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(2) If there exist
j
and
r
such that
d
wE
(
A
j
,
V
r
(
))
=
0, then let
u
rj
(
)
=
k
k
1 and
u
ij
(
)
=
=
r
.
Step 3
Calculate
k
0for
i
, where
V
V
1
(
),
V
2
(
),...,
V
c
(
(
k
+
1
)
=
k
+
1
k
+
1
k
+
1
)
V
i
(
)
=
f
(
A
w
(
i
)
(
k
+
1
,
k
+
1
)),
1
≤
i
≤
c
(2.155)
u
i
1
(
k
)
u
i
2
(
k
)
u
ip
(
k
)
w
(
i
)
(
k
+
1
)
=
)
,
)
,...,
,
1
≤
i
≤
c
p
j
=
p
j
=
p
j
=
1
u
ij
(
k
1
u
ij
(
k
1
u
ij
(
k
)
(2.156)
d
wE
Step 4
If
i
=
1
V
i
(
),
V
i
(
k
k
+
1
)
<ε
, then end the algorithm; otherwise, let
c
k
:=
k
+
1, and return to Step 2.
Example 2.6
(Xu and Wu 2010) We conduct experiments on both the real-world
and simulated data sets (Xu et al. 2008) in order to demonstrate the effectiveness of
Algorithm 2.7 for IFSs.
Below we first introduce the experimental tool, the experimental data sets, and
the cluster validity measures, respectively:
(1) Experimental tool. In the experiments, we use Algorithm 2.7 implemented by
ourselves in C language. The parameters that can be set in Algorithm 2.7 are shown
in Table
2.7
(Xu and Wu 2010).
Note that if we let
X
, then Algorithm 2.7 reduces to the
traditional fuzzy C-means (FCM) algorithm. Therefore, we can use the IFCM tool
to compare the performance of both Algorithm 2.7 and the FCM algorithm.
(2) Experimental data sets. We use two kinds of data in our experiments. The car
data set contains the information of ten new cars to be classified in the Guangzhou
car market in Guangdong, China. We also use the simulated data set for the purpose
of comparison. All these data are shown as in Example 2.2 (Table
2.2
).
(3) Cluster validitymeasure. One of the unavoidable problems for Algorithm2.7 is
the setting of the parameter
c
, i.e., the number of the clusters. To meet this challenge,
here we use two relative measures for fuzzy cluster validity given by Nasibov and
Ulutagay (2007): Partition Coefficient (PC
π(
x
)
=
0 for any
x
∈
)
and Classification Entropy (CE
)
.The
descriptions of these two measures are shown in Table
2.8
(Xu and Wu 2010).
Now we utilize Algorithm 2.7 to cluster the ten new cars
y
i
(
i
=
1
,
2
,...,
10
)
,
which involves the following steps (Xu and Wu 2010):
Step 1
Let
c
=
3 and
ε
=
0
.
005. Randomly select the initial centroid
V
(
0
)
from
the data set, say for instance,
⎡
⎤
y
9
y
10
y
8
⎣
⎦
V
(
0
)
=
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