WESML

Example of a logit model with Weighted Exogenous Sample Maximum Likelihood (WESML).

Michel Bierlaire, EPFL Wed Jun 18 2025, 11:20:51

from IPython.core.display_functions import display
from biogeme.biogeme import BIOGEME
from biogeme.expressions import Beta
from biogeme.models import loglogit
from biogeme.results_processing import get_pandas_estimated_parameters

See the data processing script: Data preparation for Swissmetro.

from swissmetro_data import (
    CAR_AV_SP,
    CAR_CO_SCALED,
    CAR_TT_SCALED,
    CHOICE,
    GROUP,
    SM_AV,
    SM_COST_SCALED,
    SM_TT_SCALED,
    TRAIN_AV_SP,
    TRAIN_COST_SCALED,
    TRAIN_TT_SCALED,
    database,
)

Parameters to be estimated.

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, None, 0)
b_cost = Beta('b_cost', 0, None, None, 0)

Definition of the utility functions.

v_train = asc_train + b_time * TRAIN_TT_SCALED + b_cost * TRAIN_COST_SCALED
v_swissmetro = asc_sm + b_time * SM_TT_SCALED + b_cost * SM_COST_SCALED
v_car = asc_car + b_time * CAR_TT_SCALED + b_cost * CAR_CO_SCALED

Associate utility functions with the numbering of alternatives.

v = {1: v_train, 2: v_swissmetro, 3: v_car}

Associate the availability conditions with the alternatives.

av = {1: TRAIN_AV_SP, 2: SM_AV, 3: CAR_AV_SP}

Definition of the model.

This is the contribution of each observation to the log likelihood function.

logprob = loglogit(v, av, CHOICE)

Definition of the weight.

WEIGHT_GROUP_2 = 8.890991e-01
WEIGHT_GROUP_3 = 1.2
weight = WEIGHT_GROUP_2 * (GROUP == 2) + WEIGHT_GROUP_3 * (GROUP == 3)

These notes will be included as such in the report file.

USER_NOTES = (
    'Example of a logit model with three alternatives: '
    'Train, Car and Swissmetro.'
    ' Weighted Exogenous Sample Maximum Likelihood estimator (WESML)'
)

Create the Biogeme object. It is possible to control the generation of the HTML and the yaml files. Note that these parameters can also be modified in the .TOML configuration file.

formulas = {'log_like': logprob, 'weight': weight}
the_biogeme = BIOGEME(
    database, formulas, user_notes=USER_NOTES, generate_html=True, generate_yaml=False
)
the_biogeme.model_name = 'b02weight'

Estimate the parameters.

results = the_biogeme.estimate()
print(results.short_summary())

Results for model b02weight
Nbr of parameters:              4
Sample size:                    6768
Excluded data:                  3960
Final log likelihood:           -5669.069
Akaike Information Criterion:   11346.14
Bayesian Information Criterion: 11373.42

Get the results in a pandas table

pandas_results = get_pandas_estimated_parameters(estimation_results=results)
display(pandas_results)

        Name     Value  Robust std err.  Robust t-stat.  Robust p-value
0  asc_train -0.795330         0.079430      -10.012971        0.000000
1     b_time -1.347405         0.096642      -13.942173        0.000000
2     b_cost -1.141353         0.064112      -17.802360        0.000000
3    asc_car -0.091280         0.053915       -1.693050        0.090446