Neural Network Modelling of Constrained Spatial Interaction Flows

_{b =}_____¹H_____

b r Σ  »", d1,"

i j ≠ i j ij

b(_i) is the origin specific balancing factor, a reflects the relationship of s_j with τ^ga
and β is the distance sensitivity parameter, β > 0. s_j is measured in terms of the gross
regional product in j, d_ij in terms of distances from i to j, while t_i, in terms of erlang.
We utilised the Alopex procedure for ML-estimation¹⁴ with T = 1,000; N = 10,
δ =0.0075 and the termination criterion к = 40,000 iterations.

The two-stage neural network modelling approach serves as second benchmark model.
In the first stage the classical unconstrained neural spatial interaction model, Ω_l (see
Equation (7)) is used. The input data were preprocessed to logarithmically transformed
data scaled to [0.1, 0.9]. The number of hidden summation units is 16. Least squares
learning and the Alopex procedure (T = 1,000; N = 10; δ = 0.001,⅛∙ = 40,000) were
used to determine the 81 model parameters. The parameters were randomly initialised
in the range of [-0.3, 0.3]. In the second stage the following constraint mechanism is
used to obtain origin constrained flows

5.5 Performance Tests and Results

Table 3 summarises the simulation results for the modular product unit network model
¹Ω_ll in comparison with the two-stage neural network approach Ω'^s and the origin
constrained gravity model τ ^a. Out-of-sample performance is measured in terms of
ARV(M3) and SRMSE(M3). In addition, training performance values are displayed for
matters of completeness. The figures represent averages taken over 60 simulations
differing in the initial parameter values randomly chosen from [-0.3, 0.3]. Note that this
random initialisation puts ¹Ω_ll in contrast to Ω^o' at a slight disadvantage as its
optimal range is [-2, +2].

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