# 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.