The

- 1.
- Initialize the set to contain
*N*units

with reference vectors chosen randomly according to .Initialize the time parameter

*t*:

- 2.
- Generate at random an input signal according to .
- 3.
- Order all elements of 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 . - 4.
- Adapt the reference vectors according to

with the following time-dependencies:

- 5.
- Increase the time parameter
*t*:

- 6.
- If 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).

Sat Apr 5 18:17:58 MET DST 1997