“

Mixture of logit models

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

import biogeme.biogeme_logging as blog
from IPython.core.display_functions import display
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 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.modelName = '06unif_mixture_integral'

Biogeme parameters read from biogeme.toml.
/Users/bierlair/MyFiles/github/biogeme/docs/source/examples/swissmetro/plot_b06unif_mixture_integral.py:116: DeprecationWarning: 'modelName' is deprecated. Please use 'model_name' instead.
  the_biogeme.modelName = '06unif_mixture_integral'

Estimate the parameters

results = the_biogeme.estimate()

*** Initial values of the parameters are obtained from the file __06unif_mixture_integral.iter
Parameter values restored from __06unif_mixture_integral.iter
Starting values for the algorithm: {'asc_train': -0.385071663361878, 'b_time': -2.3205753430924987, 'b_time_s': 2.8759594278547964, 'b_cost': -1.2779262496068977, 'asc_car': 0.14496871481580123}
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
Optimization algorithm has converged.
Relative gradient: 2.010606846718136e-06
Cause of termination: Relative gradient = 2e-06 <= 6.1e-06
Number of function evaluations: 1
Number of gradient evaluations: 1
Number of hessian evaluations: 0
Algorithm: BFGS with trust region for simple bound constraints
Number of iterations: 0
Optimization time: 0:00:00.377077
Calculate second derivatives and BHHH
File 06unif_mixture_integral~00.html has been generated.
File 06unif_mixture_integral~00.yaml has been generated.

print(results.short_summary())

Results for model 06unif_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