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Nested logit model normalized from bottom
- Example of a nested logit model where the normalization is done at the
bottom level.
- author:
Michel Bierlaire, EPFL
- date:
Tue Oct 24 13:40:46 2023
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 b10nested_bottom.py')
Example b10nested_bottom.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)
This is the scale parameter of the choice model. It is usually normalized to one. In this example, we normalize the nest parameter instead, and estimate the scale parameter for the model. If the lower bound is set to zero, the model cannot be evaluated. Therefore, we set the lower bound to a small number, strictly larger than zero.
MU = Beta('MU', 0.5, 0.000001, 1.0, 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 trival nests must be defined. A trivial nest is a nest containing exactly one alternative.
existing = OneNestForNestedLogit(
nest_param=1.0, 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.
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, where the scale parameter mu is explicitly involved.
logprob = models.lognested_mev_mu(V, av, nests, CHOICE, MU)
Create the Biogeme object.
the_biogeme = bio.BIOGEME(database, logprob)
the_biogeme.modelName = 'b10nested_bottom'
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 __b10nested_bottom.iter
Cannot read file __b10nested_bottom.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 0.4 -0.85 -0.96 -0.86 1 5.7e+03 0.13 1 0.74 +
1 -0.069 -0.81 -1 -1.2 1 5.3e+03 0.019 10 1 ++
2 -0.069 -0.81 -1 -1.2 1 5.3e+03 0.019 0.5 -2.3 -
3 -0.19 -0.93 -1.2 -1.5 0.5 5.3e+03 0.017 0.5 0.74 +
4 -0.3 -0.99 -1.6 -1.8 0.54 5.2e+03 0.00089 5 0.97 ++
5 -0.35 -1.1 -1.8 -1.9 0.48 5.2e+03 0.00059 5 0.87 +
6 -0.35 -1.1 -1.8 -1.9 0.48 5.2e+03 5.7e-05 5 1 +
Results saved in file b10nested_bottom.html
Results saved in file b10nested_bottom.pickle
print(results.short_summary())
Results for model b10nested_bottom
Nbr of parameters: 5
Sample size: 6768
Excluded data: 3960
Final log likelihood: -5236.9
Akaike Information Criterion: 10483.8
Bayesian Information Criterion: 10517.9
pandas_results = results.get_estimated_parameters()
pandas_results
Total running time of the script: (0 minutes 0.457 seconds)