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There is now wide agreement that to develop effective simulations of urban structure, urban models must be explicitly dynamic and must contain mechanisms for linking macrostructure to micro-behaviour. In this paper, the author poses the need to develop simple simulations which emphasise the conditions under which spontaneous growth, such as that which characterises the regeneration of inner cities and the location of edge cities, can be modelled. The model that is presented is based on the integration of positive feedback with the effects of local interaction, all set within a context in which innovations are produced randomly as spatial noise. The framework adopted to make this model operational is based on an extended cellular automaton in which demand for location, conceived of as potential for development, drives the system which responds through the supply of actual development. Various structures for such models are considered. The simulations are set up within a desktop environment by using the package StarLogo as the simulator, DataDesk as the means for exploring model results graphically and statistically, and Avid VideoShop as the moviemaker for simulating the dynamics of locational change. Experiments are conducted which involve systematically varying the weights or parameters of the feedback, interaction, and innovation effects, and then examining how the growth rate of the system interacts with locational change. The model is generalised to other situations where the growth of systems of cities are simulated and where the effects of self-organisation within the simulated patterns of locational change are examined. In essence, the model is able to predict the spontaneous growth of centres but under conditions where there is a high degree of noise or innovation in the system and where the growth rate is also high.

In this paper, a formal model of diffusion-limited aggregation is presented which can be used to generate a continuum of urban forms from linear to concentric. A brief derivation of this continuum model is presented in terms of its relationship to potential theory, and a method for its solution is outlined. The forms produced by the model are subject to fractional power laws which relate the occupancy of sites and densities to distances. Fractal dimensions can be derived from these fractional powers, and conventional and fast methods of estimation are introduced. A large-scale simulation provides the baseline for comparison and its average fractal dimension is estimated as 1.701 ± 0.025 from thirty aggregates. The effect of constraining the physical limits of the lattice on which such aggregates are grown is explored and the model is then used to generate a continuum of forms determined by a control parameter which distorts the potential field of the model. A series of forms from the linear with fractal dimension close to 1 to the concentric with a fractal dimension near to 2, are grown. The model is then applied to urban development in the town of Cardiff and various simulations are attempted which point the way to further research.

This essay is an attempt to describe from three related perspectives the successes and failures characterising the short but turbulent history of urban modelling: from the broader perspectives of knowledge or scientific theory, and of action or design; and from the narrower perspective of the modelling activity itself. It is argued that modelling is concerned solely neither with science nor with design but with both; that is, it is concerned with the relationship between science and design and must be examined accordingly. Various arguments pertaining to these themes are elaborated in terms of the inadequacy of its theory and the dictates imposed by policy. Viewed from the individual perspectives of science or design, modelling is often judged a dismal failure but in this essay it is argued that as such views are necessarily incomplete, some compromise must be sought. In these terms, the field manifests a limited success. Much has been learned about the activity itself but, as in all situations involving immature science, the real value of these experiences may be in raising awareness of the conflicts and dilemmas which occur when uncertain knowledge is applied to problems whose perception is continually changing.

This paper presents a formal derivation of the condition identified by Schinnar (1978) in which the prediction of nonbasic employment in models of the Garin–Lowry type can be independent of the spatial distribution of basic employment. The distributional invariance which determines this result is also a feature of the series-expansion form of model, which is characterised by the Markov property. It is shown that, in the limit, the invariant distribution is equivalent to the steady state of a finite Markov chain, and convergence to this limit is then traced by use of the spectral decomposition of the distribution. This leads to a new interpretation of the model based on spatially dependent and independent components, and statistics are suggested which measure the degree of invariance in any application. Finally these ideas are given substance by recomputing Schinnar's (1978) example in terms of its spectral decomposition, and this leads to several suggestions for future research which are drawn together in the conclusion.

This paper attempts a reformulation and generalisation of Reilly's (1931) law of retail gravitation. Reilly himself challenged workers in the field to produce new evidence which would refute or strengthen his law, and developments in spatial-interaction theory during the last decade are used here in taking up this challenge. A critique of Reilly's law sets the scene: By adopting a gravity model more general than the Newtonian model used by Reilly, it is shown how the limitations of the law with respect to hierarchy, spatial competition, locational size, and the symmetry of trade flows, are overcome. In particular the notion of Reilly's law as a special case of the market-area analysis originating from Fetter (1924) and Hotelling (1929) is demonstrated in terms of a theory of the breakpoint implying spatial price–cost indifference. Another approach, through entropy-maximisation and its dual problem, leads to similar conclusions with regard to prices, and it also serves to introduce multicentred spatial competition. These ideas are then generalised in several ways: through notions about the influence of prior spatial information, through concepts of consumer as well as producer market areas or fields, and through the implications of the analysis for the family of spatial-interaction models. A speculation on the relationship of price differentials to Tobler's (1975) interaction winds is made, and the paper is concluded with an application of these models to the definition of an urban hierarchy in the Reading subregion.

This paper presents some experiments in calibrating a retail location model designed for the Kristiansand region in southern Norway. The form and behaviour of the model under extremes of its parameter values is first investigated, and then some calibration methods proposed by Hyman are tested. The need for a more general method is evident and a search procedure based on the Fibonacci numbers is outlined. A generalisation of this method, based on the golden section number, is derived and this method is used to explore the sensitivity of the model's goodness-of-fit to changes in parameter values. A fundamental result of these explorations shows that there is no unique fit for retail models with two parameters, and this result is likely to hold for other models of spatial interaction.

This paper describes some problems encountered in developing and calibrating a version of the Lowry model in the Central Lancashire and Nottinghamshire-Derbyshire subregions. The formal structure of the model is presented first, and this provides the background to a discussion of the problems involved in estimating the model's parameter values, and in choosing sensitive and meaningful statistics to measure the model's goodness of fit. It is shown that statistics such as R2 measuring the fit of the distributions of population and employment, are extremely insensitive to changes in the parameter values, and that the most sensitive and useful statistic for calibration is the mean trip length. Certain shortcuts in calibrating the model are also suggested by these results and these can be used to get good first approximations of the parameter values.

In this paper, we argue that the geometry of urban residential development is fractal. Both the degree to which space is filled and the rate at which it is filled follow scaling laws which imply invariance of function, and self-similarity of urban form across scale. These characteristics are captured in population density functions based on inverse power laws whose parameters are fractal dimensions. First we outline the relevant elements of the theory in terms of scaling relations and then we introduce two methods for estimating fractal dimension based on varying the size of cities and the scale at which their form is detected. Exact and statistical estimation techniques are applied to each method respectively generating dimensions which measure the extent and the rate of space filling. These methods are then applied to residential development patterns in six industrial cities in the northeastern United States, with an innovative data source from the TIGER/Line files. The results support the theory of the fractal city outlined in books by Batty and Longley and Frankhauser, but with the clear conclusion that different scale and estimation techniques generate different types of fractal dimension.

Since mathematical models came to be applied to problems of architectural and urban form, new concepts based on predicting large-scale structure from local rules have emerged through insights originating in computation and biology. The clearest of these are computer models based on cellular automata (CA) and their recent generalization in evolutionary biology and artificial life. Here we show how such models can be used to simulate urban growth and form, thus linking our exposition to the longer tradition of ideas in studies of built form emanating from the 'Cambridge School'. We first review developments of CA in general and then in urban systems in particular. We propose a general class of CA models for urban simulation and illustrate two simple applications, the first a simulation of the development of the historical 'cell' city of Savannah, Georgia, the second, a generic hypothetical application. We then show how this generic model can be used to simulate the growth dynamics of a suburban area of a mid-sized North American city, thus illustrating how this approach provides insights into the way microprocesses lead to aggregate development patterns.

This paper is addressed to the problem of developing realistic-looking patterns of land use and activity from predictions generated by conventional urban models. The method developed is based on a geometric model of irregularity involving hierarchical cascading and recursion, whose rationale lies in the emergent field of fractal geometry. First the idea of fractals—shapes with fractional dimension—is introduced and then it is shown how two-dimensional patterns whose dimensions are consistent with a large city such as London can be simulated at different levels of detail or recursion. It is then argued that the hierarchical structure of cities should be exploited as a basis for such simulation, and it is argued that discrete choice models of individual spatial behaviour have excellent properties which enable their embedding into such simulations. The standard multinomial logit model is presented and then applied to house-type data in Greater London. A variety of models are estimated, house-type choice being related to age and distance from the centre of the city, and the spatial biases in these predictions are then mapped using prediction success statistics. These models are then used at the base level of a fractal simulation of house type and location in London. Random and deterministic model simulations are developed, and an unusual and possibly innovative feature of these simulations involves the way the inputs and outputs, data and predictions, are simultaneously displayed on the graphics device used. Conclusions for further research, particularly in spatial hierarchical modelling, are then sketched.

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.