Neural Network Modelling of Constrained Spatial Interaction Flows



constrained case, for example, this conservation principle is enforced from the

viewpoint of origins only: b^i> = ti, /^rij*isj fjj guarantees that ti. = ɪjτij.

Alternative forms of the general gravity model (5) can be specified by imposing
different constraints on
{τij } (see, for example Senior 1979, Fotheringham and O’Kelly
1989, Sen and Smith 1995). In the
globally constrained case the only condition
specified is that the total estimated interaction
τ.. equals the total observed interaction
t„, in the origin constrained case the estimated outflows τi, from each i have to match
the observed outflows
ti, (origin constraint), in the destination constrained case the
estimated inflows
τ.j to each region j must equal the observed inflows t.j (destination
constraint), and in the
doubly constrained case the estimated inflows τ. j and outflows
τ
i, have to match their observed counterparts. Note that in the constrained case b ij ^ = 1.

It is worth noting that in the origin-constrained [also called production constrained]
case the origin factor is linearly dependent with the origin specific balancing factor
bii i ^,
and in the destination-constrained [also termed attraction-constrained] case the
destination factor with the destination-specific balancing factor
bj^, while in the
doubly-constrained case, the constant of proportionality
biij^ depends on both origin and
destination. The origin constraint and the destination constraint are isomorph.

There are different approaches to estimating the generic spatial interaction model (5):
the maximum entropy approach developed by Wilson (1967) and the log-linear
approach which is a special case of Poisson regression (see, for example, Aufhauser
and Fischer 1985). These approaches yield identical estimates of the interaction flows
in the case where the interacting units are measured on the same level of aggregation,
and identical sets of independent variables are used to calibrate the model.

2.2 The Classical Neural Network Approach to Spatial Interaction Modelling

The neural network approach to model spatial interactions departs from Equation (5) by
viewing spatial interaction models as a particular type of input-output model. Given an
input vector
x, the network model produces an output vector y, say y = g ( x) . The
function
g is not explicitly known, but given by a finite set of samples, say



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