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(4) Update the weight vectors of the winner and its neighboring neurons, with the
rate according to the closeness to the winner neuron, as follows:
W
i
(
t
+1)=
W
i
(
t
)+
α
(
t
)Λ(
i
,
i
∗
)(
x
−
W
i
(
t
))
,
i
=1
,
2
,...,
L
(5.20)
where
α
(
t
) is the learning rate, which is a monotonically decreasing function
of the iteration step t so that in the beginning iterations, the SOM has a fast
learning rate, and slower rate later on. Function Λ(
i
,
i
∗
) is a neighborhood
function, which is also a monotonically decreasing function of the closeness
between neuron
i
and the winner neuron
i
∗
. A frequently used neighborhood
function is:
Λ(
i
,
i
∗
)=
exp
(
2
/
(2
σ
2
(
t
))
−
r
i
−
r
i
∗
(5.21)
2
is the closeness of neuron
i
to the winner neuron
i
∗
.One
example of the closeness of the neighborhood is shown in Fig. 5.6, where we
can code the closeness of the neurons in black as 0, the closeness of neurons in
gray as 1, the closeness of neurons in slash lines as 2, and etc. Term
σ
2
(
t
) is a
scale parameter, it is also a monotonically decreasing function of the iteration
number
t
.
(5) Compute the amount of the weight vectors updates
Term
r
i
−
r
i
∗
L
i
=1
E
t
=
W
i
(
t
+1)
−
W
i
(
t
)
2
(5.22)
If
E
t
is not greater than a threshold
ε
, stop; otherwise let
t
=
t
+1, go back
to (2).
The output of the SOM algorithm will be a set of neurons, where in some
regions of the map, the neighboring neurons will have weight vectors with high
similarity or small dissimilarity, and have weight vectors with small similarity or
high dissimilarity with neurons in other regions. Usually, we map the distance
between two neighboring neurons into a gray scale or a color map, and the output
can be visualized as in Fig. 5.7.
Figure 5.7 can be interpreted in the following way. In Fig. 5.7(a), it clear
that the whole map is divided by a bell of dark neurons into two parts. From the
vertical bar in Fig. 5.7(a), we can see that the darker gray means longer distance.
So, Fig. 5.7(a) shows that the whole dataset is divided into two dense areas by
a sparse area shown by the dark colored neurons. Similarly, Fig. 5.7(b) tells us
that the whole dataset has three clusters since the light grayed neurons divide the
whole map into three dark grayed regions.
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