Information Technology Reference
In-Depth Information
term is what is called the precision
p
. Thus, for a perfect fit,
P
(
ij
)
=
T
j
for all
fitness cases and
f
i
=
f
max
=
n·R
.
Mean Squared Error
The
mean squared error
fitness function is based on the standard mean
squared error, which, on its turn, is based on the absolute error. But obviously
the relative error can also be used and two different fitness functions can be
designed: one based on the absolute error and the other on the relative.
More formally, the mean squared error
E
i
of an individual program
i
is
expressed by equation (3.4a) if the error chosen is the absolute error, and by
equation (3.4b) if the error chosen is the relative:
1
n
¦
2
E
P
T
(3.4a)
i
(
ij
)
j
n
j
1
2
§
P
T
·
1
n
¦
¨
©
(
ij
)
j
¸
¹
E
(3.4b)
i
n
T
j
1
j
where
P
(
ij
)
is the value predicted by the individual program
i
for fitness case
j
(out of
n
fitness cases) and
T
j
is the target value for fitness case
j
. Thus, for
a perfect fit,
P
(
ij
)
=
T
j
for all fitness cases and
E
i
= 0. So, the mean squared
error ranges from 0 to infinity, with 0 corresponding to the ideal.
As it stands,
E
i
cannot be used directly to measure the fitness of the evolved
models since, for fitness proportionate selection, the fitness must increase
with efficiency. Thus, to evaluate the fitness
f
i
of an individual program
i
, the
following equation is used:
1
f
1000
(3.5)
i
1
E
i
which obviously ranges from 0 to 1000, with 1000 corresponding to the ideal.
The fact that both these fitness functions are not only based on standard
indicators but also very easy to implement, makes them very attractive; they
can indeed be used to find good solutions to virtually all problems. Further-
more, they can also be used as a basis for designing other fitness functions.
For instance, a slightly different kind of fitness function can be designed by
taking the square root of the mean squared error. By doing this we are giving
it the same dimensions as the quantity being predicted, so it might be inter-
esting to use this measure instead of the straight mean squared error.
