Evolving Simple Symbolic Regression Models 9
Ta b l e 1 Algorithm settings for the performed experiments (multiple values indicate
alternatives)
Standard GP NSGA-II
Population size 1000 1000
Maximum generations 500 500
Maximum evaluations 500,000 500,000
Objective function(s) maxR^2 maxR^2 , min length
R^2 , min visitation length
R^2 , min variables count
R^2 , min complexity
Maximum tree length 20 100
50
100
Terminal symbols constant, weightvariable
Function symbols C;;;=;sin;cos;tan;exp;log;x^2 ;p
xreplacement is repeated until a specified number of generations are reached. We
choose 500 generations as termination which in combination with a population
size of 1000 results in500;000model evaluations. For every algorithm variant and
every problem 50 independent algorithm executions have been conducted to take
the stochasticity of the algorithms into account.
4.1 Problems
We have used a wide variety of benchmark problems to test the suitability and the
effects of the presented approach. The first experiments were conducted on newly
defined benchmark problems (Table 2 , ProblemF 1 F 5 ) that have been designed
to include polynomial terms and more complex ones containing trigonometric or
exponential functions. All input variablesxiwere sampled uniformly fromUŒ5; 5.
Due to the fact that these problems do not contain any noise, a model representing
the data generating formula can be found and the effects of multi-objective symbolic
regression and the new complexity measures can be studied.
In addition, more complex, well known problems, which have been recom-
mended as benchmark symbolic regression problems (White et al. 2013 ) have been
used for testing. The first two problems, Breiman et al. ( 1984 ) and Friedman ( 1991 ),
contain superficial features and have noise added to the dependent variable. The
remaining three problems consist of real-world data available at the HeuristicLab
website.^2 Hence, these problems cannot be solved exactly and simulate a more
practically relevant setting.
(^2) http://dev.heuristiclab.com/AdditionalMaterial#Real-worlddatasets