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From simple structure to sparse components: a review

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  • Nickolay Trendafilov

Abstract

The article begins with a review of the main approaches for interpretation the results from principal component analysis (PCA) during the last 50–60 years. The simple structure approach is compared to the modern approach of sparse PCA where interpretable solutions are directly obtained. It is shown that their goals are identical but they differ by the way they are realized. Next, the most popular and influential methods for sparse PCA are briefly reviewed. In the remaining part of the paper, a new approach to define sparse PCA is introduced. Several alternative definitions are considered and illustrated on a well-known data set. Finally, it is demonstrated, how one of these possible versions of sparse PCA can be used as a sparse alternative to the classical rotation methods. Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • Nickolay Trendafilov, 2014. "From simple structure to sparse components: a review," Computational Statistics, Springer, vol. 29(3), pages 431-454, June.
    • Handle: RePEc:spr:compst:v:29:y:2014:i:3:p:431-454
    DOI: 10.1007/s00180-013-0434-5
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    Cited by:

    1. Kohei Adachi & Nickolay T. Trendafilov, 2016. "Sparse principal component analysis subject to prespecified cardinality of loadings," Computational Statistics, Springer, vol. 31(4), pages 1403-1427, December.
    2. Nickolay Trendafilov & Martin Kleinsteuber & Hui Zou, 2014. "Sparse matrices in data analysis," Computational Statistics, Springer, vol. 29(3), pages 403-405, June.
    3. Thomas Despois & Catherine Doz, 2021. "Identifying and interpreting the factors in factor models via sparsity: Different approaches," PSE Working Papers halshs-02235543, HAL.
    4. 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.
    5. Thomas Despois & Catherine Doz, 2022. "Identifying and interpreting the factors in factor models via sparsity : Different approaches," Working Papers halshs-03626503, HAL.
    6. Thomas Despois & Catherine Doz, 2023. "Identifying and interpreting the factors in factor models via sparsity: Different approaches," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 38(4), pages 533-555, June.
    7. 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.
    8. Shaobo Jin & Irini Moustaki & Fan Yang-Wallentin, 2018. "Approximated Penalized Maximum Likelihood for Exploratory Factor Analysis: An Orthogonal Case," Psychometrika, Springer;The Psychometric Society, vol. 83(3), pages 628-649, September.
    9. Ikemoto, Hiroki & Adachi, Kohei, 2016. "Sparse Tucker2 analysis of three-way data subject to a constrained number of zero elements in a core array," Computational Statistics & Data Analysis, Elsevier, vol. 98(C), pages 1-18.
    10. Liu, Xinyi Lin & Wallin, Gabriel & Chen, Yunxiao & Moustaki, Irini, 2023. "Rotation to sparse loadings using Lp losses and related inference problems," LSE Research Online Documents on Economics 118349, London School of Economics and Political Science, LSE Library.
    11. Rosember Guerra-Urzola & Katrijn Van Deun & Juan C. Vera & Klaas Sijtsma, 2021. "A Guide for Sparse PCA: Model Comparison and Applications," Psychometrika, Springer;The Psychometric Society, vol. 86(4), pages 893-919, December.
    12. Rosember Guerra-Urzola & Niek C. Schipper & Anya Tonne & Klaas Sijtsma & Juan C. Vera & Katrijn Deun, 2023. "Sparsifying the least-squares approach to PCA: comparison of lasso and cardinality constraint," 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 269-286, March.
    13. Xinyi Liu & Gabriel Wallin & Yunxiao Chen & Irini Moustaki, 2023. "Rotation to Sparse Loadings Using $$L^p$$ L p Losses and Related Inference Problems," Psychometrika, Springer;The Psychometric Society, vol. 88(2), pages 527-553, June.
    14. Guerra Urzola, Rosember & Van Deun, Katrijn & Vera, J. C. & Sijtsma, K., 2021. "A guide for sparse PCA : Model comparison and applications," Other publications TiSEM 4d35b931-7f49-444b-b92f-a, Tilburg University, School of Economics and Management.
    15. Thomas Despois & Catherine Doz, 2022. "Identifying and interpreting the factors in factor models via sparsity : Different approaches," PSE Working Papers halshs-03626503, HAL.
    16. Thomas Despois & Catherine Doz, 2021. "Identifying and interpreting the factors in factor models via sparsity: Different approaches," Working Papers halshs-02235543, HAL.
    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. Jin, Shaobo & Moustaki, Irini & Yang-Wallentin, Fan, 2018. "Approximated penalized maximum likelihood for exploratory factor analysis: an orthogonal case," LSE Research Online Documents on Economics 88118, London School of Economics and Political Science, LSE Library.
    19. Nickolay T. Trendafilov & Sara Fontanella & Kohei Adachi, 2017. "Sparse Exploratory Factor Analysis," Psychometrika, Springer;The Psychometric Society, vol. 82(3), pages 778-794, September.
    20. 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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