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Mixture of logit with Halton draws
Example of a mixture of logit models, using quasi Monte-Carlo integration with Halton draws (base 5). The mixing distribution is normal.
- author:
Michel Bierlaire, EPFL
- date:
Wed Apr 12 18:21:13 2023
import biogeme.biogeme_logging as blog
import biogeme.biogeme as bio
from biogeme import models
from biogeme.expressions import Beta, bioDraws, MonteCarlo, log
See the data processing script: Data preparation for Swissmetro.
from swissmetro_data import (
database,
CHOICE,
CAR_AV_SP,
TRAIN_AV_SP,
TRAIN_TT_SCALED,
TRAIN_COST_SCALED,
SM_TT_SCALED,
SM_COST_SCALED,
CAR_TT_SCALED,
CAR_CO_SCALED,
SM_AV,
)
logger = blog.get_screen_logger(level=blog.INFO)
logger.info('Example b24halton_mixture.py')
Example b24halton_mixture.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)
Define a random parameter with a normal distribution, designed to be used for quasi Monte-Carlo simulation with Halton draws (base 5).
B_TIME_RND = B_TIME + B_TIME_S * bioDraws('B_TIME_RND', 'NORMAL_HALTON5')
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 on 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 with Halton draws.'
)
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 = 'b24halton_mixture'
File few_draws.toml has been parsed.
Estimate the parameters
results = the_biogeme.estimate()
*** Initial values of the parameters are obtained from the file __b24halton_mixture.iter
Cannot read file __b24halton_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 -0.082 -0.79 -0.32 -1 0.87 5.4e+03 0.046 10 1 ++
1 0.018 -0.56 -0.99 -1.6 0.91 5.2e+03 0.008 1e+02 1.1 ++
2 0.098 -0.42 -1.2 -2 1.4 5.2e+03 0.0039 1e+03 1.2 ++
3 0.13 -0.4 -1.3 -2.2 1.6 5.2e+03 0.00081 1e+04 1.1 ++
4 0.14 -0.4 -1.3 -2.3 1.7 5.2e+03 2.7e-05 1e+05 1 ++
5 0.14 -0.4 -1.3 -2.3 1.7 5.2e+03 1.4e-08 1e+05 1 ++
Results saved in file b24halton_mixture.html
Results saved in file b24halton_mixture.pickle
print(results.short_summary())
Results for model b24halton_mixture
Nbr of parameters: 5
Sample size: 6768
Excluded data: 3960
Final log likelihood: -5215.687
Akaike Information Criterion: 10441.37
Bayesian Information Criterion: 10475.47
pandas_results = results.getEstimatedParameters()
pandas_results
Total running time of the script: (0 minutes 8.699 seconds)