.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_reliability_sensitivity/central_dispersion/plot_expectation_simulation_algorithm.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_reliability_sensitivity_central_dispersion_plot_expectation_simulation_algorithm.py>`
        to download the full example code

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_reliability_sensitivity_central_dispersion_plot_expectation_simulation_algorithm.py:


Evaluate the mean of a random vector by simulations
===================================================

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Abstract
--------
We introduce the :class:`ExpectationSimulationAlgorithm` class which implements an incremental Monte Carlo sampling algorithm to estimate the mean of a random vector.

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.. code-block:: Python

    import openturns as ot
    import openturns.viewer as otv
    from matplotlib import pylab as plt








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We shall use this algorithm for the :ref:`Ishigami function <use-case-ishigami>` that we load from the `usecases` module :


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.. code-block:: Python

    from openturns.usecases import ishigami_function

    im = ishigami_function.IshigamiModel()









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The Ishigami `model` and the distribution of the input variables are stored in
the `im` object :

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.. code-block:: Python

    model = im.model
    distribution = im.distributionX








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We create a random vector that follows the distribution of the input variables.

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.. code-block:: Python

    inputVector = ot.RandomVector(distribution)








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The output vector is a :class:`~openturns.CompositeRandomVector`.

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.. code-block:: Python

    outputVector = ot.CompositeRandomVector(model, inputVector)








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The mean of the output vector is

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.. code-block:: Python

    print("Mean of the output random vector : %.5f" % im.expectation)






.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    Mean of the output random vector : 3.50000




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We define the algorithm simply by calling it with the output vector :

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.. code-block:: Python

    algo = ot.ExpectationSimulationAlgorithm(outputVector)








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We can also set the algorithm parameters :

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.. code-block:: Python

    algo.setMaximumOuterSampling(80000)
    algo.setBlockSize(1)
    algo.setCoefficientOfVariationCriterionType("NONE")









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We are then ready to launch the algorithm and store the result.

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.. code-block:: Python

    algo.run()
    result = algo.getResult()









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As usual for Monte Carlo estimation we can draw the convergence history.

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.. code-block:: Python

    graphConvergence = algo.drawExpectationConvergence()
    view = otv.View(graphConvergence)





.. image-sg:: /auto_reliability_sensitivity/central_dispersion/images/sphx_glr_plot_expectation_simulation_algorithm_001.png
   :alt: Expectation convergence graph at level 0.95
   :srcset: /auto_reliability_sensitivity/central_dispersion/images/sphx_glr_plot_expectation_simulation_algorithm_001.png
   :class: sphx-glr-single-img





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The result obtained with the previous algorithm is an instance of the
:class:~openturns.ExpectationSimulationResult` class.

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The expected value of the mean is given by the `getExpectationEstimate` method :

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.. code-block:: Python

    expectation = result.getExpectationEstimate()
    print("Estimated mean of the output random vector : %.5f" % expectation[0])





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    Estimated mean of the output random vector : 3.50261




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The variance and standard deviation of the estimated mean are respectively given by `getVarianceEstimate` and `getStandardDeviation`:

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.. code-block:: Python

    expectationVariance = result.getVarianceEstimate()
    print(
        "Variance of the estimated mean of the output random vector : %.5f"
        % expectationVariance[0]
    )
    standardDeviation = result.getStandardDeviation()
    print("Standard deviation : %.5f" % standardDeviation[0])





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    Variance of the estimated mean of the output random vector : 0.00017
    Standard deviation : 0.01316




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This variance and this standard deviation must not to be confused with the variance and the standard deviation of the Ishigami model!

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.. code-block:: Python

    print("Ishigami variance : %.5f" % im.variance)
    print("Ishigami standard deviation : %.5f" % im.variance ** (1 / 2))






.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    Ishigami variance : 13.84459
    Ishigami standard deviation : 3.72083




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The asymptotic confidence distribution of the output random vector mean estimate is

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.. code-block:: Python

    expectationDistribution = result.getExpectationDistribution()
    print(expectationDistribution)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    Normal(mu = 3.50261, sigma = 0.0131575)




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Let us draw it:

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.. code-block:: Python

    graphExpectationDistribution = expectationDistribution.drawPDF()
    graphExpectationDistribution.setTitle(
        "Normal asymptotic distribution of the mean estimate"
    )
    view = otv.View(graphExpectationDistribution)




.. image-sg:: /auto_reliability_sensitivity/central_dispersion/images/sphx_glr_plot_expectation_simulation_algorithm_002.png
   :alt: Normal asymptotic distribution of the mean estimate
   :srcset: /auto_reliability_sensitivity/central_dispersion/images/sphx_glr_plot_expectation_simulation_algorithm_002.png
   :class: sphx-glr-single-img





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Display all figures

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.. code-block:: Python

    plt.show()








.. _sphx_glr_download_auto_reliability_sensitivity_central_dispersion_plot_expectation_simulation_algorithm.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_expectation_simulation_algorithm.ipynb <plot_expectation_simulation_algorithm.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_expectation_simulation_algorithm.py <plot_expectation_simulation_algorithm.py>`