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member of any cluster nor are they in the boundary of any cluster, but they are far
apart from any cluster. These can be easily detected from the partition matrix of a
possibilistic partition, as the member element of a particular cluster will have
degree of membership 1.0 and the boundary points of two clusters may have a
degree of membership close to 0.5 for both the two clusters, whereas outliers may
have degree of membership as low as 0.01 in all clusters, indicating that the said
data point (probably noise) is far off from all clusters.
4.7.2 Fuzzy c-means Clustering
The fuzzy
c
-means clustering algorithm is one of the most popular clustering
algorithms used for data-driven automated fuzzy rules generation. The
minimization of the
c
-means functional (4.21) represents a nonlinear optimization
problem that can efficiently be solved using genetic algorithms; here, however the
method chosen is a simple Picard iteration through the first-order conditions for
stationary points of (4.21), known as the fuzzy
c
-means (FCM) algorithm.
The stationary points of the objective function (4.21) can be found by adjoining
the constraint (4.20b) to
J
by means of Lagrange multipliers
ª
º
cN
m
N
c
JZUV
;,,
O
P
2
P
1,
¼
(4.21)
¦¦
¦
O
¦
D
gs
«
»
gsA
s
gs
¬
gs
11
s
1
g
1
and by setting the gradients of
J
with respect to
U
,
V
and
Ȝ
to zero. It can be proven
that if
!
then
M
u
nc
D
!
2
0,
gs
,
and
m
1,
UV
,
u\ may minimisz (4.21)
gsA
fc
only if
1
P
,1
d d
g
c
;1
d d
s
N
;
(4.22a)
gs
c
2
m
1
¦
DD
gsA
hsA
h
1
and
m
N
Z
¦
P
gs
s
v
s
1
;1
d d
g
c
.
(4.22b)
g
N
m
¦
P
gs
s
1
It is to be noted that this solution also satisfies the remaining constraints (4.20a)
and (4.20c). Equations (4.22a) and (4.22b) are first-order necessary conditions for
stationary points of the functional (4.21). The FCM algorithm iterates through
Equations (4.22a) and (4.22b). Sufficiency of (4.21) and the convergence of the
FCM algorithm is reported by Bezdek (1980). Also, note that Equation (4.22b)
gives
V
g
as the weighted mean of the data items that belong to a cluster, where the
weights are the membership degrees in the clusters. This being the reason why the
algorithm is called “fuzzy
c
-means.” The FCM algorithm is described next.
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