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The quality of the self-organizing map can be measured by the distortion
measure,
E
d
,
N
i
=
1
j
=
1
h
b
i
j
x
i
−
w
j
C
2
E
d
=
(11.18)
where
h
b
i
j
is the value of the neighborhood function between the map unit
j
and
b
i
, the BMU of the input sample vector
x
i
:
b
i
=
x
i
−
w
j
2
,
N
is the
number of input samples, and
C
is the number of units on the map. It is shown
in [
362
] that the distortion can be decomposed into the following components that
evaluate the quantization quality and topological presentation separation. That is
E
d
=
arg min
j
{
}
E
qx
+
E
nb
+
E
nv
:
C
j
=
1
N
j
H
j
Va r
{
x
|
j
}
j
=
1
N
j
H
j
n
j
−
w
j
C
C
j
=
1
N
j
H
j
Va r
h
{
w
|
j
}
E
d
=
+
+
(11.19)
E
qx
E
nb
E
nv
}
=
∑
x
∈
V
j
x
n
j
/
where Var
{
x
|
j
}
is the local variance of the data Var
{
x
|
j
−
N
j
,
w
j
is the weighted mean and Var
h
{
|
}
w
j
is the weighted variance of the prototype
}
=
∑
k
h
jk
w
k
−
w
j
2
. In addition,
N
j
is
vectors:
w
j
=
∑
k
h
jk
w
k
/
H
j
and Var
h
{
w
|
j
the number of data samples in Voronoi set
V
j
,
n
j
=
∑
x
∈
V
j
x
/
N
j
is their centroid, and
H
j
is the number of prototype vectors.
Equation (
11.19
) examines the contribution of each variable and each map unit to
the distortion measure. The first term,
E
qx
, measures the quantization quality as the
variance of the data vectors within each Voronoi set. The second terms,
E
nb
is the
neighborhood bias. The last term,
E
nv
, is the neighborhood variance which measures
the topological quality in terms of the closeness of prototype vectors close to each
other on the map grid.
In the calculation of the variance Var
, the data vectors are compared to the
centroid of the Voronio set,
n
j
. We observe that if all data vectors
x
{
x
|
j
}
V
j
are drawn
from the same class in the input space, their variance is small, and thus, reducing the
overall distortion
E
d
. In order to reduce the sample variance, in the current work, all
input data vectors used for construction of the Voronoi cells in a sub-codebook are
collected from the same class. This results in Voronoi cells that include data samples
with small variance.
The measurement of the distortion of the SOM was considered in the following
example. Input vectors were drawn from four Gaussian distributions. A single-
codebook SOM of size 2
,
x
∈
2 was trained by all input vectors. Figure
11.9
ashows
the plot of all data samples and the resulting prototype vectors in the 2-D feature
space. It is observed that each prototype vector converged to the centroid of the
corresponding class. Figure
11.9
b shows the Voronoi cells and the classification of
the input vectors, obtained by the single codebook. In is observed that the sample
variance in the Voronoi cells is high. In general, based on the learning procedure of
the SOM, if the number of prototypes increases, more Voronoi cells are generated
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