Mixture of logit models

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

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

import biogeme.biogeme_logging as blog
from IPython.core.display_functions import display
from biogeme.biogeme import BIOGEME
from biogeme.expressions import Beta, Integrate, IntegrateNormal, RandomVariable, 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 b05normal_mixture_integral.py')

Example b05normal_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 = 'b05normal_mixture_integral'

Biogeme parameters read from biogeme.toml.
/Users/bierlair/MyFiles/github/biogeme/docs/source/examples/swissmetro/plot_b05normal_mixture_integral.py:86: DeprecationWarning: 'modelName' is deprecated. Please use 'model_name' instead.
  the_biogeme.modelName = 'b05normal_mixture_integral'

Estimate the parameters

results = the_biogeme.estimate()

*** Initial values of the parameters are obtained from the file __b05normal_mixture_integral.iter
Parameter values restored from __b05normal_mixture_integral.iter
Starting values for the algorithm: {'asc_train': -0.3959009752580716, 'b_time': -2.2783610724991346, 'b_time_s': 1.675031645294375, 'b_cost': -1.288166620582709, 'asc_car': 0.14282068945852677}
Optimization algorithm: BFGS with simple bounds [simple_bounds_BFGS].
Optimization algorithm: hybrid Newton/BFGS with simple bounds [simple_bounds]
** Optimization: BFGS with trust region for simple bounds
Optimization algorithm has converged.
Relative gradient: 5.669145273606387e-06
Cause of termination: Relative gradient = 5.7e-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: 0:00:00.381158
Calculate second derivatives and BHHH
File b05normal_mixture_integral~00.html has been generated.
File b05normal_mixture_integral~00.yaml has been generated.

print(results.short_summary())

Results for model b05normal_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