Neural Network Modelling of Constrained Spatial Interaction Flows

they may be of little practical value in situations where the set of row totals or the set of
column totals of the spatial interaction matrix TI_xJ is known a priori. For such
situations, the families of production constrained and attraction-constrained models of
the gravity type had been developed.

The question arises how to build neural network models for the case of singly
constrained spatial interactions. Following Openshaw (1998) constrained spatial
interaction modelling may be viewed as consisting of two parts:

• The prediction of flows, and

• the imposing of accounting constraints.

These two parts can be treated separately or simultaneously. The question that follows
is whether to embed the constraint-handling mechanism within the neural network
approach [one-stage modelling approach] or whether to estimate the unconstrained
neural spatial interaction model first and then to apply the accounting constraints
subsequently [two-stage modelling approach]. The one-stage modelling approach is
harder, requiring major changes to the model structure, while the two-stage approach is
much simpler [for an application see Mozolin, Thill and Usery (2000)].

3 The One-Stage Modelling Approach: The Modular Product Unit Network
Model

3.1 Why Product rather than Summation Unit Networks?

Classical neural spatial interaction models, such as Ω (x, w) and ΩL (x, w), are
constructed using a single hidden layer of summation units. In these networks each
input to the hidden node is multiplied by a weight and then summed. A non-linear
transfer function, such as the logistic function, is used to squash the sum. Neural
network approximation theory has shown the attractivity of such summation networks
for unconstrained spatial interaction contexts. But these networks require a larger
number of hidden summation units when approximating complex functions g , such as
those for mapping constrained interaction phenomena.

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