Sum Epsilon-Tube Error Fitness Function Design for GP Symbolic Regression: Preliminary Study

Abstract

Symbolic Regression (SR) is a well-studied method in Genetic Programming (GP) for discovering free-form mathematical models from observed data, which includes not only the model parameters but also its innate structure. Another level of the regression problem is the design of an appropriate fitness function, by which are individual solutions judged. This paper proposes a new fitness function design for symbolic regression problems called a Sum epsilon-Tube Error (STE). The function of this criterion can be visualized as a tube with a small radius that stretches along the entire domain of the approximated function. The middle of the tube is defined by points that match approximated valued (in the so-called control points). The evaluation function then compares, whether each approximated point does or does not belong to the area of the tube and counts the number of points outside of the epsilon-Tube. The proposed method is compared with the standard sum square error in several test cases, where the advantages and disadvantages of the design are discussed. The obtained results show great promise for the further development of the STE design and implementation.