Note
Go to the end to download the full example code.
17a. Mixture with lognormal distributionΒΆ
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 = 'b17a_lognormal_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 __b17a_lognormal_mixture.iter
Cannot read file __b17a_lognormal_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 0 0 1 0 0 5.7e+03 0.096 0.5 -0.00064 -
1 -0.5 0.5 1.5 -0.5 0.5 5.4e+03 0.042 0.5 0.39 +
2 -0.34 0.69 1.5 -1 0 5.3e+03 0.04 0.5 0.34 +
3 -0.34 0.69 1.5 -1 0 5.3e+03 0.04 0.25 -0.63 -
4 -0.39 0.44 1.3 -1.2 0.25 5.2e+03 0.017 0.25 0.49 +
5 -0.39 0.61 1.3 -1.5 0.084 5.2e+03 0.0097 0.25 0.25 +
6 -0.39 0.61 1.3 -1.5 0.084 5.2e+03 0.0097 0.12 -2.4 -
7 -0.39 0.61 1.3 -1.5 0.084 5.2e+03 0.0097 0.062 0.017 -
8 -0.32 0.55 1.2 -1.4 0.15 5.2e+03 0.0067 0.062 0.58 +
9 -0.39 0.61 1.2 -1.4 0.17 5.2e+03 0.0065 0.062 0.15 +
10 -0.33 0.57 1.3 -1.3 0.17 5.2e+03 0.0029 0.062 0.39 +
11 -0.33 0.57 1.3 -1.3 0.17 5.2e+03 0.0029 0.031 -1.2 -
12 -0.36 0.57 1.2 -1.4 0.2 5.2e+03 0.0036 0.031 0.23 +
13 -0.34 0.58 1.2 -1.4 0.17 5.2e+03 0.0014 0.031 0.61 +
14 -0.34 0.58 1.2 -1.4 0.17 5.2e+03 0.0014 0.016 -3.7 -
15 -0.34 0.58 1.2 -1.4 0.17 5.2e+03 0.0014 0.0078 -0.65 -
16 -0.35 0.57 1.2 -1.4 0.17 5.2e+03 0.00074 0.0078 0.49 +
17 -0.35 0.58 1.2 -1.4 0.17 5.2e+03 0.00022 0.0078 0.65 +
18 -0.35 0.58 1.2 -1.4 0.17 5.2e+03 0.00022 0.0039 -0.37 -
19 -0.35 0.58 1.2 -1.4 0.18 5.2e+03 0.00023 0.0039 0.34 +
20 -0.35 0.58 1.2 -1.4 0.18 5.2e+03 0.00023 0.002 -0.78 -
21 -0.35 0.57 1.2 -1.4 0.17 5.2e+03 0.0002 0.002 0.13 +
22 -0.35 0.58 1.2 -1.4 0.17 5.2e+03 0.00013 0.002 0.51 +
23 -0.35 0.58 1.2 -1.4 0.17 5.2e+03 0.00013 0.00098 -0.59 -
24 -0.35 0.58 1.2 -1.4 0.17 5.2e+03 0.00013 0.00049 -0.037 -
25 -0.35 0.57 1.2 -1.4 0.17 5.2e+03 3.6e-05 0.00049 0.65 +
26 -0.35 0.57 1.2 -1.4 0.17 5.2e+03 3.6e-05 0.00024 0.074 -
27 -0.35 0.57 1.2 -1.4 0.17 5.2e+03 1.6e-05 0.00024 0.61 +
28 -0.35 0.57 1.2 -1.4 0.17 5.2e+03 2e-05 0.00024 0.41 +
29 -0.35 0.58 1.2 -1.4 0.17 5.2e+03 7.6e-06 0.00024 0.37 +
30 -0.35 0.58 1.2 -1.4 0.17 5.2e+03 4.8e-06 0.00024 0.43 +
Optimization algorithm has converged.
Relative gradient: 4.8025028162905605e-06
Cause of termination: Relative gradient = 4.8e-06 <= 6.1e-06
Number of function evaluations: 70
Number of gradient evaluations: 39
Number of hessian evaluations: 0
Algorithm: BFGS with trust region for simple bound constraints
Number of iterations: 31
Proportion of Hessian calculation: 0/19 = 0.0%
Optimization time: 0:01:21.466994
Calculate second derivatives and BHHH
File b17a_lognormal_mixture.html has been generated.
File b17a_lognormal_mixture.yaml has been generated.
print(results.short_summary())
Results for model b17a_lognormal_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
Total running time of the script: (3 minutes 0.119 seconds)