Evolving Simple Symbolic Regression Models 7
Standard Discrete020406080Number of Models in the Pareto frontFig. 2 Number of models in the final Pareto front of 50 repetitions for problemF 1 of NSGA-II
with standard and discretized objective functions
resulting in more models having the same prediction accuracy and therefore, a
higher selection pressure is applied to build simple models.
Furthermore, the generated Pareto fronts contain fewer models as minor improve-
ments in prediction accuracy are neglected. The differences between a discrete
objective function and the standard definition of the Pearson’sR^2 are shown in
Fig. 2 , where the number of models in the final Pareto front of 50 algorithm
repetitions on problemF 1 (see Sect.4.1) are displayed as box plots. By using
discrete objective values the size of the Pareto front is almost halved compared to
using the exact numeric value.
Illustrative examples of two Pareto fronts extracted from the performed algorithm
repetitions are displayed in Fig. 3. The Pareto fronts are shown as the models’
normalized mean squared error NMSE (Eq. ( 5 )) and their tree lengths. The NMSE
has been used for describing the results, while the Pearson’sR^2 is used as an
objective during optimization. The reason therefore is that the NMSE is not invariant
to translation and scaling (contrary to theR^2 ) and allows an unbiased comparison
on different data partitions such as training and test.
NMSE.y;y^0 /D1=nP
.yiy^0 i/^2
var.y/(5)While the Pareto front generated with a discretized objective function (Fig. 3 )
contains only 11 models, the standard one includes 33 models. The most accurate
prediction models have a length of 24 or 91 tree nodes respectively. Another aspect
is that the accuracy in the Pareto front without discretized objective values for
models larger than 40 nodes increases only by3:5 10^5 and can be regarded as
irrelevant.