Neural Network Modelling of Constrained Spatial Interaction Flows

where b_ii _i ^ is the bias signal that can be thought as being generated by a ’dummy unit’
whose output is clamped at the scalar 1/ t_i, The relation of (22) to the generic spatial
interaction model (5) becomes evident when setting x_2j_₁ = s_j and x_2j = f_jj :

H

∑ ■■ hS f ^h

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

∑∑ 7 ,'?^h f^h

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 x_2i_ ₁ = r and x_2j = f_j then (24)
becomes

∑7h Г^h1 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
Ω_sl 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

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