Spatial Aggregation in Gravity Models: 4. Generalisations and Large-Scale Applications

This paper is concerned with applying and extending a methodology for analysing spatial aggregation in gravity models developed in three earlier papers to a larger scale and hence more realistic example of spatial interaction than has been treated so far. Problems of model performance and the analysis of spatial aggregation effects identified in earlier papers are first described, and accordingly the spatial information (entropy) function used previously is then generalised to enable a wider set of models to be applied. Seven models are generated by means of spatial and generalised entropy functions subject to a standard set of model constraints, and the properties of the models in terms of their canonical forms are presented. The models are then applied to four levels of aggregation (234, 121, 58, and 22 zones) of the spatial interaction pattern in Edmonton, Alberta. As in previous papers, information in the data set at the four levels of aggregation is first measured and interpreted, the models are then fitted to these four levels, and relationships between information and parameter values sought. The approximation theory developed earlier is then used to predict parameter values of such models directly from observed spatial information. The results are only fair, better than those of part 3, but worse than those of part 2; although in terms of predicted parameter shift between levels of aggregation, the shifts associated with the doubly constrained model are accurately predicted. The various themes in these papers are then drawn together, conclusions with respect to the value of the insights gained are made, and speculations as to the most fruitful lines for future research outlined.