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

.. only:: html

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

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

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

.. _sphx_glr_auto_meta_modeling_polynomial_chaos_metamodel_plot_chaos_build_distribution.py:


Fit a distribution from an input sample
=======================================

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In this example we show how to use the
:meth:`~openturns.MetaModelAlgorithm.BuildDistribution` function to fit a distribution to an
input sample.
This function is used by the :class:`~openturns.FunctionalChaosAlgorithm`
class when we want to create a polynomial chaos metamodel and we have a
design of experiments which have been computed beforehand.
In this case, we have to identify the distributions which best fit to the
input sample in order to define the input probabilistic model.
This is, in turn, used by in the polynomial chaos to create the orthogonal basis.


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.. code-block:: Python

    import openturns as ot

    ot.Log.Show(ot.Log.NONE)








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We first create the function `model`.

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.. code-block:: Python

    ot.RandomGenerator.SetSeed(0)
    dimension = 2
    input_names = ["x1", "x2"]
    formulas = ["cos(x1 + x2)", "(x2 + 1) * exp(x1)"]
    model = ot.SymbolicFunction(input_names, formulas)








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Then we create a sample `x` and compute the corresponding output sample `y`.

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.. code-block:: Python

    distribution = ot.Normal(dimension)
    samplesize = 1000
    inputSample = distribution.getSample(samplesize)
    outputSample = model(inputSample)








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Create a functional chaos model.
The algorithm used by :meth:`~openturns.MetaModelAlgorithm.BuildDistribution`
fits a distribution on the input sample.
This is done with the Lilliefors test.
Please read :ref:`kolmogorov_smirnov_test` for more details on this topic.

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The Lilliefors test is based on sampling the distribution of the
Kolmogorov-Smirnov statistics.
The sample size corresponding to this algorithm is configured
by the `"FittingTest-LillieforsMaximumSamplingSize"` :class:`~openturns.ResourceMap`
key.
In order to get satisfactory results, the default value of this
key is relatively large.

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.. code-block:: Python

    ot.ResourceMap.GetAsUnsignedInteger("FittingTest-LillieforsMaximumSamplingSize")





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

 .. code-block:: none


    100000



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In order to speed this example up, let us reduce this value.

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.. code-block:: Python

    ot.ResourceMap.SetAsUnsignedInteger("FittingTest-LillieforsMaximumSamplingSize", 100)








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Then we fit the distribution.

.. GENERATED FROM PYTHON SOURCE LINES 70-72

.. code-block:: Python

    distribution = ot.FunctionalChaosAlgorithm.BuildDistribution(inputSample)








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Let us explore the distribution with its fitted parameters.

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.. code-block:: Python

    distribution






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    JointDistribution
    <ul>
      <li>name=JointDistribution</li>
      <li>dimension: 2</li>
      <li>description=[X0,X1]
      <li>copula: IndependentCopula(dimension = 2)</li>
    </ul>

    <table>
      <tr>
        <th>Index</th>
        <th>Variable</th>
        <th>Distribution</th>
      </tr>
      <tr>
        <td>0</td>
        <td>X0</td>
        <td>Normal(mu = -0.0268588, sigma = 0.985838)</td>
      </tr>
      <tr>
        <td>1</td>
        <td>X1</td>
        <td>Normal(mu = 0.00246667, sigma = 1.00015)</td>
      </tr>
    </table>

    </div>
    <br />
    <br />

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We can also analyze its properties in more details.

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.. code-block:: Python

    for i in range(dimension):
        marginal = distribution.getMarginal(i)
        marginalname = marginal.getImplementation().getClassName()
        print("Marginal #", i, ":", marginalname)
    distribution.getCopula()





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

 .. code-block:: none

    Marginal # 0 : Normal
    Marginal # 1 : Normal


.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    IndependentCopula
    <ul>
      <li>name=IndependentCopula</li>
      <li>dimension=2</li>
      <li>weight=1</li>
      <li>range=[0, 1]
    [0, 1]</li>
      <li>description=[X0,X1]</li>
      <li>isParallel=true</li>
      <li>isCopula=true</li>
    </ul>

    </div>
    <br />
    <br />

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The previous call to :meth:`~openturns.MetaModelAlgorithm.BuildDistribution` is what is done internally by the
following constructor of :class:`~openturns.FunctionalChaosAlgorithm`.

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.. code-block:: Python

    algo = ot.FunctionalChaosAlgorithm(inputSample, outputSample)








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The previous constructor is the main topic of the example
:doc:`Create a polynomial chaos metamodel from a data set </auto_meta_modeling/polynomial_chaos_metamodel/plot_functional_chaos>`.

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.. code-block:: Python

    ot.ResourceMap.Reload()  # reset default settings








.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 2.367 seconds)


.. _sphx_glr_download_auto_meta_modeling_polynomial_chaos_metamodel_plot_chaos_build_distribution.py:

.. only:: html

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

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

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

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

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

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_chaos_build_distribution.zip <plot_chaos_build_distribution.zip>`