Positive spatial autocorrelation, mixture distributions, and geospatial data histograms

Researchers commonly construct histograms as a first step in representing and visualizing their geospatial data. Because of the presence of spatial autocorrelation in these data, these graphs usually fail to closely align with any of the several hundred existing ideal frequency distributions. The purpose of this paper is to address how positive spatial autocorrelation — the most frequently encountered in practice — can distort histograms constructed with geospatial data. Following the auto-normal parameter specification employed in WinBUGS for Bayesian analysis, this paper summarizes results for normal, Poisson, and binomial random variables (RVs) — three of the most commonly employed ones by geospatial scientists — in terms of mixture distributions. A spatial filter description of positive spatial autocorrelation is shown to approximate a normal distribution in its initial form, a gamma distribution when exponentiated, and a beta distribution when embedded in a logistic equation. In turn, these conceptualizations allow: the mean for a normal distribution to be distributed as a normal random variable (RV) with a zero mean and a specific variance; the mean for a Poisson distribution to be distributed as a gamma RV with specific parameters (i.e., a negative binomial distribution); and, the probability for a binomial distribution to be distributed as a beta RV with specific parameters (i.e., a beta-binomial distribution). Results allow impacts of positive spatial autocorrelation on histograms to be better understood. A methodology is outlined for recovering the underlying unautocorrelated frequency distributions.