Efficient prediction designs for random fields

Müller, Werner G.; Pronzato, Luc; Rendas, João; Waldl, Helmut

doi:10.1002/asmb.2084

Cited by 9 publications

(10 citation statements)

References 39 publications

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“…The term spatial simulated annealing (SSA) finds its first manifestation in the work of Groenigen et al (1999). Trujillo-Ventura and Ellis (1991) and Müller et al (2015) consider multiobjective sampling design optimization. Recently published articles that are relevant to spatial sampling design are (Hu and Wang, 2011;Li and Bardossy, 2011).…”

Section: A Review On Methods and Software For Spatial Designmentioning

confidence: 99%

Incorporating covariance estimation uncertainty in spatial sampling design for prediction with trans-Gaussian random fields

Spöck

Pilz

2015

Front. Environ. Sci.

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Recently, Spöck and Pilz (2010), 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 Spöck and Pilz (2010) 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, 2004) 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.

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Section: A Review On Methods and Software For Spatial Designmentioning

confidence: 99%

Incorporating covariance estimation uncertainty in spatial sampling design for prediction with trans-Gaussian random fields

Spöck

Pilz

2015

Front. Environ. Sci.

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“…(5.20) underestimates the true MSE, and discusses alternative estimators of the true MSE. Jones et al (1998) and Müller et al (2015) also imply that the plug-in estimator underestimates the true variance. Stein (1999) gives asymptotic results for Kriging with ψ.…”

Section: Compute the Generalized Least Squares (Gls) Estimator Of Thementioning

confidence: 99%

Design and Analysis of Simulation Experiments

Kleijnen

2015

International Series in Operations Research &Amp; Management Science

215

138

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This chapter is organized as follows. Section 5.1 introduces Kriging, which is also called Gaussian process (GP) or spatial correlation modeling. Section 5.2 details so-called ordinary Kriging (OK), including the basic Kriging assumptions and formulas assuming deterministic simulation. Section 5.3 discusses parametric bootstrapping and conditional simulation for estimating the variance of the OK predictor. Section 5.4 discusses universal Kriging (UK) in deterministic simulation. Section 5.5 surveys designs for selecting the input combinations that gives input/output data to which Kriging metamodels can be fitted; this section focuses on Latin hypercube sampling (LHS) and customized sequential designs. Section 5.6 presents stochastic Kriging (SK) for random simulations. Section 5.7 discusses bootstrapping with acceptance/rejection for obtaining Kriging predictors that are monotonic functions of their inputs. Section 5.8 discusses sensitivity analysis of Kriging models through functional analysis of variance (FANOVA) using Sobol's indexes. Section 5.9 discusses risk analysis (RA) or uncertainty analysis (UA). Section 5.10 discusses several remaining issues. Section 5.11 summarizes the major conclusions of this chapter, and suggests topics for future research. The chapter ends with Solutions of exercises, and a long list of references.

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“…The assumption of a known covariance function is in most cases unrealistic (Müller, ). Usually, we have to use the same data for estimation of covariance parameters and for spatial prediction, and effective prediction requires good estimates of the second order characteristics (Guttorp and Sampson, (); Müller, Pronzato, Rendas, and Helmut, ()). Recent work on construction of designs that focuses on the goals of efficient spatial prediction in conjunction with parameter estimation includes Zhu (); Zhu and Stein (); Diggle and Lophaven (); Pilz and Spöck (); Zimmerman (); Banerjee, Gelfand, Finley, and Sang (); Bijleveld et al (); Müller et al () and Chipeta et al ().…”

Section: Non‐adaptive Geostatistical Design Strategiesmentioning

confidence: 99%

“…In our simulation study (Section 5), we compare the performance of inhibitory plus close pairs design with some of the optimal designs we have reviewed in Section 1, such as EK designs implemented by Zimmerman () and Müller et al (). These designs minimize the EK criterion:

E K false(scriptX false) = \max_{x \in scriptD} false{Var false[Ŷ false(x false) - Y false(x false) false] + tr false{M_{θ} Var false[\partial Ŷ false(x false) false/ \partial θ false] false} false}

This adds an explicit additive correction term to the normalized classical prediction variance.…”

Section: Empirical Kriging Optimal Designsmentioning

confidence: 99%

“…These methods are much more computationally expensive, and the resulting designs depend on the spatial locations of a set of specified potential sampling points in a more complicated way, than do our proposed ICP ( n , k , δ , ζ ) designs. In our simulation study in Section 5.3, we follow the spatial simulated annealing algorithm outlined in Müller et al ().…”

Section: Empirical Kriging Optimal Designsmentioning

confidence: 99%

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Inhibitory geostatistical designs for spatial prediction taking account of uncertain covariance structure

et al. 2016

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The problem of choosing spatial sampling designs for investigating an unobserved spatial phenomenon scriptS arises in many contexts, for example, in identifying households to select for a prevalence survey to study disease burden and heterogeneity in a study region scriptD. We studied randomized inhibitory spatial sampling designs to address the problem of spatial prediction while taking account of the need to estimate covariance structure. Two specific classes of design are inhibitory designs and inhibitory designs plus close pairs. In an inhibitory design, any pair of sample locations must be separated by at least an inhibition distance δ. In an inhibitory plus close pairs design, n − k sample locations in an inhibitory design with inhibition distance δ are augmented by k locations each positioned close to one of the randomly selected n − k locations in the inhibitory design, uniformly distributed within a disk of radius ζ. We present simulation results for the Matérn class of covariance structures. When the nugget variance is non‐negligible, inhibitory plus close pairs designs demonstrate improved predictive efficiency over designs without close pairs. We illustrate how these findings can be applied to the design of a rolling Malaria Indicator Survey that forms part of an ongoing large‐scale, 5‐year malaria transmission reduction project in Malawi.

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Efficient prediction designs for random fields

Cited by 9 publications

References 39 publications

Incorporating covariance estimation uncertainty in spatial sampling design for prediction with trans-Gaussian random fields

Incorporating covariance estimation uncertainty in spatial sampling design for prediction with trans-Gaussian random fields

Design and Analysis of Simulation Experiments

Inhibitory geostatistical designs for spatial prediction taking account of uncertain covariance structure

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