.. 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.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.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_swissmetro_plot_b17lognormal_mixture.py:


.. _plot_b17lognormal_mixture:

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

Example of a mixture of logit models, using Monte-Carlo integration.
The mixing distribution is distributed as a log normal.

:author: Michel Bierlaire, EPFL
:date: Mon Apr 10 12:11:53 2023

.. GENERATED FROM PYTHON SOURCE LINES 14-27

.. code-block:: Python


    import biogeme.biogeme_logging as blog
    import biogeme.biogeme as bio
    from biogeme import models
    from biogeme.expressions import (
        Beta,
        exp,
        log,
        MonteCarlo,
        bioDraws,
    )
    from biogeme.parameters import Parameters








.. GENERATED FROM PYTHON SOURCE LINES 28-29

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

.. GENERATED FROM PYTHON SOURCE LINES 29-46

.. 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.py')





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

 .. code-block:: none

    Example b17lognormal_mixture.py 




.. GENERATED FROM PYTHON SOURCE LINES 47-48

Parameters to be estimated.

.. GENERATED FROM PYTHON SOURCE LINES 48-53

.. 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 54-56

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

.. GENERATED FROM PYTHON SOURCE LINES 56-58

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 59-60

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

.. GENERATED FROM PYTHON SOURCE LINES 60-62

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 63-65

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

.. GENERATED FROM PYTHON SOURCE LINES 65-67

.. code-block:: Python

    B_TIME_RND = -exp(B_TIME + B_TIME_S * bioDraws('b_time_rnd', 'NORMAL'))








.. GENERATED FROM PYTHON SOURCE LINES 68-69

Definition of the utility functions.

.. GENERATED FROM PYTHON SOURCE LINES 69-73

.. 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 74-75

Associate utility functions with the numbering of alternatives.

.. GENERATED FROM PYTHON SOURCE LINES 75-77

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 78-79

Associate the availability conditions with the alternatives.

.. GENERATED FROM PYTHON SOURCE LINES 79-81

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 82-83

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

.. GENERATED FROM PYTHON SOURCE LINES 83-85

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 86-87

We integrate over b_time_rnd using Monte-Carlo.

.. GENERATED FROM PYTHON SOURCE LINES 87-89

.. code-block:: Python

    logprob = log(MonteCarlo(prob))








.. GENERATED FROM PYTHON SOURCE LINES 90-93

As the objective is to illustrate the
syntax, we calculate the Monte-Carlo approximation with a small
number of draws.

.. GENERATED FROM PYTHON SOURCE LINES 93-96

.. code-block:: Python

    the_biogeme = bio.BIOGEME(database, logprob, number_of_draws=100, seed=1223)
    the_biogeme.modelName = '17lognormal_mixture'





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

 .. code-block:: none

    Biogeme parameters read from biogeme.toml. 




.. GENERATED FROM PYTHON SOURCE LINES 97-98

Estimate the parameters.

.. GENERATED FROM PYTHON SOURCE LINES 98-100

.. 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 __17lognormal_mixture.iter 
    Cannot read file __17lognormal_mixture.iter. Statement is ignored. 
    The number of draws (100) is low. The results may not be meaningful. 
    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  -0.0079    - 
        1             0.5            -0.5            -0.5             0.5             1.5      5.4e+03      0.042      0.5     0.38    + 
        2               0           -0.36              -1            0.67             1.5      5.3e+03      0.038      0.5     0.37    + 
        3               0           -0.36              -1            0.67             1.5      5.3e+03      0.038     0.25    -0.71    - 
        4            0.25           -0.38            -1.2            0.42             1.2      5.3e+03      0.018     0.25     0.45    + 
        5           0.091           -0.46            -1.5            0.62             1.2      5.2e+03      0.015     0.25     0.16    + 
        6           0.091           -0.46            -1.5            0.62             1.2      5.2e+03      0.015     0.12     -1.4    - 
        7           0.095           -0.33            -1.5             0.5             1.2      5.2e+03      0.011     0.12     0.23    + 
        8            0.22           -0.42            -1.4            0.53             1.2      5.2e+03      0.012     0.12     0.17    + 
        9           0.095           -0.36            -1.3            0.57             1.2      5.2e+03       0.01     0.12     0.12    + 
       10           0.095           -0.36            -1.3            0.57             1.2      5.2e+03       0.01    0.062     -2.2    - 
       11            0.15           -0.34            -1.4            0.51             1.2      5.2e+03     0.0071    0.062     0.44    + 
       12            0.15           -0.34            -1.4            0.51             1.2      5.2e+03     0.0071    0.031    0.083    - 
       13            0.13           -0.38            -1.3            0.54             1.2      5.2e+03     0.0021    0.031     0.77    + 
       14            0.13           -0.38            -1.3            0.54             1.2      5.2e+03     0.0021    0.016    -0.47    - 
       15            0.15           -0.38            -1.4            0.53             1.2      5.2e+03     0.0022    0.016     0.32    + 
       16            0.14           -0.38            -1.4            0.54             1.2      5.2e+03    0.00078    0.016     0.66    + 
       17            0.14           -0.38            -1.4            0.54             1.2      5.2e+03    0.00078   0.0078    -0.29    - 
       18            0.14           -0.37            -1.4            0.54             1.2      5.2e+03     0.0004   0.0078     0.47    + 
       19            0.14           -0.37            -1.4            0.54             1.2      5.2e+03     0.0004   0.0039    0.054    - 
       20            0.14           -0.38            -1.4            0.54             1.2      5.2e+03    0.00019   0.0039     0.69    + 
       21            0.14           -0.37            -1.4            0.54             1.2      5.2e+03    0.00023   0.0039     0.14    + 
       22            0.14           -0.37            -1.4            0.54             1.2      5.2e+03    0.00023    0.002     -1.2    - 
       23            0.15           -0.37            -1.4            0.54             1.2      5.2e+03    0.00018    0.002     0.24    + 
       24            0.15           -0.37            -1.4            0.54             1.2      5.2e+03    9.4e-05    0.002     0.63    + 
    Results saved in file 17lognormal_mixture.html 
    Results saved in file 17lognormal_mixture.pickle 




.. GENERATED FROM PYTHON SOURCE LINES 101-103

.. code-block:: Python

    print(results.short_summary())





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

 .. code-block:: none

    Results for model 17lognormal_mixture
    Nbr of parameters:              5
    Sample size:                    6768
    Excluded data:                  3960
    Final log likelihood:           -5239.843
    Akaike Information Criterion:   10489.69
    Bayesian Information Criterion: 10523.78





.. GENERATED FROM PYTHON SOURCE LINES 104-106

.. 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>0.144193</td>
          <td>0.058552</td>
          <td>2.462670</td>
          <td>1.379066e-02</td>
        </tr>
        <tr>
          <th>ASC_TRAIN</th>
          <td>-0.373108</td>
          <td>0.071619</td>
          <td>-5.209652</td>
          <td>1.891950e-07</td>
        </tr>
        <tr>
          <th>B_COST</th>
          <td>-1.357899</td>
          <td>0.092837</td>
          <td>-14.626662</td>
          <td>0.000000e+00</td>
        </tr>
        <tr>
          <th>B_TIME</th>
          <td>0.543249</td>
          <td>0.068822</td>
          <td>7.893515</td>
          <td>2.886580e-15</td>
        </tr>
        <tr>
          <th>B_TIME_S</th>
          <td>1.161160</td>
          <td>0.100647</td>
          <td>11.536921</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 11.626 seconds)


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