.. only:: html

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

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

    .. _sphx_glr_auto_data_analysis_distribution_fitting_plot_maximumlikelihood_estimator.py:


Fit a distribution by maximum likelihood
========================================

In this example we are going to estimate the parameters of a parametric by generic numerical optimization of the likelihood.

The likelihood of a sample :math:`X` according to a parametric density function :math:`p_{\underline{\theta}}` is:

.. math::
   likelihood(\underline{x}_1, \dots, \underline{x}_n,\underline{\theta}) = \prod_{i=1}^n p_{\underline{\theta}}(\underline{x}_i)



.. code-block:: default

    from __future__ import print_function
    import openturns as ot
    import math as m
    ot.Log.Show(ot.Log.NONE)








Create data from a gaussian pdf with mu=4, sigma=1.5


.. code-block:: default

    sample = ot.Normal(4.0, 1.5).getSample(200)








Create the search interval of (mu, sigma): the constraint is sigma>0


.. code-block:: default

    lowerBound = [-1.0, 1.0e-4]
    upperBound = [-1.0, -1.0]
    finiteLowerBound = [False, True]
    finiteUpperBound = [False, False]
    bounds = ot.Interval(lowerBound, upperBound, finiteLowerBound, finiteUpperBound)








Create the starting point of the research
For mu : the first point
For sigma : a value evaluated from the two first points


.. code-block:: default

    mu0 = sample[0][0]
    sigma0 = m.sqrt((sample[1][0] - sample[0][0]) * (sample[1][0] - sample[0][0]))
    startingPoint = [mu0, sigma0]
    ot.Point(startingPoint)






.. raw:: html

    <p>[2.39784,4.01969]</p>
    <br />
    <br />

Create the estimator from a parametric pdf


.. code-block:: default

    pdf = ot.Normal()
    factory = ot.MaximumLikelihoodFactory(pdf)
    factory.setOptimizationBounds(bounds)








Set the starting point via the solver


.. code-block:: default

    solver = factory.getOptimizationAlgorithm()
    solver.setStartingPoint(startingPoint)
    factory.setOptimizationAlgorithm(solver)








Estimate the parametric model


.. code-block:: default

    distribution = factory.build(sample)
    str(distribution)





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

 Out:

 .. code-block:: none


    'Normal(mu = 3.94775, sigma = 1.49821)'



Retrieve the estimated parameters


.. code-block:: default

    distribution.getParameter()





.. raw:: html

    <p>[3.94775,1.49821]</p>
    <br />
    <br />


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

   **Total running time of the script:** ( 0 minutes  0.004 seconds)


.. _sphx_glr_download_auto_data_analysis_distribution_fitting_plot_maximumlikelihood_estimator.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_maximumlikelihood_estimator.py <plot_maximumlikelihood_estimator.py>`



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

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


.. only:: html

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

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