Design and Analysis for Modeling and Predicting Spatial Contamination

Sampling and prediction strategies relevant at the planning stage of the cleanup of environmental hazards are discussed. Sampling designs and models are compared using an extensive set of data on dioxin contamination at Piazza Road, Missouri. To meet the assumptions of the statistical model, such data are often transformed by taking logarithms. Predicted values may be required on the untransformed scale, however, and several predictors are also compared. Fairly small designs turn out to be sufficient for model fitting and for predicting. For fitting, taking replicates ensures a positive measurement error variance and smooths the predictor. This is strongly advised for standard predictors. Alternatively, we propose a predictor linear in the untransformed data, with coefficients derived from a model fitted to the logarithms of the data. It performs well on the Piazza Road data, even with no replication.     

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.     

附有等式约束病态模型正则化解的单位权中误差无偏估计

利用平差参数间合理的等式约束虽能提升病态模型解的精度,但其本质仍是通过引入正则化参数来改善模型的病态性,由于改变了观测方程的结构,所得的估值残差及单位权中误差均有偏。针对这一不足,在病态模型正则化解的无偏单位权方差估计式基础上引入等式约束条件,根据约束正则化解的残差二次型期望公式,导出约束正则化解的无偏单位权中误差估计式,并用数值算例和病态测边网算例验证其正确性。结果表明,本文公式所估的单位权中误差精度优于传统公式所估结果。     

关于奇异线性模型最好线性无偏估计的计算

讨论了奇异线性模型最好线性无偏估计(BLUE)的计算问题。利用分块求逆的技术，给出BLUE的一种新的表达式，其计算量比直接利用C．R．Rao所给公式的小。对新表达式的计算，给出半正定矩阵广义逆的分块求逆算法及消去变换算法。     

Covariance function diagnostics for spatial linear models

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.     

Geostatistical interpolation of chemical concentration

Measurements of contaminant concentration at a hazardous waste site typically vary over many orders of magnitude and have highly skewed distributions. This work presents a practical methodology for the estimation of solute concentration contour maps and volume averages (needed for mass calculations) from data obtained from the analysis of water and soil samples. The methodology, which is an extension of linear geostatistics, produces a point estimate, i.e., a representative value, as well as a confidence interval, which contains the true value with a given probability. The approach uses a parsimonious model that accounts for the skewness by adding only one parameter to those used in linear geostatistics (variograms or generalized covariances). The resulting nonlinear kriging method is not substantially more difficult to use than linear geostatistics. The methodology is most appropriate when concentration measurements are available on a reasonably dense grid and no additional information (based on modeling flow and transport) can be used. We present and illustrate through an application, a practical approach to estimate all the parameters needed and to select and test the model.     

Entropy principles in the prediction of water quality values at discontinued monitoring stations

A new methodology for predicting water quality values at discontinued water quality monitoring stations is proposed. The method is based upon the Principle of Maximum Entropy (POME) and provides unbiased predictions of water quality levels at upstream tributaries and on the mainstem of a river given observed changes in the distribution of the same water quality parameter at a downstream location. Changes in the values of water quality parameters which are known a priori to have occurred upstream, but which are not sufficiently large to account for all the observed change in the same water quality parameter at the downstream location are able to be incorporated in the method through the introduction of a new term in the basic entropy expression. Application of the procedure to water quality monitoring on the Mackenzie River in Queensland, Australia indicates the method has considerable potential for prediction of water quality at discontinued stations. The method also has potential for identifying the location of causes of observed changes in water quality at a downstream station.     

稳健最优不变二次无偏估计

导出了稳健最优不变二次无偏估计、稳健最小范数二次无偏估计、稳健Helmert估计,并说明了最优不变二次无偏估计、最小范数二次无偏估计以及Helmert估计等均是稳健最优不变二次无偏估计的特例。     

A STATISTICAL PROCEDURE TO EVALUATE CLEAN-UP STANDARDS

The object of this paper is to develop a suitable statistical procedure to evaluate clean-up standards athazardous waste sites.Under the assumptions that contaminant masses at a site follow a gammadistribution and that the data from the pre-remediation baseline sample as well as from the interim orfinal sample taken after a certain period of operation are both distributed as gamma with the same shapeparameter but different scale parameters,we derive a uniformly most powerful unbiased test of thehypothesis that a specified percentage of contaminant mass has been reduced.A large-sampleapproximation of the exact test procedure and a comparison with the likelihood ratio test are provided.     

Spatial interpolation of marine environment data using P-MSN

ABSTRACT

When a marine study area is large, the environmental variables often present spatially stratified non-homogeneity, violating the spatial second-order stationary assumption. The stratified non-homogeneous surface can be divided into several stationary strata with different means or variances, but still with close relationships between neighboring strata. To give the best linear-unbiased estimator for those environmental variables, an interpolated version of the mean of the surface with stratified non-homogeneity (MSN) method called point mean of the surface with stratified non-homogeneity (P-MSN) was derived. P-MSN distinguishes the spatial mean and variogram in different strata and borrows information from neighboring strata to improve the interpolation precision near the strata boundary. This paper also introduces the implementation of this method, and its performance is demonstrated in two case studies, one using ocean color remote sensing data, and the other using marine environment monitoring data. The predictions of P-MSN were compared with ordinary kriging, stratified kriging, kriging with an external drift, and empirical Bayesian kriging, the most frequently used methods that can handle some extent of spatial non-homogeneity. The results illustrated that for spatially stratified non-homogeneous environmental variables, P-MSN outperforms other methods by simultaneously improving interpolation precision and avoiding artificially abrupt changes along the strata boundaries.