.. 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_maximumlikelihood_estimator.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_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
========================================

.. GENERATED FROM PYTHON SOURCE LINES 6-13

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:`(\vect{x}_1, \dots, \vect{x}_n)` according to a parametric density function :math:`p_{\vect{\theta}}` is:

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


.. GENERATED FROM PYTHON SOURCE LINES 15-20

.. code-block:: Python

    import openturns as ot
    import math as m

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








.. GENERATED FROM PYTHON SOURCE LINES 21-22

Create data from a normal PDF with :math:`\mu=4`, :math:`\sigma=1.5`.

.. GENERATED FROM PYTHON SOURCE LINES 22-24

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 25-26

Create the search interval of (:math:`\mu`, :math:`\sigma`) : the constraint is :math:`\sigma>0`.

.. GENERATED FROM PYTHON SOURCE LINES 26-32

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 33-37

Create the starting point of the research:

- for :math:`\mu` : the first point,
- for :math:`\sigma` : a value evaluated from the two first points.

.. GENERATED FROM PYTHON SOURCE LINES 37-41

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 42-43

Create the estimator from a parametric PDF.

.. GENERATED FROM PYTHON SOURCE LINES 43-47

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 48-49

Set the starting point via the solver.

.. GENERATED FROM PYTHON SOURCE LINES 49-53

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 54-55

Estimate the parametric model.

.. GENERATED FROM PYTHON SOURCE LINES 55-58

.. code-block:: Python

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





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

 .. code-block:: none


    'Normal(mu = 3.94738, sigma = 1.52392)'



.. GENERATED FROM PYTHON SOURCE LINES 59-60

Retrieve the estimated parameters.

.. GENERATED FROM PYTHON SOURCE LINES 60-61

.. code-block:: Python

    print(distribution.getParameter())




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

 .. code-block:: none

    [3.94738,1.52392]





.. _sphx_glr_download_auto_data_analysis_distribution_fitting_plot_maximumlikelihood_estimator.py:

.. only:: html

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

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

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

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

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