Note
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17b. Mixture with lognormal distribution and numerical integrationΒΆ
Example of a mixture of logit models. The mixing distribution is distributed as a log normal. Compared to 17a. Mixture with lognormal distribution, the integration is performed using numerical integration instead of Monte-Carlo approximation.
Michel Bierlaire, EPFL Thu Jun 26 2025, 15:49:37
from IPython.core.display_functions import display
import biogeme.biogeme_logging as blog
from biogeme.biogeme import BIOGEME
from biogeme.expressions import (
Beta,
IntegrateNormal,
RandomVariable,
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 b17b_lognormal_mixture_integral.py')
Example b17b_lognormal_mixture_integral.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 numerical integration.
omega = RandomVariable('omega')
B_TIME_RND = -exp(b_time + b_time_s * omega)
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 omega, we have a logit model (called the kernel).
conditional_probability = logit(v, av, CHOICE)
We integrate over omega using numerical integration.
log_probability = log(IntegrateNormal(conditional_probability, 'omega'))
Create the Biogeme object.
the_biogeme = BIOGEME(database, log_probability)
the_biogeme.model_name = 'b17b_lognormal_mixture_integral'
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 __b17b_lognormal_mixture_integral.iter
Cannot read file __b17b_lognormal_mixture_integral.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 -0.00031 -
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 -0.38 0.61 1.2 -1.5 0.081 5.2e+03 0.0098 0.25 0.25 +
6 -0.38 0.61 1.2 -1.5 0.081 5.2e+03 0.0098 0.12 -2.2 -
7 -0.26 0.56 1.2 -1.4 0.15 5.2e+03 0.012 0.12 0.13 +
8 -0.26 0.56 1.2 -1.4 0.15 5.2e+03 0.012 0.062 0.075 -
9 -0.32 0.62 1.2 -1.3 0.22 5.2e+03 0.0043 0.062 0.35 +
10 -0.34 0.56 1.2 -1.4 0.2 5.2e+03 0.0045 0.062 0.36 +
11 -0.34 0.56 1.2 -1.4 0.2 5.2e+03 0.0045 0.031 -0.23 -
12 -0.33 0.59 1.2 -1.4 0.17 5.2e+03 0.0024 0.031 0.35 +
13 -0.36 0.57 1.2 -1.4 0.17 5.2e+03 0.0019 0.031 0.39 +
14 -0.36 0.57 1.2 -1.4 0.17 5.2e+03 0.0019 0.016 -0.98 -
15 -0.36 0.57 1.2 -1.4 0.17 5.2e+03 0.0019 0.0078 -0.11 -
16 -0.36 0.57 1.2 -1.4 0.17 5.2e+03 0.00021 0.0078 0.71 +
17 -0.36 0.57 1.2 -1.4 0.17 5.2e+03 0.00021 0.0039 -0.022 -
18 -0.35 0.57 1.2 -1.4 0.17 5.2e+03 0.00039 0.0039 0.26 +
19 -0.35 0.57 1.2 -1.4 0.17 5.2e+03 0.00026 0.0039 0.43 +
20 -0.35 0.57 1.2 -1.4 0.17 5.2e+03 0.00012 0.0039 0.49 +
21 -0.35 0.57 1.2 -1.4 0.17 5.2e+03 0.00012 0.002 -0.53 -
22 -0.35 0.57 1.2 -1.4 0.17 5.2e+03 0.00012 0.00098 0.00052 -
23 -0.35 0.57 1.2 -1.4 0.17 5.2e+03 6.7e-05 0.00098 0.61 +
24 -0.35 0.57 1.2 -1.4 0.17 5.2e+03 3e-05 0.00098 0.55 +
25 -0.35 0.57 1.2 -1.4 0.17 5.2e+03 3e-05 0.00049 -0.13 -
26 -0.35 0.57 1.2 -1.4 0.17 5.2e+03 3e-05 0.00049 0.28 +
27 -0.35 0.57 1.2 -1.4 0.17 5.2e+03 3e-05 0.00024 0.065 -
28 -0.35 0.57 1.2 -1.4 0.17 5.2e+03 6.5e-06 0.00024 0.78 +
29 -0.35 0.57 1.2 -1.4 0.17 5.2e+03 6.5e-06 0.00012 -0.78 -
30 -0.35 0.57 1.2 -1.4 0.17 5.2e+03 3.5e-06 0.00012 0.34 -
Optimization algorithm has converged.
Relative gradient: 3.531190814365559e-06
Cause of termination: Relative gradient = 3.5e-06 <= 6.1e-06
Number of function evaluations: 68
Number of gradient evaluations: 37
Number of hessian evaluations: 0
Algorithm: BFGS with trust region for simple bound constraints
Number of iterations: 31
Proportion of Hessian calculation: 0/18 = 0.0%
Optimization time: 0:00:00.557149
Calculate second derivatives and BHHH
File b17b_lognormal_mixture_integral.html has been generated.
File b17b_lognormal_mixture_integral.yaml has been generated.
print(results.short_summary())
Results for model b17b_lognormal_mixture_integral
Nbr of parameters: 5
Sample size: 6768
Excluded data: 3960
Final log likelihood: -5231.506
Akaike Information Criterion: 10473.01
Bayesian Information Criterion: 10507.11
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 -0.350867 0.073180 -4.794581 1.630150e-06
1 b_time 0.569948 0.070411 8.094634 6.661338e-16
2 b_time_s 1.213820 0.141278 8.591690 0.000000e+00
3 b_cost -1.376255 0.096032 -14.331261 0.000000e+00
4 asc_car 0.167804 0.063408 2.646410 8.135116e-03
Total running time of the script: (0 minutes 1.818 seconds)