Information Technology Reference
In-Depth Information
6
6
x
2
x
2
5
4
4
2
3
0
2
1
−2
0
−4
−1
−6
−2
−8
−3
x
1
x
1
−4
−10
−4
−3
−2
−1
0
1
2
3
−5
0
5
10
(a)
(b)
3
2
x
2
x
2
1.5
2
1
1
0.5
0
0
−1
−0.5
−2
−1
−3
−1.5
x
1
x
1
−4
−2
−3
−2
−1
0
1
2
3
−1.5
−1
−0.5
0
0.5
1
1.5
(c)
(d)
Fig. 3.17 Decision borders obtained with empirical MEE (dashed), theoretical
MEE (dotted), and
min
P
e
(solid) for WDBC
2
(a), Thyroid
2
(b), Wine
2
(c), and
PB12 (d) datasets.
p
t
1
V
R
2
(
E
)=
t
σ
t
4
√
π
2
√
πσ
t
(1 +
m
t
)+
(3.46)
μ
t
+
w
0
,and
σ
t
=
w
T
Σ
t
w
.
where
p
t
are the class priors,
m
t
=
w
T
Proof.
Let us denote the class-conditional sum of the independent Gaussian
inputs by
U
|
t
. We know that:
μ
t
+
w
0
,
w
T
Σ
t
w
)=
g
(
u
;
m
t
,σ
t
)
.
g
(
u
;
w
T
U
|
t
∈
]
−∞
,
+
∞
[
∼
(3.47)
The sum of the inputs is submitted to the atan(
·
) sigmoid, i.e., the per-
ceptron output is
Y
=
ϕ
(
U
)=atan(
U
)
∈
]
−
π/
2
,π/
2[. We could define
2
Y
=
1
,
1[, but it turns out that this would unnecessarily com-
plicate the following computations. We then also take
T
=
π
atan(
U
)
∈
]
−
;
although different from what we are accustomed to (real-valued instead of
integer-valued targets) it is perfectly legitimate to do so. When using
t
sub-
scripts, we still assume
t
{−
π/
2
,π/
2
}
∈{−
1
,
1
}
for notational simplicity (the class names
1”and“1”), but what is really meant for the targets is
t
2
.
We now determine
f
Y |t
(
y
) using Theorem 3.2. We have:
are still “
−
1
1+
u
2
;
ϕ
(
ϕ
−
1
(
y
)) =
1
1+tan
2
(
y
)
u
=
ϕ
−
1
(
y
)=tan(
y
);
ϕ
(
u
)=
.
(3.48)
