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In-Depth Information
4.4.6
Evolutionary Design
Evolutionary Design (ED) [
1
] is an effective implementation of Genetic Program-
ming (GP) methodology [
6
] for symbolic regression.
In general the goal of the regression task is to discover the relationship between
a set of inputs, or independent variables
x
given an observable output, or dependent
variable
y
. In standard regression techniques the model functional form
(
x
,
w
)is
known beforehand. The only unknown values are some coefficients
w
, i.e., the free
parameters of the model.
ED uses low-level primitive functions. These functions can be combined to spec-
ify the full function. Given a set of functions, the overall functional form induced
by genetic programming can take a variety of forms. The primitive functions are
usually standard arithmetical functions such as addition, subtraction, multiplication
and division but could also include trigonometric and transcendental functions. Any
legal combination of functions and variables can be obtained.
Each individual in GP corresponds to a given mathematical expression, and it is
represented by means of a parse tree. For example, the expression
1)
2
f
(
x
,
y
)
=
2
xy
−
(
x
+
(4.19)
is represented by the parse tree depicted in Fig.
4.5
.
Symbolic regression is then a composition of input variables, coefficients and
primitive functions such that the error of the function with respect to the expected
output is minimized. Shape and size of the solution is not specified before the regres-
sion. Number of coefficients and their values are issues that are determined in the
search process itself. By the use of such primitive functions, genetic programming
is in principle capable of expressing any functional form that use the primitive func-
tions provided by the user. Unlike the traditional methods, the Evolutionary Design
process automatically optimizes both the functional form and the coefficient values.
ED is capable of providing answers in the symbolic language of mathematics,
while others methods only provide answers in the form of sets of numbers, weights,
valid in the context of a model defined beforehand. So, after the training, the explicit
formula of the regression model is promptly available.
-
*
*
Fig. 4.5
Parse tree
representing the mathematical
expression
f
(
x
,
y
)
+
*
+
+
1
1
x
y
x
1
x
1
1)
2
=
2
xy
−
(
x
+
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