.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/swissmetro/plot_b06a_unif_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_b06a_unif_mixture.py: 6a. Mixture of logit models with uniform distribution ===================================================== Example of a uniform mixture of logit models, using Monte-Carlo integration. Michel Bierlaire, EPFL Fri Jun 20 2025, 10:43:05 .. GENERATED FROM PYTHON SOURCE LINES 12-24 .. 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 25-26 See the data processing script: :ref:`swissmetro_data`. .. GENERATED FROM PYTHON SOURCE LINES 26-43 .. 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 b06a_unif_mixture.py') .. rst-class:: sphx-glr-script-out .. code-block:: none Example b06a_unif_mixture.py .. GENERATED FROM PYTHON SOURCE LINES 44-45 Parameters to be estimated. .. GENERATED FROM PYTHON SOURCE LINES 45-50 .. 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 51-53 Define a random parameter, uniformly distributed, designed to be used for Monte-Carlo simulation. .. GENERATED FROM PYTHON SOURCE LINES 53-57 .. code-block:: Python b_time = Beta('b_time', 0, None, None, 0) b_time_s = Beta('b_time_s', 1, None, None, 0) b_time_rnd = b_time + b_time_s * Draws('b_time_rnd', 'UNIFORMSYM') .. GENERATED FROM PYTHON SOURCE LINES 58-59 Definition of the utility functions. .. GENERATED FROM PYTHON SOURCE LINES 59-63 .. 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 64-65 Associate utility functions with the numbering of alternatives. .. GENERATED FROM PYTHON SOURCE LINES 65-67 .. code-block:: Python v = {1: v_train, 2: v_swissmetro, 3: v_car} .. GENERATED FROM PYTHON SOURCE LINES 68-69 Associate the availability conditions with the alternatives. .. GENERATED FROM PYTHON SOURCE LINES 69-71 .. code-block:: Python av = {1: TRAIN_AV_SP, 2: SM_AV, 3: CAR_AV_SP} .. GENERATED FROM PYTHON SOURCE LINES 72-73 Conditional to b_time_rnd, we have a logit model (called the kernel). .. GENERATED FROM PYTHON SOURCE LINES 73-77 .. code-block:: Python conditional_probability = logit(v, av, CHOICE) # We integrate over b_time_rnd using Monte-Carlo log_probability = log(MonteCarlo(conditional_probability)) .. GENERATED FROM PYTHON SOURCE LINES 78-79 Create the Biogeme object. .. GENERATED FROM PYTHON SOURCE LINES 79-82 .. code-block:: Python the_biogeme = BIOGEME(database, log_probability, number_of_draws=10000, seed=1223) the_biogeme.model_name = 'b06a_unif_mixture' .. rst-class:: sphx-glr-script-out .. code-block:: none Biogeme parameters read from biogeme.toml. .. GENERATED FROM PYTHON SOURCE LINES 83-84 Estimate the parameters. .. GENERATED FROM PYTHON SOURCE LINES 84-91 .. 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 __b06a_unif_mixture.iter Cannot read file __b06a_unif_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 5.6e+03 0.09 1 0.39 + 1 -1 -1 2 -1 -1 5.6e+03 0.09 0.5 -0.11 - 2 -1.5 -1.3 1.5 -1.5 -0.5 5.4e+03 0.074 0.5 0.32 + 3 -1.5 -1.3 1.5 -1.5 -0.5 5.4e+03 0.074 0.25 -0.17 - 4 -1.2 -1 1.8 -1.2 -0.25 5.3e+03 0.028 0.25 0.47 + 5 -1 -1.3 1.5 -1 -0.5 5.3e+03 0.04 0.25 0.16 + 6 -1 -1.3 1.5 -1 -0.5 5.3e+03 0.04 0.12 -0.064 - 7 -0.88 -1.2 1.5 -1.1 -0.38 5.3e+03 0.024 0.12 0.63 + 8 -1 -1.3 1.4 -1.2 -0.25 5.3e+03 0.017 0.12 0.33 + 9 -0.88 -1.4 1.5 -1.1 -0.24 5.3e+03 0.0092 0.12 0.88 + 10 -0.75 -1.5 1.7 -1.3 -0.11 5.2e+03 0.012 0.12 0.71 + 11 -0.68 -1.6 1.7 -1.1 -0.079 5.2e+03 0.0073 0.12 0.71 + 12 -0.56 -1.7 1.9 -1.2 -0.043 5.2e+03 0.0092 1.2 0.91 ++ 13 -0.56 -1.7 1.9 -1.2 -0.043 5.2e+03 0.0092 0.62 -2.1 - 14 -0.56 -1.7 1.9 -1.2 -0.043 5.2e+03 0.0092 0.31 -1.4 - 15 -0.56 -1.7 1.9 -1.2 -0.043 5.2e+03 0.0092 0.16 -0.13 - 16 -0.59 -1.9 2 -1.2 0.049 5.2e+03 0.01 0.16 0.38 + 17 -0.53 -1.9 2.2 -1.2 -0.011 5.2e+03 0.0045 0.16 0.81 + 18 -0.53 -1.9 2.2 -1.2 -0.011 5.2e+03 0.0045 0.078 -0.45 - 19 -0.52 -1.9 2.2 -1.2 0.067 5.2e+03 0.0078 0.078 0.31 + 20 -0.48 -2 2.3 -1.2 0.04 5.2e+03 0.0037 0.78 0.95 ++ 21 -0.48 -2 2.3 -1.2 0.04 5.2e+03 0.0037 0.39 -1.2 - 22 -0.48 -2 2.3 -1.2 0.04 5.2e+03 0.0037 0.2 -0.73 - 23 -0.48 -2 2.3 -1.2 0.04 5.2e+03 0.0037 0.098 0.068 - 24 -0.5 -2.1 2.4 -1.3 0.063 5.2e+03 0.0065 0.098 0.48 + 25 -0.46 -2.1 2.5 -1.2 0.097 5.2e+03 0.0048 0.098 0.82 + 26 -0.44 -2.2 2.6 -1.2 0.11 5.2e+03 0.0033 0.98 0.94 ++ 27 -0.44 -2.2 2.6 -1.2 0.11 5.2e+03 0.0033 0.48 -20 - 28 -0.44 -2.2 2.6 -1.2 0.11 5.2e+03 0.0033 0.24 -4.1 - 29 -0.44 -2.2 2.6 -1.2 0.11 5.2e+03 0.0033 0.12 -0.64 - 30 -0.41 -2.2 2.7 -1.3 0.11 5.2e+03 0.003 0.12 0.34 + 31 -0.41 -2.2 2.7 -1.3 0.11 5.2e+03 0.003 0.06 -0.52 - 32 -0.41 -2.2 2.7 -1.3 0.12 5.2e+03 0.00083 0.06 0.77 + 33 -0.38 -2.3 2.8 -1.3 0.15 5.2e+03 0.0023 0.06 0.13 + 34 -0.41 -2.3 2.8 -1.3 0.13 5.2e+03 0.0024 0.06 0.33 + 35 -0.41 -2.3 2.8 -1.3 0.13 5.2e+03 0.0024 0.03 -1.7 - 36 -0.41 -2.3 2.8 -1.3 0.13 5.2e+03 0.0024 0.015 -0.6 - 37 -0.41 -2.3 2.8 -1.3 0.13 5.2e+03 0.00081 0.015 0.43 + 38 -0.39 -2.3 2.8 -1.3 0.13 5.2e+03 0.00082 0.015 0.83 + 39 -0.39 -2.3 2.8 -1.3 0.13 5.2e+03 0.00082 0.0075 -0.55 - 40 -0.39 -2.3 2.8 -1.3 0.14 5.2e+03 0.00058 0.0075 0.37 + 41 -0.39 -2.3 2.8 -1.3 0.14 5.2e+03 0.00023 0.0075 0.65 + 42 -0.39 -2.3 2.8 -1.3 0.14 5.2e+03 0.0002 0.0075 0.62 + 43 -0.39 -2.3 2.8 -1.3 0.14 5.2e+03 0.00015 0.0075 0.78 + 44 -0.39 -2.3 2.9 -1.3 0.14 5.2e+03 0.00036 0.0075 0.49 + 45 -0.39 -2.3 2.9 -1.3 0.14 5.2e+03 7.7e-05 0.075 0.92 ++ 46 -0.39 -2.3 2.9 -1.3 0.14 5.2e+03 6.5e-05 0.075 0.37 + 47 -0.39 -2.3 2.9 -1.3 0.14 5.2e+03 6.8e-06 0.75 0.98 ++ 48 -0.39 -2.3 2.9 -1.3 0.14 5.2e+03 2.3e-06 0.75 0.66 ++ Optimization algorithm has converged. Relative gradient: 2.346420034070932e-06 Cause of termination: Relative gradient = 2.3e-06 <= 6.1e-06 Number of function evaluations: 114 Number of gradient evaluations: 65 Number of hessian evaluations: 0 Algorithm: BFGS with trust region for simple bound constraints Number of iterations: 49 Proportion of Hessian calculation: 0/32 = 0.0% Optimization time: 0:01:53.544645 Calculate second derivatives and BHHH File b06a_unif_mixture.html has been generated. File b06a_unif_mixture.yaml has been generated. .. GENERATED FROM PYTHON SOURCE LINES 92-94 .. code-block:: Python print(results.short_summary()) .. rst-class:: sphx-glr-script-out .. code-block:: none Results for model b06a_unif_mixture Nbr of parameters: 5 Sample size: 6768 Excluded data: 3960 Final log likelihood: -5215.805 Akaike Information Criterion: 10441.61 Bayesian Information Criterion: 10475.71 .. GENERATED FROM PYTHON SOURCE LINES 95-97 .. 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 -0.386220 0.066029 -5.849265 4.937510e-09 1 b_time -2.316295 0.125946 -18.391184 0.000000e+00 2 b_time_s 2.868552 0.199638 14.368781 0.000000e+00 3 b_cost -1.277277 0.086562 -14.755635 0.000000e+00 4 asc_car 0.143914 0.053299 2.700124 6.931367e-03 .. rst-class:: sphx-glr-timing **Total running time of the script:** (3 minutes 23.102 seconds) .. _sphx_glr_download_auto_examples_swissmetro_plot_b06a_unif_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_b06a_unif_mixture.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_b06a_unif_mixture.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_b06a_unif_mixture.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_