Neural Network Modelling of Constrained Spatial Interaction Flows



Residual Analysis

One means of further investigating the predictive power of the modular product unit
neural network model 1Ω
SL (x, w) in comparison to the two benchmark models
ΩL'''t. (x, w) and τra is the use of residual analysis. Figure 4 displays these in terms
of

the absolute residuals of the individual flows ^tj - 1ΩSL (x, w)j j compared to
( t-ΩL-∙∙' ( x, w )J and ( t,-rg- ), and

the relative residuals of the individual flows ^tij -1 Ωsl (x, w)j j/ tj compared to
( t,- Ω s ( x, w )..) /t, and ( t,- τ     I /1,,

ij L                         ij        ij                  ij ij             ij

where both absolute and relative residuals are ordered by the size of the observed
flows,
tij . The main conclusion from this analysis can be summarised as follows:

First, all three models show a tendency to underpredict larger flows. The neural
network models underpredict 17 out of 25 flows in the largest decile, compared to 16
gravity model underpredictions. The benchmark models tend to give results with a little
larger absolute deviation.

Second, all three models show a tendency to overpredict smaller flows. This is
evidenced in the smallest decile by 23 overpredictions in the smallest decile, obtained
by the modular product unit network, compared to 24 overpredictions of the benchmark
models.

Third, the modular product unit neural network model and the benchmark models show
a relatively similar pattern of residuals. Despite this similarity the modular model tends
to produce slightly more accurate predictions in the case of larger flows, but slightly
less accurate ones in the case of smaller flows.

In summary, the analysis unequivocally shows that the modular product unit network
outperforms the benchmark models in terms of both the ARV(
M3) and SRMSE(M3)
prediction performance, as well as the prediction accuracy, but the latter to a lesser
degree than previously expected. One reason for this might be that the method of

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