.. _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::
   \model(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]^3`
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 \otimes \mathcal{U}(-\pi,\pi).

We consider the random variable :math:`Y` defined by:

.. math::
    Y = \model(X_1,X_2,X_3)

Analysis
--------

The moments of any order of the random variable  :math:`Y` are finite. 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

