.. 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.

Michel Bierlaire, EPFL
Sat Jun 28 2025, 21:09:58

.. GENERATED FROM PYTHON SOURCE LINES 12-32

.. code-block:: Python


    from IPython.core.display_functions import display
    from biogeme.biogeme import BIOGEME
    from biogeme.expressions import IntegrateNormal, RandomVariable
    from biogeme.models import logit

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








.. 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 44-47

.. code-block:: Python

    omega = RandomVariable('omega')
    b_time_rnd = b_time + b_time_s * omega








.. GENERATED FROM PYTHON SOURCE LINES 48-49

Definition of the utility functions

.. GENERATED FROM PYTHON SOURCE LINES 49-53

.. code-block:: Python

    v_train = asc_train + b_time_rnd * TRAIN_TT_SCALED + b_cost * TRAIN_COST_SCALED
    v_swissmetro = asc_sm + b_time_rnd * SM_TT_SCALED + b_cost * SM_COST_SCALED
    v_car = asc_car + b_time_rnd * CAR_TT_SCALED + b_cost * CAR_CO_SCALED








.. GENERATED FROM PYTHON SOURCE LINES 54-55

Associate utility functions with the numbering of alternatives

.. GENERATED FROM PYTHON SOURCE LINES 55-57

.. code-block:: Python

    util = {1: v_train, 2: v_swissmetro, 3: v_car}








.. GENERATED FROM PYTHON SOURCE LINES 58-59

Associate the availability conditions with the alternatives

.. GENERATED FROM PYTHON SOURCE LINES 59-61

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 62-63

The choice model is a logit, with availability conditions

.. GENERATED FROM PYTHON SOURCE LINES 63-66

.. code-block:: Python

    integrand = logit(util, av, CHOICE)
    numerical_integral = IntegrateNormal(integrand, 'omega')








.. GENERATED FROM PYTHON SOURCE LINES 67-69

.. code-block:: Python

    simulate = {'Numerical': numerical_integral}








.. GENERATED FROM PYTHON SOURCE LINES 70-72

.. code-block:: Python

    biosim = BIOGEME(database, simulate)








.. GENERATED FROM PYTHON SOURCE LINES 73-76

.. code-block:: Python

    results = biosim.simulate(the_beta_values={})
    display(results)





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

 .. code-block:: none

       Numerical
    0    0.63785




.. GENERATED FROM PYTHON SOURCE LINES 77-78

.. 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:  0.6378498356723438





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

   **Total running time of the script:** (0 minutes 0.178 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>`_