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In-Depth Information
For a grid of
μ
1
values in [0
,
4] and a variable number of instances per class,
n
, 400 experiments were performed by generating the class distributions with
the above settings and determining the empirical MEE point; this allowed
determining for each
μ
1
value the percentage of MEE points falling at an
interval end, and the average and standard deviation of the error rate. For
n
= 120 instances per class, Fig. 4.9a shows the interval-end hit rate; Fig. 4.9b
shows the average error rate (
standard deviation) with the superimposed
curve of min
P
e
(the Bayes error rate) computed with formula (3.34).
±
0.5
I
n
terval−end hits
P
e
0.5
0.45
0.4
0.4
0.35
0.3
0.3
0.25
0.2
0.2
0.15
0.1
0.1
0.05
μ
1
μ
1
0
0
0
0.5
1
1.5
2
2.5
3
3.5
4
0
0.5
1
1.5
2
2.5
3
3.5
4
Fig. 4.9 Empirical MEE for a two-class setting with Gaussian distributions,
g
0
(
x
;0
,
1)
and
g
1
(
x
;
μ
1
,
1)
: a) interval-end hit rate; b) Average error rate (solid
line) with
±
standard deviation (shadowed area) with the theoretical
min
P
e
(dot-
ted line).
From this and other experiments with smaller values of
n
, the following
conclusions can be drawn:
1. The experimental findings confirm the theoretical turn-about value for
symmetrical Gaussian distributions (Sect. 4.1.2), corresponding here to
μ
1
=1
.
405.Below
μ
1
the interval-end hit rate increases steadily, as shown
in Fig. 4.9a, reaching values near 50% or even above (for small values of
n
).
The average error rate also exhibits the turn-about effect: the estimated
derivative (using a moving-average filtered version of the average error
rate, with filter lengths of 3 to 9), exhibits an inflection point at 1
.
41,
close to
μ
1
. For smaller values of
n
the inflected aspect of the average
error rate is also present but in a less pronounced way.
2. Above
μ
1
the average error rate is close to the Bayes error with fast
decreasing standard deviation. This behavior is also observed for small
values of
n
.
The same experiments carried out on mutually symmetric triangular distri-
butions with unit interval (
b
a
=1; see 2.47) and lognormal distributions
with
σ
=0
.
5 lead to the same conclusions. Figure 4.10 shows the results for
these distributions obtained in 400 experiments with
n
=50. The empirical
−
