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Matching by Soft Intervals
1
cl. 1
cl. 2
cl. 3
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
x
Fig. 8.3. Matching probability for matching by soft interval for different parameters.
Classifiers 1 and 2 are adjacent as l 1 =0, u 1 = l 2 =0 . 2, and u 2 =0 . 5. The area where
these two classifiers overlap shows that the classifiers do not match their full interval
with probability 1 due to the soft boundaries of the intervals. Nonetheless, 95% of
the area beneath the matching function are within the specified interval. Classifier 3
matches the interval l 3 =0 . 7, u 3 =0 . 9. Comparing the boundary of classifier 2 and
3 shows that the spread of the boundary grows with the width of the interval that it
matches.
respectively. A plot of the classifiers' means, their generated function mean, and
the available data can be found in Fig. 8.4.
Both GA and MCMC model structure search were tested, where the GA is
in this and all other experiments initialised with a population of size P = 20,
crossover and mutation probability p c = p m =0 . 4, and tournament size t s =5.
The number of classifiers in each of the individuals is sampled from the binomial
distribution
(8 , 0 . 5), such that, on average, an individual has 4 classifiers. The
performance of the GA model structure search is not sensitive to the initial
size of the individuals and gives similar results for different initialisations of its
population.
The result after a single run with 250 GA iterations are shown in Fig. 8.5. As
can be seen, the model was not correctly identified as the number of classifiers
of the best found individual is 2 rather than the desired 3, with
B
118 . 81. Nonetheless, the generated function mean is still within the first standard
deviation of the predicted mean.
The MCMC model structure search was applied to the same data, using for
this and all further experiments 10 restarts with 500 steps each, and p ( A =
add )= p ( A = remove )=1 / 4. Thus, MCMC search uses the same number
L
( q )
ln K !
 
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