Acknowledgement: The authors gratefully acknowledge the grant no. P12681-INF provided by the
Austrian Fonds zur Forderung der Wissenschaftlichen Forschung (FWF).
Endnotes
Alternative models based on additive adjustment formulations were introduced by Tobler (1983).
Sigmoid transfer functions are somewhat better behaved than many other functions with respect to the
smoothness of the error surface. They are well behaved outside of their local region in that they
saturate and are constant at zero or one outside the training region. Sigmoidal units are roughly linear
for small weights [net input near zero] and get increasingly non-linear in their response as they
approach their points of maximum curvature on either side of the midpoint.
In addition one property is being lost in comparison to summation units, namely that product units are
vulnerable to translation and rotation of the input space, in the sense that a learnable problem may no
longer be learnable after translation. Rotational and translational vulnerability of single product units
is in part compensated for, if a number of them are being used in parallel.
In the production constrained case the conservation principle is enforced from the viewpoint of origins
of spatial interactions, and in the attraction constrained case from the viewpoint of destinations only
(see Ledent 1985).
Bishop (1995) has shown that the least square error function can be derived from the principle of
maximum likelihood on the assumption of Gaussian distributed target data. Of course, the use of the
error function does not require the target data to have a Gaussian distribution.
Alopex is an acronym for algorithm for pattern extraction.
For the first two iterations, the weights are chosen randomly.
ARV-values greater than one arise whenever the average error is greater than the mean.
SRMSE-values greater the one arise whenever the average error is greater than the mean.
except for the standard origin constrained model
Flows are discrete counts, but note that flows are measured here in terms of erlang, a metric variable.
This static approach for evaluating the performance of a neural model has been used for many years in
the connectionist community in general and in neural spatial interaction modelling in particular (see
Fischer and Gopal 1994). Recent experience has found this approach to be rather sensitive to the
specific splitting of the data. Thus, usual tests of forecast reliability may appear over-optimistic in
general. Fischer and Reismann (2000) suggest an approach that combines the purity of splitting the
data into three disjoint sets with the power of bootstrapping to get a better statistical picture of forecast
variability, including the ability to estimate the effect of the randomness of the splits of the data.
There is virtual unanimity of opinion that site specific variables, such as sj in this case, are generally
best represented as power functions. The specification of fij is consistent with general consensus that
the power function is more appropriate for analyzing longer distance interactions (Fotheringham and
O’Kelly 1989).
Assuming that each tij has a Poisson distribution and the tij’s are independent leads to the following
objective function: ɪij(tjj lnτgrav - ∙,ga ) that has to be maximized. This distributional assumption often
considered to be realistic is open to question in our context in view of the fact that the measurements
of flows are not discrete counts.
ARV-Out-of-sample variance of ⅛ varies between 0.1725 and 0.2361, that of n" between 0.1852
and 0.2502 and that of j between 0.2225 and 0.2327.
assessed by means of the Mann-Whitney-U-Test (pairwise comparison of two independent samples).
The differences are statistically significant at the 1 % significance level. (U=280, Sig. 0.0 [compared
to the standard origin constrained gravity model] and U=144, Sig. 0.009 [compared to the two-stage
approach] ).
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