.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/bayesian_swissmetro/plot_b08_boxcox.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_bayesian_swissmetro_plot_b08_boxcox.py: 8. Box-Cox transforms ===================== Bayesian estimation of a logit model, with a Box-Cox transform of variables. Michel Bierlaire, EPFL Mon Nov 03 2025, 13:41:40 .. GENERATED FROM PYTHON SOURCE LINES 11-19 .. code-block:: Python from IPython.core.display_functions import display import biogeme.biogeme_logging as blog from biogeme.bayesian_estimation import BayesianResults, get_pandas_estimated_parameters from biogeme.biogeme import BIOGEME from biogeme.expressions import Beta from biogeme.models import boxcox, loglogit .. GENERATED FROM PYTHON SOURCE LINES 20-21 See the data processing script: :ref:`swissmetro_data`. .. GENERATED FROM PYTHON SOURCE LINES 21-37 .. 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) .. GENERATED FROM PYTHON SOURCE LINES 38-39 Parameters to be estimated. .. GENERATED FROM PYTHON SOURCE LINES 39-46 .. 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_time = Beta('b_time', 0, None, 0, 0) b_cost = Beta('b_cost', 0, None, 0, 0) boxcox_parameter = Beta('boxcox_parameter', 1, -10, 10, 0) .. GENERATED FROM PYTHON SOURCE LINES 47-48 Definition of the utility functions. .. GENERATED FROM PYTHON SOURCE LINES 48-60 .. code-block:: Python v_train = ( asc_train + b_time * boxcox(TRAIN_TT_SCALED, boxcox_parameter) + b_cost * TRAIN_COST_SCALED ) v_swissmetro = ( asc_sm + b_time * boxcox(SM_TT_SCALED, boxcox_parameter) + b_cost * SM_COST_SCALED ) v_car = ( asc_car + b_time * boxcox(CAR_TT_SCALED, boxcox_parameter) + b_cost * CAR_CO_SCALED ) .. GENERATED FROM PYTHON SOURCE LINES 61-62 Associate utility functions with the numbering of alternatives. .. GENERATED FROM PYTHON SOURCE LINES 62-64 .. code-block:: Python v = {1: v_train, 2: v_swissmetro, 3: v_car} .. GENERATED FROM PYTHON SOURCE LINES 65-66 Associate the availability conditions with the alternatives. .. GENERATED FROM PYTHON SOURCE LINES 66-68 .. code-block:: Python av = {1: TRAIN_AV_SP, 2: SM_AV, 3: CAR_AV_SP} .. GENERATED FROM PYTHON SOURCE LINES 69-71 Definition of the model. This is the contribution of each observation to the log likelihood function. .. GENERATED FROM PYTHON SOURCE LINES 71-73 .. code-block:: Python log_probability = loglogit(v, av, CHOICE) .. GENERATED FROM PYTHON SOURCE LINES 74-75 Create the Biogeme object. .. GENERATED FROM PYTHON SOURCE LINES 75-78 .. code-block:: Python the_biogeme = BIOGEME(database, log_probability, bayesian_draws=10000, warmup=10000) the_biogeme.model_name = 'b08_boxcox' .. rst-class:: sphx-glr-script-out .. code-block:: none Biogeme parameters read from biogeme.toml. .. GENERATED FROM PYTHON SOURCE LINES 79-80 Estimate the parameters. .. GENERATED FROM PYTHON SOURCE LINES 80-87 .. code-block:: Python try: results = BayesianResults.from_netcdf( filename=f'saved_results/{the_biogeme.model_name}.nc' ) except FileNotFoundError: results = the_biogeme.bayesian_estimation() .. rst-class:: sphx-glr-script-out .. code-block:: none Loaded NetCDF file size: 511.0 MB load finished in 5887 ms (5.89 s) .. GENERATED FROM PYTHON SOURCE LINES 88-90 .. code-block:: Python print(results.short_summary()) .. rst-class:: sphx-glr-script-out .. code-block:: none posterior_predictive_loglike finished in 1561 ms (1.56 s) expected_log_likelihood finished in 54 ms best_draw_log_likelihood finished in 54 ms /Users/bierlair/python_envs/venv313/lib/python3.13/site-packages/arviz/stats/stats.py:1667: UserWarning: For one or more samples the posterior variance of the log predictive densities exceeds 0.4. This could be indication of WAIC starting to fail. See http://arxiv.org/abs/1507.04544 for details warnings.warn( waic_res finished in 3168 ms (3.17 s) waic finished in 3168 ms (3.17 s) /Users/bierlair/python_envs/venv313/lib/python3.13/site-packages/arviz/stats/stats.py:797: UserWarning: Estimated shape parameter of Pareto distribution is greater than 0.70 for one or more samples. You should consider using a more robust model, this is because importance sampling is less likely to work well if the marginal posterior and LOO posterior are very different. This is more likely to happen with a non-robust model and highly influential observations. warnings.warn( loo_res finished in 95121 ms (1.59 min) loo finished in 95121 ms (1.59 min) Sample size 6768 Sampler NUTS Number of chains 4 Number of draws per chain 10000 Total number of draws 40000 Acceptance rate target 0.9 Run time 0:01:25.194293 Posterior predictive log-likelihood (sum of log mean p) -5267.39 Expected log-likelihood E[log L(Y|θ)] -17340.91 Best-draw log-likelihood (posterior upper bound) -7689.05 WAIC (Widely Applicable Information Criterion) -25780955.15 WAIC Standard Error 22143194.30 Effective number of parameters (p_WAIC) 25775687.77 LOO (Leave-One-Out Cross-Validation) -37215.61 LOO Standard Error 12236.43 Effective number of parameters (p_LOO) 31948.23 .. GENERATED FROM PYTHON SOURCE LINES 91-93 .. code-block:: Python pandas_results = get_pandas_estimated_parameters(estimation_results=results) display(pandas_results) .. rst-class:: sphx-glr-script-out .. code-block:: none Diagnostics computation took 199.0 seconds (cached). Name Value (mean) ... ESS (bulk) ESS (tail) 0 asc_train -0.142829 ... 4.003203 4.003203 1 asc_car 0.168284 ... 4.003203 4.003203 2 b_time -1.275373 ... 4.003203 4.003203 3 boxcox_parameter 0.661758 ... 4.003203 4.003203 4 b_cost -1.377411 ... 4.003203 4.003203 [5 rows x 12 columns] .. rst-class:: sphx-glr-timing **Total running time of the script:** (5 minutes 5.254 seconds) .. _sphx_glr_download_auto_examples_bayesian_swissmetro_plot_b08_boxcox.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_b08_boxcox.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_b08_boxcox.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_b08_boxcox.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_