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.     

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.     

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.     

Generalized covariance functions in estimation

I discuss the role of generalized covariance functions in best linear unbiased estimation and methods for their selection. It is shown that the experimental variogram (or covariance function) of the detrended data can be used to obtain a preliminary estimate of the generalized covariance function without iterations and I discuss the advantages of other parameter estimation methods.     

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.     

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.     

沉降预测中的Asaoka法应用研究

系统介绍了Asaoka法的基础知识，并基于抛物插值法和直线最小二乘拟合法，编制了该法预测地基最终沉降的程序。结合工程实例，详细分析了时间间隔△t对Asaoka法的预测结果及相关指标的影响，给出了出的建议取值范围，同时对多级荷载下的某路堤沉降进行了预测。计算结果表明：程序中采用的抛物插值法等是可行的，且Asaoka法自身的优点决定了其具有较高的应用价值。     

On the condition number of covariance matrices in kriging, estimation, and simulation of random fields

The numerical stability of linear systems arising in kriging, estimation, and simulation of random fields, is studied analytically and numerically. In the state-space formulation of kriging, as developed here, the stability of the kriging system depends on the condition number of the prior, stationary covariance matrix. The same is true for conditional random field generation by the superposition method, which is based on kriging, and the multivariate Gaussian method, which requires factoring a covariance matrix. A large condition number corresponds to an ill-conditioned, numerically unstable system. In the case of stationary covariance matrices and uniform grids, as occurs in kriging of uniformly sampled data, the degree of ill-conditioning generally increases indefinitely with sampling density and, to a limit, with domain size. The precise behavior is, however, highly sensitive to the underlying covariance model. Detailed analytical and numerical results are given for five one-dimensional covariance models: (1) hole-exponential, (2) exponential, (3) linear-exponential, (4) hole-Gaussian, and (5) Gaussian. This list reflects an approximate ranking of the models, from best to worst conditioned. The methods developed in this work can be used to analyze other covariance models. Examples of such representative analyses, conducted in this work, include the spherical and periodic hole-effect (hole-sinusoidal) covariance models. The effect of small-scale variability (nugget) is addressed and extensions to irregular sampling schemes and higher dimensional spaces are discussed.     

空心包体应变计地应力分量计算方法及应用

为进一步改进和完善空心包体应变计地应力计算方法,基于地应力现场实测过程中空心包体应变计的安装方式,采用线性参数的最小二乘拟合方法对地应力分量计算公式进行了推导,得出了6个地应力分量的改进算法及其标准误差的计算公式。采用实现完全温度补偿并考虑岩体非线性的地应力测量技术,对弓长岭井下矿-160、-220、-280 m 3个水平3个测点进行了地应力现场实测,得到了3组孔壁应变数据。分别使用常规算法和改进算法对3个测点的地应力分量进行计算并分析其标准误差。结果表明：采用改进算法计算得到的地应力分量其标准误差普遍小于常规算法计算地应力分量产生的标准误差,说明改进算法比常规算法具有更高的可靠度。在此基础上,对弓长岭井下矿地应力赋存规律进行了研究。结果显示：矿区地应力场以水平构造应力为主导,最大水平主应力的走向总体上为南东东-北西西方向,最大水平主应力、最小水平主应力和垂直主应力均随深度呈增长关系。     

Inner product matrices, kriging, and nonparametric estimation of variogram

Two important problems in the practical implementation of kriging are: (1) estimation of the variogram, and (2) estimation of the prediction error. In this paper, a nonparametric estimator of the variogram to circumvent the problem of the precise choice of a variogram model is proposed. Using orthogonal decomposition of the kriging predictor and the prediction error, a method for selecting, what may be considered, a statistical neighborhood is suggested. The prediction error estimates based on this scheme, in fact, reflects the true prediction error, thus leading to proper coverage for the corresponding prediction interval. By simulations and a reanalysis of published data, it is shown that the proposals made in this paper are useful in practice.     

Estimation of linear combinations and co-kriging

Utilizing the matrix formulation of co-kriging developed previously by the author, the relationship between direct kriging of linear combinations and linear combinations of co-kriged variables is developed. Conditions for equality of the estimators and the kriging variances are examined. By presenting the problem in the context of Hilbert spaces the general relationship is clarified.     

Comparison of structural and least-squares lines for estimating geologic relations

Two different goals in fitting straight lines to data are to estimate a true linear relation (physical law) and to predict values of the dependent variable with the smallest possible error. Regarding the first goal, a Monte Carlo study indicated that the structural-analysis (SA) method of fitting straight lines to data is superior to the ordinary least-squares (OLS) method for estimating true straight-line relations. Number of data points, slope and intercept of the true relation, and variances of the errors associated with the independent (X) and dependent (Y) variables influence the degree of agreement. For example, differences between the two line-fitting methods decrease as error in X becomes small relative to error in Y. Regarding the second goal—predicting the dependent variable—OLS is better than SA. Again, the difference diminishes as X takes on less error relative to Y. With respect to estimation of slope and intercept and prediction of Y, agreement between Monte Carlo results and large-sample theory was very good for sample sizes of 100, and fair to good for sample sizes of 20. The procedures and error measures are illustrated with two geologic examples.     

新疆阿克苏河上游高寒草甸蒸散发观测与估算

蒸散发是水循环的关键环节, 是水量平衡的重要组成部分. 由于在高寒山区进行长期野外观测的难度较大, 导致对区域实际蒸散发的认识不清, 从而无法明确区域水资源分配与不同植被的生态水文功能. 在天山山区, 高寒草甸占其总面积近15%, 其对降水的调节作用巨大, 但目前高寒草甸的实际蒸散发量多用潜在蒸散发进行推算, 缺少实际观测数据. 2012年10月-2013年9月, 利用3个小型蒸渗仪观测了阿克苏河上游科其喀尔冰川综合考察站附近山区的高寒草甸的实际蒸散量, 并尝试利用最小二乘支持向量机(LS-SVM)估算实际蒸散发. 结果表明:研究区高寒草甸全年内实测蒸散量511.3 mm, 日均蒸散量为1.4 mm·d^-1; 在不同时期, 蒸散量变化剧烈, 冻结期、生长前期、生长期和生长后期的蒸散量分别为53.9、41.0、363.8和52.6 mm, 分别占全年蒸散量的10.5%、8.0%、71.2%和10.3%. 最小二乘支持向量机对实际蒸散发的估算精度较高, 对观测资料相对缺乏的高寒山区来说, 不失为一种较好的估算蒸散发方法.     

Method of Data Reduction and Uncertainty Estimation for Platinum-Group Element Data Using Inductively Coupled Plasma-Mass Spectrometry

A data reduction method is described for determining platinum-group element (PGE) abundances by inductively coupled plasma-mass spectrometry (ICP-MS) using external calibration or the method of standard addition. Gravimetric measurement of volumes, the analysis of reference materials and the use of procedural blanks were all used to minimise systematic errors. Internal standards were used to correct for instrument drift. A linear least squares regression model was used to calculate concentrations from drift-corrected counts per second (cps). Furthermore, mathematical manipulations also contribute to the uncertainty estimates of a procedure. Typical uncertainty estimate calculations for ICP-MS data manipulations involve: (1) Carrying standard deviations from the raw cps through the data reduction or (2) calculating a standard deviation from multiple final concentration calculations. It is demonstrated that method 2 may underestimate the uncertainty estimate of the calculated data. Methods 1 and 2 do not typically include an uncertainty estimate component from a regression model. As such models contribute to the uncertainty estimates affecting the calculated data, an uncertainty estimate component from the regression must be included in any final error calculations. Confidence intervals are used to account for uncertainty estimates from the regression model. These confidence intervals are simpler to calculate than uncertainty estimates from method 1, for example. The data reduction and uncertainty estimation method described here addresses problems of reporting PGE data from an article in the literature and addresses both precision and accuracy. The method can be applied to any analytical technique where drift corrections or regression models are used.     

Universal model forms for predicting the shear modulus and material damping of soils

Numerous model forms have been used to predict the dynamic behaviours of soils, mainly the shear modulus and material damping. These models are used to represent normalised shear modulus reduction and material damping curves. The majority of the existing models are empirical and were proposed for specific soil types, strain ranges, etc. Some are limited to the data used in fitting, but fail to provide a good fit to other sets of data. The available model functions are not universal, which means that a model developed for one soil type may not be applicable for use with other soil types. In this paper, two universal mathematical models were proposed to predict the normalised shear modulus reduction and material damping curves. The mathematical model forms are sufficiently flexible to be widely used with different soil types. The models were evaluated using verification data of eight different soil types, i.e., data that were not used to fit the model parameters. The results indicate that the proposed model forms can accurately model the dynamic soil properties within the typical earthquake range. The advantage of developing the two universal models are two-fold, first the forms are applicable for sand, clay, and fibre-soil composite and second with further testing of soils, the model coefficients could ultimately be used to further understand the physical processes in soil behaviour, especially damping.     

基于优化经验模态分解和聚类分析的滑坡位移智能预测研究

针对三峡库区"阶跃式"滑坡的变形特征,提出了一种新的滑坡位移预测方法。以白水河滑坡ZG118和XD-01监测点位移数据为例,采用基于软筛分停止准则的经验模态分解(SSSC-EMD)将累计位移-时间曲线和影响因子时间序列自适应地分解为多个固有模态函数(IMF),并采用K均值(K-Means)聚类法对其进行聚类累加,得到有物理含义的位移分量(趋势性位移、周期性位移以及随机性位移)和影响因子分量(高频影响因子和低频影响因子)。使用最小二乘法对趋势性位移进行拟合预测;采用果蝇优化-最小二乘支持向量机(FOA-LSSVM)模型对周期性位移和随机性位移进行预测。将各位移分量预测值进行叠加处理,实现滑坡累计位移的预测。研究结果表明,所提出的(SSSC-EMD)-K-Means-(FOA-LSSVM)模型能够预测"阶跃式"滑坡的位移变化规律,且预测精度高于传统的支持向量机回归(SVR)、最小二乘支持向量机(LSSVM)模型;并通过改变训练集长度,进行单因素分析,发现其与预测精度之间呈正相关关系。     

Comparison of Two Sample Preparation Methods for X‐Ray Fluorescence Spectrometry in the Determination of Ni and Cr

Pressed powder pellets and fused beads or glass disks are routinely used in X‐ray fluorescence spectrometry for the determination of major and trace elements, respectively, in geological materials. In order to evaluate the performance of these two sample preparation methods, we determined Ni and Cr concentrations of fourteen RMs from Japan, France and South Africa, and eighty‐five igneous and three sedimentary rock samples from Mexico in both powder pellets and glass beads. We also computed new values of statistical parameters for RMs from an outlier‐based multiple‐test method and compared them with the literature mean and confidence limit values. The results showed that the multiple‐test method provided more reliable central tendency and dispersion parameters for RMs than those obtained previously from the two or three standard deviation method, or from robust methods. The powder pellet and fused bead sample preparation methods provided consistent results for Ni and Cr at concentration levels > 50 μg g^?1 in this application; for lower concentration levels, however, these methods showed somewhat greater differences. For quantitative comparisons, both ordinary and weighted least‐squares linear regression models were used to show that the two sample preparation methods provided generally unbiased results.