Mathematical Geology

, Volume 39, Issue 3, pp 321–335 | Cite as

Markov Chain Random Fields for Estimation of Categorical Variables



Multi-dimensional Markov chain conditional simulation (or interpolation) models have potential for predicting and simulating categorical variables more accurately from sample data because they can incorporate interclass relationships. This paper introduces a Markov chain random field (MCRF) theory for building one to multi-dimensional Markov chain models for conditional simulation (or interpolation). A MCRF is defined as a single spatial Markov chain that moves (or jumps) in a space, with its conditional probability distribution at each location entirely depending on its nearest known neighbors in different directions. A general solution for conditional probability distribution of a random variable in a MCRF is derived explicitly based on the Bayes’ theorem and conditional independence assumption. One to multi-dimensional Markov chain models for prediction and conditional simulation of categorical variables can be drawn from the general solution and MCRF-based multi-dimensional Markov chain models are nonlinear.


Multi-dimensional Markov chain Markov random field Conditional simulation Interclass relationship Nonlinear Conditional independence 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Besag, J., 1974, Spatial interaction and the statistical analysis of lattice systems (with discussion): J. Roy. Stat. Soc. Ser. B, v. 36, no. 2, p. 192–236. Google Scholar
  2. Besag, J., 1986, On the statistical analysis of dirty pictures (with discussions): J. Roy. Stat. Soc. Ser. B, v. 48, no. 3, p. 259–302. Google Scholar
  3. Elfeki, A.M., Dekking, F.M., 2001, A Markov chain model for subsurface characterization: Theory and applications: Math. Geol., v. 33, no. 5, p. 569–589. CrossRefGoogle Scholar
  4. Derin, H., Elliott, H., Cristi, R., Geman, D., 1984, Bayes smoothing algorithms for segmentation of binary images modeled by Markov random fields: IEEE Trans. Pattern Anal. Mach. Intell., v. 6, no. 6, p. 707–720. CrossRefGoogle Scholar
  5. Fjortoft, R., Delignon, Y., Pieczynski, W., Sigelle, M., Tupin, F., 2003, Unsupervised classification of radar images using hidden Markov chains and hidden Markov random fields: IEEE Trans. Geosci. Remote Sens., v. 41, no. 3, p. 675–686. CrossRefGoogle Scholar
  6. Friedman, N., Geiger, D., Goldszmidt, M., 1997, Bayesian network classifiers: Mach. Learn., v. 29, no. 2-3, p. 131–163. CrossRefGoogle Scholar
  7. Gray, A.J., Kay, I.W., Titterington, D.M., 1994, An empirical study of the simulation of various models used for images: IEEE Trans. Pattern Anal. Mach. Intell., v. 16, no. 5, p. 507–513. CrossRefGoogle Scholar
  8. Haslett, J., 1985, Maximum likelihood discriminant analysis on the plane using a Markovian model of spatial context: Pattern Recognit., v. 18, no. 3-4, p. 287–296. CrossRefGoogle Scholar
  9. Idier, J., Goussard, Y., Ridolfi, A., 2001, Unsupervised image segmentation using a telegraph parameterization of Pickard random fields, in Moore, M., ed., Spatial Statistics. Methodological Aspects and Some Applications, Lecture Notes in Statistics, vol. 159: Springer, New York, p. 115–140. Google Scholar
  10. Journel, A.G., 2002, Combining knowledge from diverse sources: an alternative to traditional data independence hypothesis: Math. Geol., v. 34, no. 5, p. 573–596. CrossRefGoogle Scholar
  11. Li, W., 2006, Transiogram: a spatial relationship measure for categorical data: Int. J. Geogr. Inf. Sci., v. 20, no. 6, p. 693–699. Google Scholar
  12. Li, W., Zhang, C., 2005, Application of transiograms to Markov chain simulation and spatial uncertainty assessment of land-cover classes: GISci. Remote Sens., v. 42, no. 4, p. 297–319. Google Scholar
  13. Li, W., Zhang, C., Burt, J.E., Zhu, A.X., 2005, A Markov chain-based probability vector approach for modeling spatial uncertainty of soil classes: Soil Sci. Soc. Am. J., v. 69, no. 6, p. 1931–1942. CrossRefGoogle Scholar
  14. Li, W., Zhang, C., Burt, J.E., Zhu, A.X., Feyen, J., 2004, Two-dimensional Markov chain simulation of soil type spatial distribution: Soil Sci. Soc. Am. J., v. 68, no. 5, p. 1479–1490. CrossRefGoogle Scholar
  15. Lin, C., Harbaugh, J.W., 1984, Graphic display of two- and three-dimensional Markov computer models in geology: Van Nostrand Reinhold Company, New York, p. 180. Google Scholar
  16. Martin, R.J., 1996, Some results on unilateral ARMA lattice processes: J. Stat. Plan. Infer., v. 50, no. 3, p. 395–411. CrossRefGoogle Scholar
  17. Martin, R.J., 1997, A three dimensional unilateral autoregressive lattice process: J. Stat. Plan. Infer., v. 59, no. 1, p. 1–18. CrossRefGoogle Scholar
  18. Ortiz, J.M., Deutsch, C.V., 2004, Indicator simulation accounting for multiple-point statistics: Math. Geol., v. 36, no. 5, p. 545–565. CrossRefGoogle Scholar
  19. Pickard, D.K., 1980, Unilateral Markov fields: Adv. Appl. Probab., v. 12, no. 3, p. 655–671. CrossRefGoogle Scholar
  20. Qian, W., Titterington, D.M., 1991, Multidimensional Markov chain models for image textures: J. Roy. Stat. Soc. Ser. B, v. 53, no. 3, p. 661–674. Google Scholar
  21. Ramoni, M., Sebastiani, P., 2001, Robust Bayes classifiers: Artif. Intell., v. 125, no. 1-2, p. 209–226. CrossRefGoogle Scholar
  22. Rosholm, A., 1997, Statistical methods for segmentation and classification of images: PhD dissertation, Technical University of Denmark, Lyngby, p. 187. Google Scholar
  23. Ripley, B.D., 1990, Gibbsian interaction models, in Griffith, D.A., ed., Spatial statistics: Past, present, and future: Institute of Mathematical Geography, Syracuse University, p. 3–25. Google Scholar
  24. Sharp, W.E., Turner, B.F., 1999, The extension of a unilateral first-order autoregressive process from a square net to an isometric lattice using the herringbone method, in Lippard, S.J., Nass, A., Sinding-Larsen, R., eds., Proceedings of IMAG’99, 6–11 August 1999, Trondheim, p. 255–259. Google Scholar
  25. Turner, B.F., Sharp, W.E., 1994, Unilateral ARMA processes on a square net by the herringbone method: Math. Geol., v. 26, no. 5, p. 557–564. CrossRefGoogle Scholar
  26. Zhang, C., Li, W., 2005, Markov chain modeling of multinomial land-cover classes: GISci. Remote Sens., v. 42, no. 1, p. 1–18. Google Scholar

Copyright information

© International Association for Mathematical Geology 2007

Authors and Affiliations

  1. 1.Department of GeographyKent State UniversityKentUSA

Personalised recommendations