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Classification in High Dimension Using the Ledoit–Wolf Shrinkage Method

Author

Listed:
  • Rasoul Lotfi

    (Department of Statistics, Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood 3619995161, Iran)

  • Davood Shahsavani

    (Department of Statistics, Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood 3619995161, Iran)

  • Mohammad Arashi

    (Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad 9177948974, Iran
    Department of Statistics, Faculty of Natural and Agricultural Sciences, University of Pretoria, Pretoria 0002, South Africa)

Abstract

Classification using linear discriminant analysis (LDA) is challenging when the number of variables is large relative to the number of observations. Algorithms such as LDA require the computation of the feature vector’s precision matrices. In a high-dimension setting, due to the singularity of the covariance matrix, it is not possible to estimate the maximum likelihood estimator of the precision matrix. In this paper, we employ the Stein-type shrinkage estimation of Ledoit and Wolf for high-dimensional data classification. The proposed approach’s efficiency is numerically compared to existing methods, including LDA, cross-validation, gLasso, and SVM. We use the misclassification error criterion for comparison.

Suggested Citation

  • Rasoul Lotfi & Davood Shahsavani & Mohammad Arashi, 2022. "Classification in High Dimension Using the Ledoit–Wolf Shrinkage Method," Mathematics, MDPI, vol. 10(21), pages 1-13, November.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:21:p:4069-:d:959980
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    References listed on IDEAS

    as
    1. Ledoit, Olivier & Wolf, Michael, 2004. "A well-conditioned estimator for large-dimensional covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 365-411, February.
    2. Le, Khuyen T. & Chaux, Caroline & Richard, Frédéric J.P. & Guedj, Eric, 2020. "An adapted linear discriminant analysis with variable selection for the classification in high-dimension, and an application to medical data," Computational Statistics & Data Analysis, Elsevier, vol. 152(C).
    3. Norm A. Campbell, 1980. "Shrunken Estimators in Discriminant and Canonical Variate Analysis," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 29(1), pages 5-14, March.
    4. Jacob Bien & Robert J. Tibshirani, 2011. "Sparse estimation of a covariance matrix," Biometrika, Biometrika Trust, vol. 98(4), pages 807-820.
    5. T. Tony Cai & Linjun Zhang, 2019. "High dimensional linear discriminant analysis: optimality, adaptive algorithm and missing data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 81(4), pages 675-705, September.
    6. Jianqing Fan & Yuan Liao & Han Liu, 2016. "An overview of the estimation of large covariance and precision matrices," Econometrics Journal, Royal Economic Society, vol. 19(1), pages 1-32, February.
    7. Choi, Young-Geun & Lim, Johan & Roy, Anindya & Park, Junyong, 2019. "Fixed support positive-definite modification of covariance matrix estimators via linear shrinkage," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 234-249.
    8. Rothman, Adam J. & Levina, Elizaveta & Zhu, Ji, 2009. "Generalized Thresholding of Large Covariance Matrices," Journal of the American Statistical Association, American Statistical Association, vol. 104(485), pages 177-186.
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