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Table 4 Model equations: expressional complexity/performance characteristics (on training data) of symbolic models on the Pareto front

From: Application of multi-gene genetic programming technique for modeling and optimization of phycoremediation of Cr(VI) from wastewater

Model ID

Goodness of fit (R2)

Model complexity

Model

555

0.917

56

\(\begin{aligned} \,0.0957x_{1} x_{3}^{2} - 2.64\exp \left( { - 4} \right)x_{1}^{2} x_{3}^{3} \\ & - 7.13x_{3} - 3.47\exp \left( { - 4} \right)x_{3}^{3} x_{4} \\ & + 0.179x_{4} \left( {2x_{2} + x_{3} } \right) - 0.0641x_{3}^{2} + 8.81 \\ \end{aligned}\) (4)

560

0.915

52

\(\begin{aligned} & 0.108x_{1} x_{3}^{2} - 3.01\exp \left( { - 4} \right)x_{1}^{2} x_{3}^{3} - 10.2x_{3} \\ & - 3.97\exp \left( { - 4} \right)x_{3}^{3} x_{4} + 0.189x_{4} \left( {2x_{2} + x_{3} } \right) + 20 \\ \end{aligned}\) (5)

596

0.895

50

\(\begin{aligned} & 0.105x_{2} - 2.23\exp \left( { - 4} \right)x_{1}^{2} x_{3}^{3} + 0.0892x_{1} x_{3}^{2} \\ & + 0.174x_{4} \left( {2x_{2} + x_{3} } \right) - 0.354x_{3}^{2} - 0.0154x_{1} x_{3} x_{4} - 24.3 \\ \end{aligned}\) (6)

2584

0.944

147

\(\begin{aligned} & 0.36x_{1} + 0.18x_{3} + 2.65x_{4} \\ & - 0.526x_{4} \left( {2x_{3} - 9.19} \right) - 0.126x_{1}^{2} x_{4} \\ & + 0.18x_{1} x_{3} x_{4} \\ & + 2.34\exp \left( { - 6} \right)x_{3}^{2} x_{4}^{2} \left( {x_{3} - 18.4} \right)\left( {x_{3} - 12.7} \right) \\ & + 3.64\exp \left( { - 6} \right)x_{1}^{3} x_{3} x_{4} \left( {x_{3} - 12.7} \right)\left( {x_{2} - x_{3} } \right) + 0.394 \\ \end{aligned}\) (7)

2730

0.945

159

\(\begin{aligned} & 0.173(x_{1} + x_{2} + x_{3} ) + 2.75x_{4} - 0.508x_{4} \left( {2x_{3} - 9.19} \right) \\ & - 0.121x_{1}^{2} x_{4} + 0.173x_{1} x_{3} x_{4} \\ & + 3.87\exp \left( { - 6} \right)x_{3} x_{4}^{2} \left( {x_{3} - 18.4} \right)\left( {x_{3} - 12.7} \right)\left( {x_{3} - 12.9} \right) \\ & + 3.47\exp \left( { - 6} \right)x_{1}^{3} x_{3} x_{4} \left( {x_{3} - 12.7} \right)\left( {x_{2} - x_{3} } \right) + 0.415 \\ \end{aligned}\) (8)

2871

0.945

152

\(\begin{aligned} & 0.173(x_{1} + x_{2} + x_{3} ) + 2.74x_{4} - 0.507x_{4} \left( {2x_{3} - 9.19} \right) \\ & - 0.121x_{1}^{2} x_{4} + 0.173x_{1} x_{3} x_{4} \\ & + 3.83\exp \left( { - 6} \right)x_{3} x_{4}^{2} \left( {x_{3} - 18.4} \right)\left( {x_{3} - 12.7} \right)^{2} \\ & + 3.47\exp \left( { - 6} \right)x_{1}^{3} x_{3} x_{4} \left( {x_{3} - 12.7} \right)\left( {x_{2} - x_{3} } \right) + 0.447 \\ \end{aligned}\) (9)

4535

0.961

171

\(\begin{aligned} & 2.68\exp \left( { - 8} \right)x_{3}^{7} - 1.62\exp \left( { - 5} \right)x_{3}^{2} \left( {x_{3} + x_{3}^{3} + x_{1} x_{2} x_{4} } \right) \\ & - 1.73\exp \left( { - 4} \right)x_{1}^{4} \left( {x_{1} + 2x_{4} - 5.02} \right) + 0.0497x_{1} x_{2} x_{4} \\ & - 1.81\exp ( - 7x_{1} x_{3}^{5} \left( {x_{1} + x_{4} } \right) \\ & + 9.38\exp \left( { - 5} \right)x_{1} x_{3}^{2} \left( {x_{3} + 5.02} \right)\left( {x_{1} + x_{4} } \right) \\ & + 2.07\exp \left( { - 5} \right) \\ \end{aligned}\) (10)

4812

0.96

165

\(\begin{gathered} 2.86\exp \left( { - 8} \right)x_{3}^{7} - 1.64\exp \left( { - 5} \right)x_{3}^{2} \left( {x_{3} + x_{3}^{3} x_{1} x_{2} x_{4} } \right) \hfill \\ - 1.07\exp \left( { - 4} \right)x_{1}^{4} \left( {x_{1} + 3x_{4} } \right) + 0.0511x_{1} x_{2} x_{4} \hfill \\ + 1.28\exp \left( { - 4} \right)x_{1} x_{3}^{3} \left( {x_{1} + x_{4} } \right) \hfill \\ - 2.12\exp \left( { - 7} \right)x_{1} x_{3}^{5} \left( {x_{1} + x_{4} } \right) + 2.1\exp \left( { - 5} \right) \hfill \\ \end{gathered}\) (11)

4924

0.96

161

\(\begin{gathered} 2.69\exp \left( { - 8} \right)x_{3}^{7} - 1.64\exp \left( { - 4} \right)x_{1}^{4} \left( {x_{1} + 2x_{4} - 5.02} \right) \hfill \\ - 1.57\exp \left( { - 5} \right)x_{3}^{2} \left( {x_{1} + x_{3}^{3} - 5.02} \right) + 0.0453x_{1} x_{2} x_{4} \hfill \\ - 1.86\exp \left( { - 7x_{1} x_{3}^{5} } \right)\left( {x_{1} + x_{4} } \right) \hfill \\ + 9.24\exp \left( { - 5} \right)x_{1} x_{3}^{2} \left( {x_{3} + 5.16} \right)\left( {x_{1} + x_{4} } \right) \hfill \\ + 2.19\exp \left( { - 5} \right) \hfill \\ \end{gathered}\) (12)

7484

0.934

64

\(\begin{gathered} 0.644x_{1} + 1.77x_{2} x_{4} - 1.56x_{3} x_{4} + 0.0115x_{3}^{2} x_{4} \hfill \\ - 0.23x_{1} x_{4} \left( {x_{2} - x_{3} } \right) + 2.35\exp \left( { - 6} \right)x_{1}^{3} x_{3}^{2} x_{4} \left( {x_{2} - x_{3} } \right) + 0.0206 \hfill \\ \end{gathered}\) (13)

8003

0.935

72

\(\begin{gathered} 3.12(x_{4} - x_{1} ) - 13.4x_{3} + 0.225x_{1} x_{3}^{2} + 0.0186x_{2} x_{3} x_{4} \hfill \\ + 1.35\exp \left( { - 5} \right)x_{3}^{3} \left( {x_{3} - x_{4} } \right)\left( {x_{1} + x_{3} - 7} \right) \hfill \\ - 0.00576x_{1} x_{3}^{2} \left( {x_{1} + x_{3} } \right) + 48.9 \hfill \\ \end{gathered}\)(14)

8122

0.926

59

\(\begin{gathered} 2.92x_{4} - 11.4x_{3} + 1.27\exp \left( { - 5} \right)x_{3}^{4} \left( {x_{3} - x_{4} } \right) \hfill \\ + 0.201x_{1} x_{3}^{2} + 0.0178x_{2} x_{3} x_{4} \hfill \\ - 0.00522x_{1} x_{3}^{2} \left( {x_{1} + x_{3} } \right) + 19.1 \hfill \\ \end{gathered}\) (15)

8591

0.922

58

\(\begin{gathered} 0.706x_{4} - 10.1x_{3} - 2.83\exp \left( { - 4} \right)x_{1}^{2} x_{3}^{3} \hfill \\ + 0.103x_{1} x_{3}^{2} - 0.00238x_{2} x_{3}^{2} x_{4} \hfill \\ + 0.0661x_{2} x_{3} x_{4} + 25.7 \hfill \\ \end{gathered}\) (16)