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,))

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.

logprob = models.lognested(V, av, nests, CHOICE)

Create the Biogeme object.

the_biogeme = bio.BIOGEME(database, logprob)
the_biogeme.modelName = "b09nested"

Biogeme parameters read from biogeme.toml.

Calculate the null log likelihood for reporting.

the_biogeme.calculate_null_loglikelihood(av)

np.float64(-6964.662979191462)

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 __b09nested.iter
Cannot read file __b09nested.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             0.1           -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           -0.16           -0.51           -0.87           -0.91               2      5.2e+03    8.1e-05      8.4        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):           3455.525
Rho square (null):                      0.248
Rho bar square (null):                  0.247
Akaike Information Criterion:   10483.8
Bayesian Information Criterion: 10517.9

pandas_results = results.get_estimated_parameters()
pandas_results

	Value	Rob. Std err	Rob. t-test	Rob. p-value
ASC_CAR	-0.166892	0.054518	-3.061193	2.204573e-03
ASC_TRAIN	-0.512028	0.079123	-6.471258	9.719070e-11
B_COST	-0.857133	0.060002	-14.285004	0.000000e+00
B_TIME	-0.899360	0.107040	-8.402065	0.000000e+00
MU	2.051129	0.163477	12.546934	0.000000e+00

We calculate the correlation between the error terms of the alternatives.

corr = nests.correlation(
    parameters=results.get_beta_values(),
    alternatives_names={1: 'Train', 2: 'Swissmetro', 3: 'Car'},
)
print(corr)

               Train  Swissmetro       Car
Train       1.000000         0.0  0.762308
Swissmetro  0.000000         1.0  0.000000
Car         0.762308         0.0  1.000000