Neural Network Modelling of Constrained Spatial Interaction Flows

rather then a (3JH)-dimensional vector. Consequently, notation may be simplified as
follows for all j -1,..., J :

( j -1) H + h =: h                                                          (12)

jH =: H                                                          (13)

j β( j-1)H ₊ h,2 j-1 =: βh,2 j-1                                                                           (14)

j ^β(j-1)H+h,2j =^{: β}h,2 j                                                                          ⁽¹⁵⁾

j Y(j-1)H₊h =^: Yh                                                                                ⁽¹⁶⁾

3.3 A Mathematical Description

The network architecture described above implements the general class of neural
models of singly constrained [SL] spatial interaction

^{j 1,} ...^{, j}

with φ_h : H → H, ψ_j : K → K and a (2 J) -dimensional vector

^x   (x1, ^x2, ∙∙∙, ^x2j-1, ^x2j, ∙∙∙, ^x2J-1, ^x2J )

where x₂₁ represents a variable s_j pertaining to destination j ( j = 1, ∙∙∙, J) and x_2j a
variable fj pertaining to the separation from region i to region j ( i = 1, ∙∙∙, I ; j = 1, ∙∙∙, J )
of the spatial interaction system under scrutiny∙ β_hn (h = 1, ∙∙∙,H; n = 2j -1,2j) are the
input-to-hidden connection weights and γ_h (h = 1, ∙∙∙, H) the hidden-to-output weights
in the j-th module of the network model∙ The symbol w is a convenient shorthand
notation of the (3H)-dimensional vector of all the model parameter ψ_j (j = 1, ∙∙∙, J)
represents a non-linear summation unit and φ_h (h = 1,∙∙∙,H) a linear hidden product
unit transfer function∙

12

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