.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/montecarlo/plot_b01simple_integral.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_auto_examples_montecarlo_plot_b01simple_integral.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_montecarlo_plot_b01simple_integral.py:


Simple integral
===============

Calculation of a simple integral using Monte-Carlo integration.

:author: Michel Bierlaire, EPFL
:date: Thu Apr 13 20:42:24 2023

.. GENERATED FROM PYTHON SOURCE LINES 12-19

.. code-block:: Python


    import pandas as pd

    import biogeme.biogeme as bio
    import biogeme.database as db
    from biogeme.expressions import exp, bioDraws, MonteCarlo








.. GENERATED FROM PYTHON SOURCE LINES 20-22

We create a fake database with one entry, as it is required
to store the draws

.. GENERATED FROM PYTHON SOURCE LINES 22-26

.. code-block:: Python

    df = pd.DataFrame()
    df['FakeColumn'] = [1.0]
    database = db.Database('fake_database', df)








.. GENERATED FROM PYTHON SOURCE LINES 27-30

.. code-block:: Python

    integrand = exp(bioDraws('U', 'UNIFORM'))
    simulated_integral = MonteCarlo(integrand)








.. GENERATED FROM PYTHON SOURCE LINES 31-33

.. code-block:: Python

    true_integral = exp(1.0) - 1.0








.. GENERATED FROM PYTHON SOURCE LINES 34-60

.. code-block:: Python

    R = 200
    MULTIPLIER = 100000


    sample_variance = (
        MonteCarlo(integrand * integrand) - simulated_integral * simulated_integral
    )
    stderr = (sample_variance / R) ** 0.5
    error = simulated_integral - true_integral

    simulate = {
        'Analytical Integral': true_integral,
        'Simulated Integral': simulated_integral,
        'Sample variance   ': sample_variance,
        'Std Error         ': stderr,
        'Error             ': error,
    }

    biosim = bio.BIOGEME(database, simulate, number_of_draws=R)
    R = biosim.number_of_draws
    biosim.modelName = f'01simpleIntegral_{R}'
    results = biosim.simulate(the_beta_values={})
    print(f'Number of draws: {R}')
    for c in results.columns:
        print(f'{c}: {results.loc[0, c]}')





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

 .. code-block:: none

    Number of draws: 200
    Analytical Integral: 1.718281828459045
    Simulated Integral: 1.6914968008726095
    Sample variance   : 0.22207804709564938
    Std Error         : 0.033322518444413037
    Error             : -0.02678502758643564




.. GENERATED FROM PYTHON SOURCE LINES 61-62

Change the number of draws

.. GENERATED FROM PYTHON SOURCE LINES 62-68

.. code-block:: Python

    biogeme2 = bio.BIOGEME(database, simulate, number_of_draws=R * MULTIPLIER)
    biogeme2.modelName = '01simpleIntegral_{multiplier*R}'
    results2 = biogeme2.simulate(the_beta_values={})
    print(f'Number of draws: {MULTIPLIER * R}')
    for c in results2.columns:
        print(f'{c}: {results2.loc[0, c]}')




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

 .. code-block:: none

    Number of draws: 20000000
    Analytical Integral: 1.718281828459045
    Simulated Integral: 1.7182794027362247
    Sample variance   : 0.24205309531650032
    Std Error         : 0.0347888700101412
    Error             : -2.4257228203872216e-06





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

   **Total running time of the script:** (0 minutes 14.457 seconds)


.. _sphx_glr_download_auto_examples_montecarlo_plot_b01simple_integral.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_b01simple_integral.ipynb <plot_b01simple_integral.ipynb>`

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

      :download:`Download Python source code: plot_b01simple_integral.py <plot_b01simple_integral.py>`

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

      :download:`Download zipped: plot_b01simple_integral.zip <plot_b01simple_integral.zip>`


.. only:: html

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

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