Information Technology Reference
In-Depth Information
Example 3.2.
Let us suppose that the error PDF of Example 3.1 is convolved
with
ψ
(
e
)=
g
(
e
;0
,h
) prior to the maximization of the information potential:
1
2
(
f
E|−
1
(
e
)+
f
E|
1
(
e
))
⊗
ψ
(
e
)
.
(3.21)
Since convolution is a linear operator, we can write
1
2
(
f
E|−
1
(
e
)
⊗
ψ
(
e
)+
f
E|
1
(
e
)
⊗
ψ
(
e
))
.
(3.22)
We now apply to each
f
E|t
(
e
)
⊗
ψ
(
e
)
,t
∈{−
1
,
1
}
, the result of Proposition
ψ
(
e
)=
g
(
e
;
td,
√
σ
2
+
h
2
).
Proceeding to the computation of the information potential along the same
steps as in the previous example, we finally arrive at
3.1:
f
E|t
(
e
)
⊗
1+exp
.
d
2
σ
2
+
h
2
1
4
√
π
√
σ
2
+
h
2
V
R
2
=
−
(3.23)
Denoting
A
=exp(
−
d
2
/
(
σ
2
+
h
2
)), the first-order derivatives
2
d
2
A
σ
2
+
h
2
∂V
R
2
∂σ
σ
4
√
π
(
σ
2
+
h
2
)
3
/
2
=
−
1
−
A+
and
(3.24)
∂V
R
2
∂d
2
d
A
4
√
π
(
σ
2
+
h
2
)
3
/
2
=
−
can be used in (3.11) to perform the gradient ascent maximizing the potential
V
R
2
.
With the derivatives (3.24) one obtains, for suciently large
h
, a conver-
gent behavior similar to the one shown in Fig. 3.1. As a matter of fact, it is
enough to use
h
0
.
75 to obtain a convergence of both
d
and
σ
to zero, and
therefore to min
P
e
=0, even when the initial value of
σ
is very small (say,
0.01). It is, however, inconvenient to use a too large
h
,sinceforthesame
η
the number of needed iterations until reaching some target
d
(or
σ
)grows
with
h
according to a cubic law.
≥
Example 3.2 suggests how to use empirical EEs to our advantage.
For a given
n
,let
f
n
(
e
) denote the Parzen window estimate of the error
density obtained with the optimal bandwidth in the
IMSE
sense,
h
IMSE
,for
that
n
(see Appendix E):
f
n
(
e
)=
μ
n
⊗
G
h
IMSE
(
e
)
,
(3.25)
where
μ
n
(
e
) is the empirical density of the error. Instead of
f
n
(
e
),onecom-
putes a
fat estimate
f
fat
(
e
):
