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This paper is concerned with an inquiry into the way in which the organisation of a spatial data set affects the interpretation of the spatial phenomena which it records, in terms of its underlying pattern or density. It is argued that the number and configuration of zones affects the level of information which is imparted to the spatial analyst, and thus the quest becomes one in which the data set is to be reorganised spatially to impart maximum information. The paper sets out to define an appropriate measure of information and an algorithm designed to optimise its value. The information measure developed is in essence a modified Shannon entropy, modified to account for uneven zonal configurations and converging to Shannon entropy when the zoning system is equal-area and the distribution the best approximation to the density. The measure has some well-known aggregation properties which are presented and several interpretations of its form in spatial terms are made. Empirical measurements of this information measure on the distribution of population in seven metropolitan areas indicate that by aggregation of zones to equal areas, increases in information might be possible and a clearer picture of the underlying density perceived. Accordingly a simple algorithm embodying an heuristic to optimise towards the equal-area norm is developed and applied to the Los Angeles region. The results, in fact, indicate that information cannot be improved in this case, and this leads to a reexamination of the nature of the problem and to suggestions for further research.

This paper seeks to extend the macrostatic approach to urban modelling by treating modelling problems as many-stage processes. Within such a process the early stages are concerned with explaining the relatively trivial characteristics of the phenomena of interest, and the later stages are devoted to explaining more important behavioural issues. Coleman (1964) calls this approach the 'method of residues', and its power is first demonstrated here by a reinterpretation of the well-known gravity model. An ad hoc test of the method on the Toronto-centred region serves to emphasise the need for a more formal approach, and thus an analogy between the method and the Bayesian viewpoint is introduced. A method of information minimising, more general but consistently and unambiguously related to the method of entropy maximising, is used to make the formal approach operational, and the method is used to generate an 'extended' family of spatial-interaction models. A number of spatial-interaction models are derived, and the paper is concluded by a test of two of these models on the Toronto-centred region.

This paper presents a methodology for deriving best statistics for the calibration of spatial interaction models, and several procedures for finding best parameter values are described. The family of spatial interaction models due to Wilson is first outlined, and then some existing calibration methods are briefly reviewed. A procedure for deriving best statistics based on the principle of maximum-likelihood is then developed from the work of Hyman and Evans, and the methodology is illustrated using the example of a retail gravity model. Five methods for solving the maximum-likelihood equations are outlined: procedures based on a simple first-order iterative process, the Newton—Raphson method for several variables, multivariate Fibonacci search, search using the Simplex method, and search based on quadratic convergence, are all tested and compared. It appears that the Newton—Raphson method is the most efficient, and this is further tested in the calibration of disaggregated residential location models.

In this paper the authors introduce a method of approximating the parameter values of gravity models from measures of information or entropy associated with the observed pattern of spatial interaction. The method builds on the previous work of the authors in which parameter values were estimated in a two-stage process which involved utilising the log-linear properties of entropy models through the canonical form of entropy, together with other approximations based on Kirby's method. Here the method is elaborated by adopting a consistent set of information measures to which the parameters of the model are related and this negates the need for other approximations. The original framework is first reviewed and then elaborated through the introduction of a weighted entropy measure. The traditional family of spatial interaction models is sketched and the new method developed for each of these models. The models are then applied to various aggregations of trip data in the Reading (United Kingdom) subregion, and estimates of parameter values based on the old, new, and conventional methods are compared. The new method is demonstrably superior to the old method and various extensions through the spatial disaggregation of entropy measures are noted in conclusion.

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.

This is the third of four papers and in it the methodology for analysing spatial aggregation in gravity models outlined in the first paper is further elaborated. In the second paper, the methodology was applied to one-dimensional spatial interaction models of the population density type, with some success; and here it is proposed to apply the methodology to two-dimensional spatial interaction models using the same data base, the Reading (UK) region. Accordingly, the methodology is first stated for linking information in data measured by spatial entropy to the parameters of models generated from spatial entropy. The family of four spatial interaction models due to Cordey-Hayes and Wilson is then derived, the canonical forms of their associated spatial entropy functions presented, and the analytic properties of such models explored. These four models are then fitted to spatial aggregations of the Reading region, and various empirical relationships between their entropies and parameters described. The results are not as regular as those of the models in the second paper because of more variable model performance, but nevertheless a means of approximating scale parameters from data based on the work of Kirby is outlined. This enables estimates of the dispersion parameters to be made through the canonical forms. Although the results are poor because of model performance, the methodology outlined here serves as a basis for the more fully fledged application to be discussed in the final paper.

This paper, the second of four, is concerned with applying a methodology for analysing the spatial aggregation problem in gravity models outlined in the first paper. The methodology is based on a consistent framework for linking measures of pattern in interaction data to the derivation and estimation of related interaction models using spatial information theory. In this quest, a link is forged between information in data and the parameters of an associated model, and in part 1 it was suggested that if this link could be formalised then a means would be available for predicting changes in model parameters from different aggregations of the data, prior to the actual estimation of the models themselves. This relationship can be formalised for the case of the continuous one-dimensional interaction model such as the population density model, and this paper is concerned with demonstrating such an application to aggregations of zones in the Reading region. The framework is first described and two continuous models are presented. Then, the discrete model is estimated by means both of regression and of entropy techniques applied to various aggregations of the region, and the resulting parameters are related to the predicted and observed informations. Finally, the parameters approximated from observed information by use of the theoretical models are compared with the estimated parameters, and the approximation is deemed good, thus providing some confidence in the general concepts developed to handle these types of problem.

This is the first of four papers concerned with developing a comprehensive approach to the spatial aggregation problem in gravity models. A framework for exploring the problem is outlined in this paper, and will be applied to one-dimensional and two-dimensional spatial interaction models in the subsequent papers. Appropriate statistics for measuring changes in spatial variation due to spatial aggregation in data sets and in model predictions of spatial interaction are derived by use of information theory; and these statistics, such as spatial entropy, have excellent decomposition properties which can be readily exploited in the study of aggregation effects in data and models. These properties involve information components associated with density, level of detail or average zone size, dimension, and level of resolution. Use of the spatial entropy function, in particular, enables consistent relationships to be developed between information components and model parameters, and these relationships are examined in detail in subsequent papers.

In this paper, we propose a model of growth and form in which the processes of growth are intimately linked to the resulting geometry of the system. The model, first developed by Witten and Sander and referred to as the diffusion-limited aggregation or DLA model, generates highly ramified tree-like clusters of particles, or populations, with evident self-similarity about a fixed point. The extent to which such clusters fill space is measured by their fractal dimension which is estimated from scaling relationships linking population and density to distances within the cluster. We suggest that this model provides a suitable baseline for the development of models of urban structure and density which manifest similar scaling properties. A typical DLA simulation is presented and a variety of measures of its structure and dynamics are developed. These same measures are then applied to the urban growth and form of Taunton, a small market town in South West England, and important similarities and differences with the DLA simulation are discussed. We suggest there is much potential in extending analogies between DLA and urban form, and we also suggest future research directions involving variants of DLA and better measures of urban density.