Neural Network Modelling of Constrained Spatial Interaction Flows

6 Conclusions and Outlook

In this paper, a neural network methodology for modelling singly constrained spatial
interactions has been presented. The proposed function approximator is based on a
modular network design with functionally independent product unit network modules
where modularity refers to a decomposition on the computational level. Each module is
a feedforward network with two inputs and a hidden layer of 16 product units and
terminates with a single summation unit. The collective outputs of these modules
constitute the input to the layer of output units that perform the flow prediction by
applying some sort of the Bradley-Terry-Luce model. The paper also demonstrates a
simple way to implement the conservation rule from the viewpoint of origins
[destinations] that avoids the need to modify the parameter estimation procedure to
integrate the constraints on the predicted flows.

The Alopex procedure provides an optimisation scheme that allows to produce LS-
estimates of the model parameters. The dynamic interaction among a stochastic
exploration process, the correlation based guidance to move towards regions of higher-
quality solutions in the parameter space and the convergence-inducing process is
responsible for the attractivity of the global search process of the procedure.

The attraction of this novel model approach depends not only on the awareness of what
it can offer, but also on empirical illustrations of what can be gained in terms of out-of-
sample (testing) approximation accuracy. Benchmark comparisons against the standard
origin constrained gravity model and the two-stage neural network approach, suggested
by Openshaw (1998) and implemented by Mozolin, Thill and Usery (2000), illustrate
the superiority of the product unit neural network model, measured in terms of both the
ARV- and the SRMSE-performance over 60 simulations.

The importance of avoiding overfitting cannot be overemphasised if a good predictive
model is desired, and consequently, we believe that testing further techniques to control
the model complexity without comprising network generalisation or learning accuracy
is merited. Our research may, furthermore, be extended in two other directions in
future, first, by modifying the approach to model the issue of origin and destination

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