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Selecting the number of components in principal component analysis using cross-validation approximations

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  • Josse, Julie
  • Husson, François
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    Abstract

    Cross-validation is a tried and tested approach to select the number of components in principal component analysis (PCA), however, its main drawback is its computational cost. In a regression (or in a non parametric regression) setting, criteria such as the general cross-validation one (GCV) provide convenient approximations to leave-one-out cross-validation. They are based on the relation between the prediction error and the residual sum of squares weighted by elements of a projection matrix (or a smoothing matrix). Such a relation is then established in PCA using an original presentation of PCA with a unique projection matrix. It enables the definition of two cross-validation approximation criteria: the smoothing approximation of the cross-validation criterion (SACV) and the GCV criterion. The method is assessed with simulations and gives promising results.

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    File URL: http://www.sciencedirect.com/science/article/pii/S0167947311004099
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    Bibliographic Info

    Article provided by Elsevier in its journal Computational Statistics & Data Analysis.

    Volume (Year): 56 (2012)
    Issue (Month): 6 ()
    Pages: 1869-1879

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    Handle: RePEc:eee:csdana:v:56:y:2012:i:6:p:1869-1879

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    Web page: http://www.elsevier.com/locate/csda

    Related research

    Keywords: PCA; Number of components; Cross-validation; Smoothing matrix; Generalized cross-validation;

    References

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    1. Chunlei Ke & Yuedong Wang, 2004. "Smoothing Spline Nonlinear Nonparametric Regression Models," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 1166-1175, December.
    2. Julie Josse & Jérôme Pagès & François Husson, 2011. "Multiple imputation in principal component analysis," Advances in Data Analysis and Classification, Springer, vol. 5(3), pages 231-246, October.
    3. Peres-Neto, Pedro R. & Jackson, Donald A. & Somers, Keith M., 2005. "How many principal components? stopping rules for determining the number of non-trivial axes revisited," Computational Statistics & Data Analysis, Elsevier, vol. 49(4), pages 974-997, June.
    4. Ferre, Louis, 1995. "Selection of components in principal component analysis: A comparison of methods," Computational Statistics & Data Analysis, Elsevier, vol. 19(6), pages 669-682, June.
    5. Henk Kiers, 1997. "Weighted least squares fitting using ordinary least squares algorithms," Psychometrika, Springer, vol. 62(2), pages 251-266, June.
    6. 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.
    7. Dray, Stephane, 2008. "On the number of principal components: A test of dimensionality based on measurements of similarity between matrices," Computational Statistics & Data Analysis, Elsevier, vol. 52(4), pages 2228-2237, January.
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    Cited by:
    1. Bada, Oualid & Kneip, Alois, 2014. "Parameter cascading for panel models with unknown number of unobserved factors: An application to the credit spread puzzle," Computational Statistics & Data Analysis, Elsevier, vol. 76(C), pages 95-115.

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