Fitting Penalized Estimator for Sparse Covariance Matrix with Left-Censored Data by the EM Algorithm
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 replaci...
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2025-01-01
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| author | Shanyi Lin Qian-Zhen Zheng Laixu Shang Ping-Feng Xu Man-Lai Tang |
| author_facet | Shanyi Lin Qian-Zhen Zheng Laixu Shang Ping-Feng Xu Man-Lai Tang |
| author_sort | Shanyi Lin |
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| description | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mn>1</mn></msub></semantics></math></inline-formula> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mn>1</mn></msub></semantics></math></inline-formula> penalized estimator. Our method is used to analyze the proteomic datasets. |
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| spelling | doaj-art-5d68acad9edf40dab0a5cbc1a3b1bf6c2025-08-20T02:12:30ZengMDPI AGMathematics2227-73902025-01-0113342310.3390/math13030423Fitting Penalized Estimator for Sparse Covariance Matrix with Left-Censored Data by the EM AlgorithmShanyi Lin0Qian-Zhen Zheng1Laixu Shang2Ping-Feng Xu3Man-Lai Tang4School of Mathematics and Statistics, Changchun University of Technology, Changchun 130012, ChinaCollege of Education, Zhejiang Normal University, Jinhua 321004, ChinaCollege of Education, Zhejiang Normal University, Jinhua 321004, ChinaAcademy for Advanced Interdisciplinary Studies & Key Laboratory of Applied Statistics of MOE, Northeast Normal University, Changchun 130024, ChinaDepartment of Physics, Astronomy and Mathematics, School of Physics, Engineering & Computer Science, University of Hertfordshire, Hertfordshire AL10 9AB, UKEstimating 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mn>1</mn></msub></semantics></math></inline-formula> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mn>1</mn></msub></semantics></math></inline-formula> penalized estimator. Our method is used to analyze the proteomic datasets.https://www.mdpi.com/2227-7390/13/3/423sparse covariance matrixExpectation Maximization algorithmpenalized estimatorleft-censored data |
| spellingShingle | Shanyi Lin Qian-Zhen Zheng Laixu Shang Ping-Feng Xu Man-Lai Tang Fitting Penalized Estimator for Sparse Covariance Matrix with Left-Censored Data by the EM Algorithm Mathematics sparse covariance matrix Expectation Maximization algorithm penalized estimator left-censored data |
| title | Fitting Penalized Estimator for Sparse Covariance Matrix with Left-Censored Data by the EM Algorithm |
| title_full | Fitting Penalized Estimator for Sparse Covariance Matrix with Left-Censored Data by the EM Algorithm |
| title_fullStr | Fitting Penalized Estimator for Sparse Covariance Matrix with Left-Censored Data by the EM Algorithm |
| title_full_unstemmed | Fitting Penalized Estimator for Sparse Covariance Matrix with Left-Censored Data by the EM Algorithm |
| title_short | Fitting Penalized Estimator for Sparse Covariance Matrix with Left-Censored Data by the EM Algorithm |
| title_sort | fitting penalized estimator for sparse covariance matrix with left censored data by the em algorithm |
| topic | sparse covariance matrix Expectation Maximization algorithm penalized estimator left-censored data |
| url | https://www.mdpi.com/2227-7390/13/3/423 |
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