.. 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_non_parametric_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_estimate_non_parametric_distribution.py>`
        to download the full example code.

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

.. _sphx_glr_auto_data_analysis_distribution_fitting_plot_estimate_non_parametric_distribution.py:


Fit a non parametric distribution
=================================

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In this example we are going to estimate a non parametric distribution using the kernel smoothing method.
After a short introductory example we focus on a few basic features of the API:

- kernel selection
- bandwidth estimation
- an advanced feature such as boundary corrections


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.. code-block:: Python

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

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









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An introductory example
-----------------------


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We create the data from a Gamma distribution :

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.. code-block:: Python

    ot.RandomGenerator.SetSeed(0)
    distribution = ot.Gamma(6.0, 1.0)
    sample = distribution.getSample(800)








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We define the kernel smoother and build the smoothed estimate.

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.. code-block:: Python

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








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We can draw the original distribution vs the kernel smoothing.

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.. code-block:: Python

    graph = distribution.drawPDF()
    graph.setTitle("Kernel smoothing vs original")
    kernel_plot = estimated.drawPDF().getDrawable(0)
    graph.add(kernel_plot)
    graph.setLegends(["original", "KS"])
    graph.setLegendPosition("upper right")
    view = viewer.View(graph)




.. image-sg:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_non_parametric_distribution_001.png
   :alt: Kernel smoothing vs original
   :srcset: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_non_parametric_distribution_001.png
   :class: sphx-glr-single-img





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We can obtain the bandwdth parameter :

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.. code-block:: Python

    print(kernel.getBandwidth())





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

 .. code-block:: none

    [0.529581]




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We now compute a better bandwitdh with the Silverman rule.

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.. code-block:: Python

    bandwidth = kernel.computeSilvermanBandwidth(sample)
    print(bandwidth)





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

 .. code-block:: none

    [0.639633]




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The new bandwidth is used to regenerate another fitted distribution :

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.. code-block:: Python

    estimated = kernel.build(sample, bandwidth)








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.. code-block:: Python

    graph = distribution.drawPDF()
    graph.setTitle("Kernel smoothing vs original")
    kernel_plot = estimated.drawPDF().getDrawable(0)
    graph.add(kernel_plot)
    graph.setLegends(["original", "KS-Silverman"])
    graph.setLegendPosition("upper right")
    view = viewer.View(graph)




.. image-sg:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_non_parametric_distribution_002.png
   :alt: Kernel smoothing vs original
   :srcset: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_non_parametric_distribution_002.png
   :class: sphx-glr-single-img





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The Silverman rule of thumb to estimate the bandwidth provides a better estimate for the distribution. We can also study the impact of the kernel selection.

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Choosing a kernel
-----------------

We experiment with several kernels to perform the smoothing :

- the standard normal kernel
- the triangular kernel
- the Epanechnikov kernel
- the uniform kernel


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We first create the data from a Gamma distribution :

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.. code-block:: Python

    distribution = ot.Gamma(6.0, 1.0)
    sample = distribution.getSample(800)








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The definition of the Normal kernel :

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.. code-block:: Python

    kernelNormal = ot.KernelSmoothing(ot.Normal())
    estimatedNormal = kernelNormal.build(sample)








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The definition of the Triangular kernel :

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.. code-block:: Python

    kernelTriangular = ot.KernelSmoothing(ot.Triangular())
    estimatedTriangular = kernelTriangular.build(sample)








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The definition of the Epanechnikov kernel :

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.. code-block:: Python

    kernelEpanechnikov = ot.KernelSmoothing(ot.Epanechnikov())
    estimatedEpanechnikov = kernelEpanechnikov.build(sample)








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The definition of the Uniform kernel :

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.. code-block:: Python

    kernelUniform = ot.KernelSmoothing(ot.Uniform())
    estimatedUniform = kernelUniform.build(sample)









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We finally compare all the distributions :


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.. code-block:: Python

    graph = distribution.drawPDF()
    graph.setTitle("Different kernel smoothings vs original distribution")
    graph.setGrid(True)

    kernel_estimatedNormal_plot = estimatedNormal.drawPDF().getDrawable(0)
    graph.add(kernel_estimatedNormal_plot)

    kernel_estimatedTriangular_plot = estimatedTriangular.drawPDF().getDrawable(0)
    graph.add(kernel_estimatedTriangular_plot)

    kernel_estimatedEpanechnikov_plot = estimatedEpanechnikov.drawPDF().getDrawable(0)
    graph.add(kernel_estimatedEpanechnikov_plot)

    kernel_estimatedUniform_plot = estimatedUniform.drawPDF().getDrawable(0)
    kernel_estimatedUniform_plot.setLineStyle("dashed")
    graph.add(kernel_estimatedUniform_plot)

    graph.setLegends(
        ["original", "KS-Normal", "KS-Triangular", "KS-Epanechnikov", "KS-Uniform"]
    )

    view = viewer.View(graph)




.. image-sg:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_non_parametric_distribution_003.png
   :alt: Different kernel smoothings vs original distribution
   :srcset: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_non_parametric_distribution_003.png
   :class: sphx-glr-single-img





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We observe that all the kernels produce very similar results in practice.
The Uniform kernel may be seen as the worst of them all while the Epanechnikov one is said to be a good theoritical choice. In practice the standard normal kernel is a fine choice.
The most important aspect of kernel smoothing is the choice of the bandwidth.

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Bandwidth selection
-------------------

We reproduce a classical example of the literature : the fitting of a bimodal distribution.
We will show the result of a kernel smoothing with different bandwidth computation :

- the Silverman rule
- the Plugin bandwidth
- the Mixed bandwidth


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We define the bimodal distribution and generate a sample out of it.

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.. code-block:: Python

    X1 = ot.Normal(10.0, 1.0)
    X2 = ot.Normal(-10.0, 1.0)
    myDist = ot.Mixture([X1, X2])
    sample = myDist.getSample(2000)








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We now compare the fitted distribution :

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.. code-block:: Python

    graph = myDist.drawPDF()
    graph.setTitle("Kernel smoothing vs original")








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With the Silverman rule :

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.. code-block:: Python

    kernelSB = ot.KernelSmoothing()
    bandwidthSB = kernelSB.computeSilvermanBandwidth(sample)
    estimatedSB = kernelSB.build(sample, bandwidthSB)
    kernelSB_plot = estimatedSB.drawPDF().getDrawable(0)
    graph.add(kernelSB_plot)








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With the Plugin bandwidth :

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.. code-block:: Python

    kernelPB = ot.KernelSmoothing()
    bandwidthPB = kernelPB.computePluginBandwidth(sample)
    estimatedPB = kernelPB.build(sample, bandwidthPB)
    kernelPB_plot = estimatedPB.drawPDF().getDrawable(0)
    graph.add(kernelPB_plot)








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With the Mixed bandwidth :

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.. code-block:: Python

    kernelMB = ot.KernelSmoothing()
    bandwidthMB = kernelMB.computeMixedBandwidth(sample)
    estimatedMB = kernelMB.build(sample, bandwidthMB)
    kernelMB_plot = estimatedMB.drawPDF().getDrawable(0)
    kernelMB_plot.setLineStyle("dashed")
    graph.add(kernelMB_plot)








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.. code-block:: Python

    graph.setLegends(["original", "KS-Silverman", "KS-Plugin", "KS-Mixed"])
    graph.setLegendPosition("upper right")
    view = viewer.View(graph)




.. image-sg:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_non_parametric_distribution_004.png
   :alt: Kernel smoothing vs original
   :srcset: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_non_parametric_distribution_004.png
   :class: sphx-glr-single-img





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As expected the Silverman seriously overfit the data and the other rules are more to the point.

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Boundary corrections
--------------------

We finish this example on an advanced feature of the kernel smoothing, the boundary corrections.


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We consider a Weibull distribution :

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.. code-block:: Python

    myDist = ot.WeibullMin(2.0, 1.5, 1.0)








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We generate a sample from the defined distribution :

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.. code-block:: Python

    sample = myDist.getSample(2000)








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We draw the exact Weibull distribution :

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.. code-block:: Python

    graph = myDist.drawPDF()









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We use two different kernels :

- a standard normal kernel
- the same kernel with a boundary correction


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The first kernel without the boundary corrections.

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.. code-block:: Python

    kernel1 = ot.KernelSmoothing()
    estimated1 = kernel1.build(sample)








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The second kernel with the boundary corrections.

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.. code-block:: Python

    kernel2 = ot.KernelSmoothing()
    kernel2.setBoundaryCorrection(True)
    estimated2 = kernel2.build(sample)









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We compare the estimated PDFs :

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.. code-block:: Python

    graph.setTitle("Kernel smoothing vs original")

    kernel1_plot = estimated1.drawPDF().getDrawable(0)
    graph.add(kernel1_plot)

    kernel2_plot = estimated2.drawPDF().getDrawable(0)
    graph.add(kernel2_plot)


    graph.setLegends(["original", "KS", "KS with boundary correction"])
    graph.setLegendPosition("upper right")
    view = viewer.View(graph)




.. image-sg:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_non_parametric_distribution_005.png
   :alt: Kernel smoothing vs original
   :srcset: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_estimate_non_parametric_distribution_005.png
   :class: sphx-glr-single-img





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The boundary correction made has a remarkable impact on the quality of the estimate for the small values.

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.. code-block:: Python



    plt.show()








.. _sphx_glr_download_auto_data_analysis_distribution_fitting_plot_estimate_non_parametric_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_estimate_non_parametric_distribution.ipynb <plot_estimate_non_parametric_distribution.ipynb>`

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

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