.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/swissmetro/plot_b06unif_mixture_MHLS.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_b06unif_mixture_MHLS.py: Mixture of logit models ======================= Example of a uniform mixture of logit models, using Monte-Carlo integration. The mixing distribution is uniform. The draws are from the Modified Hypercube Latin Square. Michel Bierlaire, EPFL Fri Jun 20 2025, 11:24:34 .. GENERATED FROM PYTHON SOURCE LINES 13-21 .. 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 22-23 See the data processing script: :ref:`swissmetro_data`. .. GENERATED FROM PYTHON SOURCE LINES 23-40 .. 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 b06unif_mixture_MHLS') .. rst-class:: sphx-glr-script-out .. code-block:: none Example b06unif_mixture_MHLS .. GENERATED FROM PYTHON SOURCE LINES 41-42 Parameters to be estimated. .. GENERATED FROM PYTHON SOURCE LINES 42-47 .. 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 48-50 Define a random parameter, normally distributed, designed to be used for Monte-Carlo simulation. .. GENERATED FROM PYTHON SOURCE LINES 50-52 .. code-block:: Python b_time = Beta('b_time', 0, None, None, 0) .. GENERATED FROM PYTHON SOURCE LINES 53-54 It is advised not to use 0 as starting value for the following parameter. .. GENERATED FROM PYTHON SOURCE LINES 54-56 .. 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, uniformly distributed, designed to be used for Monte-Carlo simulation. The type of draws is set to ``NORMAL_MLHS``. .. GENERATED FROM PYTHON SOURCE LINES 59-61 .. code-block:: Python b_time_rnd = b_time + b_time_s * Draws('b_time_rnd', 'NORMAL_MLHS') .. 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 81-84 .. code-block:: Python log_probability = log(MonteCarlo(conditional_probability)) .. GENERATED FROM PYTHON SOURCE LINES 85-86 Create the Biogeme object. .. GENERATED FROM PYTHON SOURCE LINES 86-89 .. code-block:: Python the_biogeme = BIOGEME(database, log_probability, number_of_draws=10000, seed=1223) the_biogeme.model_name = '06unif_mixture_MHLS' .. rst-class:: sphx-glr-script-out .. code-block:: none Biogeme parameters read from biogeme.toml. .. GENERATED FROM PYTHON SOURCE LINES 90-91 Estimate the parameters. .. GENERATED FROM PYTHON SOURCE LINES 91-93 .. 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 __06unif_mixture_MHLS.iter Parameter values restored from __06unif_mixture_MHLS.iter Starting values for the algorithm: {'asc_train': -0.40185858631652444, 'b_time': -2.259754033646597, 'b_time_s': 1.6570229365550726, 'b_cost': -1.2854433011000461, 'asc_car': 0.13702090018261726} 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: 2.153834922568635e-06 Cause of termination: Relative gradient = 2.2e-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: 0:00:05.924978 Calculate second derivatives and BHHH File 06unif_mixture_MHLS~00.html has been generated. File 06unif_mixture_MHLS~00.yaml has been generated. .. GENERATED FROM PYTHON SOURCE LINES 94-96 .. code-block:: Python print(results.short_summary()) .. rst-class:: sphx-glr-script-out .. code-block:: none Results for model 06unif_mixture_MHLS Nbr of parameters: 5 Sample size: 6768 Excluded data: 3960 Final log likelihood: -5214.947 Akaike Information Criterion: 10439.89 Bayesian Information Criterion: 10473.99 .. GENERATED FROM PYTHON SOURCE LINES 97-99 .. 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 -0.401859 0.065945 -6.093814 1.102520e-09 1 b_time -2.259754 0.117179 -19.284630 0.000000e+00 2 b_time_s 1.657023 0.132669 12.489949 0.000000e+00 3 b_cost -1.285443 0.086294 -14.896028 0.000000e+00 4 asc_car 0.137021 0.051739 2.648291 8.089997e-03 .. rst-class:: sphx-glr-timing **Total running time of the script:** (3 minutes 2.399 seconds) .. _sphx_glr_download_auto_examples_swissmetro_plot_b06unif_mixture_MHLS.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_b06unif_mixture_MHLS.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_b06unif_mixture_MHLS.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_b06unif_mixture_MHLS.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_