Neural Network and Fuzzy Logic Models of Spatial Interaction

Stan Openshaw
School of Geography, University of Leeds, Leeds LS2 9JT, United Kingdom

Since the early work of Wilson (1967) and perhaps Fotheringham (1983) there has been very little innovatory developments in the computer modelling of spatial flows. Yet the GIS revolution has greatly increased the provision of spatial flow data and has stimulated the development of various decision support systems designed to use spatial interaction models to make intelligent decisions about the distribution of a range of public and commercial sector facilities; see for example, Birkin et al. (1996). It is believed that the increased mobility of people is resulting in a reduction of the tyranny of distance on spatial interaction making it increasingly likely that the legacy models will offer a deteriorating level of performance. The question is whether or not artificial neural networks and fuzzy modelling tools might be able to offer worthwhile improvements in the performance of spatial interaction models.

Both artificial neural networks and fuzzy logic based models offer a number of theoretical advantages: (1) they are non-linear and have been shown to be capable of representing a wide range of complex functional relationships; (2) they are equation free universal estimators with various proofs to support the belief that under favourable circumstances they can model any functional form; (3) they can be trained from observed data rather than be totally dependent on the mathematical and statistical skills of the model builder; (4) they are largely automatic with no need for extensive ‘hand tuning’ or fiddle factoring in order to boost performance; (5) they can utilise a wider range of variables than is possible with conventional models; and (6) they promise maximum levels of performance if properly constructed. The disadvantages of neural networks include: (1) they are black box models offering little or no improvement in levels of understanding; (2) they have suffered from hype and unrealistic expectations; (3) they may need large amounts of compute time; (4) training is not always easy; and (5) they may not always work. Fuzzy modelling is more flexible in that it provides a framework for the incorporation of any existing theoretical knowledge as well as an opportunity to add to it via a learning process. They are also much less of a black-box in that the models can be given a linguistic interpretation in plain english. On the other hand there are currently no fuzzy models of spatial phenomenon so it is a highly speculative and untested technology; by contrast there is a limited literature on neural net models of spatial interaction; see for example, Openshaw (1993) and Fischer & Gopal (1994).

The paper describes how to build both supervised and unsupervised neural net models of flow data. The supervised network used backpropagation whilst the unsupervised net is a modification of Kohonen’s self organising map. These models are trained to fit observed data. They are essentially black boxs and it is necessary to experiment with various neural network configuration parameters. Fuzzy logic modelling is quite different. The model is initially specified in a linguistic form; for example, if the travel cost is high then interaction intensity will be small. The linguistic terms high and small are fuzzy sets. A number of these fuzzy IF-THEN rules can be used to specify a fuzzy flow model. Two different types of fuzzy logic model are investigated: a properly fuzzy model with the fuzzy outputs having to be defuzzified to produce crisp numbers and a partly fuzzy model where only the inputs are fuzzy and the outputs are non-fuzzy. The model can be used to test the qualitative knowledge or hypotheses as expressed in the fuzzy rules. However, it is also possible to train the fuzzy model by using a genetic algorithm to modify the fuzzy set definitions and, or, the fuzzy rule base so as to optimise its performance. This can take a long time although all the software used here will run on a PC as well as Unix workstations and will be published in an easy to use windows based format see Openshaw & Glover (1997).

Both the neural network based models and the fuzzy models can be used to create a family of spatial interaction models via the incorporation of various origin and destination end constraints. In the conventional models the various constraints are embedded in the model equation, reflecting its origins in entropy maximisation where this makes considerable sense. However, there are practical advantages in seeking to separate the flow accounting constraint mechanisms found in the conventional model from the flow modelling equation. Spatial interaction modelling now becomes a two stage process; stage 1 predict the flows using a function or a neural network or a fuzzy model, stage 2 apply whatever accounting constraints are considered necessary to the predicted flows. This is a very useful means of allowing the flow modelling to be made as sophisticated and as complex as is desired. It opens up the prospect of building into spatial interaction models various multilevel micro and macro factors that are currently absent.

The results obtained so far suggest that the partly fuzzy model works exceptionally well and readily fits into a conventional theoretical framework. The neural net models also perform better than any conventional model but are much harder to explain. Both types of models are computationally efficient (once trained) and could easily be used as plug-in replacements for more conventional alternatives. They also offer the possibility that they can be run in adaptive mode whereby they change themselves to improve performance if there is a sudden and unexpected change in the data or a jump in prediction error. They have much to offer as the basis for a whole new generation of computer modelling tools relevant to many of the problems of geography and the social sciences and not just flow data.