Mixture with lognormal distribution

Example of a mixture of logit models, using Monte-Carlo integration. The mixing distribution is distributed as a log normal.

Michel Bierlaire, EPFL Thu Jun 26 2025, 15:31:41

from IPython.core.display_functions import display

import biogeme.biogeme_logging as blog
from biogeme.biogeme import BIOGEME
from biogeme.expressions import Beta, Draws, MonteCarlo, exp, 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 b17lognormal_mixture.py')

Example b17lognormal_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, normally distributed, designed to be used for Monte-Carlo simulation.

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

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

b_time_s = Beta('b_time_s', 1, -2, 2, 0)

Define a random parameter, log normally distributed, designed to be used for Monte-Carlo simulation.

b_time_rnd = -exp(b_time + b_time_s * Draws('b_time_rnd', 'NORMAL'))

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))

As the objective is to illustrate the syntax, we calculate the Monte-Carlo approximation with a small number of draws.

the_biogeme = BIOGEME(database, log_probability, number_of_draws=10_000, seed=1223)
the_biogeme.model_name = '17lognormal_mixture'

Biogeme parameters read from biogeme.toml.

Estimate the parameters.

try:
    results = EstimationResults.from_yaml_file(
        filename='saved_results/17lognormal_mixture.yaml'
    )
except FileNotFoundError:
    results = the_biogeme.estimate()

print(results.short_summary())

Results for model 17lognormal_mixture
Nbr of parameters:              5
Sample size:                    6768
Excluded data:                  3960
Final log likelihood:           -5231.272
Akaike Information Criterion:   10472.54
Bayesian Information Criterion: 10506.64

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 -0.346430         0.073266       -4.728357    2.263435e-06
1     b_time  0.575014         0.071294        8.065422    6.661338e-16
2   b_time_s  1.239151         0.128334        9.655646    0.000000e+00
3     b_cost -1.381117         0.097788      -14.123575    0.000000e+00
4    asc_car  0.173944         0.062414        2.786920    5.321162e-03