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F Y ( y )= F X ( y/Δ ) ,
f Y ( y )= 1
Δ f X ( y ) ,
(4.36)
( y )= 1
Δ
2 df X
df Y
dy
dx ( y ) .
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 )
f X ( y )
F X ( y )
( y )ln
2 F X ( y ) +
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
2 π exp
,x> 0 .
1
(ln x − μ ) 2
2 σ 2
f X ( x )=
(4.40)
 
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