A variance-ratio test for supporting a variable mean in kriging

This article discusses the issue of whether to use a variable mean and describes a test that can be used to evaluate whether it is justified to add terms to the drift (deterministic part) of a geostatistical model. The basic model could be the intrinsic one, where the deterministic part is a constant, and the alternate model could be any model that includes a constant term in the expression for the drift. Also, differences between constant- and variable-mean models 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.     

Quadratic covariance estimation and equivalence of predictions

This article illustrates the use of linear and nonlinear regression models to obtain quadratic estimates of covariance parameters. These models lead to new insights into the motivation behind estimation methods, the relationships between different methods, and the relationship of covariance estimation to prediction. In particular, we derive the standard estimating equations for minimum norm quadratic unbiased translation invariant estimates (MINQUEs) from an appropriate linear model. Connections between the linear model, minimum variance quadratic unbiased translation invariant estimates (MIVQUEs), and MINQUEs are examined and we provide a minimum norm justification for the use of one-step normal theory maximum likelihood estimates. A nonlinear regression model is used to define MINQUEs for nonlinear covariance structures and obtain REML estimates. Finally, the equivalence of predictions under various models is examined when covariance parameters are estimated. In particular, we establish that when using MINQUE, iterative MINQUE, or restricted maximum likelihood (REML) estimates, the choice between a stationary covariance function and an intrinsically stationary semivariogram is irrelevant to predictions and estimated prediction variances.     

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.     

The equivalence of predictions from universal kriging and intrinsic random-function kriging

A proof is provided that the predictions obtained from kriging based on intrinsic random functions of orderk are identical to those obtained from anappropriate universal kriging model. This is a theoretical result based on known variability measures. It does not imply that people performing traditional universal kriging will get the same predictions as those using intrinsic random functions, because traditionally these methods differ in how variability is modeled. For intrinsic random functions, the same proof shows that predictions do not depend on the specific choice of the generalized covariance function. It is argued that the choice between these methods is really one of modeling and estimating the variability in the data.     

On least squares estimation of generalized covariance functions

Least squares estimation (LSE) is theoretically related to quadratic unbiased estimation of variance components. It is argued that these methods of estimation of variance components essentially generalize LSE though they are not formally equivalent.     

On the estimation of the generalized covariance function

The estimation of the generalized covariance function, K, is a major problem in the use of intrinsic random functions of order k to obtain kriging estimates. The precise estimation by least-squares regression of the parameters in polynomial models for K is made difficult by the nature of the distribution of the dependent variable and the multicollinearity of the independent variables.     

Generalized cross-validation for covariance model selection

A weighted cross-validation technique known in the spline literature as generalized cross-validation (GCV), is proposed for covariance model selection and parameter estimation. Weights for prediction errors are selected to give more importance to a cluster of points than isolated points. Clustered points are estimated better by their neighbors and are more sensitive to model parameters. This rational weighting scheme also provides a simplifying significantly the computation of the cross-validation mean square error of prediction. With small- to medium-size datasets, GCV is performed in a global neighborhood. Optimization of usual isotropic models requires only a small number of matrix inversions. A small dataset and a simulation are used to compare performances of GCV to ordinary cross-validation (OCV) and least-squares filling (LS).     

用加权多项式回归进行球状模型变差图的最优拟合

本文提出了一种用加权多项式回归对球状模型和二级套合球状模型的变差函数进行最优拟合的方法。在已经较好地算出各实验变差函数值并选定球状模型或二级套合球状模型为理论模型的条件下,应用这种方法可以编出程序,并在计算机上快速、自动地算出最优拟合的变差图。其结果唯一确定,不因人而异,可避免人为的误差,又可为地质统计学计算的全盘自动化创造良好条件。经过几个实例验算表明,该方法简单,效果良好。     

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.     

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.     

Using exploration history to estimate undiscovered resources

Inference about deposits left to be discovered in a partially explored oil field that require only assumptions about the randomness in the exploration procedure are considered. Unbiased estimators, estimators based on partial likelihood methods and confidence procedures are proposed.     

Estimation of covariance matrix from geochemical data with observations below detection limits

Multivariate statistical analyses have been extensively applied to geochemical measurements to analyze and aid interpretation of the data. Estimation of the covariance matrix of multivariate observations is the first task in multivariate analysis. However, geochemical data for the rare elements, especially Ag, Au, and platinum-group elements, usually contain observations the below detection limits. In particular, Instrumental Neutron Activation Analysis (INAA) for the rare elements produces multilevel and possibly extremely high detection limits depending on the sample weight. Traditionally, in applying multivariate analysis to such incomplete data, the observations below detection limits are first substituted, for example, each observation below the detection limit is replaced by a certain percentage of that limit, and then the standard statistical computer packages or techniques are used to obtain the analysis of the data. If a number of samples with observations below detection limits is small, or the detection limits are relatively near zero, the results may be reasonable and most geological interpretations or conclusions are probably valid. In this paper, a new method is proposed to estimate the covariance matrix from a dataset containing observations below multilevel detection limits by using the marginal maximum likelihood estimation (MMLE) method. For each pair of variables, sayY andZ whose observations containing below detection limits, the proposed method consists of three steps: (i) for each variable separately obtaining the marginal MLE for the means and the variances, , , , and forY andZ: (ii) defining new variables by and and lettingA=C+D andB=C–D, and obtaining MLE for variances, and forA andB; (iii) estimating the correlation coefficient _YZ by and the covariance _YZ by . The procedure is illustrated by using a precious metal geochemical data set from the Fox River Sill, Manitoba, Canada.     

Maximum Likelihood Estimation of Spatial Covariance Parameters

In this paper, the maximum likelihood method for inferring the parameters of spatial covariances is examined. The advantages of the maximum likelihood estimation are discussed and it is shown that this method, derived assuming a multivariate Gaussian distribution for the data, gives a sound criterion of fitting covariance models irrespective of the multivariate distribution of the data. However, this distribution is impossible to verify in practice when only one realization of the random function is available. Then, the maximum entropy method is the only sound criterion of assigning probabilities in absence of information. Because the multivariate Gaussian distribution has the maximum entropy property for a fixed vector of means and covariance matrix, the multinormal distribution is the most logical choice as a default distribution for the experimental data. Nevertheless, it should be clear that the assumption of a multivariate Gaussian distribution is maintained only for the inference of spatial covariance parameters and not necessarily for other operations such as spatial interpolation, simulation or estimation of spatial distributions. Various results from simulations are presented to support the claim that the simultaneous use of maximum likelihood method and the classical nonparametric method of moments can considerably improve results in the estimation of geostatistical parameters.     

Estimation of covariance parameters in kriging via restricted maximum likelihood

In kriging, parametric approaches to covariance (or variogram) estimation require that unknown parameters be inferred from a single realization of the underlying random field. An approach to such an estimation problem is to assume the field to be Gaussian and iteratively minimize a (restricted) negative loglikelihood over the parameter space. In doing so, the associated computational burden can be considerable. Also, it is usually not easy to check whether or not the minimum achieved is global. In this note, we show that in many practical cases, the structure of the covariance (or variogram) function can be exploited so that iterative minimizing algorithms may be advantageously replaced by a procedure that requires the computation of the roots of a simple rational function and the search for the minimum of a function depending on one variable only. As a consequence, our approach allows one to observe in a straightforward fashion the presence of local minima. Furthermore, it is shown that insensitivity of the likelihood function to changes in parameter value can be easily detected. The note concludes with numerical simulations that illustrate some key features of our estimation procedure.     

地质统计学在桂中三水铝矿床勘查评价中的应用

桂中三水铝矿床是我国唯一的大型三水铝矿床,初步查明(333+334)铝土矿资源量7.7亿吨。本文介绍地质统计学方法在桂中三水铝土矿床勘查和资源量评价中的应用,包括桂中三水铝矿床典型矿区变异函数的计算和模型拟合;最优变异函数模型的确定;利用变异函数变程确定最佳勘查工程间距的实例分析;利用地质统计学方法进行矿体边界和资源量类别的自动化划分以及资源量估算。     

Orthonormal residuals in geostatistics: Model criticism and parameter estimation

In linear geostatistics, models for the mean function (drift) and the variogram or generalized covariance function are selected on the basis of the modeler's understanding of the phenomenon studied as well as data. One can seldom be assured that the most appropriate model has been selected; however, analysis of residuals is helpful in diagnosing whether some important characteristic of the data has been neglected and, ultimately, in providing a reasonable degree of assurance that the selected model is consistent with the available information. The orthonormal residuals presented in this work are kriging errors constructed so that, when the correct model is used, they are uncorrelated and have zero mean and unit variance. It is suggested that testing of orthonormal residuals is a practical way for evaluating the agreement of the model with the data and for diagnosing model deficiencies. Their advantages over the usually employed standardized residuals are discussed. A set of tests are presented. Orthonormal residuals can also be useful in the estimation of the covariance (or variogram) parameters for a model that is considered correct.     

The combination of sampling and kriging in the regional estimation of coal resources

Two-dimensional systematic sampling of small plots followed by the kriging of those plots may be employed to obtain regional estimates of coal resources and measures of the accuracy of the estimates. The use of sampling makes large savings in computation possible. Two case studies involving the estimation of coal tonnage are discussed.     

云南大红山铁矿床三维数学模型探讨

为实现矿山的生产动态管理与矿床的三维可视化，应用地质统计学的理论和方法，尤其是应用能够反映区域化变量特征的变异函数，借助计算机和Micromine软件包，建立了大红山铁矿床的三维矿床数学模型。结果表明，该模型可以使矿山很清楚地了解矿体的形态、产状和化学组分分布特征等。此外，通过Kriging估值计算的储量与用传统方法计算的储量很接近。     

两种方法在地下水位估值中的应用

对于许多区域水资源问题,用数值方法进行潜水水流模拟时,需要给出每个节点上地下水位值.本文首先简单介绍了趋势面方法,然后着重阐述了泛克里格方法的基本原理及它们在地下水位估值中的应用,通过比较两种方法的计算结果可以得出泛克里格方法是进行地下水位估值的空间最优估计方法.