Bratley et al. (1992) A function

Bratley et al. (1992) A function#

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

The Bratley et al. (1992) A function (or Bratley1992a function for short), is an \(M\)-dimensional scalar-valued function. The function was introduced in [BFN92] as a test function for multi-dimensional numerical integration using low discrepancy sequences.

Note

There are four other test functions used in Bratley et al. [BFN92]:

This function was reintroduced in [SSobol95] with additional parameters for global sensitivity analysis purposes. The “generalized” function became known as the Sobol’-G.

The plots for one-dimensional and two-dimensional Bratley1992a functions are shown below.

../_images/8f147de9f950c395883eb031270a34be3ed7c33d213a41f6b84d14ef67fcc655.png

Test function instance#

To create a default instance of the test function:

my_testfun = uqtf.Bratley1992a()

Check if it has been correctly instantiated:

print(my_testfun)

Function ID      : Bratley1992a
Input Dimension  : 2 (variable)
Output Dimension : 1
Parameterized    : False
Description      : Integration test function #1 from Bratley et al. (1992)
Applications     : integration, sensitivity

By default, the input dimension is set to \(2\)[1]. To create an instance with another value of input dimension, pass an integer to the parameter input_dimension (keyword only). For example, to create an instance of 10-dimensional Bratley1992a function, type:

my_testfun = uqtf.Bratley1992a(input_dimension=10)

Description#

The Bratley1992a function is defined as follows[2]:

\[ \mathcal{M}(\boldsymbol{x}) = \prod_{m = 1}^{M} \lvert 4 x_m - 2 \rvert, \]

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

Probabilistic input#

Based on [BFN92], the test function is integrated over the hypercube domain of \([0, 1]^M\). Such an input specification can be modeled using an \(M\) independent uniform random variables as shown in the table below.

No.

Name

Distribution

Parameters

Description

1

\(x_1\)

uniform

[0.0 1.0]

N/A

\(\vdots\)

\(\vdots\)

\(\vdots\)

\(\vdots\)

\(\vdots\)

M

\(x_M\)

uniform

[0.0 1.0]

N/A

Reference results#

This section provides several reference results of typical UQ analyses involving the test function.

Definite integration#

The integral value of the function over the domain of \([0.0, 1.0]^M\) is analytical:

\[ I[\mathcal{M}] (M) \equiv \int_{[0, 1]^M} \mathcal{M}(\boldsymbol{x}) \; d\boldsymbol{x} = 1.0. \]

Moments#

The moments of the test function are analytically known and the first two moments are given below.

Expected value#

Due to the domain being a hypercube, the above integral value over the domain is the same as the expected value:

\[ \mathbb{E}[\mathcal{M}](M) = 1.0. \]

Variance#

The analytical value for the variance is given as follows:

\[ \mathbb{V}[\mathcal{M}](M) = \left( \frac{4}{3} \right)^M - 1 \]

The table below shows the numerical values of the variance for several selected dimensions.

Dimension

\(I[\mathcal{M}]\)

1

\(3.3333333 \times 10^{-1}\)

2

\(7.7777778 \times 10^{-1}\)

3

\(1.3703704 \times 10^{0}\)

4

\(2.1604938 \times 10^{0}\)

5

\(3.2139918 \times 10^{0}\)

6

\(4.6186557 \times 10^{0}\)

7

\(6.4915409 \times 10^{0}\)

8

\(8.9887212 \times 10^{0}\)

9

\(1.2318295 \times 10^{1}\)

10

\(1.6757727 \times 10^{1}\)

The variance grows as the number of dimensions and becomes unbounded for a very large dimension.

References#

[SSobol95]

Andrea Saltelli and Ilya M. Sobol'. About the use of rank transformation in sensitivity analysis of model output. Reliability Engineering & System Safety, 50(3):225–239, 1995. doi:10.1016/0951-8320(95)00099-2.

[BFN92] (1,2,3,4)

Paul Bratley, Bennett L. Fox, and Harald Niederreiter. Implementation and tests of low-discrepancy sequences. ACM Transactions on Modeling and Computer Simulation, 2(3):195–213, 1992. doi:10.1145/146382.146385.