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

../_images/mclain-s4_3_0.png

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)

Name              : McLainS4
Spatial dimension : 2
Description       : McLain S4 function from McLain (1974)

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.

my_testfun.prob_input

Name: McLain-1974

Spatial 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.] None
2 X2 uniform [ 1. 10.] None

Copulas: None

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:

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);
../_images/mclain-s4_11_0.png

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.