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cluster prototypes
v
g
. In such case, the membership degree cannot be
computed at all and, therefore, zero is assigned to that
g
P
and the
memberships are distributed arbitrarily among other clusters subject to the
constraint that the sum of the degree of membership in each column of the
U
partition matrix must be one.
2.
The fuzzy
c
-means algorithm converges to a local minimum of the
c
-means
functional. Therefore, different initialization may lead to different results.
3.
While steps 1 and 2 are straightforward, step 3 is a bit more complicated, as
a singularity in the fuzzy
c
-means occurs when distance
gs
D
for some
0
Z
s
and one or more
v
g
,
though it is very rare in practice.
4.
In the above iterative optimization scheme used by fuzzy
c
-means loops
through the estimates,
U
(
l-1
)
Æ
v
(
l
)
Æ
U
(
l
)
and terminates as soon as
U
(
l
)
- U
(
l-1
)
< H.
Alternatively, the algorithm can be initialized with
v
(0)
, loop through
l
1
oo
l
l
,
v
U
v
and terminate when
vv
H
l
l
1
.
The error norm (termination tolerance) in the termination criterion is
usually chosen as
l
l
abs
PP
.
Max
gs
gs
gs
Different results may be obtained with the same values of termination
tolerance, since the termination criterion used in the algorithm requires that
more parameters become close to one another.
4.7.2.1.1 Parameters of Fuzzy c-means Algorithm
The following parameters must be specified before the fuzzy
c
-means algorithm is
executed: the number of clusters
c
, the fuzziness exponent
m
, the termination
tolerance and norm-inducing matrix
A
. Moreover, the fuzzy partition matrix
U
must be initialized. The choices for these parameters are now described next.
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