.. 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:: Python


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

    from biogeme.data.optima import (
        read_data,
        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:: Python

    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:: Python

    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:: Python

    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:: Python

    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:: Python

    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:: Python

    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:: Python

    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:: Python

    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:: Python

    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:: Python

    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

Read the data

.. GENERATED FROM PYTHON SOURCE LINES 129-131

.. code-block:: Python

    database = read_data()








.. GENERATED FROM PYTHON SOURCE LINES 132-133

Create the Biogeme object

.. GENERATED FROM PYTHON SOURCE LINES 133-136

.. code-block:: Python

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





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

 .. code-block:: none

    Default values of the Biogeme parameters are used. 
    File biogeme.toml has been created 




.. GENERATED FROM PYTHON SOURCE LINES 137-138

Estimate the parameters

.. GENERATED FROM PYTHON SOURCE LINES 138-140

.. code-block:: Python

    results = biogeme.estimate()





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

 .. code-block:: none

    As the model is not too complex, we activate the calculation of second derivatives. If you want to change it, 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 __m01_latent_variable.iter 
    Cannot read file __m01_latent_variable.iter. Statement is ignored. 
    As the model is not too complex, we activate the calculation of second derivatives. If you want to change it, 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: Newton with trust region for simple bounds 
    Iter.     Function    Relgrad   Radius      Rho      
        0      2.2e+04        1.2      0.5        0    - 
        1      1.6e+04       0.38        5      1.1   ++ 
        2      1.6e+04       0.38      2.5      1.1    - 
        3      1.6e+04       0.38      1.2      1.1    - 
        4      1.6e+04       0.38     0.62      1.1    - 
        5      1.6e+04       0.38     0.31      1.1    - 
        6      1.5e+04       0.12     0.31     0.67    + 
        7      1.5e+04      0.055      3.1      1.2   ++ 
        8      1.5e+04      0.055      1.6      -20    - 
        9      1.5e+04      0.055     0.78     -1.5    - 
       10      1.4e+04      0.025     0.78      0.9    + 
       11      1.4e+04      0.009     0.78      0.6    + 
       12      1.4e+04      0.017     0.78     0.47    + 
       13      1.4e+04      0.017     0.39     -0.3    - 
       14      1.4e+04      0.011     0.39     0.46    + 
       15      1.4e+04    0.00068      3.9     0.99   ++ 
       16      1.4e+04    0.00068        2 -1.2e+02    - 
       17      1.4e+04    0.00068     0.98      -11    - 
       18      1.4e+04    0.00068     0.49    -0.79    - 
       19      1.4e+04    0.00061     0.49     0.72    + 
       20      1.4e+04    2.3e-05     0.49        1    + 
    Results saved in file m01_latent_variable.html 
    Results saved in file m01_latent_variable.pickle 




.. GENERATED FROM PYTHON SOURCE LINES 141-142

.. 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>coef_age_30_less</th>
          <td>0.411378</td>
          <td>0.148961</td>
          <td>2.761647</td>
          <td>0.005751</td>
        </tr>
        <tr>
          <th>coef_artisans</th>
          <td>-0.104168</td>
          <td>0.059503</td>
          <td>-1.750638</td>
          <td>0.080008</td>
        </tr>
        <tr>
          <th>coef_child_center</th>
          <td>0.188734</td>
          <td>0.050024</td>
          <td>3.772861</td>
          <td>0.000161</td>
        </tr>
        <tr>
          <th>coef_child_suburb</th>
          <td>0.114693</td>
          <td>0.037611</td>
          <td>3.049471</td>
          <td>0.002292</td>
        </tr>
        <tr>
          <th>coef_employees</th>
          <td>-0.048415</td>
          <td>0.032817</td>
          <td>-1.475281</td>
          <td>0.140137</td>
        </tr>
        <tr>
          <th>coef_haveChildren</th>
          <td>-0.054407</td>
          <td>0.027774</td>
          <td>-1.958916</td>
          <td>0.050123</td>
        </tr>
        <tr>
          <th>coef_highEducation</th>
          <td>-0.066652</td>
          <td>0.043719</td>
          <td>-1.524551</td>
          <td>0.127371</td>
        </tr>
        <tr>
          <th>coef_intercept</th>
          <td>-0.550604</td>
          <td>0.050882</td>
          <td>-10.821125</td>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>coef_male</th>
          <td>0.114986</td>
          <td>0.051108</td>
          <td>2.249873</td>
          <td>0.024457</td>
        </tr>
        <tr>
          <th>coeff_LifSty07</th>
          <td>1.141445</td>
          <td>0.361607</td>
          <td>3.156588</td>
          <td>0.001596</td>
        </tr>
        <tr>
          <th>coeff_LifSty10</th>
          <td>0.690129</td>
          <td>0.241688</td>
          <td>2.855456</td>
          <td>0.004298</td>
        </tr>
        <tr>
          <th>coeff_ResidCh04</th>
          <td>0.243809</td>
          <td>0.206352</td>
          <td>1.181520</td>
          <td>0.237396</td>
        </tr>
        <tr>
          <th>coeff_ResidCh05</th>
          <td>2.274781</td>
          <td>0.613173</td>
          <td>3.709854</td>
          <td>0.000207</td>
        </tr>
        <tr>
          <th>coeff_ResidCh06</th>
          <td>1.676242</td>
          <td>0.727597</td>
          <td>2.303804</td>
          <td>0.021234</td>
        </tr>
        <tr>
          <th>delta_1</th>
          <td>0.484421</td>
          <td>0.012086</td>
          <td>40.080263</td>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>delta_2</th>
          <td>1.017231</td>
          <td>0.024393</td>
          <td>41.701887</td>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>inter_LifSty07</th>
          <td>-0.416229</td>
          <td>0.171191</td>
          <td>-2.431372</td>
          <td>0.015042</td>
        </tr>
        <tr>
          <th>inter_LifSty10</th>
          <td>0.019140</td>
          <td>0.113146</td>
          <td>0.169159</td>
          <td>0.865672</td>
        </tr>
        <tr>
          <th>inter_ResidCh04</th>
          <td>0.185378</td>
          <td>0.096960</td>
          <td>1.911893</td>
          <td>0.055890</td>
        </tr>
        <tr>
          <th>inter_ResidCh05</th>
          <td>-0.810642</td>
          <td>0.287186</td>
          <td>-2.822704</td>
          <td>0.004762</td>
        </tr>
        <tr>
          <th>inter_ResidCh06</th>
          <td>0.168830</td>
          <td>0.336571</td>
          <td>0.501618</td>
          <td>0.615937</td>
        </tr>
        <tr>
          <th>sigma_star_LifSty07</th>
          <td>1.120757</td>
          <td>0.032652</td>
          <td>34.324640</td>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>sigma_star_LifSty10</th>
          <td>0.975216</td>
          <td>0.026142</td>
          <td>37.304279</td>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>sigma_star_ResidCh04</th>
          <td>0.939158</td>
          <td>0.024710</td>
          <td>38.007332</td>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>sigma_star_ResidCh05</th>
          <td>1.369354</td>
          <td>0.052638</td>
          <td>26.014470</td>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>sigma_star_ResidCh06</th>
          <td>1.361454</td>
          <td>0.038189</td>
          <td>35.649960</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 1.273 seconds)


.. _sphx_glr_download_auto_examples_latentbis_plot_m01_latent_variable.py:

.. only:: html

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

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

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

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

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

      :download:`Download zipped: plot_m01_latent_variable.zip <plot_m01_latent_variable.zip>`


.. only:: html

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