# Composite Gaussians

The composite Gaussians function is a two-dimensional test function used in metamodeling and reliability analysis exercises (Bichon et al., 2011).

## Description

The composite Gaussians function consists of the following components:

\begin{align} g_1(\mathbf{x}) & = x_1^2 + x_2 - 8, \\ g_2(\mathbf{x}) & = \frac{x_1}{5} + x_2 - 6, \end{align}

where \mathbf{x} = \{x_1, x_2\}​ are input variables.

The overall (composite) limit state function is defined as:

g(\mathbf{x}) = \max{(g_1(\mathbf{x}), g_2(\mathbf{x}))}.

Figure 1 and 2 show the surface and contour plots of the composite Gaussians function, respectively.

Figure 1: Surface plot of the two-dimensional composite Gaussians function.

Figure 2: Contour plot of the two-dimensional composite Gaussians function.

The failure event and the failure probability are defined as g(\mathbf{x}) \leq 0 and P_f = \mathbb{P}[g(\mathbf{x}) \leq 0], respectively. Figure 3 shows the two limit state lines (g_i(\mathbf{x}) = 0, \; i = 1,2) of the composite function and the corresponding failure domain.

Figure 3: Limit state lines of the composite Gaussians function. The shaded area indicates the failure domain.

## Inputs

The input variables x_1 and x_2 are modeled as two independent Gaussian random variables.

No Variable Distribution Parameters
1 x_1 Gaussian \mu_{x_1} = 5.5,
\sigma_{x_1} = 1.0
2 x_2 Gaussian \mu_{x_2} = 5.0,
\sigma_{x_2} = 1.0

Note: These values are chosen such that the failure probability is small (\sim 10^{-4}). In Bichon et al. (2011), the function is not used as a test function for reliability analysis methods and therefore, the random inputs were not specified.

## Reference values

Some reference values for the failure probability are given in the table below.

Method N \hat{P}_f \text{CoV}[\hat{P}_f] Source
MCS 10^{4} 2.00 \times 10^{-4} 70.7\% UQLab v1.1
MCS 10^{5} 1.00 \times 10^{-4} 31.6\% UQLab v1.1
MCS 10^{6} 1.42 \times 10^{-4} 8.4\% UQLab v1.1
MCS 10^{7} 1.26 \times 10^{-4} 2.8\% UQLab v1.1

## Resources

The vectorized implementation of the composite Gaussians function in MATLAB as well as the script file with the model and probabilistic inputs definitions for the function in UQLAB can be downloaded below:

uq_compositeGaussians.zip (2.3 KB)

The contents of the file are:

Filename Description
uq_compositeGaussians.m vectorized implementation of the composite Gaussians function
uq_Example_compositeGaussians.m definitions for the model and probabilistic inputs in UQLab
LICENSE license for the function (BSD 3-Clause)

## References

• B. J. Bichon, J. M. McFarland, and S. Mahadevan, “Efficient surrogate models for system reliability analysis of systems with multiple failure modes,” Reliability Engineering and System Safety, vol. 96, pp. 1386–1395, 2011. DOI:10.1016/j.ress.2011.05.008