Beyond space-filling: An illustrative case

Abstract

International audienceWhen collecting spatial data, it has become a standard practice to position the measurement points spread out uniformly across the available space. These so-called space-filling designs are now ubiquitous in corresponding publications and conferences. The statistical folklore is that such designs have superior properties when it comes to prediction and estimation of response functions. In this presentation we want to review the circumstances under which this superiority holds, provide some new arguments and clarify the motives to go beyond space-filling. We will accompany these findings with a simple two-dimensional example with seven observations

Learning Functions and Approximate Bayesian Computation Design: ABCD

“…, when the prior distribution of θ is the point prior at θ = log(100). It can be seen that the minimum of the criterion is attained at about (z 1 , z 3 ) = (0.9, 0.1) or (z 1 , z 3 ) = (0.1, 0.9), which is comparable to the the results obtained in Müller et al [32] for empirical kriging optimal designs. Note that the diverging criterion values at the diagonal and at z 1 = 0.5 and z 3 = 0.5 are attributable to a specific feature of the ABC method used.…”

“…In order to avoid this effect, the parameter ϵ would have to be adapted in these cases. Alternatively, one could use other variants of ABC rejection, where the fixed number of N elements of I = {x Due to the uncertainty of the prior parameter θ, the optimal design points for z 1 and z 3 slightly move to the edges, which is also in accordance with the findings of Müller et al [32]. …”

“…A simple example is adopted from Müller et al [32]. The observations (x 1 (z 1 ), x 2 (z 2 ), x 3 (z 3 ), x 4 (z 4 )) are assumed to be distributed according to a one-dimensional Gaussian random field with mean zero, a marginal variance of one and z i ∈ [0, 1].…”

“…The first one is a point prior at θ = log(100), so that ρ(h) = ρ(h; log(100)) = 0.01 h . This is the correlation function used by Müller et al [32] in their study of empirical kriging optimal designs. The second prior distribution is an exponential prior for θ with scale parameter λ = 10 (i.e., θ ∼ Exp(10)).…”

Active Learning: Any Value for Classification of Remotely Sensed Data?

“…In geostatistics, a large body of work deals with space-filling methods, aiming to fill the input space, often characterized by the spatial location of training samples [18]. In remote sensing, the focus has often been on systematic methods in which the samples are acquired on a regular grid or on stratified methods, where the number of samples is balanced according to an estimate of the abundance of the classes present in the image or to another relevant parameter for which greater variability corresponds to a larger number of required samples [19]- [21].…”