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(
)
is allowed to decay indefinitely, and there is no limitation on the number
of nodes that may be allocated to the network, then it follows that the network
will continue to grow indefinitely. Intuitively, this growth should be limited by the
smallest resolution of the feature points (i.e., the Euclidean sum of the smallest
resolutions of all feature sets included in the feature space). At this limit, the purpose
of clustering to provide a compact representation of the dominant patterns or
redundancies in the data also becomes meaningless, since the network will continue
to grow until the number of classes equals the number of feature data points (or
greater).
The adaptation parameter
If
H
t
ʱ
(
)
controls the learning rate, which decreases with
time as the weight vectors approach the cluster centers. It is given by either a linear
function:
t
t
˄
ʱ
(
t
)=(
1
−
1
)
(3.6)
or an exponential function:
t
˄
2
e
−
ʱ
(
)=
t
(3.7)
where
2
are constants which determine the decreasing rate. During the
locating phase, global topological adjustment of the weight vectors
w
j
takes place.
ʱ
(
˄
1
and
˄
)
ʱ
(
)
t
stays relatively large during this phase. Initially,
t
can be set as 0.8 and
ʱ
(
)
it decreases with time. After the locating phase, a small
t
for the convergence
phase is needed for the fine turning of the map.
3.2.2.2
Visual Experiments on Synthetic 2D Data
As a means of understanding the important properties of the SOTM, the ability of
SOTM to cluster synthetic data is demonstrated. A comparison is made between the
self-organizing feature map (SOFM) [
72
] and the SOTM, highlighting the results
for clustering the same synthetic dataset at various node capacities. In the SOFM,
maps of sizes 2
5 are considered. As a direct comparison,
the SOTM is run a single time without any stop criteria, pausing at the equivalent
number of nodes (4, 9, 16, and 25) for comparison with SOFM.
There are two primary factors evident in this simulation (Fig.
3.2
). Firstly, the
SOFM is more constrained by the natural rigidity of its imposed grid topology. Since
this is not a natural fit to the underlying topology, some distortion ensues: In the 2
×
2, 3
×
3, 4
×
4, and 5
×
2
case, the SOTM is shown with five nodes (however this is at the point of insertion of
the fifth node, thus the positions of the other 4 may be compared), and has already
distinguished between the most separated regions in the underlying density. In the
SOFM 3
×
3 case, some distortion becomes evident, as partitioning has favored the
subdivision of dense clusters, over locating other quite clearly distinct regions. As
a result of this and the imposed topology, some nodes have become trapped in low
×
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