Cross-nested logit

Example of a cross-nested logit model with two nests:

• one with existing alternatives (car and train),

• one with public transportation alternatives (train and Swissmetro)

This illustrates the possibility to ignore all membership parameters that are 0.

Michel Bierlaire, EPFL Sat Jun 21 2025, 16:50:19

from IPython.core.display_functions import display

import biogeme.biogeme_logging as blog
from biogeme.biogeme import BIOGEME
from biogeme.expressions import Beta
from biogeme.models import logcnl
from biogeme.nests import OneNestForCrossNestedLogit, NestsForCrossNestedLogit
from biogeme.results_processing import get_pandas_estimated_parameters

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 b11cnl.py')

Example b11cnl.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)

existing_nest_parameter = Beta('existing_nest_parameter', 1, 1, 5, 0)
public_nest_parameter = Beta('public_nest_parameter', 1, 1, 5, 0)

Nest membership parameters.

alpha_existing = Beta('alpha_existing', 0.5, 0, 1, 0)
alpha_public = 1 - alpha_existing

Definition of the utility functions

v_train = asc_train + b_time * TRAIN_TT_SCALED + b_cost * TRAIN_COST_SCALED
v_swissmetro = asc_sm + b_time * SM_TT_SCALED + b_cost * SM_COST_SCALED
v_car = asc_car + b_time * CAR_TT_SCALED + b_cost * CAR_CO_SCALED

Associate utility functions with the numbering of alternatives

v = {1: v_train, 2: v_swissmetro, 3: v_car}

Associate the availability conditions with the alternatives

av = {1: TRAIN_AV_SP, 2: SM_AV, 3: CAR_AV_SP}

Definition of nests.

The parameter for alternative 2 is omitted, which is equivalent to sez it to zero.

nest_existing = OneNestForCrossNestedLogit(
    nest_param=existing_nest_parameter,
    dict_of_alpha={1: alpha_existing, 3: 1.0},
    name='existing',
)

The parameter for alternative 3 is omitted, which is equivalent to sez it to zero.

nest_public = OneNestForCrossNestedLogit(
    nest_param=public_nest_parameter,
    dict_of_alpha={1: alpha_public, 2: 1.0},
    name='public',
)

nests = NestsForCrossNestedLogit(
    choice_set=[1, 2, 3], tuple_of_nests=(nest_existing, nest_public)
)

The choice model is a cross-nested logit, with availability conditions.

log_probability = logcnl(v, av, nests, CHOICE)

Create the Biogeme object

the_biogeme = BIOGEME(database, log_probability)
the_biogeme.model_name = 'b11cnl_sparse'

Biogeme parameters read from biogeme.toml.

Estimate the parameters.

results = the_biogeme.estimate()

*** Initial values of the parameters are obtained from the file __b11cnl_sparse.iter
Parameter values restored from __b11cnl_sparse.iter
Starting values for the algorithm: {'asc_train': 0.09826835034457922, 'b_time': -0.7768534634860053, 'b_cost': -0.8188920096594592, 'alpha_existing': 0.49508388715964563, 'existing_nest_parameter': 2.5148602962187683, 'asc_car': -0.24044087243791165, 'public_nest_parameter': 4.113502745252501}
As the model is rather complex, we cancel the calculation of second derivatives. If you want to control the parameters, change the algorithm from "automatic" to "simple_bounds" in the TOML file.
Optimization algorithm: hybrid Newton/BFGS with simple bounds [simple_bounds]
** Optimization: BFGS with trust region for simple bounds
Optimization algorithm has converged.
Relative gradient: 3.699384197972795e-06
Cause of termination: Relative gradient = 3.7e-06 <= 6.1e-06
Number of function evaluations: 1
Number of gradient evaluations: 1
Number of hessian evaluations: 0
Algorithm: BFGS with trust region for simple bound constraints
Number of iterations: 0
Optimization time: 0:00:01.349750
Calculate second derivatives and BHHH
File b11cnl_sparse~00.html has been generated.
File b11cnl_sparse~00.yaml has been generated.

print(results.short_summary())

Results for model b11cnl_sparse
Nbr of parameters:              7
Sample size:                    6768
Excluded data:                  3960
Final log likelihood:           -5214.049
Akaike Information Criterion:   10442.1
Bayesian Information Criterion: 10489.84

pandas_results = get_pandas_estimated_parameters(estimation_results=results)
display(pandas_results)

                      Name     Value  ...  Robust t-stat.  Robust p-value
0                asc_train  0.098268  ...        1.404207    1.602572e-01
1                   b_time -0.776853  ...       -7.587858    3.241851e-14
2                   b_cost -0.818892  ...      -13.886190    0.000000e+00
3           alpha_existing  0.495084  ...       14.245344    0.000000e+00
4  existing_nest_parameter  2.514860  ...       10.127306    0.000000e+00
5                  asc_car -0.240441  ...       -4.498402    6.846623e-06
6    public_nest_parameter  4.113503  ...        8.281134    2.220446e-16

[7 rows x 5 columns]