A Fuzzy Order for Graphs Based on the Continuous Entropy of Gaussian Markov Random Fields

  1. Baz, Juan 1
  2. Díaz, Irene 1
  3. Montes, Susana 1
  4. Pérez-Fernández, Raúl 1
  1. 1 Universidad de Oviedo
    info

    Universidad de Oviedo

    Oviedo, España

    ROR https://ror.org/006gksa02

Actas:
Joint Proceedings of the 19th World Congress of the International Fuzzy Systems Association (IFSA), the 12th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT), and the 11th International Summer School on Aggregation Operators (AGOP) - Atlantis Studies in Uncertainty Modelling

Editorial: Atlantis Press

ISSN: 2589-6644

Año de publicación: 2021

Congreso: 19th World Congress of the International Fuzzy Systems Association (IFSA), the 12th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT), and the 11th International Summer School on Aggregation Operators (AGOP)

Tipo: Aportación congreso

DOI: 10.2991/ASUM.K.210827.067 GOOGLE SCHOLAR lock_openAcceso abierto editor

Resumen

Gaussian Markov Random Fields overgraphs have been widely used in the context of both theoretical and applied Statistics.In this paper, we study the influence of thegraph on the continuous entropy of the distribution. In particular, since the continuousentropy is highly dependant on the correlations between adjacent nodes in the graph,we consider the particular case of GaussianMarkov Random Fields with uniform correlation, i.e., Gaussian Markov Random Fieldsin which such correlations are equal. Wedefine a partial order relation on the set ofgraphs that orders the graphs according totheir contribution to the continuous entropy.We also present a graded version of this order relation that allows to compare incomparable graphs with respect to the original order relation. Finally, an example for the illustration of the graded order relation is provided.

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Información de financiación

This research has been partially supported by the Spanish Ministry of Science and Innovation (TIN2017-87600-P and PGC2018-098623-B-I00) and FICYT (FC-GRUPIN-IDI/2018/000176).

Financiadores

Referencias bibliográficas

  • A. H. Bajgiran, M. Mardikoraem, E. S. Soofi, Maximum entropy distributions with quantile information, European Journal of Operational Research 290 (1) (2021) 196–209.
  • D. A. Brown, C. S. McMahan, S. Watson Self, Sampling strategies for fast updating of Gaussian Markov random fields, The American Statistician (2019) 1–24.
  • A. P. Dempster, Covariance selection, Biometrics 28 (1) (1972) 157–175.
  • D. Dowson, A. Wragg, Maximum-entropy distributions having prescribed first and second moments (corresp.), IEEE Transactions on Information Theory 19 (5) (1973) 689–693.
  • J. C. Fodor, M. Roubens, Fuzzy preference modelling and multicriteria decision support, Vol. 14, Springer Science & Business Media, Luxembourg, 1994.
  • R. Grone, C. R. Johnson, E. M. Sá, H. Wolkowicz, Positive definite completions of partial hermitian matrices, Linear Algebra and its Applications 58 (1984) 109–124.
  • Y. Guo, H. Xiong, N. Ruozzi, Marginal inference in continuous Markov random fields using mixtures, in: Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 33, 2019, pp. 7834–7841.
  • P. Hénon, P. Ramet, J. Roman, Pastix: a highperformance parallel direct solver for sparse symmetric positive definite systems, Parallel Computing 28 (2) (2002) 301–321.
  • S. Hong, W. M. Moon, Application of Gaussian Markov random field model to unsupervised classification in polarimetric sar image, in: Proceedings of the 2003 IEEE International Geoscience and Remote Sensing Symposium, Vol. 2, 2003, pp. 929–931.
  • R. Kindermann, J. Snell, Markov random fields and their applications, American Mathematical Society, Providence, 1980.
  • E. P. Klement, R. Mesiar, E. Pap, Triangular norms, Vol. 8, Springer Science & Business Media, Luxembourg, 2013.
  • L. Knorr-Held, H. Rue, On block updating in Markov random field models for disease mapping, Scandinavian Journal of Statistics 29 (4) (2002) 597–614.
  • W. Kocay, D. L. Kreher, Graphs, algorithms, and optimization, CRC Press, Boca Raton, 2016.
  • S. Lee, P. Sobczyk, M. Bogdan, Structure learning of Gaussian Markov random fields with false discovery rate control, Symmetry 11 (10 (2019) 1311.
  • Y. C. MacNab, On Gaussian Markov random fields and Bayesian disease mapping, Statistical Methods in Medical Research 20 (1) (2011) 49– 68.
  • K. V. Mardia, J. T. Kent, J. M. Bibby, Multivariate Analysis, Academic Press INC, San Diego, 1979.
  • J. V. Michalowicz, J. M. Nichols, F. Bucholtz, Handbook of differential entropy, Crc Press, Boca Raton, 2013.
  • R. Molina, A. K. Katsaggelos, J. Mateos, J. Abad, Compound Gauss Markov random fields for astronomical image restoration, Vistas in Astronomy 40 (4) (1996) 539–546.
  • M. Pereira, N. Desassis, Efficient simulation of Gaussian Markov random fields by chebyshev polynomial approximation, Spatial Statistics 31 (2019) 100359.
  • V. K. Rohatgi, A. M. E. Saleh, An introduction to probability and statistics, John Wiley & Sons, New York, 2015.
  • D. Rose, G. Whitten, A. Sherman, R. Tarjan, Algorithms and software for in-core factorization of sparse symmetric positive definite matrices, Computers & Structures 11 (6) (1980) 597–608.
  • H. Rue, L. Held, Gaussian Markov Random Fields: Theory and Applications, CRC press, Boca Ratón, 2005.
  • P. Salemi, First-order intrinsic Gaussian Markov random fields for discrete optimisation via simulation, Journal of Simulation 13 (4) (2019) 272–285.
  • P. Salemi, E. Song, B. L. Nelson, J. Staum, Gaussian Markov random fields for discrete optimization via simulation: Framework and algorithms, Operations Research 67 (1) (2019) 250–266.
  • C. E. Shannon, A mathematical theory of communication, The Bell System Technical Journal 27 (3) (1948) 379–423.
  • T. P. Speed, H. T. Kiiveri, Gaussian Markov distributions over finite graphs, The Annals of Statistics (1986) 138–150.
  • B. Szegedy, On sidorenko’s conjecture for determinants and gaussian markov random fields, arXiv preprint arXiv:1701.03632.
  • N. Wermuth, E. Scheidt, Algorithm as 105: Fitting a covariance selection model to a matrix, Journal of the Royal Statistical Society. Series C (Applied Statistics) 26 (1) (1977) 88–92.
  • S. Zhang, S. Mallat, Maximum entropy models from phase harmonic covariances, Applied and Computational Harmonic Analysis.