12. Mixture of logit with panel data

Example of a mixture of logit models, using Monte-Carlo integration.

The datafile is organized as panel data.

Michel Bierlaire, EPFL Sat Jun 21 2025, 16:54:51

import biogeme.biogeme_logging as blog
from IPython.core.display_functions import display
from biogeme.biogeme import BIOGEME
from biogeme.expressions import Beta, Draws, MonteCarlo, PanelLikelihoodTrajectory, log
from biogeme.models import logit
from biogeme.results_processing import (
    EstimationResults,
    get_pandas_estimated_parameters,
)

See the data processing script: Panel data preparation for Swissmetro.

from swissmetro_panel 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,
)

# from biogeme.results_processing import get_pandas_estimated_parameters

logger = blog.get_screen_logger(level=blog.INFO)
logger.info('Example b12_panel.py')

Example b12_panel.py

Parameters to be estimated.

b_cost = Beta('b_cost', 0, None, 0, 0)

Define a random parameter, normally distributed across individuals, designed to be used for Monte-Carlo simulation.

b_time = Beta('b_time', 0, None, 0, 0)

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

b_time_s = Beta('b_time_s', 1, 1.0e-5, None, 0)
b_time_rnd = b_time + b_time_s * Draws('b_time_rnd', 'NORMAL_ANTI')

We do the same for the constants, to address serial correlation.

asc_car = Beta('asc_car', 0, None, None, 0)
asc_car_s = Beta('asc_car_s', 1, 1.0e-5, None, 0)
asc_car_rnd = asc_car + asc_car_s * Draws('asc_car_rnd', 'NORMAL_ANTI')

asc_train = Beta('asc_train', 0, None, None, 0)
asc_train_s = Beta('asc_train_s', 1, 1.0e-5, None, 0)
asc_train_rnd = asc_train + asc_train_s * Draws('asc_train_rnd', 'NORMAL_ANTI')

asc_sm = Beta('asc_sm', 0, None, None, 0)
asc_sm_s = Beta('asc_sm_s', 1, 1.0e-5, None, 0)
asc_sm_rnd = asc_sm + asc_sm_s * Draws('asc_sm_rnd', 'NORMAL_ANTI')

Definition of the utility functions.

v_train = asc_train_rnd + b_time_rnd * TRAIN_TT_SCALED + b_cost * TRAIN_COST_SCALED
v_swissmetro = asc_sm_rnd + b_time_rnd * SM_TT_SCALED + b_cost * SM_COST_SCALED
v_car = asc_car_rnd + 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 the random parameters, the likelihood of one observation is given by the logit model (called the kernel).

choice_probability_one_observation = logit(v, av, CHOICE)

Conditional on the random parameters, the likelihood of all observations for one individual (the trajectory) is the product of the likelihood of each observation.

conditional_trajectory_probability = PanelLikelihoodTrajectory(
    choice_probability_one_observation
)

We integrate over the random parameters using Monte-Carlo

log_probability = log(MonteCarlo(conditional_trajectory_probability))

As the objective is to illustrate the syntax, we calculate the Monte-Carlo approximation with a small number of draws.

the_biogeme = BIOGEME(
    database,
    log_probability,
    number_of_draws=5_000,
    seed=1223,
    calculating_second_derivatives='never',
)
the_biogeme.model_name = 'b12_panel'

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()

print(results.short_summary())

Results for model b12_panel
Nbr of parameters:              9
Sample size:                    752
Observations:                   6768
Excluded data:                  0
Final log likelihood:           -3574.944
Akaike Information Criterion:   7167.887
Bayesian Information Criterion: 7209.492

pandas_results = get_pandas_estimated_parameters(estimation_results=results)
display(pandas_results)

          Name     Value  ...  BHHH p-value  Active bound
0    asc_train -0.035216  ...  8.155440e-01         False
1  asc_train_s  2.763781  ...  0.000000e+00         False
2       b_time -6.045210  ...  0.000000e+00         False
3     b_time_s  3.547604  ...  0.000000e+00         False
4       b_cost -3.580028  ...  0.000000e+00         False
5       asc_sm  0.510528  ...  9.472799e-07         False
6     asc_sm_s  0.000010  ...  9.999822e-01          True
7      asc_car  0.904044  ...  2.848728e-10         False
8    asc_car_s  3.957354  ...  0.000000e+00         False

[9 rows x 6 columns]

Here is a list of the columns of the “flat” database fthat has been generated by Biogeme

for col in the_biogeme.model_elements.database.dataframe.columns:
    print(col)

GROUP
SURVEY
SP
ID
PURPOSE
FIRST
TICKET
WHO
LUGGAGE
AGE
MALE
INCOME
GA
ORIGIN
DEST
TRAIN_AV
CAR_AV
SM_AV
TRAIN_TT
TRAIN_CO
TRAIN_HE
SM_TT
SM_CO
SM_HE
SM_SEATS
CAR_TT
CAR_CO
CHOICE
SM_COST
TRAIN_COST
CAR_AV_SP
TRAIN_AV_SP
TRAIN_TT_SCALED
TRAIN_COST_SCALED
SM_TT_SCALED
SM_COST_SCALED
CAR_TT_SCALED
CAR_CO_SCALED

And here is a list of the variables that have been renamed in the model specification.

for var in the_biogeme.model_elements.expressions_registry.variables:
    print(var)

GROUP
SURVEY
SP
ID
PURPOSE
FIRST
TICKET
WHO
LUGGAGE
AGE
MALE
INCOME
GA
ORIGIN
DEST
TRAIN_AV
CAR_AV
SM_AV
TRAIN_TT
TRAIN_CO
TRAIN_HE
SM_TT
SM_CO
SM_HE
SM_SEATS
CAR_TT
CAR_CO
CHOICE
SM_COST
TRAIN_COST
CAR_AV_SP
TRAIN_AV_SP
TRAIN_TT_SCALED
TRAIN_COST_SCALED
SM_TT_SCALED
SM_COST_SCALED
CAR_TT_SCALED
CAR_CO_SCALED