Advertisement

Asymptotic Inference for Markov Random Fields on ℤd

  • J. Demongeot
Part of the Springer Series in Synergetics book series (SSSYN, volume 9)

Abstract

The purpose of this paper is to generalize results of D.K. PICKARD [1,2,3] on asymptotic inference for the Ising model. Using properties of Gibbs measures on ℤd, we give the canonical exponential structure in which we can study the estimation problem, in the case of vertical sampling. This exponential structure depends on the range of the interaction potential of the underlying Markov random field; we construct, therefore, a test to measure this range: this test is based on the markovian character of the associated Gibbs measure; next we present the properties and the asymptotic laws of the estimators and we compare horizontal to vertical sampling [4]. Certain properties depend on the criticality of the interaction potential [5,6]: consequently, we construct a test of criticality for the interaction potential. Finally, we set the problem of estimation in a random Ising model.

Keywords

Random Field Ising Model Convex Combination Accumulation Point Gibbs Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D.K. Pickard: J. Appl. Prob. 13, 486 (1976)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    D.K. Pickard: Adv. Appl. Prob. 9, 476 (1977)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    D.K. Pickard: J. Appl. Prob. 16, 12 (1979)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    J. Demongeot: Séminaire de Statistiques de Grenoble 2, 135 (1980)Google Scholar
  5. 5.
    J. Demongeot: Thesis, Grenoble (1975)Google Scholar
  6. 6.
    J. Demongeot: Commun. Math. Phys. (submitted)Google Scholar
  7. 7.
    K. Schürger, P. Tautu: Lecture Notes in Biomath. 11, 92 (1976)Google Scholar
  8. 8.
    A.M. Wartell, E.W. Montroll: Adv. Chem. Phys. 22, 129 (1972)CrossRefGoogle Scholar
  9. 9.
    P.A. Vuillermot: Phys. Letters 61A, 9 (1977)ADSGoogle Scholar
  10. 10.
    C. Preston: Lecture Notes in Math. 534 (1976)Google Scholar
  11. 11.
    M.B. Averintsev: Theory of Prob. and its Appl. 17, 20 (1972)CrossRefGoogle Scholar
  12. 12.
    J. Neveu: Martingales à temps discret, (Masson, Paris 1972)Google Scholar
  13. 13.
    C. Preston: Gibbs states on countable sets (Cambridge Univ. Press 1974)CrossRefMATHGoogle Scholar
  14. 14.
    O.E. Lanford: Lecture Notes in Physics 20, 1 (1973)CrossRefADSGoogle Scholar
  15. 15.
    F. Ledrappier: Commun. Math. Phys. 33, 119 (1973)CrossRefMATHADSMathSciNetGoogle Scholar
  16. 16.
    O.E. Lanford, D. Ruelle: Commun. Math. Phys. 13, 194 (1969)CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Yu.V. Prokhorov: Theory of Prob. and its Appl. 1, 158 (1956)Google Scholar
  18. 18.
    D. Ruelle: Annals of Phys. 69, 364 (1972)CrossRefADSGoogle Scholar
  19. 19.
    F. Spitzer: Lecture Notes in Math. 390, 114 (1974)CrossRefMathSciNetGoogle Scholar
  20. 20.
    K. Krickeberg: Ecole d’été due St-Flour X, Lecture Notes in Math. (to appear)Google Scholar
  21. 21.
    E. Glötzl, B. Rauchenschwandtner, R. Takacs: Preprint Univ. Linz (1979)Google Scholar
  22. 22.
    J.N. Darroch, S.L. Lauritzen, T.P. Speed: Annals of Prob. 8, 522 (1980)MATHMathSciNetGoogle Scholar
  23. 23.
    T.W. Anderson: An introduction to multivariate analysis (J. Wiley, New York 1958)MATHGoogle Scholar
  24. 24.
    C.C. Neaderhouser: Annals of Prob. 6, 207 (1978)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    J.L. Lebowitz: Commun. Math. Phys. 33, 313 (1973)Google Scholar
  26. 26.
    G. Gallavotti, A. Martin-Löf: Nuovo Cimento 25, 425 (1975)CrossRefGoogle Scholar
  27. 27.
    J.R. Barra: Notions fondamentales de statistique mathématique (Dunod, Paris 1971)MATHGoogle Scholar
  28. 28.
    J. Demongeot: Séminaire de Statistiques de Grenoble (to appear)Google Scholar
  29. 29.
    X. Guyon: Preprint Univ. Orsay (1980)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1981

Authors and Affiliations

  • J. Demongeot
    • 1
  1. 1.Laboratoire d’Informatique et de Mathématiques AppliquéesUniversité de Grenoble IGrenoble CédexFrance

Personalised recommendations