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6c. Mixture of logit models with uniform distribution and numerical integration¶
Example of a mixture of logit models, using numerical integration. The mixing distribution is uniform.
Michel Bierlaire, EPFL Fri Jun 20 2025, 10:47:24
from IPython.core.display_functions import display
import biogeme.biogeme_logging as blog
from biogeme.biogeme import BIOGEME
from biogeme.distributions import normalpdf
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 b06unif_mixture_integral.py')
Example b06unif_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 numerical integration
b_time = Beta('b_time', 0, None, None, 0)
b_time_s = Beta('b_time_s', 1, None, None, 0)
omega = RandomVariable('omega')
As the numerical integration ranges from -∞ to +∞, we need to perform a change of variable in order to integrate between -1 and 1.
LOWER_BND = -1
UPPER_BND = 1
x = LOWER_BND + (UPPER_BND - LOWER_BND) / (1 + exp(-omega))
dx = (UPPER_BND - LOWER_BND) * exp(-omega) / ((1 + exp(-omega)) ** 2)
b_time_rnd = b_time + b_time_s * x
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 omega, we have a logit model (called the kernel).
conditional_probability = logit(v, av, CHOICE)
pdf of the uniform distribution
pdf_uniform = 1 / (UPPER_BND - LOWER_BND)
As the IntegrateNormal expression is designed for a normal distribution, we need to divide by the pdf of the normal distribution, and multiply by the pdf of the uniform distribution, after applying the change of variable.
new_integrand = conditional_probability * dx * pdf_uniform / normalpdf(omega)
We integrate over omega using numerical integration. To illustrate the syntax, we specific the number of quadrature points to be used.
log_probability = log(
IntegrateNormal(
new_integrand,
'omega',
number_of_quadrature_points=60,
)
)
Create the Biogeme object.
the_biogeme = BIOGEME(database, log_probability)
the_biogeme.model_name = 'b06c_unif_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 __b06c_unif_mixture_integral.iter
Cannot read file __b06c_unif_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 -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 -0.18 -
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 -0.059 -
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 0.0093 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 0.0094 1.2 0.91 ++
13 -0.56 -1.7 1.9 -1.2 -0.043 5.2e+03 0.0094 0.62 -2.1 -
14 -0.56 -1.7 1.9 -1.2 -0.043 5.2e+03 0.0094 0.31 -1.5 -
15 -0.56 -1.7 1.9 -1.2 -0.043 5.2e+03 0.0094 0.16 -0.15 -
16 -0.59 -1.9 2 -1.2 0.05 5.2e+03 0.01 0.16 0.38 +
17 -0.52 -1.9 2.2 -1.2 -0.011 5.2e+03 0.0047 0.16 0.81 +
18 -0.52 -1.9 2.2 -1.2 -0.011 5.2e+03 0.0047 0.078 -0.45 -
19 -0.52 -1.9 2.2 -1.2 0.067 5.2e+03 0.0079 0.078 0.31 +
20 -0.48 -2 2.3 -1.2 0.04 5.2e+03 0.0038 0.78 0.94 ++
21 -0.48 -2 2.3 -1.2 0.04 5.2e+03 0.0038 0.39 -1.3 -
22 -0.48 -2 2.3 -1.2 0.04 5.2e+03 0.0038 0.2 -0.86 -
23 -0.48 -2 2.3 -1.2 0.04 5.2e+03 0.0038 0.098 0.032 -
24 -0.5 -2.1 2.4 -1.3 0.062 5.2e+03 0.0066 0.098 0.47 +
25 -0.45 -2.1 2.5 -1.2 0.095 5.2e+03 0.0047 0.098 0.83 +
26 -0.44 -2.2 2.6 -1.2 0.11 5.2e+03 0.0032 0.98 0.94 ++
27 -0.44 -2.2 2.6 -1.2 0.11 5.2e+03 0.0032 0.49 -23 -
28 -0.44 -2.2 2.6 -1.2 0.11 5.2e+03 0.0032 0.24 -3.8 -
29 -0.44 -2.2 2.6 -1.2 0.11 5.2e+03 0.0032 0.12 -0.66 -
30 -0.41 -2.2 2.7 -1.3 0.1 5.2e+03 0.0028 0.12 0.28 +
31 -0.41 -2.2 2.7 -1.3 0.1 5.2e+03 0.0028 0.061 -0.31 -
32 -0.41 -2.2 2.7 -1.3 0.12 5.2e+03 0.00091 0.061 0.75 +
33 -0.38 -2.3 2.8 -1.3 0.13 5.2e+03 0.0022 0.061 0.29 +
34 -0.38 -2.3 2.8 -1.3 0.13 5.2e+03 0.0022 0.031 -0.061 -
35 -0.4 -2.3 2.8 -1.3 0.13 5.2e+03 0.0013 0.031 0.82 +
36 -0.4 -2.3 2.8 -1.3 0.13 5.2e+03 0.0013 0.015 -0.7 -
37 -0.4 -2.3 2.8 -1.3 0.13 5.2e+03 0.0013 0.0076 -3.2 -
38 -0.4 -2.3 2.8 -1.3 0.13 5.2e+03 0.0013 0.0038 -0.56 -
39 -0.4 -2.3 2.8 -1.3 0.13 5.2e+03 0.00038 0.0038 0.44 +
40 -0.4 -2.3 2.8 -1.3 0.13 5.2e+03 0.00042 0.038 0.9 ++
41 -0.4 -2.3 2.8 -1.3 0.13 5.2e+03 0.00042 0.019 -0.73 -
42 -0.38 -2.3 2.8 -1.3 0.14 5.2e+03 0.0004 0.019 0.4 +
43 -0.39 -2.3 2.8 -1.3 0.14 5.2e+03 0.00074 0.019 0.44 +
44 -0.38 -2.3 2.9 -1.3 0.15 5.2e+03 0.00021 0.019 0.8 +
45 -0.38 -2.3 2.9 -1.3 0.15 5.2e+03 0.00011 0.019 0.81 +
46 -0.38 -2.3 2.9 -1.3 0.15 5.2e+03 0.00011 0.0065 -2.8 -
47 -0.38 -2.3 2.9 -1.3 0.15 5.2e+03 0.00011 0.0032 -0.35 -
48 -0.38 -2.3 2.9 -1.3 0.15 5.2e+03 8.5e-05 0.0032 0.56 +
49 -0.38 -2.3 2.9 -1.3 0.15 5.2e+03 2.8e-05 0.0032 0.45 +
50 -0.38 -2.3 2.9 -1.3 0.15 5.2e+03 2.8e-05 0.0012 -0.29 -
51 -0.39 -2.3 2.9 -1.3 0.14 5.2e+03 3.5e-05 0.0012 0.33 +
52 -0.39 -2.3 2.9 -1.3 0.14 5.2e+03 1.5e-06 0.0012 0.99 +
Optimization algorithm has converged.
Relative gradient: 1.5147804784176e-06
Cause of termination: Relative gradient = 1.5e-06 <= 6.1e-06
Number of function evaluations: 116
Number of gradient evaluations: 63
Number of hessian evaluations: 0
Algorithm: BFGS with trust region for simple bound constraints
Number of iterations: 53
Proportion of Hessian calculation: 0/31 = 0.0%
Optimization time: 0:00:01.077594
Calculate second derivatives and BHHH
File b06c_unif_mixture_integral.html has been generated.
File b06c_unif_mixture_integral.yaml has been generated.
print(results.short_summary())
Results for model b06c_unif_mixture_integral
Nbr of parameters: 5
Sample size: 6768
Excluded data: 3960
Final log likelihood: -5215.061
Akaike Information Criterion: 10440.12
Bayesian Information Criterion: 10474.22
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 -0.385072 0.065992 -5.835159 5.373928e-09
1 b_time -2.320575 0.126118 -18.400027 0.000000e+00
2 b_time_s 2.875959 0.200170 14.367615 0.000000e+00
3 b_cost -1.277926 0.086624 -14.752624 0.000000e+00
4 asc_car 0.144969 0.053308 2.719456 6.538948e-03
Total running time of the script: (0 minutes 2.402 seconds)