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Model selection and estimation in the matrix normal graphical model

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  • Yin, Jianxin
  • Li, Hongzhe

Abstract

Motivated by analysis of gene expression data measured over different tissues or over time, we consider matrix-valued random variable and matrix-normal distribution, where the precision matrices have a graphical interpretation for genes and tissues, respectively. We present a l1 penalized likelihood method and an efficient coordinate descent-based computational algorithm for model selection and estimation in such matrix normal graphical models (MNGMs). We provide theoretical results on the asymptotic distributions, the rates of convergence of the estimates and the sparsistency, allowing both the numbers of genes and tissues to diverge as the sample size goes to infinity. Simulation results demonstrate that the MNGMs can lead to a better estimate of the precision matrices and better identifications of the graph structures than the standard Gaussian graphical models. We illustrate the methods with an analysis of mouse gene expression data measured over ten different tissues.

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  • 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.
    • Handle: RePEc:eee:jmvana:v:107:y:2012:i:c:p:119-140
    DOI: 10.1016/j.jmva.2012.01.005
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    Cited by:

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    6. Yin Xia & Lexin Li, 2017. "Hypothesis testing of matrix graph model with application to brain connectivity analysis," Biometrics, The International Biometric Society, vol. 73(3), pages 780-791, September.
    7. Anestis Touloumis & Simon Tavaré & John C. Marioni, 2015. "Testing the mean matrix in high-dimensional transposable data," Biometrics, The International Biometric Society, vol. 71(1), pages 157-166, March.
    8. Yang, Guangren & Liu, Yiming & Pan, Guangming, 2019. "Weighted covariance matrix estimation," Computational Statistics & Data Analysis, Elsevier, vol. 139(C), pages 82-98.
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    10. Niu, Lu & Liu, Xiumin & Zhao, Junlong, 2020. "Robust estimator of the correlation matrix with sparse Kronecker structure for a high-dimensional matrix-variate," Journal of Multivariate Analysis, Elsevier, vol. 177(C).
    11. Wang, Dong & Liu, Xialu & Chen, Rong, 2019. "Factor models for matrix-valued high-dimensional time series," Journal of Econometrics, Elsevier, vol. 208(1), pages 231-248.
    12. Seongoh Park & Johan Lim & Xinlei Wang & Sanghan Lee, 2019. "Permutation based testing on covariance separability," Computational Statistics, Springer, vol. 34(2), pages 865-883, June.
    13. He, Shiyuan & Yin, Jianxin & Li, Hongzhe & Wang, Xing, 2014. "Graphical model selection and estimation for high dimensional tensor data," Journal of Multivariate Analysis, Elsevier, vol. 128(C), pages 165-185.
    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.

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