.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/montecarlo/plot_b03antithetic.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_b03antithetic.py>`
        to download the full example code

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

.. _sphx_glr_auto_examples_montecarlo_plot_b03antithetic.py:


Antithetic draws
================

Calculation of a simple integral using Monte-Carlo integration. It
illustrates how to use antithetic draws.

:author: Michel Bierlaire, EPFL
:date: Thu Apr 13 20:48:02 2023

.. GENERATED FROM PYTHON SOURCE LINES 12-20

.. code-block:: default


    import numpy as np
    import pandas as pd
    import biogeme.database as db
    import biogeme.biogeme as bio
    from biogeme import draws
    from biogeme.expressions import exp, bioDraws, MonteCarlo








.. GENERATED FROM PYTHON SOURCE LINES 21-23

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

.. GENERATED FROM PYTHON SOURCE LINES 23-28

.. code-block:: default

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









.. GENERATED FROM PYTHON SOURCE LINES 29-47

.. code-block:: default

    def halton13_anti(sample_size: int, number_of_draws: int) -> np.array:
        """The user can define new draws. For example, antithetic Halton
        draws with base 13, skipping the first 10 draws.

        :param sample_size: number of observations for which draws must be
                           generated.
        :param number_of_draws: number of draws to generate.

        """

        # We first generate half of the number of requested draws.
        the_draws = draws.getHaltonDraws(
            sample_size, int(number_of_draws / 2.0), base=13, skip=10
        )
        # We complement them with their antithetic version.
        return np.concatenate((the_draws, 1 - the_draws), axis=1)









.. GENERATED FROM PYTHON SOURCE LINES 48-56

.. code-block:: default

    mydraws = {
        'HALTON13_ANTI': (
            halton13_anti,
            'Antithetic Halton draws, base 13, skipping 10',
        )
    }
    database.setRandomNumberGenerators(mydraws)








.. GENERATED FROM PYTHON SOURCE LINES 57-58

Integrate with antithetic pseudo-random number generator.

.. GENERATED FROM PYTHON SOURCE LINES 58-61

.. code-block:: default

    integrand = exp(bioDraws('U', 'UNIFORM_ANTI'))
    simulatedI = MonteCarlo(integrand)








.. GENERATED FROM PYTHON SOURCE LINES 62-63

Integrate with antithetic Halton draws, base 13.

.. GENERATED FROM PYTHON SOURCE LINES 63-66

.. code-block:: default

    integrand_halton13 = exp(bioDraws('U_halton13', 'HALTON13_ANTI'))
    simulatedI_halton13 = MonteCarlo(integrand_halton13)








.. GENERATED FROM PYTHON SOURCE LINES 67-68

Integrate with antithetic MLHS.

.. GENERATED FROM PYTHON SOURCE LINES 68-71

.. code-block:: default

    integrand_mlhs = exp(bioDraws('U_mlhs', 'UNIFORM_MLHS_ANTI'))
    simulatedI_mlhs = MonteCarlo(integrand_mlhs)








.. GENERATED FROM PYTHON SOURCE LINES 72-73

True value

.. GENERATED FROM PYTHON SOURCE LINES 73-75

.. code-block:: default

    trueI = exp(1.0) - 1.0








.. GENERATED FROM PYTHON SOURCE LINES 76-77

Number of draws.

.. GENERATED FROM PYTHON SOURCE LINES 77-79

.. code-block:: default

    R = 20000








.. GENERATED FROM PYTHON SOURCE LINES 80-82

.. code-block:: default

    error = simulatedI - trueI








.. GENERATED FROM PYTHON SOURCE LINES 83-85

.. code-block:: default

    error_halton13 = simulatedI_halton13 - trueI








.. GENERATED FROM PYTHON SOURCE LINES 86-88

.. code-block:: default

    error_mlhs = simulatedI_mlhs - trueI








.. GENERATED FROM PYTHON SOURCE LINES 89-100

.. code-block:: default

    simulate = {
        'Analytical Integral': trueI,
        'Simulated Integral': simulatedI,
        'Error             ': error,
        'Simulated Integral (Halton13)': simulatedI_halton13,
        'Error (Halton13)             ': error_halton13,
        'Simulated Integral (MLHS)': simulatedI_mlhs,
        'Error (MLHS)             ': error_mlhs,
    }









.. GENERATED FROM PYTHON SOURCE LINES 101-106

.. code-block:: default

    biosim = bio.BIOGEME(database, simulate)
    biosim.modelName = 'b03antithetic'
    results = biosim.simulate(theBetaValues={})
    results






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }

        .dataframe tbody tr th {
            vertical-align: top;
        }

        .dataframe thead th {
            text-align: right;
        }
    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>Analytical Integral</th>
          <th>Simulated Integral</th>
          <th>Error</th>
          <th>Simulated Integral (Halton13)</th>
          <th>Error (Halton13)</th>
          <th>Simulated Integral (MLHS)</th>
          <th>Error (MLHS)</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>0</th>
          <td>1.718282</td>
          <td>1.718272</td>
          <td>-0.00001</td>
          <td>1.718282</td>
          <td>-4.449617e-09</td>
          <td>1.718282</td>
          <td>-1.322897e-11</td>
        </tr>
      </tbody>
    </table>
    </div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 107-108

Reorganize the results.

.. GENERATED FROM PYTHON SOURCE LINES 108-122

.. code-block:: default

    print(f"Analytical integral: {results.iloc[0]['Analytical Integral']:.6f}")
    print(
        f"         \t{'Uniform (Anti)':>15}\t{'Halton13 (Anti)':>15}\t{'MLHS (Anti)':>15}"
    )
    print(
        f"Simulated\t{results.iloc[0]['Simulated Integral']:>15.6g}\t"
        f"{results.iloc[0]['Simulated Integral (Halton13)']:>15.6g}\t"
        f"{results.iloc[0]['Simulated Integral (MLHS)']:>15.6g}"
    )
    print(
        f"Error\t\t{results.iloc[0]['Error             ']:>15.6g}\t"
        f"{results.iloc[0]['Error (Halton13)             ']:>15.6g}\t"
        f"{results.iloc[0]['Error (MLHS)             ']:>15.6g}"
    )




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

 .. code-block:: none

    Analytical integral: 1.718282
                     Uniform (Anti) Halton13 (Anti)     MLHS (Anti)
    Simulated               1.71827         1.71828         1.71828
    Error              -9.65275e-06    -4.44962e-09     -1.3229e-11





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

   **Total running time of the script:** (0 minutes 21.911 seconds)


.. _sphx_glr_download_auto_examples_montecarlo_plot_b03antithetic.py:

.. only:: html

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




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

      :download:`Download Python source code: plot_b03antithetic.py <plot_b03antithetic.py>`

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

      :download:`Download Jupyter notebook: plot_b03antithetic.ipynb <plot_b03antithetic.ipynb>`


.. only:: html

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

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