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Simple principal components

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  • S. K. Vines

Abstract

We introduce an algorithm for producing simple approximate principal components directly from a variance–covariance matrix. At the heart of the algorithm is a series of ‘simplicity preserving’ linear transformations. Each transformation seeks a direction within a two‐dimensional subspace that has maximum variance. However, the choice of directions is limited so that the direction can be represented by a vector of integers whenever the subspace can also be represented by vector if integers. The resulting approximate components can therefore always be represented by integers. Furthermore the elements of these integer vectors are often small, particularly for the first few components. We demonstrate the performance of this algorithm on two data sets and show that good approximations to the principal components that are also clearly simple and interpretable can result.

Suggested Citation

  • S. K. Vines, 2000. "Simple principal components," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 49(4), pages 441-451.
    • Handle: RePEc:bla:jorssc:v:49:y:2000:i:4:p:441-451
    DOI: 10.1111/1467-9876.00204
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    Cited by:

    1. Hugh Chipman & Hong Gu, 2005. "Interpretable dimension reduction," Journal of Applied Statistics, Taylor & Francis Journals, vol. 32(9), pages 969-987.
    2. Jolliffe, Ian, 2022. "A 50-year personal journey through time with principal component analysis," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    3. Kim, Hyun Hak & Swanson, Norman R., 2018. "Mining big data using parsimonious factor, machine learning, variable selection and shrinkage methods," International Journal of Forecasting, Elsevier, vol. 34(2), pages 339-354.
    4. 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.
    5. Hyun Hak Kim & Norman Swanson, 2013. "Mining Big Data Using Parsimonious Factor and Shrinkage Methods," Departmental Working Papers 201316, Rutgers University, Department of Economics.
    6. 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.
    7. Norman R. Swanson, 2016. "Comment," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 34(3), pages 348-353, July.
    8. Trendafilov, Nickolay T. & Vines, Karen, 2009. "Simple and interpretable discrimination," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 979-989, February.
    9. 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.
    10. 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.
    11. T. F. Cox & D. S. Arnold, 2018. "Simple components," Journal of Applied Statistics, Taylor & Francis Journals, vol. 45(1), pages 83-99, January.
    12. Nickolay Trendafilov, 2014. "From simple structure to sparse components: a review," Computational Statistics, Springer, vol. 29(3), pages 431-454, June.
    13. E. Raffinetti & I. Romeo, 2015. "Dealing with the biased effects issue when handling huge datasets: the case of INVALSI data," Journal of Applied Statistics, Taylor & Francis Journals, vol. 42(12), pages 2554-2570, December.
    14. Antonello D’Ambra & Pietro Amenta, 2023. "An extension of correspondence analysis based on the multiple Taguchi’s index to evaluate the relationships between three categorical variables graphically: an application to the Italian football cham," Annals of Operations Research, Springer, vol. 325(1), pages 219-244, June.
    15. Choulakian, V. & Allard, J. & Almhana, J., 2006. "Robust centroid method," Computational Statistics & Data Analysis, Elsevier, vol. 51(2), pages 737-746, November.
    16. Sabatier, Robert & Reynès, Christelle, 2008. "Extensions of simple component analysis and simple linear discriminant analysis using genetic algorithms," Computational Statistics & Data Analysis, Elsevier, vol. 52(10), pages 4779-4789, June.
    17. Luca Scrucca, 2006. "Subset selection in dimension reduction methods," Quaderni del Dipartimento di Economia, Finanza e Statistica 23/2006, Università di Perugia, Dipartimento Economia.

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