.. only:: html

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

        Click :ref:`here <sphx_glr_download_auto_probabilistic_modeling_distributions_plot_minimum_volume_level_set_2D.py>`     to download the full example code
    .. rst-class:: sphx-glr-example-title

    .. _sphx_glr_auto_probabilistic_modeling_distributions_plot_minimum_volume_level_set_2D.py:


Draw minimum volume level set in 2D
===================================

In this example, we compute the minimum volume level set of a bivariate distribution.


.. code-block:: default

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








Create a gaussian


.. code-block:: default

    corr = ot.CorrelationMatrix(2)
    corr[0, 1] = 0.2
    copula = ot.NormalCopula(corr)
    x1 = ot.Normal(-1., 1)
    x2 = ot.Normal(2, 1)
    x_funk = ot.ComposedDistribution([x1, x2], copula)

    # Create a second gaussian
    x1 = ot.Normal(1.,1)
    x2 = ot.Normal(-2,1)
    x_punk = ot.ComposedDistribution([x1, x2], copula)

    # Mix the distributions
    mixture = ot.Mixture([x_funk, x_punk], [0.5,1.])









.. code-block:: default

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




.. image:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_minimum_volume_level_set_2D_001.png
    :alt: [X0,X1] iso-PDF
    :class: sphx-glr-single-img





For a multivariate distribution (with dimension greater than 1), the `computeMinimumVolumeLevelSetWithThreshold` uses Monte-Carlo sampling. 


.. code-block:: default

    ot.ResourceMap.SetAsUnsignedInteger("Distribution-MinimumVolumeLevelSetSamplingSize",1000)








We want to compute the minimum volume LevelSet which contains `alpha`=90% of the distribution. The `threshold` is the value of the PDF corresponding the `alpha`-probability: the points contained in the LevelSet have a PDF value lower or equal to this threshold.


.. code-block:: default

    alpha = 0.9
    levelSet, threshold = mixture.computeMinimumVolumeLevelSetWithThreshold(alpha)
    threshold






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

 Out:

 .. code-block:: none


    0.009132175837384555




.. code-block:: default

    def drawLevelSetContour2D(distribution, numberOfPointsInXAxis, alpha, threshold, sampleSize= 500):
        '''
        Compute the minimum volume LevelSet of measure equal to alpha and get the 
        corresponding density value (named threshold).
        Generate a sample of the distribution and draw it. 
        Draw a contour plot for the distribution, where the PDF is equal to threshold. 
        '''
        sample = distribution.getSample(sampleSize)
        X1min = sample[:, 0].getMin()[0]
        X1max = sample[:, 0].getMax()[0]
        X2min = sample[:, 1].getMin()[0]
        X2max = sample[:, 1].getMax()[0]
        xx = ot.Box([numberOfPointsInXAxis],
                    ot.Interval([X1min], [X1max])).generate()
        yy = ot.Box([numberOfPointsInXAxis],
                    ot.Interval([X2min], [X2max])).generate()
        xy = ot.Box([numberOfPointsInXAxis, numberOfPointsInXAxis],
                    ot.Interval([X1min, X2min], [X1max, X2max])).generate()
        data = distribution.computePDF(xy)
        graph = ot.Graph('', 'X1', 'X2', True, 'topright')
        labels = ["%.2f%%" % (100*alpha)]
        contour = ot.Contour(xx, yy, data, ot.Point([threshold]), ot.Description(labels))
        contour.setColor('black')
        graph.setTitle("%.2f%% of the distribution, sample size = %d" % (100*alpha,sampleSize))
        graph.add(contour)
        cloud = ot.Cloud(sample)
        graph.add(cloud)
        return graph









The following plot shows that 90% of the sample is contained in the LevelSet.


.. code-block:: default

    numberOfPointsInXAxis = 50
    graph = drawLevelSetContour2D(mixture, numberOfPointsInXAxis, alpha, threshold)
    view = viewer.View(graph)
    plt.show()



.. image:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_minimum_volume_level_set_2D_002.png
    :alt: 90.00% of the distribution, sample size = 500
    :class: sphx-glr-single-img






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

   **Total running time of the script:** ( 0 minutes  0.222 seconds)


.. _sphx_glr_download_auto_probabilistic_modeling_distributions_plot_minimum_volume_level_set_2D.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_minimum_volume_level_set_2D.py <plot_minimum_volume_level_set_2D.py>`



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

     :download:`Download Jupyter notebook: plot_minimum_volume_level_set_2D.ipynb <plot_minimum_volume_level_set_2D.ipynb>`


.. only:: html

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

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