The Ishigami function¶
The Ishigami function of Ishigami & Homma (1990) is recurrent test case for sensitivity analysis methods and uncertainty.
Let and
(see Crestaux et al. (2007) and Marrel et al. (2009)). We consider the function
for any
We assume that the random variables are independent and have the uniform marginal distribution in the interval from
to
:
Analysis¶
The expectation and the variance of are
and
The Sobol’ decomposition variances are
and .
This leads to the following first order Sobol’ indices:
and the following total order indices:
The third variable has no effect at first order (because
it is multiplied by
), but has a total effet because of the interactions with
. On the other hand, the second variable
has no interactions which implies that the first order indice is equal to the total order indice for this input variable.
References¶
Ishigami, T., & Homma, T. (1990, December). An importance quantification technique in uncertainty analysis for computer models. In Uncertainty Modeling and Analysis, 1990. Proceedings., First International Symposium on (pp. 398-403). IEEE.
Sobol’, I. M., & Levitan, Y. L. (1999). On the use of variance reducing multipliers in Monte Carlo computations of a global sensitivity index. Computer Physics Communications, 117(1), 52-61.
Saltelli, A., Chan, K., & Scott, E. M. (Eds.). (2000). Sensitivity analysis (Vol. 134). New York: Wiley.
Crestaux, T., Martinez, J.-M., Le Maitre, O., & Lafitte, O. (2007). Polynomial chaos expansion for uncertainties quantification and sensitivity analysis. SAMO 2007, http://samo2007.chem.elte.hu/lectures/Crestaux.pdf.
Load the use case¶
We can load this classical model from the use cases module as follows :
>>> from openturns.usecases import ishigami_function as ishigami_function
>>> # Load the Ishigami use case
>>> im = ishigami_function.IshigamiModel()
API documentation¶
See IshigamiModel
.
Examples based on this use case¶
![](../_images/sphx_glr_plot_sensitivity_sobol_thumb.png)
Estimate Sobol’ indices for the Ishigami function by a sampling method: a quick start guide to sensitivity analysis¶