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Sparse estimation of a covariance matrix


  • Jacob Bien
  • Robert J. Tibshirani


We suggest a method for estimating a covariance matrix on the basis of a sample of vectors drawn from a multivariate normal distribution. In particular, we penalize the likelihood with a lasso penalty on the entries of the covariance matrix. This penalty plays two important roles: it reduces the effective number of parameters, which is important even when the dimension of the vectors is smaller than the sample size since the number of parameters grows quadratically in the number of variables, and it produces an estimate which is sparse. In contrast to sparse inverse covariance estimation, our method's close relative, the sparsity attained here is in the covariance matrix itself rather than in the inverse matrix. Zeros in the covariance matrix correspond to marginal independencies; thus, our method performs model selection while providing a positive definite estimate of the covariance. The proposed penalized maximum likelihood problem is not convex, so we use a majorize-minimize approach in which we iteratively solve convex approximations to the original nonconvex problem. We discuss tuning parameter selection and demonstrate on a flow-cytometry dataset how our method produces an interpretable graphical display of the relationship between variables. We perform simulations that suggest that simple elementwise thresholding of the empirical covariance matrix is competitive with our method for identifying the sparsity structure. Additionally, we show how our method can be used to solve a previously studied special case in which a desired sparsity pattern is prespecified. Copyright 2011, Oxford University Press.

Suggested Citation

  • Jacob Bien & Robert J. Tibshirani, 2011. "Sparse estimation of a covariance matrix," Biometrika, Biometrika Trust, vol. 98(4), pages 807-820.
  • Handle: RePEc:oup:biomet:v:98:y:2011:i:4:p:807-820

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    References listed on IDEAS

    1. Joseph P. Romano & Michael Wolf, 2005. "Exact and Approximate Stepdown Methods for Multiple Hypothesis Testing," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 94-108, March.
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    Cited by:

    1. Natalia Bailey & M. Hashem Pesaran & L. Vanessa Smith, 2014. "A Multiple Testing Approach to the Regularisation of Large Sample Correlation Matrices," CESifo Working Paper Series 4834, CESifo Group Munich.
    2. Bai, Jushan & Liao, Yuan, 2012. "Efficient Estimation of Approximate Factor Models," MPRA Paper 41558, University Library of Munich, Germany.
    3. Cui, Ying & Leng, Chenlei & Sun, Defeng, 2016. "Sparse estimation of high-dimensional correlation matrices," Computational Statistics & Data Analysis, Elsevier, vol. 93(C), pages 390-403.
    4. Ollier, Edouard & Samson, Adeline & Delavenne, Xavier & Viallon, Vivian, 2016. "A SAEM algorithm for fused lasso penalized NonLinear Mixed Effect Models: Application to group comparison in pharmacokinetics," Computational Statistics & Data Analysis, Elsevier, vol. 95(C), pages 207-221.
    5. Kenneth Lange & Eric C. Chi & Hua Zhou, 2014. "A Brief Survey of Modern Optimization for Statisticians," International Statistical Review, International Statistical Institute, vol. 82(1), pages 46-70, April.
    6. repec:spr:compst:v:32:y:2017:i:2:d:10.1007_s00180-016-0676-0 is not listed on IDEAS
    7. Paola Stolfi & Mauro Bernardi & Lea Petrella, 2016. "Multivariate Method Of Simulated Quantiles," Departmental Working Papers of Economics - University 'Roma Tre' 0212, Department of Economics - University Roma Tre.
    8. Wang, Kaibo & Yeh, Arthur B. & Li, Bo, 2014. "Simultaneous monitoring of process mean vector and covariance matrix via penalized likelihood estimation," Computational Statistics & Data Analysis, Elsevier, vol. 78(C), pages 206-217.
    9. Bai, Jushan & Liao, Yuan, 2016. "Efficient estimation of approximate factor models via penalized maximum likelihood," Journal of Econometrics, Elsevier, vol. 191(1), pages 1-18.
    10. repec:bla:jorssb:v:79:y:2017:i:4:p:1269-1292 is not listed on IDEAS
    11. Ryan J. Parker & Brian J. Reich & Jo Eidsvik, 2016. "A Fused Lasso Approach to Nonstationary Spatial Covariance Estimation," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 21(3), pages 569-587, September.

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