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Some results on the Gaussian Markov Random Field construction problem based on the use of invariant subgraphs

Author

Listed:
  • Juan Baz

    (University of Oviedo)

  • Irene Díaz

    (University of Oviedo)

  • Susana Montes

    (University of Oviedo)

  • Raúl Pérez-Fernández

    (University of Oviedo)

Abstract

The study of Gaussian Markov Random Fields has attracted the attention of a large number of scientific areas due to its increasing usage in several fields of application. Here, we consider the construction of Gaussian Markov Random Fields from a graph and a positive-definite matrix, which is closely related to the problem of finding the Maximum Likelihood Estimator of the covariance matrix of the underlying distribution. In particular, it is simultaneously required that the variances and the covariances between variables associated with adjacent nodes in the graph are fixed by the positive-definite matrix and that pairs of variables associated with non-adjacent nodes in the graph are conditionally independent given all other variables. The solution to this construction problem exists and is unique up to the choice of a vector of means. In this paper, some results focusing on a certain type of subgraphs (invariant subgraphs) and a representation of the Gaussian Markov Random Field as a Multivariate Gaussian Markov Random Field are presented. These results ease the computation of the solution to the aforementioned construction problem.

Suggested Citation

  • Juan Baz & Irene Díaz & Susana Montes & Raúl Pérez-Fernández, 2022. "Some results on the Gaussian Markov Random Field construction problem based on the use of invariant subgraphs," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(3), pages 856-874, September.
    • Handle: RePEc:spr:testjl:v:31:y:2022:i:3:d:10.1007_s11749-022-00804-3
    DOI: 10.1007/s11749-022-00804-3
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    References listed on IDEAS

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    1. Ying C. MacNab, 2018. "Rejoinder on: Some recent work on multivariate Gaussian Markov random fields," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(3), pages 554-569, September.
    2. Ying C. MacNab, 2018. "Some recent work on multivariate Gaussian Markov random fields," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(3), pages 497-541, September.
    3. Nanny Wermuth & Eberhard Scheidt, 1977. "Fitting a Covariance Selection Model to a Matrix," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 26(1), pages 88-92, March.
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    Cited by:

    1. Hentschel, Manuel & Engelke, Sebastian & Segers, Johan, 2022. "Statistical Inference for Hüsler–Reiss Graphical Models Through Matrix Completions," LIDAM Discussion Papers ISBA 2022032, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Liangkun Fang & Zhangjie Wu & Yuan Tao & Jinfeng Gao, 2023. "Light Pollution Index System Model Based on Markov Random Field," Mathematics, MDPI, vol. 11(13), pages 1-18, July.

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