.. only:: html

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

        Click :ref:`here <sphx_glr_download_auto_probabilistic_modeling_distributions_plot_distribution_transformation.py>`     to download the full example code
    .. rst-class:: sphx-glr-example-title

    .. _sphx_glr_auto_probabilistic_modeling_distributions_plot_distribution_transformation.py:


Transform a distribution
========================

In this example we are going to use distribution algebra and distribution transformation via functions.


.. code-block:: default

    from __future__ import print_function
    import openturns as ot
    import openturns.viewer as viewer
    from matplotlib import pylab as plt
    ot.Log.Show(ot.Log.NONE)








We define some (classical) distribution :


.. code-block:: default

    distribution1 = ot.Uniform(0.0, 1.0)
    distribution2 = ot.Uniform(0.0, 2.0)
    distribution3 = ot.WeibullMin(1.5, 2.0)








Sum & difference of distributions
---------------------------------

It is easy to compute the sum of distributions. For example:


.. code-block:: default

    distribution = distribution1 + distribution2
    print(distribution)
    graph = distribution.drawPDF()
    view = viewer.View(graph)




.. image:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_distribution_transformation_001.png
    :alt: plot distribution transformation
    :class: sphx-glr-single-img


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

 Out:

 .. code-block:: none

    Trapezoidal(a = 0, b = 1, c = 2, d = 3)




We might also use substraction even with scalar values: 


.. code-block:: default

    distribution = 3.0 - distribution3
    print(distribution)
    graph = distribution.drawPDF()
    view = viewer.View(graph)




.. image:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_distribution_transformation_002.png
    :alt: plot distribution transformation
    :class: sphx-glr-single-img


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

 Out:

 .. code-block:: none

    RandomMixture(3 - WeibullMin(beta = 1.5, alpha = 2, gamma = 0))




Product & inverse
-----------------

We might also compute the product of two (or more) distributions. For example:


.. code-block:: default

    distribution = distribution1 * distribution2
    print(distribution)
    graph = distribution.drawPDF()
    view = viewer.View(graph)




.. image:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_distribution_transformation_003.png
    :alt: plot distribution transformation
    :class: sphx-glr-single-img


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

 Out:

 .. code-block:: none

    ProductDistribution(Uniform(a = 0, b = 1) * Uniform(a = 0, b = 2))




We could also inverse a distribution : 


.. code-block:: default

    distribution = 1 / distribution1
    print(distribution)
    graph = distribution.drawPDF()
    view = viewer.View(graph)




.. image:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_distribution_transformation_004.png
    :alt: plot distribution transformation
    :class: sphx-glr-single-img


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

 Out:

 .. code-block:: none

    CompositeDistribution=f(Uniform(a = 0, b = 1)) with f=[x]->[1.0 / x]




Or compute a ratio distrobution:


.. code-block:: default

    ratio = distribution2 / distribution1
    print(ratio)
    graph = ratio.drawPDF()
    view = viewer.View(graph)




.. image:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_distribution_transformation_005.png
    :alt: plot distribution transformation
    :class: sphx-glr-single-img


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

 Out:

 .. code-block:: none

    ProductDistribution(Uniform(a = 0, b = 2) * CompositeDistribution=f(Uniform(a = 0, b = 1)) with f=[x]->[1.0 / x])




Transformation using functions
------------------------------

The library provides methods to get the full distributions of `f(x)` where `f` can be equal to :

 - `sin`, 
 - `asin`,
 - `cos`, 
 - `acos`, 
 - `tan`,
 - `atan`,
 - `sinh`, 
 - `asinh`,
 - `cosh`, 
 - `acosh`, 
 - `tanh`,
 - `atanh`,
 - `sqr` (for square), 
 - `inverse`, 
 - `sqrt`,
 - `exp`,
 - `log`/`ln`,
 - `abs`,
 - `cbrt`.

If one wants a specific method, user might rely on `CompositeDistribution`. 

For example for the usual `log` transformation:


.. code-block:: default

    graph =distribution1.log().drawPDF()
    view = viewer.View(graph)




.. image:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_distribution_transformation_006.png
    :alt: plot distribution transformation
    :class: sphx-glr-single-img





And for the `log2` function:


.. code-block:: default

    f = ot.SymbolicFunction(['x'], ['log2(x)'])
    f.setDescription(["X","ln(X)"])
    graph = ot.CompositeDistribution(f, distribution1).drawPDF()
    view = viewer.View(graph)
    plt.show()



.. image:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_distribution_transformation_007.png
    :alt: plot distribution transformation
    :class: sphx-glr-single-img






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

   **Total running time of the script:** ( 0 minutes  0.641 seconds)


.. _sphx_glr_download_auto_probabilistic_modeling_distributions_plot_distribution_transformation.py:


.. only :: html

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



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

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



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

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


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_