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  • 1
    Electronic Resource
    Electronic Resource
    Mean squared prediction error in the spatial linear model with estimated covariance parameters (1992)
    Springer
    Annals of the Institute of Statistical Mathematics 44 (1992), S. 27-43 
    Details
    ISSN: 1572-9052
    Keywords: Best linear unbiased prediction ; generalized covariances ; geostatistics ; kriging ; spatial models
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The problem considered is that of predicting the value of a linear functional of a random field when the parameter vector θ of the covariance function (or generalized covariance function) is unknown. The customary predictor when θ is unknown, which we call the EBLUP, is obtained by substituting an estimator Ĝj for θ in the expression for the best linear unbiased predictor (BLUP). Similarly, the customary estimator of the mean squared prediction error (MSPE) of the EBLUP is obtained by substituting Ĝj for θ in the expression f for the BLUP's MSPE; we call this the EMSPE. In this article, the appropriateness of the EMSPE as an estimator of the EBLUP's MSPE is examined, and alternative estimators of the EBLUP's MSPE for use when the EMSPE is inappropriate are suggested. Several illustrative examples show that the performance of the EMSPE depends on the strength of spatial correlation; the EMSPE is at its best when the spatial correlation is strong.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00048668
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  • 2
    Electronic Resource
    Electronic Resource
    Robust kriging—A proposal (1984)
    Springer
    Mathematical geology 16 (1984), S. 3-18 
    Details
    ISSN: 1573-8868
    Keywords: Geostatistics ; kriging ; robust estimation ; time series
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Geological data frequently have a heavy-tailed normal-in-the-middle distribution, which gives rise to grade distributions that appear to be normal except for the occurrence of a few outliers. This same situation also applies to log-transformed data to which lognormal kriging is to be applied. For such data, linear kriging is nonrobust in that (1)kriged estimates tend to infinity as the outliers do, and (2)it is also not minimum mean squared error. The more general nonlinear method of disjunctive kriging is even more nonrobust, computationally more laborious, and in the end need not produce better practical answers. We propose a robust kriging method for such nearly normal data based on linear kriging of an editing of the data. It is little more laborious than conventional linear kriging and, used in conjunction with a robust estimator of the variogram, provides good protection against the effects of data outliers. The method is also applicable to time series analysis.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01036237
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  • 3
    Electronic Resource
    Electronic Resource
    Spatial prediction and ordinary kriging (1988)
    Springer
    Mathematical geology 20 (1988), S. 405-421 
    Details
    ISSN: 1573-8868
    Keywords: kriging ; nugget effect ; range ; sill ; variogram
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Suppose data {Z(s i ):i=1, ..., n} are observed at spatial locations {s i :i=1, ..., n}. From these data, an unknownZ(s 0) is to be predicted at a known locations 0c, or, ifZ(s0) has a component of measurement error, then a smooth versionS(s 0) should be predicted. This article considers the assumptions needed to carry out the spatial prediction using ordinary kriging, and looks at how nugget effect, range, and sill of the variogram affect the predictor. It is concluded that certain commonly held interpretations of these variogram parameters should be modified.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00892986
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  • 4
    Electronic Resource
    Electronic Resource
    On the stability of the geostatistical method (1992)
    Springer
    Mathematical geology 24 (1992), S. 45-59 
    Details
    ISSN: 1573-8868
    Keywords: kriging ; mean squared prediction error ; resistance ; robustness ; spatial dependence ; spatial prediction ; trend ; variogram
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract A thorough geostatistical analysis of spatial data, observed at given spatial locations, includes exploratory data analysis, spatial-model building, diagnosing the model fit, and inference on unknown model parameters or unobserved values (at known locations). Using results from mathematical analysis, exact and asymptotic distribution theory, and simulation studies, we argue that, when used sensibly, the geostatistical method is reassuringly stable.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00890087
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  • 5
    Electronic Resource
    Electronic Resource
    Robust estimation of the variogram: I (1980)
    Springer
    Mathematical geology 12 (1980), S. 115-125 
    Details
    ISSN: 1573-8868
    Keywords: Geostatistics ; kriging ; robust estimation ; variogram
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract It is a matter of common experience that ore values often do not follow the normal (or lognormal) distributions assumed for them, but, instead, follow some other heavier-tailed distribution. In this paper we discuss the robust estimation of the variogram when the distribution is normal-like in the central region but heavier than normal in the tails. It is shown that the use of a fourth-root transformation with or without the use of M-estimation yields stable robust estimates of the variogram.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01035243
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  • 6
    Electronic Resource
    Electronic Resource
    Fitting variogram models by weighted least squares (1985)
    Springer
    Mathematical geology 17 (1985), S. 563-586 
    Details
    ISSN: 1573-8868
    Keywords: generalized least squares ; kriging ; median polish ; robustness ; stationarity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract The method of weighted least squares is shown to be an appropriate way of fitting variogram models. The weighting scheme automatically gives most weight to early lags and down-weights those lags with a small number of pairs. Although weights are derived assuming the data are Gaussian (normal), they are shown to be still appropriate in the setting where data are a (smooth) transform of the Gaussian case. The method of (iterated) generalized least squares, which takes into account correlation between variogram estimators at different lags, offer more statistical efficiency at the price of more complexity. Weighted least squares for the robust estimator, based on square root differences, is less of a compromise.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01032109
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  • 7
    Electronic Resource
    Electronic Resource
    Multivariable spatial prediction (1993)
    Springer
    Mathematical geology 25 (1993), S. 219-240 
    Details
    ISSN: 1573-8868
    Keywords: geostatistics ; kriging ; cokriging ; cross-variogram ; best linear unbiased prediction ; generalized least squares
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract For spatial prediction, it has been usual to predict one variable at a time, with the predictor using data from the same type of variable (kriging) or using additional data from auxiliary variables (cokriging). Optimal predictors can be expressed in terms of covariance functions or variograms. In earth science applications, it is often desirable to predict the joint spatial abundance of variables. A review of cokriging shows that a new cross-variogram allows optimal prediction without any symmetry condition on the covariance function. A bivariate model shows that cokriging with previously used cross-variograms can result in inferior prediction. The simultaneous spatial prediction of several variables, based on the new cross-variogram, is then developed. Multivariable spatial prediction yields the mean-squared prediction error matrix, and so allows the construction of multivariate prediction regions. Relationships between cross-variograms, between single-variable and multivariable spatial prediction, and between generalized least squares estimation and spatial prediction are also given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00893273
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