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

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

.. _sphx_glr_auto_data_analysis_distribution_fitting_plot_estimate_conditional_quantile.py:


Estimate a conditional quantile
===============================

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.. code-block:: Python

    # sphinx_gallery_thumbnail_number = 8







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From a multivariate data sample, we estimate a distribution with kernel smoothing.
Here we present a bivariate distribution :math:`X= (X_1, X_2)`.
We use the `computeConditionalQuantile` method to estimate the 90% quantile :math:`Q_1` of the conditional variable :math:`X_2|X_1` :

.. math::
   Q_2 : x_1 \mapsto q_{0.9}(X_2|X_1=x_1)

We then draw the curve :math:`Q_2 : x_1 \mapsto Q_2(x_1)`.
We first start with independent normals then we consider dependent marginals with a Clayton copula.


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.. code-block:: Python

    import openturns as ot
    import openturns.viewer as viewer
    from matplotlib import pylab as plt
    import numpy as np

    ot.Log.Show(ot.Log.NONE)








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Set the random generator seed

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.. code-block:: Python

    ot.RandomGenerator.SetSeed(0)








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Defining the marginals
----------------------

We consider two independent normal marginals :

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.. code-block:: Python

    X1 = ot.Normal(0.0, 1.0)
    X2 = ot.Normal(0.0, 3.0)









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Independent marginals
---------------------

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.. code-block:: Python


    distX = ot.ComposedDistribution([X1, X2])
    sample = distX.getSample(1000)









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Let's see the data

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.. code-block:: Python

    graph = ot.Graph("2D-Normal sample", "x1", "x2", True, "")
    cloud = ot.Cloud(sample, "blue", "fsquare", "My Cloud")
    graph.add(cloud)
    graph.setXTitle("$X_1$")
    graph.setYTitle("$X_2$")
    graph.setTitle("A sample from $X=(X_1, X_2)$")
    view = viewer.View(graph)




.. image-sg:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_conditional_quantile_001.png
   :alt: A sample from $X=(X_1, X_2)$
   :srcset: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_conditional_quantile_001.png
   :class: sphx-glr-single-img





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We draw the isolines of the PDF of :math:`X` :

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.. code-block:: Python

    graph = distX.drawPDF()
    graph.setXTitle("$X_1$")
    graph.setYTitle("$X_2$")
    graph.setTitle("iso-PDF of $X=(X_1, X_2)$")
    view = viewer.View(graph)




.. image-sg:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_conditional_quantile_002.png
   :alt: iso-PDF of $X=(X_1, X_2)$
   :srcset: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_conditional_quantile_002.png
   :class: sphx-glr-single-img





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We estimate the density with kernel smoothing :

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.. code-block:: Python

    kernel = ot.KernelSmoothing()
    estimated = kernel.build(sample)









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We draw the isolines of the estimated PDF of :math:`X` :

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.. code-block:: Python

    graph = estimated.drawPDF()
    graph.setXTitle("$X_1$")
    graph.setYTitle("$X_2$")
    graph.setTitle("iso-PDF of $X=(X_1, X_2)$ estimated by kernel smoothing")
    view = viewer.View(graph)





.. image-sg:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_conditional_quantile_003.png
   :alt: iso-PDF of $X=(X_1, X_2)$ estimated by kernel smoothing
   :srcset: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_conditional_quantile_003.png
   :class: sphx-glr-single-img





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We can compute the conditional quantile of :math:`X_2 | X_1` with the `computeConditionalQuantile` method and draw it after.


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We first create N observation points in :math:`[-3.0, 3.0]` :

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.. code-block:: Python

    N = 301
    xobs = np.linspace(-3.0, 3.0, N)
    sampleObs = ot.Sample([[xi] for xi in xobs])








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We create curves of the exact and approximated quantile :math:`Q_1`

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.. code-block:: Python

    x = [xi for xi in xobs]
    yapp = [estimated.computeConditionalQuantile(0.9, sampleObs[i]) for i in range(N)]
    yex = [distX.computeConditionalQuantile(0.9, sampleObs[i]) for i in range(N)]








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.. code-block:: Python

    cxy_app = ot.Curve(x, yapp)
    cxy_ex = ot.Curve(x, yex)
    graph = ot.Graph("90% quantile of $X_2 | X_1=x_1$", "$x_1$", "$Q_2(x_1)$", True, "")
    graph.add(cxy_app)
    graph.add(cxy_ex)
    graph.setLegends(["$Q_2$ kernel smoothing", "$Q_2$ exact"])
    graph.setLegendPosition("lower right")
    graph.setColors(["red", "blue"])
    view = viewer.View(graph)




.. image-sg:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_conditional_quantile_004.png
   :alt: 90% quantile of $X_2 | X_1=x_1$
   :srcset: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_conditional_quantile_004.png
   :class: sphx-glr-single-img





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In this case the :math:`Q_2` quantile is constant because of the independence of the marginals.


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Dependence through a Clayton copula
-----------------------------------

We now define a Clayton copula to model the dependence between our marginals.
The Clayton copula is a bivariate asymmmetric Archimedean copula, exhibiting greater dependence
in the negative tail than in the positive.


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.. code-block:: Python

    copula = ot.ClaytonCopula(2.5)
    distX = ot.ComposedDistribution([X1, X2], copula)








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We generate a sample from the distribution :

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.. code-block:: Python

    sample = distX.getSample(1000)








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Let's see the data

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.. code-block:: Python

    graph = ot.Graph("2D-Normal sample", "x1", "x2", True, "")
    cloud = ot.Cloud(sample, "blue", "fsquare", "My Cloud")
    graph.add(cloud)
    graph.setXTitle("$X_1$")
    graph.setYTitle("$X_2$")
    graph.setTitle("A sample from $X=(X_1, X_2)$")
    view = viewer.View(graph)




.. image-sg:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_conditional_quantile_005.png
   :alt: A sample from $X=(X_1, X_2)$
   :srcset: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_conditional_quantile_005.png
   :class: sphx-glr-single-img





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We draw the isolines of the PDF of :math:`X` :

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.. code-block:: Python

    graph = distX.drawPDF()
    graph.setXTitle("$X_1$")
    graph.setYTitle("$X_2$")
    graph.setTitle("iso-PDF of $X=(X_1, X_2)$")
    view = viewer.View(graph)




.. image-sg:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_conditional_quantile_006.png
   :alt: iso-PDF of $X=(X_1, X_2)$
   :srcset: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_conditional_quantile_006.png
   :class: sphx-glr-single-img





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We estimate the density with kernel smoothing :

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.. code-block:: Python

    kernel = ot.KernelSmoothing()
    estimated = kernel.build(sample)








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We draw the isolines of the estimated PDF of :math:`X` :

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.. code-block:: Python

    graph = estimated.drawPDF()
    graph.setXTitle("$X_1$")
    graph.setYTitle("$X_2$")
    graph.setTitle("iso-PDF of $X=(X_1, X_2)$ estimated by kernel smoothing")
    view = viewer.View(graph)





.. image-sg:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_conditional_quantile_007.png
   :alt: iso-PDF of $X=(X_1, X_2)$ estimated by kernel smoothing
   :srcset: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_conditional_quantile_007.png
   :class: sphx-glr-single-img





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We can compute the conditional quantile of :math:`X_2 | X_1=x1` with the `computeConditionalQuantile` method and draw it after.


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We first create N observation points in :math:`[-3.0, 3.0]` :

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.. code-block:: Python

    N = 301
    xobs = np.linspace(-3.0, 3.0, N)
    sampleObs = ot.Sample([[xi] for xi in xobs])








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We create curves of the exact and approximated quantile :math:`Q_1`

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.. code-block:: Python

    x = [xi for xi in xobs]
    yapp = [estimated.computeConditionalQuantile(0.9, sampleObs[i]) for i in range(N)]
    yex = [distX.computeConditionalQuantile(0.9, sampleObs[i]) for i in range(N)]








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.. code-block:: Python

    cxy_app = ot.Curve(x, yapp)
    cxy_ex = ot.Curve(x, yex)
    graph = ot.Graph("90% quantile of $X_2 | X_1=x_1$", "$x_1$", "$Q_2(x_1)$", True, "")
    graph.add(cxy_app)
    graph.add(cxy_ex)
    graph.setLegends(["$Q_2$ kernel smoothing", "$Q_2$ exact"])
    graph.setLegendPosition("lower right")
    graph.setColors(["red", "blue"])
    view = viewer.View(graph)




.. image-sg:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_conditional_quantile_008.png
   :alt: 90% quantile of $X_2 | X_1=x_1$
   :srcset: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_conditional_quantile_008.png
   :class: sphx-glr-single-img





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Our estimated conditional quantile is a good approximate and should be better the more data we have. We can observe it by increasing the number of samples.


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.. code-block:: Python

    plt.show()








.. _sphx_glr_download_auto_data_analysis_distribution_fitting_plot_estimate_conditional_quantile.py:

.. only:: html

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

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

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

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

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