Mathematical Geology

, Volume 22, Issue 7, pp 763–777 | Cite as

A Bayesian/maximum-entropy view to the spatial estimation problem

  • George Christakos
Articles

Abstract

The purpose of this paper is to stress the importance of a Bayesian/maximum-entropy view toward the spatial estimation problem. According to this view, the estimation equations emerge through a process that balances two requirements: High prior information about the spatial variability and high posterior probability about the estimated map. The first requirement uses a variety of sources of prior information and involves the maximization of an entropy function. The second requirement leads to the maximization of a so-called Bayes function. Certain fundamental results and attractive features of the proposed approach in the context of the random field theory are discussed, and a systematic spatial estimation scheme is presented. The latter satisfies a variety of useful properties beyond those implied by the traditional stochastic estimation methods.

Key words

spatial estimation entropy Bayes law information geostatistics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Burg, J. P., 1972, The Relationship Between Maximum Entropy Spectra and Maximum Likelihood Spectra: Geophysics, v. 38, p. 375–376.Google Scholar
  2. Carnap, R., 1950, Logical Foundations of Probability: University of Chicago Press, Chicago.Google Scholar
  3. Christakos, G., 1986, Recursive Estimation of Nonlinear State-Nonlinear Observation Systems (Part 1: On-Line Data): Res. Rep. OF.86-29, Kansas Geological Survey, Lawrence, Kansas, 85 p.Google Scholar
  4. Christakos, G., 1989, Optimal Estimation of Nonlinear State Nonlinear-Observation Systems: J. Optimization Theory Appl., v. 62, p. 29–48.Google Scholar
  5. Christakos, G., 1990, Some Applications of the Bayesian, Maximum-Entropy Concepts in Geostatistics (invited paper),in T. Grandy (Ed.), Proceedings of the 10th International Max Ent workshop, University of Wyoming, Laramie, Wyoming: Kluwer, The Netherlands.Google Scholar
  6. Ewing, G. M., 1969, Calculus of Variations with Applications: Norton, New York, 343.Google Scholar
  7. Gandin, L. S., 1963, Objective Analysis of Meteorological Fields: Gidrometerologischeskoe Izdatel'stvo, Leningrad, 242 p.Google Scholar
  8. Jaynes, E. T., 1982, On the Rationale of Maximum-Entropy Methods: Proceed. IEEE, v. 70, p. 939–952.Google Scholar
  9. Johnson, N. L., and Kotz, S., 1972, Distributions in Statistics: Continuous Multivariate Distributions: Wiley, New York, 333 p.Google Scholar
  10. Journel, A. G., and Huijbregts, Ch., 1978, Mining Geostatistics: Academic Press, London, 600 p.Google Scholar
  11. Journel, A. G., 1986, Constrained Interpolation and Qualitative Information: The Soft Kriging Approach: Math. Geol., v. 18, p. 269–286.Google Scholar
  12. Journel, A. G., 1989, Fundamentals of Geostatistics in Five Lessons: American Geophysical Union, Washington, D.C., 40 p.Google Scholar
  13. Kullback, S., 1968, Information Theory and Statistics: Dover, New York, 399 p.Google Scholar
  14. Matheron, 1965, Les Variables Regionalisees at Leur Estimation: Masson, Paris, 212 p.Google Scholar
  15. Matheron, G., 1984, Isofactorial Models and Change of Support, in Verly et al (Eds.), Adv. Geostatistics for the Characterization of Natural Resources, Vol. 1: Reidel, p. 449–468.Google Scholar
  16. Popper, K. R., 1934,Logik der Forschung: Springer, Vienna, 1934 (English translation, The Logic of Scientific Discovery: Hutchinson, London, 1959).Google Scholar
  17. Popper, K. R., 1972, Objective Knowledge: An Evolutionary Approach: Oxford University Press, Oxford, 395 p.Google Scholar
  18. Shannon, C. E., 1948, A Mathematical Theory of Communication: Bell System Tech. J., v. 27, p. 379–423, 623–656.Google Scholar
  19. Shore, J. E., and Johnson, R. W., 1980, Axiomatic Derivation of the Principle of Maximum Entropy and the Principle of Minimum Cross-Entropy: IEEE Trans. Inform. Theory, v. 26, p. 26–37.Google Scholar
  20. Wiener, N., 1949. Time Series: MIT Press, Cambridge, Massachusetts, 163 p.Google Scholar

Copyright information

© International Association for Mathematical Geology 1990

Authors and Affiliations

  • George Christakos
    • 1
  1. 1.Department of Environmental Sciences and EngineeringThe University of North CarolinaChapel Hill

Personalised recommendations