Sine Function from Higdon (2002)

Sine Function from Higdon (2002)#

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

The function is a simple one-dimensional, scalar-valued test function. It was featured in [Hig02] as an example for illustrating a multi-resolution spatial modeling technique.

A plot of the function is shown below..

../_images/01993fdc6392235fc4833e342d72a0e1791b85599c1369833a6ee7338c8d082d.png

Note

In the original paper, the function was evaluated at 30 equispaced points in \([1.0, 10.0]\) with added i.i.d noise from \(\mathcal{N} \sim (0, 0.1)\); these points are shown in the above plot.

Test function instance#

To create a default instance of the test function:

my_testfun = uqtf.HigdonSine()

Check if it has been correctly instantiated:

print(my_testfun)

Function ID      : HigdonSine
Input Dimension  : 1 (fixed)
Output Dimension : 1
Parameterized    : False
Description      : Sine function from Higdon (2002)
Applications     : metamodeling

Description#

The test function is analytically defined as follows[1]:

\[ \mathcal{M}(x) = \sin{\left(2 \pi \frac{x}{10} \right)} + 0.2 \, \sin{\left(2 \pi \frac{x}{2.5} \right)}, \]

where \(x\) is further defined below.

Notice that the second term of the equation gives variation 5 times smaller but 4 times faster.

Probabilistic input#

The probabilistic input model for the test function is shown below.

Hide code cell source 
print(my_testfun.prob_input)

Function ID     : HigdonSine
Input ID        : Higdon2002
Input Dimension : 1
Description     : Input model for the sine function from Higdon (2002)
Marginals       :

 No.    Name    Distribution    Parameters    Description
-----  ------  --------------  ------------  -------------
  1      x        uniform       [ 1. 10.]          -

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 
my_testfun.prob_input.reset_rng(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/dcede6784acce53440442e39c2501b47aec9250470ad5e35948698e783ba5e81.png

References#

[Hig02] (1,2)

Dave Higdon. Quantitative methods for current environmental issues, chapter Space and space-time modeling using process convolutions, pages 37–56. Springer London, 2002. doi:10.1007/978-1-4471-0657-9_2.