The spatial statistic trinity: A generic framework for spatial sampling and inference

Geospatial referenced environmental data are extensively used in environmental assessment, prediction, and management. Data are commonly obtained by nonrandom surveys or monitoring networks, whereas spatial sampling and inference affect the accuracy of subsequent applications. Design-based and model-based procedures (DB and MB for short) both allow one to address the gap between statistical inference and spatial data. Creating independence by sampling implies that DB may neglect spatial autocorrelation (SAC) if the sampling interval is beyond the SAC range. In MB, however, a particular sampling design can be irrelevant for inferential results. Empirical studies further showed that MSE (mean squared error) values for both DB and MB are affected by SAC and spatial stratified heterogeneity (SSH). We propose a novel framework for integrating SAC and SSH into DB and MB. We do so by distinguishing the spatial population from the spatial sample. We show that spatial independence in a spatial population results in independence in a spatial sample, whereas SAC in a spatial population is reflected in a spatial sample if sampling distances are within the range of dependence; otherwise, SAC is absent in the spatial sample. Similarly, SSH in a population may or may not be inherited in data, and this depends on the sampling method. Thus, the population, sample, and inference constitute a so-called spatial statistic trinity (SST), providing a new framework for spatial statistics, including sampling and inference. This paper shows that it greatly simplifies the choice of method in spatial sampling and inferences. Two empirical examples and various citations illustrate the theory.