Mixture of logit models

Example of a normal mixture of logit models, using Monte-Carlo integration.

Michel Bierlaire, EPFL Wed Jun 18 2025, 11:28:46

import biogeme.biogeme_logging as blog
from IPython.core.display_functions import display
from biogeme.biogeme import BIOGEME
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 b05normal_mixtures.py')

Example b05normal_mixtures.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, None, None, 0)
b_time_rnd = 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).

prob = logit(v, av, CHOICE)

We integrate over b_time_rnd using Monte-Carlo.

logprob = log(MonteCarlo(prob))

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

USER_NOTES = (
    'Example of a mixture of logit models with three alternatives, '
    'approximated using Monte-Carlo integration.'
)

Create the Biogeme object.

the_biogeme = BIOGEME(
    database, logprob, user_notes=USER_NOTES, number_of_draws=10000, seed=1223
)
the_biogeme.model_name = 'b05normal_mixture'

Biogeme parameters read from biogeme.toml.

print(f'Number of draws: {the_biogeme.number_of_draws}')

Number of draws: 10000

Estimate the parameters

results = the_biogeme.estimate()

*** Initial values of the parameters are obtained from the file __b05normal_mixture.iter
Parameter values restored from __b05normal_mixture.iter
Starting values for the algorithm: {'asc_train': -0.4026421286359191, 'b_time': -2.256833276642447, 'b_time_s': 1.654339197707796, 'b_cost': -1.2846226483353265, 'asc_car': 0.13632028087892392}
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: 5.906997030385587e-06
Cause of termination: Relative gradient = 5.9e-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: 0:00:02.141733
Calculate second derivatives and BHHH
File b05normal_mixture~00.html has been generated.
File b05normal_mixture~00.yaml has been generated.

print(results.short_summary())

Results for model b05normal_mixture
Nbr of parameters:              5
Sample size:                    6768
Excluded data:                  3960
Final log likelihood:           -5215.694
Akaike Information Criterion:   10441.39
Bayesian Information Criterion: 10475.49

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 -0.402642         0.065846       -6.114866    9.663816e-10
1     b_time -2.256833         0.117184      -19.258931    0.000000e+00
2   b_time_s  1.654339         0.131836       12.548455    0.000000e+00
3     b_cost -1.284623         0.086250      -14.894150    0.000000e+00
4    asc_car  0.136320         0.051755        2.633961    8.439514e-03