The kriging update equations and their application to the selection of neighboring data

Emery, Xavier

doi:10.1007/s10596-008-9116-8

Cited by 67 publications

(63 citation statements)

References 18 publications

(21 reference statements)

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“…The simplest version of such a scheme would be via nearest neighbors (NN): X n (x) comprised of closest elements of X N to x. Emory (2009) showed that this works well for many common choices of K θ . However, NN designs are known to be sub-optimal (Vecchia 1988;Stein, Chi, and Welty 2004) as it pays to have some spread in X n (x) in order to obtain good estimates of correlation hyperparameters like θ.…”

Section: Local Approximate Gaussian Process Modelsmentioning

confidence: 99%

laGP: Large-Scale Spatial Modeling via Local Approximate Gaussian Processes inR

Gramacy¹

2016

J. Stat. Soft.

157

142

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Gaussian process (GP) regression models make for powerful predictors in out of sample exercises, but cubic runtimes for dense matrix decompositions severely limit the size of data -training and testing -on which they can be deployed. That means that in computer experiment, spatial/geo-physical, and machine learning contexts, GPs no longer enjoy privileged status as data sets continue to balloon in size. We discuss an implementation of local approximate Gaussian process models, in the laGP package for R, that offers a particular sparse-matrix remedy uniquely positioned to leverage modern parallel computing architectures. The laGP approach can be seen as an update on the spatial statistical method of local kriging neighborhoods. We briefly review the method, and provide extensive illustrations of the features in the package through worked-code examples. The appendix covers custom building options for symmetric multi-processor and graphical processing units, and built-in wrapper routines that automate distribution over a simple network of workstations.

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Section: Local Approximate Gaussian Process Modelsmentioning

confidence: 99%

laGP: Large-Scale Spatial Modeling via Local Approximate Gaussian Processes inR

Gramacy¹

2016

J. Stat. Soft.

157

142

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“…Now, using the kriging update formula (see, e.g., Barnes and Watson (1992); Gao et al (1996);Emery (2009), as well as Chevalier et al (2013a)), we obtain:…”

Section: Of the Centered Bivariate Gaussian With Covariance Matrixmentioning

confidence: 99%

Fast Parallel Kriging-Based Stepwise Uncertainty Reduction With Application to the Identification of an Excursion Set

Chevalier¹,

Bect²,

Ginsbourger³

et al. 2014

Technometrics

115

132

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Stepwise Uncertainty Reduction (SUR) strategies aim at constructing a sequence of points for evaluating a function f in such a way that the residual uncertainty about a quantity of interest progressively decreases to zero. Using such strategies in the framework of Gaussian process modeling has been shown to be efficient for estimating the volume of excursion of f above a fixed threshold. However, SUR strategies remain cumbersome to use in practice because of their high computational complexity, and the fact that they deliver a single point at each iteration. In this paper we introduce several multi-point sampling criteria, allowing the selection of batches of points at which f can be evaluated in parallel. Such criteria are of particular interest when f is costly to evaluate and several CPUs are simultaneously available. We also manage to drastically reduce the computational cost of these strategies through the use of closed form formulas. We illustrate their performances in various numerical experiments, including a nuclear safety test case. Basic notions about kriging, auxiliary problems, complexity calculations, R code and datas are available online as supplementary materials.

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“…, Z(x n+q )) = Z(X q ). One of the key ingredient to obtain the formulas and set up the algorithm happens to be the batch-sequential kriging update formulas of Emery (2009);Chevalier et al (2014), as detailed next. The paper is organized as follows: in Sect.…”

Section: Introductionmentioning

confidence: 99%

Fast Update of Conditional Simulation Ensembles

2014

Self Cite

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Gaussian random fields (GRF) conditional simulation is a key ingredient in many spatial statistics problems for computing Monte-Carlo estimators and quantifying uncertainties on non-linear functionals of GRFs conditional on data. Conditional simulations are known to often be computer intensive, especially when appealing to matrix decomposition approaches with a large number of simulation points. Here we study the settings where conditioning observations are assimilated batch-sequentially, i.e. one point or batch of points at each stage. Assuming that conditional simulations have been performed at a previous stage, we aim at taking advantage of already available sample paths and by-products in order to produce updated conditional simulations at minimal cost. We provide explicit formulas allowing to update an ensemble of sample paths conditioned on n ≥ 0 observations to an ensemble conditioned on n + q observations, for arbitrary q ≥ 1. Compared to direct approaches, the proposed formulas prove to substantially reduce computational complexity. Moreover, these formulas enable explicitly exhibiting how the q "new" observations are updating the "old" sample paths. Detailed complexity calculations highlighting the benefits of our approach with respect to state-of-the-art algorithms are provided and are complemented by numerical experiments.

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The kriging update equations and their application to the selection of neighboring data

Cited by 67 publications

References 18 publications

laGP: Large-Scale Spatial Modeling via Local Approximate Gaussian Processes inR

laGP: Large-Scale Spatial Modeling via Local Approximate Gaussian Processes inR

Fast Parallel Kriging-Based Stepwise Uncertainty Reduction With Application to the Identification of an Excursion Set

Fast Update of Conditional Simulation Ensembles

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