McLain S4 Function

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

The McLain S4 function is a two-dimensional scalar-valued function. The function was introduced in [McL74] as a test function for procedures to construct contours from a given set of points.

Note

The McLain’s test functions are a set of five two-dimensional functions that mathematically defines surfaces. The functions are:

• S1: A part of a sphere

• S2: A steep hill rising from a plain

• S3: A less steep hill

• S4: A long narrow hill (this function)

• S5: A plateau and plain separated by a steep cliff

As shown in the plots above, the resulting surface consists of a long narrow hill running diagonally from \((0.0, 10.0)\) to \((10.0, 0.0)\). The maximum height is \(1.0\) at \((5.5, 5.5)\).

Test function instance

To create a default instance of the McLain S4 function:

my_testfun = uqtf.McLainS4()

Check if it has been correctly instantiated:

print(my_testfun)

Function ID      : McLainS4
Input Dimension  : 2 (fixed)
Output Dimension : 1
Parameterized    : False
Description      : McLain S4 function from McLain (1974)
Applications     : metamodeling

Description

The McLain S4 function is defined as follows:

\[ \mathcal{M}(\boldsymbol{x}) = \exp{\left[ -1 \left( (x_1 + x_2 - 11)^2 + \frac{(x_1 - x_2)^2}{10} \right) \right]} \]

where \(\boldsymbol{x} = \{ x_1, x_2 \}\) is the two-dimensional vector of input variables further defined below.

Probabilistic input

Based on [McL74], 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     : McLain
Input ID        : McLain1974
Input Dimension : 2
Description     : Input specification for the McLain's test functions from
                  McLain (1974).
Marginals       :

 No.    Name    Distribution    Parameters    Description
-----  ------  --------------  ------------  -------------
  1      X1       uniform       [ 1. 10.]          -
  2      X2       uniform       [ 1. 10.]          -

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

References

[McL74] (1,2)

D. H. McLain. Drawing contours from arbitrary data points. The Computer Journal, 17(4):318–324, 1974. doi:10.1093/comjnl/17.4.318.