Entropy-Based Spatial Sampling

Entropy is not only a useful measure in descriptive studies, but may also be employed as auxiliary information for properly sampling from georeferenced environmental data. Techniques are borrowed from the branch of spatial sampling, a collection of methods for extracting subsets of observations from a population, where the spatial location of occurrences is considered relevant for estimating the target characteristics, such as the mean or total of some population quantity. In sampling, it is common to deal with the concept of sampling entropy, which must be separated from the idea of entropy-based spatial sampling: in the present Chapter, we outline both concepts and remark the differences between the two. We also summarize the necessary spatial sampling background and introduce a selection of the most common spatial sampling techniques. We focus, in particular, on sequential procedures based on a system of weights, assigned according to the distance between units, for the inclusion probabilities, i.e. the probabilities to include units in the sample. Such procedures are adapted to include the spatial entropy of the study variable in the weighting system. We show how the traditional spatial sampling methods perform well over data with a compact spatial structure, while different spatial patterns, such as random or repulsive behaviours, benefit from the inclusion of spatial entropy in the weighting system of the inclusion probabilities.