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.. _use-case-ishigami:
The Ishigami function
=====================
The Ishigami function of Ishigami & Homma (1990) is recurrent test case for sensitivity analysis methods and uncertainty.
Let :math:`a=7` and :math:`b=0.1` (see Crestaux et al. (2007) and Marrel et al. (2009)). We consider the function
.. math::
g(X_1,X_2,X_3) = \sin(X_1)+a \sin (X_2)^2 + b X_3^4 \sin(X_1)
for any :math:`X_1,X_2,X_3\in[-\pi,\pi]`
We assume that the random variables :math:`X_1,X_2,X_3` are independent and have the uniform marginal distribution in the interval from :math:`-\pi` to :math:`\pi`:
.. math::
X_1,X_2,X_3\sim \mathcal{U}(-\pi,\pi).
Analysis
--------
The expectation and the variance of :math:`Y` are
.. math::
\Expect{Y} = \frac{a}{2}
and
.. math::
\Var{Y} = \frac{1}{2} + \frac{a^2}{8} + \frac{b^2 \pi^8}{18} + \frac{b\pi^4}{5}.
The Sobol' decomposition variances are
.. math::
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 :math:`V_3=V_{1,2} = V_{2,3}=V_{1,2,3} = 0`.
This leads to the following first order Sobol' indices:
.. math::
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:
.. math::
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 :math:`X_3` has no effect at first order (because :math:`X_3^4` it is multiplied
by :math:`\sin(X_1)`), but has a total effet because of the interactions with :math:`X_1`.
On the other hand, the second variable :math:`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.
* [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.
Load the use case
-----------------
We can load this model from the use cases module as follows :
.. code-block:: python
>>> from openturns.usecases import ishigami_function
>>> # Load the Ishigami use case
>>> im = ishigami_function.IshigamiModel()
API documentation
-----------------
.. currentmodule:: openturns.usecases.ishigami_function
.. autoclass:: IshigamiModel
:noindex:
Examples based on this use case
-------------------------------
.. minigallery:: openturns.usecases.ishigami_function.IshigamiModel
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