IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v116y2013icp365-381.html
   My bibliography  Save this article

Adjusting for high-dimensional covariates in sparse precision matrix estimation by ℓ1-penalization

Author

Listed:
  • Yin, Jianxin
  • Li, Hongzhe

Abstract

Motivated by the analysis of genetical genomic data, we consider the problem of estimating high-dimensional sparse precision matrix adjusting for possibly a large number of covariates, where the covariates can affect the mean value of the random vector. We develop a two-stage estimation procedure to first identify the relevant covariates that affect the means by a joint ℓ1 penalization. The estimated regression coefficients are then used to estimate the mean values in a multivariate sub-Gaussian model in order to estimate the sparse precision matrix through a ℓ1-penalized log-determinant Bregman divergence. Under the multivariate normal assumption, the precision matrix has the interpretation of a conditional Gaussian graphical model. We show that under some regularity conditions, the estimates of the regression coefficients are consistent in element-wise ℓ∞ norm, Frobenius norm and also spectral norm even when p≫n and q≫n. We also show that with probability converging to one, the estimate of the precision matrix correctly specifies the zero pattern of the true precision matrix. We illustrate our theoretical results via simulations and demonstrate that the method can lead to improved estimate of the precision matrix. We apply the method to an analysis of a yeast genetical genomic data.

Suggested Citation

  • Yin, Jianxin & Li, Hongzhe, 2013. "Adjusting for high-dimensional covariates in sparse precision matrix estimation by ℓ1-penalization," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 365-381.
    • Handle: RePEc:eee:jmvana:v:116:y:2013:i:c:p:365-381
    DOI: 10.1016/j.jmva.2013.01.005
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047259X13000067
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jmva.2013.01.005?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Peng, Jie & Wang, Pei & Zhou, Nengfeng & Zhu, Ji, 2009. "Partial Correlation Estimation by Joint Sparse Regression Models," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 735-746.
    2. Cai, Tony & Liu, Weidong & Luo, Xi, 2011. "A Constrained â„“1 Minimization Approach to Sparse Precision Matrix Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 106(494), pages 594-607.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Ding, Wenliang & Shu, Lianjie & Gu, Xinhua, 2023. "A robust Glasso approach to portfolio selection in high dimensions," Journal of Empirical Finance, Elsevier, vol. 70(C), pages 22-37.
    2. Avagyan, Vahe & Nogales, Francisco J., 2015. "D-trace Precision Matrix Estimation Using Adaptive Lasso Penalties," DES - Working Papers. Statistics and Econometrics. WS 21775, Universidad Carlos III de Madrid. Departamento de Estadística.
    3. Vahe Avagyan, 2022. "Precision matrix estimation using penalized Generalized Sylvester matrix equation," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(4), pages 950-967, December.
    4. Avagyan, Vahe & Nogales, Francisco J., 2014. "Improving the graphical lasso estimation for the precision matrix through roots ot the sample convariance matrix," DES - Working Papers. Statistics and Econometrics. WS ws141208, Universidad Carlos III de Madrid. Departamento de Estadística.
    5. Ting Wang & Zhao Ren & Ying Ding & Zhou Fang & Zhe Sun & Matthew L MacDonald & Robert A Sweet & Jieru Wang & Wei Chen, 2016. "FastGGM: An Efficient Algorithm for the Inference of Gaussian Graphical Model in Biological Networks," PLOS Computational Biology, Public Library of Science, vol. 12(2), pages 1-16, February.
    6. Fan, Xinyan & Zhang, Qingzhao & Ma, Shuangge & Fang, Kuangnan, 2021. "Conditional score matching for high-dimensional partial graphical models," Computational Statistics & Data Analysis, Elsevier, vol. 153(C).
    7. Avagyan, Vahe, 2016. "D-Trace precision matrix estimator with eigenvalue control," DES - Working Papers. Statistics and Econometrics. WS 23410, Universidad Carlos III de Madrid. Departamento de Estadística.
    8. Vahe Avagyan & Andrés M. Alonso & Francisco J. Nogales, 2018. "D-trace estimation of a precision matrix using adaptive Lasso penalties," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 12(2), pages 425-447, June.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Luo, Shan & Chen, Zehua, 2014. "Edge detection in sparse Gaussian graphical models," Computational Statistics & Data Analysis, Elsevier, vol. 70(C), pages 138-152.
    2. Guanghui Cheng & Zhengjun Zhang & Baoxue Zhang, 2017. "Test for bandedness of high-dimensional precision matrices," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 29(4), pages 884-902, October.
    3. Huihang Liu & Xinyu Zhang, 2023. "Frequentist model averaging for undirected Gaussian graphical models," Biometrics, The International Biometric Society, vol. 79(3), pages 2050-2062, September.
    4. Tan, Kean Ming & Witten, Daniela & Shojaie, Ali, 2015. "The cluster graphical lasso for improved estimation of Gaussian graphical models," Computational Statistics & Data Analysis, Elsevier, vol. 85(C), pages 23-36.
    5. Jie Cheng & Elizaveta Levina & Pei Wang & Ji Zhu, 2014. "A sparse ising model with covariates," Biometrics, The International Biometric Society, vol. 70(4), pages 943-953, December.
    6. Hirose, Kei & Fujisawa, Hironori & Sese, Jun, 2017. "Robust sparse Gaussian graphical modeling," Journal of Multivariate Analysis, Elsevier, vol. 161(C), pages 172-190.
    7. Bailey, Natalia & Pesaran, M. Hashem & Smith, L. Vanessa, 2019. "A multiple testing approach to the regularisation of large sample correlation matrices," Journal of Econometrics, Elsevier, vol. 208(2), pages 507-534.
    8. Ines Wilms & Jacob Bien, 2021. "Tree-based Node Aggregation in Sparse Graphical Models," Papers 2101.12503, arXiv.org.
    9. Jianqing Fan & Han Liu & Yang Ning & Hui Zou, 2017. "High dimensional semiparametric latent graphical model for mixed data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(2), pages 405-421, March.
    10. repec:cte:wsrepe:24534 is not listed on IDEAS
    11. Banerjee, Sayantan & Akbani, Rehan & Baladandayuthapani, Veerabhadran, 2019. "Spectral clustering via sparse graph structure learning with application to proteomic signaling networks in cancer," Computational Statistics & Data Analysis, Elsevier, vol. 132(C), pages 46-69.
    12. Zamar, Rubén, 2015. "Ranking Edges and Model Selection in High-Dimensional Graphs," DES - Working Papers. Statistics and Econometrics. WS ws1511, Universidad Carlos III de Madrid. Departamento de Estadística.
    13. Xiao Guo & Hai Zhang, 2020. "Sparse directed acyclic graphs incorporating the covariates," Statistical Papers, Springer, vol. 61(5), pages 2119-2148, October.
    14. Chang, Jinyuan & Qiu, Yumou & Yao, Qiwei & Zou, Tao, 2018. "Confidence regions for entries of a large precision matrix," LSE Research Online Documents on Economics 87513, London School of Economics and Political Science, LSE Library.
    15. Wang, Ke & Franks, Alexander & Oh, Sang-Yun, 2023. "Learning Gaussian graphical models with latent confounders," Journal of Multivariate Analysis, Elsevier, vol. 198(C).
    16. 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.
    17. Hokeun Sun & Hongzhe Li, 2012. "Robust Gaussian Graphical Modeling Via l 1 Penalization," Biometrics, The International Biometric Society, vol. 68(4), pages 1197-1206, December.
    18. Jiadong Ji & Yong He & Lei Liu & Lei Xie, 2021. "Brain connectivity alteration detection via matrix‐variate differential network model," Biometrics, The International Biometric Society, vol. 77(4), pages 1409-1421, December.
    19. Byrd, Michael & Nghiem, Linh H. & McGee, Monnie, 2021. "Bayesian regularization of Gaussian graphical models with measurement error," Computational Statistics & Data Analysis, Elsevier, vol. 156(C).
    20. Jianqing Fan & Xu Han, 2017. "Estimation of the false discovery proportion with unknown dependence," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(4), pages 1143-1164, September.
    21. Seunghwan Lee & Sang Cheol Kim & Donghyeon Yu, 2023. "An efficient GPU-parallel coordinate descent algorithm for sparse precision matrix estimation via scaled lasso," Computational Statistics, Springer, vol. 38(1), pages 217-242, March.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:jmvana:v:116:y:2013:i:c:p:365-381. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.