Hock

In this case the true underlying function HyGP had to approximate is the 2D Hock function (Hock and Schittkowski 1981):

f(z_{1}, z_{2}) = \left[30 + z_{1}sin(z_{1})\right](4+e^{-z_{2}})

Alvarez_3D

BUILDING DATA SET:
20-point Optimal Latin Hypercube DoE in [0, 5] x [0, 5]

Alvarez_2D_points_png
VALIDATION DATA SET:
441-point Full Factorial DoE  [-0.5: 0.3 : 5.5] x [-0.5: 0.3 : 5.5]

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

Results:
The best model returned by HyGP was:

\tilde{f}(z_{1},z_{2}) = {\left(3.59644475784\, \mathrm{z_{1}} - 0.770501844171\, {\mathrm{z_{1}}}^2\right)}^2 - 24.074516819\, \mathrm{z_{2}}  - 4.30155851714\, \mathrm{z_{1}} + 7.02267630102\, {\mathrm{z_{2}}}^2 - 0.681383038881\, {\mathrm{z_{2}}}^3 + 148.962136216

which corresponds to a coefficient of determination R^2 = 0.97672 on the validation data set.

wp_Hock_comparison_r9_gen50
Hock function (black) and model generated by HyGP (red)

 

References:

  • W. Hock and K. Schittkowski. Test examples for nonlinear programming codes. Lecture notes in economics and mathematical systems, 187, 1981. http://www.klaus-schittkowski.de.

 

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