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

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

.. _sphx_glr_auto_data_analysis_distribution_fitting_plot_model_singular_multivariate_distribution.py:


Model a singular multivariate distribution
==========================================

.. GENERATED FROM PYTHON SOURCE LINES 7-11

From time to time we need to model singular :math:`n_D` distributions
(e.g. the joint distribution of KL coefficients for curves resulting from the transport of a low dimensional random vector).
A way to do that is to use an :class:`~openturns.EmpiricalBernsteinCopula` with a bin number equal to the sample size
(also called the empirical beta copula in this case).

.. GENERATED FROM PYTHON SOURCE LINES 11-19

.. code-block:: Python


    import openturns as ot
    import openturns.viewer as viewer
    import math as m

    ot.Log.Show(ot.Log.NONE)
    ot.RandomGenerator.SetSeed(0)








.. GENERATED FROM PYTHON SOURCE LINES 20-21

routine to draw a distribution cloud and a sample

.. GENERATED FROM PYTHON SOURCE LINES 21-43

.. code-block:: Python



    def draw(dist, Y):
        g = ot.Graph()
        g.setAxes(True)
        g.setGrid(True)
        c = ot.Cloud(dist.getSample(10000))
        c.setColor("red")
        c.setPointStyle("bullet")
        g.add(c)
        c = ot.Cloud(Y)
        c.setColor("black")
        c.setPointStyle("bullet")
        g.add(c)
        g.setBoundingBox(
            ot.Interval(
                Y.getMin() - 0.5 * Y.computeRange(), Y.getMax() + 0.5 * Y.computeRange()
            )
        )
        return g









.. GENERATED FROM PYTHON SOURCE LINES 44-45

generate some multivariate data to estimate, with correlation

.. GENERATED FROM PYTHON SOURCE LINES 45-55

.. code-block:: Python

    f = ot.SymbolicFunction(
        ["U", "xi1", "xi2"],
        ["sin(U)/(1+cos(U)^2)+0.05*xi1", "sin(U)*cos(U)/(1+cos(U)^2)+0.05*xi2"],
    )
    U = ot.Uniform(-0.85 * m.pi, 0.85 * m.pi)
    xi = ot.Normal(2)
    X = ot.BlockIndependentDistribution([U, xi])
    N = 200
    Y = f(X.getSample(N))








.. GENERATED FROM PYTHON SOURCE LINES 56-57

estimation by multivariate kernel smoothing

.. GENERATED FROM PYTHON SOURCE LINES 57-60

.. code-block:: Python

    multi_ks = ot.KernelSmoothing().build(Y)
    view = viewer.View(draw(multi_ks, Y))




.. image-sg:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_model_singular_multivariate_distribution_001.png
   :alt: plot model singular multivariate distribution
   :srcset: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_model_singular_multivariate_distribution_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 61-62

estimation by empirical beta copula

.. GENERATED FROM PYTHON SOURCE LINES 62-70

.. code-block:: Python

    beta_copula = ot.EmpiricalBernsteinCopula(Y, len(Y))
    marginals = [
        ot.KernelSmoothing().build(Y.getMarginal(j)) for j in range(Y.getDimension())
    ]
    beta_dist = ot.ComposedDistribution(marginals, beta_copula)
    view = viewer.View(draw(beta_dist, Y))

    viewer.View.ShowAll()



.. image-sg:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_model_singular_multivariate_distribution_002.png
   :alt: plot model singular multivariate distribution
   :srcset: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_model_singular_multivariate_distribution_002.png
   :class: sphx-glr-single-img






.. _sphx_glr_download_auto_data_analysis_distribution_fitting_plot_model_singular_multivariate_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_model_singular_multivariate_distribution.ipynb <plot_model_singular_multivariate_distribution.ipynb>`

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

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