Triangular mixture with panel data

Example of a mixture of logit models, using Monte-Carlo integration. The mixing distribution is user-defined (triangular, here). The datafile is organized as panel data.

Michel Bierlaire, EPFL

import numpy as np
from IPython.core.display_functions import display

import biogeme.biogeme_logging as blog
from biogeme.biogeme import BIOGEME
from biogeme.draws import RandomNumberGeneratorTuple
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,
)

logger = blog.get_screen_logger(level=blog.INFO)
logger.info('Example b26triangular_panel_mixture.py')

Example b26triangular_panel_mixture.py

Function generating the draws.

def the_triangular_generator(sample_size: int, number_of_draws: int) -> np.ndarray:
    """
    Provide my own random number generator to the database.
    See the `numpy.random` documentation to obtain a list of other distributions.
    """
    return np.random.triangular(-1, 0, 1, (sample_size, number_of_draws))

Associate the function with a name.

my_random_number_generators = {
    'TRIANGULAR': RandomNumberGeneratorTuple(
        the_triangular_generator,
        'Draws from a triangular distribution',
    )
}

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.

Mean of the distribution.

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

Scale of the distribution. 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', 'TRIANGULAR')

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', 'TRIANGULAR')

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', 'TRIANGULAR')

asc_sm = Beta('asc_sm', 0, None, None, 1)
asc_sm_s = Beta('asc_sm_s', 1, None, None, 0)
asc_sm_rnd = asc_sm + asc_sm_s * Draws('asc_sm_rnd', 'TRIANGULAR')

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 to the random parameters, the likelihood of one observation is given by the logit model (called the kernel).

one_observation_conditional_probability = 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.

trajectory_conditional_probability = PanelLikelihoodTrajectory(
    one_observation_conditional_probability
)

We integrate over the random parameters using Monte-Carlo

log_probability = log(MonteCarlo(trajectory_conditional_probability))

the_biogeme = BIOGEME(
    database,
    log_probability,
    random_number_generators=my_random_number_generators,
    number_of_draws=10_000,
    seed=1223,
)
the_biogeme.model_name = 'b26triangular_panel_mixture'

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)]

Estimate the parameters.

try:
    results = EstimationResults.from_yaml_file(
        filename='saved_results/b26triangular_panel_mixture.yaml'
    )
except FileNotFoundError:
    results = the_biogeme.estimate()

print(results.short_summary())

Results for model b26triangular_panel_mixture
Nbr of parameters:              8
Sample size:                    752
Observations:                   6768
Excluded data:                  0
Final log likelihood:           -3593.921
Akaike Information Criterion:   7203.842
Bayesian Information Criterion: 7240.824

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 -0.357907         0.232508       -1.539332    1.237234e-01
1  asc_train_s  5.672924         0.756461        7.499296    6.417089e-14
2       b_time -6.078874         0.359765      -16.896772    0.000000e+00
3     b_time_s  8.941038         0.538493       16.603824    0.000000e+00
4       b_cost -3.282904         0.426908       -7.689959    1.465494e-14
5     asc_sm_s  3.492176         0.666815        5.237095    1.631240e-07
6      asc_car  0.355111         0.245255        1.447926    1.476377e-01
7    asc_car_s  9.534559         0.836263       11.401385    0.000000e+00