Note
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25. Triangular mixture of logitΒΆ
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 numpy as np
from IPython.core.display_functions import display
import biogeme.biogeme_logging as blog
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 (
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 b25_triangular_mixture.py')
Example b25_triangular_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 = 'b25_triangular_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 __b25_triangular_mixture.iter
Cannot read file __b25_triangular_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 5.5e+03 0.092 1 0.39 +
1 -1 -1 2 -1 -1 5.5e+03 0.092 0.5 -0.57 -
2 -1.5 -0.82 1.9 -1.5 -0.5 5.4e+03 0.045 0.5 0.27 +
3 -1 -1.2 1.9 -1.2 -0.54 5.3e+03 0.035 0.5 0.88 +
4 -0.93 -1.3 2 -1.3 -0.038 5.3e+03 0.029 0.5 0.31 +
5 -0.52 -1.8 2 -1.1 -0.073 5.3e+03 0.025 0.5 0.42 +
6 -0.5 -1.8 2.5 -1.4 0.026 5.2e+03 0.019 0.5 0.41 +
7 -0.5 -1.8 2.5 -1.4 0.026 5.2e+03 0.019 0.25 0.045 -
8 -0.45 -1.8 2.6 -1.2 -0.043 5.2e+03 0.0094 0.25 0.6 +
9 -0.45 -1.8 2.6 -1.2 -0.043 5.2e+03 0.0094 0.12 -1 -
10 -0.45 -1.8 2.6 -1.2 -0.043 5.2e+03 0.0094 0.062 -0.057 -
11 -0.51 -1.9 2.7 -1.2 0.019 5.2e+03 0.0067 0.062 0.8 +
12 -0.48 -1.9 2.7 -1.2 0.017 5.2e+03 0.0046 0.062 0.84 +
13 -0.51 -1.9 2.8 -1.2 0.076 5.2e+03 0.0057 0.062 0.26 +
14 -0.46 -1.9 2.9 -1.2 0.048 5.2e+03 0.0068 0.062 0.84 +
15 -0.48 -1.9 2.9 -1.2 0.063 5.2e+03 0.0048 0.62 1 ++
16 -0.48 -1.9 2.9 -1.2 0.063 5.2e+03 0.0048 0.31 -0.0074 -
17 -0.47 -2 3.2 -1.2 0.11 5.2e+03 0.0044 0.31 0.6 +
18 -0.47 -2 3.2 -1.2 0.11 5.2e+03 0.0044 0.16 -1.4 -
19 -0.45 -2.1 3.3 -1.3 0.091 5.2e+03 0.0043 0.16 0.51 +
20 -0.41 -2.1 3.5 -1.2 0.14 5.2e+03 0.0068 0.16 0.59 +
21 -0.41 -2.1 3.5 -1.2 0.14 5.2e+03 0.0068 0.078 -0.85 -
22 -0.41 -2.1 3.5 -1.2 0.14 5.2e+03 0.0068 0.039 -0.73 -
23 -0.44 -2.1 3.5 -1.2 0.099 5.2e+03 0.0037 0.039 0.29 +
24 -0.42 -2.1 3.5 -1.2 0.11 5.2e+03 0.0018 0.39 0.95 ++
25 -0.36 -2.3 3.9 -1.3 0.15 5.2e+03 0.0034 0.39 0.17 +
26 -0.36 -2.3 3.9 -1.3 0.15 5.2e+03 0.0034 0.2 -2 -
27 -0.36 -2.3 3.9 -1.3 0.15 5.2e+03 0.0034 0.098 -0.66 -
28 -0.4 -2.3 4 -1.2 0.16 5.2e+03 0.0027 0.098 0.36 +
29 -0.4 -2.3 4 -1.2 0.16 5.2e+03 0.0027 0.049 -1.1 -
30 -0.4 -2.2 4 -1.3 0.12 5.2e+03 0.0026 0.049 0.4 +
31 -0.37 -2.3 4 -1.3 0.15 5.2e+03 0.00081 0.049 0.45 +
32 -0.37 -2.3 4 -1.3 0.15 5.2e+03 0.00081 0.024 -0.51 -
33 -0.4 -2.3 4 -1.3 0.15 5.2e+03 0.0013 0.024 0.12 +
34 -0.4 -2.3 4 -1.3 0.14 5.2e+03 0.00055 0.024 0.63 +
35 -0.4 -2.3 4 -1.3 0.14 5.2e+03 0.00055 0.012 -0.65 -
36 -0.4 -2.3 4 -1.3 0.14 5.2e+03 0.00055 0.0061 0.018 -
37 -0.39 -2.3 4 -1.3 0.14 5.2e+03 0.00059 0.0061 0.6 +
38 -0.39 -2.3 4 -1.3 0.14 5.2e+03 0.00018 0.0061 0.63 +
39 -0.39 -2.3 4 -1.3 0.14 5.2e+03 3.4e-05 0.0061 0.78 +
40 -0.39 -2.3 4 -1.3 0.14 5.2e+03 1e-05 0.0061 0.81 +
41 -0.39 -2.3 4 -1.3 0.14 5.2e+03 1e-05 0.0023 -1.4 -
42 -0.39 -2.3 4 -1.3 0.14 5.2e+03 1e-05 0.0012 -0.13 -
43 -0.39 -2.3 4 -1.3 0.14 5.2e+03 6.3e-06 0.0012 0.19 +
44 -0.39 -2.3 4 -1.3 0.14 5.2e+03 7.3e-06 0.0012 0.5 +
45 -0.39 -2.3 4 -1.3 0.14 5.2e+03 2e-06 0.0012 0.74 +
Optimization algorithm has converged.
Relative gradient: 1.9709841856068635e-06
Cause of termination: Relative gradient = 2e-06 <= 6.1e-06
Number of function evaluations: 107
Number of gradient evaluations: 61
Number of hessian evaluations: 0
Algorithm: BFGS with trust region for simple bound constraints
Number of iterations: 46
Proportion of Hessian calculation: 0/30 = 0.0%
Optimization time: 0:01:45.055364
Calculate second derivatives and BHHH
File b25_triangular_mixture.html has been generated.
File b25_triangular_mixture.yaml has been generated.
print(results.short_summary())
Results for model b25_triangular_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
Total running time of the script: (3 minutes 13.301 seconds)