# 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.

• [saltelli2000]

• 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.

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)

distributionXJointDistribution

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.

## Examples based on this use case¶

Estimate correlation coefficients

Estimate correlation coefficients

Visualize sensitivity

Visualize sensitivity

Compute grouped indices for the Ishigami function

Compute grouped indices for the Ishigami function

Validate a polynomial chaos

Validate a polynomial chaos

Create a polynomial chaos for the Ishigami function: a quick start guide to polynomial chaos

Create a polynomial chaos for the Ishigami function: a quick start guide to polynomial chaos

Polynomial chaos expansion cross-validation

Polynomial chaos expansion cross-validation

Create a sparse chaos by integration

Create a sparse chaos by integration

Kriging: propagate uncertainties

Kriging: propagate uncertainties

Evaluate the mean of a random vector by simulations

Evaluate the mean of a random vector by simulations

Sobol’ sensitivity indices using rank-based algorithm

Sobol' sensitivity indices using rank-based algorithm

FAST sensitivity indices

FAST sensitivity indices

Estimate Sobol’ indices for the Ishigami function by a sampling method: a quick start guide to sensitivity analysis

Estimate Sobol' indices for the Ishigami function by a sampling method: a quick start guide to sensitivity analysis

The HSIC sensitivity indices: the Ishigami model

The HSIC sensitivity indices: the Ishigami model

Compute the L2 error between two functions

Compute the L2 error between two functions

Compute leave-one-out error of a polynomial chaos expansion

Compute leave-one-out error of a polynomial chaos expansion