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

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  • 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
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    References listed on IDEAS

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    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.
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    Citations

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    Cited by:

    1. 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.
    2. Avagyan, Vahe & Alonso, Andrés M. & 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. Avagyan, Vahe & Alonso, Andrés M. & 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.

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