In the following, the abbreviation GP refers to tree-based GP without local optimization and GPC refers to tree-based GP with local optimization. Both algorithms can be used with and without shape constraints. Table 2 shows the parameter values that have been used for the experiments with GP and GPC. ITEA refers to the Interaction-Transformation Evolutionary Algorithm and FI-2POP-IT (short form: FIIT) refers to the two-population version of ITEA which supports shape constraints. Table 3 shows the parameters values for ITEA and FI-2POP-IT.

Table
2:

Parameter . | Value . |
---|---|

Population size | 1000 |

Generations | 200 |

20 (for GPC with memetic optimization) | |

Initialization | PTC2 |

$Lmax$, $Dmax$ | 50, 20 |

Fitness evaluation | NMSE with linear scaling |

Selection | Tournament (group size = 5) |

Crossover | Subtree crossover |

Mutation (one of) | Replace subtree with random branch |

Add $x\u223cN(0,1)$ to all numeric parameters | |

Add $x\u223cN(0,1)$ to a single numeric parameter | |

Change a single function symbol | |

Crossover rate | 100% |

Mutation rate | 15% |

Replacement | Generational with a single elite |

Terminal set | real-valued parameters and input variables |

Function set | $+,*,%,log,exp,sin,cos,tanh,x2,x$ |

GPC | max. 10 iterations of Levenberg-Marquardt (LM) |

Parameter . | Value . |
---|---|

Population size | 1000 |

Generations | 200 |

20 (for GPC with memetic optimization) | |

Initialization | PTC2 |

$Lmax$, $Dmax$ | 50, 20 |

Fitness evaluation | NMSE with linear scaling |

Selection | Tournament (group size = 5) |

Crossover | Subtree crossover |

Mutation (one of) | Replace subtree with random branch |

Add $x\u223cN(0,1)$ to all numeric parameters | |

Add $x\u223cN(0,1)$ to a single numeric parameter | |

Change a single function symbol | |

Crossover rate | 100% |

Mutation rate | 15% |

Replacement | Generational with a single elite |

Terminal set | real-valued parameters and input variables |

Function set | $+,*,%,log,exp,sin,cos,tanh,x2,x$ |

GPC | max. 10 iterations of Levenberg-Marquardt (LM) |

Table 3:

Parameter . | Value . |
---|---|

Population size | 200 |

Number of Iterations | 500 |

Function set | $sin,cos,tanh,,log,log1p,exp$ |

Fitness evaluation | RMSE |

Maximum number of terms (init. pop.) | 4 |

Range of strength values (init. pop.) | $[-4,4]$ |

Min. and max. term length | $2,6$ |

Regression model | OLS |

Parameter . | Value . |
---|---|

Population size | 200 |

Number of Iterations | 500 |

Function set | $sin,cos,tanh,,log,log1p,exp$ |

Fitness evaluation | RMSE |

Maximum number of terms (init. pop.) | 4 |

Range of strength values (init. pop.) | $[-4,4]$ |

Min. and max. term length | $2,6$ |

Regression model | OLS |

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