.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/swissmetro/plot_b05normal_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_b05normal_mixture_integral.py: Mixture of logit models ======================= Example of a normal mixture of logit models, using numerical integration. Michel Bierlaire, EPFL Fri Jun 20 2025, 10:25:34 .. GENERATED FROM PYTHON SOURCE LINES 11-19 .. 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, Integrate, IntegrateNormal, RandomVariable, log from biogeme.models import logit from biogeme.results_processing import get_pandas_estimated_parameters .. GENERATED FROM PYTHON SOURCE LINES 20-21 See the data processing script: :ref:`swissmetro_data`. .. GENERATED FROM PYTHON SOURCE LINES 21-38 .. 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 b05normal_mixture_integral.py') .. rst-class:: sphx-glr-script-out .. code-block:: none Example b05normal_mixture_integral.py .. GENERATED FROM PYTHON SOURCE LINES 39-40 Parameters to be estimated. .. GENERATED FROM PYTHON SOURCE LINES 40-45 .. 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 46-48 Define a random parameter, normally distributed, designed to be used for numerical integration. .. GENERATED FROM PYTHON SOURCE LINES 48-50 .. code-block:: Python b_time = Beta('b_time', 0, None, None, 0) .. GENERATED FROM PYTHON SOURCE LINES 51-52 It is advised not to use 0 as starting value for the following parameter. .. GENERATED FROM PYTHON SOURCE LINES 52-56 .. code-block:: Python b_time_s = Beta('b_time_s', 1, None, None, 0) omega = RandomVariable('omega') b_time_rnd = b_time + b_time_s * omega .. GENERATED FROM PYTHON SOURCE LINES 57-58 Definition of the utility functions. .. GENERATED FROM PYTHON SOURCE LINES 58-62 .. 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 63-64 Associate utility functions with the numbering of alternatives. .. GENERATED FROM PYTHON SOURCE LINES 64-66 .. code-block:: Python v = {1: v_train, 2: v_swissmetro, 3: v_car} .. GENERATED FROM PYTHON SOURCE LINES 67-68 Associate the availability conditions with the alternatives. .. GENERATED FROM PYTHON SOURCE LINES 68-70 .. code-block:: Python av = {1: TRAIN_AV_SP, 2: SM_AV, 3: CAR_AV_SP} .. GENERATED FROM PYTHON SOURCE LINES 71-72 Conditional on omega, we have a logit model (called the kernel). .. GENERATED FROM PYTHON SOURCE LINES 72-74 .. code-block:: Python conditional_probability = logit(v, av, CHOICE) .. GENERATED FROM PYTHON SOURCE LINES 75-76 We integrate over omega using numerical integration .. GENERATED FROM PYTHON SOURCE LINES 76-78 .. code-block:: Python log_probability = log(IntegrateNormal(conditional_probability, 'omega')) .. GENERATED FROM PYTHON SOURCE LINES 79-80 Create the Biogeme object .. GENERATED FROM PYTHON SOURCE LINES 80-88 .. code-block:: Python the_biogeme = BIOGEME( database, log_probability, optimization_algorithm='simple_bounds_BFGS', ) # the_biogeme = BIOGEME(database, logprob) the_biogeme.modelName = 'b05normal_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_b05normal_mixture_integral.py:86: DeprecationWarning: 'modelName' is deprecated. Please use 'model_name' instead. the_biogeme.modelName = 'b05normal_mixture_integral' .. GENERATED FROM PYTHON SOURCE LINES 89-90 Estimate the parameters .. GENERATED FROM PYTHON SOURCE LINES 90-92 .. 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 __b05normal_mixture_integral.iter Parameter values restored from __b05normal_mixture_integral.iter Starting values for the algorithm: {'asc_train': -0.3959009752580716, 'b_time': -2.2783610724991346, 'b_time_s': 1.675031645294375, 'b_cost': -1.288166620582709, 'asc_car': 0.14282068945852677} Optimization algorithm: BFGS with simple bounds [simple_bounds_BFGS]. Optimization algorithm: hybrid Newton/BFGS with simple bounds [simple_bounds] ** Optimization: BFGS with trust region for simple bounds Optimization algorithm has converged. Relative gradient: 5.669145273606387e-06 Cause of termination: Relative gradient = 5.7e-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: 0:00:00.381158 Calculate second derivatives and BHHH File b05normal_mixture_integral~00.html has been generated. File b05normal_mixture_integral~00.yaml has been generated. .. GENERATED FROM PYTHON SOURCE LINES 93-94 .. code-block:: Python print(results.short_summary()) .. rst-class:: sphx-glr-script-out .. code-block:: none Results for model b05normal_mixture_integral Nbr of parameters: 5 Sample size: 6768 Excluded data: 3960 Final log likelihood: -5213.725 Akaike Information Criterion: 10437.45 Bayesian Information Criterion: 10471.55 .. 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 -0.395901 0.063674 -6.217619 5.047569e-10 1 b_time -2.278361 0.117234 -19.434326 0.000000e+00 2 b_time_s 1.675032 0.102317 16.370945 0.000000e+00 3 b_cost -1.288167 0.086419 -14.906055 0.000000e+00 4 asc_car 0.142821 0.051744 2.760128 5.777867e-03 .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 1.982 seconds) .. _sphx_glr_download_auto_examples_swissmetro_plot_b05normal_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_b05normal_mixture_integral.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_b05normal_mixture_integral.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_b05normal_mixture_integral.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_