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In-Depth Information
•
Huber's function:
y
2
2
|
y
| ≤
β
for
f
(
y
)
=
(3.7)
| −
β
2
2
β
|
y
for
|
y
|
> β
•
Talvar's function:
y
2
2
for
|
y
| ≤
β
f
(
y
)
=
(3.8)
2
β
for
|
y
|
> β
2
•
Hampel's Function:
1
−
cos
π
y
β
for
2
β
|
y
| ≤
β
π
f
(
y
)
=
(3.9)
2
2
β
|
y
|
>
β
for
π
where
β
is a problem-dependent control parameter.
Replacing the finite sum in eq. (3.2) by the theoretical expectation of
f
gives
the following cost function:
J
robust EXIN
(w)
=
E
f
w
T
x
√
w
(3.10)
T
w
Then
w
w
x
−
w
T
x
w
g
w
T
T
x
√
w
dJ
robust EXIN
(w)
d
w
=
E
w
w
3
2
T
w
T
is odd and assuming that
w
T
x
<ε
∀
ε >
Considered that
g
(
y
)
0, the following
approximation is valid:
g
w
T
x
g
w
T
x
√
w
√
w
(3.11)
∼
T
w
T
w
which, in a stochastic approximation-type gradient algorithm in which the expec-
tations are replaced by the instantaneous values of the variables, gives the
robust
MCA EXIN learning law
(NMCA EXIN):
g
(3.12)
α (
t
)
g
(
y
(
t
))
y
(
t
) w (
t
)
w
w (
t
+
1
)
=
w (
t
)
−
(
y
(
t
))
x
(
t
)
−
w
T
(
t
) w (
t
)
T
(
t
) w (
t
)
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