Note
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Mixture of logit with Halton drawsΒΆ
Example of a mixture of logit models, using quasi Monte-Carlo integration with Halton draws (base 5). The mixing distribution is normal.
Michel Bierlaire, EPFL Sat Jun 28 2025, 12:45:21
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 b24halton_mixture.py')
Example b24halton_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, None, None, 0)
Define a random parameter with a normal distribution, designed to be used for quasi Monte-Carlo simulation with Halton draws (base 5).
b_time_rnd = b_time + b_time_s * Draws('b_time_rnd', 'NORMAL_HALTON5')
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 on 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))
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 with Halton draws.'
)
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, user_notes=USER_NOTES, number_of_draws=10_000, seed=1223
)
the_biogeme.model_name = 'b24halton_mixture'
Biogeme parameters read from biogeme.toml.
Estimate the parameters
results = the_biogeme.estimate()
*** Initial values of the parameters are obtained from the file __b24halton_mixture.iter
Parameter values restored from __b24halton_mixture.iter
Starting values for the algorithm: {'asc_train': -0.40195185364181046, 'b_time': -2.2595838469305267, 'b_time_s': 1.6573078631733604, 'b_cost': -1.2852992156480805, 'asc_car': 0.1370262574783267}
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
Iter. asc_train b_time b_time_s b_cost asc_car Function Relgrad Radius Rho
0 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 7e-06 0.018 -8.5e+02 -
1 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 7e-06 0.0091 -4.3e+02 -
2 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 7e-06 0.0045 -1.8e+02 -
3 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 7e-06 0.0023 -76 -
4 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 7e-06 0.0011 -35 -
5 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 7e-06 0.00057 -17 -
6 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 7e-06 0.00028 -7.6 -
7 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 7e-06 0.00014 -3.3 -
8 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 7e-06 7.1e-05 -1.1 -
9 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 7e-06 3.6e-05 -0.065 -
10 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 4.4e-06 3.6e-05 0.47 -
Optimization algorithm has converged.
Relative gradient: 4.391778983297403e-06
Cause of termination: Relative gradient = 4.4e-06 <= 6.1e-06
Number of function evaluations: 14
Number of gradient evaluations: 3
Number of hessian evaluations: 0
Algorithm: BFGS with trust region for simple bound constraints
Number of iterations: 11
Proportion of Hessian calculation: 0/1 = 0.0%
Optimization time: 0:00:28.744647
Calculate second derivatives and BHHH
File b24halton_mixture~00.html has been generated.
File b24halton_mixture~00.yaml has been generated.
print(results.short_summary())
Results for model b24halton_mixture
Nbr of parameters: 5
Sample size: 6768
Excluded data: 3960
Final log likelihood: -5214.905
Akaike Information Criterion: 10439.81
Bayesian Information Criterion: 10473.91
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 -0.401916 0.065838 -6.104587 1.030669e-09
1 b_time -2.259619 0.117085 -19.299005 0.000000e+00
2 b_time_s 1.657343 0.131714 12.582927 0.000000e+00
3 b_cost -1.285264 0.086295 -14.893876 0.000000e+00
4 asc_car 0.136991 0.051721 2.648645 8.081517e-03
Total running time of the script: (2 minutes 32.643 seconds)