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Graphical Gaussian models with edge and vertex symmetries

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  • Søren Højsgaard
  • Steffen L. Lauritzen

Abstract

Summary. We introduce new types of graphical Gaussian models by placing symmetry restrictions on the concentration or correlation matrix. The models can be represented by coloured graphs, where parameters that are associated with edges or vertices of the same colour are restricted to being identical. We study the properties of such models and derive the necessary algorithms for calculating maximum likelihood estimates. We identify conditions for restrictions on the concentration and correlation matrices being equivalent. This is for example the case when symmetries are generated by permutation of variable labels. For such models a particularly simple maximization of the likelihood function is available.

Suggested Citation

  • Søren Højsgaard & Steffen L. Lauritzen, 2008. "Graphical Gaussian models with edge and vertex symmetries," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(5), pages 1005-1027, November.
    • Handle: RePEc:bla:jorssb:v:70:y:2008:i:5:p:1005-1027
    DOI: 10.1111/j.1467-9868.2008.00666.x
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    References listed on IDEAS

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    1. Mathias Drton, 2004. "Multimodality of the likelihood in the bivariate seemingly unrelated regressions model," Biometrika, Biometrika Trust, vol. 91(2), pages 383-392, June.
    2. Mathias Drton & Michael Eichler, 2006. "Maximum Likelihood Estimation in Gaussian Chain Graph Models under the Alternative Markov Property," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(2), pages 247-257, June.
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    Cited by:

    1. Saverio Ranciati & Alberto Roverato & Alessandra Luati, 2021. "Fused graphical lasso for brain networks with symmetries," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 70(5), pages 1299-1322, November.
    2. Vinciotti Veronica & Augugliaro Luigi & Abbruzzo Antonino & Wit Ernst C., 2016. "Model selection for factorial Gaussian graphical models with an application to dynamic regulatory networks," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 15(3), pages 193-212, June.
    3. Bernd Sturmfels & Caroline Uhler, 2010. "Multivariate Gaussians, semidefinite matrix completion, and convex algebraic geometry," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(4), pages 603-638, August.
    4. Alessandro Casa & Andrea Cappozzo & Michael Fop, 2022. "Group-Wise Shrinkage Estimation in Penalized Model-Based Clustering," Journal of Classification, Springer;The Classification Society, vol. 39(3), pages 648-674, November.
    5. Kiiveri, Harri & de Hoog, Frank, 2012. "Fitting very large sparse Gaussian graphical models," Computational Statistics & Data Analysis, Elsevier, vol. 56(9), pages 2626-2636.
    6. Abbruzzo, Antonino & Fasone, Vincenzo & Scuderi, Raffaele, 2016. "Operational and financial performance of Italian airport companies: A dynamic graphical model," Transport Policy, Elsevier, vol. 52(C), pages 231-237.

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