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Fitting Penalized Estimator for Sparse Covariance Matrix with Left-Censored Data by the EM Algorithm

Author

Listed:
  • Shanyi Lin

    (School of Mathematics and Statistics, Changchun University of Technology, Changchun 130012, China)

  • Qian-Zhen Zheng

    (College of Education, Zhejiang Normal University, Jinhua 321004, China)

  • Laixu Shang

    (College of Education, Zhejiang Normal University, Jinhua 321004, China)

  • Ping-Feng Xu

    (Academy for Advanced Interdisciplinary Studies & Key Laboratory of Applied Statistics of MOE, Northeast Normal University, Changchun 130024, China
    Shanghai Zhangjiang Institute of Mathematics, Shanghai 201203, China)

  • Man-Lai Tang

    (Department of Physics, Astronomy and Mathematics, School of Physics, Engineering & Computer Science, University of Hertfordshire, Hertfordshire AL10 9AB, UK)

Abstract

Estimating the sparse covariance matrix can effectively identify important features and patterns, and traditional estimation methods require complete data vectors on all subjects. When data are left-censored due to detection limits, common strategies such as excluding censored individuals or replacing censored values with suitable constants may result in large biases. In this paper, we propose two penalized log-likelihood estimators, incorporating the L 1 penalty and SCAD penalty, for estimating the sparse covariance matrix of a multivariate normal distribution in the presence of left-censored data. However, the fitting of these penalized estimators poses challenges due to the observed log-likelihood involving high-dimensional integration over the censored variables. To address this issue, we treat censored data as a special case of incomplete data and employ the Expectation Maximization algorithm combined with the coordinate descent algorithm to efficiently fit the two penalized estimators. Through simulation studies, we demonstrate that both penalized estimators achieve greater estimation accuracy compared to methods that replace censored values with constants. Moreover, the SCAD penalized estimator generally outperforms the L 1 penalized estimator. Our method is used to analyze the proteomic datasets.

Suggested Citation

  • Shanyi Lin & Qian-Zhen Zheng & Laixu Shang & Ping-Feng Xu & Man-Lai Tang, 2025. "Fitting Penalized Estimator for Sparse Covariance Matrix with Left-Censored Data by the EM Algorithm," Mathematics, MDPI, vol. 13(3), pages 1-17, January.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:3:p:423-:d:1578467
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    References listed on IDEAS

    as
    1. Wang, Xin & Kong, Lingchen & Wang, Liqun, 2024. "Estimation of sparse covariance matrix via non-convex regularization," Journal of Multivariate Analysis, Elsevier, vol. 202(C).
    2. Adam J. Rothman, 2012. "Positive definite estimators of large covariance matrices," Biometrika, Biometrika Trust, vol. 99(3), pages 733-740.
    3. Kasper Johansson & Mehmet Giray Ogut & Markus Pelger & Thomas Schmelzer & Stephen Boyd, 2023. "A Simple Method for Predicting Covariance Matrices of Financial Returns," Papers 2305.19484, arXiv.org, revised Nov 2023.
    4. Pesonen, Maiju & Pesonen, Henri & Nevalainen, Jaakko, 2015. "Covariance matrix estimation for left-censored data," Computational Statistics & Data Analysis, Elsevier, vol. 92(C), pages 13-25.
    5. Sung, Bongjung & Lee, Jaeyong, 2023. "Covariance structure estimation with Laplace approximation," Journal of Multivariate Analysis, Elsevier, vol. 198(C).
    6. Jacob Bien & Robert J. Tibshirani, 2011. "Sparse estimation of a covariance matrix," Biometrika, Biometrika Trust, vol. 98(4), pages 807-820.
    7. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    8. Lingzhou Xue & Shiqian Ma & Hui Zou, 2012. "Positive-Definite ℓ 1 -Penalized Estimation of Large Covariance Matrices," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(500), pages 1480-1491, December.
    9. Horrace, William C., 2005. "Some results on the multivariate truncated normal distribution," Journal of Multivariate Analysis, Elsevier, vol. 94(1), pages 209-221, May.
    10. Cai, Tony & Liu, Weidong, 2011. "Adaptive Thresholding for Sparse Covariance Matrix Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 106(494), pages 672-684.
    11. Li, Degui, 2024. "Estimation of Large Dynamic Covariance Matrices: A Selective Review," Econometrics and Statistics, Elsevier, vol. 29(C), pages 16-30.
    12. Runmin Wei & Jingye Wang & Erik Jia & Tianlu Chen & Yan Ni & Wei Jia, 2018. "GSimp: A Gibbs sampler based left-censored missing value imputation approach for metabolomics studies," PLOS Computational Biology, Public Library of Science, vol. 14(1), pages 1-14, January.
    13. 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.
    14. Jason Xu & Kenneth Lange, 2022. "A proximal distance algorithm for likelihood-based sparse covariance estimation [Estimating large correlation matrices for international migration]," Biometrika, Biometrika Trust, vol. 109(4), pages 1047-1066.
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