Note
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Nested logit with corrections for endogeneous sampling
The sample is said to be endogenous if the probability for an individual to be in the sample depends on the choice that has been made. In that case, the ESML estimator is not appropriate anymore, and corrections need to be made. See Bierlaire, bolduc, McFadden (2008).
This is illustrated in this example.
- author:
Michel Bierlaire, EPFL
- date:
Sun Apr 9 18:25:03 2023
import numpy as np
import biogeme.biogeme_logging as blog
import biogeme.biogeme as bio
from biogeme import models
from biogeme.expressions import Beta
from biogeme.nests import OneNestForNestedLogit, NestsForNestedLogit
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 b14nested_endogenous_sampling.py')
Example b14nested_endogenous_sampling.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_TIME = Beta('B_TIME', 0, None, None, 0)
B_COST = Beta('B_COST', 0, None, None, 0)
MU = Beta('MU', 1, 1, 10, 0)
In this example, we assume that the three modes exist, and that the sampling protocol is choice-based. The probability that a respondent belongs to the sample is R_i.
R_TRAIN = 4.42e-2
R_SM = 3.36e-3
R_CAR = 7.5e-3
The correction terms are the log of these quantities
correction = {1: np.log(R_TRAIN), 2: np.log(R_SM), 3: np.log(R_CAR)}
Definition of the utility functions.
V1 = ASC_TRAIN + B_TIME * TRAIN_TT_SCALED + B_COST * TRAIN_COST_SCALED
V2 = ASC_SM + B_TIME * SM_TT_SCALED + B_COST * SM_COST_SCALED
V3 = ASC_CAR + B_TIME * 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}
Definition of nests. Only the non-trivial nests must be defined. A trivial nest is a nest containing exactly one alternative. In this example, we create a nest for the existing modes, that is train (1) and car (3).
existing = OneNestForNestedLogit(
nest_param=MU, list_of_alternatives=[1, 3], name='existing'
)
nests = NestsForNestedLogit(choice_set=list(V), tuple_of_nests=(existing,))
The following elements do not appear in any nest and are assumed each to be alone in a separate nest: {2}. If it is not the intention, check the assignment of alternatives to nests.
The choice model is a nested logit, with corrections for endogenous sampling We first obtain the expression of the Gi function for nested logit.
Gi = models.get_mev_for_nested(V, av, nests)
Then we calculate the MEV log probability, accounting for the correction.
logprob = models.logmev_endogenous_sampling(V, Gi, av, correction, CHOICE)
Create the Biogeme object.
the_biogeme = bio.BIOGEME(database, logprob)
the_biogeme.modelName = 'b14nested_endogenous_eampling'
Biogeme parameters read from biogeme.toml.
Estimate the parameters.
results = the_biogeme.estimate()
As the model is not too complex, we activate the calculation of second derivatives. If you want to change it, change the name of the algorithm in the TOML file from "automatic" to "simple_bounds"
*** Initial values of the parameters are obtained from the file __b14nested_endogenous_eampling.iter
Cannot read file __b14nested_endogenous_eampling.iter. Statement is ignored.
As the model is not too complex, we activate the calculation of second derivatives. If you want to change it, change the name of the algorithm in the TOML file from "automatic" to "simple_bounds"
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 MU Function Relgrad Radius Rho
0 1 -1 0.47 -1 2 8.1e+03 0.15 1 0.56 +
1 0 -1.1 0.0036 -2 1.6 6.1e+03 0.068 10 1.1 ++
2 -1.5 -3 -0.95 -0.12 2.2 5.4e+03 0.036 10 0.71 +
3 -1.4 -2.9 -0.91 -0.46 1.9 5.3e+03 0.015 1e+02 1.3 ++
4 -1.2 -2.8 -0.98 -0.86 1.7 5.2e+03 0.0048 1e+03 1.1 ++
5 -1.1 -2.8 -1 -0.97 1.6 5.2e+03 0.00035 1e+04 1 ++
6 -1.1 -2.8 -1 -0.97 1.6 5.2e+03 2e-06 1e+04 1 ++
Results saved in file b14nested_endogenous_eampling.html
Results saved in file b14nested_endogenous_eampling.pickle
print(results.short_summary())
Results for model b14nested_endogenous_eampling
Nbr of parameters: 5
Sample size: 6768
Excluded data: 3960
Final log likelihood: -5202.916
Akaike Information Criterion: 10415.83
Bayesian Information Criterion: 10449.93
pandas_results = results.get_estimated_parameters()
pandas_results
Total running time of the script: (0 minutes 0.383 seconds)