.. 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 14-23

.. code-block:: Python


    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
    from biogeme.native_draws import RandomNumberGeneratorTuple








.. GENERATED FROM PYTHON SOURCE LINES 24-26

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

.. GENERATED FROM PYTHON SOURCE LINES 26-31

.. code-block:: Python

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









.. GENERATED FROM PYTHON SOURCE LINES 32-45

.. code-block:: Python

    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.get_halton_draws(sample_size, number_of_draws, base=13, skip=10)









.. GENERATED FROM PYTHON SOURCE LINES 46-53

.. code-block:: Python

    mydraws = {
        'HALTON13': RandomNumberGeneratorTuple(
            generator=halton13, description='Halton draws, base 13, skipping 10'
        )
    }
    database.set_random_number_generators(mydraws)








.. GENERATED FROM PYTHON SOURCE LINES 54-55

Integrate with pseudo-random number generator.

.. GENERATED FROM PYTHON SOURCE LINES 55-58

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 59-60

Integrate with Halton draws, base 2.

.. GENERATED FROM PYTHON SOURCE LINES 60-63

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 64-65

Integrate with Halton draws, base 13.

.. GENERATED FROM PYTHON SOURCE LINES 65-68

.. code-block:: Python

    integrand_halton13 = exp(bioDraws('U_halton13', 'HALTON13'))
    simulated_integral_halton13 = MonteCarlo(integrand_halton13)








.. GENERATED FROM PYTHON SOURCE LINES 69-70

Integrate with MLHS.

.. GENERATED FROM PYTHON SOURCE LINES 70-73

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 74-75

True value

.. GENERATED FROM PYTHON SOURCE LINES 75-77

.. code-block:: Python

    true_integral = exp(1.0) - 1.0








.. GENERATED FROM PYTHON SOURCE LINES 78-79

Number of draws.

.. GENERATED FROM PYTHON SOURCE LINES 79-81

.. code-block:: Python

    R = 20000








.. GENERATED FROM PYTHON SOURCE LINES 82-88

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 89-96

.. code-block:: Python

    sample_variance_halton = (
        MonteCarlo(integrand_halton * integrand_halton)
        - simulated_integral_halton * simulated_integral_halton
    )
    stderr_halton = (sample_variance_halton / R) ** 0.5
    error_halton = simulated_integral_halton - true_integral








.. GENERATED FROM PYTHON SOURCE LINES 97-104

.. code-block:: Python

    sampleVariance_halton13 = (
        MonteCarlo(integrand_halton13 * integrand_halton13)
        - simulated_integral_halton13 * simulated_integral_halton13
    )
    stderr_halton13 = (sampleVariance_halton13 / R) ** 0.5
    error_halton13 = simulated_integral_halton13 - true_integral








.. GENERATED FROM PYTHON SOURCE LINES 105-112

.. code-block:: Python

    sampleVariance_mlhs = (
        MonteCarlo(integrand_mlhs * integrand_mlhs)
        - simulated_integral_mlhs * simulated_integral_mlhs
    )
    stderr_mlhs = (sampleVariance_mlhs / R) ** 0.5
    error_mlhs = simulated_integral_mlhs - true_integral








.. GENERATED FROM PYTHON SOURCE LINES 113-133

.. code-block:: Python

    simulate = {
        'Analytical Integral': true_integral,
        'Simulated Integral': simulated_integral,
        'Sample variance   ': sample_variance,
        'Std Error         ': stderr,
        'Error             ': error,
        'Simulated Integral (Halton)': simulated_integral_halton,
        'Sample variance (Halton)   ': sample_variance_halton,
        'Std Error (Halton)         ': stderr_halton,
        'Error (Halton)             ': error_halton,
        'Simulated Integral (Halton13)': simulated_integral_halton13,
        'Sample variance (Halton13)   ': sampleVariance_halton13,
        'Std Error (Halton13)         ': stderr_halton13,
        'Error (Halton13)             ': error_halton13,
        'Simulated Integral (MLHS)': simulated_integral_mlhs,
        'Sample variance (MLHS)   ': sampleVariance_mlhs,
        'Std Error (MLHS)         ': stderr_mlhs,
        'Error (MLHS)             ': error_mlhs,
    }








.. GENERATED FROM PYTHON SOURCE LINES 134-139

.. code-block:: Python

    biosim = bio.BIOGEME(database, simulate)
    biosim.modelName = 'b02simple_integral'
    results = biosim.simulate(the_beta_values={})
    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>1.703441</td>
          <td>0.222157</td>
          <td>0.003333</td>
          <td>-0.01484</td>
          <td>1.712366</td>
          <td>0.23553</td>
          <td>0.003432</td>
          <td>-0.005916</td>
          <td>1.682423</td>
          <td>0.235354</td>
          <td>0.00343</td>
          <td>-0.035859</td>
          <td>1.718692</td>
          <td>0.242166</td>
          <td>0.00348</td>
          <td>0.000411</td>
        </tr>
      </tbody>
    </table>
    </div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 140-141

Reorganize the results.

.. GENERATED FROM PYTHON SOURCE LINES 141-167

.. code-block:: Python

    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               1.70344         1.71237         1.68242         1.71869
    Sample var.            0.222157         0.23553        0.235354        0.242166
    Std error            0.00333285      0.00343169      0.00343041       0.0034797
    Error                -0.0148405     -0.00591567      -0.0358592     0.000410515





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

   **Total running time of the script:** (0 minutes 0.014 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-jupyter

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

      :download:`Download Python source code: plot_b02simple_integral.py <plot_b02simple_integral.py>`

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      :download:`Download zipped: plot_b02simple_integral.zip <plot_b02simple_integral.zip>`


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

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