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Sparse HJ Biplot: A New Methodology via Elastic Net

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  • Mitzi Cubilla-Montilla

    (Departamento de Estadística, Facultad de Ciencias Naturales, Exactas y Tecnología, Universidad de Panamá, Panama City 0824, Panama
    Sistema Nacional de Investigación, Secretaría Nacional de Ciencia, Tecnología e Innovación (SENACYT), Panama City 0824, Panama)

  • Ana Belén Nieto-Librero

    (Department of Statistics, University of Salamanca, 37008 Salamanca, Spain
    Institute of Biomedical Research of Salamanca, 37008 Salamanca, Spain)

  • M. Purificación Galindo-Villardón

    (Department of Statistics, University of Salamanca, 37008 Salamanca, Spain
    Institute of Biomedical Research of Salamanca, 37008 Salamanca, Spain)

  • Carlos A. Torres-Cubilla

    (Department of Data Analytics, Banco General, Panama City 07096, Panama)

Abstract

The HJ biplot is a multivariate analysis technique that allows us to represent both individuals and variables in a space of reduced dimensions. To adapt this approach to massive datasets, it is necessary to implement new techniques that are capable of reducing the dimensionality of the data and improving interpretation. Because of this, we propose a modern approach to obtaining the HJ biplot called the elastic net HJ biplot, which applies the elastic net penalty to improve the interpretation of the results. It is a novel algorithm in the sense that it is the first attempt within the biplot family in which regularisation methods are used to obtain modified loadings to optimise the results. As a complement to the proposed method, and to give practical support to it, a package has been developed in the R language called SparseBiplots. This package fills a gap that exists in the context of the HJ biplot through penalized techniques since in addition to the elastic net, it also includes the ridge and lasso to obtain the HJ biplot. To complete the study, a practical comparison is made with the standard HJ biplot and the disjoint biplot, and some results common to these methods are analysed.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:11:p:1298-:d:569636
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    References listed on IDEAS

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    1. Alessio Farcomeni, 2009. "An exact approach to sparse principal component analysis," Computational Statistics, Springer, vol. 24(4), pages 583-604, December.
    2. Vichi, Maurizio & Saporta, Gilbert, 2009. "Clustering and disjoint principal component analysis," Computational Statistics & Data Analysis, Elsevier, vol. 53(8), pages 3194-3208, June.
    3. Cinthia Leonora Murillo‐Avalos & Mitzi Cubilla‐Montilla & Miguel Ángel Celestino Sánchez & Purificación Vicente‐Galindo, 2021. "What environmental social responsibility practices do large companies manage for sustainable development?," Corporate Social Responsibility and Environmental Management, John Wiley & Sons, vol. 28(1), pages 153-168, January.
    4. 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.
    5. 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.
    6. 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.
    7. Nickolay Trendafilov, 2014. "From simple structure to sparse components: a review," Computational Statistics, Springer, vol. 29(3), pages 431-454, June.
    8. Li, Baibing & Martin, Elaine B. & Morris, A. Julian, 2002. "On principal component analysis in L1," Computational Statistics & Data Analysis, Elsevier, vol. 40(3), pages 471-474, September.
    9. Ledyard Tucker, 1966. "Some mathematical notes on three-mode factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 31(3), pages 279-311, September.
    10. Lavit, Christine & Escoufier, Yves & Sabatier, Robert & Traissac, Pierre, 1994. "The ACT (STATIS method)," Computational Statistics & Data Analysis, Elsevier, vol. 18(1), pages 97-119, August.
    11. Qi, Xin & Luo, Ruiyan & Zhao, Hongyu, 2013. "Sparse principal component analysis by choice of norm," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 127-160.
    12. Hui Zou & Trevor Hastie, 2005. "Addendum: Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(5), pages 768-768, November.
    13. Hui Zou & Trevor Hastie, 2005. "Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(2), pages 301-320, April.
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    3. Wilson Rojas-Preciado & Mauricio Rojas-Campuzano & Purificación Galindo-Villardón & Omar Ruiz-Barzola, 2023. "Control Chart T2Qv for Statistical Control of Multivariate Processes with Qualitative Variables," Mathematics, MDPI, vol. 11(12), pages 1-32, June.

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