Neural Network Modelling of Constrained Spatial Interaction Flows

where δ is the step size that has to be chosen a priori and ε an uniformly distributed
random value with ε ∈ [0,1]. The probability of change of the parameter is calculated
as

p_k ( n ) = (1+^exp ( Ck (ⁿ ) / T (ⁿ ) ) )^{1                                               (29)}

with C_k (n) given by the correlation

Ck (ⁿ) = [w_k (ⁿ -¹) - w_k (ⁿ - ²)] ^[Q (^x, У^, wk (ⁿ -¹)) - Q (^x, У^, wk (ⁿ - ²))]     ⁽³⁰⁾

= [∆w_k (n)] [∆Q (^χ, y^, w_k (n))]

The weight will be incremented in a given fixed magnitude δ, when Δw_k > 0, and the
opposite when it is less than zero. The sign of C_k indicates whether Q varies in the
same way as w_k. If C_k > 0, both Q and w_k will be raised or lowered. If C_k < 0, one
will be lowered and the other one raised.

If T is too small, the algorithm gets trapped into local minima of Q. Thus, the value of T
for each iteration, T(n), is chosen using the following heuristic ’annealing schedule’:
where 3H denotes the number of weights. The annealing schedule controls the
randomness of the algorithm. When T is small, the probability of changing the
parameters is around zero if C_k is negative and around one if C_k is positive. If T is
large, then p_k ≡ 0.5. This means that there is the same probability to increment or
decrement the weights and that the direction of the steps is now random. In other
words, high values of T imply a random walk, while low values cause a better
correlation guidance (see Bia 2000). The effectiveness of Alopex in locating global
minima and its speed of convergence critically depends on the balance of the size of the
feedback term Δw_k ΔQ and the temperature T. If T is very large compared to Δw_k ΔQ

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