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

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

.. _sphx_glr_auto_examples_programmers_plot_biogeme.py:


biogeme.biogeme
===============

Examples of use of several functions.

This is designed for programmers who need examples of use of the
functions of the module. The examples are designed to illustrate the
syntax. They do not correspond to any meaningful model.

Michel Bierlaire
Sun Jun 29 2025, 01:14:48

.. GENERATED FROM PYTHON SOURCE LINES 15-32

.. code-block:: Python


    import pandas as pd
    from IPython.core.display_functions import display

    import biogeme.biogeme_logging as blog
    from biogeme.biogeme import BIOGEME
    from biogeme.calculator import evaluate_formula
    from biogeme.database import Database
    from biogeme.expressions import Beta, Variable, exp
    from biogeme.function_output import FunctionOutput
    from biogeme.results_processing import get_pandas_estimated_parameters
    from biogeme.second_derivatives import SecondDerivativesMode
    from biogeme.tools import CheckDerivativesResults
    from biogeme.tools.files import files_of_type
    from biogeme.validation import ValidationResult
    from biogeme.version import get_text








.. GENERATED FROM PYTHON SOURCE LINES 33-34

Version of Biogeme.

.. GENERATED FROM PYTHON SOURCE LINES 34-37

.. code-block:: Python

    print(get_text())






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

 .. code-block:: none

    biogeme 3.3.1 [2025-09-03]
    Home page: http://biogeme.epfl.ch
    Submit questions to https://groups.google.com/d/forum/biogeme
    Michel Bierlaire, Transport and Mobility Laboratory, Ecole Polytechnique Fédérale de Lausanne (EPFL)





.. GENERATED FROM PYTHON SOURCE LINES 38-39

Logger.

.. GENERATED FROM PYTHON SOURCE LINES 39-43

.. code-block:: Python

    logger = blog.get_screen_logger(level=blog.INFO)
    logger.info('Logger initialized')






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

 .. code-block:: none

    Logger initialized 




.. GENERATED FROM PYTHON SOURCE LINES 44-45

Definition of a database

.. GENERATED FROM PYTHON SOURCE LINES 45-59

.. code-block:: Python

    df = pd.DataFrame(
        {
            'Person': [1, 1, 1, 2, 2],
            'Exclude': [0, 0, 1, 0, 1],
            'Variable1': [1, 2, 3, 4, 5],
            'Variable2': [10, 20, 30, 40, 50],
            'Choice': [1, 2, 3, 1, 2],
            'Av1': [0, 1, 1, 1, 1],
            'Av2': [1, 1, 1, 1, 1],
            'Av3': [0, 1, 1, 1, 1],
        }
    )
    my_data = Database('test', df)








.. GENERATED FROM PYTHON SOURCE LINES 60-61

Data

.. GENERATED FROM PYTHON SOURCE LINES 61-64

.. code-block:: Python

    display(my_data.dataframe)






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

 .. code-block:: none

       Person  Exclude  Variable1  Variable2  Choice  Av1  Av2  Av3
    0     1.0      0.0        1.0       10.0     1.0  0.0  1.0  0.0
    1     1.0      0.0        2.0       20.0     2.0  1.0  1.0  1.0
    2     1.0      1.0        3.0       30.0     3.0  1.0  1.0  1.0
    3     2.0      0.0        4.0       40.0     1.0  1.0  1.0  1.0
    4     2.0      1.0        5.0       50.0     2.0  1.0  1.0  1.0




.. GENERATED FROM PYTHON SOURCE LINES 65-66

Definition of various expressions.

.. GENERATED FROM PYTHON SOURCE LINES 66-74

.. code-block:: Python

    Variable1 = Variable('Variable1')
    Variable2 = Variable('Variable2')
    beta1 = Beta('beta1', -1.0, -3, 3, 0)
    beta2 = Beta('beta2', 2.0, -3, 10, 0)
    likelihood = -(beta1**2) * Variable1 - exp(beta2 * beta1) * Variable2 - beta2**4
    simul = beta1 / Variable1 + beta2 / Variable2
    dict_of_expressions = {'log_like': likelihood, 'beta1': beta1, 'simul': simul}








.. GENERATED FROM PYTHON SOURCE LINES 75-76

Creation of the BIOGEME object.

.. GENERATED FROM PYTHON SOURCE LINES 76-80

.. code-block:: Python

    my_biogeme = BIOGEME(my_data, dict_of_expressions)
    my_biogeme.model_name = 'simple_example'
    print(my_biogeme)





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

 .. code-block:: none

    Biogeme parameters read from biogeme.toml. 
    simple_example: database [test]{'log_like': (((UnaryMinus(PowerConstant(<Beta name=beta1 value=-1.0 status=0>, 2.0)) * <Variable name=Variable1>) - (exp((<Beta name=beta2 value=2.0 status=0> * <Beta name=beta1 value=-1.0 status=0>)) * <Variable name=Variable2>)) - PowerConstant(<Beta name=beta2 value=2.0 status=0>, 4.0)), 'beta1': <Beta name=beta1 value=-1.0 status=0>, 'simul': ((<Beta name=beta1 value=-1.0 status=0> / <Variable name=Variable1>) + (<Beta name=beta2 value=2.0 status=0> / <Variable name=Variable2>))}




.. GENERATED FROM PYTHON SOURCE LINES 81-82

The data is stored in the Biogeme object.

.. GENERATED FROM PYTHON SOURCE LINES 82-84

.. code-block:: Python

    display(my_biogeme.database.dataframe)





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

 .. code-block:: none

       Person  Exclude  Variable1  Variable2  Choice  Av1  Av2  Av3
    0     1.0      0.0        1.0       10.0     1.0  0.0  1.0  0.0
    1     1.0      0.0        2.0       20.0     2.0  1.0  1.0  1.0
    2     1.0      1.0        3.0       30.0     3.0  1.0  1.0  1.0
    3     2.0      0.0        4.0       40.0     1.0  1.0  1.0  1.0
    4     2.0      1.0        5.0       50.0     2.0  1.0  1.0  1.0




.. GENERATED FROM PYTHON SOURCE LINES 85-86

Log likelihood with the initial values of the parameters.

.. GENERATED FROM PYTHON SOURCE LINES 86-88

.. code-block:: Python

    my_biogeme.calculate_init_likelihood()





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

 .. code-block:: none


    -115.30029248549191



.. GENERATED FROM PYTHON SOURCE LINES 89-91

Calculate the log-likelihood with a different value of the
parameters. We retrieve the current value and add 1 to each of them.

.. GENERATED FROM PYTHON SOURCE LINES 91-95

.. code-block:: Python

    x = my_biogeme.expressions_registry.free_betas_init_values
    x_plus = {key: value + 1.0 for key, value in x.items()}
    print(x_plus)





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

 .. code-block:: none

    {'beta1': 0.0, 'beta2': 3.0}




.. GENERATED FROM PYTHON SOURCE LINES 96-104

.. code-block:: Python

    log_likelihood_x_plus = evaluate_formula(
        model_elements=my_biogeme.model_elements,
        the_betas=x_plus,
        second_derivatives_mode=SecondDerivativesMode.NEVER,
        numerically_safe=False,
    )
    print(log_likelihood_x_plus)





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

 .. code-block:: none

    -555.0




.. GENERATED FROM PYTHON SOURCE LINES 105-106

Calculate the log-likelihood function and its derivatives.

.. GENERATED FROM PYTHON SOURCE LINES 106-113

.. code-block:: Python

    the_function_output: FunctionOutput = my_biogeme.function_evaluator.evaluate(
        the_betas=x_plus,
        gradient=True,
        hessian=True,
        bhhh=True,
    )








.. GENERATED FROM PYTHON SOURCE LINES 114-116

.. code-block:: Python

    print(f'f = {the_function_output.function}')





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

 .. code-block:: none

    f = -555.0




.. GENERATED FROM PYTHON SOURCE LINES 117-119

.. code-block:: Python

    print(f'g = {the_function_output.gradient}')





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

 .. code-block:: none

    g = [-450. -540.]




.. GENERATED FROM PYTHON SOURCE LINES 120-122

.. code-block:: Python

    pd.DataFrame(the_function_output.hessian)






.. 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>0</th>
          <th>1</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>0</th>
          <td>-1380.0</td>
          <td>-150.0</td>
        </tr>
        <tr>
          <th>1</th>
          <td>-150.0</td>
          <td>-540.0</td>
        </tr>
      </tbody>
    </table>
    </div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 123-126

.. code-block:: Python

    pd.DataFrame(the_function_output.bhhh)







.. 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>0</th>
          <th>1</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>0</th>
          <td>49500.0</td>
          <td>48600.0</td>
        </tr>
        <tr>
          <th>1</th>
          <td>48600.0</td>
          <td>58320.0</td>
        </tr>
      </tbody>
    </table>
    </div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 127-130

Check numerically the derivatives' implementation. The analytical
derivatives are compared to the numerical derivatives obtains by
finite differences.

.. GENERATED FROM PYTHON SOURCE LINES 130-132

.. code-block:: Python

    check_results: CheckDerivativesResults = my_biogeme.check_derivatives(verbose=True)





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

 .. code-block:: none

    Comparing first derivatives 
    x        Gradient    FinDiff    Difference
    beta1    -10.6006   -10.6006  -9.40832e-07
    beta2   -139.7     -139.7      3.80828e-06 
    Comparing second derivatives 
    Row    Col      Hessian    FinDiff    Difference
    beta1  beta1  -111.201   -111.201   -9.27991e-07
    beta2  beta1    20.3003    20.3003   1.42409e-06
    beta1  beta2    20.3003    20.3003  -2.4484e-07
    beta2  beta2  -260.3     -260.3      3.34428e-06 




.. GENERATED FROM PYTHON SOURCE LINES 133-134

.. code-block:: Python

    print(f'f = {check_results.function}')




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

 .. code-block:: none

    f = -115.30029248549191




.. GENERATED FROM PYTHON SOURCE LINES 135-136

.. code-block:: Python

    print(f'g = {check_results.analytical_gradient}')




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

 .. code-block:: none

    g = [ -10.60058497 -139.69970751]




.. GENERATED FROM PYTHON SOURCE LINES 137-138

.. code-block:: Python

    display(pd.DataFrame(check_results.analytical_hessian))




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

 .. code-block:: none

                0           1
    0 -111.201170   20.300292
    1   20.300292 -260.300292




.. GENERATED FROM PYTHON SOURCE LINES 139-140

.. code-block:: Python

    display(pd.DataFrame(check_results.finite_differences_gradient))




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

 .. code-block:: none

                0
    0  -10.600584
    1 -139.699711




.. GENERATED FROM PYTHON SOURCE LINES 141-143

.. code-block:: Python

    display(pd.DataFrame(check_results.finite_differences_hessian))





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

 .. code-block:: none

                0           1
    0 -111.201169   20.300293
    1   20.300291 -260.300296




.. GENERATED FROM PYTHON SOURCE LINES 144-146

Estimation
----------

.. GENERATED FROM PYTHON SOURCE LINES 148-149

Estimation of the parameters, with bootstrapping

.. GENERATED FROM PYTHON SOURCE LINES 149-151

.. code-block:: Python

    results = my_biogeme.estimate(run_bootstrap=True)





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

 .. code-block:: none

    *** Initial values of the parameters are obtained from the file __simple_example.iter 
    Parameter values restored from __simple_example.iter 
    Starting values for the algorithm: {'beta1': -1.273263987213694, 'beta2': 1.2487688099301162} 
    As the model is not too complex, we activate the calculation of second derivatives. To change this behavior, modify the algorithm to "simple_bounds" in the TOML file. 
    Optimization algorithm: hybrid Newton/BFGS with simple bounds [simple_bounds] 
    ** Optimization: Newton with trust region for simple bounds 
    Optimization algorithm has converged. 
    Relative gradient: 2.1439291582969576e-09 
    Cause of termination: Relative gradient = 2.1e-09 <= 6.1e-06 
    Number of function evaluations: 1 
    Number of gradient evaluations: 1 
    Number of hessian evaluations: 0 
    Algorithm: Newton with trust region for simple bound constraints 
    Number of iterations: 0 
    Optimization time: 0:00:00.001324 
    Calculate second derivatives and BHHH 
    Re-estimate the model 100 times for bootstrapping 

    Bootstraps:   0%|          | 0/100 [00:00<?, ?it/s]

      0%|          | 0/100 [00:00<?, ?it/s]

      1%|          | 1/100 [00:01<02:22,  1.44s/it]

      3%|▎         | 3/100 [00:01<00:41,  2.34it/s]

      6%|▌         | 6/100 [00:01<00:17,  5.29it/s]

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     10%|█         | 10/100 [00:01<00:10,  8.98it/s]

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    100%|██████████| 100/100 [00:06<00:00, 17.77it/s]
    100%|██████████| 100/100 [00:06<00:00, 14.43it/s]
    File simple_example~01.html has been generated. 
    File simple_example~01.yaml has been generated. 




.. GENERATED FROM PYTHON SOURCE LINES 152-155

.. code-block:: Python

    estimated_parameters = get_pandas_estimated_parameters(estimation_results=results)
    display(estimated_parameters)





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

 .. code-block:: none

        Name     Value  Bootstrap std err.  Bootstrap t-stat.  Bootstrap p-value
    0  beta1 -1.273264            0.012234        -104.079084                0.0
    1  beta2  1.248769            0.052923          23.595900                0.0




.. GENERATED FROM PYTHON SOURCE LINES 156-159

If the model has already been estimated, it is possible to recycle
the estimation results. In that case, the other arguments are
ignored, and the results are whatever is in the file.

.. GENERATED FROM PYTHON SOURCE LINES 161-163

.. code-block:: Python

    recycled_results = my_biogeme.estimate(recycle=True, run_bootstrap=True)





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

 .. code-block:: none

    Several files .yaml are available for this model: ['simple_example.yaml', 'simple_example~00.yaml', 'simple_example~01.yaml']. The file simple_example~01.yaml is used to load the results. 

    Bootstraps:   0%|          | 0/100 [00:06<?, ?it/s]
    Estimation results read from simple_example~01.yaml. There is no guarantee that they correspond to the specified model. 




.. GENERATED FROM PYTHON SOURCE LINES 164-166

.. code-block:: Python

    print(recycled_results.short_summary())





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

 .. code-block:: none

    Results for model simple_example
    Nbr of parameters:              2
    Sample size:                    5
    Excluded data:                  0
    Final log likelihood:           -67.06549
    Akaike Information Criterion:   138.131
    Bayesian Information Criterion: 137.3499





.. GENERATED FROM PYTHON SOURCE LINES 167-173

.. code-block:: Python

    recycled_parameters = get_pandas_estimated_parameters(
        estimation_results=recycled_results
    )
    display(recycled_parameters)






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

 .. code-block:: none

        Name     Value  Bootstrap std err.  Bootstrap t-stat.  Bootstrap p-value
    0  beta1 -1.273264            0.012234        -104.079084                0.0
    1  beta2  1.248769            0.052923          23.595900                0.0




.. GENERATED FROM PYTHON SOURCE LINES 174-176

Simulation
----------

.. GENERATED FROM PYTHON SOURCE LINES 178-179

Simulate with the estimated values for the parameters.

.. GENERATED FROM PYTHON SOURCE LINES 181-183

.. code-block:: Python

    display(results.get_beta_values())





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

 .. code-block:: none

    {'beta1': -1.273263987213694, 'beta2': 1.2487688099301162}




.. GENERATED FROM PYTHON SOURCE LINES 184-187

.. code-block:: Python

    simulation_with_estimated_betas = my_biogeme.simulate(results.get_beta_values())
    display(simulation_with_estimated_betas)





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

 .. code-block:: none

        log_like     beta1     simul
    0  -6.092234 -1.273264 -1.148387
    1  -9.752666 -1.273264 -0.574194
    2 -13.413098 -1.273264 -0.382796
    3 -17.073530 -1.273264 -0.287097
    4 -20.733962 -1.273264 -0.229677




.. GENERATED FROM PYTHON SOURCE LINES 188-190

Confidence intervals.
First, we extract the values of betas from the bootstrapping draws.

.. GENERATED FROM PYTHON SOURCE LINES 190-194

.. code-block:: Python

    draws_from_betas = results.get_betas_for_sensitivity_analysis()
    for draw in draws_from_betas:
        print(draw)





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

 .. code-block:: none

    {'beta1': -1.2504079848560654, 'beta2': 1.346111255894925}
    {'beta1': -1.2649797742013058, 'beta2': 1.2842631765105266}
    {'beta1': -1.2690260405244644, 'beta2': 1.2669668838392423}
    {'beta1': -1.2649797742013058, 'beta2': 1.2842631765105266}
    {'beta1': -1.287332533620085, 'beta2': 1.1875198317198776}
    {'beta1': -1.273263987213694, 'beta2': 1.2487688099301162}
    {'beta1': -1.287332533620085, 'beta2': 1.1875198317198776}
    {'beta1': -1.2777126116342759, 'beta2': 1.2295568140731905}
    {'beta1': -1.287332533620085, 'beta2': 1.1875198317198776}
    {'beta1': -1.282393736283639, 'beta2': 1.209195603860136}
    {'beta1': -1.2649797742013058, 'beta2': 1.2842631765105266}
    {'beta1': -1.2925578214686664, 'beta2': 1.164322217510277}
    {'beta1': -1.2573978799513064, 'beta2': 1.3165120810083486}
    {'beta1': -1.287332533620085, 'beta2': 1.1875198317198776}
    {'beta1': -1.2611085813010015, 'beta2': 1.3007517002098443}
    {'beta1': -1.2573978799513064, 'beta2': 1.3165120810083486}
    {'beta1': -1.2611085813010015, 'beta2': 1.3007517002098443}
    {'beta1': -1.2573978799513064, 'beta2': 1.3165120810083486}
    {'beta1': -1.2471072737805293, 'beta2': 1.3600596419678224}
    {'beta1': -1.2649797742013058, 'beta2': 1.2842631765105266}
    {'beta1': -1.2611085813010015, 'beta2': 1.3007517002098443}
    {'beta1': -1.2690260405244644, 'beta2': 1.2669668838392423}
    {'beta1': -1.282393736283639, 'beta2': 1.209195603860136}
    {'beta1': -1.2690260405244644, 'beta2': 1.2669668838392423}
    {'beta1': -1.282393736283639, 'beta2': 1.209195603860136}
    {'beta1': -1.2611085813010015, 'beta2': 1.3007517002098443}
    {'beta1': -1.2573978799513064, 'beta2': 1.3165120810083486}
    {'beta1': -1.287332533620085, 'beta2': 1.1875198317198776}
    {'beta1': -1.287332533620085, 'beta2': 1.1875198317198776}
    {'beta1': -1.2573978799513064, 'beta2': 1.3165120810083486}
    {'beta1': -1.2649797742013058, 'beta2': 1.2842631765105266}
    {'beta1': -1.2777126116342759, 'beta2': 1.2295568140731905}
    {'beta1': -1.2611085813010015, 'beta2': 1.3007517002098443}
    {'beta1': -1.2573978799513064, 'beta2': 1.3165120810083486}
    {'beta1': -1.273263987213694, 'beta2': 1.2487688099301162}
    {'beta1': -1.2649797742013058, 'beta2': 1.2842631765105266}
    {'beta1': -1.273263987213694, 'beta2': 1.2487688099301162}
    {'beta1': -1.2573978799513064, 'beta2': 1.3165120810083486}
    {'beta1': -1.2925578214686664, 'beta2': 1.164322217510277}
    {'beta1': -1.2611085813010015, 'beta2': 1.3007517002098443}
    {'beta1': -1.2690260405244644, 'beta2': 1.2669668838392423}
    {'beta1': -1.2649797740194353, 'beta2': 1.2842631510962423}
    {'beta1': -1.287332533620085, 'beta2': 1.1875198317198776}
    {'beta1': -1.287332533620085, 'beta2': 1.1875198317198776}
    {'beta1': -1.2649797742013058, 'beta2': 1.2842631765105266}
    {'beta1': -1.2732639874527638, 'beta2': 1.2487688498678546}
    {'beta1': -1.2690260405244644, 'beta2': 1.2669668838392423}
    {'beta1': -1.2732639874991436, 'beta2': 1.2487688117902658}
    {'beta1': -1.2777126116342759, 'beta2': 1.2295568140731905}
    {'beta1': -1.2873325027193596, 'beta2': 1.1875194143603178}
    {'beta1': -1.2777126116342759, 'beta2': 1.2295568140731905}
    {'beta1': -1.2690260418393766, 'beta2': 1.2669669182885877}
    {'beta1': -1.2690260405244644, 'beta2': 1.2669668838392423}
    {'beta1': -1.2732639874991436, 'beta2': 1.2487688117902658}
    {'beta1': -1.273263987213694, 'beta2': 1.2487688099301162}
    {'beta1': -1.2649797742013058, 'beta2': 1.2842631765105266}
    {'beta1': -1.282393736283639, 'beta2': 1.209195603860136}
    {'beta1': -1.2777126118096789, 'beta2': 1.229556814569854}
    {'beta1': -1.2573978799513064, 'beta2': 1.3165120810083486}
    {'beta1': -1.277712593402649, 'beta2': 1.2295586250129056}
    {'beta1': -1.287332533620085, 'beta2': 1.1875198317198776}
    {'beta1': -1.2732640117680483, 'beta2': 1.2487690786447256}
    {'beta1': -1.2649797742013058, 'beta2': 1.2842631765105266}
    {'beta1': -1.2823937666074936, 'beta2': 1.2091966201417774}
    {'beta1': -1.282393736283639, 'beta2': 1.209195603860136}
    {'beta1': -1.2925579457232135, 'beta2': 1.1643223694950584}
    {'beta1': -1.2611085813010015, 'beta2': 1.3007517002098443}
    {'beta1': -1.2573978837738118, 'beta2': 1.3165118213080087}
    {'beta1': -1.273263987213694, 'beta2': 1.2487688099301162}
    {'beta1': -1.2649797742013058, 'beta2': 1.2842631765105266}
    {'beta1': -1.298103506705179, 'beta2': 1.1393434542527645}
    {'beta1': -1.2649797740398232, 'beta2': 1.2842631494771497}
    {'beta1': -1.2504079848560654, 'beta2': 1.346111255894925}
    {'beta1': -1.2823937314558564, 'beta2': 1.2091955251477786}
    {'beta1': -1.2925578214686664, 'beta2': 1.164322217510277}
    {'beta1': -1.2873324933528516, 'beta2': 1.187519309859021}
    {'beta1': -1.273263987213694, 'beta2': 1.2487688099301162}
    {'beta1': -1.2925578214686664, 'beta2': 1.164322217510277}
    {'beta1': -1.2777126116342759, 'beta2': 1.2295568140731905}
    {'beta1': -1.2649797777101763, 'beta2': 1.2842632913325391}
    {'beta1': -1.304007541673126, 'beta2': 1.1122455742287705}
    {'beta1': -1.25383482258777, 'beta2': 1.3316118845742497}
    {'beta1': -1.2611085813010015, 'beta2': 1.3007517002098443}
    {'beta1': -1.26497977397441, 'beta2': 1.2842631509636993}
    {'beta1': -1.2777126116342759, 'beta2': 1.2295568140731905}
    {'beta1': -1.2649797777101763, 'beta2': 1.2842632913325391}
    {'beta1': -1.2690260405244644, 'beta2': 1.2669668838392423}
    {'beta1': -1.2690260405244644, 'beta2': 1.2669668838392423}
    {'beta1': -1.2777126116342759, 'beta2': 1.2295568140731905}
    {'beta1': -1.2823937318606777, 'beta2': 1.2091955303777047}
    {'beta1': -1.2573978799513064, 'beta2': 1.3165120810083486}
    {'beta1': -1.2732639720980743, 'beta2': 1.2487693919282534}
    {'beta1': -1.2649797742013058, 'beta2': 1.2842631765105266}
    {'beta1': -1.2981035067539988, 'beta2': 1.1393434544494636}
    {'beta1': -1.2777126116342759, 'beta2': 1.2295568140731905}
    {'beta1': -1.2611085873267327, 'beta2': 1.3007519381922703}
    {'beta1': -1.282393736283639, 'beta2': 1.209195603860136}
    {'beta1': -1.287332492839996, 'beta2': 1.1875193084801916}
    {'beta1': -1.287332533620085, 'beta2': 1.1875198317198776}
    {'beta1': -1.2649798555111198, 'beta2': 1.2842643064389756}




.. GENERATED FROM PYTHON SOURCE LINES 195-197

Then, we calculate the confidence intervals. The default interval
size is 0.9. Here, we use a different one.

.. GENERATED FROM PYTHON SOURCE LINES 197-200

.. code-block:: Python

    left, right = my_biogeme.confidence_intervals(draws_from_betas, interval_size=0.95)
    display(left)





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

 .. code-block:: none


      0%|          | 0/100 [00:00<?, ?it/s]
      4%|▍         | 4/100 [00:00<00:02, 38.62it/s]
      9%|▉         | 9/100 [00:00<00:02, 39.43it/s]
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     87%|████████▋ | 87/100 [00:02<00:00, 36.30it/s]
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    100%|██████████| 100/100 [00:02<00:00, 38.69it/s]
        log_like     beta1     simul
    0  -6.654739 -1.295469 -1.180349
    1 -10.092196 -1.295469 -0.590174
    2 -13.576352 -1.295469 -0.393450
    3 -17.474367 -1.295469 -0.295087
    4 -21.403557 -1.295469 -0.236070




.. GENERATED FROM PYTHON SOURCE LINES 201-203

.. code-block:: Python

    display(right)





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

 .. code-block:: none

        log_like     beta1     simul
    0  -5.686797 -1.252036 -1.118113
    1  -9.619739 -1.252036 -0.559057
    2 -13.413098 -1.252036 -0.372704
    3 -16.968836 -1.252036 -0.279528
    4 -20.404569 -1.252036 -0.223623




.. GENERATED FROM PYTHON SOURCE LINES 204-214

Validation
----------
The validation consists in organizing the data into several slices
of about the same size, randomly defined.  Each slide is considered
as a validation dataset. The model is then re-estimated using all
the data except the slice, and the estimated model is applied on the
validation set (i.e. the slice). The value of the log likelihood for
each observation in the validation set is reported in a
dataframe. As this is done for each slice, the output is a list of
dataframes, each corresponding to one of these exercises.

.. GENERATED FROM PYTHON SOURCE LINES 216-218

.. code-block:: Python

    validation_results: list[ValidationResult] = my_biogeme.validate(results, slices=5)





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

 .. code-block:: none

    Optimization algorithm: hybrid Newton/BFGS with simple bounds [simple_bounds] 
    ** Optimization: Newton with trust region for simple bounds 
    Iter.           beta1           beta2     Function    Relgrad   Radius      Rho      
        0            -1.3             1.2           46     0.0022       10        1   ++ 
        1            -1.3             1.2           46      6e-07       10        1   ++ 
    Optimization algorithm: hybrid Newton/BFGS with simple bounds [simple_bounds] 
    ** Optimization: Newton with trust region for simple bounds 
    Iter.           beta1           beta2     Function    Relgrad   Radius      Rho      
        0            -1.3             1.2           50    0.00051       10        1   ++ 
        1            -1.3             1.2           50    3.1e-08       10        1   ++ 
    Optimization algorithm: hybrid Newton/BFGS with simple bounds [simple_bounds] 
    ** Optimization: Newton with trust region for simple bounds 
    Iter.           beta1           beta2     Function    Relgrad   Radius      Rho      
        0            -1.3             1.3           61     0.0028       10     0.99   ++ 
        1            -1.3             1.3           61      8e-07       10        1   ++ 
    Optimization algorithm: hybrid Newton/BFGS with simple bounds [simple_bounds] 
    ** Optimization: Newton with trust region for simple bounds 
    Iter.           beta1           beta2     Function    Relgrad   Radius      Rho      
        0            -1.3             1.2           54     0.0015       10        1   ++ 
        1            -1.3             1.2           54    2.6e-07       10        1   ++ 
    Optimization algorithm: hybrid Newton/BFGS with simple bounds [simple_bounds] 
    ** Optimization: Newton with trust region for simple bounds 
    Iter.           beta1           beta2     Function    Relgrad   Radius      Rho      
        0            -1.3             1.3           57    0.00035       10        1   ++ 
        1            -1.3             1.3           57    1.3e-08       10        1   ++ 




.. GENERATED FROM PYTHON SOURCE LINES 219-226

.. code-block:: Python

    for one_validation_result in validation_results:
        print(
            f'Log likelihood for {one_validation_result.validation_modeling_elements.sample_size} '
            f'validation data: {one_validation_result.simulated_values.iloc[0].sum()}'
        )






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

 .. code-block:: none

    Log likelihood for 1 validation data: -22.555245733904922
    Log likelihood for 1 validation data: -18.713287155490335
    Log likelihood for 1 validation data: -8.737896466562866
    Log likelihood for 1 validation data: -15.069157781386547
    Log likelihood for 1 validation data: -11.656146972777968




.. GENERATED FROM PYTHON SOURCE LINES 227-229

The following tools is used to obtain the list of files with a given extension in the
local directory.

.. GENERATED FROM PYTHON SOURCE LINES 229-230

.. code-block:: Python

    display(files_of_type(extension='yaml', name=my_biogeme.model_name))




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

 .. code-block:: none

    ['simple_example.yaml', 'simple_example~00.yaml', 'simple_example~01.yaml']





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

   **Total running time of the script:** (0 minutes 10.865 seconds)


.. _sphx_glr_download_auto_examples_programmers_plot_biogeme.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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