Gibbs classifiers
Author:
B. A. Zalesky
Translated by:
The author
Journal:
Theor. Probability and Math. Statist. 70 (2005), 41-51
MSC (2000):
Primary 60G60, 62H30; Secondary 90C27
DOI:
https://doi.org/10.1090/S0094-9000-05-00629-0
Published electronically:
August 5, 2005
MathSciNet review:
2109821
Full-text PDF Free Access
Abstract |
References |
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Additional Information
Abstract:
New statistical classifiers for dependent observations with Gibbs prior distributions of the exponential or Gaussian type are presented. It is assumed that the observations are characterized by feature functions that assume values in finite sets of rational numbers. The distributions of observations are either Gibbs exponential or Gibbs Gaussian. Arbitrary neighborhoods on a completely connected graph are considered instead of local neighborhoods of the nearest observation.
The models studied in this paper can be used for some problems of the classification of random fields, in statistical physics, and for image processing.
A method of finding an optimal Bayes decision rule is described. The method is based on the reduction of the problem to the evaluation of the minimal cut of an appropriate graph. The method can be used for the fast evaluation of optimal Bayes decision rules for large samples.
References
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References
- S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Trans. Pattern Anal. Mach. Intel. PAMI-6 6 (1984), 721–741.
- B. Gidas, Metropolis Type Monte Carlo Simulation Algorithm and Simulated Annealing, Topics in Contemp. Probab. Appl., Stochastics Ser., CRC, Boca Raton, Fl, 1995. MR 1410538 (98b:60128)
- B. A. Zalesky, Sthochastic relaxation for buildind some classes of piecewise linear regression functions, Monte-Carlo Meth. Appl. 6 (2000), no. 2, 141–157. MR 1773375
- D. M. Greig, B. T. Porteous, and A. H. Seheult, Exact maximum a posteriori estimation for binary images, J. R. Statist. Soc. B. 58 (1989), 271–279.
- Yu. Boykov, O. Veksler, and R. Zabih, Fast approximate energy minimization via graph cuts, IEEE Trans. Pattern Anal. Machine Intel. 23 (2001), no. 11, 1222–1239.
- V. Kolmogorov and R. Zabih, What Energy Functions Can Be Minimized Via Graph Cuts?, Cornell CS Technical Report, TR2001-1857 (2001).
- B. A. Zalesky, Computation of Gibbs estimates of gray-scale images by discrete optimization methods, Proceedings of the Sixth International Conference PRIP’2001 (Minsk, May 18–20, 2001), pp. 81–85.
- B. A. Zalesskiĭ, Efficient minimization of quadratic polynomials in integer variables with quadratic monomials $b^2_{i,j}(x_i-x_j)^2$, Dokl. Nats. Akad. Nauk Belarusi 45 (2001), no. 6, 9–11. (Russian) MR 1984687 (2003m:90073)
- B. A. Zalesky, Network Flow Optimization for Restoration of Images, Preprint AMS, Mathematics ArXiv, math.OC/0106180.
- S. A. Aivazyan, B. M. Bukhshtaber, I. S. Enyukov, and L. D. Meshalkin, Applied Statistics: Classification and Dimensionality Reduction, “Finansy i statistika”, Moscow, 1989. (Russian)
- G. J. McLachlan, Discriminant Analysis and Statistical Pattern Recognition, John Wiley, New York, 1992. MR 1190469 (94a:62091)
- V. V. Mottl’ and I. B. Muchnik, Hidden Markov Models in Structural Analysis of Signals, Fizmatlit, Moscow, 1999. (Russian) MR 1778152 (2001m:94014)
- E. E. Zhuk and Yu. S. Kharin, Stability in the Cluster Analysis of Multivariate Data, Belgosuniversitet, Minsk, 1998. (Russian)
- B. V. Cherkassky and A. V. Goldberg, On Implementing Push-Relabel Method for the Maximum Flow Problem, Technical Report STAN-CS-94-1523, Department of Computer Science, Stanford University, 1994.
- J. C. Picard and H. D. Ratliff, Minimal cuts and related problems, Networks 5 (1975), 357–370. MR 0391960 (52:12778)
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Additional Information
B. A. Zalesky
Affiliation:
Joint Institute of Problems in Informatics, National Academy of Sciences, Surganov Street 6, Minsk 220012, Belarus
Email:
zalesky@newman.bas-net.by
Received by editor(s):
April 4, 2002
Published electronically:
August 5, 2005
Additional Notes:
Supported by a grant of MNTC B–517.
Article copyright:
© Copyright 2005
American Mathematical Society
