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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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    8. 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.
    9. Soloveychik, I. & Trushin, D., 2016. "Gaussian and robust Kronecker product covariance estimation: Existence and uniqueness," Journal of Multivariate Analysis, Elsevier, vol. 149(C), pages 92-113.
    10. Hafner, C. M. & Linton, O., 2016. "Estimation of a Multiplicative Covariance Structure in the Large Dimensional Case," Cambridge Working Papers in Economics 1664, Faculty of Economics, University of Cambridge.
    11. 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).
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