.. 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:: Python


    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:: Python

    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:: Python

    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:: Python

    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:: Python

    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:: Python

    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:: 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 77-78

Associate utility functions with the numbering of alternatives.

.. GENERATED FROM PYTHON SOURCE LINES 78-80

.. code-block:: Python

    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:: Python

    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:: Python

    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:: Python

    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:: Python

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





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

 .. code-block:: none

    Biogeme parameters read from biogeme.toml. 




.. GENERATED FROM PYTHON SOURCE LINES 98-99

Estimate the parameters

.. GENERATED FROM PYTHON SOURCE LINES 99-101

.. code-block:: Python

    results = the_biogeme.estimate()





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

 .. code-block:: none

    As the model is rather complex, we cancel the calculation of second derivatives. If you want to control the parameters, change the name of the algorithm in the TOML file from "automatic" to "simple_bounds" 
    *** Initial values of the parameters are obtained from the file __b17lognormal_mixture_integral.iter 
    Cannot read file __b17lognormal_mixture_integral.iter. Statement is ignored. 
    As the model is rather complex, we cancel the calculation of second derivatives. If you want to control the parameters, change the name of the algorithm in the TOML file from "automatic" to "simple_bounds" 
    Optimization algorithm: hybrid Newton/BFGS with simple bounds [simple_bounds] 
    ** Optimization: BFGS with trust region for simple bounds 
    Iter.         ASC_CAR       ASC_TRAIN          B_COST          B_TIME        B_TIME_S     Function    Relgrad   Radius      Rho      
        0               0               0               0               0               1      5.7e+03      0.096      0.5 -0.00035    - 
        1             0.5            -0.5            -0.5             0.5             1.5      5.4e+03      0.042      0.5     0.39    + 
        2               0           -0.34              -1            0.69             1.5      5.3e+03       0.04      0.5     0.34    + 
        3               0           -0.34              -1            0.69             1.5      5.3e+03       0.04     0.25    -0.63    - 
        4            0.25           -0.39            -1.2            0.44             1.3      5.2e+03      0.017     0.25     0.49    + 
        5           0.084           -0.38            -1.5            0.61             1.3      5.2e+03     0.0097     0.25     0.25    + 
        6           0.084           -0.38            -1.5            0.61             1.3      5.2e+03     0.0097     0.12     -2.2    - 
        7           0.084           -0.38            -1.5            0.61             1.3      5.2e+03     0.0097    0.062    0.093    - 
        8            0.15           -0.32            -1.4            0.55             1.2      5.2e+03     0.0068    0.062     0.57    + 
        9            0.17           -0.38            -1.4            0.61             1.2      5.2e+03     0.0065    0.062     0.18    + 
       10            0.18           -0.34            -1.3            0.56             1.3      5.2e+03     0.0023    0.062     0.44    + 
       11            0.18           -0.35            -1.4            0.57             1.2      5.2e+03     0.0018    0.062     0.42    + 
       12            0.18           -0.35            -1.4            0.57             1.2      5.2e+03     0.0018    0.031     -4.6    - 
       13            0.18           -0.35            -1.4            0.57             1.2      5.2e+03     0.0018    0.016     -1.7    - 
       14            0.18           -0.35            -1.4            0.57             1.2      5.2e+03     0.0018   0.0078    0.051    - 
       15            0.17           -0.35            -1.4            0.57             1.2      5.2e+03    0.00057   0.0078     0.55    + 
       16            0.17           -0.35            -1.4            0.57             1.2      5.2e+03    0.00057   0.0039    0.042    - 
       17            0.17           -0.35            -1.4            0.57             1.2      5.2e+03     0.0004   0.0039     0.56    + 
       18            0.17           -0.35            -1.4            0.57             1.2      5.2e+03    0.00014   0.0039     0.74    + 
       19            0.17           -0.35            -1.4            0.57             1.2      5.2e+03    0.00014    0.002    -0.22    - 
       20            0.17           -0.35            -1.4            0.57             1.2      5.2e+03    0.00024    0.002      0.2    + 
       21            0.17           -0.35            -1.4            0.57             1.2      5.2e+03    4.9e-05    0.002     0.77    + 
    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:: Python

    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:           -5231.42
    Akaike Information Criterion:   10472.84
    Bayesian Information Criterion: 10506.94





.. GENERATED FROM PYTHON SOURCE LINES 105-107

.. code-block:: Python

    pandas_results = results.get_estimated_parameters()
    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>0.174040</td>
          <td>0.040594</td>
          <td>4.287292</td>
          <td>1.808646e-05</td>
        </tr>
        <tr>
          <th>ASC_TRAIN</th>
          <td>-0.346716</td>
          <td>0.048707</td>
          <td>-7.118431</td>
          <td>1.091571e-12</td>
        </tr>
        <tr>
          <th>B_COST</th>
          <td>-1.380908</td>
          <td>0.057160</td>
          <td>-24.158547</td>
          <td>0.000000e+00</td>
        </tr>
        <tr>
          <th>B_TIME</th>
          <td>0.575354</td>
          <td>0.048100</td>
          <td>11.961535</td>
          <td>0.000000e+00</td>
        </tr>
        <tr>
          <th>B_TIME_S</th>
          <td>1.241199</td>
          <td>0.074153</td>
          <td>16.738374</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 11.245 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-jupyter

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

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

      :download:`Download Python source code: plot_b17lognormal_mixture_integral.py <plot_b17lognormal_mixture_integral.py>`

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

      :download:`Download zipped: plot_b17lognormal_mixture_integral.zip <plot_b17lognormal_mixture_integral.zip>`


.. only:: html

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

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