.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/swissmetro/plot_b17lognormal_mixture_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_swissmetro_plot_b17lognormal_mixture_integral.py>`
        to download the full example code

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

.. _sphx_glr_auto_examples_swissmetro_plot_b17lognormal_mixture_integral.py:


Mixture with lognormal distribution
===================================

Example of a mixture of logit models.  The mixing distribution is
distributed as a log normal.  Compared to
:ref:`plot_b17lognormal_mixture`, the integration is performed using
numerical integration instead of Monte-Carlo approximation.

:author: Michel Bierlaire, EPFL
:date: Mon Apr 10 12:13:23 2023

.. GENERATED FROM PYTHON SOURCE LINES 15-28

.. code-block:: default


    import biogeme.biogeme_logging as blog
    import biogeme.biogeme as bio
    import biogeme.distributions as dist
    from biogeme import models
    from biogeme.expressions import (
        Beta,
        RandomVariable,
        exp,
        log,
        Integrate,
    )








.. GENERATED FROM PYTHON SOURCE LINES 29-30

See the data processing script: :ref:`swissmetro_data`.

.. GENERATED FROM PYTHON SOURCE LINES 30-47

.. code-block:: default

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

    logger = blog.get_screen_logger(level=blog.INFO)
    logger.info('Example b17lognormal_mixture_integral.py')





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

 .. code-block:: none

    Example b17lognormal_mixture_integral.py 




.. GENERATED FROM PYTHON SOURCE LINES 48-49

Parameters to be estimated.

.. GENERATED FROM PYTHON SOURCE LINES 49-54

.. code-block:: default

    ASC_CAR = Beta('ASC_CAR', 0, None, None, 0)
    ASC_TRAIN = Beta('ASC_TRAIN', 0, None, None, 0)
    ASC_SM = Beta('ASC_SM', 0, None, None, 1)
    B_COST = Beta('B_COST', 0, None, None, 0)








.. GENERATED FROM PYTHON SOURCE LINES 55-57

Define a random parameter, normally distributed, designed to be used.
for Monte-Carlo simulation

.. GENERATED FROM PYTHON SOURCE LINES 57-59

.. code-block:: default

    B_TIME = Beta('B_TIME', 0, None, None, 0)








.. GENERATED FROM PYTHON SOURCE LINES 60-61

It is advised not to use 0 as starting value for the following parameter..

.. GENERATED FROM PYTHON SOURCE LINES 61-63

.. code-block:: default

    B_TIME_S = Beta('B_TIME_S', 1, -2, 2, 0)








.. GENERATED FROM PYTHON SOURCE LINES 64-66

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

.. GENERATED FROM PYTHON SOURCE LINES 66-70

.. code-block:: default

    omega = RandomVariable('omega')
    B_TIME_RND = -exp(B_TIME + B_TIME_S * omega)
    density = dist.normalpdf(omega)








.. GENERATED FROM PYTHON SOURCE LINES 71-72

Definition of the utility functions.

.. GENERATED FROM PYTHON SOURCE LINES 72-76

.. code-block:: default

    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 77-78

Associate utility functions with the numbering of alternatives.

.. GENERATED FROM PYTHON SOURCE LINES 78-80

.. code-block:: default

    V = {1: V1, 2: V2, 3: V3}








.. GENERATED FROM PYTHON SOURCE LINES 81-82

Associate the availability conditions with the alternatives.

.. GENERATED FROM PYTHON SOURCE LINES 82-84

.. code-block:: default

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








.. GENERATED FROM PYTHON SOURCE LINES 85-86

Conditional to omega, we have a logit model (called the kernel).

.. GENERATED FROM PYTHON SOURCE LINES 86-88

.. code-block:: default

    condprob = models.logit(V, av, CHOICE)








.. GENERATED FROM PYTHON SOURCE LINES 89-90

We integrate over omega using numerical integration.

.. GENERATED FROM PYTHON SOURCE LINES 90-92

.. code-block:: default

    logprob = log(Integrate(condprob * density, 'omega'))








.. GENERATED FROM PYTHON SOURCE LINES 93-94

Create the Biogeme object.

.. GENERATED FROM PYTHON SOURCE LINES 94-97

.. code-block:: default

    the_biogeme = bio.BIOGEME(database, logprob)
    the_biogeme.modelName = 'b17lognormal_mixture_integral'





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

 .. code-block:: none

    File biogeme.toml has been parsed. 




.. GENERATED FROM PYTHON SOURCE LINES 98-99

Estimate the parameters

.. GENERATED FROM PYTHON SOURCE LINES 99-101

.. code-block:: default

    results = the_biogeme.estimate()





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

 .. code-block:: none

    *** Initial values of the parameters are obtained from the file __b17lognormal_mixture_integral.iter 
    Cannot read file __b17lognormal_mixture_integral.iter. Statement is ignored. 
    Optimization algorithm: hybrid Newton/BFGS with simple bounds [simple_bounds] 
    ** Optimization: Newton with trust region for simple bounds 
    Iter.         ASC_CAR       ASC_TRAIN          B_COST          B_TIME        B_TIME_S     Function    Relgrad   Radius      Rho      
        0            0.18            -0.4              -1            0.36               1      5.3e+03      0.018       10        1   ++ 
        1            0.18           -0.34            -1.3            0.57             1.1      5.2e+03     0.0026    1e+02      1.1   ++ 
        2            0.17           -0.34            -1.4            0.58             1.2      5.2e+03    0.00017    1e+03        1   ++ 
        3            0.17           -0.34            -1.4            0.58             1.2      5.2e+03    2.9e-06    1e+03        1   ++ 
    Results saved in file b17lognormal_mixture_integral.html 
    Results saved in file b17lognormal_mixture_integral.pickle 




.. GENERATED FROM PYTHON SOURCE LINES 102-104

.. code-block:: default

    print(results.short_summary())





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

 .. code-block:: none

    Results for model b17lognormal_mixture_integral
    Nbr of parameters:              5
    Sample size:                    6768
    Excluded data:                  3960
    Final log likelihood:           -5231.419
    Akaike Information Criterion:   10472.84
    Bayesian Information Criterion: 10506.94





.. GENERATED FROM PYTHON SOURCE LINES 105-107

.. code-block:: default

    pandas_results = results.getEstimatedParameters()
    pandas_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>Value</th>
          <th>Rob. Std err</th>
          <th>Rob. t-test</th>
          <th>Rob. p-value</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>ASC_CAR</th>
          <td>0.174559</td>
          <td>0.062655</td>
          <td>2.786034</td>
          <td>5.335733e-03</td>
        </tr>
        <tr>
          <th>ASC_TRAIN</th>
          <td>-0.345857</td>
          <td>0.073252</td>
          <td>-4.721472</td>
          <td>2.341435e-06</td>
        </tr>
        <tr>
          <th>B_COST</th>
          <td>-1.380475</td>
          <td>0.097826</td>
          <td>-14.111584</td>
          <td>0.000000e+00</td>
        </tr>
        <tr>
          <th>B_TIME</th>
          <td>0.575771</td>
          <td>0.071228</td>
          <td>8.083497</td>
          <td>6.661338e-16</td>
        </tr>
        <tr>
          <th>B_TIME_S</th>
          <td>1.239158</td>
          <td>0.132469</td>
          <td>9.354327</td>
          <td>0.000000e+00</td>
        </tr>
      </tbody>
    </table>
    </div>
    </div>
    <br />
    <br />


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

   **Total running time of the script:** (0 minutes 8.113 seconds)


.. _sphx_glr_download_auto_examples_swissmetro_plot_b17lognormal_mixture_integral.py:

.. only:: html

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




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

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

      :download:`Download Jupyter notebook: plot_b17lognormal_mixture_integral.ipynb <plot_b17lognormal_mixture_integral.ipynb>`


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