Error and Correlation as fitness functions for Scaled Symbolic Regression in Grammatical Evolution

Linear scaling has greatly improved the performance of genetic programming when performing symbolic regression. Linear scaling transforms the output of an expression to reduce its error. Mean squared error and correlation have been used with scaling, often interchangeably and with assumed equivalence. We examine if this equivalence is justified by investigating the differences between an error-based metric and a correlation-based metric on 11 well-known symbolic regression benchmarks. We investigate the effect a change of fitness function has on performance, individuals size and diversity. Error-based scaling and Correlation were seen to attain equivalent performance and found solutions with very similar size and diversity on the majority of problem, but not all. In order to ascertain if the strengths of both approaches could be combined, we explored a double tournament selection strategy, where two tournaments are conducted sequentially to select individuals for recombination. Double tournament selection found smaller solutions and the best solution in five benchmarks, including finding the best solutions on both real-world dataset used in our experiments.