Neural Network Modelling of Constrained Spatial Interaction Flows

of two overall performance measures. The first performance measure are the average
relative variances, ARV ( M3 ), a normalised mean squared error metric that is widely
utilised in the neural network community:

where ( x^{u 3}, y^{u 3} j denotes the u 3-th pattern of the testing set M₃, y^{u 3}the average over
the U₃ = 248 desired values. The averaging, that is the division by U₃ [the number of
patterns in M₃], makes ARV independent of the size of M₃ . The division by the
estimated σ² of the data removes the dependence on the dynamic range of the data.
This implies that if the estimated mean of the observed data would be taken as
predictor, ARV would equal to one (Weigend, Rumelhart and Hubermann 1991). The
statistic has a lower limit of zero indicating perfectly accurate predictions and an upper
limit that is in practice one.⁸

The second performance measure is the standardised root mean square error (SRMSE)
that is widely utilised by spatial analysts (see Fotheringham and Knudsen 1987):

1

. . 1 Γ , ... .      .... ..2       1³

SRMSE(M.) = —   X (y^u3 - ¹ ⅛_l (x^u3, w)) /U3

^y ( x^{u 3}, y^{u 3} ) M₃

This statistic - closely related to the ARV-measure - has a lower limit of zero
indicating perfectly accurate predictions and an upper limit that is variable⁹ and
depends on the distribution of the y^u3 .

5.2 The Data

The testbed for the benchmark comparisons uses interregional telecommunication
traffic data for Austria. From three Austrian data sources - a (32, 32)-interregional

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