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Clustering and disjoint principal component analysis

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  • Vichi, Maurizio
  • Saporta, Gilbert

Abstract

A constrained principal component analysis, which aims at a simultaneous clustering of objects and a partitioning of variables, is proposed. The new methodology allows us to identify components with maximum variance, each one a linear combination of a subset of variables. All the subsets form a partition of variables. Simultaneously, a partition of objects is also computed maximizing the between cluster variance. The methodology is formulated in a semi-parametric least-squares framework as a quadratic mixed continuous and integer problem. An alternating least-squares algorithm is proposed to solve the clustering and disjoint PCA. Two applications are given to show the features of the methodology.

Suggested Citation

  • Vichi, Maurizio & Saporta, Gilbert, 2009. "Clustering and disjoint principal component analysis," Computational Statistics & Data Analysis, Elsevier, vol. 53(8), pages 3194-3208, June.
    • Handle: RePEc:eee:csdana:v:53:y:2009:i:8:p:3194-3208
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    3. Nickolay T. Trendafilov & Tsegay Gebrehiwot Gebru, 2016. "Recipes for sparse LDA of horizontal data," METRON, Springer;Sapienza Università di Roma, vol. 74(2), pages 207-221, August.
    4. 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.
    5. 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.
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    8. Carlos Martin-Barreiro & John A. Ramirez-Figueroa & Xavier Cabezas & Victor Leiva & Ana Martin-Casado & M. Purificación Galindo-Villardón, 2021. "A New Algorithm for Computing Disjoint Orthogonal Components in the Parallel Factor Analysis Model with Simulations and Applications to Real-World Data," Mathematics, MDPI, vol. 9(17), pages 1-22, August.
    9. Yannis Yatracos, 2013. "Detecting Clusters in the Data from Variance Decompositions of Its Projections," Journal of Classification, Springer;The Classification Society, vol. 30(1), pages 30-55, April.
    10. Gianluca Sottile & Giada Adelfio, 2019. "Clusters of effects curves in quantile regression models," Computational Statistics, Springer, vol. 34(2), pages 551-569, June.
    11. Fidele Tugizimana & Paul A Steenkamp & Lizelle A Piater & Ian A Dubery, 2014. "Multi-Platform Metabolomic Analyses of Ergosterol-Induced Dynamic Changes in Nicotiana tabacum Cells," PLOS ONE, Public Library of Science, vol. 9(1), pages 1-18, January.
    12. Nickolay Trendafilov, 2014. "From simple structure to sparse components: a review," Computational Statistics, Springer, vol. 29(3), pages 431-454, June.
    13. Jérome SARACCO & Marie CHAVENT & Vanessa KUENTZ, 2010. "Clustering of categorical variables around latent variables," Cahiers du GREThA (2007-2019) 2010-02, Groupe de Recherche en Economie Théorique et Appliquée (GREThA).
    14. Blasius, J. & Greenacre, M. & Groenen, P.J.F. & van de Velden, M., 2009. "Special issue on correspondence analysis and related methods," Computational Statistics & Data Analysis, Elsevier, vol. 53(8), pages 3103-3106, June.
    15. Vanessa Kuentz-Simonet & Amaury Labenne & Tina Rambonilaza, 2017. "Using ClustOfVar to Construct Quality of Life Indicators for Vulnerability Assessment Municipality Trajectories in Southwest France from 1999 to 2009," Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, Springer, vol. 131(3), pages 973-997, April.
    16. 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.
    17. 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.
    18. José Fernando Romero Cañizares & Purificación Vicente Galindo & Yannis Phillis & Evangelos Grigoroudis, 2022. "Graphical sustainability analysis using disjoint biplots," Operational Research, Springer, vol. 22(2), pages 1575-1596, April.
    19. Carlos Martin-Barreiro & John A. Ramirez-Figueroa & Ana B. Nieto-Librero & Víctor Leiva & Ana Martin-Casado & M. Purificación Galindo-Villardón, 2021. "A New Algorithm for Computing Disjoint Orthogonal Components in the Three-Way Tucker Model," Mathematics, MDPI, vol. 9(3), pages 1-22, January.
    20. Bauer, Jan O. & Drabant, Bernhard, 2023. "Regression based thresholds in principal loading analysis," Journal of Multivariate Analysis, Elsevier, vol. 193(C).
    21. 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.
    22. 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.
    23. Donatella Vicari & Paolo Giordani, 2023. "CPclus: Candecomp/Parafac Clustering Model for Three-Way Data," Journal of Classification, Springer;The Classification Society, vol. 40(2), pages 432-465, July.
    24. 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.

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