Modelling spatial processes: The identification and analysis of spatial relationships in regression residuals by means of Moran\u27s I (Germany)

This thesis deals with the implementation, identification and analysis of relationships between geo-referenced objects. Significant spatial relationships manifest themselves in a spatial process which is modelled on a combination of spatial structure and spatial autocorrelation levels. Simple spatial structures reflect plain spill-over effects between adjacent spatial objects, whereas theoretical guided spatial structure mirror functional exchange relationships within the system of spatial objects. In empirical work, the underlying spatial process is unknown. The linking mechanism between a hypothetical spatial structure and a stochastic input must be identified from the observed data. This is accomplished here by means of the test statistic Moran\u27s I for regression residuals from a Gaussian spatial process. A closed statistical theory on the conditional distribution of Moran\u27s I under the influence of an hypothetical underlying spatial process is developed. In contrast to simulation experiments or the asymptotic maximum likelihood approach, the presented results are exact for samples of any size and they are not necessarily normally distributed. A direct link of an observed value Io of Moran\u27s I to the autocorrelation level po of an underlying spatial process is derived by means of the conditional expectation of Moran’s I. Furthermore, the distribution of local Moran\u27s I; conditional to forces of a global spatial process is developed. It permits identification of local heterogeneity in a global spatial process. Finally, a procedure based on the conditional significance is presented to distinguish between two competing hypothetical spatial processes. A preliminary model to describe the empirical spatial distribution of bladder cancer incidence rats in 219 counties of the former German Democratic Republic is used to demonstrate the feasibility and flexibility of the proposed exact approach. A theoretical basis to address the migration problem in spatial epidemiology, which blurs the observed local disease rats, is presented and tested by the proposed methodology against a simple spatial clustering process