Neural Network Modelling of Constrained Spatial Interaction Flows

J

t_i. = Σ ^tH     ⁱ = ^1,∙∙∙^, I

j=1

I

^t. j = Σ ^tij     ^j=^1,∙∙∙^, J

i=1

respectively. In turn these marginal sums can be utilised to calculate the overall level of
interaction that is defined as

The Generic Interaction Model of the Gravity Type

The distribution of interactions within such a system can be described by the generic
interaction model of the gravity type that asserts a multiplicative relationship between
the interaction frequency and the effects of origin, destination and separation attributes,
respectively. In general form it may be written as (see Wilson 1967, Alonso 1978)¹

^τi,= ^b(i₁ ) ^ιis₁^fij       ⁱ=^1,∙∙∙^, I ;   ^j=^1,∙∙∙^, J
where τ_ij is the estimated flow from i to j∙ r_i is an origin factor characterising i
[=measure of origin propulsiveness], s_j a destination factor characterising j [=measure
of destination attractiveness], and f_ij a separation factor that measures separation from
i to j∙ The separation factor f_jj is generally - but not necessarily - assumed to be a
function of some univariate measure d_ij of separation from i to j∙ The exact functional
form of each of these three variables is subject to varying degrees of conjecture (see
Fotheringham and O’Kelly 1989)∙ b^i_j^ is a balancing factor with varying subscripts
depending on which constraints {τ_ij } has to obey concerning t_i,, t,_j or t_n ∙ In the origin

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