Simulation of panel model

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

author:

Michel Bierlaire, EPFL

date:

Sun Apr 9 18:16:29 2023

import sys

import biogeme.biogeme as bio
import biogeme.biogeme_logging as blog
import biogeme.exceptions as excep
import biogeme.results as res
from biogeme import models
from biogeme.expressions import (
    Beta,
    bioDraws,
    PanelLikelihoodTrajectory,
    MonteCarlo,
    log,
)
from biogeme.parameters import Parameters

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

from swissmetro_panel import (
    database,
    CHOICE,
    SM_AV,
    CAR_AV_SP,
    TRAIN_AV_SP,
    TRAIN_TT_SCALED,
    TRAIN_COST_SCALED,
    SM_TT_SCALED,
    SM_COST_SCALED,
    CAR_TT_SCALED,
    CAR_CO_SCALED,
)

print(f'Samples size = {database.get_sample_size()}')

Samples size = 752

We use a low number of draws, as the objective is to illustrate the syntax. In practice, this value is insufficient to have a good approximation of the integral.

NUMBER_OF_DRAWS = 50

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

Example b13panel_simul.py

Parameters.

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)
B_TIME_S = Beta('B_TIME_S', 1, None, None, 0)
B_TIME_RND = B_TIME + B_TIME_S * bioDraws('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 * bioDraws('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 * bioDraws('ASC_TRAIN_RND', 'NORMAL_ANTI')

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 * bioDraws('ASC_SM_RND', 'NORMAL_ANTI')

Definition of the utility functions.

V1 = ASC_TRAIN_RND + B_TIME_RND * TRAIN_TT_SCALED + B_COST * TRAIN_COST_SCALED
V2 = ASC_SM_RND + B_TIME_RND * SM_TT_SCALED + B_COST * SM_COST_SCALED
V3 = ASC_CAR_RND + B_TIME_RND * CAR_TT_SCALED + B_COST * CAR_CO_SCALED

Associate utility functions with the numbering of alternatives.

V = {1: V1, 2: V2, 3: V3}

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).

obsprob = models.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.

condprobIndiv = PanelLikelihoodTrajectory(obsprob)

We integrate over the random parameters using Monte-Carlo

logprob = log(MonteCarlo(condprobIndiv))

We retrieve the parameters estimates.

try:
    results = res.bioResults(pickle_file='saved_results/b12panel.pickle')
except excep.BiogemeError:
    sys.exit(
        'Run first the script b12panel.py '
        'in order to generate the '
        'file b12panel.pickle.'
    )

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

simulated_loglike = logprob.get_value_c(
    database=database,
    betas=results.get_beta_values(),
    number_of_draws=NUMBER_OF_DRAWS,
    aggregation=True,
    prepare_ids=True,
)

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

Simulated log likelihood: -3746.058773770821

We also calculate the individual parameters for the time coefficient.

numerator = MonteCarlo(B_TIME_RND * condprobIndiv)
denominator = MonteCarlo(condprobIndiv)

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

Creation of the Biogeme object.

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

Biogeme parameters read from biogeme.toml.

Suimulation.

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

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

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

sim.shape=(752, 3)

sim

	Numerator	Denominator	Individual-level parameters
1	-0.062321	0.009942	-6.268342
2	-5.534657	0.742892	-7.450155
3	-2.157851	0.241836	-8.922785
4	-3.585943	0.446293	-8.034958
5	-3.211187	0.455042	-7.056910
...	...	...	...
935	-0.006234	0.001198	-5.201502
936	-0.000400	0.000205	-1.949459
937	-0.000012	0.000006	-1.942529
938	-5.267642	0.630936	-8.348929
939	0.017225	0.045669	0.377173

752 rows × 3 columns