.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/swissmetro/plot_b05normal_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_b05normal_mixture.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_swissmetro_plot_b05normal_mixture.py:


.. _plot_b05normal_mixture:

Mixture of logit models
=======================

Example of a normal mixture of logit models, using Monte-Carlo integration.

:author: Michel Bierlaire, EPFL
:date: Sun Apr  9 17:30:14 2023

.. GENERATED FROM PYTHON SOURCE LINES 12-19

.. code-block:: Python


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








.. GENERATED FROM PYTHON SOURCE LINES 20-21

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

.. GENERATED FROM PYTHON SOURCE LINES 21-38

.. 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 b05normal_mixtures.py')





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

 .. code-block:: none

    Example b05normal_mixtures.py 




.. GENERATED FROM PYTHON SOURCE LINES 39-40

Parameters to be estimated

.. GENERATED FROM PYTHON SOURCE LINES 40-45

.. 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 46-48

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

.. GENERATED FROM PYTHON SOURCE LINES 48-50

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 51-52

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

.. GENERATED FROM PYTHON SOURCE LINES 52-55

.. code-block:: Python

    B_TIME_S = Beta('B_TIME_S', 1, None, None, 0)
    B_TIME_RND = B_TIME + B_TIME_S * bioDraws('b_time_rnd', 'NORMAL')








.. GENERATED FROM PYTHON SOURCE LINES 56-57

Definition of the utility functions.

.. GENERATED FROM PYTHON SOURCE LINES 57-61

.. 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 62-63

Associate utility functions with the numbering of alternatives.

.. GENERATED FROM PYTHON SOURCE LINES 63-65

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 66-67

Associate the availability conditions with the alternatives.

.. GENERATED FROM PYTHON SOURCE LINES 67-69

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 70-71

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

.. GENERATED FROM PYTHON SOURCE LINES 71-73

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 74-75

We integrate over b_time_rnd using Monte-Carlo.

.. GENERATED FROM PYTHON SOURCE LINES 75-77

.. code-block:: Python

    logprob = log(MonteCarlo(prob))








.. GENERATED FROM PYTHON SOURCE LINES 78-79

These notes will be included as such in the report file.

.. GENERATED FROM PYTHON SOURCE LINES 79-84

.. code-block:: Python

    USER_NOTES = (
        'Example of a mixture of logit models with three alternatives, '
        'approximated using Monte-Carlo integration.'
    )








.. GENERATED FROM PYTHON SOURCE LINES 85-86

Create the Biogeme object.

.. GENERATED FROM PYTHON SOURCE LINES 86-91

.. code-block:: Python

    the_biogeme = bio.BIOGEME(
        database, logprob, user_notes=USER_NOTES, number_of_draws=100, seed=1223
    )
    the_biogeme.modelName = 'b05normal_mixture'





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

 .. code-block:: none

    Biogeme parameters read from biogeme.toml. 




.. GENERATED FROM PYTHON SOURCE LINES 92-94

.. code-block:: Python

    print(f'Number of draws: {the_biogeme.number_of_draws}')





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

 .. code-block:: none

    Number of draws: 100




.. GENERATED FROM PYTHON SOURCE LINES 95-96

Estimate the parameters

.. GENERATED FROM PYTHON SOURCE LINES 96-97

.. 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 __b05normal_mixture.iter 
    Cannot read file __b05normal_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               1              -1              -1              -1               2      6.1e+03       0.16        1     0.25    + 
        1               0           -0.75           -0.35              -2               3      5.5e+03      0.049        1     0.35    + 
        2               0           -0.75           -0.35              -2               3      5.5e+03      0.049      0.5   -0.051    - 
        3             0.5           -0.94           -0.85            -2.4             2.6      5.4e+03      0.044      0.5     0.48    + 
        4               0           -0.44            -1.3            -2.9             2.1      5.3e+03      0.073      0.5     0.34    + 
        5               0           -0.44            -1.3            -2.9             2.1      5.3e+03      0.073     0.25     -0.6    - 
        6            0.25           -0.19            -1.6            -2.7             2.3      5.2e+03      0.025     0.25     0.26    + 
        7            0.25           -0.19            -1.6            -2.7             2.3      5.2e+03      0.025     0.12    0.099    - 
        8            0.24           -0.32            -1.5            -2.8             2.2      5.2e+03      0.011      1.2     0.92   ++ 
        9            0.24           -0.32            -1.5            -2.8             2.2      5.2e+03      0.011     0.62     -4.4    - 
       10            0.24           -0.32            -1.5            -2.8             2.2      5.2e+03      0.011     0.31     -1.4    - 
       11            0.24           -0.32            -1.5            -2.8             2.2      5.2e+03      0.011     0.16    -0.34    - 
       12            0.27            -0.2            -1.3            -2.6             2.1      5.2e+03     0.0082     0.16     0.41    + 
       13            0.27            -0.2            -1.3            -2.6             2.1      5.2e+03     0.0082    0.078     0.09    - 
       14            0.25           -0.28            -1.3            -2.7               2      5.2e+03      0.019    0.078      0.2    + 
       15            0.26           -0.25            -1.3            -2.6               2      5.2e+03     0.0033     0.78     0.93   ++ 
       16            0.26           -0.25            -1.3            -2.6               2      5.2e+03     0.0033     0.39    -0.98    - 
       17            0.26           -0.25            -1.3            -2.6               2      5.2e+03     0.0033      0.2   -0.058    - 
       18            0.17           -0.45            -1.3            -2.5             1.9      5.2e+03     0.0093      0.2     0.38    + 
       19            0.17           -0.45            -1.3            -2.5             1.9      5.2e+03     0.0093    0.098    -0.19    - 
       20            0.17           -0.45            -1.3            -2.5             1.9      5.2e+03     0.0093    0.049     -1.1    - 
       21            0.12            -0.4            -1.3            -2.4             1.9      5.2e+03     0.0061    0.049      0.2    + 
       22            0.17           -0.42            -1.3            -2.4             1.9      5.2e+03      0.003    0.049     0.76    + 
       23            0.15           -0.37            -1.3            -2.4             1.8      5.2e+03     0.0033    0.049     0.76    + 
       24            0.18           -0.38            -1.3            -2.4             1.8      5.2e+03     0.0024    0.049     0.59    + 
       25            0.16           -0.37            -1.3            -2.4             1.7      5.2e+03     0.0019    0.049     0.86    + 
       26            0.16           -0.38            -1.3            -2.3             1.7      5.2e+03     0.0016    0.049     0.67    + 
       27            0.16           -0.38            -1.3            -2.3             1.7      5.2e+03     0.0016    0.024    0.099    - 
       28            0.14            -0.4            -1.3            -2.3             1.7      5.2e+03     0.0028    0.024     0.59    + 
       29            0.14            -0.4            -1.3            -2.3             1.7      5.2e+03     0.0028    0.012   -0.056    - 
       30            0.15           -0.39            -1.3            -2.3             1.7      5.2e+03     0.0011    0.012     0.64    + 
       31            0.14           -0.39            -1.3            -2.3             1.7      5.2e+03     0.0012     0.12     0.98   ++ 
       32            0.14           -0.39            -1.3            -2.3             1.7      5.2e+03     0.0012    0.061    -0.74    - 
       33            0.12            -0.4            -1.3            -2.2             1.6      5.2e+03     0.0016    0.061     0.33    + 
       34            0.12            -0.4            -1.3            -2.2             1.6      5.2e+03     0.0016    0.031     -1.3    - 
       35            0.12            -0.4            -1.3            -2.2             1.6      5.2e+03     0.0016    0.015    -0.61    - 
       36            0.13           -0.41            -1.3            -2.2             1.6      5.2e+03    0.00092    0.015     0.42    + 
       37            0.13           -0.41            -1.3            -2.2             1.6      5.2e+03    0.00067    0.015     0.72    + 
       38            0.13           -0.41            -1.3            -2.2             1.6      5.2e+03    0.00067   0.0076     -1.4    - 
       39            0.13           -0.41            -1.3            -2.2             1.6      5.2e+03    0.00067   0.0038    -0.38    - 
       40            0.12           -0.41            -1.3            -2.2             1.6      5.2e+03    0.00036   0.0038     0.42    + 
       41            0.13           -0.41            -1.3            -2.2             1.6      5.2e+03    0.00013   0.0038     0.63    + 
       42            0.13           -0.41            -1.3            -2.2             1.6      5.2e+03    0.00013   0.0019    -0.42    - 
       43            0.13           -0.41            -1.3            -2.2             1.6      5.2e+03    5.7e-05   0.0019     0.51    - 
    Results saved in file b05normal_mixture.html 
    Results saved in file b05normal_mixture.pickle 




.. GENERATED FROM PYTHON SOURCE LINES 98-99

.. code-block:: Python

    print(results.short_summary())




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

 .. code-block:: none

    Results for model b05normal_mixture
    Nbr of parameters:              5
    Sample size:                    6768
    Excluded data:                  3960
    Final log likelihood:           -5216.341
    Akaike Information Criterion:   10442.68
    Bayesian Information Criterion: 10476.78





.. GENERATED FROM PYTHON SOURCE LINES 100-102

.. 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>0.126124</td>
          <td>0.051603</td>
          <td>2.444141</td>
          <td>1.451974e-02</td>
        </tr>
        <tr>
          <th>ASC_TRAIN</th>
          <td>-0.409875</td>
          <td>0.065874</td>
          <td>-6.222127</td>
          <td>4.904608e-10</td>
        </tr>
        <tr>
          <th>B_COST</th>
          <td>-1.278857</td>
          <td>0.085489</td>
          <td>-14.959347</td>
          <td>0.000000e+00</td>
        </tr>
        <tr>
          <th>B_TIME</th>
          <td>-2.220810</td>
          <td>0.115148</td>
          <td>-19.286529</td>
          <td>0.000000e+00</td>
        </tr>
        <tr>
          <th>B_TIME_S</th>
          <td>1.602677</td>
          <td>0.122264</td>
          <td>13.108344</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 16.532 seconds)


.. _sphx_glr_download_auto_examples_swissmetro_plot_b05normal_mixture.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_b05normal_mixture.ipynb <plot_b05normal_mixture.ipynb>`

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

      :download:`Download Python source code: plot_b05normal_mixture.py <plot_b05normal_mixture.py>`

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

      :download:`Download zipped: plot_b05normal_mixture.zip <plot_b05normal_mixture.zip>`


.. only:: html

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

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