考虑权值非负约束的泛克立格算法

泛克立格法是地质统计学的一种重要估值方法,当区域化变量在空间变异几何域内非平稳时常用泛克立格法来估值。然而,用泛克立格方程组计算估计权值时由于未对权的符号作任何限制,从而使得计算出的权值经常出现负权现象,而负权的存在有许多弊端,所以在许多应用中有必要对权值作非负要求。本文基于线性规划方法提出了一种考虑权值非负约束的泛克立格算法,该算法既考虑到了权值的非负约束条件,又利用了线性规划方法求解简便快捷的优点。     

Notes on the robustness of the kriging system

The robustness of the kriging system with respect to uncertainty of the theoretical variogram is investigated. Inequalities for possible changes of the kriging estimator and the estimation variance are derived. Results of a numerical study show that changes of kriging weights can be predicted partly with the help of the maximal kriging weight.     

Is Lognormal Kriging Suitable for Local Estimation?

Lognormal kriging was developed early in geostatistics to take account of the often seen skewed distribution of the experimental mining data. Intuitively, taking the distribution of the data into account should lead to a better local estimate than that which would have been obtained when it is ignored. In practice however, the results obtained are sometimes disappointing. This paper tries to explain why this is so from the behavior of the lognormal kriging estimator. The estimator is shown to respect certain unbiasedness properties when considering the whole working field using the regression curve and its confidence interval for both simple or ordinary kriging. When examined locally, however, the estimator presents a behavior that is neither expected nor intuitive. These results lead to the question: is the theoretically correct lognormal kriging estimator suited to the practical problem of local estimation?     

Kriging in terms of projections

In the last few years, an increasing number of practical studies using so-called kriging estimation procedures have been published. Various terms, such as universal kriging, lognormal kriging, ordinary kriging, etc., are used to define different estimation procedures, leaving a certain confusion about what kriging really is. The object of this paper is to show what is the common backbone of all these estimation procedures, thus justifying the common name of kriging procedures. The word kriging (in French krigeage) is a concise and convenient term to designate the classical procedure of selecting, within agiven class of possible estimators, the estimator with a minimum estimation variance (i.e., the estimator which leads to a minimum variance of the resulting estimation error). This estimation variance can be seen as a squared distance between the unknown value and its estimator; the process of minimization of this distance can then be seen as the projection of the unknown value onto the space within which the search for an estimator is carried out.     

Kriging in a finite domain

Adopting a random function model {Z(u),u study areaA} and using the normal equations (kriging) for estimation amounts to assume that the study areaA is embedded within a infinite domain. At first glance, this assumption has no inherent limitations since all locations outsideA are of no interest and simply not considered. However, there is an interesting and practically important consequence that is reflected in the kriging weights assigned to data contiguously aligned along finite strings; the weights assigned to the end points of a string are large since the end points inform the infinite half-space beyond the string. These large weights are inappropriate when the finite string has been created by either stratigraphic/geological limits or a finite search neighborhood. This problem will be demonstrated with numerical examples and some partial solutions will be proposed.     

Kriging Regionalized Positive Variables Revisited: Sample Space and Scale Considerations

Frequently, regionalized positive variables are treated by preliminarily applying a logarithm, and kriging estimates are back-transformed using classical formulae for the expectation of a lognormal random variable. This practice has several problems (lack of robustness, non-optimal confidence intervals, etc.), particularly when estimating block averages. Therefore, many practitioners take exponentials of the kriging estimates, although the final estimations are deemed as non-optimal. Another approach arises when the nature of the sample space and the scale of the data are considered. Since these concepts can be suitably captured by an Euclidean space structure, we may define an optimal kriging estimator for positive variables, with all properties analogous to those of linear geostatistical techniques, even for the estimation of block averages. In this particular case, no assumption on preservation of lognormality is needed. From a practical point of view, the proposed method coincides with the median estimator and offers theoretical ground to this extended practice. Thus, existing software and routines remain fully applicable.     

Comparative performance of indicator algorithms for modeling conditional probability distribution functions

This paper compares the performance of four algorithms (full indicator cokriging. adjacent cutoffs indicator cokriging, multiple indicator kriging, median indicator kriging) for modeling conditional cumulative distribution functions (ccdf).The latter three algorithms are approximations to the theoretically better full indicator cokriging in the sense that they disregard cross-covariances between some indicator variables or they consider that all covariances are proportional to the same function. Comparative performance is assessed using a reference soil data set that includes 2649 locations at which both topsoil copper and cobalt were measured. For all practical purposes, indicator cokriging does not perform better than the other simpler algorithms which involve less variogram modeling effort and smaller computational cost. Furthermore, the number of order relation deviations is found to be higher for cokriging algorithms, especially when constraints on the kriging weights are applied.     

An Alternative Measure of the Reliability of Ordinary Kriging Estimates

This paper presents an interpolation variance as an alternative to the measure of the reliability of ordinary kriging estimates. Contrary to the traditional kriging variance, the interpolation variance is data-values dependent, variogram dependent, and a measure of local accuracy. Natural phenomena are not homogeneous; therefore, local variability as expressed through data values must be recognized for a correct assessment of uncertainty. The interpolation variance is simply the weighted average of the squared differences between data values and the retained estimate. Ordinary kriging or simple kriging variances are the expected values of interpolation variances; therefore, these traditional homoscedastic estimation variances cannot properly measure local data dispersion. More precisely, the interpolation variance is an estimate of the local conditional variance, when the ordinary kriging weights are interpreted as conditional probabilities associated to the n neighboring data. This interpretation is valid if, and only if, all ordinary kriging weights are positive or constrained to be such. Extensive tests illustrate that the interpolation variance is a useful alternative to the traditional kriging variance.     

用线性规划法求解克立格价值权系数的研究

基于线性规划原理，针对各种克立格法提出了相应的能考虑到权值非负约束的求解权系数的线性规划方法，用该方法求解估值权系数具有以下优点：（1）与克立格方程法相比，可考虑到估值权系数的非负约束条件；（2）与二次规划法相比，不仅计算原理比较简单，而且还可大大减少计算工作量，具有实用价值。     

Kriging in a global neighborhood

The kriging estimator is usually computed in a moving neighborhood; only the data near the point to be estimated are used. This moving neighborhood approach creates discontinuities in mapping applications. An alternative approach is presented here, whereby all points are estimated using all the available data. To solve the resulting large linear system the kriging estimator is expressed in terms of the inverse of the covariance matrix. The covariance matrix has the advantage of being positive definite and the size of system which can be solved without encountering numerical instability is substantially increased. Because the kriging matrix does not change, the estimator can be written in terms of scalar products, thus avoiding the more time-consuming matrix multiplications of the standard approach. In the particular case of a covariance which is zero for distances greater than a fixed value (the range), the resulting banded structure of the covariance matrix is shown to lead to substantial computational savings in both run time and storage space. In this case the calculation time for the kriging variance is also substantially reduced. The present method is extended to the nonstationary case.     

A Simple and Robust Lognormal Estimator

In an open pit mine, the selection of blocks for mill feed necessitates the use of a conditionally unbiased estimator not only to maximize profits, but also to predict precisely the grades at the mill. Estimation of blocks usually is done using a series of blasthole assays on a regular grid. In many instances, the blasthole grades show a lognormal-like distribution. This study examines an estimator based on the hypothesis of bilognormality between the true block grade and the estimate obtained using the blastholes. The properties of the estimator are established and the estimator is proven to be conditionally unbiased. It is almost as precise as the lognormal kriging estimator when the points are multilognormal. However, it is more precise than lognormal krigings when only univariate lognormality is present or when the distribution is not exactly lognormal. The estimator also is shown to be robust to errors in the specifications of the variogram model or of the expectation of Z. Contrary to lognormal krigings, the estimator does only a slight correction to the original estimate obtained using the blastholes assays.     

The integral of the semivariogram: A powerful method for adjusting the semivariogram in geostatistics

A good fining of the structural junction that describes the variability of a spatial phenomenon is an essential stage in the building of an accurate estimator by kriging. The technique of the integral of the semivariogram (ISV) makes it possible to find this structural function while overcoming the problem of grouping together the pairs of experimental points into classes of distances when the data are not sampled on a regular grid. The ISV is particularly useful when the dispersion of the values of the classical Semivariogram (SV) makes it difficult to fit a model. Since the ISV is composed of a large number of values, it is more continuous than a SV and therefore easier to fit analytically. In fact, when the general shape of the SV is known, the ISV method proves its worth in finding the parameters that best fit a given variogram model. The analytical models of ISV which will be used, are the integral expressions of the traditional analytical SV. In this paper and on the basis of hydrogeological examples, we propose a method to adjust all the parameters of each model. The first derivative of a filled ISV, used in the kriging equations, appears to be systematically the best SV for a cross-validation on the data. This is why we think that the ISV technique should be used when the strong spatial variability of a parameter spreads out the values of the experimental SV.     

Disjunctive Kriging with Hard and Imprecise Data

This paper presents a methodology for assessing local probability distributions by disjunctive kriging when the available data set contains some imprecise measurements, like noisy or soft information or interval constraints. The basic idea consists in replacing the set of imprecise data by a set of pseudohard data simulated from their posterior distribution; an iterative algorithm based on the Gibbs sampler is proposed to achieve such a simulation step. The whole procedure is repeated many times and the final result is the average of the disjunctive kriging estimates computed from each simulated data set. Being data-independent, the kriging weights need to be calculated only once, which enables fast computing. The simulation procedure requires encoding each datum as a pre-posterior distribution and assuming a Markov property to allow the updating of pre-posterior distributions into posterior ones. Although it suffers some imperfections, disjunctive kriging turns out to be a much more flexible approach than conditional expectation, because of the vast class of models that allows its computation, namely isofactorial models.     

一种考虑权值非负约束的克立格算法

基于线性规划方法提出了一种考虑权值非负约束的克立格算法,该算法既利用了线性规划求解简便、快捷的优点,又克服了其它正克立格法计算工作量大以及受主观因素影响的缺点。     

On Visualization for Assessing Kriging Outcomes

Extant opinion about kriging is that all weights should be positive. Visualizations rendered by converting kriged grids to digital images are presented to show that negative weights may be beneficial to some spatial problems. In particular, variogram models with zero-valued nuggets, already well known to minimize smoothing through kriging, result in a visual resolution substantially superior to that from kriging with a variogram model having a nonzero nugget value in application to satellite acquired data. Negative weights are more likely when using variogram models with zero-valued nuggets, but resultant visualizations often show a smoother transition between extreme data values. This is true even when a variogram model having a nugget value of zero is not optimum with respect to mean square error, as is demonstrated using a nitrate data set. An analogy to digital image processing is used to suggest that the influence of negative weights in kriging is similar to a high-boost kernel.     

Correcting the Smoothing Effect of Estimators: A Spectral Postprocessor

The postprocessing algorithm introduced by Yao for imposing the spectral amplitudes of a target covariance model is shown to be efficient in correcting the smoothing effect of estimation maps, whether obtained by kriging or any other interpolation technique. As opposed to stochastic simulation, Yao's algorithm yields a unique map starting from an original, typically smooth, estimation map. Most importantly it is shown that reproduction of a covariance/semivariogram model (global accuracy) is necessarily obtained at the cost of local accuracy reduction and increase in conditional bias. When working on one location at a time, kriging remains the most accurate (in the least squared error sense) estimator. However, kriging estimates should only be listed, not mapped, since they do not reflect the correct (target) spatial autocorrelation. This mismatch in spatial autocorrelation can be corrected via stochastic simulation, or can be imposed a posteriori via Yao's algorithm.     

Interval-valued random functions and the kriging of intervals

Estimation procedures using data that include some values known to lie within certain intervals are usually regarded as problems of constrained optimization. A different approach is used here. Intervals are treated as elements of a positive cone, obeying the arithmetic of interval analysis, and positive interval-valued random functions are discussed. A kriging formalism for interval-valued data is developed. It provides estimates that are themselves intervals. In this context, the condition that kriging weights be positive is seen to arise in a natural way. A numerical example is given, and the extension to universal kriging is sketched.     

On the equivalence of kriging and maximum entropy estimators

This study compares kriging and maximum entropy estimators for spatial estimation and monitoring network design. For second-order stationary random fields (a subset of Gaussian fields) the estimators and their associated interpolation error variances are identical. Simple lognormal kriging differs from the lognormal maximum entropy estimator, however, in both mathematical formulation and estimation error variances. Two numerical examples are described that compare the two estimators. Simple lognormal kriging yields systematically higher estimates and smoother interpolation surfaces compared to those produced by the lognormal maximum entropy estimator. The second empirical comparison applies kriging and entropy-based models to the problem of optimizing groundwater monitoring network design, using six alternative objective functions. The maximum entropy-based sampling design approach is shown to be the more computationally efficient of the two.     

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。