Simulation of panel model

Calculates each contribution to the log likelihood function using simulation. We also calculate the individual parameters.

Michel Bierlaire, EPFL Sat Jun 21 2025, 17:06:31

import sys

from IPython.core.display_functions import display

import biogeme.biogeme_logging as blog
from biogeme.biogeme import BIOGEME
from biogeme.calculator.single_formula import calculate_single_formula_from_expression
from biogeme.expressions import Beta, Draws, MonteCarlo, PanelLikelihoodTrajectory, log
from biogeme.models import logit
from biogeme.results_processing import EstimationResults
from biogeme.second_derivatives import SecondDerivativesMode

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,
)

Define the number of draws to be used for Monte-Carlo integration.

NUMBER_OF_DRAWS = 100_000

logger = blog.get_screen_logger(level=blog.INFO)
logger.info('Example b13panel_simul.py')

Example b13panel_simul.py

Parameters to be estimated.

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

Define a random parameter, normally distributed across individuals, 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, None, 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, None, 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, None, 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, None, 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))

We retrieve the parameters estimates.

try:
    results = EstimationResults.from_yaml_file(filename='saved_results/b12panel.yaml')
except FileNotFoundError:
    sys.exit(
        'Run first the script b12panel.py '
        'in order to generate the '
        'file b12panel.yaml and move it to the directory saved_results.'
    )

Simulate to recalculate the log likelihood directly from the formula, without the Biogeme object

simulated_loglike = calculate_single_formula_from_expression(
    expression=log_probability,
    database=database,
    number_of_draws=NUMBER_OF_DRAWS,
    the_betas=results.get_beta_values(),
    numerically_safe=False,
    second_derivatives_mode=SecondDerivativesMode.NEVER,
    use_jit=True,
)

Flattening database [(6768, 38)].
Database flattened [(752, 362)]
Flattening database [(6768, 38)].
Database flattened [(752, 362)]
Flattening database [(6768, 38)].
Database flattened [(752, 362)]
Flattening database [(6768, 38)].
Database flattened [(752, 362)]

print(f'Simulated log likelihood: {simulated_loglike}')

Simulated log likelihood: -3574.7230792117853

We also calculate the individual parameters for the time coefficient.

numerator = MonteCarlo(b_time_rnd * conditional_trajectory_probability)
denominator = MonteCarlo(conditional_trajectory_probability)

simulate = {
    'Numerator': numerator,
    'Denominator': denominator,
}

Creation of the Biogeme object.

biosim = BIOGEME(database, simulate, number_of_draws=NUMBER_OF_DRAWS, seed=1223)

Biogeme parameters read from biogeme.toml.
Flattening database [(6768, 38)].
Database flattened [(752, 362)]
Flattening database [(6768, 38)].
Database flattened [(752, 362)]
Flattening database [(6768, 38)].
Database flattened [(752, 362)]
Flattening database [(6768, 38)].
Database flattened [(752, 362)]

Simulation.

sim = biosim.simulate(results.get_beta_values())

sim['Individual-level parameters'] = sim['Numerator'] / sim['Denominator']

print(f'{sim.shape=}')

sim.shape=(752, 3)

display(sim)

     Numerator  Denominator  Individual-level parameters
0    -0.071506     0.010421                    -6.861496
1    -5.400202     0.762721                    -7.080176
2    -2.116223     0.269996                    -7.837982
3    -3.576361     0.463302                    -7.719288
4    -3.351181     0.476229                    -7.036911
..         ...          ...                          ...
747  -0.008062     0.001617                    -4.987226
748  -0.000451     0.000129                    -3.498607
749  -0.000016     0.000007                    -2.303046
750  -5.102442     0.697186                    -7.318624
751  -0.034959     0.024362                    -1.434957

[752 rows x 3 columns]