.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/latent/plot_b04latent_choice_seq_mc.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_latent_plot_b04latent_choice_seq_mc.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_latent_plot_b04latent_choice_seq_mc.py:


Choice model with a latent variable: sequential estimation (Monte-Carlo)
========================================================================

Mixture of logit, with Monte-Carlo integration
Measurement equation for the indicators.
Sequential estimation.

:author: Michel Bierlaire, EPFL
:date: Thu Apr 13 18:00:05 2023

.. GENERATED FROM PYTHON SOURCE LINES 14-55

.. code-block:: Python


    import sys

    import biogeme.biogeme as bio
    import biogeme.biogeme_logging as blog
    from biogeme.exceptions import BiogemeError
    from biogeme import models
    from biogeme.data.optima import (
        read_data,
        age_65_more,
        moreThanOneCar,
        moreThanOneBike,
        individualHouse,
        male,
        haveChildren,
        haveGA,
        highEducation,
        WaitingTimePT,
        Choice,
        TimePT_scaled,
        TimeCar_scaled,
        MarginalCostPT_scaled,
        CostCarCHF_scaled,
        distance_km_scaled,
        PurpHWH,
        PurpOther,
        ScaledIncome,
    )
    from biogeme.expressions import (
        Beta,
        bioDraws,
        MonteCarlo,
        exp,
        log,
    )
    from biogeme.results import bioResults
    from read_or_estimate import read_or_estimate

    logger = blog.get_screen_logger(level=blog.INFO)
    logger.info('Example b04latent_choice_seq_mc.py')





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

 .. code-block:: none

    Example b04latent_choice_seq_mc.py 




.. GENERATED FROM PYTHON SOURCE LINES 56-57

Read the estimates from the structural equation estimation.

.. GENERATED FROM PYTHON SOURCE LINES 57-68

.. code-block:: Python

    MODELNAME = 'b02one_latent_ordered'
    try:
        struct_results = bioResults(pickle_file=f'saved_results/{MODELNAME}.pickle')
    except BiogemeError:
        print(
            f'Run first the script {MODELNAME}.py in order to generate the '
            f'file {MODELNAME}.pickle, and move it to the directory saved_results'
        )
        sys.exit()
    struct_betas = struct_results.get_beta_values()








.. GENERATED FROM PYTHON SOURCE LINES 69-70

Coefficients.

.. GENERATED FROM PYTHON SOURCE LINES 70-80

.. code-block:: Python

    coef_intercept = struct_betas['coef_intercept']
    coef_age_65_more = struct_betas['coef_age_65_more']
    coef_haveGA = struct_betas['coef_haveGA']
    coef_moreThanOneCar = struct_betas['coef_moreThanOneCar']
    coef_moreThanOneBike = struct_betas['coef_moreThanOneBike']
    coef_individualHouse = struct_betas['coef_individualHouse']
    coef_male = struct_betas['coef_male']
    coef_haveChildren = struct_betas['coef_haveChildren']
    coef_highEducation = struct_betas['coef_highEducation']








.. GENERATED FROM PYTHON SOURCE LINES 81-82

Latent variable: structural equation.

.. GENERATED FROM PYTHON SOURCE LINES 84-86

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

.. GENERATED FROM PYTHON SOURCE LINES 86-90

.. code-block:: Python


    sigma_s = Beta('sigma_s', 1, None, None, 0)
    error_component = sigma_s * bioDraws('EC', 'NORMAL_MLHS')








.. GENERATED FROM PYTHON SOURCE LINES 91-92

Piecewise linear specification for income.

.. GENERATED FROM PYTHON SOURCE LINES 92-120

.. code-block:: Python

    thresholds = [None, 4, 6, 8, 10, None]
    formula_income = models.piecewise_formula(
        variable=ScaledIncome,
        thresholds=thresholds,
        betas=[
            struct_betas['beta_ScaledIncome_minus_inf_4'],
            struct_betas['beta_ScaledIncome_4_6'],
            struct_betas['beta_ScaledIncome_6_8'],
            struct_betas['beta_ScaledIncome_8_10'],
            struct_betas['beta_ScaledIncome_10_inf'],
        ],
    )


    CARLOVERS = (
        coef_intercept
        + coef_age_65_more * age_65_more
        + formula_income
        + coef_moreThanOneCar * moreThanOneCar
        + coef_moreThanOneBike * moreThanOneBike
        + coef_individualHouse * individualHouse
        + coef_male * male
        + coef_haveChildren * haveChildren
        + coef_haveGA * haveGA
        + coef_highEducation * highEducation
        + error_component
    )








.. GENERATED FROM PYTHON SOURCE LINES 121-122

Choice model.

.. GENERATED FROM PYTHON SOURCE LINES 122-132

.. code-block:: Python

    ASC_CAR = Beta('ASC_CAR', 0, None, None, 0)
    ASC_PT = Beta('ASC_PT', 0, None, None, 1)
    ASC_SM = Beta('ASC_SM', 0, None, None, 0)
    BETA_COST_HWH = Beta('BETA_COST_HWH', 0, None, None, 0)
    BETA_COST_OTHER = Beta('BETA_COST_OTHER', 0, None, None, 0)
    BETA_DIST = Beta('BETA_DIST', 0, None, None, 0)
    BETA_TIME_CAR_REF = Beta('BETA_TIME_CAR_REF', 0, None, 0, 0)
    BETA_TIME_PT_REF = Beta('BETA_TIME_PT_REF', 0, None, 0, 0)
    BETA_WAITING_TIME = Beta('BETA_WAITING_TIME', 0, None, None, 0)








.. GENERATED FROM PYTHON SOURCE LINES 133-136

The coefficient of the latent variable should be initialized to
something different from zero. If not, the algorithm may be trapped
in a local optimum, and never change the value.

.. GENERATED FROM PYTHON SOURCE LINES 136-139

.. code-block:: Python

    BETA_TIME_PT_CL = Beta('BETA_TIME_PT_CL', -0.01, None, None, 0)
    BETA_TIME_CAR_CL = Beta('BETA_TIME_CAR_CL', -0.01, None, None, 0)








.. GENERATED FROM PYTHON SOURCE LINES 140-141

Definition of utility functions:.

.. GENERATED FROM PYTHON SOURCE LINES 141-162

.. code-block:: Python

    BETA_TIME_PT = BETA_TIME_PT_REF * exp(BETA_TIME_PT_CL * CARLOVERS)

    V0 = (
        ASC_PT
        + BETA_TIME_PT * TimePT_scaled
        + BETA_WAITING_TIME * WaitingTimePT
        + BETA_COST_HWH * MarginalCostPT_scaled * PurpHWH
        + BETA_COST_OTHER * MarginalCostPT_scaled * PurpOther
    )

    BETA_TIME_CAR = BETA_TIME_CAR_REF * exp(BETA_TIME_CAR_CL * CARLOVERS)

    V1 = (
        ASC_CAR
        + BETA_TIME_CAR * TimeCar_scaled
        + BETA_COST_HWH * CostCarCHF_scaled * PurpHWH
        + BETA_COST_OTHER * CostCarCHF_scaled * PurpOther
    )

    V2 = ASC_SM + BETA_DIST * distance_km_scaled








.. GENERATED FROM PYTHON SOURCE LINES 163-164

Associate utility functions with the numbering of alternatives.

.. GENERATED FROM PYTHON SOURCE LINES 164-166

.. code-block:: Python

    V = {0: V0, 1: V1, 2: V2}








.. GENERATED FROM PYTHON SOURCE LINES 167-168

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

.. GENERATED FROM PYTHON SOURCE LINES 168-170

.. code-block:: Python

    condprob = models.logit(V, None, Choice)








.. GENERATED FROM PYTHON SOURCE LINES 171-172

We integrate over omega using numerical integration

.. GENERATED FROM PYTHON SOURCE LINES 172-174

.. code-block:: Python

    loglike = log(MonteCarlo(condprob))








.. GENERATED FROM PYTHON SOURCE LINES 175-176

Read the data

.. GENERATED FROM PYTHON SOURCE LINES 176-178

.. code-block:: Python

    database = read_data()








.. GENERATED FROM PYTHON SOURCE LINES 179-182

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

.. GENERATED FROM PYTHON SOURCE LINES 182-185

.. code-block:: Python

    the_biogeme = bio.BIOGEME(database, loglike, number_of_draws=100, seed=1223)
    the_biogeme.modelName = 'b04latent_choice_seq_mc'





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

 .. code-block:: none

    Biogeme parameters read from biogeme.toml. 




.. GENERATED FROM PYTHON SOURCE LINES 186-188

If estimation results are saved on file, we read them to speed up the process.
If not, we estimate the parameters.

.. GENERATED FROM PYTHON SOURCE LINES 188-190

.. code-block:: Python

    results = read_or_estimate(the_biogeme=the_biogeme, directory='saved_results')








.. GENERATED FROM PYTHON SOURCE LINES 191-197

.. code-block:: Python

    print(f'Estimated betas: {len(results.data.betaValues)}')
    print(f'Final log likelihood: {results.data.logLike:.3f}')
    print(f'Output file: {results.data.htmlFileName}')
    results.write_latex()
    print(f'LaTeX file: {results.data.latexFileName}')





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

 .. code-block:: none

    Estimated betas: 11
    Final log likelihood: -1103.506
    Output file: b04latent_choice_seq_mc~01.html
    Results saved in file b04latent_choice_seq_mc.tex 
    LaTeX file: b04latent_choice_seq_mc.tex




.. GENERATED FROM PYTHON SOURCE LINES 198-199

.. code-block:: Python

    results.get_estimated_parameters()





.. 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>0.860275</td>
          <td>0.113300</td>
          <td>7.592904</td>
          <td>3.130829e-14</td>
        </tr>
        <tr>
          <th>ASC_SM</th>
          <td>1.937109</td>
          <td>0.231571</td>
          <td>8.365062</td>
          <td>0.000000e+00</td>
        </tr>
        <tr>
          <th>BETA_COST_HWH</th>
          <td>-1.720510</td>
          <td>0.376961</td>
          <td>-4.564162</td>
          <td>5.014930e-06</td>
        </tr>
        <tr>
          <th>BETA_COST_OTHER</th>
          <td>-0.933392</td>
          <td>0.231425</td>
          <td>-4.033233</td>
          <td>5.501473e-05</td>
        </tr>
        <tr>
          <th>BETA_DIST</th>
          <td>-5.457147</td>
          <td>0.635376</td>
          <td>-8.588845</td>
          <td>0.000000e+00</td>
        </tr>
        <tr>
          <th>BETA_TIME_CAR_CL</th>
          <td>-0.133062</td>
          <td>0.007496</td>
          <td>-17.751639</td>
          <td>0.000000e+00</td>
        </tr>
        <tr>
          <th>BETA_TIME_CAR_REF</th>
          <td>-11.890599</td>
          <td>1.724086</td>
          <td>-6.896754</td>
          <td>5.320411e-12</td>
        </tr>
        <tr>
          <th>BETA_TIME_PT_CL</th>
          <td>-0.144913</td>
          <td>0.004588</td>
          <td>-31.586011</td>
          <td>0.000000e+00</td>
        </tr>
        <tr>
          <th>BETA_TIME_PT_REF</th>
          <td>-2.924784</td>
          <td>0.566019</td>
          <td>-5.167286</td>
          <td>2.375175e-07</td>
        </tr>
        <tr>
          <th>BETA_WAITING_TIME</th>
          <td>-0.040270</td>
          <td>0.012112</td>
          <td>-3.324803</td>
          <td>8.848109e-04</td>
        </tr>
        <tr>
          <th>sigma_s</th>
          <td>11.679959</td>
          <td>0.603737</td>
          <td>19.346106</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 0.654 seconds)


.. _sphx_glr_download_auto_examples_latent_plot_b04latent_choice_seq_mc.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_b04latent_choice_seq_mc.ipynb <plot_b04latent_choice_seq_mc.ipynb>`

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

      :download:`Download Python source code: plot_b04latent_choice_seq_mc.py <plot_b04latent_choice_seq_mc.py>`

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

      :download:`Download zipped: plot_b04latent_choice_seq_mc.zip <plot_b04latent_choice_seq_mc.zip>`


.. only:: html

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

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