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Mixture of logit models
Example of a normal mixture of logit models, using Monte-Carlo integration.
- author:
Michel Bierlaire, EPFL
- date:
Sun Apr 9 17:30:14 2023
import biogeme.biogeme_logging as blog
import biogeme.biogeme as bio
from biogeme import models
from biogeme.expressions import Beta, bioDraws, log, MonteCarlo
See the data processing script: Data preparation for Swissmetro.
from swissmetro_data 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,
)
logger = blog.get_screen_logger(level=blog.INFO)
logger.info('Example b05normal_mixtures.py')
Example b05normal_mixtures.py
Parameters to be estimated
ASC_CAR = Beta('ASC_CAR', 0, None, None, 0)
ASC_TRAIN = Beta('ASC_TRAIN', 0, None, None, 0)
ASC_SM = Beta('ASC_SM', 0, None, None, 1)
B_COST = Beta('B_COST', 0, None, None, 0)
Define a random parameter, normally distributed, 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 * bioDraws('B_TIME_RND', 'NORMAL')
Definition of the utility functions.
V1 = ASC_TRAIN + B_TIME_RND * TRAIN_TT_SCALED + B_COST * TRAIN_COST_SCALED
V2 = ASC_SM + B_TIME_RND * SM_TT_SCALED + B_COST * SM_COST_SCALED
V3 = ASC_CAR + 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 to B_TIME_RND, we have a logit model (called the kernel).
prob = models.logit(V, av, CHOICE)
We integrate over B_TIME_RND using Monte-Carlo.
logprob = log(MonteCarlo(prob))
These notes will be included as such in the report file.
USER_NOTES = (
'Example of a mixture of logit models with three alternatives, '
'approximated using Monte-Carlo integration.'
)
Create the Biogeme object. As the objective is to illustrate the syntax, we calculate the Monte-Carlo approximation with a small number of draws. To achieve that, we provide a parameter file different from the default one.
the_biogeme = bio.BIOGEME(
database, logprob, userNotes=USER_NOTES, parameter_file='few_draws.toml'
)
the_biogeme.modelName = 'b05normal_mixture'
File few_draws.toml has been parsed.
print(f'Number of draws: {the_biogeme.number_of_draws}')
Number of draws: 100
Estimate the parameters
results = the_biogeme.estimate()
*** Initial values of the parameters are obtained from the file __b05normal_mixture.iter
Cannot read file __b05normal_mixture.iter. Statement is ignored.
Optimization algorithm: hybrid Newton/BFGS with simple bounds [simple_bounds]
** Optimization: Newton with trust region for simple bounds
Iter. ASC_CAR ASC_TRAIN B_COST B_TIME B_TIME_S Function Relgrad Radius Rho
0 -0.083 -0.8 -0.32 -1 0.87 5.4e+03 0.046 10 1 ++
1 0.013 -0.57 -1 -1.6 0.92 5.2e+03 0.0082 1e+02 1.1 ++
2 0.096 -0.43 -1.2 -2 1.4 5.2e+03 0.0041 1e+03 1.1 ++
3 0.12 -0.41 -1.3 -2.2 1.6 5.2e+03 0.00069 1e+04 1.1 ++
4 0.13 -0.41 -1.3 -2.2 1.6 5.2e+03 9.3e-06 1e+05 1 ++
5 0.13 -0.41 -1.3 -2.2 1.6 5.2e+03 2.2e-09 1e+05 1 ++
Results saved in file b05normal_mixture.html
Results saved in file b05normal_mixture.pickle
print(results.short_summary())
Results for model b05normal_mixture
Nbr of parameters: 5
Sample size: 6768
Excluded data: 3960
Final log likelihood: -5216.34
Akaike Information Criterion: 10442.68
Bayesian Information Criterion: 10476.78
pandas_results = results.getEstimatedParameters()
pandas_results
Total running time of the script: (0 minutes 8.466 seconds)