Finite population properties of individual predictors based on spatial patterns

In this study a conceptual framework for assessing the statistical properties of a non-stochastic spatial interpolator is developed through the use of design-based finite population inference tools. By considering the observed locations as the result of a probabilistic sampling design, we propose a standardized weighted predictor for spatial data starting from a deterministic interpolator that usually does not provide uncertainty measures. The information regarding the coordinates of the spatial locations is known at the population level and is directly used in constructing the weighting system. Our procedure captures the spatial pattern by means of the Euclidean distances between locations, which are fixed and do not require any further assessment after the sample has been drawn. The predictor for any individual value turns in a ratio of design-based random quantities. We illustrate the predictor design-based statistical properties, i.e. asymptotically p-unbiasedness and p-consistency, for simple random sampling without replacement. An application to a couple of environmental datasets is presented, for assessing predictor performances in correspondence of different population characteristics. A comparison with the equivalent non-spatial predictor is presented.