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F
Y
(
y
)=
F
X
(
y/Δ
)
,
f
Y
(
y
∗
)=
1
Δ
f
X
(
y
∗
/Δ
)
,
(4.36)
(
y
∗
)=
1
Δ
2
df
X
df
Y
dy
dx
(
y
∗
/Δ
)
.
Therefore,
Q
(
y
∗
)=
df
Y
dy
F
Y
(
y
∗
)
f
Y
(
y
∗
)
(
y
∗
)ln
2
F
Y
(
y
∗
)
+
F
Y
(
y
∗
)
=0
⇔
1
−
1
Δ
2
df
X
dx
=0
F
X
(
y
∗
/Δ
)
f
X
(
y
∗
/Δ
)
F
X
(
y
∗
/Δ
)
(
y
∗
/Δ
)ln
⇔
2
F
X
(
y
∗
/Δ
)
+
⇔
1
−
1
Δ
2
Q
(
x
∗
)=0
.
⇔
(4.37)
Hence, the solution is
y
∗
=
Δ
x
∗
.
The turn-about value
t
value
for symmetric triangular distributions is
t
value
=0
.
3473
Δ
(see [216] for details). In the following examples we an-
alyze the
t
value
for the case of mutually symmetric Gaussian and lognormal
class distributions.
·
Example 4.3.
Consider two Gaussian distributed classes with scale
Δ
=
σ
±
1
=
σ
=1and distance
d
between their centers (means). Setting
p
=
q
=1
/
2,
Q
(
x
∗
) can easily be re-written as a function of
d
as follows
exp
−d
2
/
4
4
π
(1
d
2
/
8) ln
1
+
d
4
√
2
π
exp(
Φ
(
d/
2)
2
Φ
(
d/
2)
−
Q
(
d
)=
−
,
(4.38)
−
Φ
(
d/
2))
and
Φ
(
d/
2)) +
exp
−d
2
/
8
2
ln
1
(1
d
Φ
(
d/
2)
2
Φ
(
d/
2)
−
√
2
π
Q
(
d
)=0
⇔
−
=0
⇔
⇔
d
=1
.
4052
.
(4.39)
Thus, from the preceding discussion one may conclude that the turn-about
value
t
value
for a Gaussian two-class problem with scale
σ
is
t
value
=1
.
4052
σ
,
that is, one should maximize SEE when
d<t
value
.
Example 4.4.
Consider now the lognormal distribution with PDF
xσ
√
2
π
exp
,x>
0
.
1
(ln
x − μ
)
2
2
σ
2
f
X
(
x
)=
−
(4.40)