The Ishigami function¶

The Ishigami function of Ishigami & Homma (1990) is recurrent test case for sensitivity analysis methods and uncertainty.

Let $a=7$ and $b=0.1$ (see Crestaux et al. (2007) and Marrel et al. (2009)). We consider the function

$g(X_1,X_2,X_3) = \sin(X_1)+a \sin (X_2)^2 + b X_3^4 \sin(X_1)$

for any $X_1,X_2,X_3\in[-\pi,\pi]$

We assume that the random variables $X_1,X_2,X_3$ are independent and have the uniform marginal distribution in the interval from $-\pi$ to $\pi$ :

$X_1,X_2,X_3\sim \mathcal{U}(-\pi,\pi).$

Analysis¶

The expectation and the variance of $Y$ are

$E(Y) = \frac{a}{2}$

and

$V(Y) = \frac{1}{2} + \frac{a^2}{8} + \frac{b^2 \pi^8}{18} + \frac{b\pi^4}{5}.$

The Sobol’ decomposition variances are

$V_1 = \frac{1}{2} \left(1 + b\frac{\pi^4}{5} \right)^2, \qquad V_2 = \frac{a^2}{8}, \qquad V_{1,3} = b^2 \pi^8 \frac{8}{225}$

and $V_3=V_{1,2} = V_{2,3}=V_{1,2,3} = 0$ .

This leads to the following first order Sobol’ indices:

$S_1 = \frac{V_1}{V(Y)}, \qquad S_2 = \frac{V_2}{V(Y)}, \qquad S_3 = 0,$

and the following total order indices:

$ST_1 = \frac{V_1+V_{1,3}}{V(Y)}, \qquad ST_2 = S_2, \qquad ST_3 = \frac{V_{1,3}}{V(Y)}.$

The third variable $X_3$ has no effect at first order (because $X_3^4$ it is multiplied by $\sin(X_1)$ ), but has a total effet because of the interactions with $X_1$ . On the other hand, the second variable $X_2$ 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 model from the use cases module as follows :

>>> from openturns.usecases import ishigami_function
>>> # Load the Ishigami use case
>>> im = ishigami_function.IshigamiModel()

API documentation¶

class IshigamiModel

Data class for the Ishigami model.

Examples

>>> from openturns.usecases import ishigami_function
>>> # Load the Ishigami model
>>> im = ishigami_function.IshigamiModel()

Attributes:

dimThe dimension of the problem: dim = 3
aConstant: a = 7.0
bConstant: b = 0.1
X1Uniform distribution: First marginal, ot.Uniform(-np.pi, np.pi)
X2Uniform distribution: Second marginal, ot.Uniform(-np.pi, np.pi)
X3Uniform distribution: Third marginal, ot.Uniform(-np.pi, np.pi)
distributionXComposedDistribution: The joint distribution of the input parameters.
ishigamiSymbolicFunction: The Ishigami model with a, b as variables.
modelParametricFunction: The Ishigami model with the a=7.0 and b=0.1 parameters fixed.
expectationConstant: Expectation of the output variable.
varianceConstant: Variance of the output variable.
S1Constant: First order Sobol index number 1
S2Constant: First order Sobol index number 2
S3Constant: First order Sobol index number 3
S12Constant: Second order Sobol index for marginals 1 and 2.
S13Constant: Second order Sobol index for marginals 1 and 3.
S23Constant: Second order Sobol index for marginals 2 and 3.
S123Constant
ST1Constant: Total order Sobol index number 1.
ST2Constant: Total order Sobol index number 2.
ST3Constant: Total order Sobol index number 3.