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In-Depth Information
7
Polynomial Induction
and Time Series Prediction
In this chapter we are going to explore further the idea of domains of random
numerical constants in order to evolve high-order multivariate polynomials.
Then we are going to discuss the importance of these so called Kolmogorov-
Gabor polynomials in evolutionary modeling by comparing the performance
of this new algorithm, GEP-KGP (GEP for inducing Kolmogorov-Gabor poly-
nomials), with much simpler and much more intelligible GEP systems. The
chapter finishes with the detailed description of all the steps in time series
prediction, from preparing the data and building the model through to mak-
ing predictions about future events.
7.1 Evolution of Kolmogorov-Gabor Polynomials
Kolmogorov-Gabor polynomials have been widely used to evolve general
nonlinear models (Iba and Sato 1992, Iba et al. 1994, Ivakhnenko 1971,
Kargupta and Smith 1991, Nikolaev and Iba 2001). Evolving such polyno-
mials with gene expression programming is very simple and only requires
the implementation of special functions of two arguments. For instance, the
expression tree represented below:
F
x
1
x
2
corresponds to the following mathematical expression:
y
a
a
x
a
x
a
x
x
a
x
2
1
a
x
2
2
(7.1)
0
1
1
2
2
3
1
2
4
5
for the complete second-order bivariate basis polynomial function (the com-
plete series is represented in Table 7.1). The six coefficients
a
0
-
a
5
can be
