Kriging variance

The quality of a theoretical semivariogram model can be assessed by means of point kriging and cross-validation [MYERS, 1991]. Each point of the data set is deleted one after another and then newly estimated according to Eqs. 4-33 to 4-35 by means of its neighbors. Additionally, the kriging variance is calculated for each estimated point (Eq. 4-36). 

These values are squared and divided by the corresponding kriging variance ... 

Kiiging is an exact estimator, that is, the kriging estimator at a data location will be the data value. The minimized error variance or kriging variance can be calculated for all estimated locations ... 

Kriging estimates are smooth. The kriging variance is a quantitative measme of the smoothness of the kriging estimates. There is no smoothing when kriging at a data location, a =0. There is complete smoothness when kriging with data far from the location being estimated the es-... 

Go to a location u (chosen randomly from the set of locations that have not been simulated yet) and perform kriging to obtain a kiiged estimate and the corresponding kriging variance. 

Geostatistical techniques, such as variography and kriging, have been recently introduced into the environmental sciences (O Although kriging allows mapping of the pollution plume with qualification of the estimation variance, it falls short of providing a truly risk-qualified estimate of the spatial distribution of pollutants. 

Probabilistic techniques of estimation provide some Insights Into the potential error of estimation. In the case of krlglng, the variable pCic) spread over the site A is first elevated to the status of a random function PC c). An estimator P (2c) is then built to minimize the estimation variance E [P(2c)-P (2c) ], defined as the expected squared error ( ). The krlglng process not only provides the estimated values pCiyc) from which a kriged map can be produced, but also the corresponding minimum estimation variances 0 (39 ) ... 

Geostatistical processing is one of the most useful and practical methods for evaluation and estimation of a resource. Its BLUE (Best Liner Unbiased Estimator) Kriging not only can indicate the distribution and amount of ore in a resource, but also, based on variance and error of Kriging can identify some parts of ore body, that have lack of data and need more exploration. For a routine Geostatistical processing some issues should be considered ... 

To use what is termed universal kriging, it is assumed that Z(2 ) is an intrinsic random function of order k. But the problem of identifying the drift and the semi-variogram when they are both unknown is still present. However, Matheron (11) defined a family of functions called the generalized covariance, K(h). and the variance of the generalized increment of order k can be defined in terms of K(h ). That is. 

The semivariogram as a variance function can also be used to estimate the value and the variance for new points not sampled in the investigated area. The method applied for this purpose is termed kriging. Kriging is a special regression method for interpolation of spatially or temporally correlated data with minimization of variance. The normal distribution of the data is an important condition. If the original data are not normally distributed, which is often the case for trace components in environmental compartments, the logarithm of the data or otherwise transformed data have to be applied to obtain a normal distribution of the data (see also Section 9.4). 

The advantage of kriging is that it furnishes not only an estimate of the unsampled point, but also an estimate of the variance at this location. For a sampling location the exact value is estimated with the variance being zero. If the kriging method is applied, an exact and undistorted interpolation is possible. 

We view the real or the simulated system as a black box that transforms inputs into outputs. Experiments with such a system are often analyzed through an approximating regression or analysis of variance model. Other types of approximating models include those for Kriging, neural nets, radial basis functions, and various types of splines. We call such approximating models metamodels other names include auxiliary models, emulators, and response surfaces. The simulation itself is a model of some real-world system. The goal is to build a parsimonious metamodel that describes the input-output relationship in simple terms. 

In the kriging process selected data points are used for spatial weighting, which satisfy the outcome of the variogram analysis. Spatial estimates or interpolation of data is performed in such a manner that the estimated variances become minimal. Equation 12.3 demonstrates the formulation of ordinary kriging. 

Li and Der Kiureghian (1993) introduced a spectral decomposition of the nodal covariance matrix. They showed that the maximum error of the KL expansion is not always smaller than the error of Kriging for a given number of retained terms. The point-wise variance error estimator of the KL expansion for a given order of truncation is smaller than the error of Kriging in the interior of the discretization domain but larger at the boundaries. Note however that the... 

Recall that C(h) = a - y(h) therefore, knowledge of the variogram model permits calculation of all needed covariance terms. The left-hand side contains all of the iirforma-tion related to redundancy in the data, and the right-hand side contains all of the urformation related to closeness of the data to the location being estimated. Kriging is the best estimator in terms of minimum error variance. 

Draw a random residual (u) that follows a normal distribution with mean of 0.0 and a variance of (u). Add the kriging estimate and residual to get a simulated value. The independent residual (u) is drawn with classical Monte Carlo simnlatioa... 

Ordinary kriging is a best linear unbiased estimate of the parameter. It is linear because its estimates are weighted linear combinations of the available data, unbiased because it tries to have the mean error equal to zero, and best because it aims at minimizing the variance of the error (7). The kriging estimator, Z is described as. 

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