泛克立格法和指示克立格法在地球化学探矿中的应用

泛克立格法是一种非平稳随机函数的最佳线性无偏估计方法，作者将之用于处理区域地球化学探矿数据，给出被测元素的估计值，漂移植和涨落值，后者为评估元素区域北景和异常特性提供了有用信息，作者用非参数地质统计学的指示克立格法对化探元素含量进行异值的分析及大于各级下限值的概率估计。     

A modification of minimum norm quadratic estimation of a generalized covariance function for use with large data sets

Marshall and Mardia (1985) and Kitanidis (1985) have suggested using minimum norm quadratic estimation as a method to estimate parameters of a generalized covariance function. Unfortunately, this method is difficult to use with large data sets as it requires inversion of an n × n matrix, where n is number of observations. These authors suggest replacing the matrix to be inverted by the identity matrix, which eliminates the computational burden, although with a considerable loss of efficiency. As an alternative, the data set can be broken into subsets, and minimum norm quadratic estimates of parameters of the generalized covariance function can be obtained within each subset. These local estimates can be averaged to obtain global estimates. This procedure also avoids large matrix inversions, but with less loss in efficiency.     

Hybrid Estimation of Semivariogram Parameters

Two widely used methods of semivariogram estimation are weighted least squares estimation and maximum likelihood estimation. The former have certain computational advantages, whereas the latter are more statistically efficient. We introduce and study a “hybrid” semivariogram estimation procedure that combines weighted least squares estimation of the range parameter with maximum likelihood estimation of the sill (and nugget) assuming known range, in such a way that the sill-to-range ratio in an exponential semivariogram is estimated consistently under an infill asymptotic regime. We show empirically that such a procedure is nearly as efficient computationally, and more efficient statistically for some parameters, than weighted least squares estimation of all of the semivariogram’s parameters. Furthermore, we demonstrate that standard plug-in (or empirical) spatial predictors and prediction error variances, obtained by replacing the unknown semivariogram parameters with estimates in expressions for the ordinary kriging predictor and kriging variance, respectively, perform better when hybrid estimates are plugged in than when weighted least squares estimates are plugged in. In view of these results and the simplicity of computing the hybrid estimates from weighted least squares estimates, we suggest that software that currently estimates the semivariogram by weighted least squares methods be amended to include hybrid estimation as an option.     

Fitting variogram models by weighted least squares

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.     

Empirical Maximum Likelihood Kriging: The General Case

Although linear kriging is a distribution-free spatial interpolator, its efficiency is maximal only when the experimental data follow a Gaussian distribution. Transformation of the data to normality has thus always been appealing. The idea is to transform the experimental data to normal scores, krige values in the “Gaussian domain” and then back-transform the estimates and uncertainty measures to the “original domain.” An additional advantage of the Gaussian transform is that spatial variability is easier to model from the normal scores because the transformation reduces effects of extreme values. There are, however, difficulties with this methodology, particularly, choosing the transformation to be used and back-transforming the estimates in such a way as to ensure that the estimation is conditionally unbiased. The problem has been solved for cases in which the experimental data follow some particular type of distribution. In general, however, it is not possible to verify distributional assumptions on the basis of experimental histograms calculated from relatively few data and where the uncertainty is such that several distributional models could fit equally well. For the general case, we propose an empirical maximum likelihood method in which transformation to normality is via the empirical probability distribution function. Although the Gaussian domain simple kriging estimate is identical to the maximum likelihood estimate, we propose use of the latter, in the form of a likelihood profile, to solve the problem of conditional unbiasedness in the back-transformed estimates. Conditional unbiasedness is achieved by adopting a Bayesian procedure in which the likelihood profile is the posterior distribution of the unknown value to be estimated and the mean of the posterior distribution is the conditionally unbiased estimate. The likelihood profile also provides several ways of assessing the uncertainty of the estimation. Point estimates, interval estimates, and uncertainty measures can be calculated from the posterior distribution.     

Multivariable spatial prediction

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.     

Variogram Model Selection via Nonparametric Derivative Estimation

Before optimal linear prediction can be performed on spatial data sets, the variogram is usually estimated at various lags and a parametric model is fitted to those estimates. Apart from possible a priori knowledge about the process and the user's subjectivity, there is no standard methodology for choosing among valid variogram models like the spherical or the exponential ones. This paper discusses the nonparametric estimation of the variogram and its derivative, based on the spectral representation of positive definite functions. The use of the estimated derivative to help choose among valid parametric variogram models is presented. Once a model is selected, its parameters can be estimated—for example, by generalized least squares. A small simulation study is performed that demonstrates the usefulness of estimating the derivative to help model selection and illustrates the issue of aliasing. MATLAB software for nonparametric variogram derivative estimation is available at http://www-math.mit.edu/~gorsich/derivative.html. An application to the Walker Lake data set is also presented.     

The Correlation Structure of Matheron's Classical Variogram Estimator Under Elliptically Contoured Distributions

The classical variogram estimator proposed by Matheron can be written as a quadratic form of the observations. When data have an elliptically contoured distribution with constant mean, the correlation between the classical variogram estimator at two different lags is a function of the spatial design matrix, the covariance matrix, and the kurtosis. Several specific cases are studied closely. A subclass of elliptically contoured distributions with a particular family of covariance matrices is shown to possess exactly the same correlation structure for the classical variogram estimator as the multivariate independent Gaussian distribution. The consequences on variogram fitting by generalized least squares are discussed.     

Computationally efficient restricted maximum likelihood estimation of generalized covariance functions

Computational aspects of the estimation of generalized covariance functions by the method of restricted maximum likelihood (REML) are considered in detail. In general, REML estimation is computationally intensive, but significant computational savings are available in important special cases. The approach taken here restricts attention to data whose spatial configuration is a regular lattice, but makes no restrictions on the number of parameters involved in the generalized covariance nor (with the exception of one result) on the nature of the generalized covariance function's dependence on those parameters. Thus, this approach complements the recent work of L. G. Barendregt (1987), who considered computational aspects of REML estimation in the context of arbitrary spatial data configurations, but restricted attention to generalized covariances which are linear functions of only two parameters.