Note
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Mixture of logit models
- Example of a uniform mixture of logit models, using Monte-Carlo
integration. The mixing distribution is uniform. The draws are from the Modified Hypercube Latin Square.
- author:
Michel Bierlaire, EPFL
- date:
Sun Apr 9 17:50:28 2023
import biogeme.biogeme_logging as blog
import biogeme.biogeme as bio
from biogeme import models
from biogeme.expressions import (
Beta,
bioDraws,
exp,
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 b06unif_mixture_MHLS')
Example b06unif_mixture_MHLS
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, uniformly distributed, designed to be used
for Monte-Carlo simulation. The type of draws is set to NORMAL_MLHS.
B_TIME_RND = B_TIME + B_TIME_S * bioDraws('B_TIME_RND', 'NORMAL_MLHS')
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 = exp(models.loglogit(V, av, CHOICE))
We integrate over B_TIME_RND using Monte-Carlo
logprob = log(MonteCarlo(prob))
Create the Biogeme object.
the_biogeme = bio.BIOGEME(database, logprob, parameter_file='few_draws.toml')
the_biogeme.modelName = '06unif_mixture_MHLS'
File few_draws.toml has been parsed.
Estimate the parameters.
results = the_biogeme.estimate()
*** Initial values of the parameters are obtained from the file __06unif_mixture_MHLS.iter
Cannot read file __06unif_mixture_MHLS.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 -0.08 -0.8 -0.32 -1 0.87 5.4e+03 0.046 10 1 ++
1 0.0086 -0.58 -0.99 -1.6 0.92 5.2e+03 0.0096 1e+02 1.1 ++
2 0.091 -0.44 -1.2 -2 1.4 5.2e+03 0.0063 1e+03 1.1 ++
3 0.12 -0.42 -1.3 -2.2 1.5 5.2e+03 0.00062 1e+04 1.1 ++
4 0.12 -0.42 -1.3 -2.2 1.6 5.2e+03 7.4e-06 1e+05 1 ++
5 0.12 -0.42 -1.3 -2.2 1.6 5.2e+03 7.8e-10 1e+05 1 ++
Results saved in file 06unif_mixture_MHLS.html
Results saved in file 06unif_mixture_MHLS.pickle
print(results.short_summary())
Results for model 06unif_mixture_MHLS
Nbr of parameters: 5
Sample size: 6768
Excluded data: 3960
Final log likelihood: -5222.239
Akaike Information Criterion: 10454.48
Bayesian Information Criterion: 10488.58
pandas_results = results.getEstimatedParameters()
pandas_results
Total running time of the script: (0 minutes 9.122 seconds)