Neural Network Modelling of Constrained Spatial Interaction Flows



where bii i ^ is the bias signal that can be thought as being generated by a dummy unit
whose output is clamped at the scalar 1/ ti, The relation of (22) to the generic spatial
interaction model (5) becomes evident when setting
x2j_1 = sj and x2j = fjj :

H

∑ ■■ hS f h

:    " (v,wY = ⅛ Jh---------- j = 1,...,J                  (23)

∑∑ 7 ,'?h fh

1'=1 h'=1

Analogously one arrives at the modular product unit neural network for the destination
constrained case:
where
b>^ j ^ is the bias signal that can be thought as being generated by a dummy unit
whose output is clamped at the scalar 1/1,j. Set x2i_ 1 = r and x2j = fj then (24)
becomes

2Ol ( V, w )■ = Ьj )


H 2j

yhχfhn
h=1     n =2 i-1________

I H 2 i1
∑∑Yh` xh'
i '=1 h '=1      n=2 i '-1


(24)


7h Гh1 jh2

:,:= J ( V, w )i= ⅛√⅛----------- i = 1,..., I                      (25)

∑∑ 7 «Гh ∙j
i
,=1 h ,=1

3.4 Two Issues of Crucial Importance for Real-World Applications

Two major issues have to be addressed when applying the spatial interaction model
Ωsl in a real world context: first, the issue of finding a suitable number H of hidden
product units [the so-called representation problem], and
second, the issue of network
training or learning [the so-called learning problem]. The first issue is a challenging
task because the number of hidden product units affects the generalisation performance

14



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