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 =: βh,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

ωsl ( x, w ) j = Ψj


2j

π


j 1, ..., j


(17)


with φh : H → H, ψj : K → K and a (2 J) -dimensional vector

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

(18)


where x21 represents a variable sj pertaining to destination j ( j = 1, ∙∙∙, J) and x2j 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∙ βhn (h = 1, ∙∙∙,H; n = 2j -1,2j) are the
input-to-hidden connection weights and
γh (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 (3
H)-dimensional vector of all the model parameter ψj (j = 1, ∙∙∙, J)
represents a non-linear summation unit and φh (h = 1,∙∙∙,H) a linear hidden product
unit transfer function∙

12



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