.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_probabilistic_modeling/distributions/plot_compound_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_probabilistic_modeling_distributions_plot_compound_distribution.py>`
        to download the full example code.

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

.. _sphx_glr_auto_probabilistic_modeling_distributions_plot_compound_distribution.py:


Create a compound distribution
===================================

.. GENERATED FROM PYTHON SOURCE LINES 7-35

In this example we are going to build the distribution of the random vector :math:`\inputRV`
defined by the conditional distribution of:

.. math::

   \inputRV|\vect{\Theta}

where :math:`\vect{\Theta}` is the output of the random variable :math:`\vect{Y}` through the link
function :math:`f`:

.. math::

   \vect{\Theta} & = f(\vect{Y})\\
   \vect{Y} & \sim \cL_{\vect{Y}}

This example creates a :class:`~openturns.CompoundDistribution` which offers all the methods
attached to the distributions.

We consider the case where :math:`X` is of dimension 1 and follows a uniform distribution defined
by:

===============  =========================================================  ===============================
Variable         Distribution                                               Parameter
===============  =========================================================  ===============================
:math:`X`        :class:`~openturns.Uniform` (:math:`a, b`)                 :math:`(a,b) = (Y, 1+Y^2)`
:math:`Y`        :class:`~openturns.Uniform` (:math:`c, d`)                 :math:`(c,d) = (-1, 1)`
===============  =========================================================  ===============================


.. GENERATED FROM PYTHON SOURCE LINES 37-40

.. code-block:: Python

    import openturns as ot
    import openturns.viewer as otv








.. GENERATED FROM PYTHON SOURCE LINES 41-42

Create the :math:`Y` distribution.

.. GENERATED FROM PYTHON SOURCE LINES 42-44

.. code-block:: Python

    YDist = ot.Uniform(-1.0, 1.0)








.. GENERATED FROM PYTHON SOURCE LINES 45-46

Create the link function :math:`f: y \rightarrow (y, 1+y^2)`.

.. GENERATED FROM PYTHON SOURCE LINES 46-48

.. code-block:: Python

    f = ot.SymbolicFunction(["y"], ["y", "1+y^2"])








.. GENERATED FROM PYTHON SOURCE LINES 49-51

Create the conditional distribution of :math:`\vect{X}|\vect{\Theta}`: as the parameters have no
importance, we use the default distribution.

.. GENERATED FROM PYTHON SOURCE LINES 51-53

.. code-block:: Python

    XgivenThetaDist = ot.Uniform()








.. GENERATED FROM PYTHON SOURCE LINES 54-58

In this example, the range of :math:`\vect{X}` depends on its parameters which are random. Thus, the model is not regular.
We advice to change the discretization method that performs the integration.
By default, this method is the quadrature method *GaussProduct*. We prefer to use the *QMC*.
We use the :class:`~openturns.ResourceMap`.

.. GENERATED FROM PYTHON SOURCE LINES 58-60

.. code-block:: Python

    ot.ResourceMap.SetAsString('CompoundDistribution-ContinuousDiscretizationMethod', 'QMC')








.. GENERATED FROM PYTHON SOURCE LINES 61-62

Create the compound distribution of :math:`X`.

.. GENERATED FROM PYTHON SOURCE LINES 62-66

.. code-block:: Python

    XDist = ot.CompoundDistribution(XgivenThetaDist, YDist, f)
    XDist.setDescription([r"$X|\mathbf{\boldsymbol{\Theta}} = f(Y)$"])
    XDist






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    CompoundDistribution
    <ul>
      <li>name=CompoundDistribution</li>
      <li>dimension=1</li>
      <li>weight=1</li>
      <li>range=[-0.999969, 1.99997]</li>
      <li>description=[$X|\mathbf{\boldsymbol{\Theta}} = f(Y)$]</li>
      <li>isParallel=true</li>
      <li>isCopula=false</li>
    </ul>

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

.. GENERATED FROM PYTHON SOURCE LINES 67-68

Get a sample:

.. GENERATED FROM PYTHON SOURCE LINES 68-70

.. code-block:: Python

    XDist.getSample(5)






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <table>
      <tr><td></td><th>$X|\mathbf{\boldsymbol{\Theta}} = f(Y)$</th></tr>
      <tr><th>0</th><td>0.9728113</td></tr>
      <tr><th>1</th><td>-0.655941</td></tr>
      <tr><th>2</th><td>1.050775</td></tr>
      <tr><th>3</th><td>1.277257</td></tr>
      <tr><th>4</th><td>-0.101663</td></tr>
    </table>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 71-72

Draw the PDF.

.. GENERATED FROM PYTHON SOURCE LINES 72-76

.. code-block:: Python

    graph = XDist.drawPDF()
    view = otv.View(graph)





.. image-sg:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_compound_distribution_001.svg
   :alt: plot compound distribution
   :srcset: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_compound_distribution_001.svg
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 77-78

.. code-block:: Python

    view.ShowAll()








.. _sphx_glr_download_auto_probabilistic_modeling_distributions_plot_compound_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_compound_distribution.ipynb <plot_compound_distribution.ipynb>`

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

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

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

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