Mixture with lognormal distribution

Example of a mixture of logit models. The mixing distribution is distributed as a log normal. Compared to 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

import biogeme.biogeme_logging as blog
from IPython.core.display_functions import display
from biogeme.biogeme import BIOGEME
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 b17lognormal_mixture_integral.py')

Example b17lognormal_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 = 'b17lognormal_mixture_integral'

Biogeme parameters read from biogeme.toml.

Estimate the parameters

results = the_biogeme.estimate()

*** Initial values of the parameters are obtained from the file __b17lognormal_mixture_integral.iter
Parameter values restored from __b17lognormal_mixture_integral.iter
Starting values for the algorithm: {'asc_train': -0.3508670994332833, 'b_time': 0.5699481745949635, 'b_time_s': 1.2138204557723493, 'b_cost': -1.3762554626050059, 'asc_car': 0.16780371550483728}
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: 3.8488160574638475e-06
Cause of termination: Relative gradient = 3.8e-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: 0:00:00.862432
Calculate second derivatives and BHHH
File b17lognormal_mixture_integral~00.html has been generated.
File b17lognormal_mixture_integral~00.yaml has been generated.

print(results.short_summary())

Results for model b17lognormal_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