.. only:: html

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

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

    .. _sphx_glr_auto_probabilistic_modeling_distributions_plot_distribution_manipulation.py:


Distribution manipulation
=========================

In this example we are going to exhibit some of the services exposed by the distribution objects:

- ask for the dimension, with the method getDimension
- extract the marginal distributions, with the method getMarginal
- to ask for some properties, with isContinuous, isDiscrete, isElliptical
- to get the copula, with the method getCopula*
- to ask for some properties on the copula, with the methods hasIndependentCopula, hasEllipticalCopula
- to evaluate some moments, with getMean, getStandardDeviation, getCovariance, getSkewness, getKurtosis
- to evaluate the roughness, with the method getRoughness
- to get one realization or simultaneously :math:`n` realizations, with the method getRealization, getSample
- to evaluate the probability content of a given interval, with the method computeProbability
- to evaluate a quantile or a complementary quantile, with the method computeQuantile
- to evaluate the characteristic function of the distribution
- to evaluate the derivative of the CDF or PDF
- to draw some curves


.. 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)








Create an 1-d distribution


.. code-block:: default

    dist_1 = ot.Normal()

    # Create a 2-d distribution
    dist_2 = ot.ComposedDistribution([ot.Normal(), ot.Triangular(0.0, 2.0, 3.0)], ot.ClaytonCopula(2.3))

    # Create a 3-d distribution
    copula_dim3 = ot.Student(5.0, 3).getCopula()
    dist_3 = ot.ComposedDistribution([ot.Normal(), ot.Triangular(0.0, 2.0, 3.0), ot.Exponential(0.2)], copula_dim3)








Get the dimension fo the distribution


.. code-block:: default

    dist_2.getDimension()





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

 Out:

 .. code-block:: none


    2



Get the 2nd marginal


.. code-block:: default

    dist_2.getMarginal(1)






.. raw:: html

    <p>Triangular(a = 0, m = 2, b = 3)</p>
    <br />
    <br />

Get a 2-d marginal


.. code-block:: default

    dist_3.getMarginal([0, 1]).getDimension()





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

 Out:

 .. code-block:: none


    2



Ask some properties of the distribution


.. code-block:: default

    dist_1.isContinuous(), dist_1.isDiscrete(), dist_1.isElliptical()





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

 Out:

 .. code-block:: none


    (True, False, True)



Get the copula


.. code-block:: default

    copula = dist_2.getCopula()








Ask some properties on the copula


.. code-block:: default

    dist_2.hasIndependentCopula(), dist_2.hasEllipticalCopula()





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

 Out:

 .. code-block:: none


    (False, False)



mean vector of the distribution


.. code-block:: default

    dist_2.getMean()






.. raw:: html

    <p>[0,1.66667]</p>
    <br />
    <br />

standard deviation vector of the distribution


.. code-block:: default

    dist_2.getStandardDeviation()






.. raw:: html

    <p>[1,0.62361]</p>
    <br />
    <br />

covariance matrix of the distribution


.. code-block:: default

    dist_2.getCovariance()






.. raw:: html

    <p>[[ 1        0.491927 ]<br>
     [ 0.491927 0.388889 ]]</p>
    <br />
    <br />

skewness vector of the distribution


.. code-block:: default

    dist_2.getSkewness()






.. raw:: html

    <p>[0,-0.305441]</p>
    <br />
    <br />

kurtosis vector of the distribution


.. code-block:: default

    dist_2.getKurtosis()






.. raw:: html

    <p>[3,2.4]</p>
    <br />
    <br />

roughness of the distribution


.. code-block:: default

    dist_1.getRoughness()





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

 Out:

 .. code-block:: none


    0.28209479177387814



Get one realization


.. code-block:: default

    dist_2.getRealization()






.. raw:: html

    <p>[0.779052,2.64799]</p>
    <br />
    <br />

Get several realizations


.. code-block:: default

    dist_2.getSample(5)






.. raw:: html

    <TABLE><TR><TD></TD><TH>X0</TH><TH>X1</TH></TR>
    <TR><TH>0</TH><TD>0.06985929</TD><TD>1.42593</TD></TR>
    <TR><TH>1</TH><TD>-0.2847443</TD><TD>1.680554</TD></TR>
    <TR><TH>2</TH><TD>-1.842467</TD><TD>0.7630956</TD></TR>
    <TR><TH>3</TH><TD>0.3236553</TD><TD>2.048827</TD></TR>
    <TR><TH>4</TH><TD>0.1203091</TD><TD>2.285018</TD></TR>
    </TABLE>
    <br />
    <br />

Evaluate the PDF at the mean point


.. code-block:: default

    dist_2.computePDF(dist_2.getMean())





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

 Out:

 .. code-block:: none


    0.3528005531670077



Evaluate the CDF at the mean point


.. code-block:: default

    dist_2.computeCDF(dist_2.getMean())





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

 Out:

 .. code-block:: none


    0.3706626446357781



Evaluate the complementary CDF


.. code-block:: default

    dist_2.computeComplementaryCDF(dist_2.getMean())





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

 Out:

 .. code-block:: none


    0.6293373553642219



Evaluate the survival function at the mean point


.. code-block:: default

    dist_2.computeSurvivalFunction(dist_2.getMean())





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

 Out:

 .. code-block:: none


    0.4076996816728151



Evaluate the PDF on a sample


.. code-block:: default

    dist_2.computePDF(dist_2.getSample(5))






.. raw:: html

    <TABLE><TR><TD></TD><TH>v0</TH></TR>
    <TR><TH>0</TH><TD>0.2107205</TD></TR>
    <TR><TH>1</TH><TD>0.2829345</TD></TR>
    <TR><TH>2</TH><TD>0.2683372</TD></TR>
    <TR><TH>3</TH><TD>0.3159856</TD></TR>
    <TR><TH>4</TH><TD>0.3618715</TD></TR>
    </TABLE>
    <br />
    <br />

Evaluate the CDF on a sample


.. code-block:: default

    dist_2.computeCDF(dist_2.getSample(5))






.. raw:: html

    <TABLE><TR><TD></TD><TH>v0</TH></TR>
    <TR><TH>0</TH><TD>0.7326854</TD></TR>
    <TR><TH>1</TH><TD>0.5003563</TD></TR>
    <TR><TH>2</TH><TD>0.820944</TD></TR>
    <TR><TH>3</TH><TD>0.01091264</TD></TR>
    <TR><TH>4</TH><TD>0.4877803</TD></TR>
    </TABLE>
    <br />
    <br />

Evaluate the probability content of an 1-d interval


.. code-block:: default

    interval = ot.Interval(-2.0, 3.0)
    dist_1.computeProbability(interval)





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

 Out:

 .. code-block:: none


    0.9758999700201907



Evaluate the probability content of a 2-d interval


.. code-block:: default

    interval = ot.Interval([0.4, -1], [3.4, 2])
    dist_2.computeProbability(interval)





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

 Out:

 .. code-block:: none


    0.129833882783416



Evaluate the quantile of order p=90%


.. code-block:: default

    dist_2.computeQuantile(0.90)






.. raw:: html

    <p>[1.60422,2.59627]</p>
    <br />
    <br />

and the quantile of order 1-p


.. code-block:: default

    dist_2.computeQuantile(0.90, True)






.. raw:: html

    <p>[-1.10363,0.899591]</p>
    <br />
    <br />

Evaluate the quantiles of order p et q
For example, the quantile 90% and 95%


.. code-block:: default

    dist_1.computeQuantile([0.90, 0.95])






.. raw:: html

    <TABLE><TR><TD></TD><TH>v0</TH></TR>
    <TR><TH>0</TH><TD>1.281552</TD></TR>
    <TR><TH>1</TH><TD>1.644854</TD></TR>
    </TABLE>
    <br />
    <br />

and the quantile of order 1-p and 1-q


.. code-block:: default

    dist_1.computeQuantile([0.90, 0.95], True)






.. raw:: html

    <TABLE><TR><TD></TD><TH>v0</TH></TR>
    <TR><TH>0</TH><TD>-1.281552</TD></TR>
    <TR><TH>1</TH><TD>-1.644854</TD></TR>
    </TABLE>
    <br />
    <br />

Evaluate the characteristic function of the distribution (only 1-d)


.. code-block:: default

    dist_1.computeCharacteristicFunction(dist_1.getMean()[0])





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

 Out:

 .. code-block:: none


    (1+0j)



Evaluate the derivatives of the PDF with respect to the parameters at mean


.. code-block:: default

    dist_2.computePDFGradient(dist_2.getMean())






.. raw:: html

    <p>[0,-0.398942,0.12963,-0.277778,-0.185185,0]</p>
    <br />
    <br />

Evaluate the derivatives of the CDF with respect to the parameters at mean


.. code-block:: default

    dist_2.computeCDFGradient(dist_2.getMean())






.. raw:: html

    <p>[-0.398942,-0,-0.169753,-0.231481,-0.555556,0]</p>
    <br />
    <br />

draw PDF


.. code-block:: default

    graph = dist_1.drawPDF()
    view = viewer.View(graph)




.. image:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_distribution_manipulation_001.png
    :alt: plot distribution manipulation
    :class: sphx-glr-single-img





draw CDF


.. code-block:: default

    graph = dist_1.drawCDF()
    view = viewer.View(graph)




.. image:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_distribution_manipulation_002.png
    :alt: plot distribution manipulation
    :class: sphx-glr-single-img





Draw an 1-d quantile curve


.. code-block:: default


    # Define the range and the number of points
    qMin = 0.2
    qMax = 0.6
    nbrPoints = 101
    quantileGraph = dist_1.drawQuantile(qMin, qMax, nbrPoints)
    view = viewer.View(quantileGraph)




.. image:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_distribution_manipulation_003.png
    :alt: plot distribution manipulation
    :class: sphx-glr-single-img





Draw a 2-d quantile curve


.. code-block:: default


    # Define the range and the number of points
    qMin = 0.3
    qMax = 0.9
    nbrPoints = 101
    quantileGraph = dist_2.drawQuantile(qMin, qMax, nbrPoints)
    view = viewer.View(quantileGraph)
    plt.show()



.. image:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_distribution_manipulation_004.png
    :alt: [X0,X1] Quantile
    :class: sphx-glr-single-img






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

   **Total running time of the script:** ( 0 minutes  0.322 seconds)


.. _sphx_glr_download_auto_probabilistic_modeling_distributions_plot_distribution_manipulation.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_manipulation.py <plot_distribution_manipulation.py>`



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

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


.. only:: html

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

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