Exploring relationships between semi-variogram and spatial autoregressive models

Abstract

This paper seeks to continue the building of a common foundation for spatial statistics and geostatistics. Equations from the conditional autoregressive (CAR) model of spatial statistics for estimating missing geo-referenced data have been found to be exactly those best linear unbiased estimates obtained with the exponential semi-variogram model of kriging, but in terms of the inverse covariance matrix rather than the covariance matrix itself. Further articulation of such relations, between the moving average (MA) and simultaneous autoregressive (SAR) or autoregressive response (AR) models of spatial statistics, and, respectively, the linear and Gaussian semi-variogram models of kriging, is outlined. The exploratory graphical and numerical work summarized in this paper indicates the following: (a) there is evidence to pair the moving average and linear models; (b) the simultaneous autoregressive and autoregressive response model pair with a Bessel function (modified of the second kind and order one) rather than the Gaussian semi-variogram model; (c) both specification error and measurement error can give rise to the nugget effect discussed in geostatistics; (d) restricting estimation to a geographic subregion introduces edge effects that increasingly bias semi-variogram model parameter estimates as the degree of spatial autocorrelation increases toward its upper limit; and (e) the theoretical spectral density function for a simultaneous autoregressive model is a direct extension of that for the conditional autoregressive model.