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specific input pattern. The density map can be obtained by applying the
following equation to a trained SOM:
N
=Num(
i
j
)
,
j
=1
,
2
,...,l,
(1)
where
N
is an integer,
i
is the neuron of the trained SOM, and Num( )
is a function counting the number of the neuron
i
“imaged” by certain
input patterns. Every pattern in the input space has only one image, but
one neuron can be the image of many input patterns. The density map
can reveal the grouping of input patterns. From Eq. (1), the density map
can be obtained easily by labeling each grid of the map with the integer
N
.
Suppose that the number of each grid is the “elevation” of the density map.
Then the grouping of input patterns is shown by certain isolated “plateaus”
separated by “valleys” on the density map. The valleys are the clusters'
boundaries. Therefore, a proper number of clusters can be determined.
There are some neurons “imaged” by certain input patterns in the fea-
ture map. According to the aforementioned feature map and density map,
labeling the “imaged” neurons with the identity of the corresponding clus-
ter forms a part of a discrimination map. There are still some blank neurons
not labeled. A complete discrimination map can be obtained by applying
the following equation to the blank neurons:
I
=
C
(
i
)=argmi
j
w
(
i
)
−
x
j
,
j
=1
,
2
,...,p,
(2)
where
I
is the identity of a specific cluster,
C
( ) is a function indicating
the cluster to which the blank neuron
i
belongs,
w
(
i
)istheweightsofthe
neuron
i
,
x
is a input pattern, and
p
is the number of input patterns. The
identity of a blank neuron is the identity of the input pattern which best
matches the synaptic weights of the blank neuron. A complete discrimi-
nation map can then be obtained by labeling all blank neurons with their
corresponding identities
I
as shown in Eq. (2).
3. Application and Discussions
The research area in this paper is the north area of Taiwan. About 34 rain
gauges are available in the area. Among these gauges, gages 2, 6 and 31 are
selected to validate the discrimination capability of SOMCD in this paper.
The single-station design hyetographs of the 31 rain gages are obtained by
analyzing the annual maximum events till 1998.
10
The three maps (the feature, density, and discrimination maps) are
derived using the SOMCD. In conventional cluster analysis, it is di
cult to
