Keyword(s)eigenvalue and eigenvectoren_US
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Keyword(s)covariance matrixen_US
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Keyword(s)principal component analysisen_US
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Keyword(s)perturbationen_US
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Keyword(s)Rayleigh quotient.en_US
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Dewey Decimal Classification number(s)150
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TitleA comparative study of approximations for perturbation analysis of principal componentsen_US
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DRO typearticle
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Issue1
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Journal titleMethodology
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Page numbers27–45
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Volume21
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Visible tag(s)Version of Record