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A pure L1-norm principal component analysis

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  • Brooks, J.P.
  • Dulá, J.H.
  • Boone, E.L.

Abstract

The L1 norm has been applied in numerous variations of principal component analysis (PCA). An L1-norm PCA is an attractive alternative to traditional L2-based PCA because it can impart robustness in the presence of outliers and is indicated for models where standard Gaussian assumptions about the noise may not apply. Of all the previously-proposed PCA schemes that recast PCA as an optimization problem involving the L1 norm, none provide globally optimal solutions in polynomial time. This paper proposes an L1-norm PCA procedure based on the efficient calculation of the optimal solution of the L1-norm best-fit hyperplane problem. We present a procedure called L1-PCA∗ based on the application of this idea that fits data to subspaces of successively smaller dimension. The procedure is implemented and tested on a diverse problem suite. Our tests show that L1-PCA∗ is the indicated procedure in the presence of unbalanced outlier contamination.

Suggested Citation

  • Brooks, J.P. & Dulá, J.H. & Boone, E.L., 2013. "A pure L1-norm principal component analysis," Computational Statistics & Data Analysis, Elsevier, vol. 61(C), pages 83-98.
    • Handle: RePEc:eee:csdana:v:61:y:2013:i:c:p:83-98
    DOI: 10.1016/j.csda.2012.11.007
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    2. Galpin, Jacqueline S. & Hawkins, Douglas M., 1987. "Methods of L1 estimation of a covariance matrix," Computational Statistics & Data Analysis, Elsevier, vol. 5(4), pages 305-319, September.
    3. 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.
    4. Choulakian, V., 2006. "L1-norm projection pursuit principal component analysis," Computational Statistics & Data Analysis, Elsevier, vol. 50(6), pages 1441-1451, March.
    5. Croux, Christophe & Ruiz-Gazen, Anne, 2005. "High breakdown estimators for principal components: the projection-pursuit approach revisited," Journal of Multivariate Analysis, Elsevier, vol. 95(1), pages 206-226, July.
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    Cited by:

    1. Charpentier, Arthur & Mussard, Stéphane & Ouraga, Téa, 2021. "Principal component analysis: A generalized Gini approach," European Journal of Operational Research, Elsevier, vol. 294(1), pages 236-249.
    2. Kyungmin Kim, 2016. "Measuring the Informativeness of Market Statistics," Finance and Economics Discussion Series 2016-076, Board of Governors of the Federal Reserve System (U.S.).
    3. Young Woong Park, 2021. "Optimization for L 1 -Norm Error Fitting via Data Aggregation," INFORMS Journal on Computing, INFORMS, vol. 33(1), pages 120-142, January.

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    More about this item

    Keywords

    Principal component analysis; Linear programming; L1 regression;
    All these keywords.

    JEL classification:

    • L1 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance

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