Databases Reference
In-Depth Information
Y
e
(18, 11)
b
(4, 10)
d
(14, 8)
f
(21, 7)
c
(9, 6)
a
(3, 3)
X
o
Figure11.2
Data set for fuzzy clustering.
Table11.3
Intermediate Results from the First Three Iterations of Example 11.7's EM Algorithm
Iteration
E-Step M-Step
"
1
#
0
0.48
0.42
0.41
0.47
c
1
D.
/
c
2
D.10.42, 8.99/
8.47, 5.12
M
T
D
1
0
1
0.52
0.58
0.59
0.53
"
0.73
#
0.49
0.91
0.26
0.33
0.42
c
1
D.8.51, 6.11/
c
2
D.
M
T
D
2
0.27
0.51
0.09
0.74
0.67
0.58
14.42, 8.69
/
"
0.80
#
0.76
0.99
0.02
0.14
0.23
c
1
D.
/
c
2
D.16.55, 8.64/
6.40, 6.24
M
T
D
3
0.20
0.24
0.01
0.98
0.86
0.77
respectively, where
dist
.
,
/
is the Euclidean distance. The rationale is that, if
o
is close to
c
1
and
dist
is small, the membership degree of
o
with respect to
c
1
should be high.
We also normalize the membership degrees so that the sum of degrees for an object is
equal to 1.
For point
a
, we have
w
a
,
c
1
D 1 and
w
a
,
c
2
D 0. That is,
a
exclusively belongs to
c
1
. For
point
b
, we have
w
b
,
c
1
D 0 and
w
b
,
c
2
D 1. For point
c
, we have
w
c
,
c
1
D
41
.
o
,
c
1
/
45C41
D 0.48 and
w
c
,
c
2
D
45
45C41
D 0.52. The degrees of membership of the other points are shown in the
partition matrix in Table 11.3.
In the
M-step
, we recalculate the centroids according to the partition matrix,
minimizing the SSE given in Eq. (11.4). The new centroid should be adjusted to
X
w
o
,
c
j
o
each point
o
c
j
D
X
,
(11.12)
w
o
,
c
j
each point
o
where
j
D 1, 2.
