25. Triangular mixture of logit

Example of a mixture of logit models, using Monte-Carlo integration. The mixing distribution is specified by the user. Here, a triangular distribution.

Michel Bierlaire, EPFL Sat Jun 28 2025, 12:49:10

import numpy as np
from IPython.core.display_functions import display

import biogeme.biogeme_logging as blog
from biogeme.biogeme import BIOGEME
from biogeme.draws import RandomNumberGeneratorTuple
from biogeme.expressions import Beta, Draws, MonteCarlo, log
from biogeme.models import logit
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,
    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)
logger.info('Example b25_triangular_mixture.py')

Example b25_triangular_mixture.py

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

Define a random parameter with a triangular distribution, designed to be used for Monte-Carlo simulation. The triangular distribution is not directly available from Biogeme. The draws have to be generated by a function provided by the user.

Mean of the distribution.

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

Scale of the distribution. It is advised not to use 0 as starting value for the following parameter.

b_time_s = Beta('b_time_s', 1, None, None, 0)

Function generating the draws.

def the_triangular_generator(sample_size: int, number_of_draws: int) -> np.ndarray:
    """
    User-defined random number generator to the database.
    See the `numpy.random` documentation to obtain a list of other distributions.
    """
    return np.random.triangular(-1, 0, 1, (sample_size, number_of_draws))

Associate the function with a name.

my_random_number_generators = {
    'TRIANGULAR': RandomNumberGeneratorTuple(
        generator=the_triangular_generator,
        description='Draws from a triangular distribution',
    )
}

Define a random parameter with a triangular distribution, designed to be used for Monte-Carlo simulation.

b_time_rnd = b_time + b_time_s * Draws('b_time_rnd', 'TRIANGULAR')

Definition of the utility functions.

v_train = asc_train + b_time_rnd * TRAIN_TT_SCALED + b_cost * TRAIN_COST_SCALED
v_swissmetro = asc_sm + b_time_rnd * SM_TT_SCALED + b_cost * SM_COST_SCALED
v_car = asc_car + b_time_rnd * 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}

Conditional to b_time_rnd, we have a logit model (called the kernel)

conditional_probability = logit(v, av, CHOICE)

We integrate over b_time_rnd using Monte-Carlo

log_probability = log(MonteCarlo(conditional_probability))

the_biogeme = BIOGEME(
    database,
    log_probability,
    random_number_generators=my_random_number_generators,
    number_of_draws=10_000,
    seed=1223,
)
the_biogeme.model_name = 'b25_triangular_mixture'

Biogeme parameters read from biogeme.toml.

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())

Results for model b25_triangular_mixture
Nbr of parameters:              5
Sample size:                    6768
Excluded data:                  3960
Final log likelihood:           -5214.972
Akaike Information Criterion:   10439.94
Bayesian Information Criterion: 10474.04

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

{'Estimated parameters':         Name     Value  Robust std err.  Robust t-stat.  Robust p-value
0  asc_train -0.393970         0.065698       -5.996645    2.014361e-09
1     b_time -2.272662         0.119136      -19.076255    0.000000e+00
2   b_time_s  3.983067         0.306149       13.010235    0.000000e+00
3     b_cost -1.280404         0.086226      -14.849390    0.000000e+00
4    asc_car  0.140253         0.052139        2.689983    7.145576e-03}