Fig. 3.3. Two-dimensional distributions of input vectors (framed areas), and the networks of reference vectors approximating them
**Examples of Ordering**. It may be quite surprising that when starting with random mi(O), the reference vectors will attain ordered values in the long run, even in high-dimensional spaces. This ordering is first illustrated by means of two-dimensional input data […] that have some arbitrarily structured distribution. For instance, if x is a stochastic vector, its probability density function p(x) may be assumed uniform within the framed areas in Fig. 3.3 and zero outside them.
The topological relations between the neurons in a square array can be visualized by auxiliary lines that are drawn between the neighboring reference or codebook vectors (points in the signal space). The reference vectors in these graphs now correspond to the crossings and end points of this network of auxiliary lines, whereby the relative topological order becomes immediately visible.
The codebook vectors, while being ordered, also tend to approximate p(x), the probability density function of x. This approximation, however, is not quite accurate, as will be seen later.
The examples shown in Fig. 3.3 represent the approximately converged state of the weight vectors. The different units have clearly become sensitized to different domains of input vectors in an orderly fashion. There is a boundary effect visible in Fig. 3.3, a slight contraction of the edges of the maps. We shall analyze this effect in Sect. 3.4.1 On the other hand, the density of weight vectors is correspondingly higher around the contraction. The relative contraction effect diminishes with increasing size of the array.
Examples of intermediate phases that occur during the self-organizing process are given in Figs. 3.4 and 3.5. ⇒ Ordering Process
Fig. 3.5. Reference vectors during the ordering process, linear array
Fig. 3.4. Reference vectors during the ordering process, square array. The numbers at lower right-hand corner indicate learning cycles
The initial values mi(O) were selected at random from a certain (circular) support of values, and the structure of the network becomes visible after some time. Notice that the array can be, e.g., one-dimensional although the vectors are two-dimensional, as shown in Fig. 3.5. The "order" thereby created resembles a Peano curve, or fractal form.
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KOHONEN, Teuvo, 1995. Self-organizing maps. Berlin ; New York: Springer. Springer series in information sciences, 30. ISBN 978-3-540-58600-5, p. 81