IDEAS home Printed from https://ideas.repec.org/a/spr/compst/v31y2016i4d10.1007_s00180-015-0608-4.html
   My bibliography  Save this article

Sparse principal component analysis subject to prespecified cardinality of loadings

Author

Listed:
  • Kohei Adachi

    (Osaka University)

  • Nickolay T. Trendafilov

    (Open University)

Abstract

Most of the existing procedures for sparse principal component analysis (PCA) use a penalty function to obtain a sparse matrix of weights by which a data matrix is post-multiplied to produce PC scores. In this paper, we propose a new sparse PCA procedure which differs from the existing ones in two ways. First, the new procedure does not sparsify the weight matrix. Instead, the so-called loadings matrix is sparsified by which the score matrix is post-multiplied to approximate the data matrix. Second, the cardinality of the loading matrix i.e., the total number of nonzero loadings, is pre-specified to be an integer without using penalty functions. The procedure is called unpenalized sparse loading PCA (USLPCA). A desirable property of USLPCA is that the indices for the percentages of explained variances can be defined in the same form as in the standard PCA. We develop an alternate least squares algorithm for USLPCA which uses the fact that the PCA loss function can be decomposed as a sum of a term irrelevant to the loadings, and another one being easily minimized under cardinality constraints. A procedure is also presented for selecting the best cardinality using information criteria. The procedures are assessed in a simulation study and illustrated with real data examples.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:compst:v:31:y:2016:i:4:d:10.1007_s00180-015-0608-4
    DOI: 10.1007/s00180-015-0608-4
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00180-015-0608-4
    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/s00180-015-0608-4?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. Shen, Haipeng & Huang, Jianhua Z., 2008. "Sparse principal component analysis via regularized low rank matrix approximation," Journal of Multivariate Analysis, Elsevier, vol. 99(6), pages 1015-1034, July.
    2. Doyo Enki & Nickolay Trendafilov, 2012. "Sparse principal components by semi-partition clustering," Computational Statistics, Springer, vol. 27(4), pages 605-626, December.
    3. Nickolay Trendafilov, 2014. "From simple structure to sparse components: a review," Computational Statistics, Springer, vol. 29(3), pages 431-454, June.
    4. Nickolay Trendafilov & Kohei Adachi, 2015. "Sparse Versus Simple Structure Loadings," Psychometrika, Springer;The Psychometric Society, vol. 80(3), pages 776-790, September.
    5. Carl Eckart & Gale Young, 1936. "The approximation of one matrix by another of lower rank," Psychometrika, Springer;The Psychometric Society, vol. 1(3), pages 211-218, September.
    6. J. N. R. Jeffers, 1967. "Two Case Studies in the Application of Principal Component Analysis," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 16(3), pages 225-236, November.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. 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.
    3. Guerra Urzola, Rosember & Van Deun, Katrijn & Vera, J. C. & Sijtsma, K., 2021. "A guide for sparse PCA : Model comparison and applications," Other publications TiSEM 4d35b931-7f49-444b-b92f-a, Tilburg University, School of Economics and Management.
    4. Rosember Guerra-Urzola & Katrijn Van Deun & Juan C. Vera & Klaas Sijtsma, 2021. "A Guide for Sparse PCA: Model Comparison and Applications," Psychometrika, Springer;The Psychometric Society, vol. 86(4), pages 893-919, December.
    5. 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.

    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. Rosember Guerra-Urzola & Katrijn Van Deun & Juan C. Vera & Klaas Sijtsma, 2021. "A Guide for Sparse PCA: Model Comparison and Applications," Psychometrika, Springer;The Psychometric Society, vol. 86(4), pages 893-919, December.
    2. Nickolay Trendafilov, 2014. "From simple structure to sparse components: a review," Computational Statistics, Springer, vol. 29(3), pages 431-454, June.
    3. Guerra Urzola, Rosember & Van Deun, Katrijn & Vera, J. C. & Sijtsma, K., 2021. "A guide for sparse PCA : Model comparison and applications," Other publications TiSEM 4d35b931-7f49-444b-b92f-a, Tilburg University, School of Economics and Management.
    4. Jin, Shaobo & Moustaki, Irini & Yang-Wallentin, Fan, 2018. "Approximated penalized maximum likelihood for exploratory factor analysis: an orthogonal case," LSE Research Online Documents on Economics 88118, London School of Economics and Political Science, LSE Library.
    5. Shaobo Jin & Irini Moustaki & Fan Yang-Wallentin, 2018. "Approximated Penalized Maximum Likelihood for Exploratory Factor Analysis: An Orthogonal Case," Psychometrika, Springer;The Psychometric Society, vol. 83(3), pages 628-649, September.
    6. Nerea González-García & Ana Belén Nieto-Librero & Purificación Galindo-Villardón, 2023. "CenetBiplot: a new proposal of sparse and orthogonal biplots methods by means of elastic net CSVD," 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 5-19, March.
    7. Mitzi Cubilla-Montilla & Ana Belén Nieto-Librero & M. Purificación Galindo-Villardón & Carlos A. Torres-Cubilla, 2021. "Sparse HJ Biplot: A New Methodology via Elastic Net," Mathematics, MDPI, vol. 9(11), pages 1-15, June.
    8. Jushan Bai & Serena Ng, 2020. "Simpler Proofs for Approximate Factor Models of Large Dimensions," Papers 2008.00254, arXiv.org.
    9. Thomas Despois & Catherine Doz, 2022. "Identifying and interpreting the factors in factor models via sparsity : Different approaches," Working Papers halshs-03626503, HAL.
    10. Thomas Despois & Catherine Doz, 2023. "Identifying and interpreting the factors in factor models via sparsity: Different approaches," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 38(4), pages 533-555, June.
    11. Jin-Xing Liu & Yong Xu & Chun-Hou Zheng & Yi Wang & Jing-Yu Yang, 2012. "Characteristic Gene Selection via Weighting Principal Components by Singular Values," PLOS ONE, Public Library of Science, vol. 7(7), pages 1-10, July.
    12. Merola, Giovanni Maria & Chen, Gemai, 2019. "Projection sparse principal component analysis: An efficient least squares method," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 366-382.
    13. Mihee Lee & Haipeng Shen & Jianhua Z. Huang & J. S. Marron, 2010. "Biclustering via Sparse Singular Value Decomposition," Biometrics, The International Biometric Society, vol. 66(4), pages 1087-1095, December.
    14. Carrizosa, Emilio & Guerrero, Vanesa, 2014. "Biobjective sparse principal component analysis," Journal of Multivariate Analysis, Elsevier, vol. 132(C), pages 151-159.
    15. Amir Beck & Yakov Vaisbourd, 2016. "The Sparse Principal Component Analysis Problem: Optimality Conditions and Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 170(1), pages 119-143, July.
    16. Thomas Despois & Catherine Doz, 2021. "Identifying and interpreting the factors in factor models via sparsity: Different approaches," Working Papers halshs-02235543, HAL.
    17. Shen, Dan & Shen, Haipeng & Marron, J.S., 2013. "Consistency of sparse PCA in High Dimension, Low Sample Size contexts," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 317-333.
    18. Bai, Jushan & Ng, Serena, 2019. "Rank regularized estimation of approximate factor models," Journal of Econometrics, Elsevier, vol. 212(1), pages 78-96.
    19. Davood Hajinezhad & Qingjiang Shi, 2018. "Alternating direction method of multipliers for a class of nonconvex bilinear optimization: convergence analysis and applications," Journal of Global Optimization, Springer, vol. 70(1), pages 261-288, January.
    20. Thomas Despois & Catherine Doz, 2022. "Identifying and interpreting the factors in factor models via sparsity : Different approaches," PSE Working Papers halshs-03626503, HAL.

    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:compst:v:31:y:2016:i:4:d:10.1007_s00180-015-0608-4. 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.