IDEAS home Printed from https://ideas.repec.org/a/spr/advdac/v17y2023i2d10.1007_s11634-022-00503-9.html
   My bibliography  Save this article

Principal component analysis constrained by layered simple structures

Author

Listed:
  • Naoto Yamashita

    (Kansai University)

Abstract

The paper proposes a procedure for principal component analysis called layered principal component analysis (LPCA) to produce a simple and interpretable loading matrix. The novelty of LPCA is that a loading matrix is constrained as a sum of matrices with simple structures called layers, and the resulting simplicity of the LPCA solution is controlled by how many layers are used. LPCA is a generalization of disjoint PCA proposed as reported by Ferrara (in: Giommi (ed) Topics in theoretical and applied statistics, Springer, Cham 2016). The number of layers controls the balance of simplicity and the fit to the data, and the user can choose the desired level of simplicity between the most restrictive but simplest case with a single layer or multiple layers with better fit to the data. The optimal number of layers is specified in terms of explained variance and two information criteria. Two simulation studies were conducted to evaluate how accurately the LPCA procedure recovers the true parameter values. The results showed that LPCA was effective for parameter recovery. The paper presents three examples of LPCA applied to real data, which show the potential of LPCA for producing simple and interpretable loading matrices.

Suggested Citation

  • Naoto Yamashita, 2023. "Principal component analysis constrained by layered simple structures," 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. 17(2), pages 347-367, June.
    • Handle: RePEc:spr:advdac:v:17:y:2023:i:2:d:10.1007_s11634-022-00503-9
    DOI: 10.1007/s11634-022-00503-9
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11634-022-00503-9
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11634-022-00503-9?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Kohei Adachi, 2009. "Joint Procrustes Analysis for Simultaneous Nonsingular Transformation of Component Score and Loading Matrices," Psychometrika, Springer;The Psychometric Society, vol. 74(4), pages 667-683, December.
    2. Henk Kiers & Jos Berge, 1992. "Minimization of a class of matrix trace functions by means of refined majorization," Psychometrika, Springer;The Psychometric Society, vol. 57(3), pages 371-382, September.
    3. Coen Bernaards & Robert Jennrich, 2003. "Orthomax rotation and perfect simple structure," Psychometrika, Springer;The Psychometric Society, vol. 68(4), pages 585-588, December.
    4. Vichi, Maurizio & Saporta, Gilbert, 2009. "Clustering and disjoint principal component analysis," Computational Statistics & Data Analysis, Elsevier, vol. 53(8), pages 3194-3208, June.
    5. Kohei Adachi, 2011. "Constrained principal component analysis of standardized data for biplots with unit-length variable vectors," 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. 5(1), pages 23-36, April.
    6. Maurizio Vichi, 2017. "Disjoint factor analysis with cross-loadings," 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. 11(3), pages 563-591, September.
    7. Henk Kiers, 1990. "Majorization as a tool for optimizing a class of matrix functions," Psychometrika, Springer;The Psychometric Society, vol. 55(3), pages 417-428, September.
    8. Kohei Adachi & Nickolay T. Trendafilov, 2018. "Sparsest factor analysis for clustering variables: a matrix decomposition approach," 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. 12(3), pages 559-585, September.
    9. Kiers, Henk A. L., 2002. "Setting up alternating least squares and iterative majorization algorithms for solving various matrix optimization problems," Computational Statistics & Data Analysis, Elsevier, vol. 41(1), pages 157-170, November.
    10. Henry Kaiser, 1974. "An index of factorial simplicity," Psychometrika, Springer;The Psychometric Society, vol. 39(1), pages 31-36, March.
    11. Kohei Adachi & Nickolay T. Trendafilov, 2016. "Sparse principal component analysis subject to prespecified cardinality of loadings," Computational Statistics, Springer, vol. 31(4), pages 1403-1427, December.
    12. Chester Harris & Henry Kaiser, 1964. "Oblique factor analytic solutions by orthogonal transformations," Psychometrika, Springer;The Psychometric Society, vol. 29(4), pages 347-362, December.
    13. Naoto Yamashita & Shin-ichi Mayekawa, 2015. "A new biplot procedure with joint classification of objects and variables by fuzzy c-means clustering," 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. 9(3), pages 243-266, September.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Naoto Yamashita & Kohei Adachi, 2020. "A Modified k-Means Clustering Procedure for Obtaining a Cardinality-Constrained Centroid Matrix," Journal of Classification, Springer;The Classification Society, vol. 37(2), pages 509-525, July.
    2. Adelaide Freitas & Eloísa Macedo & Maurizio Vichi, 2021. "An empirical comparison of two approaches for CDPCA in high-dimensional data," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 30(3), pages 1007-1031, September.
    3. Henk Kiers, 1995. "Maximization of sums of quotients of quadratic forms and some generalizations," Psychometrika, Springer;The Psychometric Society, vol. 60(2), pages 221-245, June.
    4. Krijnen, Wim P., 2006. "Convergence of the sequence of parameters generated by alternating least squares algorithms," Computational Statistics & Data Analysis, Elsevier, vol. 51(2), pages 481-489, November.
    5. Kohei Adachi, 2009. "Joint Procrustes Analysis for Simultaneous Nonsingular Transformation of Component Score and Loading Matrices," Psychometrika, Springer;The Psychometric Society, vol. 74(4), pages 667-683, December.
    6. Robert Jennrich, 2001. "A simple general procedure for orthogonal rotation," Psychometrika, Springer;The Psychometric Society, vol. 66(2), pages 289-306, June.
    7. Kiers, Henk A. L., 2002. "Setting up alternating least squares and iterative majorization algorithms for solving various matrix optimization problems," Computational Statistics & Data Analysis, Elsevier, vol. 41(1), pages 157-170, November.
    8. Kim, Nam-Hwui & Browne, Ryan P., 2021. "In the pursuit of sparseness: A new rank-preserving penalty for a finite mixture of factor analyzers," Computational Statistics & Data Analysis, Elsevier, vol. 160(C).
    9. Henk Kiers, 1997. "Techniques for rotating two or more loading matrices to optimal agreement and simple structure: A comparison and some technical details," Psychometrika, Springer;The Psychometric Society, vol. 62(4), pages 545-568, December.
    10. Giovanni Franco, 2014. "Toward a simple structure: a comparison of different rotation techniques," Quality & Quantity: International Journal of Methodology, Springer, vol. 48(3), pages 1785-1797, May.
    11. Heij, C. & Groenen, P.J.F. & van Dijk, D.J.C., 2006. "Time series forecasting by principal covariate regression," Econometric Institute Research Papers EI 2006-37, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    12. P. Bentler, 1977. "Factor simplicity index and transformations," Psychometrika, Springer;The Psychometric Society, vol. 42(2), pages 277-295, June.
    13. Henk Kiers, 1997. "Weighted least squares fitting using ordinary least squares algorithms," Psychometrika, Springer;The Psychometric Society, vol. 62(2), pages 251-266, June.
    14. Rosember Guerra-Urzola & Niek C. Schipper & Anya Tonne & Klaas Sijtsma & Juan C. Vera & Katrijn Deun, 2023. "Sparsifying the least-squares approach to PCA: comparison of lasso and cardinality constraint," 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. 17(1), pages 269-286, March.
    15. Carlo Cavicchia & Maurizio Vichi, 2022. "Second-Order Disjoint Factor Analysis," Psychometrika, Springer;The Psychometric Society, vol. 87(1), pages 289-309, March.
    16. Carlo Cavicchia & Maurizio Vichi & Giorgia Zaccaria, 2023. "Hierarchical disjoint principal component analysis," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 107(3), pages 537-574, September.
    17. Kohei Adachi, 2011. "Three-Way Tucker2 Component Analysis Solutions of Stimuli × Responses × Individuals Data with Simple Structure and the Fewest Core Differences," Psychometrika, Springer;The Psychometric Society, vol. 76(2), pages 285-305, April.
    18. Lillemo, Shuling Chen, 2014. "Measuring the effect of procrastination and environmental awareness on households' energy-saving behaviours: An empirical approach," Energy Policy, Elsevier, vol. 66(C), pages 249-256.
    19. Xiaoxu Dong & Huawei Zhao & Tiancai Li, 2022. "The Role of Live-Streaming E-Commerce on Consumers’ Purchasing Intention regarding Green Agricultural Products," Sustainability, MDPI, vol. 14(7), pages 1-13, April.
    20. Simplice A. Asongu & Nicholas M. Odhiambo, 2019. "Governance, capital flight and industrialisation in Africa," Journal of Economic Structures, Springer;Pan-Pacific Association of Input-Output Studies (PAPAIOS), vol. 8(1), pages 1-22, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:advdac:v:17:y:2023:i:2:d:10.1007_s11634-022-00503-9. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.