Note
Go to the end to download the full example code.
24. 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
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, 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 b24_halton_mixture.py')
Example b24_halton_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 = 'b24_halton_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()
*** Initial values of the parameters are obtained from the file __b24_halton_mixture.iter
Cannot read file __b24_halton_mixture.iter. Statement is ignored.
Starting values for the algorithm: {}
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 -1 -1 2 -1 1 6.1e+03 0.16 1 0.25 +
1 -0.73 -2 3 -0.4 0 5.5e+03 0.049 1 0.36 +
2 -0.95 -2.3 2.6 -1.4 0.51 5.4e+03 0.054 1 0.39 +
3 -0.95 -2.3 2.6 -1.4 0.51 5.4e+03 0.054 0.5 -0.15 -
4 -0.45 -2.8 2.6 -1.1 0.0057 5.3e+03 0.03 0.5 0.5 +
5 -0.092 -2.6 2.5 -1.6 0.33 5.3e+03 0.046 0.5 0.14 +
6 -0.092 -2.6 2.5 -1.6 0.33 5.3e+03 0.046 0.25 -0.19 -
7 -0.34 -2.9 2.3 -1.4 0.26 5.2e+03 0.022 0.25 0.65 +
8 -0.3 -2.6 2.2 -1.2 0.22 5.2e+03 0.0084 0.25 0.51 +
9 -0.3 -2.6 2.2 -1.2 0.22 5.2e+03 0.0084 0.12 -3 -
10 -0.3 -2.6 2.2 -1.2 0.22 5.2e+03 0.0084 0.062 -0.22 -
11 -0.36 -2.6 2.1 -1.3 0.28 5.2e+03 0.0063 0.062 0.42 +
12 -0.3 -2.6 2.1 -1.4 0.22 5.2e+03 0.0065 0.062 0.54 +
13 -0.35 -2.6 2 -1.3 0.21 5.2e+03 0.0096 0.062 0.47 +
14 -0.33 -2.5 2 -1.3 0.22 5.2e+03 0.0034 0.62 0.9 ++
15 -0.33 -2.5 2 -1.3 0.22 5.2e+03 0.0034 0.31 -0.57 -
16 -0.38 -2.3 1.7 -1.2 0.12 5.2e+03 0.0045 0.31 0.47 +
17 -0.38 -2.3 1.7 -1.2 0.12 5.2e+03 0.0045 0.16 -3.1 -
18 -0.38 -2.3 1.7 -1.2 0.12 5.2e+03 0.0045 0.078 -2.2 -
19 -0.38 -2.3 1.7 -1.2 0.12 5.2e+03 0.0045 0.039 -0.99 -
20 -0.42 -2.2 1.6 -1.3 0.15 5.2e+03 0.0034 0.039 0.21 +
21 -0.4 -2.2 1.6 -1.3 0.12 5.2e+03 0.0018 0.039 0.18 +
22 -0.41 -2.2 1.6 -1.3 0.13 5.2e+03 0.00021 0.039 0.77 +
23 -0.41 -2.2 1.6 -1.3 0.13 5.2e+03 0.00021 0.02 -2.9 -
24 -0.41 -2.2 1.6 -1.3 0.13 5.2e+03 0.00021 0.0098 -0.57 -
25 -0.4 -2.3 1.6 -1.3 0.14 5.2e+03 0.00039 0.0098 0.24 +
26 -0.4 -2.3 1.6 -1.3 0.14 5.2e+03 0.00039 0.0049 -0.99 -
27 -0.4 -2.3 1.6 -1.3 0.14 5.2e+03 0.00039 0.0024 -0.33 -
28 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 0.00018 0.0024 0.46 +
29 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 0.00014 0.0024 0.69 +
30 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 0.00021 0.0024 0.15 +
31 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 0.00021 0.0012 -0.49 -
32 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 8.1e-05 0.0012 0.39 +
33 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 3.1e-05 0.0012 0.68 +
34 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 2.4e-05 0.0012 0.57 +
35 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 2.4e-05 0.00061 -1.9 -
36 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 2.4e-05 0.00031 -0.91 -
37 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 2.4e-05 0.00015 -0.17 -
38 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 1.2e-05 0.00015 0.68 +
39 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 8.1e-06 0.00015 0.74 +
40 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 5.3e-06 0.00015 0.57 +
Optimization algorithm has converged.
Relative gradient: 5.2868161203223544e-06
Cause of termination: Relative gradient = 5.3e-06 <= 6.1e-06
Number of function evaluations: 92
Number of gradient evaluations: 51
Number of hessian evaluations: 0
Algorithm: BFGS with trust region for simple bound constraints
Number of iterations: 41
Proportion of Hessian calculation: 0/25 = 0.0%
Optimization time: 0:01:30.115353
Calculate second derivatives and BHHH
File b24_halton_mixture.html has been generated.
File b24_halton_mixture.yaml has been generated.
print(results.short_summary())
Results for model b24_halton_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 -0.401952 0.065839 -6.105116 1.027262e-09
1 b_time -2.259584 0.117082 -19.299092 0.000000e+00
2 b_time_s 1.657308 0.131713 12.582698 0.000000e+00
3 b_cost -1.285299 0.086297 -14.893935 0.000000e+00
4 asc_car 0.137026 0.051721 2.649327 8.065219e-03
Total running time of the script: (3 minutes 34.189 seconds)