Note
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26. Triangular mixture with panel dataΒΆ
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 biogeme.biogeme_logging as blog
import numpy as np
from IPython.core.display_functions import display
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 b26_triangular_panel_mixture.py')
Example b26_triangular_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=5_000,
calculating_second_derivatives='never',
seed=1223,
)
the_biogeme.model_name = 'b26_triangular_panel_mixture'
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()
Flattening database [(6768, 38)].
Database flattened [(752, 362)]
*** Initial values of the parameters are obtained from the file __b26_triangular_panel_mixture.iter
Parameter values restored from __b26_triangular_panel_mixture.iter
Starting values for the algorithm: {'asc_train': -0.3579068677530871, 'asc_train_s': 5.672923903548474, 'b_time': -6.078873916291999, 'b_time_s': 8.941037943738221, 'b_cost': -3.2829038751791275, 'asc_sm_s': 3.4921755055663684, 'asc_car': 0.35511086490784627, 'asc_car_s': 9.534559216931386}
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
Iter. asc_train asc_train_s b_time b_time_s b_cost asc_sm_s asc_car asc_car_s Function Relgrad Radius Rho
0 -0.36 5.7 -6.1 8.9 -3.3 3.5 0.36 9.5 3.6e+03 0.005 0.5 -9.2 -
1 -0.36 5.7 -6.1 8.9 -3.3 3.5 0.36 9.5 3.6e+03 0.005 0.25 -3.4 -
2 -0.36 5.7 -6.1 8.9 -3.3 3.5 0.36 9.5 3.6e+03 0.005 0.12 -1.6 -
3 -0.36 5.7 -6.1 8.9 -3.3 3.5 0.36 9.5 3.6e+03 0.005 0.062 -0.74 -
4 -0.36 5.7 -6.1 8.9 -3.3 3.5 0.36 9.5 3.6e+03 0.005 0.031 -0.2 -
5 -0.33 5.7 -6.1 8.9 -3.3 3.5 0.39 9.6 3.6e+03 0.004 0.031 0.44 +
6 -0.33 5.7 -6.1 8.9 -3.3 3.5 0.39 9.6 3.6e+03 0.004 0.016 -0.036 -
7 -0.33 5.7 -6.1 8.9 -3.3 3.4 0.4 9.5 3.6e+03 0.0025 0.016 0.66 +
8 -0.34 5.7 -6.1 8.9 -3.3 3.4 0.42 9.5 3.6e+03 0.0035 0.016 0.46 +
9 -0.33 5.7 -6.1 8.9 -3.3 3.4 0.43 9.5 3.6e+03 0.0028 0.016 0.68 +
10 -0.33 5.7 -6.1 8.9 -3.2 3.4 0.42 9.5 3.6e+03 0.0014 0.16 1 ++
11 -0.35 5.7 -6 8.8 -3.2 3.2 0.41 9.5 3.6e+03 0.0021 0.16 0.68 +
12 -0.35 5.7 -6 8.8 -3.2 3.2 0.41 9.5 3.6e+03 0.0021 0.078 -4.9 -
13 -0.35 5.7 -6 8.8 -3.2 3.2 0.41 9.5 3.6e+03 0.0021 0.039 -1.1 -
14 -0.35 5.8 -6 8.8 -3.2 3.2 0.41 9.5 3.6e+03 0.0012 0.039 0.16 +
15 -0.36 5.8 -6 8.8 -3.2 3.2 0.4 9.5 3.6e+03 0.0029 0.039 0.21 +
16 -0.36 5.8 -6 8.8 -3.2 3.2 0.41 9.5 3.6e+03 0.0028 0.039 0.29 +
17 -0.35 5.8 -6 8.8 -3.2 3.1 0.4 9.4 3.6e+03 0.0004 0.039 0.74 +
18 -0.35 5.8 -6 8.8 -3.2 3.1 0.4 9.4 3.6e+03 0.0004 0.02 -1.3 -
19 -0.35 5.8 -6 8.8 -3.2 3.1 0.4 9.4 3.6e+03 0.0004 0.0098 -0.065 -
20 -0.36 5.8 -6 8.8 -3.2 3.1 0.41 9.4 3.6e+03 0.001 0.0098 0.48 +
21 -0.37 5.8 -6 8.8 -3.2 3.1 0.41 9.4 3.6e+03 0.00021 0.0098 0.61 +
22 -0.37 5.8 -6 8.8 -3.2 3.1 0.41 9.4 3.6e+03 0.00021 0.0049 -0.24 -
23 -0.36 5.8 -6 8.8 -3.2 3.1 0.4 9.4 3.6e+03 0.00047 0.0049 0.71 +
24 -0.36 5.8 -6 8.8 -3.2 3.1 0.4 9.4 3.6e+03 0.00067 0.0049 0.26 +
25 -0.37 5.8 -6 8.8 -3.2 3.1 0.4 9.4 3.6e+03 0.00022 0.0049 0.72 +
26 -0.37 5.8 -6 8.8 -3.2 3.1 0.4 9.4 3.6e+03 0.00012 0.0049 0.73 +
27 -0.37 5.8 -6 8.8 -3.2 3.1 0.4 9.4 3.6e+03 0.00011 0.049 0.95 ++
28 -0.37 5.9 -6 8.7 -3.2 3 0.4 9.4 3.6e+03 0.0013 0.049 0.42 +
29 -0.38 5.9 -6 8.8 -3.2 3 0.39 9.4 3.6e+03 0.00062 0.049 0.48 +
30 -0.38 5.9 -6 8.8 -3.2 3 0.39 9.4 3.6e+03 0.00062 0.024 -5.2 -
31 -0.38 5.9 -6 8.8 -3.2 3 0.39 9.4 3.6e+03 0.00062 0.012 -1.2 -
32 -0.37 5.9 -6 8.8 -3.2 3 0.4 9.4 3.6e+03 0.00025 0.012 0.4 +
33 -0.37 5.9 -6 8.8 -3.2 3 0.4 9.4 3.6e+03 0.00025 0.0061 -0.6 -
34 -0.38 5.9 -6 8.8 -3.2 3 0.39 9.4 3.6e+03 0.00027 0.0061 0.49 +
35 -0.38 5.9 -6 8.8 -3.2 3 0.4 9.5 3.6e+03 0.00013 0.0061 0.21 +
36 -0.37 5.9 -6 8.8 -3.2 3 0.4 9.5 3.6e+03 4.7e-05 0.0061 0.71 +
37 -0.37 5.9 -6 8.8 -3.2 3 0.4 9.5 3.6e+03 4.4e-05 0.0061 0.33 +
38 -0.37 5.9 -6 8.8 -3.2 3 0.4 9.5 3.6e+03 2.5e-05 0.0061 0.59 +
39 -0.37 5.9 -6 8.8 -3.2 3 0.4 9.5 3.6e+03 2.5e-05 0.0031 -0.51 -
40 -0.37 5.9 -6 8.8 -3.2 3 0.4 9.5 3.6e+03 1.1e-05 0.0031 0.56 +
41 -0.37 5.9 -6 8.8 -3.2 3 0.4 9.5 3.6e+03 4.1e-06 0.0031 0.75 +
Optimization algorithm has converged.
Relative gradient: 4.144730231720455e-06
Cause of termination: Relative gradient = 4.1e-06 <= 6.1e-06
Number of function evaluations: 97
Number of gradient evaluations: 55
Number of hessian evaluations: 0
Algorithm: BFGS with trust region for simple bound constraints
Number of iterations: 42
Proportion of Hessian calculation: 0/27 = 0.0%
Optimization time: 0:00:45.151425
Calculate BHHH
File b26_triangular_panel_mixture.html has been generated.
File b26_triangular_panel_mixture.yaml has been generated.
print(results.short_summary())
Results for model b26_triangular_panel_mixture
Nbr of parameters: 8
Sample size: 752
Observations: 6768
Excluded data: 0
Final log likelihood: -3599.787
Akaike Information Criterion: 7215.573
Bayesian Information Criterion: 7252.555
pandas_results = get_pandas_estimated_parameters(estimation_results=results)
display(pandas_results)
Name Value BHHH std err. BHHH t-stat. BHHH p-value
0 asc_train -0.371950 0.226528 -1.641960 0.100598
1 asc_train_s 5.890475 0.602045 9.784113 0.000000
2 b_time -5.992359 0.273432 -21.915335 0.000000
3 b_time_s 8.764962 0.615886 14.231465 0.000000
4 b_cost -3.201639 0.137116 -23.349784 0.000000
5 asc_sm_s 2.984671 0.678491 4.398982 0.000011
6 asc_car 0.396768 0.202729 1.957135 0.050332
7 asc_car_s 9.466908 0.599459 15.792424 0.000000
Total running time of the script: (0 minutes 51.658 seconds)