Digital Signal Processing Reference
In-Depth Information
monitored by the bound
|
M
i
|
=
ρ
i
,
i
>
0
,
i
=1
,
···
,n
(7.41)
R
s
beadataset.Let
C
be a fuzzy set
on
X
describing a fuzzy cluster of points in
X
, and having a cluster
substructure which is described by a fuzzy partition
P
=
Let
X
=
{
x
1
,
···
,
x
p
}
,
x
j
∈
,A
n
}
of
C
. Each fuzzy class
A
i
is described by the point prototype
L
i
∈
{
A
1
,
···
R
s
.
The local distance with respect to
A
i
is given by
d
i
(
x
j
,
L
i
)=
u
ij
(
x
j
−
L
i
)
T
M
i
(
x
j
−
L
i
)
(7.42)
As an objective function we choose
n
p
n
p
d
2
(
x
j
,
L
i
)=
u
ij
(
x
j
−
L
i
)
T
M
i
(
x
j
−
J
(
P,
L
,M
)=
L
i
)
i=1
j=1
i=1
j=1
(7.43)
where
M
=(
M
1
,
,
M
n
).
The objective function chosen is again of the least-squares error type.
We can find the optimal fuzzy partition and its representation as the
local solution of the minimization problem:
···
⎧
⎨
minimize
J
(
P,
L
,M
)
n
u
ij
=
u
C
(
x
j
)
,
j
=1
,
···
,p
(7.44)
i=1
⎩
|
M
i
|
=
ρ
i
,
i
>
0
,
i
=1
,
···
,n
R
sn
L
∈
Without proof theorem 7.3 which regards the minimization of the func-
tions
J
(
P,
L
,
·
), is given. It is known as the adaptive norm theorem.
Theorem 7.3: Assuming that the point prototype
L
i
of the fuzzy class
A
i
equals the cluster center of this class,
L
i
=
m
i
, and the determinant
of the shape matrix
M
i
is bounded,
|
M
i
|
=
ρ
i
,ρ
i
>
0
,i
=1
,
···
,n
,then
M
i
is a local minimum of the function
J
(
P,
L
,
·
)onlyif
]
s
S
−1
i
M
i
=[
ρ
i
|
S
i
|
(7.45)
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