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its mathematical expression shows, the GEP-RNC algorithm discovered the
numerical constants of function (5.1) with great precision:
2
y
2
.
7179
x
3
1415
x
(5.22b)
5.6.4 Diagnosis of Breast Cancer
In contrast to the two previous computer-generated problems, the breast cancer
problem is a complex real-world problem with nine independent variables,
and investigating whether numerical constants are essential to make an
accurate diagnosis can give us a more realistic view on the importance of
numerical constants to the design of robust mathematical models.
For this classification problem, a very simple function set consisting of
the basic arithmetic operators was chosen and used in all the three approaches,
that is, F = {+, -, *, /}, where each function was weighted four times. For the
GEA-B algorithm, the set of terminals consists obviously of the nine at-
tributes used in this problem and were represented by T = {d
0
, ..., d
8
} which
correspond, respectively, to clump thickness, uniformity of cell size, uni-
formity of cell shape, marginal adhesion, single epithelial cell size, bare nu-
clei, bland chromatin, normal nucleoli, and mitoses. For the GEP-NC algo-
rithm, besides the nine attributes, five different rational constants randomly
chosen from the interval [-2, 2] and represented by
c
0
-
c
4
were used, thus
giving T = {d
0
, d
1
, d
2
, d
3
, d
4
, d
5
, d
6
, d
7
, d
8
, c
0
, c
1
, c
2
, c
3
, c
4
}, where
c
0
= 0.938233,
c
1
= 1.859162,
c
2
= -0.190002,
c
3
= 1.498413,
c
4
= -0.242737. For the GEP-
RNC algorithm, the set of terminals consists obviously of the nine attributes
plus the ephemeral random constant “?”, thus giving T = {d
0
, ..., d
8
, ?}. Fur-
thermore, 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 rational interval [-2, 2] (the complete
list of all the parameters used per run is shown in Table 5.9).
And as you can see in Table 5.9, the inclusion of numerical constants in
the evolutionary toolkit didn't result in a better performance. Indeed, both
the GEP-NC and the GEP-RNC algorithms perform slightly worse than the
simpler GEA-B approach, as the average best-of-run fitness indicates (339.340
for the GEA-B algorithm, 339.050 for the GEP-NC algorithm, and 339.130
for the GEP-RNC algorithm). Also interesting is the fact that, even when
readily available, numerical constants were seldom integrated in the best-of-
run solutions as will next be shown.
