case with one CBD. These convex combinations we construct via distance deterrence functions.
Such functions were introduced by the authors in Thorsen et al. (1999), see this paper for a
motivation and discussion of the concept. As an example of this sort, we will make use of a logistic
function
(4.1)
with
x = 2(d0 + d»), k =
2ln(1/a -1)
(do - dɔ)
Here a is defined as the marginal level of interaction. If x is very small, D(x)'0 (i.e., no deterrence),
and if x is large, then D(x)~1 (i.e., full deterrence). The parameter d0 signifies the distance at which D
is marginally close to no deterrence, and d8 signifies the distance at which D is marginally close to
full deterrence.
Now consider the situation where there are two CBDs, CBD1 and CBD2. Consider a point in space
at distances d1 to CBD1 and d2 to CBD2. We let R1[d] and R2[d] be the functions found from (3.9)
using the construction in Section 2 and 3. R12[d] is the corresponding expression when we merge
CBD1 and CBD2 together. Now we propose to model E/L using the expression
(4.2)
R[dι, d2]= D(d)D(d2)R0 + (1 - D(d1))D(d2)R[d1]
+(1 -D(d2))D(d1)R2[d2] + (1 - D(c⅞))(1 - D(d2))R⅞2[(d + d2)/2]
As the reader may wish to verify, the four numbers D(d1)D(d2), (1-D(d1))D(d2), (1-D(d2))D(d1), and
(1-D(d1))(1-D(d2)) always add to 1. Hence E/L is modeled as a convex combination of the four states
R8,R1,R2, and R12. From (4.2) we obtain the following:
• If d1,d2 are both large, the model suggest the expression R8.
• If d1 is small and d2 is large, the model suggest the expression R1[d1]. (All other terms are small in
this case).
• If d2 is small and d1 is large, the model suggest the expression R2[d2].
• If d1 and d2 are both very small, the model suggests that CBD1 and CBD2 are acting as a single
unit.
At first sight the inclusion of a merged state R12 may seem somewhat superficial. One application of
this kind of model is, however, to provide predictions for a change in the spatial pattern of retailing
facilities when the internal distances within the system are subject to change. Consider the network
shown in Figure 7. If a new road connection is introduced between the nodes B and E, one would
expect a stronger kind of response if B and E are both CBDs than if any other pair of nodes has this
property.
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