%feature("docstring") OT::MauntzKucherenkoSensitivityAlgorithm
"Sensitivity analysis using MauntzKucherenko method.

Available constructors:
    MauntzKucherenkoSensitivityAlgorithm(*inputDesign, outputDesign, N*)

    MauntzKucherenkoSensitivityAlgorithm(*distribution, N, model, computeSecondOrder*)

    MauntzKucherenkoSensitivityAlgorithm(*experiment, model, computeSecondOrder*)

Parameters
----------
inputDesign : :class:`~openturns.Sample`
    Design for the evaluation of sensitivity indices, obtained with the
    :class:`~openturns.SobolIndicesExperiment`:code:`.`:meth:`~openturns.SobolIndicesExperiment.generate`
    method
outputDesign : :class:`~openturns.Sample`
    Design for the evaluation of sensitivity indices, obtained as the evaluation
    of a Function (model) on the previous inputDesign
distribution : :class:`~openturns.Distribution`
    Input probabilistic model.
    Should have independent copula
experiment : :class:`~openturns.WeightedExperiment`
    Experiment for the generation of two independent samples.
N : int
    Size of samples to generate
computeSecondOrder : bool
    If True, design that will be generated contains elements for the evaluation
    of second order indices.

Notes
-----
This class analyzes the influence of each component of a random vector
:math:`\vect{X} = \left( X^1, \ldots, X^{n_X} \right)` on a random vector
:math:`\vect{Y} = \left( Y^1, \ldots, Y^{n_Y} \right)` by computing Sobol' indices (see also [sobol1993]_).

The [sobol2007]_ method is used to estimate both first
and total order indices.

The class constructor ``MauntzKucherenkoSensitivityAlgorithm(inputDesign, outputDesign, N)``
requires a specific structure for the ``outputDesign``, and therefore for the  ``inputDesign``.
The latter should be generated using :class:`~openturns.SobolIndicesExperiment`
(see example below). Otherwise, results will be **worthless**.

Notations are defined in the documentation page of the :class:`~openturns.SobolIndicesAlgorithm` class.

The estimators of :math:`V_i` and :math:`V_{-i}` used by this class are respectively:

.. math::

   \begin{array}{ccc}
   \hat{V}_i & = & \frac{1}{N-1} \sum_{k=1}^{N} \tilde{\vect{g}}(\vect{B}_k) \left( \tilde{\vect{g}}(\vect{E}_k) - \tilde{\vect{g}}(\vect{A}_k) \right) \\
   \hat{VT}_{i} & = & \frac{1}{N-1} \sum_{k=1}^{N} \tilde{\vect{g}}(\vect{A}_k) \left( \tilde{\vect{g}}(\vect{A}_k) -  \tilde{\vect{g}}(\vect{E}_k) \right)
   \end{array}

where :math:`\tilde{\vect{g}}` is the centered model based on the sample. 

See also
--------
SobolIndicesAlgorithm

Examples
--------

Estimate first and total order Sobol' indices:

>>> import openturns as ot
>>> ot.RandomGenerator.SetSeed(0)
>>> formula = ['sin(pi_*X1)+7*sin(pi_*X2)^2+0.1*(pi_*X3)^4*sin(pi_*X1)']
>>> model = ot.SymbolicFunction(['X1', 'X2', 'X3'], formula)
>>> distribution = ot.ComposedDistribution([ot.Uniform(-1.0, 1.0)] * 3)
>>> # Define designs to pre-compute
>>> size = 100000
>>> inputDesign = ot.SobolIndicesExperiment(distribution, size).generate()
>>> outputDesign = model(inputDesign)
>>> # sensitivity analysis algorithm
>>> sensitivityAnalysis = ot.MauntzKucherenkoSensitivityAlgorithm(inputDesign, outputDesign, size)
>>> print(sensitivityAnalysis.getFirstOrderIndices())
[0.311678,0.439075,0.00401876]
>>> print(sensitivityAnalysis.getTotalOrderIndices())
[0.556564,0.439185,0.24036]"