Heteroscedastic specification

Illustrates a heteroscedastic specification. A different scale is associated with different segments of the sample.

Michel Bierlaire, EPFL Wed Jun 18 2025, 11:25:26

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)
scale_group3 = Beta('scale_group3', 1, 0.001, 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

Scale associated with group 3 is estimated.

scale = (GROUP != 3) + (GROUP == 3) * scale_group3

Scale the utility functions, and associate them with the numbering of alternatives.

v = {1: scale * v_train, 2: scale * v_swissmetro, 3: scale * 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)

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

USER_NOTES = (
    'Illustrates heteroscedastic specification. A different scale is'
    ' associated with different segments of the sample.'
)

Create the Biogeme object.

the_biogeme = BIOGEME(database, logprob, user_notes=USER_NOTES)
the_biogeme.model_name = 'b03scale'

Estimate the parameters.

results = the_biogeme.estimate()

print(results.short_summary())

Results for model b03scale
Nbr of parameters:              5
Sample size:                    6768
Excluded data:                  3960
Final log likelihood:           -4976.691
Akaike Information Criterion:   9963.381
Bayesian Information Criterion: 9997.481

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  scale_group3  4.177737         0.370552       11.274371        0.000000
1     asc_train -0.447096         0.041146      -10.866099        0.000000
2        b_time -0.374455         0.044514       -8.412151        0.000000
3        b_cost -0.357349         0.038418       -9.301647        0.000000
4       asc_car -0.015332         0.018508       -0.828415        0.407435