Distributional properties of georeferenced random variables based on the eigenfunction spatial filter

Abstract.

The eigenfunction spatial filter derives from the Moran Coefficient that indexes spatial autocorrelation. Mean, variance and statistical distribution characterizations and descriptions of georeferenced random variables and their interrelationships are derived in terms of the eigenfunction spatial filter. Selected comparisons are made with spatial autoregressive model results. Implications of the eigenfunction spatial filter are outlined for simulation experiments, variance components of correlation coefficients, missing values estimation, and non-normal georeferenced random variables. Particularly unanticipated findings include: spatial filtering reveals the potential need to employ a spatial autocorrelation reduction stepwise regression variable selection criterion, the standard error of a univariate mean can be expressed in terms of a conventional regression variance inflation factor that captures spatial autocorrelation effects, spatial autocorrelation can simultaneously inflate and deflate a bivariate correlation coefficient, simulation experiments can be easily designed to involve specific map patterns associated with non-zero spatial autocorrelation, spatial filtering furnishes a convenient way to estimate missing georeferenced values, and the asymptotic variance of the Moran Coefficient is unaltered by spatial filtering.