.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/swissmetro/plot_b24halton_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_b24halton_mixture.py: Mixture of logit with Halton draws ================================== Example of a mixture of logit models, using quasi Monte-Carlo integration with Halton draws (base 5). The mixing distribution is normal. Michel Bierlaire, EPFL Sat Jun 28 2025, 12:45:21 .. GENERATED FROM PYTHON SOURCE LINES 14-22 .. 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 23-24 See the data processing script: :ref:`swissmetro_data`. .. GENERATED FROM PYTHON SOURCE LINES 24-41 .. 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 b24halton_mixture.py') .. rst-class:: sphx-glr-script-out .. code-block:: none Example b24halton_mixture.py .. GENERATED FROM PYTHON SOURCE LINES 42-43 Parameters to be estimated. .. GENERATED FROM PYTHON SOURCE LINES 43-48 .. 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 49-51 Define a random parameter, normally distributed, designed to be used for Monte-Carlo simulation. .. GENERATED FROM PYTHON SOURCE LINES 51-53 .. code-block:: Python b_time = Beta('b_time', 0, None, None, 0) .. GENERATED FROM PYTHON SOURCE LINES 54-55 It is advised not to use 0 as starting value for the following parameter. .. GENERATED FROM PYTHON SOURCE LINES 55-56 .. code-block:: Python b_time_s = Beta('b_time_s', 1, None, None, 0) .. GENERATED FROM PYTHON SOURCE LINES 57-59 Define a random parameter with a normal distribution, designed to be used for quasi Monte-Carlo simulation with Halton draws (base 5). .. GENERATED FROM PYTHON SOURCE LINES 59-61 .. code-block:: Python b_time_rnd = b_time + b_time_s * Draws('b_time_rnd', 'NORMAL_HALTON5') .. GENERATED FROM PYTHON SOURCE LINES 62-63 Definition of the utility functions. .. GENERATED FROM PYTHON SOURCE LINES 63-67 .. 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 68-69 Associate utility functions with the numbering of alternatives. .. GENERATED FROM PYTHON SOURCE LINES 69-71 .. code-block:: Python v = {1: v_train, 2: v_swissmetro, 3: v_car} .. GENERATED FROM PYTHON SOURCE LINES 72-73 Associate the availability conditions with the alternatives. .. GENERATED FROM PYTHON SOURCE LINES 73-75 .. code-block:: Python av = {1: TRAIN_AV_SP, 2: SM_AV, 3: CAR_AV_SP} .. GENERATED FROM PYTHON SOURCE LINES 76-77 Conditional on b_time_rnd, we have a logit model (called the kernel) .. GENERATED FROM PYTHON SOURCE LINES 77-79 .. code-block:: Python conditional_probability = logit(v, av, CHOICE) .. GENERATED FROM PYTHON SOURCE LINES 80-81 We integrate over b_time_rnd using Monte-Carlo. .. GENERATED FROM PYTHON SOURCE LINES 81-83 .. code-block:: Python log_probability = log(MonteCarlo(conditional_probability)) .. GENERATED FROM PYTHON SOURCE LINES 84-85 These notes will be included as such in the report file. .. GENERATED FROM PYTHON SOURCE LINES 85-90 .. code-block:: Python USER_NOTES = ( 'Example of a mixture of logit models with three alternatives, ' 'approximated using Monte-Carlo integration with Halton draws.' ) .. GENERATED FROM PYTHON SOURCE LINES 91-94 As the objective is to illustrate the syntax, we calculate the Monte-Carlo approximation with a small number of draws. .. GENERATED FROM PYTHON SOURCE LINES 94-99 .. code-block:: Python the_biogeme = BIOGEME( database, log_probability, user_notes=USER_NOTES, number_of_draws=10_000, seed=1223 ) the_biogeme.model_name = 'b24halton_mixture' .. rst-class:: sphx-glr-script-out .. code-block:: none Biogeme parameters read from biogeme.toml. .. GENERATED FROM PYTHON SOURCE LINES 100-101 Estimate the parameters .. GENERATED FROM PYTHON SOURCE LINES 101-103 .. 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 __b24halton_mixture.iter Parameter values restored from __b24halton_mixture.iter Starting values for the algorithm: {'asc_train': -0.40195185364181046, 'b_time': -2.2595838469305267, 'b_time_s': 1.6573078631733604, 'b_cost': -1.2852992156480805, 'asc_car': 0.1370262574783267} 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 Iter. asc_train b_time b_time_s b_cost asc_car Function Relgrad Radius Rho 0 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 7e-06 0.018 -8.5e+02 - 1 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 7e-06 0.0091 -4.3e+02 - 2 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 7e-06 0.0045 -1.8e+02 - 3 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 7e-06 0.0023 -76 - 4 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 7e-06 0.0011 -35 - 5 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 7e-06 0.00057 -17 - 6 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 7e-06 0.00028 -7.6 - 7 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 7e-06 0.00014 -3.3 - 8 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 7e-06 7.1e-05 -1.1 - 9 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 7e-06 3.6e-05 -0.065 - 10 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 4.4e-06 3.6e-05 0.47 - Optimization algorithm has converged. Relative gradient: 4.391778983297403e-06 Cause of termination: Relative gradient = 4.4e-06 <= 6.1e-06 Number of function evaluations: 14 Number of gradient evaluations: 3 Number of hessian evaluations: 0 Algorithm: BFGS with trust region for simple bound constraints Number of iterations: 11 Proportion of Hessian calculation: 0/1 = 0.0% Optimization time: 0:00:28.744647 Calculate second derivatives and BHHH File b24halton_mixture~00.html has been generated. File b24halton_mixture~00.yaml has been generated. .. GENERATED FROM PYTHON SOURCE LINES 104-106 .. code-block:: Python print(results.short_summary()) .. rst-class:: sphx-glr-script-out .. code-block:: none Results for model b24halton_mixture Nbr of parameters: 5 Sample size: 6768 Excluded data: 3960 Final log likelihood: -5214.905 Akaike Information Criterion: 10439.81 Bayesian Information Criterion: 10473.91 .. GENERATED FROM PYTHON SOURCE LINES 107-109 .. 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 -0.401916 0.065838 -6.104587 1.030669e-09 1 b_time -2.259619 0.117085 -19.299005 0.000000e+00 2 b_time_s 1.657343 0.131714 12.582927 0.000000e+00 3 b_cost -1.285264 0.086295 -14.893876 0.000000e+00 4 asc_car 0.136991 0.051721 2.648645 8.081517e-03 .. rst-class:: sphx-glr-timing **Total running time of the script:** (2 minutes 32.643 seconds) .. _sphx_glr_download_auto_examples_swissmetro_plot_b24halton_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_b24halton_mixture.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_b24halton_mixture.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_b24halton_mixture.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_