Abstract
Case deletion diagnostics are developed for detecting observations that are influential in estimating the covariance function of a spatial random field. Diagnostics are developed within the context of universal kriging. Computational formulae are given that make the procedures feasible and the diagnostics are illustrated in an example.
Similar content being viewed by others
References
Beckman, R. J., Nachtsheim, C. J., and Cook, R. D., 1987, Diagnostics for Mixed-Model Analysis of Variance: Technometrics, v. 29, p. 413–426.
Belsley, D. A., Kuh, E., and Welsch, R. E., 1980, Regression Diagnostics: John Wiley and Sons, New York.
Christensen, R., 1987, Plane Answers to Complex Questions: The Theory of Linear Models: Springer-Verlag, New York.
Christensen, R. 1990a, The Equivalence of Predictions from Universal Kriging and Intrinsic Random Function Kriging: Math. Geo., v. 22, p. 655–664.
Christense, R., 1990b, Linear Models for Multivariate, Time Series, and Spatial Data: Springer-Verlag, New York.
Christensen, R. Pearson, L. M., and Johnson, W., 1992, Case Deletion Diagnostics for Mixed-Models: Technometrics, v. 34, p. 38–45.
Christensen, R., Johnson, W., and Pearson, L. M., 1992, Prediction Diagnostics for Spatial Linear Models: Biometrika, v. 79, p. 583–591.
Cook, R. D., 1977, Detection of Influential Observations in Linear Regression: Technometrics, v. 19, p. 15–18.
Cook, R. D., 1979, Influential Observations in Linear Regression: J. Am. Stat. Assoc., v. 74, p. 169–174.
Cook, R. D., and Weisberg, S., 1980, Characterizations of an Empirical Influence Function for Detecting Influential Cases in Regression: Technometrics, v. 22, p. 495–508.
Cook, R. D., and Weisberg, S., 1982, Residuals and Influence in Regression: Chapman and Hall, New York.
Cressie, N., 1989, Geostatistics: Am. Stat., v. 43, p. 197–202.
Davis, B. M., 1987, Uses and Abuses of Cross-Validation in Geostatistics: Math. Geo., v. 19, p. 241–248.
Delfiner, P., 1976, Linear Estimation of Nonstationary Spatial Phenomena,in M. Guarascia, M. David, and C. Hüijbregts (Eds.), Advanced Geostatistics in the Mining Industry: Reidel, Dordrecht.
Diamond, P., and Armstrong, M., 1984, Robustness of Variograms and Conditioning of Kriging Matrices: J. Int. Assoc. Math. Geo., v. 16, p. 809–822.
Goldberger, A. S., 1962, Best Linear Unbiased Prediction in the Generalized Linear Regression Model: J. Am. Stat. Assoc., v. 57, p. 369–375.
Golub, G. H., and Van Loan, C. F., 1989, Matrix Computations, 2nd ed.: The Johns Hopkins University Press, Baltimore.
Harper, W. V., and Furr, J. M., 1986, Geostatistical Analysis of Potentiometric Data in the Wolfcamp Aquifer of the Palo Duro Basin: Technical Report ONWI-587, Battelle Memorial Institute, Columbus, OH.
Harville, D. A., 1977, Maximum Likelihood Approaches to Variance Component Estimation and to Related Problems: J. Am. Stat. Assoc., v. 72, p. 320–338.
Johnson, W., 1985, Influence Measures for Logistic Regression: Another Point of View: Biometrika, v. 72, p. 59–65.
Johnson, W., and Geisser, S., 1983, A Predictive View of the Detection and Characterization of Influential Observations in Regression Analysis: J. Am. Stat. Assoc., v. 78, p. 137–144.
Journel, A. G., and Hüijbregts, Ch. J., 1978, Mining Geostatistics: Academic Press, New York.
Kitanidis, P. K., 1983, Statistical Estimation of Polynomial Generalized Covariance Functions and Hydrologic Applications: Water Res. Res., v. 19, p. 909–921.
Kitanidis, P. K., 1985, Minimum-Variance Unbiased Quadratic Estimation of Covariances of Regionalized Variables: J. Int. Assoc. Math. Geo., v. 17, p. 195–208.
Kitanidis, P. K., and Lane, R. W., 1985, Maximum Likelihood Parameter Estimation of Hydrologic Spatial Processes by the Gauss-Newton Method: J. Hydrol., v. 79, p. 53–71.
Mardia, K. V., and Marshall, R. J., 1984, Maximum Likelihood Estimation of Models for Residual Covariance in Spatial Regression: Biometrika, v. 71, p. 135–146.
Marshall, R. J., and Mardia, K. V., 1985, Minimum Norm Quadratic Estimation of Components of Spatial Covariance: J. Int. Assoc. Math. Geo., v. 17, p. 517–525.
Matheron, G., 1969, Le Krigeage Universal: Fascicule 1, Cahiers du CMMM., 82 p.
Matheron, G., 1973, Intrinsic Random Functions and Their Applications: Adv. App. Prob., v. 5, p. 439–468.
Patterson, H. D., and Thompson, R., 1974, Maximum Likelihood Estimation of Variance Components: Proc. of the 8th Int. Biomet. Conf., p. 197–207.
Pregibon, D., 1981, Logistic Regression Diagnostics: Ann. Stat., v. 9, p. 705–724.
Seber, G. A. F., and Wild, C. J., 1989, Nonlinear Regression: John Wiley and Sons, New York.
Warnes, J. J., 1986, A Sensitivity Analysis for Universal Kriging: J. Int. Assoc. Math. Geo., v. 18, p. 653–676.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Christensen, R., Johnson, W. & Pearson, L.M. Covariance function diagnostics for spatial linear models. Math Geol 25, 145–160 (1993). https://doi.org/10.1007/BF00893270
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00893270