Article Version of Record

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
  • Author(s) / Creator(s)
    Bénasséni, Jacques
  • Author(s) / Creator(s)
    Mom, Alain
  • PsychArchives acquisition timestamp
    2025-04-25T11:32:59Z
  • Made available on
    2025-04-25T11:32:59Z
  • Date of first publication
    2025-03-31
  • 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.
    en_US
  • Publication status
    publishedVersion
  • Review status
    peerReviewed
  • 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
  • ISSN
    1614-2241
  • Persistent Identifier
    https://hdl.handle.net/20.500.12034/11700
  • Persistent Identifier
    https://doi.org/10.23668/psycharchives.16288
  • Language of content
    eng
  • Publisher
    PsychOpen GOLD
  • Is version of
    https://doi.org/10.5964/meth.15357
  • Keyword(s)
    approximation
    en_US
  • Keyword(s)
    eigenvalue and eigenvector
    en_US
  • Keyword(s)
    covariance matrix
    en_US
  • Keyword(s)
    principal component analysis
    en_US
  • Keyword(s)
    perturbation
    en_US
  • Keyword(s)
    Rayleigh quotient.
    en_US
  • Dewey Decimal Classification number(s)
    150
  • Title
    A comparative study of approximations for perturbation analysis of principal components
    en_US
  • DRO type
    article
  • Issue
    1
  • Journal title
    Methodology
  • Page numbers
    27–45
  • Volume
    21
  • Visible tag(s)
    Version of Record