.. _sphx_glr_auto_examples_montecarlo:

Monte-Carlo integration with Biogeme
************************************

Example discussed in `Bierlaire (2019) Monte-Carlo integration with
Biogeme
<https://transp-or.epfl.ch/documents/technicalReports/Bier19.pdf>`_





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    <div class="sphx-glr-thumbnails">


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    <div class="sphx-glr-thumbcontainer" tooltip="Creation of the Biogeme object for a mixtures of logit models where the integral is approximate...">

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  .. image:: /auto_examples/montecarlo/images/thumb/sphx_glr_b07estimation_specification_thumb.png
    :alt:

  :ref:`sphx_glr_auto_examples_montecarlo_b07estimation_specification.py`

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      <div class="sphx-glr-thumbnail-title">Specification of the mixtures of logit</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Calculation of a simple integral using Monte-Carlo integration.">

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  .. image:: /auto_examples/montecarlo/images/thumb/sphx_glr_plot_b01simple_integral_thumb.png
    :alt:

  :ref:`sphx_glr_auto_examples_montecarlo_plot_b01simple_integral.py`

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      <div class="sphx-glr-thumbnail-title">Simple integral</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Calculation of a simple integral using numerical integration and Monte-Carlo integration with v...">

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  .. image:: /auto_examples/montecarlo/images/thumb/sphx_glr_plot_b02simple_integral_thumb.png
    :alt:

  :ref:`sphx_glr_auto_examples_montecarlo_plot_b02simple_integral.py`

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      <div class="sphx-glr-thumbnail-title">Various integration methods</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Calculation of a simple integral using Monte-Carlo integration. It illustrates how to use antit...">

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  .. image:: /auto_examples/montecarlo/images/thumb/sphx_glr_plot_b03antithetic_thumb.png
    :alt:

  :ref:`sphx_glr_auto_examples_montecarlo_plot_b03antithetic.py`

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      <div class="sphx-glr-thumbnail-title">Antithetic draws</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Calculation of a simple integral using Monte-Carlo integration. It illustrates how to use antot...">

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  .. image:: /auto_examples/montecarlo/images/thumb/sphx_glr_plot_b03antithetic_explicit_thumb.png
    :alt:

  :ref:`sphx_glr_auto_examples_montecarlo_plot_b03antithetic_explicit.py`

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      <div class="sphx-glr-thumbnail-title">Antithetic draws explicitly generated</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Calculation of a mixtures of logit models where the integral is calculated using numerical inte...">

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  .. image:: /auto_examples/montecarlo/images/thumb/sphx_glr_plot_b04normal_mixture_numerical_thumb.png
    :alt:

  :ref:`sphx_glr_auto_examples_montecarlo_plot_b04normal_mixture_numerical.py`

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      <div class="sphx-glr-thumbnail-title">Numerical integration</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Calculation of a mixtures of logit models where the integral is calculated using numerical inte...">

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  .. image:: /auto_examples/montecarlo/images/thumb/sphx_glr_plot_b05normal_mixture_monte_carlo_thumb.png
    :alt:

  :ref:`sphx_glr_auto_examples_montecarlo_plot_b05normal_mixture_monte_carlo.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Monte-Carlo integration</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Estimation of a mixtures of logit models where the integral is calculated using numerical integ...">

.. only:: html

  .. image:: /auto_examples/montecarlo/images/thumb/sphx_glr_plot_b06estimation_integral_thumb.png
    :alt:

  :ref:`sphx_glr_auto_examples_montecarlo_plot_b06estimation_integral.py`

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      <div class="sphx-glr-thumbnail-title">Estimation of mixtures of logit</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Estimation of a mixtures of logit models where the integral is approximated using MonteCarlo in...">

.. only:: html

  .. image:: /auto_examples/montecarlo/images/thumb/sphx_glr_plot_b07estimation_monte_carlo_thumb.png
    :alt:

  :ref:`sphx_glr_auto_examples_montecarlo_plot_b07estimation_monte_carlo.py`

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      <div class="sphx-glr-thumbnail-title">Mixtures of logit with Monte-Carlo 2000 draws</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Estimation of a mixtures of logit models where the integral is approximated using MonteCarlo in...">

.. only:: html

  .. image:: /auto_examples/montecarlo/images/thumb/sphx_glr_plot_b07estimation_monte_carlo_500_thumb.png
    :alt:

  :ref:`sphx_glr_auto_examples_montecarlo_plot_b07estimation_monte_carlo_500.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Mixtures of logit with Monte-Carlo 500 draws</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Estimation of a mixtures of logit models where the integral is approximated using MonteCarlo in...">

.. only:: html

  .. image:: /auto_examples/montecarlo/images/thumb/sphx_glr_plot_b07estimation_monte_carlo_anti_thumb.png
    :alt:

  :ref:`sphx_glr_auto_examples_montecarlo_plot_b07estimation_monte_carlo_anti.py`

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      <div class="sphx-glr-thumbnail-title">Mixtures of logit with Monte-Carlo 2000 antithetic draws</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Estimation of a mixtures of logit models where the integral is approximated using MonteCarlo in...">

.. only:: html

  .. image:: /auto_examples/montecarlo/images/thumb/sphx_glr_plot_b07estimation_monte_carlo_anti_500_thumb.png
    :alt:

  :ref:`sphx_glr_auto_examples_montecarlo_plot_b07estimation_monte_carlo_anti_500.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Mixtures of logit with Monte-Carlo 500 antithetic draws</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Estimation of a mixtures of logit models where the integral is approximated using MonteCarlo in...">

.. only:: html

  .. image:: /auto_examples/montecarlo/images/thumb/sphx_glr_plot_b07estimation_monte_carlo_halton_thumb.png
    :alt:

  :ref:`sphx_glr_auto_examples_montecarlo_plot_b07estimation_monte_carlo_halton.py`

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      <div class="sphx-glr-thumbnail-title">Mixtures of logit with Monte-Carlo 2000 Halton draws</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Estimation of a mixtures of logit models where the integral is approximated using MonteCarlo in...">

.. only:: html

  .. image:: /auto_examples/montecarlo/images/thumb/sphx_glr_plot_b07estimation_monte_carlo_halton_500_thumb.png
    :alt:

  :ref:`sphx_glr_auto_examples_montecarlo_plot_b07estimation_monte_carlo_halton_500.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Mixtures of logit with Monte-Carlo 500 Halton draws</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Estimation of a mixtures of logit models where the integral is approximated using MonteCarlo in...">

.. only:: html

  .. image:: /auto_examples/montecarlo/images/thumb/sphx_glr_plot_b07estimation_monte_carlo_mlhs_thumb.png
    :alt:

  :ref:`sphx_glr_auto_examples_montecarlo_plot_b07estimation_monte_carlo_mlhs.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Mixtures of logit with Monte-Carlo 2000 MLHS draws</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Estimation of a mixtures of logit models where the integral is approximated using MonteCarlo in...">

.. only:: html

  .. image:: /auto_examples/montecarlo/images/thumb/sphx_glr_plot_b07estimation_monte_carlo_mlhs_500_thumb.png
    :alt:

  :ref:`sphx_glr_auto_examples_montecarlo_plot_b07estimation_monte_carlo_mlhs_500.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Mixtures of logit with Monte-Carlo 500 MLHS draws</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Estimation of a mixtures of logit models where the integral is approximated using MonteCarlo in...">

.. only:: html

  .. image:: /auto_examples/montecarlo/images/thumb/sphx_glr_plot_b07estimation_monte_carlo_mlhs_anti_thumb.png
    :alt:

  :ref:`sphx_glr_auto_examples_montecarlo_plot_b07estimation_monte_carlo_mlhs_anti.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Mixtures of logit with Monte-Carlo 2000 antithetic MLHS draws</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Estimation of a mixtures of logit models where the integral is approximated using MonteCarlo in...">

.. only:: html

  .. image:: /auto_examples/montecarlo/images/thumb/sphx_glr_plot_b07estimation_monte_carlo_mlhs_anti_500_thumb.png
    :alt:

  :ref:`sphx_glr_auto_examples_montecarlo_plot_b07estimation_monte_carlo_mlhs_anti_500.py`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Mixtures of logit with Monte-Carlo 2000 antithetic MLHS draws</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Reading data and defining variables.">

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  .. image:: /auto_examples/montecarlo/images/thumb/sphx_glr_swissmetro_thumb.png
    :alt:

  :ref:`sphx_glr_auto_examples_montecarlo_swissmetro.py`

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      <div class="sphx-glr-thumbnail-title">Data preparation for Swissmetro</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Use only the first observation for simulation.">

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  .. image:: /auto_examples/montecarlo/images/thumb/sphx_glr_swissmetro_one_thumb.png
    :alt:

  :ref:`sphx_glr_auto_examples_montecarlo_swissmetro_one.py`

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      <div class="sphx-glr-thumbnail-title">Data preparation for Swissmetro: one observation</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /auto_examples/montecarlo/b07estimation_specification
   /auto_examples/montecarlo/plot_b01simple_integral
   /auto_examples/montecarlo/plot_b02simple_integral
   /auto_examples/montecarlo/plot_b03antithetic
   /auto_examples/montecarlo/plot_b03antithetic_explicit
   /auto_examples/montecarlo/plot_b04normal_mixture_numerical
   /auto_examples/montecarlo/plot_b05normal_mixture_monte_carlo
   /auto_examples/montecarlo/plot_b06estimation_integral
   /auto_examples/montecarlo/plot_b07estimation_monte_carlo
   /auto_examples/montecarlo/plot_b07estimation_monte_carlo_500
   /auto_examples/montecarlo/plot_b07estimation_monte_carlo_anti
   /auto_examples/montecarlo/plot_b07estimation_monte_carlo_anti_500
   /auto_examples/montecarlo/plot_b07estimation_monte_carlo_halton
   /auto_examples/montecarlo/plot_b07estimation_monte_carlo_halton_500
   /auto_examples/montecarlo/plot_b07estimation_monte_carlo_mlhs
   /auto_examples/montecarlo/plot_b07estimation_monte_carlo_mlhs_500
   /auto_examples/montecarlo/plot_b07estimation_monte_carlo_mlhs_anti
   /auto_examples/montecarlo/plot_b07estimation_monte_carlo_mlhs_anti_500
   /auto_examples/montecarlo/swissmetro
   /auto_examples/montecarlo/swissmetro_one