Abstract
The impact of using an incorrect covariance function on kriging predictors is investigated. Results of Stein (1988) show that the impact on the kriging predictor from not using the correct covariance function is asymptotically negligible as the number of observations increases if the covariance function used is “compatible” with the actual covariance function on the region of interestR. The definition and some properties of compatibility of covariance functions are given. The compatibility of generalized covariances also is defined. Compatibility supports the intuitively sensible concept that usually only the behavior near the origin of the covariance function is critical for purposes of kriging. However, the commonly used spherical covariance function is an exception: observations at a distance near the range of a spherical covariance function can have a nonnegligible effect on kriging predictors for three-dimensional processes. Finally, a comparison is made with the perturbation approach of Diamond and Armstrong (1984) and some observations of Warnes (1986) are clarified.
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References
Armstrong, M. and Myers, D., 1984, Robustness of Kriging; Variogram Modelling and its Applications: Technical Report, Centre de Geostatistique, Fontainebleau, France.
Delfiner, P., 1976, Linear Estimation of Non-Stationary Spatial Phenomena: p. 49–68in M. Gurascio, M. David, and C. Huijbregts (Eds.), Advanced Geostatistics in the Mining Industry: Reidel, Dordrecht.
Diamond, P. and Armstrong, M., 1984, Robustness of Variograms and Conditioning of Kriging Matrices: Math. Geol., v. 16, p. 809–822.
Doob, J. L., 1953, Stochastic Processes: John Wiley, New York, 654 p.
Goldberger, A., 1962, Best Linear Unbiased Prediction in the Generalized Linear Regression Model: J. Amer. Stat. Assoc., v. 57, p. 369–375.
Hajek, J., 1958, On a Property of Normal Distributions of an Arbitrary Stochastic Process: Czech. Math. J., v. 8, p. 610–618.
Ibragimov, I. A. and Rozanov, Y. A., 1978, Gaussian Random Processes: (trans. Aries, A. B.), Springer-Verlag, New York, 275 p.
Journel, A. G. and Huijbregts, C. J., 1978, Mining Geostatistics: Academic Press, London, 600 p.
Kitanidis, P. K., 1983, Statistical Estimation of Polynornial Generalized Covariance Functions and Hydrologic Applications: Water Resour. Res., v. 19, p. 909–921.
Krasnitskii, S. M., 1973, On Conditions of Equivalence and Perpendicularity of Measures Corresponding to Homogeneous Gaussian Fields: Theory Probab. Its Appl., v. 18, p. 588–592.
Matheron, G., 1973, The Intrinsic Random Functions and their Applications: Adv. Appl. Probab., v. 5, p. 437–468.
Rendu, J-M., 1978, An Introduction to Geostatistical Methods of Mineral Evaluation: Monogr. S. Afr. Inst. Mining Metall., 100 p.
Ripley, B. D., 1981, Spatial Statistics: John Wiley, New York, 252 p.
Rozanov, Y. A., 1968, Infinite Dimensional Gaussian Distributions: Proceedings of the Steklov Institute of Mathematics, Amer. Math. Soc. v. 108, p. 1–136.
Skorokhod, A. V. and Yadrenko, M. I., 1973, On Absolute Continuity of Measures Corresponding to Homogeneous Gaussian Fields: Theory Probab. Its Appl., v. 8, p. 27–40.
Stein, M., 1988, Asymptotically Efficient Spatial Interpolation with a Misspecified Covariance Function: Ann. Stat., v. 16, p. 55–63.
Sukhatme, S., 1985, Kriging with Perturbed Variogram: Proceedings of the Social Statistics Section of the American Statistical Association Annual Meetings, p. 296–299.
Warnes, J. J., 1986, A Sensitivity Analysis for Universal Kriging: Math. Geol., v. 18, p. 653–676.
Yaglom, A. M., 1962, An Introduction to the Theory of Stationary Random Functions: Prentice-Hall, New Jersey, 235 p.
Yakowitz, S. J. and Szidarovszky, F., 1985, A Comparison of Kriging with Nonparametric Regression Methods: J. Multivariate Anal. v. 16, p. 21–53.
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Stein, M.L., Handcock, M.S. Some asymptotic properties of kriging when the covariance function is misspecified. Math Geol 21, 171–190 (1989). https://doi.org/10.1007/BF00893213
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DOI: https://doi.org/10.1007/BF00893213