The neural gas algorithm (Martinetz and Schulten, 1991) sorts for each input signal
the units of the network according to the distance of their
reference vectors to 
. Based on this ``rank order'' a certain number of units
is adapted. Both the number of adapted units and the adaptation strength are decreased
according to a fixed schedule. The complete neural gas algorithm is the following:
to contain N units 
chosen randomly according to
.
Initialize the time parameter t:
according to 
.
according to their distance to 
,
i.e., find the sequence of indices 
such that
is the reference vector closest to 
, 
is the
reference vector second-closest to 
and 
is the reference vector such that k vectors 
exist with
. Following Martinetz et al. (1993) we denote
with
the number k associated with 
.
continue with step 2
For the time-dependent parameters suitable initial values
and final values 
have
to be chosen.
Figure 5.1 shows some stages of a simulation for a simple ring-shaped data distribution. Figure 5.2 displays the final results after 40000 adaptation steps for three other distribution.
Following
Martinetz et al. (1993) we used the following parameters:
.
Figure 5.1: Neural gas simulation sequence for a ring-shaped uniform probability distribution. a) Initial state. b-f) Intermediate states. g) Final state. h) Voronoi tessellation corresponding to the final state. Initially strong neighborhood interaction leads to a clustering of the reference vectors which then relaxes until at the end a rather even distribution of reference vectors is found.
Figure: Neural gas simulation results after 40000 input signals for three different probability distributions (described in the caption of figure 4.4).
