Neural Network Modelling of Constrained Spatial Interaction Flows

We restrict our attention to some specific members of the above class. First, we assume
the hidden transfer function φ_h (∙) to be the identity function. Thus, the output of the
h-th hidden product unit of the j-th network module, denoted by _jz_h, reads as

2j

j^zh = Φh(jneth) = ∏ ^{χβn   j}=^1,∙∙∙^,J;^h = ^1,∙∙∙^,H

n = 2 j-1

Second, we assume that each output unit, Ω_sl (x, w)_j =: y_j ( j = 1,..., J), uses a non-
linear normalised transfer function ψ_j (∙) :

net j

^ωsl (^{x, w} ) _j = Ψ j ( ^netj ) = ^b∖₁j )  ------- j = ^1,∙∙∙^, J                              ^(-0)

Σ ^{net j} '

f=1

that resembles the Bradley-Terry-Luce model augmented by a bias unit b. _j ^. net j is the
value of the weighted sum for the j-th output node given by

H      2j

^netj = ∑Yh ∏ ^{xβhn j} = I,-,J
h-1     n-2 j -1

The choice of the output transfer function (20) is motivated by the goal to ensure that
the network outputs satisfy the conservation principle (Ledent 1985) that is enforced
from the viewpoint of origins if b^_jj^ = b_ii.^ [origin constrained case] or from the
viewpoint of destinations if b.^ = t>^_j^ [destination constrained case]. Ω_sl (x, w)_j may be
interpreted as probability of spatial interactions, conditioned on the output ∑H1 _jz_h of
the j-th network module.

With the two above specifications, our modular product unit neural network to model
origin constrained spatial interactions reads in a compact way as

∑ r h ι²i √■

⅛ (x, w )j≈ ⅛_l J-H n------

ΣΣrh ∏ xfhh
j ^l-1 h ^l-1      n-2 j ^,-1

13

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