.. 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.

Michel Bierlaire, EPFL
Sat Jun 28 2025, 21:03:30

.. GENERATED FROM PYTHON SOURCE LINES 11-18

.. code-block:: Python


    import pandas as pd

    from biogeme.biogeme import BIOGEME
    from biogeme.database import Database
    from biogeme.expressions import Draws, MonteCarlo, exp








.. GENERATED FROM PYTHON SOURCE LINES 19-21

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

.. GENERATED FROM PYTHON SOURCE LINES 21-25

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 26-29

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 30-32

.. code-block:: Python

    true_integral = exp(1.0) - 1.0








.. GENERATED FROM PYTHON SOURCE LINES 33-59

.. code-block:: Python

    R = 2_000
    MULTIPLIER = 100_000


    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 = BIOGEME(database, simulate, number_of_draws=R)
    R = biosim.number_of_draws
    biosim.model_name = 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: 2000
    Analytical Integral: 1.718281828459045
    Simulated Integral: 1.7114018513619786
    Sample variance   : 0.24475248418333923
    Std Error         : 0.01106237958540881
    Error             : -0.0068799770970664476




.. GENERATED FROM PYTHON SOURCE LINES 60-61

Change the number of draws

.. GENERATED FROM PYTHON SOURCE LINES 61-67

.. code-block:: Python

    biogeme2 = BIOGEME(database, simulate, number_of_draws=R * MULTIPLIER)
    biogeme2.model_name = '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: 200_000_000
    Analytical Integral: 1.718281828459045
    Simulated Integral: 1.7182840012739995
    Sample variance   : 0.24201170665587718
    Std Error         : 0.011000266057143281
    Error             : 2.1728149544042452e-06





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

   **Total running time of the script:** (0 minutes 1.659 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>`_