Digital Signal Processing Reference
In-Depth Information
7.4
Fuzzy Clustering Algorithms
This section describes several well-known fuzzy clustering algorithms,
such as the generalized adaptive fuzzy
n
-means algorithm, the general-
ized adaptive fuzzy
n
-shells algorithm, the Gath-Geva algorithms, and
fuzzy learning vector quantization algorithms.
Let
X
=
define the data set, and
C
a fuzzy set, on
X
.
The following assumptions are made:
{
x
1
,
···
,
x
p
}
•
C
represents a cluster of points from
X
.
•
C
has a cluster substructure described by the fuzzy partition
P
=
{
A
1
,
···
,A
n
}
.
•
n
is the number of known subclusters in
C
.
The algorithms require a random initialization of the fuzzy partition.
In order to monitor the convergence of the algorithm, the
n
p partition
matrix
Q
i
is introduced to describe each fuzzy partition
P
i
at the
i
th iteration, and is used to determine the distance between two fuzzy
partitions. The matrix
Q
i
is defined as
×
Q
i
=
U
at
iteration
i
.
(7.38)
The termination criterion for iteration
m
is given by
d
(
P
m
,P
m−1
)=
Q
m
Q
m−1
||
||
−
<ε.
(7.39)
where
ε
defines the admissible error and
|| · ||
is any vector norm.
Generalized Adaptive Fuzzy
n
-Means Algorithm
This adaptive fuzzy technique employs different distance metrics such
that several cluster shapes, ranging from spherical to ellipsoidal, can be
detected.
To achieve this, an adaptive metric is used. We define a new distance
metric
d
(
x
j
,
L
i
), from the data point
x
j
to the cluster prototype
L
i
,as
d
2
(
x
j
,
L
i
)=(
x
j
−
L
i
)
T
M
i
(
x
j
−
L
i
)
,
(7.40)
where
M
i
is a symmetric and positive definite shape matrix and adapts
to the clusters' shape variations. The growth of the shape matrix is
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