.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/swissmetro/plot_b25triangular_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_b25triangular_mixture.py: Triangular mixture of logit =========================== Example of a mixture of logit models, using Monte-Carlo integration. The mixing distribution is specified by the user. Here, a triangular distribution. Michel Bierlaire, EPFL Sat Jun 28 2025, 12:49:10 .. GENERATED FROM PYTHON SOURCE LINES 15-25 .. code-block:: Python import biogeme.biogeme_logging as blog import numpy as np from IPython.core.display_functions import display from biogeme.biogeme import BIOGEME from biogeme.draws import RandomNumberGeneratorTuple 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 26-27 See the data processing script: :ref:`swissmetro_data`. .. GENERATED FROM PYTHON SOURCE LINES 27-44 .. 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 b25triangular_mixture.py') .. rst-class:: sphx-glr-script-out .. code-block:: none Example b25triangular_mixture.py .. GENERATED FROM PYTHON SOURCE LINES 45-46 Parameters to be estimated. .. GENERATED FROM PYTHON SOURCE LINES 46-51 .. 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 52-56 Define a random parameter with a triangular distribution, designed to be used for Monte-Carlo simulation. The triangular distribution is not directly available from Biogeme. The draws have to be generated by a function provided by the user. .. GENERATED FROM PYTHON SOURCE LINES 58-59 Mean of the distribution. .. GENERATED FROM PYTHON SOURCE LINES 59-61 .. code-block:: Python b_time = Beta('b_time', 0, None, None, 0) .. GENERATED FROM PYTHON SOURCE LINES 62-64 Scale of the distribution. It is advised not to use 0 as starting value for the following parameter. .. GENERATED FROM PYTHON SOURCE LINES 64-67 .. code-block:: Python b_time_s = Beta('b_time_s', 1, None, None, 0) .. GENERATED FROM PYTHON SOURCE LINES 68-69 Function generating the draws. .. GENERATED FROM PYTHON SOURCE LINES 69-77 .. code-block:: Python def the_triangular_generator(sample_size: int, number_of_draws: int) -> np.ndarray: """ User-defined random number generator to the database. See the `numpy.random` documentation to obtain a list of other distributions. """ return np.random.triangular(-1, 0, 1, (sample_size, number_of_draws)) .. GENERATED FROM PYTHON SOURCE LINES 78-79 Associate the function with a name. .. GENERATED FROM PYTHON SOURCE LINES 79-87 .. code-block:: Python my_random_number_generators = { 'TRIANGULAR': RandomNumberGeneratorTuple( generator=the_triangular_generator, description='Draws from a triangular distribution', ) } .. GENERATED FROM PYTHON SOURCE LINES 88-90 Define a random parameter with a triangular distribution, designed to be used for Monte-Carlo simulation. .. GENERATED FROM PYTHON SOURCE LINES 90-92 .. code-block:: Python b_time_rnd = b_time + b_time_s * Draws('b_time_rnd', 'TRIANGULAR') .. GENERATED FROM PYTHON SOURCE LINES 93-94 Definition of the utility functions. .. GENERATED FROM PYTHON SOURCE LINES 94-98 .. 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 99-100 Associate utility functions with the numbering of alternatives .. GENERATED FROM PYTHON SOURCE LINES 100-102 .. code-block:: Python v = {1: v_train, 2: v_swissmetro, 3: v_car} .. GENERATED FROM PYTHON SOURCE LINES 103-104 Associate the availability conditions with the alternatives .. GENERATED FROM PYTHON SOURCE LINES 104-106 .. code-block:: Python av = {1: TRAIN_AV_SP, 2: SM_AV, 3: CAR_AV_SP} .. GENERATED FROM PYTHON SOURCE LINES 107-108 Conditional to b_time_rnd, we have a logit model (called the kernel) .. GENERATED FROM PYTHON SOURCE LINES 108-110 .. code-block:: Python conditional_probability = logit(v, av, CHOICE) .. GENERATED FROM PYTHON SOURCE LINES 111-112 We integrate over b_time_rnd using Monte-Carlo .. GENERATED FROM PYTHON SOURCE LINES 112-114 .. code-block:: Python log_probability = log(MonteCarlo(conditional_probability)) .. GENERATED FROM PYTHON SOURCE LINES 115-124 .. code-block:: Python the_biogeme = BIOGEME( database, log_probability, random_number_generators=my_random_number_generators, number_of_draws=10_000, seed=1223, ) the_biogeme.model_name = 'b25triangular_mixture' .. rst-class:: sphx-glr-script-out .. code-block:: none Biogeme parameters read from biogeme.toml. .. GENERATED FROM PYTHON SOURCE LINES 125-126 Estimate the parameters .. GENERATED FROM PYTHON SOURCE LINES 126-128 .. 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 __b25triangular_mixture.iter Parameter values restored from __b25triangular_mixture.iter Starting values for the algorithm: {'asc_train': -0.39397048552414804, 'b_time': -2.2726620702434417, 'b_time_s': 3.983067366643079, 'b_cost': -1.280403947694777, 'asc_car': 0.14025321441389768} 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: 2.623788529923212e-06 Cause of termination: Relative gradient = 2.6e-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: 0:00:08.638604 Calculate second derivatives and BHHH File b25triangular_mixture~00.html has been generated. File b25triangular_mixture~00.yaml has been generated. .. GENERATED FROM PYTHON SOURCE LINES 129-131 .. code-block:: Python print(results.short_summary()) .. rst-class:: sphx-glr-script-out .. code-block:: none Results for model b25triangular_mixture Nbr of parameters: 5 Sample size: 6768 Excluded data: 3960 Final log likelihood: -5214.972 Akaike Information Criterion: 10439.94 Bayesian Information Criterion: 10474.04 .. GENERATED FROM PYTHON SOURCE LINES 132-134 .. 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 -0.393970 0.065698 -5.996645 2.014361e-09 1 b_time -2.272662 0.119136 -19.076255 0.000000e+00 2 b_time_s 3.983067 0.306149 13.010235 0.000000e+00 3 b_cost -1.280404 0.086226 -14.849390 0.000000e+00 4 asc_car 0.140253 0.052139 2.689983 7.145576e-03 .. rst-class:: sphx-glr-timing **Total running time of the script:** (2 minutes 50.798 seconds) .. _sphx_glr_download_auto_examples_swissmetro_plot_b25triangular_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_b25triangular_mixture.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_b25triangular_mixture.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_b25triangular_mixture.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_