  We investigate the addition of constraints on the function image and its
derivatives for the incorporation of prior knowledge in symbolic regression.
The approach is called shape-constrained symbolic regression and allows us to
enforce e.g. monotonicity of the function over selected inputs. The aim is to
find models which conform to expected behaviour and which have improved
extrapolation capabilities. We demonstrate the feasibility of the idea and
propose and compare two evolutionary algorithms for shape-constrained symbolic
regression: i) an extension of tree-based genetic programming which discards
infeasible solutions in the selection step, and ii) a two population
evolutionary algorithm that separates the feasible from the infeasible
solutions. In both algorithms we use interval arithmetic to approximate bounds
for models and their partial derivatives. The algorithms are tested on a set of
19 synthetic and four real-world regression problems. Both algorithms are able
to identify models which conform to shape constraints which is not the case for
the unmodified symbolic regression algorithms. However, the predictive accuracy
of models with constraints is worse on the training set and the test set.
Shape-constrained polynomial regression produces the best results for the test
set but also significantly larger models.
