.. 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. :author: Michel Bierlaire, EPFL :date: Sun Apr 9 17:33:39 2023 .. GENERATED FROM PYTHON SOURCE LINES 12-24 .. code-block:: Python import biogeme.biogeme_logging as blog import biogeme.biogeme as bio import biogeme.distributions as dist from biogeme import models from biogeme.expressions import ( Beta, RandomVariable, log, Integrate, ) .. 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 ( database, CHOICE, SM_AV, CAR_AV_SP, TRAIN_AV_SP, TRAIN_TT_SCALED, TRAIN_COST_SCALED, SM_TT_SCALED, SM_COST_SCALED, CAR_TT_SCALED, CAR_CO_SCALED, ) 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 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, normally distributed, designed to be used for numerical integration. .. GENERATED FROM PYTHON SOURCE LINES 53-55 .. code-block:: Python B_TIME = Beta('B_TIME', 0, None, None, 0) .. GENERATED FROM PYTHON SOURCE LINES 56-57 It is advised not to use 0 as starting value for the following parameter. .. GENERATED FROM PYTHON SOURCE LINES 57-62 .. code-block:: Python B_TIME_S = Beta('B_TIME_S', 1, None, None, 0) omega = RandomVariable('omega') density = dist.normalpdf(omega) B_TIME_RND = B_TIME + B_TIME_S * omega .. GENERATED FROM PYTHON SOURCE LINES 63-64 Definition of the utility functions. .. GENERATED FROM PYTHON SOURCE LINES 64-68 .. code-block:: Python V1 = ASC_TRAIN + B_TIME_RND * TRAIN_TT_SCALED + B_COST * TRAIN_COST_SCALED V2 = ASC_SM + B_TIME_RND * SM_TT_SCALED + B_COST * SM_COST_SCALED V3 = ASC_CAR + B_TIME_RND * CAR_TT_SCALED + B_COST * CAR_CO_SCALED .. GENERATED FROM PYTHON SOURCE LINES 69-70 Associate utility functions with the numbering of alternatives. .. GENERATED FROM PYTHON SOURCE LINES 70-72 .. code-block:: Python V = {1: V1, 2: V2, 3: V3} .. GENERATED FROM PYTHON SOURCE LINES 73-74 Associate the availability conditions with the alternatives. .. GENERATED FROM PYTHON SOURCE LINES 74-76 .. code-block:: Python av = {1: TRAIN_AV_SP, 2: SM_AV, 3: CAR_AV_SP} .. GENERATED FROM PYTHON SOURCE LINES 77-78 Conditional on omega, we have a logit model (called the kernel). .. GENERATED FROM PYTHON SOURCE LINES 78-80 .. code-block:: Python condprob = models.logit(V, av, CHOICE) .. GENERATED FROM PYTHON SOURCE LINES 81-82 We integrate over omega using numerical integration .. GENERATED FROM PYTHON SOURCE LINES 82-84 .. code-block:: Python logprob = log(Integrate(condprob * density, 'omega')) .. GENERATED FROM PYTHON SOURCE LINES 85-86 Create the Biogeme object .. GENERATED FROM PYTHON SOURCE LINES 86-89 .. code-block:: Python the_biogeme = bio.BIOGEME(database, logprob) the_biogeme.modelName = 'b05normal_mixture_integral' .. 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 As the model is rather complex, we cancel the calculation of second derivatives. If you want to control the parameters, change the name of the algorithm in the TOML file from "automatic" to "simple_bounds" *** Initial values of the parameters are obtained from the file __b05normal_mixture_integral.iter Cannot read file __b05normal_mixture_integral.iter. Statement is ignored. As the model is rather complex, we cancel the calculation of second derivatives. If you want to control the parameters, change the name of the algorithm in the TOML file from "automatic" to "simple_bounds" Optimization algorithm: hybrid Newton/BFGS with simple bounds [simple_bounds] ** Optimization: BFGS with trust region for simple bounds Iter. ASC_CAR ASC_TRAIN B_COST B_TIME B_TIME_S Function Relgrad Radius Rho 0 1 -1 -1 -1 2 6.1e+03 0.16 1 0.25 + 1 0 -0.73 -0.4 -2 3 5.5e+03 0.049 1 0.36 + 2 0.51 -0.95 -1.4 -2.3 2.6 5.4e+03 0.054 1 0.39 + 3 0.51 -0.95 -1.4 -2.3 2.6 5.4e+03 0.054 0.5 -0.15 - 4 0.0062 -0.45 -1.1 -2.8 2.6 5.3e+03 0.03 0.5 0.49 + 5 0.33 -0.089 -1.6 -2.6 2.5 5.3e+03 0.047 0.5 0.14 + 6 0.33 -0.089 -1.6 -2.6 2.5 5.3e+03 0.047 0.25 -0.17 - 7 0.26 -0.34 -1.4 -2.9 2.3 5.2e+03 0.022 0.25 0.66 + 8 0.22 -0.3 -1.2 -2.6 2.2 5.2e+03 0.0086 0.25 0.53 + 9 0.22 -0.3 -1.2 -2.6 2.2 5.2e+03 0.0086 0.12 -3.2 - 10 0.22 -0.3 -1.2 -2.6 2.2 5.2e+03 0.0086 0.062 -0.28 - 11 0.28 -0.36 -1.3 -2.6 2.1 5.2e+03 0.0065 0.062 0.38 + 12 0.22 -0.3 -1.4 -2.6 2.1 5.2e+03 0.0064 0.062 0.54 + 13 0.21 -0.36 -1.3 -2.6 2 5.2e+03 0.0094 0.062 0.49 + 14 0.22 -0.33 -1.3 -2.5 2 5.2e+03 0.0035 0.062 0.89 + 15 0.18 -0.35 -1.3 -2.5 1.9 5.2e+03 0.0055 0.062 0.85 + 16 0.21 -0.36 -1.3 -2.4 1.9 5.2e+03 0.003 0.062 0.65 + 17 0.17 -0.37 -1.3 -2.4 1.8 5.2e+03 0.0021 0.062 0.84 + 18 0.17 -0.39 -1.3 -2.3 1.8 5.2e+03 0.002 0.062 0.58 + 19 0.13 -0.4 -1.3 -2.3 1.7 5.2e+03 0.0039 0.062 0.27 + 20 0.12 -0.41 -1.3 -2.2 1.7 5.2e+03 0.0019 0.062 0.18 + 21 0.12 -0.41 -1.3 -2.2 1.7 5.2e+03 0.0019 0.031 -0.47 - 22 0.13 -0.42 -1.3 -2.3 1.6 5.2e+03 0.0012 0.031 0.73 + 23 0.13 -0.42 -1.3 -2.3 1.6 5.2e+03 0.0012 0.016 -1.3 - 24 0.13 -0.42 -1.3 -2.3 1.6 5.2e+03 0.0012 0.0078 0.066 - 25 0.13 -0.41 -1.3 -2.2 1.7 5.2e+03 0.0011 0.0078 0.36 + 26 0.13 -0.41 -1.3 -2.3 1.7 5.2e+03 0.00016 0.0078 0.81 + 27 0.13 -0.41 -1.3 -2.3 1.7 5.2e+03 0.00016 0.0039 -1.1 - 28 0.13 -0.41 -1.3 -2.3 1.7 5.2e+03 0.00016 0.002 -0.15 - 29 0.14 -0.4 -1.3 -2.3 1.7 5.2e+03 0.0003 0.002 0.36 + 30 0.14 -0.4 -1.3 -2.3 1.7 5.2e+03 7.1e-05 0.002 0.69 + Results saved in file b05normal_mixture_integral.html Results saved in file b05normal_mixture_integral.pickle .. 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 b05normal_mixture_integral Nbr of parameters: 5 Sample size: 6768 Excluded data: 3960 Final log likelihood: -5214.88 Akaike Information Criterion: 10439.76 Bayesian Information Criterion: 10473.86 .. GENERATED FROM PYTHON SOURCE LINES 97-99 .. code-block:: Python pandas_results = results.get_estimated_parameters() pandas_results .. raw:: html
Value Rob. Std err Rob. t-test Rob. p-value
ASC_CAR 0.134900 0.047631 2.832218 4.622627e-03
ASC_TRAIN -0.403873 0.058895 -6.857458 7.009726e-12
B_COST -1.284092 0.046169 -27.812680 0.000000e+00
B_TIME -2.254195 0.118544 -19.015637 0.000000e+00
B_TIME_S 1.649665 0.136564 12.079820 0.000000e+00


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