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

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

.. _sphx_glr_auto_examples_latent_plot_b04latent_choice_seq.py:


Choice model with a latent variable: sequential estimation
==========================================================

Mixture of logit.
Measurement equation for the indicators.
Sequential estimation.

:author: Michel Bierlaire, EPFL
:date: Thu Apr 13 17:32:34 2023

.. GENERATED FROM PYTHON SOURCE LINES 14-58

.. code-block:: default


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

    from read_or_estimate import read_or_estimate

    from optima import (
        database,
        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,
    )

    logger = blog.get_screen_logger(level=blog.INFO)
    logger.info('Example b04latent_choice_seq.py')






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

 .. code-block:: none

    Example b04latent_choice_seq.py 




.. GENERATED FROM PYTHON SOURCE LINES 59-60

Read the estimates from the structural equation estimation.

.. GENERATED FROM PYTHON SOURCE LINES 60-71

.. code-block:: default

    MODELNAME = 'b02one_latent_ordered'
    try:
        struct_results = res.bioResults(pickleFile=f'saved_results/{MODELNAME}.pickle')
    except excep.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.getBetaValues()








.. GENERATED FROM PYTHON SOURCE LINES 72-73

Coefficients.

.. GENERATED FROM PYTHON SOURCE LINES 73-83

.. code-block:: default

    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 84-85

Latent variable: structural equation.

.. GENERATED FROM PYTHON SOURCE LINES 87-89

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

.. GENERATED FROM PYTHON SOURCE LINES 89-122

.. code-block:: default


    omega = RandomVariable('omega')
    density = dist.normalpdf(omega)
    sigma_s = Beta('sigma_s', 1, None, None, 0)

    thresholds = [None, 4, 6, 8, 10, None]
    formula_income = models.piecewiseFormula(
        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
        + sigma_s * omega
    )








.. GENERATED FROM PYTHON SOURCE LINES 123-124

Choice model.

.. GENERATED FROM PYTHON SOURCE LINES 124-134

.. code-block:: default

    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 135-138

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 138-143

.. code-block:: default

    BETA_TIME_PT_CL = Beta('BETA_TIME_PT_CL', -0.01, None, None, 0)
    BETA_TIME_PT = BETA_TIME_PT_REF * exp(BETA_TIME_PT_CL * CARLOVERS)
    BETA_TIME_CAR_CL = Beta('BETA_TIME_CAR_CL', -0.01, None, None, 0)
    BETA_TIME_CAR = BETA_TIME_CAR_REF * exp(BETA_TIME_CAR_CL * CARLOVERS)








.. GENERATED FROM PYTHON SOURCE LINES 144-145

Definition of utility functions:.

.. GENERATED FROM PYTHON SOURCE LINES 145-164

.. code-block:: default


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


    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 165-166

Associate utility functions with the numbering of alternatives.

.. GENERATED FROM PYTHON SOURCE LINES 166-168

.. code-block:: default

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








.. GENERATED FROM PYTHON SOURCE LINES 169-170

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

.. GENERATED FROM PYTHON SOURCE LINES 170-172

.. code-block:: default

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








.. GENERATED FROM PYTHON SOURCE LINES 173-174

We integrate over omega using numerical integration.

.. GENERATED FROM PYTHON SOURCE LINES 174-176

.. code-block:: default

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








.. GENERATED FROM PYTHON SOURCE LINES 177-178

Create the Biogeme object.

.. GENERATED FROM PYTHON SOURCE LINES 178-181

.. code-block:: default

    the_biogeme = bio.BIOGEME(database, loglike)
    the_biogeme.modelName = 'b04latent_choice_seq'





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

 .. code-block:: none

    File biogeme.toml has been parsed. 




.. GENERATED FROM PYTHON SOURCE LINES 182-184

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 184-186

.. code-block:: default

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








.. GENERATED FROM PYTHON SOURCE LINES 187-191

.. code-block:: default

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





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

 .. code-block:: none

    Estimated betas: 11
    Final log likelihood: -1094.545
    Output file: b04latent_choice_seq.html




.. GENERATED FROM PYTHON SOURCE LINES 192-193

.. code-block:: default

    results.getEstimatedParameters()





.. 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>0.729886</td>
          <td>0.124347</td>
          <td>5.869732</td>
          <td>4.364996e-09</td>
        </tr>
        <tr>
          <th>ASC_SM</th>
          <td>1.774204</td>
          <td>0.232349</td>
          <td>7.635957</td>
          <td>2.242651e-14</td>
        </tr>
        <tr>
          <th>BETA_COST_HWH</th>
          <td>-1.787025</td>
          <td>0.510332</td>
          <td>-3.501691</td>
          <td>4.623148e-04</td>
        </tr>
        <tr>
          <th>BETA_COST_OTHER</th>
          <td>-0.829724</td>
          <td>0.233819</td>
          <td>-3.548573</td>
          <td>3.873240e-04</td>
        </tr>
        <tr>
          <th>BETA_DIST</th>
          <td>-5.397874</td>
          <td>0.618156</td>
          <td>-8.732226</td>
          <td>0.000000e+00</td>
        </tr>
        <tr>
          <th>BETA_TIME_CAR_CL</th>
          <td>-1.395039</td>
          <td>0.064582</td>
          <td>-21.601098</td>
          <td>0.000000e+00</td>
        </tr>
        <tr>
          <th>BETA_TIME_CAR_REF</th>
          <td>-17.348357</td>
          <td>2.309146</td>
          <td>-7.512890</td>
          <td>5.773160e-14</td>
        </tr>
        <tr>
          <th>BETA_TIME_PT_CL</th>
          <td>-1.007749</td>
          <td>0.056733</td>
          <td>-17.763026</td>
          <td>0.000000e+00</td>
        </tr>
        <tr>
          <th>BETA_TIME_PT_REF</th>
          <td>-6.329932</td>
          <td>0.909073</td>
          <td>-6.963061</td>
          <td>3.329559e-12</td>
        </tr>
        <tr>
          <th>BETA_WAITING_TIME</th>
          <td>-0.027669</td>
          <td>0.014990</td>
          <td>-1.845835</td>
          <td>6.491620e-02</td>
        </tr>
        <tr>
          <th>sigma_s</th>
          <td>0.963082</td>
          <td>0.031482</td>
          <td>30.591360</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 0.143 seconds)


.. _sphx_glr_download_auto_examples_latent_plot_b04latent_choice_seq.py:

.. only:: html

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

      :download:`Download Python source code: plot_b04latent_choice_seq.py <plot_b04latent_choice_seq.py>`

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      :download:`Download Jupyter notebook: plot_b04latent_choice_seq.ipynb <plot_b04latent_choice_seq.ipynb>`


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