Ten-dimensional (10D) Friedman Function

The 10D Friedman function (or Friedman6D function for short) is a ten-dimensional (including five dummy variables) scalar-valued function. The function features a combination of non-linearity and variable interaction.

It was originally used in [Fri91] as a test function for testing a regression spline method.

Note

The function was an extension of the six-dimensional version introduced in [FGS83] by adding four additional dummy variables (for a total of five); the function is also available in UQTestFuns.

import numpy as np
import matplotlib.pyplot as pltx
import uqtestfuns as uqtf

Test function instance

To create a default instance of the test function:

my_testfun = uqtf.Friedman10D()

Check if it has been correctly instantiated:

print(my_testfun)

Function ID      : Friedman10D
Input Dimension  : 10 (fixed)
Output Dimension : 1
Parameterized    : False
Description      : Ten-dimensional function from Friedman (1991)
Applications     : metamodeling

Description

The test function is analytically defined as follows:

\[ \mathcal{M}(\boldsymbol{x}) = 10 \sin{(\pi x_1 x_2)} + 20 (x_3 - 0.5)^2 + 10 x_4 + 5 x_5 + 0 x_6 + 0 x_7 + 0 x_8 + 0 x_9 + 0 x_{10}, \]

where \(x\) is defined below. Notice that the last five input variables are inert.

Probabilistic input

Based on [Fri91], the probabilistic input model for the function consists of two independent random variables as shown below.

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print(my_testfun.prob_input)

Function ID     : Friedman10D
Input ID        : Friedman1983
Input Dimension : 10
Description     : Input specification for the 6D test function from
                  Friedman (1991)
Marginals       :

 No.    Name    Distribution    Parameters    Description
-----  ------  --------------  ------------  -------------
  1     x_1       uniform         [0 1]            -
  2     x_2       uniform         [0 1]            -
  3     x_3       uniform         [0 1]            -
  4     x_4       uniform         [0 1]            -
  5     x_5       uniform         [0 1]            -
  6     x_6       uniform         [0 1]            -
  7     x_7       uniform         [0 1]            -
  8     x_8       uniform         [0 1]            -
  9     x_9       uniform         [0 1]            -
 10     x_10      uniform         [0 1]            -

Copulas         : Independence

Reference results

This section provides several reference results of typical UQ analyses involving the test function.

Sample histogram

Shown below is the histogram of the output based on \(100'000\) random points:

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xx_test = my_testfun.prob_input.get_sample(100000)
yy_test = my_testfun(xx_test)

plt.hist(yy_test, bins="auto", color="#8da0cb");
plt.grid();
plt.ylabel("Counts [-]");
plt.xlabel("$\mathcal{M}(\mathbf{X})$");
plt.gcf().set_dpi(150);

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Cell In[5], line 4
      1 xx_test = my_testfun.prob_input.get_sample(100000)
      2 yy_test = my_testfun(xx_test)
----> 4 plt.hist(yy_test, bins="auto", color="#8da0cb");
      5 plt.grid();
      6 plt.ylabel("Counts [-]");

NameError: name 'plt' is not defined

References

[Fri91] (1,2)

J. H. Friedman. Multivariate adaptive regression splines. The Annals of Statistics, 1991. doi:10.1214/aos/1176347963.

[FGS83]

J. H. Friedman, E. Grosse, and W. Stuetzle. Multidimensional additive spline approximation. SIAM Journal on Scientific and Statistical Computing, 1983.