One-dimensional (1D) Forrester et al. (2008) Function

One-dimensional (1D) Forrester et al. (2008) Function#

The 1D Forrester et al. (2008) function (or Forrester2008 function for short) is a one-dimensional scalar-valued function. It was used in [FSK08] as a test function for illustrating optimization using metamodels.

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

A plot of the function is shown below for \(x \in [0, 1]\).

../_images/b20888f69b30e746e1a15486363073ce2907a5b5c9d54d132cb7342650acfe01.png

As can be seen in the plot above, the function features a multimodal shape with one global minimum (\(\approx -6.02074006\) at \(x = 0.75724876\)), one global maximum (\(\approx -0.98632541\) at \(x = 014258919\)), and an inflection point with zero gradient (\(\approx -6.02074006\) at \(x = 0.5240772\)).

Test function instance#

To create a default instance of the test function:

my_testfun = uqtf.Forrester2008()

Check if it has been correctly instantiated:

print(my_testfun)

Function ID      : Forrester2008
Input Dimension  : 1 (fixed)
Output Dimension : 1
Parameterized    : False
Description      : One-dimensional function from Forrester et al. (2008)
Applications     : optimization, metamodeling

Description#

The test function is analytically defined as follows:

\[ \mathcal{M}(x) = (6 x - 2)^2 \sin{(12 x - 4)}, \]

where \(x\) is defined below.

Input#

Based on [FSK08], the search domain of the function is in \([0, 1]\). In UQTestFuns, this search domain can be represented as probabilistic input using the uniform distribution with a marginal shown in the table below.

Hide code cell source 
print(my_testfun.prob_input)

Function ID     : Forrester2008
Input ID        : Forrester2008
Input Dimension : 1
Description     : Input specification for the 1D test function from
                  Forrester et al. (2008)
Marginals       :

 No.    Name    Distribution    Parameters    Description
-----  ------  --------------  ------------  -------------
  1      x        uniform         [0 1]            -

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 
np.random.seed(42)
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("$\mathcal{M}(X)$");
plt.gcf().tight_layout(pad=3.0)
plt.gcf().set_dpi(150);
../_images/4dee5f32afdadb412c5db64e4942e074d84079b905db36c0c35cd29c58874c5a.png

Optimum values#

The optimum values of the function are:

  • Global minimum: \(\mathcal{M}(\boldsymbol{x}^*) \approx -6.02074006\) at \(x^* = 0.75724876\).

  • Local minimum: \(\mathcal{M}(\boldsymbol{x}^*) \approx -0.98632541\) at \(x^* = 014258919\).

  • Inflection: \(\mathcal{M}(\boldsymbol{x}^*) \approx -6.02074006\) at \(x^* = 0.5240772\).

References#

[FSK08] (1,2)

Alexander I. J. Forrester, András Sóbester, and Andy J. Keane. Engineering Design via Surrogate Modelling: A Practical Guide. Wiley, 1 edition, 2008. ISBN 978-0-470-06068-1 978-0-470-77080-1. doi:10.1002/9780470770801.