Self-Organizing Maps and Unsupervised Classification - Neural Networks: Methodology and Applications - page 393

Fig. 7.4. Application of k -means with 5 reference vectors on the same data distri-
bution as in Fig. 7.3; observations are distributed according two Gaussian modes
with different anisotropic covariance matrices. Crosses denote the positions of the
reference vectors. Four reference vectors are allocated to the first Gaussian mode.
The last reference vector is allocated to the second Gaussian mode
neurons are the edges of the graph. For simplicity, we denote the whole graph
and the set of its nodes with the same letter C . The graph structure allows
the definition of an integer distance δ on C as follows: the length of a path on
the graph is the number of edges of that path. For all the couple of neurons
( c,r ) of the map, δ ( c,r ) is the length of the shortest path on C between c and
r . For any neuron c , that integer distance leads to defining the neighborhood
of c of order d ,
V c ( d )= {r ∈ C,δ ( c,r ) ≤ d} .
As mentioned above, the maps that are currently used are regular lattices.
Therefore, the distance and the neighborhoods are quite easy to visualize,
and they define the discrete topology on the map in a straightforward way.
Examples of distance and neighborhoods are shown on Fig. 7.5 for a 2D grid.
For self-organizing maps, an association is sought between neurons of C
and reference vectors in data space D , similarly to k -means. Training enables
the set of reference vectors to sample the underlying probability distribution
on the data set as faithfully as possible. In the case of topological maps,