Optimal design of experiments subject to correlated errors

Pázman, Andrej; Müller, Werner G.

doi:10.1016/s0167-7152(00)00201-7

Cited by 45 publications

(21 citation statements)

References 3 publications

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“…It is based on works of Fedorov and Müller (1988), Fedorov and Hackl (1994) and Fedorov and Müller (2007) who use eigenfunction expansions of the random field under investigation to approximate it by means of a linear regression model with stochastic coefficents and then apply classical experimental design theory to this regression model. Further papers falling into this category, where it is tried to exploit classical experimental design theory for spatial sampling design, are Müller andPazman (1998, 1999), Pazman and Müller (2001), Müller and Pazman (2003) and Müller (2005). The first four papers deserve particular attention because there a new design measure and an approximate information matrix are investigated that has an interpretation as amount of added noise to design locations not considered to be important.…”

Section: Survey Of Model-based Spatial Sampling Designmentioning

confidence: 99%

Spatial sampling design and covariance-robust minimax prediction based on convex design ideas

Spöck

Pilz

2009

Stoch Environ Res Risk Assess

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This paper presents new ideas on sampling design and minimax prediction in a geostatistical model setting. Both presented methodologies are based on regression design ideas. For this reason the appendix of this paper gives an introduction to optimum Bayesian experimental design theory for linear regression models with uncorrelated errors. The presented methodologies and algorithms are then applied to the spatial setting of correlated random fields. To be specific, in Sect. 1 we will approximate an isotropic random field by means of a regression model with a large number of regression functions with random amplitudes, similarly to Fedorov and Flanagan (J Combat Inf Syst Sci: 23, 1997). These authors make use of the Karhunen Loeve approximation of the isotropic random field. We use the so-called polar spectral approximation instead; i.e. we approximate the isotropic random field by means of a regression model with sinecosine-Bessel surface harmonics with random amplitudes and then, in accordance with Fedorov and Flanagan (J Combat Inf Syst Sci: 23, 1997), apply standard Bayesian experimental design algorithms to the resulting Bayesian regression model. Section 2 deals with minimax prediction when the covariance function is known to vary in some set of a priori plausible covariance functions. Using a minimax theorem due to Sion (Pac J Math 8:171-176, 1958) we are able to formulate the minimax problem as being equivalent to an optimum experimental design problem, too. This makes the whole experimental design apparatus available for finding minimax kriging predictors. Furthermore some hints are given, how the approach to spatial sampling design with one a priori fixed covariance function may be extended by means of minimax kriging to a whole set of a priori plausible covariance functions such that the resulting designs are robust. The theoretical developments are illustrated with two examples taken from radiological monitoring and soil science.

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Section: Survey Of Model-based Spatial Sampling Designmentioning

confidence: 99%

Spatial sampling design and covariance-robust minimax prediction based on convex design ideas

Spöck

Pilz

2009

Stoch Environ Res Risk Assess

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“…First of all we may distinguish between design criteria for spatial prediction and for estimation of the covariance function and between combined criteria for both goals. Works falling into the category of criteria for prediction are (Fedorov and Flanagan, 1997;Müller andPazman, 1998, 1999;Pazman and Müller, 2001;Müller, 2005;Brus and Heuvelink, 2007). Criteria for the estimation of the covariance function are considered by Müller and Zimmerman (1999) and Zimmerman (2006).…”

Section: A Review On Methods and Software For Spatial Designmentioning

confidence: 99%

“…We can distinguish between stochastic search algorithms like simulated annealing (Aarts and Korst, 1989) or evolutionary genetic algorithms and deterministic algorithms for optimizing the investigated design criteria. With the exception of the works of Fedorov and Flanagan (1997), Müller andPazman (1998, 1999), Pazman and Müller (2001), Müller (2005), and Spöck (2011) almost all algorithms for spatial sampling design optimization use stochastic search algorithms for finding optimal configurations of sampling locations x 1 , x 2 , . .…”

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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“…A wide class of optimality criteria, such as D-, A-or E-optimality, have been proposed in the literature for independent observations in order to combine the uncertainty of the parameters of interest [27], but only recently they have been applied in the correlated setup [19][20][21][22][28][29][30][31]. For instance, assuming the D-optimality criterion, the aim is to choose the distances in order to maximize…”

Section: Optimal Designs For the Ornstein-uhlenbeck Processmentioning

confidence: 99%

Optimal designs for parameter estimation of the Ornstein–Uhlenbeck process

Zagoraiou

Antognini

2008

Appl Stoch Models Bus & Ind

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This paper deals with optimal designs for Gaussian random fields with constant trend and exponential correlation structure, widely known as the Ornstein-Uhlenbeck process. Assuming the maximum likelihood approach, we study the optimal design problem for the estimation of the trend and the correlation parameter using a criterion based on the Fisher information matrix. For the problem of trend estimation, we give a new proof of the optimality of the equispaced design for any sample size (see Statist. Probab. Lett. 2008; 78:1388-1396. We also show that for the estimation of the correlation parameter, an optimal design does not exist. Furthermore, we show that the optimal strategy for conflicts with the one for , since the equispaced design is the worst solution for estimating the correlation. Hence, when the inferential purpose concerns both the unknown parameters we propose the geometric progression design, namely a flexible class of procedures that allow the experimenter to choose a suitable compromise regarding the estimation's precision of the two unknown parameters guaranteeing, at the same time, high efficiency for both.

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Optimal design of experiments subject to correlated errors

Cited by 45 publications

References 3 publications

Spatial sampling design and covariance-robust minimax prediction based on convex design ideas

Spatial sampling design and covariance-robust minimax prediction based on convex design ideas

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

Optimal designs for parameter estimation of the Ornstein–Uhlenbeck process

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