Korns P10

In this case the true underlying function HyGP had to approximate is the 2D Kotanchek function (Korns 2011):

f(z_{1}, z_{2}, z_{3}, z_{4})= 0.81 + 24.3*\dfrac{2.0\,z_{1}+3.0\,z_{2}^2}{4.0\,z_{3}^3+5.0\,z_{4}^4}

 

BUILDING DATA SET:
300-point Optimal Latin Hypercube DoE in [-50, 50] x [-50, 50] x [-50, 50] x [-50, 50]
VALIDATION DATA SET:
4096-point Full Factorial DoE  [-50: 14.28 : 50] x [-50: 14.28 : 50] x [-50: 14.28 : 50] x [-50: 14.28 : 50]

HyGP hyperparameters:
Population size: 400
Generations: 200
Primitives: +, -, *, / (protected), ^2, ^3, sin, cos, tanh, exp, log

Results:
Using editing and factorisation bonus, the best model returned by HyGP is:

\tilde{f}(\mathrm{z_{1}},\mathrm{z_{2}}, \mathrm{z_{3}}, \mathrm{z_{4}}) = 0.809396 + (-36.6710\, \mathrm{z_{2}}) /  (((((147.049\, \mathrm{z_{4}}\, \mathrm{z_{4}} - 2.01659\, \mathrm{z_{4}})^2) - (128.716\, \mathrm{z_{4}} + (-222.307\, (\mathrm{z_{2}} / (\mathrm{z_{3}}\mathrm{z_{3}}))))) /  (-32.5875\, (263.633\, \mathrm{z_{2}}))) - (-150.732\, (\mathrm{z_{3}} / (\mathrm{z_{2}} / (\mathrm{z_{3}} (\mathrm{z_{3}} / -74.6861))))))

resulting in a coefficient of determination R^2=0.99974, max abs error = 0.28575 on the validation data set.

wp_KornsP10_fact_edit_best_act_vs_est_test
Estimated vs Actual response plot for the best metamodel returned by HyGP

 

References:

  • M. F. Korns. Accuracy in symbolic regression. In R. Riolo, E. Vladislavleva, and J. H. Moore, editors, Genetic Programming Theory and Practice IX, Genetic and Evolutionary Computation. Springer New York, 2011.
  • See also gpBenchmark page reporting a collection of challenging symbolic regression problems for GP, among which “Korn P10” was chosen (section “Difficult synthetic symbolic regression problems”):
    http://www.gpbenchmarks.org/wiki/index.php?title=Problem_Classification

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