Abstract
The purpose of this paper is to extend the locally based prediction methodology of BayMar to a global one by modelling discrete spatial structures as Markov random fields. BayMar uses one-dimensional Markov-properties for estimating spatial correlation and Bayesian updating for locally integrating prior and additional information. The methodology of this paper introduces a new estimator of the field parameters based on the maximum likelihood technique for one-dimensional Markov chains. This makes the estimator straightforward to calculate also when there is a large amount of missing observations, which often is the case in geological applications. We make simulations (both unconditional and conditional on the observed data) and maximum a posteriori predictions (restorations) of the non-observed data using Markov chain Monte Carlo methods, in the restoration case by employing simulated annealing. The described method gives satisfactory predictions, while more work is needed in order to simulate, since it appears to have a tendency to overestimate strong spatial dependence. It provides an important development compared to the BayMar-methodology by facilitating global predictions and improved use of sparse data.
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REFERENCES
Aarts, E., and Korst, J., 1989, Simulated annealing and Boltzmann machines: Wiley, New York, 242 p.
Baran, Á., and Baran, S., 1997, An application of simulated annealing to ML-estimation of a partially observed Markov chain: Report. Centre for Applied Mathematics and Statistics, Chalmers and Göteborg University, Sweden.
Baran, S., and Szabó Á., 1997, An application of simulated annealing to ML-estimation of a partially observed Markov chain, in Proceedings of the 3rd International Conference on Applied Informatics, Eger-Noszvaj, Hungary, August 24–28.
Besag, J., 1974, Spatial interaction and the statistical analysis of lattice systems (with discussion): J. Roy. Statist. Soc. B, v. 36, p. 192–236.
Corana, A., Marchesi, M., Martini, C., and Ridella, S., 1987, Minimising multimodal functions of continuous variables with the “simulated annealing” algorithm: ACM Trans. Math. Softw., v. 13, no. 3, p. 262–280.
Cressie, N., 1993, Statistics for spatial data, Rev. ed.: Wiley, New York, 900 p.
Geman, S., and Geman, D., 1984, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images: IEEE Trans., v. PAMI-6, p. 721–741.
Guyon, X., 1995, Random fields on a network: Springer, New York, 255 p.
Laarhoven, P. J. M., and Aarts, E. H. L., 1987, Simulated annealing: Theory and applications, Kluwer Academic, Dordrecht, 186 p.
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E., 1953, Equations of state calculations by fast computing machines: J. Chem. Phys., v. 21, p. 1087–1091.
Rosén, L., and Gustafson, G., 1996, A Bayesian Markov geostatistical model for estimation of hydrogeological properties: Ground Water, v. 34, p. 865–875.
Tjelmeland, H., and Besag, J., 1998, Markov random fields with higher-order interactions: Scand. J. Stat., v. 25, p. 415–433.
Wikberg, P., Gustafson, G., and Stanfors, R., 1991. Äspö Hard Rock Laboratory: Evaluation and conceptual modelling based on the pre-investigations: SKB TR 91–22.
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Norberg, T., Rosén, L., Baran, Á. et al. On Modelling Discrete Geological Structures as Markov Random Fields. Mathematical Geology 34, 63–77 (2002). https://doi.org/10.1023/A:1014079411253
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DOI: https://doi.org/10.1023/A:1014079411253