Information Technology Reference
In-Depth Information
Had we swapped the assigned labels we would obtain in general a different
value:
PE
(
p, α, β
)=
pP E
1
+
qPE
0
=
pα
+
qβ
.
•
Entropy-of-Error criterion:
P
10
≡ P
(
E
=1)=
p
(1
− α
);
P
01
≡ P
(
E
=
−
1) =
q
(1
− β
);
(4.51)
To these values we apply formula 4.46. Again, for swapped assignments
one obtains in general a different value.
The three-dimensional functions expressed by formulas (4.46), (4.47), (4.49)
and (4.50) are dicult to mathematically analyze and compare. It turns out
to be more instructive to analyze these functions by considering two categories
of configurations —
α
=
β
and
α
=
β
— and inspect the respective values.
Configurations with
α
=
β
For
α
=
β
configurations
PE
doesn't depend on
p
(
PE
=1
α
). All criteria
depend on two parameters alone,
p
and
PE
, which facilitates their inspection
as in Fig. 4.12. The behavior of
TWO
is similar to that of
IG
.Weobserve
that
SEE
is concave, symmetric in
p
but asymmetric in
PE
! In conclusion,
SEE
displays, just as
GI
and
IG
, a desirable concave behavior
but more
sensitive
to the
PE
value.
−
1.4
SEE
1.2
1
1
IG
0.8
0.5
0.6
0.4
0
0
0
0.2
0.2
0.4
p
0
0
p
0.5
0.6
0
0.5
0.8
0.5
PE
PE
1
1
1
1
IG and SEE for
α
=
β
(adapted from [152]).
Fig. 4.12
Configurations with
α
=
β
Figure 4.13 shows the four criteria represented in a (
α, β
) grid for
p
=0
.
7.
We readily see that
IG
,
TWO
and
SEE
are concave functions whereas
PE
is a linear function. Figure 4.14 shows the error PMFs for four specifications
of (
α, β
). Table 4.3 lists the respective criteria values.
IG
and
TWO
found
cases 'a' and 'b' to be similar, which seems rather inadequate, whereas both
PE
and
SEE
found 'b' to be an improvement over 'a'. Cases 'b' and 'c'
correspond to those analyzed in [33] to illustrate why
PE
should not be
used. Note that
IG
and
TWO
select 'c' with a decrease of 16% and 11%,
respectively;
SEE
also selects 'c' and with a more pronounced decrease: 22%.
Therefore, for equal probability of error (
PE
=0
.
2 in both cases),
SEE
also
