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Waterhouse et al. (1996) Function and Available Data
f(x) mean
data
1.5
1
0.5
0
-0.5
-1
-1.5
-2
0
0.5
1
1.5
2
2.5
3
3.5
4
Input x
Fig. 8.7.
Plot showing the test function used in [227], and the 200 available observations
Mixed Prediction and Prediction of Classifiers
Fitness and Average Number of Classifiers
-150
5
data
pred +/- 1sd
f(x)
cl. 1
cl. 2
max. fitness
avg. fitness
min. fitness
avg. K
1.5
-160
4
1
0.5
-170
3
0
-180
2
-0.5
-1
-190
1
-1.5
-2
-200
0
0
0.5
1
1.5
2
2.5
3
3.5
4
0
50
100
150
200
250
Input x
GA iteration
(a)
(b)
Fig. 8.8.
Plots similar to the ones in Fig. 8.5, when using a GA for model structure
search applied to the function as used by Waterhouse et al. [227]. The best discovered
model structure is given by
μ
1
=0
.
52
,σ
1
=0
.
016 and
μ
2
=3
.
32
,σ
2
=1
.
000.
classifiers, which is in both cases sampled from
(4
,
0
.
5). As before, the result
is insensitive to this number. The best discovered model structures are shown
in Fig. 8.8 for the GA, with
B
L
(
q
)
−
ln
K
!
≈−
159
.
07, and in Fig. 8.9 for the
MCMC, with
158
.
55. The MCMC search had a reject rate of
about 97
.
0% over its 5000 steps.
Both the GA and the MCMC search resulted in about the same model struc-
ture which at the first sight seems slightly surprising: looking at Fig. 8.7, one
would initially expect the function to be modelled by a flat line over 1
.
5
<x<
4,
and 2 straight lines for the bump at around
x
=0
.
4, requiring altogether 3
L
(
q
)
−
ln
K
!
≈−
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