Abstract
Recently, Spock and Pilz [38], demonstrated that the spatial sampling design problem for the Bayesian linear kriging predictor can be transformed to an equivalent experimental design problem for a linear regression model with stochastic regression coefficients and uncorrelated errors. The stochastic regression coefficients derive from the polar spectral approximation of the residual process. Thus, standard optimal convex experimental design theory can be used to calculate optimal spatial sampling designs. The design functionals considered in Spock and Pilz [38] did not take into account the fact that kriging is actually a plug-in predictor which uses the estimated covariance function. The resulting optimal designs were close to space-filling configurations, because the design criterion did not consider the uncertainty of the covariance function. In this paper we also assume that the covariance function is estimated, e.g., by restricted maximum likelihood (REML). We then develop a design criterion that fully takes account of the covariance uncertainty. The resulting designs are less regular and space- filling compared to those ignoring covariance uncertainty. The new designs, however, also require some closely spaced samples in order to improve the estimate of the covariance function. We also relax the assumption of Gaussian observations and assume that the data is transformed to Gaussianity by means of the Box-Cox transformation. The resulting prediction method is known as trans-Gaussian kriging. We apply the Smith and Zhu [37] approach to this kriging method and show that resulting optimal designs also depend on the available data. We illustrate our results with a data set of monthly rainfall measurements from Upper Austria.
Highlights
A similar approach as in Section 7.3 applies to stationary random fields Z(x, i) that are correlated in space and time and whose Gaussian correlation function can be assumed to be separable, i.e., C( x, i) = CX( x) CT( i), if we assume the Gaussian time-correlation function to be fixed and not estimated, and only the Gaussian spatial covariance function to be estimated by restricted maximum likelihood (REML)
Developments The preceding sections demonstrate that design functionals have a convenient and mathematically tractable structure and there is no need for stochastic search algorithms like simulated annealing in order to optimize them
Designs with the Smith and Zhu criterion differ from designs with the trace functional: Because the Smith and Zhu design criterion takes the uncertainty and estimation of the covariance function into account, resulting designs with this criterion must have sampling locations very close to each other as well as spacefilling locations
Summary
(Brown et al, 1994; Fuentes et al, 2007) consider the covariance function to be non-stationary and deal with an entropy based design criterion according to which the determinant of the covariance matrix between locations to be added to the design must be maximized Both make use of simulated annealing algorithms to find optimal designs obeying their criteria. In treed Gaussian random fields the area X of investigation is partitioned by means of classification trees into rectangular sub-areas with sides parallel to the coordinate axes This software is especially useful for the design of computer simulation experiments, where parameters guiding the computer simulation output are identified as spatial coordinates. Since for the practitioner there is a strong need for spatial sampling design and almost no software is freely available this was an impetus for the first author of this paper to write his own toolbox for spatial sampling design, see Section 7
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