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chosen by the trial and error approach. There are some general rules,
such as that the number of neurons in the SOM should be approximately
one to three times the number of data in the set (Chen and Gasteiger,
1997).
Training of the SOM is an iterative process, where each iteration is a
comparison of the whole data set to all neurons in the grid. This
comparison involves calculation of differences
d
pq
between each data
sample
x
pi
and the closest neuron in the grid
m
qi
[5.33]
Distances are most often expressed in terms of Euclidean metrics, but
other forms are also available (Kohonen, 1998). The SOM grid neuron
that is the most similar to the selected data sample is called the winner.
The vectors of the winner and its neighboring neurons are modifi ed
during training to represent the data set more closely, according to the
following equation:
m
i
(
t
+ 1) =
m
i
(
t
) +
h
ci
(
t
)[
x
(
t
) −
m
i
(
t
)]
[5.34]
where
t
stands for training iteration,
m
i
is the winning neuron, and
x
is a
data sample. Multiplier
h
ci
(t)
is termed as the neighborhood kernel and
its purpose is to determine which neurons are considered as neighboring
the winner neuron. Neurons that are topologically closer to the winning
neuron will be modifi ed to a greater extent in comparison to neurons that
are, topologically, further away. Such training causes the neurons to
stretch through the densely populated areas of the data space, due to
their elastic bonds between each other.
The simplest neighborhood kernel is the bubble function (Kohonen,
1994), which is nonzero for the neighborhood but zero elsewhere.
However, most frequently applied in SOMs is a Gaussian kernel
(Kohonen, 1998), defi ned as:
[5.35]
where σ
(t)
is the neighborhood radius at iteration
t
and it monotonically
decreases with time. Therefore, as time (and iterations) progresses, the
number of neurons considered as neighbors to the winning neuron
decrease. The term ⏐⏐
r
c
−
r
i
⏐⏐ represents the Euclidean distance between
the winning and neighboring neuron. Double vertical lines denote
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