IDEAS home Printed from https://ideas.repec.org/a/spr/stmapp/v32y2023i3d10.1007_s10260-023-00689-y.html
   My bibliography  Save this article

Procrustes-based distances for exploring between-matrices similarity

Author

Listed:
  • Angela Andreella

    (Ca’ Foscari University)

  • Riccardo Santis

    (University of Padova)

  • Anna Vesely

    (University of Bremen)

  • Livio Finos

    (University of Padova)

Abstract

The statistical shape analysis called Procrustes analysis minimizes the Frobenius distance between matrices by similarity transformations. The method returns a set of optimal orthogonal matrices, which project each matrix into a common space. This manuscript presents two types of distances derived from Procrustes analysis for exploring between-matrices similarity. The first one focuses on the residuals from the Procrustes analysis, i.e., the residual-based distance metric. In contrast, the second one exploits the fitted orthogonal matrices, i.e., the rotational-based distance metric. Thanks to these distances, similarity-based techniques such as the multidimensional scaling method can be applied to visualize and explore patterns and similarities among observations. The proposed distances result in being helpful in functional magnetic resonance imaging (fMRI) data analysis. The brain activation measured over space and time can be represented by a matrix. The proposed distances applied to a sample of subjects—i.e., matrices—revealed groups of individuals sharing patterns of neural brain activation. Finally, the proposed method is useful in several contexts when the aim is to analyze the similarity between high-dimensional matrices affected by functional misalignment.

Suggested Citation

  • Angela Andreella & Riccardo Santis & Anna Vesely & Livio Finos, 2023. "Procrustes-based distances for exploring between-matrices similarity," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 32(3), pages 867-882, September.
    • Handle: RePEc:spr:stmapp:v:32:y:2023:i:3:d:10.1007_s10260-023-00689-y
    DOI: 10.1007/s10260-023-00689-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10260-023-00689-y
    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/s10260-023-00689-y?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. J. Gower, 1975. "Generalized procrustes analysis," Psychometrika, Springer;The Psychometric Society, vol. 40(1), pages 33-51, March.
    2. Peter J. Green & Kanti V. Mardia, 2006. "Bayesian alignment using hierarchical models, with applications in protein bioinformatics," Biometrika, Biometrika Trust, vol. 93(2), pages 235-254, June.
    3. Bert Green, 1952. "The orthogonal approximation of an oblique structure in factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 17(4), pages 429-440, December.
    4. Angela Andreella & Livio Finos, 2022. "Procrustes Analysis for High-Dimensional Data," Psychometrika, Springer;The Psychometric Society, vol. 87(4), pages 1422-1438, December.
    5. Jos Berge, 1977. "Orthogonal procrustes rotation for two or more matrices," Psychometrika, Springer;The Psychometric Society, vol. 42(2), pages 267-276, June.
    6. de Leeuw, Jan & Mair, Patrick, 2009. "Multidimensional Scaling Using Majorization: SMACOF in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 31(i03).
    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. Angela Andreella & Livio Finos, 2022. "Procrustes Analysis for High-Dimensional Data," Psychometrika, Springer;The Psychometric Society, vol. 87(4), pages 1422-1438, December.
    2. Bennani Dosse, Mohammed & Kiers, Henk A.L. & Ten Berge, Jos M.F., 2011. "Anisotropic generalized Procrustes analysis," Computational Statistics & Data Analysis, Elsevier, vol. 55(5), pages 1961-1968, May.
    3. Joost Ginkel & Pieter Kroonenberg, 2014. "Using Generalized Procrustes Analysis for Multiple Imputation in Principal Component Analysis," Journal of Classification, Springer;The Classification Society, vol. 31(2), pages 242-269, July.
    4. Dahl, Tobias & Naes, Tormod, 2006. "A bridge between Tucker-1 and Carroll's generalized canonical analysis," Computational Statistics & Data Analysis, Elsevier, vol. 50(11), pages 3086-3098, July.
    5. Ian L. Dryden & Jonathan D. Hirst & James L. Melville, 2007. "Statistical Analysis of Unlabeled Point Sets: Comparing Molecules in Chemoinformatics," Biometrics, The International Biometric Society, vol. 63(1), pages 237-251, March.
    6. Meyners, M. & Kunert, Joachim & Qannari, El Mostafa, 1998. "Comparing Generalized Procrustes Analysis and STATIS," Technical Reports 1998,35, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    7. Mohammed Bennani Dosse & Jos Berge, 2010. "Anisotropic Orthogonal Procrustes Analysis," Journal of Classification, Springer;The Classification Society, vol. 27(1), pages 111-128, March.
    8. Kohei Adachi, 2013. "Generalized joint Procrustes analysis," Computational Statistics, Springer, vol. 28(6), pages 2449-2464, December.
    9. Meyners, Michael & Qannari, El Mostafa, 2001. "Relating principal component analysis on merged data sets to a regression approach," Technical Reports 2001,47, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    10. Edmund Peay, 1988. "Multidimensional rotation and scaling of configurations to optimal agreement," Psychometrika, Springer;The Psychometric Society, vol. 53(2), pages 199-208, June.
    11. Kensuke Okada & Shin-ichi Mayekawa, 2018. "Post-processing of Markov chain Monte Carlo output in Bayesian latent variable models with application to multidimensional scaling," Computational Statistics, Springer, vol. 33(3), pages 1457-1473, September.
    12. Balbi, S & Esposito, V, 2000. "Rotated canonical analysis onto a reference subspace," Computational Statistics & Data Analysis, Elsevier, vol. 32(3-4), pages 395-410, January.
    13. Jos Berge & Dirk Knol, 1984. "Orthogonal rotations to maximal agreement for two or more matrices of different column orders," Psychometrika, Springer;The Psychometric Society, vol. 49(1), pages 49-55, March.
    14. Juliana Martins Ruzante & Valerie J. Davidson & Julie Caswell & Aamir Fazil & John A. L. Cranfield & Spencer J. Henson & Sven M. Anders & Claudia Schmidt & Jeffrey M. Farber, 2010. "A Multifactorial Risk Prioritization Framework for Foodborne Pathogens," Risk Analysis, John Wiley & Sons, vol. 30(5), pages 724-742, May.
    15. Barbara McGillivray & Gard B. Jenset & Khalid Salama & Donna Schut, 2022. "Investigating patterns of change, stability, and interaction among scientific disciplines using embeddings," Palgrave Communications, Palgrave Macmillan, vol. 9(1), pages 1-15, December.
    16. Wei Wang & Stephen J Lycett & Noreen von Cramon-Taubadel & Jennie J H Jin & Christopher J Bae, 2012. "Comparison of Handaxes from Bose Basin (China) and the Western Acheulean Indicates Convergence of Form, Not Cognitive Differences," PLOS ONE, Public Library of Science, vol. 7(4), pages 1-7, April.
    17. Michael Browne & Walter Kristof, 1969. "On the oblique rotation of a factor matrix to a specified pattern," Psychometrika, Springer;The Psychometric Society, vol. 34(2), pages 237-248, June.
    18. W. Gibson, 1962. "On the least-squares orthogonalization of an oblique transformation," Psychometrika, Springer;The Psychometric Society, vol. 27(2), pages 193-195, June.
    19. Mardia, Kanti V. & Wiechers, Henrik & Eltzner, Benjamin & Huckemann, Stephan F., 2022. "Principal component analysis and clustering on manifolds," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    20. Ibrahim, Muhammad Sohail & Dong, Wei & Yang, Qiang, 2020. "Machine learning driven smart electric power systems: Current trends and new perspectives," Applied Energy, Elsevier, vol. 272(C).

    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:stmapp:v:32:y:2023:i:3:d:10.1007_s10260-023-00689-y. 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.