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A procedure of linear discrimination analysis with detected sparsity structure for high-dimensional multi-class classification

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  • Luo, Shan
  • Chen, Zehua

Abstract

In this article, we consider discrimination analyses in high-dimensional cases where the dimension of the predictor vector diverges with the sample size in a theoretical setting. The emphasis is on the case where the number of classes is bigger than two. We first deal with the asymptotic misclassification rates of linear discrimination rules under various conditions. In practical high-dimensional classification problems, it is reasonable to assume certain sparsity conditions on the class means and the common precision matrix. Our theoretical study reveals that with known sparsity structures an asymptotically optimal linear discrimination rule can be constructed. Motivated by the theoretical result, we propose a linear discrimination rule constructed based on estimated sparsity structures which is dubbed as linear discrimination with detected sparsity (LDwDS). The asymptotic optimality of LDwDS is established. Numerical studies are carried out for the comparison of LDwDS with other existing methods. The numerical studies include a comprehensive simulation study and two real data analyses. The numerical studies demonstrate that the LDwDS has an edge in terms of misclassification rate over all the other methods under consideration in the comparison.

Suggested Citation

  • Luo, Shan & Chen, Zehua, 2020. "A procedure of linear discrimination analysis with detected sparsity structure for high-dimensional multi-class classification," Journal of Multivariate Analysis, Elsevier, vol. 179(C).
    • Handle: RePEc:eee:jmvana:v:179:y:2020:i:c:s0047259x20302220
    DOI: 10.1016/j.jmva.2020.104641
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    References listed on IDEAS

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    1. Jiahua Chen & Zehua Chen, 2008. "Extended Bayesian information criteria for model selection with large model spaces," Biometrika, Biometrika Trust, vol. 95(3), pages 759-771.
    2. Rui Pan & Hansheng Wang & Runze Li, 2016. "Ultrahigh-Dimensional Multiclass Linear Discriminant Analysis by Pairwise Sure Independence Screening," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(513), pages 169-179, March.
    3. Trendafilov, Nickolay T. & Jolliffe, Ian T., 2007. "DALASS: Variable selection in discriminant analysis via the LASSO," Computational Statistics & Data Analysis, Elsevier, vol. 51(8), pages 3718-3736, May.
    4. Qing Mai & Hui Zou & Ming Yuan, 2012. "A direct approach to sparse discriminant analysis in ultra-high dimensions," Biometrika, Biometrika Trust, vol. 99(1), pages 29-42.
    5. Shan Luo & Zehua Chen, 2014. "Sequential Lasso Cum EBIC for Feature Selection With Ultra-High Dimensional Feature Space," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(507), pages 1229-1240, September.
    6. Daniela M. Witten & Robert Tibshirani, 2009. "Covariance‐regularized regression and classification for high dimensional problems," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(3), pages 615-636, June.
    7. Luo, Shan & Chen, Zehua, 2014. "Edge detection in sparse Gaussian graphical models," Computational Statistics & Data Analysis, Elsevier, vol. 70(C), pages 138-152.
    8. Peirong Xu & Ji Zhu & Lixing Zhu & Yi Li, 2015. "Covariance-enhanced discriminant analysis," Biometrika, Biometrika Trust, vol. 102(1), pages 33-45.
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    Cited by:

    1. Zengchao Xu & Shan Luo & Zehua Chen, 2023. "A Portmanteau Local Feature Discrimination Approach to the Classification with High-dimensional Matrix-variate Data," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(1), pages 441-467, February.

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