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corresponds to 12. For the GEP-RNC algorithm, the set of terminals consists
obviously of the three tolerances plus the ephemeral random constant “?”,
thus giving T = {a, b, c, ?}. Furthermore, a set of random numerical constants
represented by the numerals 0-9 was used, thus giving R = {0, 1, 2, 3, 4, 5, 6,
7, 8, 9}, and the ephemeral random constant “?” ranged over the integer
interval [0, 100] (the complete list of all the parameters used per run is shown
in Table 5.10).
And as you can see in Table 5.10, for this problem, the inclusion of nu-
merical constants in the evolutionary toolkit is responsible for a consider-
able increase in performance. Indeed, both the GEP-NC and the GEP-RNC
algorithms perform considerably better than the simpler GEA-B approach,
as both the average best-of-run fitness (12.9010 for the GEA-B system,
36.8972 for the GEP-NC algorithm, and 38.7630 for the GEP-RNC algo-
rithm) and average best-of-run R-square (0.6871088316 for the GEA-B al-
gorithm, 0.9097327677 for the GEP-NC algorithm, and 0.9141464770 for
the GEP-RNC algorithm) indicate. Notice also that, of the two algorithms
with numerical constants, the GEP-RNC algorithm performs considerably
better than the GEP-NC algorithm, showing again that the facility for han-
dling random numerical constants in gene expression programming is in-
deed extremely efficient and, therefore, the ideal tool for dealing with great
quantities of numerical constants.
The best solution designed with the GEA-B algorithm was found in gen-
eration 2696 of run 62 (the sub-ETs are linked by addition):
0123456789012345678901234
EEQQEQQ/EEQ+aabacaacaacaa
EEQQQQQ/EEQ+aacaaaccabbab
EEQQQQQ/EE-+abcaabbaacbbb
EEQQEQ/Q++EEbbcabbaacbbba
EEQQQQQ/*E**aaaaabcbbbbbc
EEQQQQQ/EE-+bacbcccaccbca
(5.26a)
It has a fitness of 48.4756 and an R-square of 0.9394514385 evaluated over
the training set of 40 fitness cases and is therefore a good solution to the
problem at hand. Note that the R-square of this best-of-experiment solution
is much higher than the average best-of-run R-square obtained in this experi-
ment (0.6871088316), showing that, although harder to find, good solutions
can nevertheless be designed without numerical constants. More formally,
the model (5.26) can be expressed by the following C++ function:
