.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/swissmetro/plot_b24_halton_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_b24_halton_mixture.py: 24. 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 14-26 .. code-block:: Python from IPython.core.display_functions import display import biogeme.biogeme_logging as blog from biogeme.biogeme import BIOGEME from biogeme.expressions import Beta, Draws, MonteCarlo, log from biogeme.models import logit from biogeme.results_processing import ( EstimationResults, get_pandas_estimated_parameters, ) .. GENERATED FROM PYTHON SOURCE LINES 27-28 See the data processing script: :ref:`swissmetro_data`. .. GENERATED FROM PYTHON SOURCE LINES 28-45 .. 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 b24_halton_mixture.py') .. rst-class:: sphx-glr-script-out .. code-block:: none Example b24_halton_mixture.py .. GENERATED FROM PYTHON SOURCE LINES 46-47 Parameters to be estimated. .. GENERATED FROM PYTHON SOURCE LINES 47-52 .. 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 53-55 Define a random parameter, normally distributed, designed to be used for Monte-Carlo simulation. .. GENERATED FROM PYTHON SOURCE LINES 55-57 .. code-block:: Python b_time = Beta('b_time', 0, None, None, 0) .. GENERATED FROM PYTHON SOURCE LINES 58-59 It is advised not to use 0 as starting value for the following parameter. .. GENERATED FROM PYTHON SOURCE LINES 59-60 .. code-block:: Python b_time_s = Beta('b_time_s', 1, None, None, 0) .. GENERATED FROM PYTHON SOURCE LINES 61-63 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 63-65 .. code-block:: Python b_time_rnd = b_time + b_time_s * Draws('b_time_rnd', 'NORMAL_HALTON5') .. GENERATED FROM PYTHON SOURCE LINES 66-67 Definition of the utility functions. .. GENERATED FROM PYTHON SOURCE LINES 67-71 .. 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 72-73 Associate utility functions with the numbering of alternatives. .. GENERATED FROM PYTHON SOURCE LINES 73-75 .. code-block:: Python v = {1: v_train, 2: v_swissmetro, 3: v_car} .. GENERATED FROM PYTHON SOURCE LINES 76-77 Associate the availability conditions with the alternatives. .. GENERATED FROM PYTHON SOURCE LINES 77-79 .. code-block:: Python av = {1: TRAIN_AV_SP, 2: SM_AV, 3: CAR_AV_SP} .. GENERATED FROM PYTHON SOURCE LINES 80-81 Conditional on b_time_rnd, we have a logit model (called the kernel) .. GENERATED FROM PYTHON SOURCE LINES 81-83 .. code-block:: Python conditional_probability = logit(v, av, CHOICE) .. GENERATED FROM PYTHON SOURCE LINES 84-85 We integrate over b_time_rnd using Monte-Carlo. .. GENERATED FROM PYTHON SOURCE LINES 85-87 .. code-block:: Python log_probability = log(MonteCarlo(conditional_probability)) .. GENERATED FROM PYTHON SOURCE LINES 88-89 These notes will be included as such in the report file. .. GENERATED FROM PYTHON SOURCE LINES 89-94 .. 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 95-98 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 98-103 .. code-block:: Python the_biogeme = BIOGEME( database, log_probability, user_notes=USER_NOTES, number_of_draws=10_000, seed=1223 ) the_biogeme.model_name = 'b24_halton_mixture' .. rst-class:: sphx-glr-script-out .. code-block:: none Biogeme parameters read from biogeme.toml. .. GENERATED FROM PYTHON SOURCE LINES 104-105 Estimate the parameters. .. GENERATED FROM PYTHON SOURCE LINES 105-112 .. code-block:: Python try: results = EstimationResults.from_yaml_file( filename=f'saved_results/{the_biogeme.model_name}.yaml' ) except FileNotFoundError: results = the_biogeme.estimate() .. rst-class:: sphx-glr-script-out .. code-block:: none *** Initial values of the parameters are obtained from the file __b24_halton_mixture.iter Cannot read file __b24_halton_mixture.iter. Statement is ignored. Starting values for the algorithm: {} 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 -1 -1 2 -1 1 6.1e+03 0.16 1 0.25 + 1 -0.73 -2 3 -0.4 0 5.5e+03 0.049 1 0.36 + 2 -0.95 -2.3 2.6 -1.4 0.51 5.4e+03 0.054 1 0.39 + 3 -0.95 -2.3 2.6 -1.4 0.51 5.4e+03 0.054 0.5 -0.15 - 4 -0.45 -2.8 2.6 -1.1 0.0057 5.3e+03 0.03 0.5 0.5 + 5 -0.092 -2.6 2.5 -1.6 0.33 5.3e+03 0.046 0.5 0.14 + 6 -0.092 -2.6 2.5 -1.6 0.33 5.3e+03 0.046 0.25 -0.19 - 7 -0.34 -2.9 2.3 -1.4 0.26 5.2e+03 0.022 0.25 0.65 + 8 -0.3 -2.6 2.2 -1.2 0.22 5.2e+03 0.0084 0.25 0.51 + 9 -0.3 -2.6 2.2 -1.2 0.22 5.2e+03 0.0084 0.12 -3 - 10 -0.3 -2.6 2.2 -1.2 0.22 5.2e+03 0.0084 0.062 -0.22 - 11 -0.36 -2.6 2.1 -1.3 0.28 5.2e+03 0.0063 0.062 0.42 + 12 -0.3 -2.6 2.1 -1.4 0.22 5.2e+03 0.0065 0.062 0.54 + 13 -0.35 -2.6 2 -1.3 0.21 5.2e+03 0.0096 0.062 0.47 + 14 -0.33 -2.5 2 -1.3 0.22 5.2e+03 0.0034 0.62 0.9 ++ 15 -0.33 -2.5 2 -1.3 0.22 5.2e+03 0.0034 0.31 -0.57 - 16 -0.38 -2.3 1.7 -1.2 0.12 5.2e+03 0.0045 0.31 0.47 + 17 -0.38 -2.3 1.7 -1.2 0.12 5.2e+03 0.0045 0.16 -3.1 - 18 -0.38 -2.3 1.7 -1.2 0.12 5.2e+03 0.0045 0.078 -2.2 - 19 -0.38 -2.3 1.7 -1.2 0.12 5.2e+03 0.0045 0.039 -0.99 - 20 -0.42 -2.2 1.6 -1.3 0.15 5.2e+03 0.0034 0.039 0.21 + 21 -0.4 -2.2 1.6 -1.3 0.12 5.2e+03 0.0018 0.039 0.18 + 22 -0.41 -2.2 1.6 -1.3 0.13 5.2e+03 0.00021 0.039 0.77 + 23 -0.41 -2.2 1.6 -1.3 0.13 5.2e+03 0.00021 0.02 -2.9 - 24 -0.41 -2.2 1.6 -1.3 0.13 5.2e+03 0.00021 0.0098 -0.57 - 25 -0.4 -2.3 1.6 -1.3 0.14 5.2e+03 0.00039 0.0098 0.24 + 26 -0.4 -2.3 1.6 -1.3 0.14 5.2e+03 0.00039 0.0049 -0.99 - 27 -0.4 -2.3 1.6 -1.3 0.14 5.2e+03 0.00039 0.0024 -0.33 - 28 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 0.00018 0.0024 0.46 + 29 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 0.00014 0.0024 0.69 + 30 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 0.00021 0.0024 0.15 + 31 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 0.00021 0.0012 -0.49 - 32 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 8.1e-05 0.0012 0.39 + 33 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 3.1e-05 0.0012 0.68 + 34 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 2.4e-05 0.0012 0.57 + 35 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 2.4e-05 0.00061 -1.9 - 36 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 2.4e-05 0.00031 -0.91 - 37 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 2.4e-05 0.00015 -0.17 - 38 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 1.2e-05 0.00015 0.68 + 39 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 8.1e-06 0.00015 0.74 + 40 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 5.3e-06 0.00015 0.57 + Optimization algorithm has converged. Relative gradient: 5.2868161203223544e-06 Cause of termination: Relative gradient = 5.3e-06 <= 6.1e-06 Number of function evaluations: 92 Number of gradient evaluations: 51 Number of hessian evaluations: 0 Algorithm: BFGS with trust region for simple bound constraints Number of iterations: 41 Proportion of Hessian calculation: 0/25 = 0.0% Optimization time: 0:01:30.115353 Calculate second derivatives and BHHH File b24_halton_mixture.html has been generated. File b24_halton_mixture.yaml has been generated. .. GENERATED FROM PYTHON SOURCE LINES 113-115 .. code-block:: Python print(results.short_summary()) .. rst-class:: sphx-glr-script-out .. code-block:: none Results for model b24_halton_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 116-118 .. 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 -0.401952 0.065839 -6.105116 1.027262e-09 1 b_time -2.259584 0.117082 -19.299092 0.000000e+00 2 b_time_s 1.657308 0.131713 12.582698 0.000000e+00 3 b_cost -1.285299 0.086297 -14.893935 0.000000e+00 4 asc_car 0.137026 0.051721 2.649327 8.065219e-03 .. rst-class:: sphx-glr-timing **Total running time of the script:** (3 minutes 34.189 seconds) .. _sphx_glr_download_auto_examples_swissmetro_plot_b24_halton_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_b24_halton_mixture.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_b24_halton_mixture.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_b24_halton_mixture.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_