2. Estimation with weights: 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 (
    EstimationResults,
    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. Here, we need to provide both the formula for the log likelihood function, and the formula for the weights. This is done via a dict with keys log_like and weight. 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 = 'b02_weight'

Estimate the parameters.

try:
    results = EstimationResults.from_yaml_file(
        filename=f'saved_results/{the_biogeme.model_name}.yaml'
    )
except FileNotFoundError:
    results = the_biogeme.estimate()

print(results.short_summary())

Get the results in a pandas table

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