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 biogeme.biogeme_logging as blog
import numpy as np
from IPython.core.display_functions import display
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 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 b25triangular_mixture.py')

Example b25triangular_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 = 'b25triangular_mixture'

Biogeme parameters read from biogeme.toml.

Estimate the parameters

results = the_biogeme.estimate()

*** Initial values of the parameters are obtained from the file __b25triangular_mixture.iter
Parameter values restored from __b25triangular_mixture.iter
Starting values for the algorithm: {'asc_train': -0.39397048552414804, 'b_time': -2.2726620702434417, 'b_time_s': 3.983067366643079, 'b_cost': -1.280403947694777, 'asc_car': 0.14025321441389768}
As the model is rather complex, we cancel the calculation of second derivatives. If you want to control the parameters, change the algorithm from "automatic" to "simple_bounds" in the TOML file.
Optimization algorithm: hybrid Newton/BFGS with simple bounds [simple_bounds]
** Optimization: BFGS with trust region for simple bounds
Optimization algorithm has converged.
Relative gradient: 2.623788529923212e-06
Cause of termination: Relative gradient = 2.6e-06 <= 6.1e-06
Number of function evaluations: 1
Number of gradient evaluations: 1
Number of hessian evaluations: 0
Algorithm: BFGS with trust region for simple bound constraints
Number of iterations: 0
Optimization time: 0:00:08.638604
Calculate second derivatives and BHHH
File b25triangular_mixture~00.html has been generated.
File b25triangular_mixture~00.yaml has been generated.

print(results.short_summary())

Results for model b25triangular_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)

        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