Kotanchek

In this case the true underlying function HyGP had to approximate is the 2D Kotanchek function (Keijzer 2003):

f_{1}(z_{1}, z_{2}) = \dfrac{e^{-(z_{1}-1)^{2}}}{1.2+(z_{2}-2.5)^{2}}

Kotanchek_3D

BUILDING DATA SET:
40-point Optimal Latin Hypercube DoE in [0, 4] x [0, 4]
Available here: kotanchek_input_file

Kotanchek_2D_points_png
VALIDATION DATA SET:
2025-point Full Factorial DoE  [-0.2: 0.1: 4.2] x [-0.2: 0.1: 4.2]
Available here: kotanchek_test_dataset

HyGP hyperparameters (see kotanchek_input_file):
Population size: 200
Generations: 50
Primitives: +, -, *, / (protected), ^2, ^3, sin, cos, exp, shift

Results:
Using the penalisation approach (p=3, a5=0.0001), the best symbolic expression returned by HyGP was:

\tilde{f}(z_{1}, z_{2}) = - 0.0241637 + \dfrac{ \left[33.5706\, \sin\!\left(1.44049\, \mathrm{z_{1}}\right) - 0.923777\, \mathrm{z_{2}} + 62.1925\right]^2  + 265.5989} {{\left(68.3857\, \mathrm{z_{2}} - 105.061\right)}^2  + {\left(60.4282\, \mathrm{z_{2}} - 228.862\right)}^2  + {\left(60.0122\, \mathrm{z_{1}} - 44.1352\right)}^2 \, - 269.67 }

resulting in R^2 = 0.99819, max abs error = 0.03177 on the validation data set.
Using instead editing and factorisation bonus (omegalim_shif_Ed_F), the best metamodel returned was:

\tilde{f}(z_{1},z_{2}) = -0.0103772 +  \dfrac{ (-36.3242\, \mathrm{z_{1}} - 48.3108 + 15.5212\, \mathrm{z_{1}}\mathrm{z_{1}})^2 }{   ((40.8610\, \mathrm{z_{1}} -90.0379 - 25.8195\, \mathrm{z_{1}}\mathrm{z_{1}})^2 + (-69.1777 \, \mathrm{z_{2}} -0.256008\, \mathrm{z_{2}}\mathrm{z_{1}} + 17403.4)^2)}

returning R^2=0.99452, max abs error = 0.05174 on the validation data set.

Video of the HyGP run leading to the generation of symbolic expression reported above (with penalisation approach (p=3, a5=0.0001)):

 

References:

  • M. Keijzer. Improving symbolic regression with interval arithmetic and linear scaling. In C. Ryan, T. Soule, M. Keijzer, E. Tsang, R. Poli, and E. Costa, editors, Proceedings of EuroGP 2003, volume 2610 of LNCS, pages 70–82. Springer-Verlag, 2003.

 

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