Neural Network Modelling of Constrained Spatial Interaction Flows

presents the results of a benchmark comparison of the model against the standard origin
constrained gravity model and the two-stage neural network approach that treats the
prediction of flows and the imposing of accounting constraints as two independent
issues. The testbed for the evaluation uses interregional telecommunication traffic data
from Austria. Section 6 summarises the results achieved, and outlines some directions
for future research.

2 Background

2.1 Definitions and the Generic Interaction Model of the Gravity Type

Suppose we have a spatial system consisting of I origins and J destinations and let tij
denote the volume of interaction from spatial unit (region) i to j ( i = 1,..., I ; j = 1,..., J ).
This information may be displayed in the form of an interaction matrix of the following
kind

In some cases the sets of origins and destinations are the same and, thus, T_{1 x}J is a
squared matrix. The interpretation of the main diagonal of TI _xj depends on the specific
application. For instance, it might represent internal telecommunication flows within
region i (i = 1,...I). Often such values are not recorded. In other applications, for
example shopping trips from residential areas to individual shopping malls, the number
of origins and destinations may differ and TI_xj will not be square.

For all applications, the i-th row of the matrix TI_xj describes the outflows from region i
to each of the J destinations, while inflows from each of the I origins into destination j
are described by the j-th column. From TI_xj we can calculate the volume of interaction
originating from region i or terminating in region j, that is

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