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

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

.. _sphx_glr_auto_data_analysis_statistical_tests_plot_kolmogorov_distribution.py:


Kolmogorov-Smirnov : get the statistics distribution
====================================================

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In this example, we draw the Kolmogorov-Smirnov (KS) distribution for a sample size 10.
We want to test the hypothesis that this sample has the `Uniform(0, 1)`
distribution.
The K.S. distribution is first plotted in the case where the
parameters of the uniform distribution are known.
Then we plot the distribution when the parameters of the uniform
distribution are estimated from the sample.

*Reference* : Hovhannes Keutelian, "The Kolmogorov-Smirnov test when
parameters are estimated from data", 30 April 1991, Fermilab

Note: There is a sign error in the paper; the equation:
`D[i]=max(abs(S+step),D[i])` must be replaced with `D[i]=max(abs(S-step),D[i])`.

.. GENERATED FROM PYTHON SOURCE LINES 24-28

.. code-block:: Python

    import openturns as ot
    import openturns.viewer as viewer









.. GENERATED FROM PYTHON SOURCE LINES 29-32

.. code-block:: Python

    x = [0.9374, 0.7629, 0.4771, 0.5111, 0.8701, 0.0684, 0.7375, 0.5615, 0.2835, 0.2508]
    sample = ot.Sample([[xi] for xi in x])








.. GENERATED FROM PYTHON SOURCE LINES 33-36

.. code-block:: Python

    samplesize = sample.getSize()
    samplesize





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

 .. code-block:: none


    10



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Plot the empirical distribution function.

.. GENERATED FROM PYTHON SOURCE LINES 40-50

.. code-block:: Python

    graph = ot.UserDefined(sample).drawCDF()
    graph.setLegends(["Sample"])
    curve = ot.Curve([0, 1], [0, 1])
    curve.setLegend("Uniform")
    graph.add(curve)
    graph.setXTitle("X")
    graph.setTitle("Cumulated distribution function")
    view = viewer.View(graph)





.. image-sg:: /auto_data_analysis/statistical_tests/images/sphx_glr_plot_kolmogorov_distribution_001.svg
   :alt: Cumulated distribution function
   :srcset: /auto_data_analysis/statistical_tests/images/sphx_glr_plot_kolmogorov_distribution_001.svg
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 51-55

The `computeKSStatisticsIndex` function computes the Kolmogorov-Smirnov
distance between the sample and the distribution.
The following function is for teaching purposes only: use `FittingTest`
for real applications.

.. GENERATED FROM PYTHON SOURCE LINES 58-73

.. code-block:: Python

    def computeKSStatistics(sample, distribution):
        sample = sample.sort()
        n = sample.getSize()
        D = 0.0
        D_previous = 0.0
        for i in range(n):
            F = distribution.computeCDF(sample[i])
            Fminus = F - float(i) / n
            Fplus = float(i + 1) / n - F
            D = max(Fminus, Fplus, D)
            if D > D_previous:
                D_previous = D
        return D









.. GENERATED FROM PYTHON SOURCE LINES 74-77

.. code-block:: Python

    dist = ot.Uniform(0, 1)
    dist






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    Uniform
    <ul>
      <li>name=Uniform</li>
      <li>dimension=1</li>
      <li>weight=1</li>
      <li>range=[0, 1]</li>
      <li>description=[X0]</li>
      <li>isParallel=true</li>
      <li>isCopula=false</li>
    </ul>

    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 78-81

.. code-block:: Python

    computeKSStatistics(sample, dist)






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

 .. code-block:: none


    0.17710000000000004



.. GENERATED FROM PYTHON SOURCE LINES 82-83

The following function generates a sample of K.S. distances when the tested distribution is the `Uniform(0,1)` distribution.

.. GENERATED FROM PYTHON SOURCE LINES 86-99

.. code-block:: Python

    def generateKSSampleKnownParameters(nrepeat, samplesize):
        """
        nrepeat : Number of repetitions, size of the table
        samplesize : the size of each sample to generate from the Uniform distribution
        """
        dist = ot.Uniform(0, 1)
        D = ot.Sample(nrepeat, 1)
        for i in range(nrepeat):
            sample = dist.getSample(samplesize)
            D[i, 0] = computeKSStatistics(sample, dist)
        return D









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Generate a sample of KS distances.

.. GENERATED FROM PYTHON SOURCE LINES 103-107

.. code-block:: Python

    nrepeat = 10000  # Size of the KS distances sample
    sampleD = generateKSSampleKnownParameters(nrepeat, samplesize)









.. GENERATED FROM PYTHON SOURCE LINES 108-109

Compute exact Kolmogorov CDF.

.. GENERATED FROM PYTHON SOURCE LINES 112-117

.. code-block:: Python

    def pKolmogorovPy(x):
        y = ot.DistFunc.pKolmogorov(samplesize, x[0])
        return [y]









.. GENERATED FROM PYTHON SOURCE LINES 118-121

.. code-block:: Python

    pKolmogorov = ot.PythonFunction(1, 1, pKolmogorovPy)









.. GENERATED FROM PYTHON SOURCE LINES 122-137

.. code-block:: Python

    def dKolmogorov(x, samplesize):
        """
        Compute Kolmogorov PDF for given x.
        x : an array, the points where the PDF must be evaluated
        samplesize : the size of the sample
        Reference
        Numerical Derivatives in Scilab, Michael Baudin, May 2009
        """
        n = x.getSize()
        y = ot.Sample(n, 1)
        for i in range(n):
            y[i, 0] = pKolmogorov.gradient(x[i])[0, 0]
        return y









.. GENERATED FROM PYTHON SOURCE LINES 138-147

.. code-block:: Python

    def linearSample(xmin, xmax, npoints):
        """Returns a sample created from a regular grid
        from xmin to xmax with npoints points."""
        step = (xmax - xmin) / (npoints - 1)
        rg = ot.RegularGrid(xmin, step, npoints)
        vertices = rg.getVertices()
        return vertices









.. GENERATED FROM PYTHON SOURCE LINES 148-152

.. code-block:: Python

    n = 1000  # Number of points in the plot
    s = linearSample(0.001, 0.999, n)
    y = dKolmogorov(s, samplesize)








.. GENERATED FROM PYTHON SOURCE LINES 153-163

.. code-block:: Python

    curve = ot.Curve(s, y)
    curve.setLegend("Exact distribution")
    graph = ot.HistogramFactory().build(sampleD).drawPDF()
    graph.setLegends(["Empirical distribution"])
    graph.add(curve)
    graph.setTitle("Kolmogorov-Smirnov distribution (known parameters)")
    graph.setXTitle("KS-Statistics")
    view = viewer.View(graph)





.. image-sg:: /auto_data_analysis/statistical_tests/images/sphx_glr_plot_kolmogorov_distribution_002.svg
   :alt: Kolmogorov-Smirnov distribution (known parameters)
   :srcset: /auto_data_analysis/statistical_tests/images/sphx_glr_plot_kolmogorov_distribution_002.svg
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 164-166

Known parameters versus estimated parameters
--------------------------------------------

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The following function generates a sample of K.S. distances when the tested
distribution is the `Uniform(a,b)` distribution, where the `a` and `b`
parameters are estimated from the sample.

.. GENERATED FROM PYTHON SOURCE LINES 174-189

.. code-block:: Python

    def generateKSSampleEstimatedParameters(nrepeat, samplesize):
        """
        nrepeat : Number of repetitions, size of the table
        samplesize : the size of each sample to generate from the Uniform distribution
        """
        distfactory = ot.UniformFactory()
        refdist = ot.Uniform(0, 1)
        D = ot.Sample(nrepeat, 1)
        for i in range(nrepeat):
            sample = refdist.getSample(samplesize)
            trialdist = distfactory.build(sample)
            D[i, 0] = computeKSStatistics(sample, trialdist)
        return D









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Generate a sample of KS distances.

.. GENERATED FROM PYTHON SOURCE LINES 193-195

.. code-block:: Python

    sampleDP = generateKSSampleEstimatedParameters(nrepeat, samplesize)








.. GENERATED FROM PYTHON SOURCE LINES 196-205

.. code-block:: Python

    graph = ot.KernelSmoothing().build(sampleD).drawPDF()
    graph.setLegends(["Known parameters"])
    graphP = ot.KernelSmoothing().build(sampleDP).drawPDF()
    graphP.setLegends(["Estimated parameters"])
    graph.add(graphP)
    graph.setTitle("Kolmogorov-Smirnov distribution")
    graph.setXTitle("KS-Statistics")
    view = viewer.View(graph)




.. image-sg:: /auto_data_analysis/statistical_tests/images/sphx_glr_plot_kolmogorov_distribution_003.svg
   :alt: Kolmogorov-Smirnov distribution
   :srcset: /auto_data_analysis/statistical_tests/images/sphx_glr_plot_kolmogorov_distribution_003.svg
   :class: sphx-glr-single-img





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Display the graphs

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.. code-block:: Python

    view.ShowAll()









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We see that the distribution of the KS distances when the parameters are
estimated is shifted towards the left: smaller distances occur more often.
This is a consequence of the fact that the estimated parameters tend to make
the estimated distribution closer to the empirical sample.


.. _sphx_glr_download_auto_data_analysis_statistical_tests_plot_kolmogorov_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_kolmogorov_distribution.ipynb <plot_kolmogorov_distribution.ipynb>`

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

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

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_kolmogorov_distribution.zip <plot_kolmogorov_distribution.zip>`