Optimizing stratification and allocation for design-based estimation of spatial means using predictions with error

We present a novel method (Ospats) to optimize spatial stratification and allocation for stratified random sampling of points in the plane. Our quality criterion is the sampling variance under Neyman allocation given a sample size. The method uses a grid of points with uncertain predictions of the target variable. The difference with existing techniques is that we account for the uncertainties. From the quality criterion, we derive an objective function defined by generalized distances between pairs of grid points, determined by the difference between the predictions, the variances of the prediction errors, and their covariance as a function of the geographical distance. Iterative reallocation is used to minimize the function. Resulting stratifications typically represent solutions on a continuous scale between two extremes: for errorless predictions, a stratification close to those by the cum-root-f method, and for entirely uninformative or missing predictions, a compact geographical stratification based only on the locations of the grid points. From a simulation study, we conclude that Ospats performs as expected: the stratifications that it produces are more efficient than the two extreme solutions. A case study showed that the method can be successfully applied at farm scale. Extensions to larger-scales and 1D or 3D spaces are straightforward.