# Benchmark case - Ishigami function

The original post can be found here.

# Ishigami Function

The Ishigami function is commonly used as a test function to benchmark global sensitivity analysis methods (Ishigami and Homma, 1990; Sobol’ and Levitan, 1999; Marrel et al., 2009).

## Description

The analytic expression of the Ishigami function is given as:

f(\mathbf{x}) = \sin{x_1} + a \sin^2{x_2} + b x_3^4 \sin{x_1}

where \mathbf{x} = \{x_1, x_2,x_3\} \in [-\pi, \pi]^3 are input variables; while a and b are parameters.

Figure 1: The histogram of the Ishigami function output when using uniform input distributions (see details below) based on 10^6 sample points.

## Inputs

For computer experiment purposes, the inputs x_1, x_2, x_3 are modeled as three independent uniform random variables.

No Variable Distribution Parameters
1 X_1 Uniform [-\pi, \pi]
2 X_2 Uniform [-\pi, \pi]
3 X_3 Uniform [-\pi, \pi]

## Parameters

The parameters of the Ishigami function differ according to the literature, as shown in the table below.

No Value Source
1 a = 7, \; b = 0.1 Marrel et al. (2009)
2 a = 7, \; b = 0.05 Sobol’ and Levitan (1999)

## Reference values

### First-order Sobol’ indices

The analytical solution for the first-order Sobol’ indices of the Ishigami function as described above is:

• S_1 = \frac{V_1}{\mathbb{V}[Y]}
• S_2 = \frac{V_2}{\mathbb{V}[Y]}
• S_3 = \frac{V_3}{\mathbb{V}[Y]} = 0

where:

• \mathbb{V}[Y] = \frac{a^2}{8} + \frac{b\pi^4}{5} + \frac{b^2\pi^8}{18} + \frac{1}{2}
• V_1 = \frac{1}{2}(1 + \frac{b\pi^4}{5})^2
• V_2 = \frac{a^2}{8}
• V_3 = 0

### Total-effect Sobol’ indices

The analytical solution for the total-effect Sobol’ indices of the Ishigami function as described above is:

• S_{T1} = \frac{V_{T1}}{\mathbb{V}[Y]}
• S_{T2} = \frac{V_{T2}}{\mathbb{V}[Y]}
• S_{T3} = \frac{V_{T3}}{\mathbb{V}[Y]}

where:

• V_{T1} = \frac{1}{2} (1 + \frac{b\pi^4}{5})^2 + \frac{8b^2\pi^8}{225}
• V_{T2}= \frac{a^2}{8}
• V_{T3} = \frac{8b^2\pi^8}{225}

## Resources

The vectorized implementation of the Ishigami 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_ishigami.zip (2.1 KB)

The contents of the file are:

Filename Description
uq_ishigami.m vectorized implementation of the Ishigami function in MATLAB
uq_Example_ishigami.m definitions for the model and probabilistic inputs in UQLab
LICENSE license for the function (BSD 3-Clause)

## Open-access repository

The dataset used in this benchmark study is titled “Benchmark case datasets - Ishigami function” and is authored by Adéla Hlobilová, Stefano Marelli, and Bruno Sudret. It was published in 2024 and is available on Zenodo. The dataset can be accessed directly via the following DOI link: 10.5281/zenodo.12684378.

The experimental designs include datasets with 40, 80, 120, 160, and 200 samples, each generated using optimized Latin Hypercube Sampling (LHS) with 1,000 iterations to improve the maximin criterion. Each dataset is replicated 20 times. The validation set contains 100,000 samples generated by Monte Carlo simulation. Each dataset contains samples and responses of the computational model.

For citation purposes, please use the following format:
Hlobilová, A., Marelli, S., and Sudret, B. (2024). Benchmark case datasets - Ishigami function. Zenodo. https://doi.org/10.5281/zenodo.12684378.

This project was supported by the Open Research Data Program of the ETH Board under Grant number EPFL SCR0902285.

## References

• T. Ishigami and T. Homma, “An importance quantification technique in uncertainty analysis for computer models,” In the First International Symposium on Uncertainty Modeling and Analysis, Maryland, USA, Dec. 3–5, 1990. DOI:10.1109/SUMA.1990.151285
• I. M. Sobol’ and Y. L. Levitan, “On the use of variance reducing multipliers in Monte Carlo computations of a global sensitivity index,” Computer Physics Communications, vol. 117, no. 1, pp. 52–61, 1999. DOI:10.1016/S0010-4655(98)00156-8
• A. Marrel, B. Iooss, B. Laurent, and O. Roustant, “Calculations of Sobol indices for the Gaussian process metamodel,” Reliability Engineering & System Safety, vol. 94, no. 3, pp. 742–751, 2009. DOI:10.1016/j.ress.2008.07.008