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...ibution of data locations, one possibility is to select knots on a uniform grid overlaid on the domain. Selection can then be achieved through a formal design-based approach that minimizes a spatially averaged predictive variance criterion (see, e.g., Diggle and Lophaven, 2006). However, in general the locations are highly irregular, generating substantial areas of sparse observations where we wish to avoid placing knots, since they would be “wasted” and possibly lead to inflated predictive process variances and slower convergence. Here we implement the space-covering design algorithms (e.g., Royle and Nychka, 1998) that yield a representative collection of knots that better cover the domain. Other alternatives include popular clustering algorithms such as k-means or more robust median-based partitioning around medoids algorithms (e.g., Kaufman and Rosseeauw, 1990). User-friendly implementations of these algorithms are available in R packages such as fields and cluster. 2.3.3 Gradients on predictive process surfaces. We assume a stationary and separable cross-covariance function Γw(s, s′; Bayesian Point Process Wombling 1247 θ) = ρ(s, s′; θ)Σ, where ρ(s, s′; θ) is a stationary univariate correlation func...

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... some remarks on knot selection. With evenly distributed locations, knots on a uniform grid (see, e.g., Diggle and Lophaven, 2006) may suffice. With irregular locations, space-covering designs (e.g., =-=Royle and Nychka, 1998-=-) yield a more representative set. In general, different knots selection methods still produce robust results in predictive process models (see, e.g., Finley et al., 2009; Banerjee et al., 2010). And ...

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...often investigate sensitivity of inference to different choices of n∗, which entails separately estimating a number of low-rank models. Typically, for each n∗ we use some space-covering design (e.g., =-=Royle and Nychka, 1998-=-) to fix the knots. In chapter 2, our modeling framework expands existing hierarchical low-rank models, as explored in the aforementioned references, to accommodate modeling knot locations as random p...

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...here is no dependence on the spatial covariance or the stochastic model. The design criterion is based on geometry and the distance between both current and potential sample locations. Royle & Nychka =-=[70]-=- and Nychka & Salzman [61] describe space-filling designs. Dixon et al. [23] describe an approach for the selection of river sampling sites that considers potential locations in a river networkM. Dob...

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...h to avoid placing knots, since they would be “wasted” and possibly lead to inflated predictive process variances and slower convergence. Here we implement the space-covering design algorithms (e.g., =-=Royle and Nychka, 1998-=-) that yield a representative collection of knots that better cover the domain. Other alternatives include popular clustering algorithms such as k-means or more robust median-based partitioning around...

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...here is no dependence on the spatial covariance or the stochastic model. The design criterion is based on geometry and the distance between both current and potential sample locations. Royle & Nychka =-=[17]-=- and Nychka & Salzman [18] describe space-filling designs. Probability-based designs assume a fixed underlying process and use probability sampling to select the monitoring sites. This contrasts with ...