.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/swissmetro/plot_b05normal_mixture.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_swissmetro_plot_b05normal_mixture.py: .. _plot_b05normal_mixture: Mixture of logit models ======================= Example of a normal mixture of logit models, using Monte-Carlo integration. Michel Bierlaire, EPFL Wed Jun 18 2025, 11:28:46 .. GENERATED FROM PYTHON SOURCE LINES 12-20 .. code-block:: Python import biogeme.biogeme_logging as blog from IPython.core.display_functions import display from biogeme.biogeme import BIOGEME from biogeme.expressions import Beta, Draws, MonteCarlo, log from biogeme.models import logit from biogeme.results_processing import get_pandas_estimated_parameters .. GENERATED FROM PYTHON SOURCE LINES 21-22 See the data processing script: :ref:`swissmetro_data`. .. GENERATED FROM PYTHON SOURCE LINES 22-39 .. code-block:: Python from swissmetro_data import ( CAR_AV_SP, CAR_CO_SCALED, CAR_TT_SCALED, CHOICE, SM_AV, SM_COST_SCALED, SM_TT_SCALED, TRAIN_AV_SP, TRAIN_COST_SCALED, TRAIN_TT_SCALED, database, ) logger = blog.get_screen_logger(level=blog.INFO) logger.info('Example b05normal_mixtures.py') .. rst-class:: sphx-glr-script-out .. code-block:: none Example b05normal_mixtures.py .. GENERATED FROM PYTHON SOURCE LINES 40-41 Parameters to be estimated .. GENERATED FROM PYTHON SOURCE LINES 41-46 .. code-block:: Python asc_car = Beta('asc_car', 0, None, None, 0) asc_train = Beta('asc_train', 0, None, None, 0) asc_sm = Beta('asc_sm', 0, None, None, 1) b_cost = Beta('b_cost', 0, None, None, 0) .. GENERATED FROM PYTHON SOURCE LINES 47-49 Define a random parameter, normally distributed, designed to be used for Monte-Carlo simulation. .. GENERATED FROM PYTHON SOURCE LINES 49-51 .. code-block:: Python b_time = Beta('b_time', 0, None, None, 0) .. GENERATED FROM PYTHON SOURCE LINES 52-53 It is advised *not* to use 0 as starting value for the following parameter. .. GENERATED FROM PYTHON SOURCE LINES 53-56 .. code-block:: Python b_time_s = Beta('b_time_s', 1, None, None, 0) b_time_rnd = b_time + b_time_s * Draws('b_time_rnd', 'NORMAL') .. GENERATED FROM PYTHON SOURCE LINES 57-58 Definition of the utility functions. .. GENERATED FROM PYTHON SOURCE LINES 58-62 .. code-block:: Python v_train = asc_train + b_time_rnd * TRAIN_TT_SCALED + b_cost * TRAIN_COST_SCALED v_swissmetro = asc_sm + b_time_rnd * SM_TT_SCALED + b_cost * SM_COST_SCALED v_car = asc_car + b_time_rnd * CAR_TT_SCALED + b_cost * CAR_CO_SCALED .. GENERATED FROM PYTHON SOURCE LINES 63-64 Associate utility functions with the numbering of alternatives. .. GENERATED FROM PYTHON SOURCE LINES 64-66 .. code-block:: Python v = {1: v_train, 2: v_swissmetro, 3: v_car} .. GENERATED FROM PYTHON SOURCE LINES 67-68 Associate the availability conditions with the alternatives. .. GENERATED FROM PYTHON SOURCE LINES 68-70 .. code-block:: Python av = {1: TRAIN_AV_SP, 2: SM_AV, 3: CAR_AV_SP} .. GENERATED FROM PYTHON SOURCE LINES 71-72 Conditional to b_time_rnd, we have a logit model (called the kernel). .. GENERATED FROM PYTHON SOURCE LINES 72-74 .. code-block:: Python prob = logit(v, av, CHOICE) .. GENERATED FROM PYTHON SOURCE LINES 75-76 We integrate over b_time_rnd using Monte-Carlo. .. GENERATED FROM PYTHON SOURCE LINES 76-78 .. code-block:: Python logprob = log(MonteCarlo(prob)) .. GENERATED FROM PYTHON SOURCE LINES 79-80 These notes will be included as such in the report file. .. GENERATED FROM PYTHON SOURCE LINES 80-85 .. code-block:: Python USER_NOTES = ( 'Example of a mixture of logit models with three alternatives, ' 'approximated using Monte-Carlo integration.' ) .. GENERATED FROM PYTHON SOURCE LINES 86-87 Create the Biogeme object. .. GENERATED FROM PYTHON SOURCE LINES 87-92 .. code-block:: Python the_biogeme = BIOGEME( database, logprob, user_notes=USER_NOTES, number_of_draws=10000, seed=1223 ) the_biogeme.model_name = 'b05normal_mixture' .. rst-class:: sphx-glr-script-out .. code-block:: none Biogeme parameters read from biogeme.toml. .. GENERATED FROM PYTHON SOURCE LINES 93-95 .. code-block:: Python print(f'Number of draws: {the_biogeme.number_of_draws}') .. rst-class:: sphx-glr-script-out .. code-block:: none Number of draws: 10000 .. GENERATED FROM PYTHON SOURCE LINES 96-97 Estimate the parameters .. GENERATED FROM PYTHON SOURCE LINES 97-99 .. code-block:: Python results = the_biogeme.estimate() .. rst-class:: sphx-glr-script-out .. code-block:: none *** Initial values of the parameters are obtained from the file __b05normal_mixture.iter Parameter values restored from __b05normal_mixture.iter Starting values for the algorithm: {'asc_train': -0.4026421286359191, 'b_time': -2.256833276642447, 'b_time_s': 1.654339197707796, 'b_cost': -1.2846226483353265, 'asc_car': 0.13632028087892392} As the model is rather complex, we cancel the calculation of second derivatives. If you want to control the parameters, change the algorithm from "automatic" to "simple_bounds" in the TOML file. Optimization algorithm: hybrid Newton/BFGS with simple bounds [simple_bounds] ** Optimization: BFGS with trust region for simple bounds Optimization algorithm has converged. Relative gradient: 5.906997030385587e-06 Cause of termination: Relative gradient = 5.9e-06 <= 6.1e-06 Number of function evaluations: 1 Number of gradient evaluations: 1 Number of hessian evaluations: 0 Algorithm: BFGS with trust region for simple bound constraints Number of iterations: 0 Optimization time: 0:00:02.141733 Calculate second derivatives and BHHH File b05normal_mixture~00.html has been generated. File b05normal_mixture~00.yaml has been generated. .. GENERATED FROM PYTHON SOURCE LINES 100-102 .. code-block:: Python print(results.short_summary()) .. rst-class:: sphx-glr-script-out .. code-block:: none Results for model b05normal_mixture Nbr of parameters: 5 Sample size: 6768 Excluded data: 3960 Final log likelihood: -5215.694 Akaike Information Criterion: 10441.39 Bayesian Information Criterion: 10475.49 .. GENERATED FROM PYTHON SOURCE LINES 103-105 .. code-block:: Python pandas_results = get_pandas_estimated_parameters(estimation_results=results) display(pandas_results) .. rst-class:: sphx-glr-script-out .. code-block:: none Name Value Robust std err. Robust t-stat. Robust p-value 0 asc_train -0.402642 0.065846 -6.114866 9.663816e-10 1 b_time -2.256833 0.117184 -19.258931 0.000000e+00 2 b_time_s 1.654339 0.131836 12.548455 0.000000e+00 3 b_cost -1.284623 0.086250 -14.894150 0.000000e+00 4 asc_car 0.136320 0.051755 2.633961 8.439514e-03 .. rst-class:: sphx-glr-timing **Total running time of the script:** (1 minutes 15.146 seconds) .. _sphx_glr_download_auto_examples_swissmetro_plot_b05normal_mixture.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_b05normal_mixture.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_b05normal_mixture.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_b05normal_mixture.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_