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3.4 Solving a Simple Problem with GEP
The aim of this section consists of studying a successful run in its entirety in
order to understand how populations adapt, finding in the process perfect or
good solutions to the problem at hand.
And the problem we are going to solve with gene expression program-
ming, is a very simple problem of symbolic regression (also called function
finding), where the goal is to find an expression that satisfactorily explains
the dependent variable. The input into the system is a set of fitness cases in
the form (
a
(
i
,1)
,
a
(
i
,2)
,...,
a
(
i
,
n
)
,
y
i
) where
a
(
i
,1)
-
a
(
i
,
n
)
are the
n
independent vari-
ables and
y
i
is the dependent variable. We have already seen that the set of
fitness cases consists of the environment in which populations of candidate
solutions (computer programs that are, in fact, the output of the system) adapt,
discovering, with time, good or perfect solutions to the problem at hand.
In the simple example of this section, the fitness cases were computer
generated using a test function. This means that, for this particular problem,
we know exactly what the perfect solution to our problem should look like.
And this is indeed a good thing because then we will be able to undoubtedly
recognize and rightly appreciate the amazing results produced by the blind
action of the genetic operators. Keep in mind, however, that in real-world
problems the target function is what we are trying to discover and is obvi-
ously unknown.
So, let's suppose that we are given a sampling of the numerical values
from the curve:
a
2
y
3
a
(3.19)
2
over 10 randomly chosen points in the real interval [-10, +10] and we wanted
to find a function fitting those values within a certain error. In this case, we
are given a sample of data in the form of 10 pairs (
a
i
,
y
i
), where
a
i
is the value
of the independent variable in the given interval and
y
i
is the respective value
of the dependent variable (Table 3.2). These 10 pairs are the fitness cases
(the input to the system) that will be used as the selection environment, against
which the fitness of each individual will be evaluated generation after gen-
eration. And, hopefully, after a certain number of generations, the individual
better adapted to this environment will be a perfect solution to our problem.
There are five major steps in preparing to use gene expression program-
ming, and the first is to choose the fitness function. For this problem we will
measure the fitness by equation (3.3a), using an absolute error of 100 as the
