.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/swissmetro/plot_b17lognormal_mixture_integral.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_b17lognormal_mixture_integral.py: Mixture with lognormal distribution =================================== Example of a mixture of logit models. The mixing distribution is distributed as a log normal. Compared to :ref:`plot_b17lognormal_mixture`, the integration is performed using numerical integration instead of Monte-Carlo approximation. Michel Bierlaire, EPFL Thu Jun 26 2025, 15:49:37 .. GENERATED FROM PYTHON SOURCE LINES 15-29 .. 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, IntegrateNormal, RandomVariable, exp, log, ) from biogeme.models import logit from biogeme.results_processing import get_pandas_estimated_parameters .. GENERATED FROM PYTHON SOURCE LINES 30-31 See the data processing script: :ref:`swissmetro_data`. .. GENERATED FROM PYTHON SOURCE LINES 31-48 .. 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 b17lognormal_mixture_integral.py') .. rst-class:: sphx-glr-script-out .. code-block:: none Example b17lognormal_mixture_integral.py .. GENERATED FROM PYTHON SOURCE LINES 49-50 Parameters to be estimated. .. GENERATED FROM PYTHON SOURCE LINES 50-55 .. 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 56-58 Define a random parameter, normally distributed, designed to be used. for Monte-Carlo simulation .. GENERATED FROM PYTHON SOURCE LINES 58-60 .. code-block:: Python b_time = Beta('b_time', 0, None, None, 0) .. GENERATED FROM PYTHON SOURCE LINES 61-62 It is advised not to use 0 as starting value for the following parameter.. .. GENERATED FROM PYTHON SOURCE LINES 62-64 .. code-block:: Python b_time_s = Beta('b_time_s', 1, -2, 2, 0) .. GENERATED FROM PYTHON SOURCE LINES 65-67 Define a random parameter, log normally distributed, designed to be used for numerical integration. .. GENERATED FROM PYTHON SOURCE LINES 67-70 .. code-block:: Python omega = RandomVariable('omega') B_TIME_RND = -exp(b_time + b_time_s * omega) .. GENERATED FROM PYTHON SOURCE LINES 71-72 Definition of the utility functions. .. GENERATED FROM PYTHON SOURCE LINES 72-76 .. 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 77-78 Associate utility functions with the numbering of alternatives. .. GENERATED FROM PYTHON SOURCE LINES 78-80 .. code-block:: Python v = {1: v_train, 2: v_swissmetro, 3: v_car} .. GENERATED FROM PYTHON SOURCE LINES 81-82 Associate the availability conditions with the alternatives. .. GENERATED FROM PYTHON SOURCE LINES 82-84 .. code-block:: Python av = {1: TRAIN_AV_SP, 2: SM_AV, 3: CAR_AV_SP} .. GENERATED FROM PYTHON SOURCE LINES 85-86 Conditional to omega, we have a logit model (called the kernel). .. GENERATED FROM PYTHON SOURCE LINES 86-88 .. code-block:: Python conditional_probability = logit(v, av, CHOICE) .. GENERATED FROM PYTHON SOURCE LINES 89-90 We integrate over omega using numerical integration. .. GENERATED FROM PYTHON SOURCE LINES 90-92 .. code-block:: Python log_probability = log(IntegrateNormal(conditional_probability, 'omega')) .. GENERATED FROM PYTHON SOURCE LINES 93-94 Create the Biogeme object. .. GENERATED FROM PYTHON SOURCE LINES 94-97 .. code-block:: Python the_biogeme = BIOGEME(database, log_probability) the_biogeme.model_name = 'b17lognormal_mixture_integral' .. rst-class:: sphx-glr-script-out .. code-block:: none Biogeme parameters read from biogeme.toml. .. GENERATED FROM PYTHON SOURCE LINES 98-99 Estimate the parameters .. GENERATED FROM PYTHON SOURCE LINES 99-101 .. 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 __b17lognormal_mixture_integral.iter Parameter values restored from __b17lognormal_mixture_integral.iter Starting values for the algorithm: {'asc_train': -0.3508670994332833, 'b_time': 0.5699481745949635, 'b_time_s': 1.2138204557723493, 'b_cost': -1.3762554626050059, 'asc_car': 0.16780371550483728} 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: 3.8488160574638475e-06 Cause of termination: Relative gradient = 3.8e-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: 0:00:00.862432 Calculate second derivatives and BHHH File b17lognormal_mixture_integral~00.html has been generated. File b17lognormal_mixture_integral~00.yaml has been generated. .. GENERATED FROM PYTHON SOURCE LINES 102-104 .. code-block:: Python print(results.short_summary()) .. rst-class:: sphx-glr-script-out .. code-block:: none Results for model b17lognormal_mixture_integral Nbr of parameters: 5 Sample size: 6768 Excluded data: 3960 Final log likelihood: -5231.506 Akaike Information Criterion: 10473.01 Bayesian Information Criterion: 10507.11 .. GENERATED FROM PYTHON SOURCE LINES 105-107 .. 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 -0.350867 0.073180 -4.794581 1.630150e-06 1 b_time 0.569948 0.070411 8.094634 6.661338e-16 2 b_time_s 1.213820 0.141278 8.591690 0.000000e+00 3 b_cost -1.376255 0.096032 -14.331261 0.000000e+00 4 asc_car 0.167804 0.063408 2.646410 8.135116e-03 .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 4.534 seconds) .. _sphx_glr_download_auto_examples_swissmetro_plot_b17lognormal_mixture_integral.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_b17lognormal_mixture_integral.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_b17lognormal_mixture_integral.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_b17lognormal_mixture_integral.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_