Digital Signal Processing Reference
In-Depth Information
Proposition 7.2 is the proposition for optimal prototype centers.
Proposition 7.2:
The optimal value of
V
with respect to the function
J
(
P,
·
,R,M
)isgivenby
p
d
ij
q
ij
u
ij
(
x
j
−
v
i
)=0
,
i
=1
,
···
,n,
(7.62)
j=1
where
q
ij
is given by
v
i
)
T
M
i
(
x
j
−
q
ij
=(
x
j
−
v
i
)
(7.63)
Proposition 7.3 is the proposition for optimal prototype radii.
Proposition 7.3:
The optimal value of
R
with respect to the function
J
(
P, V,
·
,M
)isgivenby
p
u
ij
d
ij
=0
,
i
=1
,
···
,n.
(7.64)
j=1
To ensure that the adaptive norm is bounded, we impose the con-
straint
|
M
i
|
=
ρ
i
,
where
ρ
i
>
0
,
i
=1
,
···
,n
(7.65)
The norm is given by theorem 7.4, the adaptive norm theorem [71].
Theorem 7.4:
Let
X
R
s
. Suppose the objective function
J
already contains
the optimal
P, V
,and
R
. If the determinant of the shape matrix
M
i
is bounded,
⊂
|
M
i
|
=
ρ
i
,
i
>
0
,
i
=1
,
···
,n
,then
M
i
is a local
minimum of the function
J
(
P, V, R,
·
)onlyif
]
s
S
−1
si
M
i
=[
ρ
i
|
S
si
|
,
(7.66)
where
S
si
represents the nonsingular shell scatter matrix of the fuzzy
class
A
i
:
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