Bratley et al. (1992) C function#

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

The Bratley et al. (1992) C function (or Bratley1992c 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]:

Bratley et al. (1992) A: A product of an absolute function
Bratley et al. (1992) B: A product of a trigonometric function
Bratley et al. (1992) C: A product of the Chebyshev polynomial of the first kind (this function)
Bratley et al. (1992) D: A sum of product

The 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 Bratley1992c functions are shown below.

Test function instance#

To create a default instance of the test function:

my_testfun = uqtf.Bratley1992c()

Check if it has been correctly instantiated:

print(my_testfun)

Name              : Bratley1992c
Spatial dimension : 2
Description       : Integration test function #3 from Bratley et al. (1992)

By default, the spatial dimension is set to \(2\)1. To create an instance with another value of spatial dimension, pass an integer to the parameter spatial_dimension (keyword only). For example, to create an instance of 10-dimensional Bratley1992c function, type:

my_testfun = uqtf.Bratley1992c(spatial_dimension=10)

Description#

The Bratley1992c function is defined as follows2:

\[ \mathcal{M}(\boldsymbol{x}) = \prod_{m = 1}^M T_{n_m} (2 x_m - 1) \]

where \(T_{n_m}\) is the Chebyshev polynomial (of the first kind) of degree \(n_m\), \(n_m = m \bmod 4 + 1\), and \(\boldsymbol{x} = \{ x_1, \ldots, x_M \}\) 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:

\[\begin{split} I[\mathcal{M}] (M) \equiv \int_{[0, 1]^M} \mathcal{M}(\boldsymbol{x}) \; d\boldsymbol{x} = \begin{cases} -\frac{1}{3}, & M = 1 \\ 0, & M \neq 1 \end{cases}. \end{split}\]

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

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.

1: This default dimension applies to all variable dimension test functions. It will be used if the spatial_dimension argument is not given.
2: see Section 5.1, p. 207 (test function no. 3) in [BFN92].

UQTestFuns

Bratley et al. (1992) C function

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