Mathematical Geology

, Volume 30, Issue 4, pp 323–345 | Cite as

Variogram Fitting by Generalized Least Squares Using an Explicit Formula for the Covariance Structure

  • Marc G. Genton

Abstract

In the context of spatial statistics, the classical variogram estimator proposed by Matheron can be written as a quadratic form of the observations. If data are Gaussian with constant mean, then the correlation between the classical variogram estimator at two different lags is a function of the spatial design matrix and the variance matrix. When data are independent with unidimensional and regular support, an explicit formula for this correlation is available. The same is true for a multidimensional and regular support as can be shown by using Kronecker products of matrices. As variogram fitting is a crucial stage for correct spatial prediction, it is proposed to use a generalized least squares method with an explicit formula for the covariance structure (GLSE). A good approximation of the covariance structure is achieved by taking account of the explicit formula for the correlation in the independent situation. Simulations are carried out with several types of underlying variograms, as well as with outliers in the data. Results show that this technique (GLSE), combined with a robust estimator of the variogram, improves the fit significantly.

spatial statistics scale estimation robustness dependent data correlation structure ordinary kriging 

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REFERENCES

  1. Beran, J., 1994, Statistics for long-memory processes: Chapman & Hall, New York, 315 p.Google Scholar
  2. Cressie, N., 1985, Fitting variogram models by weighted least squares: Math. Geology, v. 17,no. 5, p. 563–586.Google Scholar
  3. Cressie, N., 1991, Statistics for spatial data: John Wiley & Sons, New York, 900 p.Google Scholar
  4. Cressie, N., and Hawkins, D. M., 1980, Robust estimation of the variogram, I: Math. Geology, v. 12,no. 2, p. 115–125.Google Scholar
  5. Fang, K., and Zhang, Y., 1990, Generalized multivariate analysis: Springer, Beijing, 220 p.Google Scholar
  6. Fedorov, V. V., 1974, Regression problems with controllable variables subject to error: Biometrika, v. 61,no. 1, p. 49–56.Google Scholar
  7. Genton, M. G., 1995, Robustesse dans l'estimation du variogramme: Bulletin de l'Institut International de Statistique, Beijing, China, v. 1, p. 400–401.Google Scholar
  8. Genton, M. G., 1996, Robustness in variogram estimation and fitting: unpubl. doctoral dissertation, no. 1595, Swiss Federal Inst. Technology, Lausanne, 132 p.Google Scholar
  9. Genton, M. G., 1998, Highly robust variogram estimation: Math. Geology, v. 30,no. 2, p. 213–221.Google Scholar
  10. Genton, M. G., and Rousseeuw, P. J., 1995, The change-of-variance function of M-estimators of scale under general contamination: Jour. Computational and Applied Mathematics, v. 64,no. 1, p. 69–80.Google Scholar
  11. Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J., and Stahel, W. A., 1986, Robust statistics, the approach based on influence functions: John Wiley & Sons, New York, 502 p.Google Scholar
  12. Horn, R. A., and Johnson, C. R., 1991, Topics in matrix analysis: Cambridge Univ. Press, Cambridge, 607 p.Google Scholar
  13. Journel, A. G., and Huijbregts, Ch. J., 1978, Mining geostatistics: Academic Press, London, 600 p.Google Scholar
  14. Matheron, G., 1962, Traité de géostatistique appliquée, Tome I: Mémoires du Bureau de Recherches Géologiques et Minières, no. 14, Editions Technip, Paris, 333 p.Google Scholar
  15. Portnoy, S., 1977, Robust estimation in dependent situations: The Annals of Statistics, v. 1, p. 22–43.Google Scholar
  16. Rousseeuw, P. J., and Croux, C., 1992, Explicit scale estimators with high breakdown point: L 1 Statistical Analyses and Related Methods, p. 77–92.Google Scholar
  17. Rousseeuw, P. J., and Croux, C., 1993, Alternatives to the median absolute deviation: Jour. Am. Stat. Assoc., v. 88,no. 424, p. 1273–1283.Google Scholar
  18. Seber, G. A. F., and Wild, C. J., 1989, Nonlinear regression: John Wiley & Sons, New York, 768 p.Google Scholar

Copyright information

© International Association for Mathematical Geology 1998

Authors and Affiliations

  • Marc G. Genton
    • 1
  1. 1.Department of MathematicsSwiss Federal Institute of TechnologyLausanneSwitzerland

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