Note
Go to the end to download the full example code.
6a. Mixture of logit models with uniform distributionΒΆ
Example of a uniform mixture of logit models, using Monte-Carlo integration.
Michel Bierlaire, EPFL Fri Jun 20 2025, 10:43:05
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 b06a_unif_mixture.py')
Example b06a_unif_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, uniformly distributed, designed to be used for Monte-Carlo simulation.
b_time = Beta('b_time', 0, None, None, 0)
b_time_s = Beta('b_time_s', 1, None, None, 0)
b_time_rnd = b_time + b_time_s * Draws('b_time_rnd', 'UNIFORMSYM')
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))
Create the Biogeme object.
the_biogeme = BIOGEME(database, log_probability, number_of_draws=10000, seed=1223)
the_biogeme.model_name = 'b06a_unif_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 __b06a_unif_mixture.iter
Cannot read file __b06a_unif_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 5.6e+03 0.09 1 0.39 +
1 -1 -1 2 -1 -1 5.6e+03 0.09 0.5 -0.11 -
2 -1.5 -1.3 1.5 -1.5 -0.5 5.4e+03 0.074 0.5 0.32 +
3 -1.5 -1.3 1.5 -1.5 -0.5 5.4e+03 0.074 0.25 -0.17 -
4 -1.2 -1 1.8 -1.2 -0.25 5.3e+03 0.028 0.25 0.47 +
5 -1 -1.3 1.5 -1 -0.5 5.3e+03 0.04 0.25 0.16 +
6 -1 -1.3 1.5 -1 -0.5 5.3e+03 0.04 0.12 -0.064 -
7 -0.88 -1.2 1.5 -1.1 -0.38 5.3e+03 0.024 0.12 0.63 +
8 -1 -1.3 1.4 -1.2 -0.25 5.3e+03 0.017 0.12 0.33 +
9 -0.88 -1.4 1.5 -1.1 -0.24 5.3e+03 0.0092 0.12 0.88 +
10 -0.75 -1.5 1.7 -1.3 -0.11 5.2e+03 0.012 0.12 0.71 +
11 -0.68 -1.6 1.7 -1.1 -0.079 5.2e+03 0.0073 0.12 0.71 +
12 -0.56 -1.7 1.9 -1.2 -0.043 5.2e+03 0.0092 1.2 0.91 ++
13 -0.56 -1.7 1.9 -1.2 -0.043 5.2e+03 0.0092 0.62 -2.1 -
14 -0.56 -1.7 1.9 -1.2 -0.043 5.2e+03 0.0092 0.31 -1.4 -
15 -0.56 -1.7 1.9 -1.2 -0.043 5.2e+03 0.0092 0.16 -0.13 -
16 -0.59 -1.9 2 -1.2 0.049 5.2e+03 0.01 0.16 0.38 +
17 -0.53 -1.9 2.2 -1.2 -0.011 5.2e+03 0.0045 0.16 0.81 +
18 -0.53 -1.9 2.2 -1.2 -0.011 5.2e+03 0.0045 0.078 -0.45 -
19 -0.52 -1.9 2.2 -1.2 0.067 5.2e+03 0.0078 0.078 0.31 +
20 -0.48 -2 2.3 -1.2 0.04 5.2e+03 0.0037 0.78 0.95 ++
21 -0.48 -2 2.3 -1.2 0.04 5.2e+03 0.0037 0.39 -1.2 -
22 -0.48 -2 2.3 -1.2 0.04 5.2e+03 0.0037 0.2 -0.73 -
23 -0.48 -2 2.3 -1.2 0.04 5.2e+03 0.0037 0.098 0.068 -
24 -0.5 -2.1 2.4 -1.3 0.063 5.2e+03 0.0065 0.098 0.48 +
25 -0.46 -2.1 2.5 -1.2 0.097 5.2e+03 0.0048 0.098 0.82 +
26 -0.44 -2.2 2.6 -1.2 0.11 5.2e+03 0.0033 0.98 0.94 ++
27 -0.44 -2.2 2.6 -1.2 0.11 5.2e+03 0.0033 0.48 -20 -
28 -0.44 -2.2 2.6 -1.2 0.11 5.2e+03 0.0033 0.24 -4.1 -
29 -0.44 -2.2 2.6 -1.2 0.11 5.2e+03 0.0033 0.12 -0.64 -
30 -0.41 -2.2 2.7 -1.3 0.11 5.2e+03 0.003 0.12 0.34 +
31 -0.41 -2.2 2.7 -1.3 0.11 5.2e+03 0.003 0.06 -0.52 -
32 -0.41 -2.2 2.7 -1.3 0.12 5.2e+03 0.00083 0.06 0.77 +
33 -0.38 -2.3 2.8 -1.3 0.15 5.2e+03 0.0023 0.06 0.13 +
34 -0.41 -2.3 2.8 -1.3 0.13 5.2e+03 0.0024 0.06 0.33 +
35 -0.41 -2.3 2.8 -1.3 0.13 5.2e+03 0.0024 0.03 -1.7 -
36 -0.41 -2.3 2.8 -1.3 0.13 5.2e+03 0.0024 0.015 -0.6 -
37 -0.41 -2.3 2.8 -1.3 0.13 5.2e+03 0.00081 0.015 0.43 +
38 -0.39 -2.3 2.8 -1.3 0.13 5.2e+03 0.00082 0.015 0.83 +
39 -0.39 -2.3 2.8 -1.3 0.13 5.2e+03 0.00082 0.0075 -0.55 -
40 -0.39 -2.3 2.8 -1.3 0.14 5.2e+03 0.00058 0.0075 0.37 +
41 -0.39 -2.3 2.8 -1.3 0.14 5.2e+03 0.00023 0.0075 0.65 +
42 -0.39 -2.3 2.8 -1.3 0.14 5.2e+03 0.0002 0.0075 0.62 +
43 -0.39 -2.3 2.8 -1.3 0.14 5.2e+03 0.00015 0.0075 0.78 +
44 -0.39 -2.3 2.9 -1.3 0.14 5.2e+03 0.00036 0.0075 0.49 +
45 -0.39 -2.3 2.9 -1.3 0.14 5.2e+03 7.7e-05 0.075 0.92 ++
46 -0.39 -2.3 2.9 -1.3 0.14 5.2e+03 6.5e-05 0.075 0.37 +
47 -0.39 -2.3 2.9 -1.3 0.14 5.2e+03 6.8e-06 0.75 0.98 ++
48 -0.39 -2.3 2.9 -1.3 0.14 5.2e+03 2.3e-06 0.75 0.66 ++
Optimization algorithm has converged.
Relative gradient: 2.346420034070932e-06
Cause of termination: Relative gradient = 2.3e-06 <= 6.1e-06
Number of function evaluations: 114
Number of gradient evaluations: 65
Number of hessian evaluations: 0
Algorithm: BFGS with trust region for simple bound constraints
Number of iterations: 49
Proportion of Hessian calculation: 0/32 = 0.0%
Optimization time: 0:01:53.544645
Calculate second derivatives and BHHH
File b06a_unif_mixture.html has been generated.
File b06a_unif_mixture.yaml has been generated.
print(results.short_summary())
Results for model b06a_unif_mixture
Nbr of parameters: 5
Sample size: 6768
Excluded data: 3960
Final log likelihood: -5215.805
Akaike Information Criterion: 10441.61
Bayesian Information Criterion: 10475.71
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 -0.386220 0.066029 -5.849265 4.937510e-09
1 b_time -2.316295 0.125946 -18.391184 0.000000e+00
2 b_time_s 2.868552 0.199638 14.368781 0.000000e+00
3 b_cost -1.277277 0.086562 -14.755635 0.000000e+00
4 asc_car 0.143914 0.053299 2.700124 6.931367e-03
Total running time of the script: (3 minutes 23.102 seconds)