.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/montecarlo/plot_b02simple_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_b02simple_integral.py>`
        to download the full example code

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

.. _sphx_glr_auto_examples_montecarlo_plot_b02simple_integral.py:


Various integration methods
===========================

Calculation of a simple integral using numerical integration and
Monte-Carlo integration with various types of draws, including Halton
draws base 13. It illustrates how to use draws that are not directly
available in Biogeme.

:author: Michel Bierlaire, EPFL
:date: Thu Apr 13 20:46:01 2023

.. GENERATED FROM PYTHON SOURCE LINES 14-22

.. 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 23-25

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

.. GENERATED FROM PYTHON SOURCE LINES 25-30

.. code-block:: default

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









.. GENERATED FROM PYTHON SOURCE LINES 31-44

.. code-block:: default

    def halton13(sample_size: int, number_of_draws: int) -> np.array:
        """
        The user can define new draws. For example, 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.

        """
        return draws.getHaltonDraws(sample_size, number_of_draws, base=13, skip=10)









.. GENERATED FROM PYTHON SOURCE LINES 45-48

.. code-block:: default

    mydraws = {'HALTON13': (halton13, 'Halton draws, base 13, skipping 10')}
    database.setRandomNumberGenerators(mydraws)








.. GENERATED FROM PYTHON SOURCE LINES 49-50

Integrate with pseudo-random number generator.

.. GENERATED FROM PYTHON SOURCE LINES 50-53

.. code-block:: default

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








.. GENERATED FROM PYTHON SOURCE LINES 54-55

Integrate with Halton draws, base 2.

.. GENERATED FROM PYTHON SOURCE LINES 55-58

.. code-block:: default

    integrand_halton = exp(bioDraws('U_halton', 'UNIFORM_HALTON2'))
    simulatedI_halton = MonteCarlo(integrand_halton)








.. GENERATED FROM PYTHON SOURCE LINES 59-60

Integrate with Halton draws, base 13.

.. GENERATED FROM PYTHON SOURCE LINES 60-63

.. code-block:: default

    integrand_halton13 = exp(bioDraws('U_halton13', 'HALTON13'))
    simulatedI_halton13 = MonteCarlo(integrand_halton13)








.. GENERATED FROM PYTHON SOURCE LINES 64-65

Integrate with MLHS.

.. GENERATED FROM PYTHON SOURCE LINES 65-68

.. code-block:: default

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








.. GENERATED FROM PYTHON SOURCE LINES 69-70

True value

.. GENERATED FROM PYTHON SOURCE LINES 70-72

.. code-block:: default

    trueI = exp(1.0) - 1.0








.. GENERATED FROM PYTHON SOURCE LINES 73-74

Number of draws.

.. GENERATED FROM PYTHON SOURCE LINES 74-76

.. code-block:: default

    R = 20000








.. GENERATED FROM PYTHON SOURCE LINES 77-81

.. code-block:: default

    sampleVariance = MonteCarlo(integrand * integrand) - simulatedI * simulatedI
    stderr = (sampleVariance / R) ** 0.5
    error = simulatedI - trueI








.. GENERATED FROM PYTHON SOURCE LINES 82-89

.. code-block:: default

    sampleVariance_halton = (
        MonteCarlo(integrand_halton * integrand_halton)
        - simulatedI_halton * simulatedI_halton
    )
    stderr_halton = (sampleVariance_halton / R) ** 0.5
    error_halton = simulatedI_halton - trueI








.. GENERATED FROM PYTHON SOURCE LINES 90-97

.. code-block:: default

    sampleVariance_halton13 = (
        MonteCarlo(integrand_halton13 * integrand_halton13)
        - simulatedI_halton13 * simulatedI_halton13
    )
    stderr_halton13 = (sampleVariance_halton13 / R) ** 0.5
    error_halton13 = simulatedI_halton13 - trueI








.. GENERATED FROM PYTHON SOURCE LINES 98-104

.. code-block:: default

    sampleVariance_mlhs = (
        MonteCarlo(integrand_mlhs * integrand_mlhs) - simulatedI_mlhs * simulatedI_mlhs
    )
    stderr_mlhs = (sampleVariance_mlhs / R) ** 0.5
    error_mlhs = simulatedI_mlhs - trueI








.. GENERATED FROM PYTHON SOURCE LINES 105-125

.. code-block:: default

    simulate = {
        'Analytical Integral': trueI,
        'Simulated Integral': simulatedI,
        'Sample variance   ': sampleVariance,
        'Std Error         ': stderr,
        'Error             ': error,
        'Simulated Integral (Halton)': simulatedI_halton,
        'Sample variance (Halton)   ': sampleVariance_halton,
        'Std Error (Halton)         ': stderr_halton,
        'Error (Halton)             ': error_halton,
        'Simulated Integral (Halton13)': simulatedI_halton13,
        'Sample variance (Halton13)   ': sampleVariance_halton13,
        'Std Error (Halton13)         ': stderr_halton13,
        'Error (Halton13)             ': error_halton13,
        'Simulated Integral (MLHS)': simulatedI_mlhs,
        'Sample variance (MLHS)   ': sampleVariance_mlhs,
        'Std Error (MLHS)         ': stderr_mlhs,
        'Error (MLHS)             ': error_mlhs,
    }








.. GENERATED FROM PYTHON SOURCE LINES 126-131

.. code-block:: default

    biosim = bio.BIOGEME(database, simulate)
    biosim.modelName = 'b02simple_integral'
    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>Sample variance</th>
          <th>Std Error</th>
          <th>Error</th>
          <th>Simulated Integral (Halton)</th>
          <th>Sample variance (Halton)</th>
          <th>Std Error (Halton)</th>
          <th>Error (Halton)</th>
          <th>Simulated Integral (Halton13)</th>
          <th>Sample variance (Halton13)</th>
          <th>Std Error (Halton13)</th>
          <th>Error (Halton13)</th>
          <th>Simulated Integral (MLHS)</th>
          <th>Sample variance (MLHS)</th>
          <th>Std Error (MLHS)</th>
          <th>Error (MLHS)</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>0</th>
          <td>1.718282</td>
          <td>1.718309</td>
          <td>0.242098</td>
          <td>0.003479</td>
          <td>0.000027</td>
          <td>1.718282</td>
          <td>0.242036</td>
          <td>0.003479</td>
          <td>-2.442439e-07</td>
          <td>1.718281</td>
          <td>0.242035</td>
          <td>0.003479</td>
          <td>-6.103944e-07</td>
          <td>1.718282</td>
          <td>0.242036</td>
          <td>0.003479</td>
          <td>-4.164225e-12</td>
        </tr>
      </tbody>
    </table>
    </div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 132-133

Reorganize the results.

.. GENERATED FROM PYTHON SOURCE LINES 133-159

.. code-block:: default

    print(f'Analytical integral: {results.iloc[0]["Analytical Integral"]:.6g}')
    print(f"         \t{'Uniform':>15}\t{'Halton':>15}\t{'Halton13':>15}\t{'MLHS':>15}")
    print(
        f'Simulated\t{results.iloc[0]["Simulated Integral"]:>15.6g}\t'
        f'{results.iloc[0]["Simulated Integral (Halton)"]:>15.6g}\t'
        f'{results.iloc[0]["Simulated Integral (Halton13)"]:>15.6g}\t'
        f'{results.iloc[0]["Simulated Integral (MLHS)"]:>15.6g}'
    )
    print(
        f'Sample var.\t{results.iloc[0]["Sample variance   "]:>15.6g}\t'
        f'{results.iloc[0]["Sample variance (Halton)   "]:>15.6g}\t'
        f'{results.iloc[0]["Sample variance (Halton13)   "]:>15.6g}\t'
        f'{results.iloc[0]["Sample variance (MLHS)   "]:>15.6g}'
    )
    print(
        f'Std error\t{results.iloc[0]["Std Error         "]:>15.6g}\t'
        f'{results.iloc[0]["Std Error (Halton)         "]:>15.6g}\t'
        f'{results.iloc[0]["Std Error (Halton13)         "]:>15.6g}\t'
        f'{results.iloc[0]["Std Error (MLHS)         "]:>15.6g}'
    )
    print(
        f'Error\t\t{results.iloc[0]["Error             "]:>15.6g}\t'
        f'{results.iloc[0]["Error (Halton)             "]:>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: 1.71828
                            Uniform          Halton        Halton13            MLHS
    Simulated               1.71831         1.71828         1.71828         1.71828
    Sample var.            0.242098        0.242036        0.242035        0.242036
    Std error            0.00347921      0.00347876      0.00347876      0.00347876
    Error               2.71241e-05    -2.44244e-07    -6.10394e-07    -4.16422e-12





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

   **Total running time of the script:** (0 minutes 48.422 seconds)


.. _sphx_glr_download_auto_examples_montecarlo_plot_b02simple_integral.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_b02simple_integral.py <plot_b02simple_integral.py>`

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

      :download:`Download Jupyter notebook: plot_b02simple_integral.ipynb <plot_b02simple_integral.ipynb>`


.. only:: html

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

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