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Discrete mixture with panel dataΒΆ
Example of a discrete mixture of logit models, also called latent_old class model. The class membership model includes socio-economic variables. The datafile is organized as panel data.
Michel Bierlaire, EPFL Mon Jun 23 2025, 16:29:45
from IPython.core.display_functions import display
See the data processing script: Panel data preparation for Swissmetro.
from swissmetro_panel import (
CAR_AV_SP,
CAR_CO_SCALED,
CAR_TT_SCALED,
CHOICE,
INCOME,
SM_AV,
SM_COST_SCALED,
SM_TT_SCALED,
TRAIN_AV_SP,
TRAIN_COST_SCALED,
TRAIN_TT_SCALED,
database,
)
import biogeme.biogeme_logging as blog
from biogeme.biogeme import BIOGEME
from biogeme.expressions import (
Beta,
Draws,
ExpressionOrNumeric,
MonteCarlo,
PanelLikelihoodTrajectory,
log,
)
from biogeme.models import logit
from biogeme.results_processing import get_pandas_estimated_parameters
logger = blog.get_screen_logger(level=blog.INFO)
logger.info('Example b16panel_discrete_socio_eco.py')
Example b16panel_discrete_socio_eco.py
Parameters to be estimated. One version for each latent_old class.
NUMBER_OF_CLASSES = 2
b_cost = [Beta(f'b_cost_class{i}', 0, None, None, 0) for i in range(NUMBER_OF_CLASSES)]
Define a random parameter, normally distributed across individuals, designed to be used for Monte-Carlo simulation.
b_time = [Beta(f'b_time_class{i}', 0, None, None, 0) for i in range(NUMBER_OF_CLASSES)]
It is advised not to use 0 as starting value for the following parameter.
b_time_s = [
Beta(f'b_time_s_class{i}', 1, None, None, 0) for i in range(NUMBER_OF_CLASSES)
]
b_time_rnd: list[ExpressionOrNumeric] = [
b_time[i] + b_time_s[i] * Draws(f'b_time_rnd_class{i}', 'NORMAL_ANTI')
for i in range(NUMBER_OF_CLASSES)
]
We do the same for the constants, to address serial correlation.
asc_car = [
Beta(f'asc_car_class{i}', 0, None, None, 0) for i in range(NUMBER_OF_CLASSES)
]
asc_car_s = [
Beta(f'asc_car_s_class{i}', 1, None, None, 0) for i in range(NUMBER_OF_CLASSES)
]
asc_car_rnd = [
asc_car[i] + asc_car_s[i] * Draws(f'asc_car_rnd_class{i}', 'NORMAL_ANTI')
for i in range(NUMBER_OF_CLASSES)
]
asc_train = [
Beta(f'asc_train_class{i}', 0, None, None, 0) for i in range(NUMBER_OF_CLASSES)
]
asc_train_s = [
Beta(f'asc_train_s_class{i}', 1, None, None, 0) for i in range(NUMBER_OF_CLASSES)
]
asc_train_rnd = [
asc_train[i] + asc_train_s[i] * Draws(f'asc_train_rnd_class{i}', 'NORMAL_ANTI')
for i in range(NUMBER_OF_CLASSES)
]
asc_sm = [Beta(f'asc_sm_class{i}', 0, None, None, 1) for i in range(NUMBER_OF_CLASSES)]
asc_sm_s = [
Beta(f'asc_sm_s_class{i}', 1, None, None, 0) for i in range(NUMBER_OF_CLASSES)
]
asc_sm_rnd = [
asc_sm[i] + asc_sm_s[i] * Draws(f'asc_sm_rnd_class{i}', 'NORMAL_ANTI')
for i in range(NUMBER_OF_CLASSES)
]
Parameters for the class membership model.
class_cte = Beta('class_cte', 0, None, None, 0)
class_inc = Beta('class_inc', 0, None, None, 0)
In class 0, it is assumed that the time coefficient is zero
b_time_rnd[0] = 0
Utility functions.
v_train_per_class = [
asc_train_rnd[i] + b_time_rnd[i] * TRAIN_TT_SCALED + b_cost[i] * TRAIN_COST_SCALED
for i in range(NUMBER_OF_CLASSES)
]
v_swissmetro_per_class = [
asc_sm_rnd[i] + b_time_rnd[i] * SM_TT_SCALED + b_cost[i] * SM_COST_SCALED
for i in range(NUMBER_OF_CLASSES)
]
v_car_per_class = [
asc_car_rnd[i] + b_time_rnd[i] * CAR_TT_SCALED + b_cost[i] * CAR_CO_SCALED
for i in range(NUMBER_OF_CLASSES)
]
v = [
{1: v_train_per_class[i], 2: v_swissmetro_per_class[i], 3: v_car_per_class[i]}
for i in range(NUMBER_OF_CLASSES)
]
Associate the availability conditions with the alternatives
av = {1: TRAIN_AV_SP, 2: SM_AV, 3: CAR_AV_SP}
The choice model is a discrete mixture of logit, with availability conditions We calculate the conditional probability for each class.
choice_probability_per_class = [
PanelLikelihoodTrajectory(logit(v[i], av, CHOICE)) for i in range(NUMBER_OF_CLASSES)
]
Class membership model.
score_class_0 = class_cte + class_inc * INCOME
prob_class0 = logit({0: score_class_0, 1: 0}, None, 0)
prob_class1 = logit({0: score_class_0, 1: 0}, None, 1)
Conditional on the random variables, likelihood for the individual.
conditional_choice_probability = (
prob_class0 * choice_probability_per_class[0]
+ prob_class1 * choice_probability_per_class[1]
)
We integrate over the random variables using Monte-Carlo
log_probability = log(MonteCarlo(conditional_choice_probability))
The model is complex, and there are numerical issues when calculating the second derivatives. Therefore, we instruct Biogeme not to evaluate the second derivatives. As a consequence, the statistics reported after estimation are based on the BHHH matrix instead of the Rao-Cramer bound.
the_biogeme = BIOGEME(
database,
log_probability,
number_of_draws=10_000,
seed=1223,
calculating_second_derivatives='never',
)
the_biogeme.model_name = 'b16panel_discrete_socio_eco'
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.
results = the_biogeme.estimate()
*** Initial values of the parameters are obtained from the file __b16panel_discrete_socio_eco.iter
Parameter values restored from __b16panel_discrete_socio_eco.iter
Starting values for the algorithm: {'class_cte': -1.1838573632904563, 'class_inc': -0.2168566401672651, 'asc_train_class0': -1.0070579844765999, 'asc_train_s_class0': 2.9246102810862666, 'b_cost_class0': -1.245143131711365, 'asc_sm_s_class0': -0.5275443379396421, 'asc_car_class0': -4.8274919022341685, 'asc_car_s_class0': 5.957644303222493, 'asc_train_class1': -0.2685221222610353, 'asc_train_s_class1': 1.5792373811714624, 'b_time_class1': -7.051757160900264, 'b_time_s_class1': 3.3013552142580487, 'b_cost_class1': -4.793999737865342, 'asc_sm_s_class1': 1.942567308396997, 'asc_car_class1': 1.0720186428091785, 'asc_car_s_class1': 2.729276132405476}
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: 5.147837856960197e-06
Cause of termination: Relative gradient = 5.1e-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: 0:00:22.909314
Calculate BHHH
File b16panel_discrete_socio_eco~00.html has been generated.
File b16panel_discrete_socio_eco~00.yaml has been generated.
print(results.short_summary())
Results for model b16panel_discrete_socio_eco
Nbr of parameters: 16
Sample size: 752
Observations: 6768
Excluded data: 0
Final log likelihood: -3525.907
Akaike Information Criterion: 7083.813
Bayesian Information Criterion: 7157.777
pandas_results = get_pandas_estimated_parameters(estimation_results=results)
display(pandas_results)
Name Value BHHH std err. BHHH t-stat. BHHH p-value
0 class_cte -1.183857 0.468198 -2.528537 1.145389e-02
1 class_inc -0.216857 0.174246 -1.244541 2.133006e-01
2 asc_train_class0 -1.007058 0.638902 -1.576231 1.149725e-01
3 asc_train_s_class0 2.924610 1.040826 2.809894 4.955783e-03
4 b_cost_class0 -1.245143 0.549631 -2.265415 2.348721e-02
5 asc_sm_s_class0 -0.527544 5.292939 -0.099669 9.206068e-01
6 asc_car_class0 -4.827492 1.697726 -2.843505 4.462035e-03
7 asc_car_s_class0 5.957644 1.800618 3.308666 9.374171e-04
8 asc_train_class1 -0.268522 0.287712 -0.933302 3.506638e-01
9 asc_train_s_class1 1.579237 0.505302 3.125331 1.776050e-03
10 b_time_class1 -7.051757 0.380884 -18.514210 0.000000e+00
11 b_time_s_class1 3.301355 0.402533 8.201461 2.220446e-16
12 b_cost_class1 -4.794000 0.254748 -18.818578 0.000000e+00
13 asc_sm_s_class1 1.942567 0.315657 6.154051 7.552836e-10
14 asc_car_class1 1.072019 0.218995 4.895179 9.821623e-07
15 asc_car_s_class1 2.729276 0.267099 10.218233 0.000000e+00
Total running time of the script: (1 minutes 20.455 seconds)