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4.7.2.1 Fuzzy c-means Algorithm
Given the data set
^
`
the
weighting exponent (also called fuzziness exponent)
m
> 1, the termination
tolerance
ZZZ
,
,
" , select the number of clusters 1
,
Z
cN
,
12
N
H! and the norm-inducing matrix
A
. Initialize the partition matrix
randomly, such that
0
l
.
0
U
M
fc
Repeat for iterations
l
= 1, 2, 3, ....
x
Step 1:
compute the cluster prototypes or cluster centres (means)
m
§
l
gs
1
·
N
Z
¦
¨
P
¸
s
©
¹
s
1
g
;1
d d
g
c
.
v
m
§
l
gs
1
·
N
P
¦
¨
¸
©
¹
s
1
x
Step 2:
compute the distances:
T
§ ·
l
2
Z
A Z
dddd
l
,1
g
c
;1
s
N
;
D
v
v
¨
¸
s
g
gsA
s
g
©
¹
x
Step 3:
update the partition matrix
if
gs
D
!
for all
0,
g
= 1, 2, 3, ....,
c
.
1
l
gs
P
,1
d d
g
c
;1
d d
s
N
,
c
2
m
1
¦
DD
gsA
hsA
h
1
else,
c
>@
l
l
l
gs
P
0 and
P
0,1 , with
¦
P
1
gs
gs
g
1
until
1
UU
H
l
l
.
Listed below are a few general remarks on the fuzzy
c
-means algorithm.
1.
The “if and else” branch at step 3 takes care of singularity that occurs in
fuzzy
c
-means when the distance term
for some
Z
s
and certain
0
D
gsA
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