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from its centre is 0
.
6827
√
2
πσ
. Solving these equations with respect to
b
k
and
σ
k
for a given
b
k
results in
1
0
.
95
−
1
1
√
2
π
b
k
≈
σ
k
=
0
.
0662
b
k
,
(8.11)
1
−
0
.
6827
0
.
6827
√
2
πσ
k
≈
b
k
=
b
k
−
0
.
8868
b
k
.
(8.12)
Thus, about 89% of the specified interval are matched with probability 1, and
the leftover 5.5% to either side are matched according to one standard deviation
of a Gaussian. Therefore, the matching function for soft interval matching is
given by
⎧
⎨
exp
l
k
)
2
1
if
x<l
k
,
−
2
σ
k
(
x
−
exp
u
k
)
2
m
k
(
x
)=
1
(8.13)
−
2
σ
k
(
x
−
if
x>u
k
⎩
1
otherwise
,
where
l
k
and
u
k
are the lower and upper bound of the interval that the classifier
matches with probability 1, and are given by
l
k
l
k
+0
.
0566
b
k
and
u
k
≈
≈
l
k
=
b
k
. Fig. 8.3 shows examples for the shape of
the matching function for soft interval matching.
Classifier
k
is initialised as by Stone and Bull, by sampling
l
k
and
u
k
from
by a uniform distribution over [
l, u
], which is the range of
x
.If
l
k
>u
k
,then
their values are swapped. While Stone and Bull [203] and Wilson [239] mutate
the boundary values a uniform random variable, here the changes are sampled
from a Gaussian to make small changes more likely than large changes. Thus,
the boundaries after mutation are given by perturbing both bounds by
0
.
0566
b
k
, such that
u
k
−
u
k
−
−
l
)
/
10), that is, a sample from a zero-mean Gaussian with a standard deviation
that is a 10th of the range of
x
. After that, it is again made sure that
l
N
(0
,
(
u
≤
l
k
<
u
k
≤
u
by swapping and bounding their values if required.
Even though both matching functions are only introduced for the case when
D
X
= 1, they can be easily extended to higher-dimensional input spaces. In
the case of radial-basis function matching, the matching function is specified
by a multivariate Gaussian, analogous to the hyper-ellipsoidal conditions for
XCS [41, 52]. Matching by a soft interval becomes slightly more complex due
to the interval-specification of the matching function, but its computation can
be simplified by defining the matching function as the product of one single-
dimensional matching function per dimension of the input space.
8.3.2
Generated Function
To see if the optimality criterion is correct if the data conforms to the underlying
assumptions of the model, it is firstly tested on a function that was generated to
satisfy these assumptions. The data is generated by taking 300 samples from 3
linear classifiers with models
|−
1
.
5+2
.
5
x,
0
.
1) which use radial-basis function matching with (
μ, σ
2
) parameters
(0.2, 0.05), (0.5, 0.01), (0.8, 0.05) and mixing weights
v
1
=0
.
5
,v
2
=1
.
0
,v
3
=0
.
4,
N
(
y
|
0
.
05 + 0
.
5
x,
0
.
1),
N
(
y
|
2
−
4
x,
0
.
1), and
N
(
y
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