.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/latentbis/plot_m01_latent_variable.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_latentbis_plot_m01_latent_variable.py>`
        to download the full example code

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

.. _sphx_glr_auto_examples_latentbis_plot_m01_latent_variable.py:


Measurement equations: discrete indicators
==========================================

Ordered probit.

:author: Michel Bierlaire, EPFL
:date: Fri Apr 14 09:39:10 2023

.. GENERATED FROM PYTHON SOURCE LINES 12-38

.. code-block:: default


    import biogeme.biogeme_logging as blog
    import biogeme.biogeme as bio
    from biogeme.expressions import (
        Beta,
        log,
        Elem,
        bioNormalCdf,
        Variable,
        bioMultSum,
    )

    from optima import (
        database,
        male,
        age,
        haveChildren,
        highEducation,
        childCenter,
        childSuburb,
        SocioProfCat,
    )

    logger = blog.get_screen_logger(level=blog.INFO)
    logger.info('Example m01_latent_variable.py')





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

 .. code-block:: none

    Example m01_latent_variable.py 




.. GENERATED FROM PYTHON SOURCE LINES 39-40

Parameters for the structural equation

.. GENERATED FROM PYTHON SOURCE LINES 40-50

.. code-block:: default

    coef_intercept = Beta('coef_intercept', 0.0, None, None, 0)
    coef_age_30_less = Beta('coef_age_30_less', 0.0, None, None, 0)
    coef_male = Beta('coef_male', 0.0, None, None, 0)
    coef_haveChildren = Beta('coef_haveChildren', 0.0, None, None, 0)
    coef_highEducation = Beta('coef_highEducation', 0.0, None, None, 0)
    coef_artisans = Beta('coef_artisans', 0.0, None, None, 0)
    coef_employees = Beta('coef_employees', 0.0, None, None, 0)
    coef_child_center = Beta('coef_child_center', 0.0, None, None, 0)
    coef_child_suburb = Beta('coef_child_suburb', 0.0, None, None, 0)








.. GENERATED FROM PYTHON SOURCE LINES 51-52

Latent variable: structural equation

.. GENERATED FROM PYTHON SOURCE LINES 52-64

.. code-block:: default

    ACTIVELIFE = (
        coef_intercept
        + coef_child_center * childCenter
        + coef_child_suburb * childSuburb
        + coef_highEducation * highEducation
        + coef_artisans * (SocioProfCat == 5)
        + coef_employees * (SocioProfCat == 6)
        + coef_age_30_less * (age <= 30)
        + coef_male * male
        + coef_haveChildren * haveChildren
    )








.. GENERATED FROM PYTHON SOURCE LINES 65-66

Measurement equations

.. GENERATED FROM PYTHON SOURCE LINES 66-75

.. code-block:: default

    indicators = [
        'ResidCh01',
        'ResidCh04',
        'ResidCh05',
        'ResidCh06',
        'LifSty07',
        'LifSty10',
    ]








.. GENERATED FROM PYTHON SOURCE LINES 76-77

We define the intercept parameters. The first one is normalized to 0.

.. GENERATED FROM PYTHON SOURCE LINES 77-80

.. code-block:: default

    inter = {k: Beta(f'inter_{k}', 0, None, None, 0) for k in indicators[1:]}
    inter[indicators[0]] = Beta(f'INTER_{indicators[0]}', 0, None, None, 1)








.. GENERATED FROM PYTHON SOURCE LINES 81-82

We define the coefficients. The first one is normalized to 1.

.. GENERATED FROM PYTHON SOURCE LINES 82-85

.. code-block:: default

    coefficients = {k: Beta(f'coeff_{k}', 0, None, None, 0) for k in indicators[1:]}
    coefficients[indicators[0]] = Beta(f'B_{indicators[0]}', 1, None, None, 1)








.. GENERATED FROM PYTHON SOURCE LINES 86-87

We define the measurement equations for each indicator

.. GENERATED FROM PYTHON SOURCE LINES 87-89

.. code-block:: default

    models = {k: inter[k] + coefficients[k] * ACTIVELIFE for k in indicators}








.. GENERATED FROM PYTHON SOURCE LINES 90-91

We define the scale parameters of the error terms.

.. GENERATED FROM PYTHON SOURCE LINES 91-94

.. code-block:: default

    sigma_star = {k: Beta(f'sigma_star_{k}', 1, 1.0e-5, None, 0) for k in indicators[1:]}
    sigma_star[indicators[0]] = Beta(f'sigma_star_{indicators[0]}', 1, None, None, 1)








.. GENERATED FROM PYTHON SOURCE LINES 95-96

Symmetric threshold.

.. GENERATED FROM PYTHON SOURCE LINES 96-103

.. code-block:: default

    delta_1 = Beta('delta_1', 0.1, 1.0e-5, None, 0)
    delta_2 = Beta('delta_2', 0.2, 1.0e-5, None, 0)
    tau_1 = -delta_1 - delta_2
    tau_2 = -delta_1
    tau_3 = delta_1
    tau_4 = delta_1 + delta_2








.. GENERATED FROM PYTHON SOURCE LINES 104-105

Ordered probit models.

.. GENERATED FROM PYTHON SOURCE LINES 105-123

.. code-block:: default

    tau_1_residual = {k: (tau_1 - models[k]) / sigma_star[k] for k in indicators}
    tau_2_residual = {k: (tau_2 - models[k]) / sigma_star[k] for k in indicators}
    tau_3_residual = {k: (tau_3 - models[k]) / sigma_star[k] for k in indicators}
    tau_4_residual = {k: (tau_4 - models[k]) / sigma_star[k] for k in indicators}
    dict_prob_indicators = {
        k: {
            1: bioNormalCdf(tau_1_residual[k]),
            2: bioNormalCdf(tau_2_residual[k]) - bioNormalCdf(tau_1_residual[k]),
            3: bioNormalCdf(tau_3_residual[k]) - bioNormalCdf(tau_2_residual[k]),
            4: bioNormalCdf(tau_4_residual[k]) - bioNormalCdf(tau_3_residual[k]),
            5: 1 - bioNormalCdf(tau_4_residual[k]),
            6: 1.0,
            -1: 1.0,
            -2: 1.0,
        }
        for k in indicators
    }








.. GENERATED FROM PYTHON SOURCE LINES 124-127

.. code-block:: default

    log_proba = {k: log(Elem(dict_prob_indicators[k], Variable(k))) for k in indicators}
    loglike = bioMultSum(log_proba)








.. GENERATED FROM PYTHON SOURCE LINES 128-129

Create the Biogeme object

.. GENERATED FROM PYTHON SOURCE LINES 129-132

.. code-block:: default

    biogeme = bio.BIOGEME(database, loglike)
    biogeme.modelName = 'm01_latent_variable'





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

 .. code-block:: none

    File biogeme.toml has been created 




.. GENERATED FROM PYTHON SOURCE LINES 133-134

Estimate the parameters

.. GENERATED FROM PYTHON SOURCE LINES 134-136

.. code-block:: default

    results = biogeme.estimate()





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

 .. code-block:: none

    *** Initial values of the parameters are obtained from the file __m01_latent_variable.iter 
    Cannot read file __m01_latent_variable.iter. Statement is ignored. 
    Optimization algorithm: hybrid Newton/BFGS with simple bounds [simple_bounds] 
    ** Optimization: Newton with trust region for simple bounds 
    Iter.     Function    Relgrad   Radius      Rho      
        0      2.2e+04        1.2      0.5 -2.1e+304    - 
        1      1.6e+04       0.38        5      1.1   ++ 
        2      1.6e+04       0.38     0.39 -8.9e+304    - 
        3      1.5e+04       0.23     0.39     0.35    + 
        4      1.5e+04      0.083      3.9      1.1   ++ 
        5      1.5e+04      0.083        2 -2.9e+305    - 
        6      1.5e+04      0.083     0.98       -6    - 
        7      1.5e+04       0.05     0.98      0.4    + 
        8      1.4e+04      0.022     0.98     0.61    + 
        9      1.4e+04      0.022     0.49    -0.38    - 
       10      1.4e+04       0.03     0.49     0.73    + 
       11      1.4e+04      0.017     0.49     0.74    + 
       12      1.4e+04      0.016     0.49     0.33    + 
       13      1.4e+04    0.00098      4.9        1   ++ 
       14      1.4e+04    0.00069      4.9     0.78    + 
       15      1.4e+04    5.9e-05       49     0.97   ++ 
       16      1.4e+04    8.5e-06  4.9e+02     0.99   ++ 
       17      1.4e+04    3.9e-06  4.9e+02        1   ++ 




.. GENERATED FROM PYTHON SOURCE LINES 137-138

.. 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>coef_age_30_less</th>
          <td>0.423594</td>
          <td>0.142712</td>
          <td>2.968171</td>
          <td>0.002996</td>
        </tr>
        <tr>
          <th>coef_artisans</th>
          <td>-0.105012</td>
          <td>0.061111</td>
          <td>-1.718386</td>
          <td>0.085726</td>
        </tr>
        <tr>
          <th>coef_child_center</th>
          <td>0.192517</td>
          <td>0.048943</td>
          <td>3.933479</td>
          <td>0.000084</td>
        </tr>
        <tr>
          <th>coef_child_suburb</th>
          <td>0.116267</td>
          <td>0.038895</td>
          <td>2.989274</td>
          <td>0.002796</td>
        </tr>
        <tr>
          <th>coef_employees</th>
          <td>-0.049326</td>
          <td>0.033410</td>
          <td>-1.476388</td>
          <td>0.139840</td>
        </tr>
        <tr>
          <th>coef_haveChildren</th>
          <td>-0.054718</td>
          <td>0.028698</td>
          <td>-1.906638</td>
          <td>0.056568</td>
        </tr>
        <tr>
          <th>coef_highEducation</th>
          <td>-0.068243</td>
          <td>0.043947</td>
          <td>-1.552854</td>
          <td>0.120458</td>
        </tr>
        <tr>
          <th>coef_intercept</th>
          <td>-0.552730</td>
          <td>0.050422</td>
          <td>-10.962108</td>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>coef_male</th>
          <td>0.117892</td>
          <td>0.050425</td>
          <td>2.337988</td>
          <td>0.019388</td>
        </tr>
        <tr>
          <th>coeff_LifSty07</th>
          <td>1.103010</td>
          <td>0.331175</td>
          <td>3.330590</td>
          <td>0.000867</td>
        </tr>
        <tr>
          <th>coeff_LifSty10</th>
          <td>0.669214</td>
          <td>0.224133</td>
          <td>2.985791</td>
          <td>0.002828</td>
        </tr>
        <tr>
          <th>coeff_ResidCh04</th>
          <td>0.240344</td>
          <td>0.202376</td>
          <td>1.187612</td>
          <td>0.234986</td>
        </tr>
        <tr>
          <th>coeff_ResidCh05</th>
          <td>2.208125</td>
          <td>0.548170</td>
          <td>4.028177</td>
          <td>0.000056</td>
        </tr>
        <tr>
          <th>coeff_ResidCh06</th>
          <td>1.624885</td>
          <td>0.673324</td>
          <td>2.413230</td>
          <td>0.015812</td>
        </tr>
        <tr>
          <th>delta_1</th>
          <td>0.484309</td>
          <td>0.012076</td>
          <td>40.105274</td>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>delta_2</th>
          <td>1.017076</td>
          <td>0.024382</td>
          <td>41.713938</td>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>inter_LifSty07</th>
          <td>-0.434497</td>
          <td>0.157059</td>
          <td>-2.766458</td>
          <td>0.005667</td>
        </tr>
        <tr>
          <th>inter_LifSty10</th>
          <td>0.009327</td>
          <td>0.105072</td>
          <td>0.088768</td>
          <td>0.929266</td>
        </tr>
        <tr>
          <th>inter_ResidCh04</th>
          <td>0.183459</td>
          <td>0.095071</td>
          <td>1.929707</td>
          <td>0.053643</td>
        </tr>
        <tr>
          <th>inter_ResidCh05</th>
          <td>-0.841766</td>
          <td>0.257460</td>
          <td>-3.269495</td>
          <td>0.001077</td>
        </tr>
        <tr>
          <th>inter_ResidCh06</th>
          <td>0.144285</td>
          <td>0.311306</td>
          <td>0.463483</td>
          <td>0.643018</td>
        </tr>
        <tr>
          <th>sigma_star_LifSty07</th>
          <td>1.120425</td>
          <td>0.032638</td>
          <td>34.328743</td>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>sigma_star_LifSty10</th>
          <td>0.974981</td>
          <td>0.026120</td>
          <td>37.326387</td>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>sigma_star_ResidCh04</th>
          <td>0.938857</td>
          <td>0.024690</td>
          <td>38.026541</td>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>sigma_star_ResidCh05</th>
          <td>1.368679</td>
          <td>0.052609</td>
          <td>26.016010</td>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>sigma_star_ResidCh06</th>
          <td>1.361357</td>
          <td>0.038153</td>
          <td>35.681186</td>
          <td>0.000000</td>
        </tr>
      </tbody>
    </table>
    </div>
    </div>
    <br />
    <br />


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 1.237 seconds)


.. _sphx_glr_download_auto_examples_latentbis_plot_m01_latent_variable.py:

.. only:: html

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




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

      :download:`Download Python source code: plot_m01_latent_variable.py <plot_m01_latent_variable.py>`

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      :download:`Download Jupyter notebook: plot_m01_latent_variable.ipynb <plot_m01_latent_variable.ipynb>`


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