.. only:: html

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

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

    .. _sphx_glr_auto_data_analysis_manage_data_and_samples_plot_quantile_estimation_wilks.py:


Estimate Wilks and empirical quantile
=====================================

In this example we want to evaluate a particular quantile, with the empirical estimator or the Wilks one, from a sample of a random variable.


Let us suppose we want to estimate the quantile :math:`q_{\alpha}` of order :math:`\alpha` of the variable :math:`Y`:
:math:`P(Y \leq q_{\alpha}) = \alpha`, from the sample :math:`(Y_1, ..., Y_n)`
of size :math:`n`, with a confidence level equal to :math:`\beta`.

We note :math:`(Y^{(1)}, ..., Y^{(n)})` the sample where the values are sorted in ascending order.
The empirical estimator, noted :math:`q_{\alpha}^{emp}`, and its confidence interval, is defined by the expressions:

.. math::
    \left\{
     \begin{array}{lcl}
       q_{\alpha}^{emp} & = & Y^{(E[n\alpha])} \\
       P(q_{\alpha} \in [Y^{(i_n)}, Y^{(j_n)}]) & = & \beta \\
       i_n & = & E[n\alpha - a_{\alpha}\sqrt{n\alpha(1-\alpha)}] \\
       i_n & = & E[n\alpha + a_{\alpha}\sqrt{n\alpha(1-\alpha)}]
     \end{array}
    \right\}

The Wilks estimator, noted :math:`q_{\alpha, \beta}^{Wilks}`, and its confidence interval, is defined by the expressions:

.. math::
  \left\{
  \begin{array}{lcl}
    q_{\alpha, \beta}^{Wilks} & = & Y^{(n-i)} \\
    P(q_{\alpha}  \leq q_{\alpha, \beta}^{Wilks}) & \geq & \beta \\
    i\geq 0 \, \, /  \, \, n \geq N_{Wilks}(\alpha, \beta,i)
  \end{array}
  \right\}

Once the order :math:`i` has been chosen, the Wilks number :math:`N_{Wilks}(\alpha, \beta,i)` is evaluated,
thanks to the static method :math:`ComputeSampleSize(\alpha, \beta, i)` of the Wilks object.

In the example, we want to evaluate a quantile :math:`\alpha = 95\%`,
with a confidence level of :math:`\beta = 90\%` thanks to the :math:`4`th maximum of
the ordered sample (associated to the order :math:`i = 3`).

Be careful: :math:`i=0` means that the Wilks estimator is the maximum of the sample:
it corresponds to the first maximum of the sample.


.. code-block:: default

    from __future__ import print_function
    import openturns as ot
    import math as m
    import openturns.viewer as viewer
    from matplotlib import pylab as plt
    ot.Log.Show(ot.Log.NONE)









.. code-block:: default

    model = ot.SymbolicFunction(['x1', 'x2'], ['x1^2+x2'])
    R = ot.CorrelationMatrix(2)
    R[0,1] = -0.6
    inputDist = ot.Normal([0.,0.], R)
    inputDist.setDescription(['X1','X2'])
    inputVector = ot.RandomVector(inputDist)

    # Create the output random vector Y=model(X)
    output = ot.CompositeRandomVector(model, inputVector)








Quantile level


.. code-block:: default

    alpha = 0.95

    # Confidence level of the estimation
    beta = 0.90








Get a sample of the variable


.. code-block:: default

    N = 10**4
    sample = output.getSample(N)
    graph = ot.UserDefined(sample).drawCDF()
    view = viewer.View(graph)




.. image:: /auto_data_analysis/manage_data_and_samples/images/sphx_glr_plot_quantile_estimation_wilks_001.png
    :alt: y0 CDF
    :class: sphx-glr-single-img





Empirical Quantile Estimator


.. code-block:: default

    empiricalQuantile = sample.computeQuantile(alpha)

    # Get the indices of the confidence interval bounds
    aAlpha = ot.Normal(1).computeQuantile((1.0+beta)/2.0)[0]
    min_i = int(N*alpha - aAlpha*m.sqrt(N*alpha*(1.0-alpha)))
    max_i = int(N*alpha + aAlpha*m.sqrt(N*alpha*(1.0-alpha)))
    #print(min_i, max_i)

    # Get the sorted sample
    sortedSample = sample.sort()

    # Get the Confidence interval of the Empirical Quantile Estimator [infQuantile, supQuantile]
    infQuantile = sortedSample[min_i-1]
    supQuantile = sortedSample[max_i-1]
    print(infQuantile, empiricalQuantile, supQuantile)





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

 Out:

 .. code-block:: none

    [4.13903] [4.28037] [4.35925]




Wilks number


.. code-block:: default

    i = N - (min_i+max_i)//2 # compute wilks with the same sample size
    wilksNumber = ot.Wilks.ComputeSampleSize(alpha, beta, i)
    print('wilksNumber =', wilksNumber)





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

 Out:

 .. code-block:: none

    wilksNumber = 10604




Wilks Quantile Estimator


.. code-block:: default

    algo = ot.Wilks(output)
    wilksQuantile = algo.computeQuantileBound(alpha, beta, i)
    print('wilks Quantile 0.95 =', wilksQuantile)




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

 Out:

 .. code-block:: none

    wilks Quantile 0.95 = [4.37503]





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

   **Total running time of the script:** ( 0 minutes  0.108 seconds)


.. _sphx_glr_download_auto_data_analysis_manage_data_and_samples_plot_quantile_estimation_wilks.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_quantile_estimation_wilks.py <plot_quantile_estimation_wilks.py>`



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

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


.. only:: html

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

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