5b. Mixture of logit models with numerical integration

Example of a normal mixture of logit models, using numerical integration.

Michel Bierlaire, EPFL Fri Jun 20 2025, 10:25:34

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, 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 b05b_normal_mixture_integral.py')

Example b05b_normal_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)

It is advised not to use 0 as starting value for the following parameter.

b_time_s = Beta('b_time_s', 1, None, None, 0)
omega = RandomVariable('omega')
b_time_rnd = 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 on 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,
    optimization_algorithm='simple_bounds_BFGS',
)
# the_biogeme = BIOGEME(database, logprob)
the_biogeme.modelName = 'b05b_normal_mixture_integral'

Biogeme parameters read from biogeme.toml.
/Users/bierlair/MyFiles/github/biogeme/docs/source/examples/swissmetro/plot_b05b_normal_mixture_integral.py:90: DeprecationWarning: 'modelName' is deprecated. Please use 'model_name' instead.
  the_biogeme.modelName = 'b05b_normal_mixture_integral'

Estimate the parameters.

try:
    results = EstimationResults.from_yaml_file(
        filename=f'saved_results/{the_biogeme.model_name}.yaml'
    )
except FileNotFoundError:
    results = the_biogeme.estimate()

print(results.short_summary())

Results for model b05b_normal_mixture_integral
Nbr of parameters:              5
Sample size:                    6768
Excluded data:                  3960
Final log likelihood:           -5213.725
Akaike Information Criterion:   10437.45
Bayesian Information Criterion: 10471.55

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 -0.395901         0.063674       -6.217619    5.047569e-10
1     b_time -2.278361         0.117234      -19.434326    0.000000e+00
2   b_time_s  1.675032         0.102317       16.370945    0.000000e+00
3     b_cost -1.288167         0.086419      -14.906055    0.000000e+00
4    asc_car  0.142821         0.051744        2.760128    5.777867e-03