IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v176y2020ics0047259x19302830.html
   My bibliography  Save this article

Regularized estimation of precision matrix for high-dimensional multivariate longitudinal data

Author

Listed:
  • Fang, Qian
  • Yu, Chen
  • Weiping, Zhang

Abstract

Estimating covariance matrix is one of the most important analytical tasks in analyzing multivariate longitudinal data, which provides a unique opportunity in studying the joint evolution of multiple response variables over time. This paper introduces a BiConvex Blockwise Regularization (BCBR) sparse estimator for the precision matrix (inverse of the covariance matrix) of high dimensional multivariate longitudinal responses. Using the modified Cholesky block decomposition, we impose a block banded structure on the Cholesky factor and sparsity on the innovation variance matrices via a novel convex hierarchical penalty and lasso penalty, respectively. The blockwise banding structure is a generalization of the existing banding structure for univariate longitudinal data. An efficient alternative convex optimization algorithm is developed by using ADMM algorithm. The resulting estimators are shown to converge in an optimal rate of Frobenius norm, and the exact bandwidth recovery is established for the precision matrix. Simulations and real-life data analysis show that the proposed estimator outperforms its competitors.

Suggested Citation

  • Fang, Qian & Yu, Chen & Weiping, Zhang, 2020. "Regularized estimation of precision matrix for high-dimensional multivariate longitudinal data," Journal of Multivariate Analysis, Elsevier, vol. 176(C).
    • Handle: RePEc:eee:jmvana:v:176:y:2020:i:c:s0047259x19302830
    DOI: 10.1016/j.jmva.2019.104580
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047259X19302830
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jmva.2019.104580?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. 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. P. Tseng, 2001. "Convergence of a Block Coordinate Descent Method for Nondifferentiable Minimization," Journal of Optimization Theory and Applications, Springer, vol. 109(3), pages 475-494, June.
    3. Jochen Gorski & Frank Pfeuffer & Kathrin Klamroth, 2007. "Biconvex sets and optimization with biconvex functions: a survey and extensions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 66(3), pages 373-407, December.
    4. Yin, Jianxin & Li, Hongzhe, 2012. "Model selection and estimation in the matrix normal graphical model," Journal of Multivariate Analysis, Elsevier, vol. 107(C), pages 119-140.
    5. Jing Xu & Gilbert Mackenzie, 2012. "Modelling covariance structure in bivariate marginal models for longitudinal data," Biometrika, Biometrika Trust, vol. 99(3), pages 649-662.
    6. Jianhua Z. Huang & Naiping Liu & Mohsen Pourahmadi & Linxu Liu, 2006. "Covariance matrix selection and estimation via penalised normal likelihood," Biometrika, Biometrika Trust, vol. 93(1), pages 85-98, March.
    7. Jacob Bien & Florentina Bunea & Luo Xiao, 2016. "Convex Banding of the Covariance Matrix," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(514), pages 834-845, April.
    8. Steffen Fieuws & Geert Verbeke, 2006. "Pairwise Fitting of Mixed Models for the Joint Modeling of Multivariate Longitudinal Profiles," Biometrics, The International Biometric Society, vol. 62(2), pages 424-431, June.
    9. 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.
    10. Adam J. Rothman & Elizaveta Levina & Ji Zhu, 2010. "A new approach to Cholesky-based covariance regularization in high dimensions," Biometrika, Biometrika Trust, vol. 97(3), pages 539-550.
    11. Lam, Clifford & Fan, Jianqing, 2009. "Sparsistency and rates of convergence in large covariance matrix estimation," LSE Research Online Documents on Economics 31540, London School of Economics and Political Science, LSE Library.
    12. Furrer, Reinhard & Bengtsson, Thomas, 2007. "Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants," Journal of Multivariate Analysis, Elsevier, vol. 98(2), pages 227-255, February.
    13. Teng Zhang & Hui Zou, 2014. "Sparse precision matrix estimation via lasso penalized D-trace loss," Biometrika, Biometrika Trust, vol. 101(1), pages 103-120.
    14. Chenlei Leng & Cheng Yong Tang, 2012. "Sparse Matrix Graphical Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(499), pages 1187-1200, September.
    15. Liam M. O'Brien & Garrett M. Fitzmaurice, 2004. "Analysis of longitudinal multiple‐source binary data using generalized estimating equations," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 53(1), pages 177-193, January.
    16. Fan, Jianqing & Fan, Yingying & Lv, Jinchi, 2008. "High dimensional covariance matrix estimation using a factor model," Journal of Econometrics, Elsevier, vol. 147(1), pages 186-197, November.
    17. Ming Yuan & Yi Lin, 2006. "Model selection and estimation in regression with grouped variables," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(1), pages 49-67, February.
    Full references (including those not matched with items on IDEAS)

    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. Lam, Clifford, 2020. "High-dimensional covariance matrix estimation," LSE Research Online Documents on Economics 101667, London School of Economics and Political Science, LSE Library.
    2. Wei Lan & Ronghua Luo & Chih-Ling Tsai & Hansheng Wang & Yunhong Yang, 2015. "Testing the Diagonality of a Large Covariance Matrix in a Regression Setting," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 33(1), pages 76-86, January.
    3. Gautam Sabnis & Debdeep Pati & Anirban Bhattacharya, 2019. "Compressed Covariance Estimation with Automated Dimension Learning," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 81(2), pages 466-481, December.
    4. Pan, Yuqing & Mai, Qing, 2020. "Efficient computation for differential network analysis with applications to quadratic discriminant analysis," Computational Statistics & Data Analysis, Elsevier, vol. 144(C).
    5. Benjamin Poignard & Manabu Asai, 2023. "Estimation of high-dimensional vector autoregression via sparse precision matrix," The Econometrics Journal, Royal Economic Society, vol. 26(2), pages 307-326.
    6. Hafner, Christian M. & Linton, Oliver B. & Tang, Haihan, 2020. "Estimation of a multiplicative correlation structure in the large dimensional case," Journal of Econometrics, Elsevier, vol. 217(2), pages 431-470.
    7. Kang, Xiaoning & Wang, Mingqiu, 2021. "Ensemble sparse estimation of covariance structure for exploring genetic disease data," Computational Statistics & Data Analysis, Elsevier, vol. 159(C).
    8. Chen, Xin & Yang, Dan & Xu, Yan & Xia, Yin & Wang, Dong & Shen, Haipeng, 2023. "Testing and support recovery of correlation structures for matrix-valued observations with an application to stock market data," Journal of Econometrics, Elsevier, vol. 232(2), pages 544-564.
    9. 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.
    10. Xue, Lingzhou & Zou, Hui, 2013. "Minimax optimal estimation of general bandable covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 45-51.
    11. Chen, Songxi, 2012. "Two Sample Tests for High Dimensional Covariance Matrices," MPRA Paper 46026, University Library of Munich, Germany.
    12. Abadir, Karim M. & Distaso, Walter & Žikeš, Filip, 2014. "Design-free estimation of variance matrices," Journal of Econometrics, Elsevier, vol. 181(2), pages 165-180.
    13. Lam, Clifford, 2008. "Estimation of large precision matrices through block penalization," LSE Research Online Documents on Economics 31543, London School of Economics and Political Science, LSE Library.
    14. Christian M. Hafner & Oliver Linton & Haihan Tang, 2016. "Estimation of a multiplicative covariance structure in the large dimensional case," CeMMAP working papers 52/16, Institute for Fiscal Studies.
    15. HAFNER, Christian & LINTON, Oliver B. & TANG, Haihan, 2016. "Estimation of a Multiplicative Covariance Structure in the Large Dimensional Case," LIDAM Discussion Papers CORE 2016044, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    16. Pesaran, M. Hashem & Yamagata, Takashi, 2012. "Testing CAPM with a Large Number of Assets," IZA Discussion Papers 6469, Institute of Labor Economics (IZA).
    17. Qiu, Yumou & Chen, Songxi, 2012. "Test for Bandedness of High Dimensional Covariance Matrices with Bandwidth Estimation," MPRA Paper 46242, University Library of Munich, Germany.
    18. Chen, Jia & Li, Degui & Linton, Oliver, 2019. "A new semiparametric estimation approach for large dynamic covariance matrices with multiple conditioning variables," Journal of Econometrics, Elsevier, vol. 212(1), pages 155-176.
    19. Xi Luo, 2011. "Recovering Model Structures from Large Low Rank and Sparse Covariance Matrix Estimation," Papers 1111.1133, arXiv.org, revised Mar 2013.
    20. Wang, Luheng & Chen, Zhao & Wang, Christina Dan & Li, Runze, 2020. "Ultrahigh dimensional precision matrix estimation via refitted cross validation," Journal of Econometrics, Elsevier, vol. 215(1), pages 118-130.

    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:176:y:2020:i:c:s0047259x19302830. 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.