IDEAS home Printed from https://ideas.repec.org/p/ems/eureir/77757.html
   My bibliography  Save this paper

Least-squares Bilinear Clustering of Three-way Data

Author

Listed:
  • Schoonees, P.C.
  • Groenen, P.J.F.
  • van de Velden, M.

Abstract

__Abstract__ A least-squares bilinear clustering framework for modelling three-way data, where each observation consists of an ordinary two-way matrix, is introduced. The method combines bilinear decompositions of the two-way matrices into overall means, row margins, column margins and row-column interactions with clustering along the third way. Different clusterings are defined for each part of the decomposition, so that up to four different classifications are defined jointly. The computational burden is greatly reduced by the orthogonality of the bilinear model, such that the joint clustering problem reduces to separate ones which can be handled independently. Three of these sub-problems are specific cases of $k$-means clustering; a special algorithm is formulated for the row-column interactions, which are displayed in clusterwise biplots. The method is illustrated via two empirical examples and interpreting the interaction biplots are discussed.

Suggested Citation

  • Schoonees, P.C. & Groenen, P.J.F. & van de Velden, M., 2015. "Least-squares Bilinear Clustering of Three-way Data," Econometric Institute Research Papers EI2014-23, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    • Handle: RePEc:ems:eureir:77757
    as

    Download full text from publisher

    File URL: https://repub.eur.nl/pub/77757/EI2014-23.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Henk Kiers & Wim Krijnen, 1991. "An efficient algorithm for PARAFAC of three-way data with large numbers of observation units," Psychometrika, Springer;The Psychometric Society, vol. 56(1), pages 147-152, March.
    2. Pieter Kroonenberg & Jan Leeuw, 1980. "Principal component analysis of three-mode data by means of alternating least squares algorithms," Psychometrika, Springer;The Psychometric Society, vol. 45(1), pages 69-97, March.
    3. Grunert, Susanne C. & Scherlorn, Gerhard, 1990. "Consumer values in West Germany underlying dimensions and cross-cultural comparison with North America," Journal of Business Research, Elsevier, vol. 20(2), pages 97-107, March.
    4. Robert Tibshirani & Guenther Walther & Trevor Hastie, 2001. "Estimating the number of clusters in a data set via the gap statistic," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 411-423.
    5. Brunso, Karen & Scholderer, Joachim & Grunert, Klaus G., 2004. "Closing the gap between values and behavior--a means-end theory of lifestyle," Journal of Business Research, Elsevier, vol. 57(6), pages 665-670, June.
    6. Sara Dolnicar & Friedrich Leisch, 2010. "Evaluation of structure and reproducibility of cluster solutions using the bootstrap," Marketing Letters, Springer, vol. 21(1), pages 83-101, March.
    7. van Herk, H., 2000. "Equivalence in a Cross-National Context : Methodological & Empirical Issues in Marketing Research," Other publications TiSEM ef813d35-1129-4db2-9fb8-3, Tilburg University, School of Economics and Management.
    8. Jan Schepers & Eva Ceulemans & Iven Mechelen, 2008. "Selecting Among Multi-Mode Partitioning Models of Different Complexities: A Comparison of Four Model Selection Criteria," Journal of Classification, Springer;The Classification Society, vol. 25(1), pages 67-85, June.
    9. J. Carroll & Jih-Jie Chang, 1970. "Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart-Young” decomposition," Psychometrika, Springer;The Psychometric Society, vol. 35(3), pages 283-319, September.
    10. 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.
    11. Glenn Milligan & Martha Cooper, 1985. "An examination of procedures for determining the number of clusters in a data set," Psychometrika, Springer;The Psychometric Society, vol. 50(2), pages 159-179, June.
    12. Kamakura, Wagner A & Novak, Thomas P, 1992. "Value-System Segmentation: Exploring the Meaning of LOV," Journal of Consumer Research, Journal of Consumer Research Inc., vol. 19(1), pages 119-132, 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.
    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. van Rosmalen, J.M. & van Herk, H. & Groenen, P.J.F., 2007. "Identifying Unknown Response Styles: A Latent-Class Bilinear Multinomial Logit Model," ERIM Report Series Research in Management ERS-2007-045-MKT, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    2. Pieter C. Schoonees & Patrick J. F. Groenen & Michel Velden, 2022. "Least-squares bilinear clustering of three-way data," 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. 16(4), pages 1001-1037, December.
    3. Paolo Giordani & Roberto Rocci & Giuseppe Bove, 2020. "Factor Uniqueness of the Structural Parafac Model," Psychometrika, Springer;The Psychometric Society, vol. 85(3), pages 555-574, September.
    4. Alwin Stegeman & Tam Lam, 2014. "Three-Mode Factor Analysis by Means of Candecomp/Parafac," Psychometrika, Springer;The Psychometric Society, vol. 79(3), pages 426-443, July.
    5. Alwin Stegeman, 2018. "Simultaneous Component Analysis by Means of Tucker3," Psychometrika, Springer;The Psychometric Society, vol. 83(1), pages 21-47, March.
    6. Andreas Falke & Nadine Schröder & Claudia Hofmann, 2022. "The influence of values in sustainable consumption among millennials," Journal of Business Economics, Springer, vol. 92(6), pages 899-928, August.
    7. Mariela González-Narváez & María José Fernández-Gómez & Susana Mendes & José-Luis Molina & Omar Ruiz-Barzola & Purificación Galindo-Villardón, 2021. "Study of Temporal Variations in Species–Environment Association through an Innovative Multivariate Method: MixSTATICO," Sustainability, MDPI, vol. 13(11), pages 1-25, May.
    8. Henk Kiers, 1991. "Hierarchical relations among three-way methods," Psychometrika, Springer;The Psychometric Society, vol. 56(3), pages 449-470, September.
    9. Li, Pai-Ling & Chiou, Jeng-Min, 2011. "Identifying cluster number for subspace projected functional data clustering," Computational Statistics & Data Analysis, Elsevier, vol. 55(6), pages 2090-2103, June.
    10. Willem Kloot & Pieter Kroonenberg, 1985. "External analysis with three-mode principal component models," Psychometrika, Springer;The Psychometric Society, vol. 50(4), pages 479-494, December.
    11. Elisa Frutos-Bernal & Ángel Martín del Rey & Irene Mariñas-Collado & María Teresa Santos-Martín, 2022. "An Analysis of Travel Patterns in Barcelona Metro Using Tucker3 Decomposition," Mathematics, MDPI, vol. 10(7), pages 1-17, March.
    12. 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.
    13. Giuseppe Brandi & Ruggero Gramatica & Tiziana Di Matteo, 2019. "Unveil stock correlation via a new tensor-based decomposition method," Papers 1911.06126, arXiv.org, revised Apr 2020.
    14. Cyril Atkinson-Clement & Eléonore Pigalle, 2021. "What can we learn from Covid-19 pandemic’s impact on human behaviour? The case of France’s lockdown," Palgrave Communications, Palgrave Macmillan, vol. 8(1), pages 1-12, December.
    15. Wilderjans, Tom & Ceulemans, Eva & Van Mechelen, Iven, 2009. "Simultaneous analysis of coupled data blocks differing in size: A comparison of two weighting schemes," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 1086-1098, February.
    16. Öttl, Gerald & Böck, Philipp & Werpup, Nadja & Schwarze, Malte, 2013. "Derivation of representative air traffic peaks as standard input for airport related simulation," Journal of Air Transport Management, Elsevier, vol. 28(C), pages 31-39.
    17. J. Fernando Vera & Rodrigo Macías, 2021. "On the Behaviour of K-Means Clustering of a Dissimilarity Matrix by Means of Full Multidimensional Scaling," Psychometrika, Springer;The Psychometric Society, vol. 86(2), pages 489-513, June.
    18. Peter Schönemann, 1970. "On metric multidimensional unfolding," Psychometrika, Springer;The Psychometric Society, vol. 35(3), pages 349-366, September.
    19. Piet Brouwer & Pieter Kroonenberg, 1991. "Some notes on the diagonalization of the extended three-mode core matrix," Journal of Classification, Springer;The Classification Society, vol. 8(1), pages 93-98, January.
    20. Henner Gimpel & Daniel Rau & Maximilian Röglinger, 2018. "Understanding FinTech start-ups – a taxonomy of consumer-oriented service offerings," Electronic Markets, Springer;IIM University of St. Gallen, vol. 28(3), pages 245-264, August.

    More about this item

    Keywords

    hree-way data; bilinear decomposition; k-means cluster analysis; least-squares; estimation; biplots.;
    All these keywords.

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    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:ems:eureir:77757. 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: RePub (email available below). General contact details of provider: https://edirc.repec.org/data/feeurnl.html .

    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.