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:

GP parameter configuration used for the experiments (% refers to protected division). For the GPC runs, we need fewer generations because of the faster convergence with local optimization.

ParameterValue
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 xN(0,1) to all numeric parameters 
 Add xN(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) 
ParameterValue
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 xN(0,1) to all numeric parameters 
 Add xN(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 values for the ITEA algorithm that have been used for all experiments.

ParameterValue
Population size 200 
Number of Iterations 500 
Function set sin,cos,tanh,,log,log1p,exp 
Fitness evaluation RMSE 
Maximum number of terms (init. pop.) 
Range of strength values (init. pop.) [-4,4] 
Min. and max. term length 2,6 
Regression model OLS 
ParameterValue
Population size 200 
Number of Iterations 500 
Function set sin,cos,tanh,,log,log1p,exp 
Fitness evaluation RMSE 
Maximum number of terms (init. pop.) 
Range of strength values (init. pop.) [-4,4] 
Min. and max. term length 2,6 
Regression model OLS 

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