.. 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_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_b06unif_mixture_integral.py: " Mixture of logit models ======================= Example of a mixture of logit models, using numerical integration. The mixing distribution is uniform. Michel Bierlaire, EPFL Fri Jun 20 2025, 10:47:24 .. GENERATED FROM PYTHON SOURCE LINES 13-28 .. code-block:: Python import biogeme.biogeme_logging as blog from IPython.core.display_functions import display from biogeme.biogeme import BIOGEME from biogeme.distributions import normalpdf 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 29-30 See the data processing script: :ref:`swissmetro_data`. .. GENERATED FROM PYTHON SOURCE LINES 30-47 .. 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_integral.py') .. rst-class:: sphx-glr-script-out .. code-block:: none Example b06unif_mixture_integral.py .. GENERATED FROM PYTHON SOURCE LINES 48-49 Parameters to be estimated. .. GENERATED FROM PYTHON SOURCE LINES 49-54 .. 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 55-57 Define a random parameter, normally distributed, designed to be used for numerical integration .. GENERATED FROM PYTHON SOURCE LINES 57-61 .. code-block:: Python b_time = Beta('b_time', 0, None, None, 0) b_time_s = Beta('b_time_s', 1, None, None, 0) omega = RandomVariable('omega') .. GENERATED FROM PYTHON SOURCE LINES 62-68 .. |infinity| unicode:: U+221E :trim: As the numerical integration ranges from -|infinity| \ to + |infinity| , we need to perform a change of variable in order to integrate between -1 and 1. .. GENERATED FROM PYTHON SOURCE LINES 68-74 .. code-block:: Python LOWER_BND = -1 UPPER_BND = 1 x = LOWER_BND + (UPPER_BND - LOWER_BND) / (1 + exp(-omega)) dx = (UPPER_BND - LOWER_BND) * exp(-omega) / ((1 + exp(-omega)) ** 2) b_time_rnd = b_time + b_time_s * x .. GENERATED FROM PYTHON SOURCE LINES 75-76 Definition of the utility functions. .. GENERATED FROM PYTHON SOURCE LINES 76-80 .. 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 81-82 Associate utility functions with the numbering of alternatives. .. GENERATED FROM PYTHON SOURCE LINES 82-84 .. code-block:: Python v = {1: v_train, 2: v_swissmetro, 3: v_car} .. GENERATED FROM PYTHON SOURCE LINES 85-86 Associate the availability conditions with the alternatives. .. GENERATED FROM PYTHON SOURCE LINES 86-88 .. code-block:: Python av = {1: TRAIN_AV_SP, 2: SM_AV, 3: CAR_AV_SP} .. GENERATED FROM PYTHON SOURCE LINES 89-90 Conditional on omega, we have a logit model (called the kernel). .. GENERATED FROM PYTHON SOURCE LINES 90-92 .. code-block:: Python conditional_probability = logit(v, av, CHOICE) .. GENERATED FROM PYTHON SOURCE LINES 93-94 pdf of the uniform distribution .. GENERATED FROM PYTHON SOURCE LINES 94-96 .. code-block:: Python pdf_uniform = 1 / (UPPER_BND - LOWER_BND) .. GENERATED FROM PYTHON SOURCE LINES 97-99 As the `IntegrateNormal` expression is designed for a normal distribution, we need to divide by the pdf of the normal distribution, and multiply by the pdf of the uniform distribution, after applying the change of variable. .. GENERATED FROM PYTHON SOURCE LINES 99-102 .. code-block:: Python new_integrand = conditional_probability * dx * pdf_uniform / normalpdf(omega) .. GENERATED FROM PYTHON SOURCE LINES 103-105 We integrate over omega using numerical integration. To illustrate the syntax, we specific the number of quadrature points to be used. .. GENERATED FROM PYTHON SOURCE LINES 105-113 .. code-block:: Python log_probability = log( IntegrateNormal( new_integrand, 'omega', number_of_quadrature_points=60, ) ) .. GENERATED FROM PYTHON SOURCE LINES 114-115 Create the Biogeme object. .. GENERATED FROM PYTHON SOURCE LINES 115-118 .. code-block:: Python the_biogeme = BIOGEME(database, log_probability) the_biogeme.modelName = '06unif_mixture_integral' .. rst-class:: sphx-glr-script-out .. code-block:: none Biogeme parameters read from biogeme.toml. /Users/bierlair/MyFiles/github/biogeme/docs/source/examples/swissmetro/plot_b06unif_mixture_integral.py:116: DeprecationWarning: 'modelName' is deprecated. Please use 'model_name' instead. the_biogeme.modelName = '06unif_mixture_integral' .. GENERATED FROM PYTHON SOURCE LINES 119-120 Estimate the parameters .. GENERATED FROM PYTHON SOURCE LINES 120-122 .. 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_integral.iter Parameter values restored from __06unif_mixture_integral.iter Starting values for the algorithm: {'asc_train': -0.385071663361878, 'b_time': -2.3205753430924987, 'b_time_s': 2.8759594278547964, 'b_cost': -1.2779262496068977, 'asc_car': 0.14496871481580123} 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: 2.010606846718136e-06 Cause of termination: Relative gradient = 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: 0:00:00.377077 Calculate second derivatives and BHHH File 06unif_mixture_integral~00.html has been generated. File 06unif_mixture_integral~00.yaml has been generated. .. GENERATED FROM PYTHON SOURCE LINES 123-125 .. code-block:: Python print(results.short_summary()) .. rst-class:: sphx-glr-script-out .. code-block:: none Results for model 06unif_mixture_integral Nbr of parameters: 5 Sample size: 6768 Excluded data: 3960 Final log likelihood: -5215.061 Akaike Information Criterion: 10440.12 Bayesian Information Criterion: 10474.22 .. GENERATED FROM PYTHON SOURCE LINES 126-128 .. 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 -0.385072 0.065992 -5.835159 5.373928e-09 1 b_time -2.320575 0.126118 -18.400027 0.000000e+00 2 b_time_s 2.875959 0.200170 14.367615 0.000000e+00 3 b_cost -1.277926 0.086624 -14.752624 0.000000e+00 4 asc_car 0.144969 0.053308 2.719456 6.538948e-03 .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 2.681 seconds) .. _sphx_glr_download_auto_examples_swissmetro_plot_b06unif_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_b06unif_mixture_integral.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_b06unif_mixture_integral.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_b06unif_mixture_integral.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_