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Symbolic Regression Benchmark Functions
At last year's GECCO in Dublin a discussion revolved around the fact that the genetic programming community needs a set of suitable benchmark problems. Many experiments presented in the GP literature are based on very simple toy problems and thus the results are often unconvincing. The whole topic is summarized on http://gpbenchmarks.org.
This page also lists benchmark problems for symbolic regression from a number of different papers. Thanks to our new developer Stefan Forstenlechner most of these problems are now available in HeuristicLab and will be included in the next release. The benchmark problems can be easily loaded directly in the GUI through problem instance providers. Additionally, it is very simple to create an experiment to execute an algorithm on all instances using the new 'Create Experiment' dialog implemented by Andreas (see the previous blog post)
I used these new features to quickly apply a random forest regression algorithm (R=0.7, Number of trees=50) on all regression benchmark problems and got the following results. Let's see how symbolic regression with GP will perform...
| Problem instance | Avg. R² (test) |
| Keijzer 4 f(x) = 0.3 * x *sin(2 * PI * x) | 0.984 |
| Keijzer 5 f(x) = x ^ 3 * exp(-x) * cos(x) * sin(x) * (sin(x) ^ 2 * cos(x) - 1) | 1.000 |
| Keijzer 6 f(x) = (30 * x * z) / ((x - 10) * y^2) | 0.956 |
| Keijzer 7 f(x) = Sum(1 / i) From 1 to X | 0.911 |
| Keijzer 8 f(x) = log(x) | 1.000 |
| Keijzer 9 f(x) = sqrt(x) | 1.000 |
| Keijzer 11 f(x, y) = x ^ y | 0.957 |
| Keijzer 12 f(x, y) = xy + sin((x - 1)(y - 1)) | 0.267 |
| Keijzer 13 f(x, y) = x^4 - x^3 + y^2 / 2 - y | 0.610 |
| Keijzer 14 f(x, y) = 6 * sin(x) * cos(y) | 0.321 |
| Keijzer 15 f(x, y) = 8 / (2 + x^2 + y^2) | 0.484 |
| Keijzer 16 f(x, y) = x^3 / 5 + y^3 / 2 - y - x | 0.599 |
| Korns 1 y = 1.57 + (24.3 * X3) | 0.998 |
| Korns 2 y = 0.23 + (14.2 * ((X3 + X1) / (3.0 * X4))) | 0.009 |
| Korns 3 y = -5.41 + (4.9 * (((X3 - X0) + (X1 / X4)) / (3 * X4))) | 0.023 |
| Korns 4 y = -2.3 + (0.13 * sin(X2)) | 0.384 |
| Korns 5 y = 3.0 + (2.13 * log(X4)) | 0.977 |
| Korns 6 y = 1.3 + (0.13 * sqrt(X0)) | 0.997 |
| Korns 7 y = 213.80940889 - (213.80940889 * exp(-0.54723748542 * X0)) | 0.000 |
| Korns 8 y = 6.87 + (11 * sqrt(7.23 * X0 * X3 * X4)) | 0.993 |
| Korns 9 y = ((sqrt(X0) / log(X1)) * (exp(X2) / square(X3))) | 0.000 |
| Korns 10 y = 0.81 + (24.3 * (((2.0 * X1) + (3.0 * square(X2))) / ((4.0 * cube(X3)) + (5.0 * quart(X4))))) | 0.003 |
| Korns 11 y = 6.87 + (11 * cos(7.23 * X0 * X0 * X0)) | 0.000 |
| Korns 12 y = 2.0 - (2.1 * (cos(9.8 * X0) * sin(1.3 * X4))) | 0.001 |
| Korns 13 y = 32.0 - (3.0 * ((tan(X0) / tan(X1)) * (tan(X2) / tan(X3)))) | 0.000 |
| Korns 14 y = 22.0 + (4.2 * ((cos(X0) - tan(X1)) * (tanh(X2) / sin(X3)))) | 0.000 |
| Korns 15 y = 12.0 - (6.0 * ((tan(X0) / exp(X1)) * (log(X2) - tan(X3)))) | 0.000 |
| Nguyen F1 = x^3 + x^2 + x | 0.944 |
| Nguyen F2 = x^4 + x^3 + x^2 + x | 0.992 |
| Nguyen F3 = x^5 + x^4 + x^3 + x^2 + x | 0.983 |
| Nguyen F4 = x^6 + x^5 + x^4 + x^3 + x^2 + x | 0.960 |
| Nguyen F5 = sin(x^2)cos(x) - 1 | 0.975 |
| Nguyen F6 = sin(x) + sin(x + x^2) | 0.997 |
| Nguyen F7 = log(x + 1) + log(x^2 + 1) | 0.977 |
| Nguyen F8 = Sqrt(x) | 0.966 |
| Nguyen F9 = sin(x) + sin(y^2) | 0.988 |
| Nguyen F10 = 2sin(x)cos(y) | 0.986 |
| Nguyen F11 = x^y | 0.961 |
| Nguyen F12 = x^4 - x^3 + y^2/2 - y | 0.979 |
| Spatial co-evolution F(x,y) = 1/(1+power(x,-4)) + 1/(1+pow(y,-4)) | 0.983 |
| TowerData | 0.972 |
| Vladislavleva Kotanchek | 0.854 |
| Vladislavleva RatPol2D | 0.785 |
| Vladislavleva RatPol3D | 0.795 |
| Vladislavleva Ripple | 0.951 |
| Vladislavleva Salutowicz | 0.996 |
| Vladislavleva Salutowicz2D | 0.960 |
| Vladislavleva UBall5D | 0.892 |
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