Fast Selection of Spatially Balanced Samples

Sampling from very large spatial populations is challenging. The solutions suggested in recent literature on this subject often require that the randomly selected units are well distributed across the study region by using complex algorithms that have the feature, essential in a design-based framework, to respect the fixed first-order inclusion probabilities for every unit of the population. The size of the frame, $N$, often causes some problems to these algorithms since, being based on the distance matrix between the units of the population, have at least a computational cost of order $N^2$. In this paper we propose a draw-by-draw algorithm that randomly selects a sample of size $n$ in exactly $n$ steps, updating at each step the selection probability of not-selected units depending on their distance from the units already selected in the previous steps. The performance of this solution is compared with those of other methods derived from the {\it spatially balanced sampling} literature in terms of their root mean squared error (RMSE) using the simple random sampling (SRS) without replacement as benchmark. The fundamental interest is not only to evaluate the efficiency of a such different procedure, but also to understand if similar results can be obtained even with a notable reduction in the computational burden needed to obtain more efficient sampling designs. Repeated sample selections on real and simulated populations support this perspective. An application to the Land Use and Land Cover Survey (LUCAS) 2012 data-set in an Italian region is presented as a concrete and practical illustration of the capabilities of the proposed sample selection method.