Digital Signal Processing Reference
In-Depth Information
To allow the detection of cluster shapes ranging from spherical to
ellipsoidal, different metrics have to be used. Usually, an adaptive metric
is used. In general a distance metric
d
(
x
j
,
L
i
) from the data point
x
j
to
the cluster prototype
L
i
is defined as
d
2
(
x
j
,
L
i
)=(
x
j
−
L
i
)
T
F
−1
(
x
j
−
L
i
)
,
(7.76)
i
where
F
i
is a symmetric and positive definite shape matrix, and adapts
to the clusters' shape variations.
Due to this exponential distance, the Gath-Geva algorithm seeks an
optimum in a narrow local region. Its major advantage is obtaining good
partition results in cases of unequally variable features and densities, but
only when the starting cluster prototypes are properly chosen.
An algorithmic description of the Gath-Geva algorithm is given
below [89]:
1. Initialization and adaptation, part I:
These are similar to the fuzzy
n
-means algorithm.
3. Adaptation, part II:
Determine the fuzzy covariance matrix
F
i
,i
=
1
,
···
,c
by using
N
u
ik
(
x
k
−
L
i
)
T
L
i
)(
x
k
−
k=1
F
i
=
(7.77)
N
u
ik
k=1
4. Adaptation, part III:
Compute the exponential distance
d
e
:
d
e
(
x
j
,
L
i
)=
|
F
i
|
α
i
e
[(
x
j
−
L
i
)
T
F
−1
(
x
j
−
L
i
)/2]
,
(7.78)
i
N
k=1
u
l−1
with the a priori probability
α
i
=
,where
l
−
1isthe
ik
previous iteration.
5. Adaptation, part IV:
Update the membership degrees according to
1
u
ij
=
,
∀
1
≤
i
≤
c
;
1
≤
j
≤
N.
(7.79)
k=1
c
d
e
(
x
j
,
L
i
)
d
e
(
x
j
,
L
k
)
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