Six-dimensional (6D) Friedman Function

Six-dimensional (6D) Friedman Function#

The 6D Friedman function (or Friedman6D function for short) is a six-dimensional (including one dummy variable) scalar-valued function. The function features a combination of non-linearity and variable interaction.

It was originally used in [FGS83] as a test function for testing a spline approximation method. In [SCJR22] and [HPS21] (albeit in a modified form) the function was employed as a test function in the context of sensitivity analysis.

Note

The function was later extended to ten dimension by incorporating four additional dummy variables (for a total of five) in [Fri91]; the function is also available in UQTestFuns.

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

Test function instance#

To create a default instance of the test function:

my_testfun = uqtf.Friedman6D()

Check if it has been correctly instantiated:

print(my_testfun)

Function ID      : Friedman6D
Input Dimension  : 6 (fixed)
Output Dimension : 1
Parameterized    : False
Description      : Six-dimensional function from Friedman et al. (1983)
Applications     : metamodeling, sensitivity

Description#

The test function is analytically defined as follows:

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

where \(x\) is defined below. Notice that the sixth input variable is inert.

Probabilistic input#

Based on [FGS83], the probabilistic input model for the function consists of two independent random variables as shown below.

Hide code cell source 
print(my_testfun.prob_input)

Function ID     : Friedman6D
Input ID        : Friedman1983
Input Dimension : 6
Description     : Input specification for the 6D test function from
                  Friedman et al. (1983)
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]            -

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:

Hide code cell source 
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);
../_images/c2de3ea08d0f6accc75b99fce28d1c927e34fcda4ce749db9f08bc5276f21921.png

References#

[Fri91]

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

[FGS83] (1,2)

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

[HPS21]

Akira Horiguchi, Matthew T. Pratola, and Thomas J. Santner. Assessing variable activity for Bayesian regression trees. Reliability Engineering & System Safety, 207:107391, March 2021. doi:10.1016/j.ress.2020.107391.

[SCJR22]

Xifu Sun, Barry Croke, Anthony Jakeman, and Stephen Roberts. Benchmarking Active Subspace methods of global sensitivity analysis against variance-based Sobol’ and Morris methods with established test functions. Environmental Modelling & Software, 149:105310, 2022. doi:10.1016/j.envsoft.2022.105310.