IDEAS home Printed from https://ideas.repec.org/a/spr/jclass/v39y2022i2d10.1007_s00357-022-09412-0.html
   My bibliography  Save this article

Complex Principal Component Analysis: Theory and Geometrical Aspects

Author

Listed:
  • Jean-Jacques Denimal

    (Université des Sciences et Technologies de Lille)

  • Sergio Camiz

    (National Researches Council-ISPC)

Abstract

The extension of exploratory use of real principal component analysis (RPCA) to complex data tables clears the deficiency existing in complex principal component analysis (CPCA), a method mainly developed in the statistical framework, but lacking effective interpretation tools. While often used in climatology, oceanography, and signal analysis among others, its use raises understanding difficulties, due to an intrinsic indeterminacy, which gets more tortuous its use. In this paper, a real framework in which CPCA may be embedded is proposed in order to solve its intricacies. This is obtained through the RPCA of a particular real table, derived from the complex at hand, whose double eigenvalues correspond to eigenplanes, that are proved to be both holomorphic and isoclinic. Relations existing between the two analyses lead to fix the intrinsic CPCA indetermination through a second RPCA, that optimizes the complex principal components inner structure. As a spin-off, appropriate interpretation aids derive, associated with statistics describing the structure of the clouds of units associated with the complex variables, which may get meaningful the issued graphical results. Eventually, CPCA is applied to a small wind speeds data table, to show both its use and the effectiveness of its interpretation aids, allowing an easier understanding of the CPCA’s abilities in the exploratory framework.

Suggested Citation

  • Jean-Jacques Denimal & Sergio Camiz, 2022. "Complex Principal Component Analysis: Theory and Geometrical Aspects," Journal of Classification, Springer;The Classification Society, vol. 39(2), pages 376-408, July.
    • Handle: RePEc:spr:jclass:v:39:y:2022:i:2:d:10.1007_s00357-022-09412-0
    DOI: 10.1007/s00357-022-09412-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00357-022-09412-0
    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/s00357-022-09412-0?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. 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.
    2. R. Gupta, 1972. "Principal components analysis in the complex case," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 19(1), pages 150-155, December.
    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. Sewell, Daniel K., 2018. "Visualizing data through curvilinear representations of matrices," Computational Statistics & Data Analysis, Elsevier, vol. 128(C), pages 255-270.
    2. Jushan Bai & Serena Ng, 2020. "Simpler Proofs for Approximate Factor Models of Large Dimensions," Papers 2008.00254, arXiv.org.
    3. Alfredo García-Hiernaux & José Casals & Miguel Jerez, 2012. "Estimating the system order by subspace methods," Computational Statistics, Springer, vol. 27(3), pages 411-425, September.
    4. Mitzi Cubilla‐Montilla & Ana‐Belén Nieto‐Librero & Ma Purificación Galindo‐Villardón & Ma Purificación Vicente Galindo & Isabel‐María Garcia‐Sanchez, 2019. "Are cultural values sufficient to improve stakeholder engagement human and labour rights issues?," Corporate Social Responsibility and Environmental Management, John Wiley & Sons, vol. 26(4), pages 938-955, July.
    5. Jos Berge & Henk Kiers, 1993. "An alternating least squares method for the weighted approximation of a symmetric matrix," Psychometrika, Springer;The Psychometric Society, vol. 58(1), pages 115-118, March.
    6. Shimeng Huang & Henry Wolkowicz, 2018. "Low-rank matrix completion using nuclear norm minimization and facial reduction," Journal of Global Optimization, Springer, vol. 72(1), pages 5-26, September.
    7. Antti J. Tanskanen & Jani Lukkarinen & Kari Vatanen, 2016. "Random selection of factors preserves the correlation structure in a linear factor model to a high degree," Papers 1604.05896, arXiv.org, revised Dec 2018.
    8. 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.
    9. Kargin, V. & Onatski, A., 2008. "Curve forecasting by functional autoregression," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2508-2526, November.
    10. Yoshio Takane & Forrest Young & Jan Leeuw, 1977. "Nonmetric individual differences multidimensional scaling: An alternating least squares method with optimal scaling features," Psychometrika, Springer;The Psychometric Society, vol. 42(1), pages 7-67, March.
    11. W. Gibson, 1962. "On the least-squares orthogonalization of an oblique transformation," Psychometrika, Springer;The Psychometric Society, vol. 27(2), pages 193-195, June.
    12. Walter Kristof, 1967. "Orthogonal inter-battery factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 32(2), pages 199-227, June.
    13. Willem E. Saris & Marius de Pijper & Jan Mulder, 1978. "Optimal Procedures for Estimation of Factor Scores," Sociological Methods & Research, , vol. 7(1), pages 85-106, August.
    14. 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.
    15. Juan Carlos Carrasco Baquero & Verónica Lucía Caballero Serrano & Fernando Romero Cañizares & Daisy Carolina Carrasco López & David Alejandro León Gualán & Rufino Vieira Lanero & Fernando Cobo-Gradín, 2023. "Water Quality Determination Using Soil and Vegetation Communities in the Wetlands of the Andes of Ecuador," Land, MDPI, vol. 12(8), pages 1-18, August.
    16. 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.
    17. Johannes Burge & Priyank Jaini, 2017. "Accuracy Maximization Analysis for Sensory-Perceptual Tasks: Computational Improvements, Filter Robustness, and Coding Advantages for Scaled Additive Noise," PLOS Computational Biology, Public Library of Science, vol. 13(2), pages 1-32, February.
    18. Elvin Isufi & Andreas Loukas & Nathanael Perraudin & Geert Leus, 2018. "Forecasting Time Series with VARMA Recursions on Graphs," Papers 1810.08581, arXiv.org, revised Jul 2019.
    19. Naccarato, Alessia & Zurlo, Davide & Pieraccini, Luciano, 2012. "Least Orthogonal Distance Estimator and Total Least Square," MPRA Paper 42365, University Library of Munich, Germany.
    20. Michael Greenacre, 2009. "Contribution biplots," Economics Working Papers 1162, Department of Economics and Business, Universitat Pompeu Fabra, revised Jan 2011.

    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:jclass:v:39:y:2022:i:2:d:10.1007_s00357-022-09412-0. 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.