.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/montecarlo/plot_b04normal_mixture_numerical.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_b04normal_mixture_numerical.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_montecarlo_plot_b04normal_mixture_numerical.py:


Numerical integration
=====================

Calculation of a mixtures of logit models where the integral is
calculated using numerical integration.

:author: Michel Bierlaire, EPFL
:date: Thu Apr 13 20:51:32 2023

.. GENERATED FROM PYTHON SOURCE LINES 12-32

.. code-block:: Python


    import biogeme.biogeme as bio
    import biogeme.distributions as dist
    from biogeme.expressions import RandomVariable, Integrate
    from biogeme import models

    from swissmetro_one import (
        database,
        TRAIN_TT_SCALED,
        TRAIN_COST_SCALED,
        SM_TT_SCALED,
        SM_COST_SCALED,
        CAR_TT_SCALED,
        CAR_CO_SCALED,
        TRAIN_AV_SP,
        SM_AV,
        CAR_AV_SP,
        CHOICE,
    )








.. GENERATED FROM PYTHON SOURCE LINES 33-34

Parameters

.. GENERATED FROM PYTHON SOURCE LINES 34-41

.. code-block:: Python

    ASC_CAR = 0.137
    ASC_TRAIN = -0.402
    ASC_SM = 0
    B_TIME = -2.26
    B_TIME_S = 1.66
    B_COST = -1.29








.. GENERATED FROM PYTHON SOURCE LINES 42-44

Define a random parameter, normally distributed,
designed to be used for integration

.. GENERATED FROM PYTHON SOURCE LINES 44-48

.. code-block:: Python

    omega = RandomVariable('omega')
    density = dist.normalpdf(omega)
    b_time_rnd = B_TIME + B_TIME_S * omega








.. GENERATED FROM PYTHON SOURCE LINES 49-50

Definition of the utility functions

.. GENERATED FROM PYTHON SOURCE LINES 50-54

.. code-block:: Python

    v1 = ASC_TRAIN + b_time_rnd * TRAIN_TT_SCALED + B_COST * TRAIN_COST_SCALED
    v2 = ASC_SM + b_time_rnd * SM_TT_SCALED + B_COST * SM_COST_SCALED
    v3 = ASC_CAR + b_time_rnd * CAR_TT_SCALED + B_COST * CAR_CO_SCALED








.. GENERATED FROM PYTHON SOURCE LINES 55-56

Associate utility functions with the numbering of alternatives

.. GENERATED FROM PYTHON SOURCE LINES 56-58

.. code-block:: Python

    util = {1: v1, 2: v2, 3: v3}








.. GENERATED FROM PYTHON SOURCE LINES 59-60

Associate the availability conditions with the alternatives

.. GENERATED FROM PYTHON SOURCE LINES 60-62

.. code-block:: Python

    av = {1: TRAIN_AV_SP, 2: SM_AV, 3: CAR_AV_SP}








.. GENERATED FROM PYTHON SOURCE LINES 63-64

The choice model is a logit, with availability conditions

.. GENERATED FROM PYTHON SOURCE LINES 64-67

.. code-block:: Python

    integrand = models.logit(util, av, CHOICE)
    numerical_integral = Integrate(integrand * density, 'omega')








.. GENERATED FROM PYTHON SOURCE LINES 68-70

.. code-block:: Python

    simulate = {'Numerical': numerical_integral}








.. GENERATED FROM PYTHON SOURCE LINES 71-73

.. code-block:: Python

    biosim = bio.BIOGEME(database, simulate)








.. GENERATED FROM PYTHON SOURCE LINES 74-77

.. code-block:: Python

    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>Numerical</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>0</th>
          <td>0.63785</td>
        </tr>
      </tbody>
    </table>
    </div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 78-79

.. code-block:: Python

    print('Mixture of logit - numerical integration: ', results.iloc[0]['Numerical'])




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

 .. code-block:: none

    Mixture of logit - numerical integration:  0.6378498355784457





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

   **Total running time of the script:** (0 minutes 0.023 seconds)


.. _sphx_glr_download_auto_examples_montecarlo_plot_b04normal_mixture_numerical.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_b04normal_mixture_numerical.ipynb <plot_b04normal_mixture_numerical.ipynb>`

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

      :download:`Download Python source code: plot_b04normal_mixture_numerical.py <plot_b04normal_mixture_numerical.py>`

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

      :download:`Download zipped: plot_b04normal_mixture_numerical.zip <plot_b04normal_mixture_numerical.zip>`


.. only:: html

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

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