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Nested logit model
Example of a nested logit model.
- author:
Michel Bierlaire, EPFL
- date:
Tue Oct 24 13:37:32 2023
from biogeme import 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 b09nested')
Example b09nested
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)
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,))
Definition of the model. This is the contribution of each observation to the log likelihood function. The choice model is a nested logit, with availability conditions.
logprob = models.lognested(V, av, nests, CHOICE)
Create the Biogeme object.
the_biogeme = bio.BIOGEME(database, logprob)
the_biogeme.modelName = "b09nested"
File biogeme.toml has been parsed.
Calculate the null log likelihood for reporting.
the_biogeme.calculateNullLoglikelihood(av)
-6964.662979191462
Estimate the parameters
results = the_biogeme.estimate()
*** Initial values of the parameters are obtained from the file __b09nested.iter
Cannot read file __b09nested.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 MU Function Relgrad Radius Rho
0 0.1 -0.75 -1 -0.8 1.5 5.4e+03 0.082 10 0.92 ++
1 -0.22 -0.28 -0.82 -0.86 2.2 5.3e+03 0.076 10 0.44 +
2 -0.17 -0.52 -0.7 -0.7 2.7 5.3e+03 0.023 10 0.72 +
3 -0.17 -0.52 -0.7 -0.7 2.7 5.3e+03 0.023 0.84 -4.6 -
4 -0.14 -0.52 -0.87 -0.92 1.8 5.2e+03 0.006 0.84 0.63 +
5 -0.16 -0.51 -0.87 -0.91 2 5.2e+03 0.0014 8.4 1.1 ++
6 -0.17 -0.51 -0.86 -0.9 2.1 5.2e+03 8.1e-05 84 1 ++
7 -0.17 -0.51 -0.86 -0.9 2.1 5.2e+03 2.8e-07 84 1 ++
Results saved in file b09nested.html
Results saved in file b09nested.pickle
print(results.short_summary())
Results for model b09nested
Nbr of parameters: 5
Sample size: 6768
Excluded data: 3960
Null log likelihood: -6964.663
Final log likelihood: -5236.9
Likelihood ratio test (null): 3455.526
Rho square (null): 0.248
Rho bar square (null): 0.247
Akaike Information Criterion: 10483.8
Bayesian Information Criterion: 10517.9
pandas_results = results.getEstimatedParameters()
pandas_results
We calculate the correlation between the error terms of the alternatives.
corr = nests.correlation(
parameters=results.getBetaValues(),
alternatives_names={1: 'Train', 2: 'Swissmetro', 3: 'Car'},
)
print(corr)
Train Swissmetro Car
Train 1.0 0.0 0.0
Swissmetro 0.0 1.0 0.0
Car 0.0 0.0 1.0
Total running time of the script: (0 minutes 0.411 seconds)