.. 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_quantilematching_estimator.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_quantilematching_estimator.py>`
        to download the full example code

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

.. _sphx_glr_auto_data_analysis_distribution_fitting_plot_quantilematching_estimator.py:


Define a distribution from quantiles
====================================

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In this example we are going to estimate a parametric distribution by numerical optimization of some quantiles.

.. GENERATED FROM PYTHON SOURCE LINES 10-13

.. code-block:: Python

    import openturns as ot
    from openturns.viewer import View








.. GENERATED FROM PYTHON SOURCE LINES 14-21

We need as many quantile values as there are parameters
of the distribution.
For example, for a normal distribution, the two parameters are the mean and the
standard deviation.
This implies that two quantiles are required to estimate the parameters
of a normal distribution.
The values of the parameters :math:`\mu`, :math:`\sigma` will be used as the reference to assess the optimization.

.. GENERATED FROM PYTHON SOURCE LINES 21-29

.. code-block:: Python

    dist_ref = ot.Normal(17.0, 34.0)
    dist_ref.setDescription(["reference"])
    p1 = 0.05
    p2 = 0.95
    q1 = dist_ref.computeQuantile(p1)[0]
    q2 = dist_ref.computeQuantile(p2)[0]
    print(q1, q2)





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

 .. code-block:: none

    -38.92502331635007 72.92502331635006




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Fit a normal distribution from these quantiles by numerical optimization.
Ensure we get the same values as the reference.

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.. code-block:: Python

    factory = ot.QuantileMatchingFactory(ot.Normal(), [p1, p2])
    dist_estim = factory.buildFromQuantiles([q1, q2])
    dist_estim.setDescription(["estimated"])
    print(dist_estim)





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

 .. code-block:: none

    Normal(mu = 17, sigma = 34)




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Graphically validate if the estimated distribution verifies the imposed quantiles.

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.. code-block:: Python

    graph = dist_estim.drawCDF()
    curve_q1 = ot.Curve([q1, q1, 0.0], [0.0, p1, p1])
    curve_q2 = ot.Curve([q2, q2, 0.0], [0.0, p2, p2])
    graph.add(curve_q1)
    graph.add(curve_q2)
    text_q1 = ot.Text([[q1, -0.1]], [r"$q_1$"])
    text_p1 = ot.Text([[0.0, p1]], [r"$p_1$"])
    graph.add(text_q1)
    graph.add(text_p1)
    text_q2 = ot.Text([[q2, -0.1]], [r"$q_2$"])
    text_p2 = ot.Text([[0.0, p2]], [r"$p_2$"])
    graph.add(text_q2)
    graph.add(text_p2)
    _ = View(graph)




.. image-sg:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_quantilematching_estimator_001.png
   :alt: plot quantilematching estimator
   :srcset: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_quantilematching_estimator_001.png
   :class: sphx-glr-single-img





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It is also possible to define a Histogram distribution from quantiles.
As it is a non-parametric distribution we can define as many quantiles as needed.

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.. code-block:: Python

    N = 10
    probabilities = [(i + 1) / N for i in range(N)]
    probabilities[-1] = 1.0 - 1e-3
    quantiles = [dist_ref.computeQuantile(pi)[0] for pi in probabilities]








.. GENERATED FROM PYTHON SOURCE LINES 63-65

The service is part of the :class:`~openturns.HistogramFactory` class.
We also need to define the lower bound of the Histogram to build the distribution.

.. GENERATED FROM PYTHON SOURCE LINES 65-70

.. code-block:: Python

    lowerBound = quantiles[0] - 10.0
    histo_quant = ot.HistogramFactory().buildFromQuantiles(
        lowerBound, probabilities, quantiles
    )








.. GENERATED FROM PYTHON SOURCE LINES 71-73

Graphically validate if the estimated distribution verifies the imposed quantiles.
We can see that the CDF of the estimated Histogram matches all the quantile dots.

.. GENERATED FROM PYTHON SOURCE LINES 73-81

.. code-block:: Python

    histo_quant.setDescription(["estimated"])
    graph = histo_quant.drawCDF()
    curve_qi = ot.Cloud(quantiles, probabilities)
    curve_qi.setLegend("quantiles")
    graph.add(curve_qi)
    _ = View(graph)

    View.ShowAll()



.. image-sg:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_quantilematching_estimator_002.png
   :alt: plot quantilematching estimator
   :srcset: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_quantilematching_estimator_002.png
   :class: sphx-glr-single-img






.. _sphx_glr_download_auto_data_analysis_distribution_fitting_plot_quantilematching_estimator.py:

.. only:: html

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

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

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

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

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