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Cross-nested logitΒΆ
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)
Michel Bierlaire, EPFL Sat Jun 21 2025, 16:33:38
import biogeme.biogeme_logging as blog
from IPython.core.display_functions import display
from biogeme.biogeme import BIOGEME
from biogeme.expressions import Beta
from biogeme.models import logcnl
from biogeme.nests import NestsForCrossNestedLogit, OneNestForCrossNestedLogit
from biogeme.results_processing import get_pandas_estimated_parameters
See the data processing script: Data preparation for Swissmetro.
from swissmetro_data import (
CAR_AV_SP,
CAR_CO_SCALED,
CAR_TT_SCALED,
CHOICE,
GA,
SM_AV,
SM_COST_SCALED,
SM_HE,
SM_TT_SCALED,
TRAIN_AV_SP,
TRAIN_COST_SCALED,
TRAIN_HE,
TRAIN_TT_SCALED,
database,
)
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_swissmetro = Beta('b_time_swissmetro', 0, None, None, 0)
b_time_train = Beta('b_time_train', 0, None, None, 0)
b_time_car = Beta('b_time_car', 0, None, None, 0)
b_cost = Beta('b_cost', 0, None, None, 0)
b_headway_swissmetro = Beta('b_headway_swissmetro', 0, None, None, 0)
b_headway_train = Beta('b_headway_train', 0, None, None, 0)
ga_train = Beta('ga_train', 0, None, None, 0)
ga_swissmetro = Beta('ga_swissmetro', 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 * TRAIN_TT_SCALED
+ b_cost * TRAIN_COST_SCALED
+ b_headway_train * TRAIN_HE
+ ga_train * GA
)
v_swissmetro = (
asc_sm
+ b_time_swissmetro * SM_TT_SCALED
+ b_cost * SM_COST_SCALED
+ b_headway_swissmetro * SM_HE
+ ga_swissmetro * GA
)
v_car = asc_car + b_time_car * 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.
nest_existing = OneNestForCrossNestedLogit(
nest_param=existing_nest_parameter,
dict_of_alpha={1: alpha_existing, 2: 0.0, 3: 1.0},
name='existing',
)
nest_public = OneNestForCrossNestedLogit(
nest_param=public_nest_parameter,
dict_of_alpha={1: alpha_public, 2: 1.0, 3: 0.0},
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'
Biogeme parameters read from biogeme.toml.
Estimate the parameters.
results = the_biogeme.estimate()
*** Initial values of the parameters are obtained from the file __b11cnl.iter
Parameter values restored from __b11cnl.iter
Starting values for the algorithm: {'ASC_TRAIN': -2.9802322387695312e-08, 'B_TIME': -2.9802322387695312e-08, 'B_COST': -2.9802322387695312e-08, 'ALPHA_EXISTING': 0.5, 'MU_EXISTING': 1.0000000298023224, 'ASC_CAR': -2.9802322387695312e-08, 'MU_PUBLIC': 1.0000000298023224}
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
Iter. asc_train b_time_train b_cost b_headway_train ga_train alpha_existing existing_nest_p b_time_swissmet b_headway_swiss ga_swissmetro asc_car b_time_car public_nest_par Function Relgrad Radius Rho
0 0 0 0 0 0 0.5 1 0 0 0 0 0 1 7e+03 16 0.5 -0.35 -
1 0 0 0 0 0 0.5 1 0 0 0 0 0 1 7e+03 16 0.25 -0.33 -
2 0 0 0 0 0 0.5 1 0 0 0 0 0 1 7e+03 16 0.12 -0.29 -
3 0 0 0 0 0 0.5 1 0 0 0 0 0 1 7e+03 16 0.062 -0.21 -
4 0 0 0 0 0 0.5 1 0 0 0 0 0 1 7e+03 16 0.031 -0.075 -
5 -0.031 -0.031 -0.031 -0.031 0.031 0.5 1 0.031 0.031 0.031 -0.031 -0.031 1 6.4e+03 4.1 0.031 0.12 +
6 -0.031 -0.031 -0.031 -0.031 0.031 0.5 1 0.031 0.031 0.031 -0.031 -0.031 1 6.4e+03 4.1 0.016 -0.071 -
7 -0.031 -0.031 -0.031 -0.031 0.031 0.5 1 0.031 0.031 0.031 -0.031 -0.031 1 6.4e+03 4.1 0.0078 -0.064 -
8 -0.031 -0.031 -0.031 -0.031 0.031 0.5 1 0.031 0.031 0.031 -0.031 -0.031 1 6.4e+03 4.1 0.0039 -1 -
9 -0.031 -0.031 -0.031 -0.031 0.031 0.5 1 0.031 0.031 0.031 -0.031 -0.031 1 6.4e+03 4.1 0.002 -0.83 -
10 -0.031 -0.031 -0.031 -0.031 0.031 0.5 1 0.031 0.031 0.031 -0.031 -0.031 1 6.4e+03 4.1 0.00098 -0.74 -
11 -0.031 -0.031 -0.031 -0.031 0.031 0.5 1 0.031 0.031 0.031 -0.031 -0.031 1 6.4e+03 4.1 0.00049 -0.69 -
12 -0.031 -0.031 -0.031 -0.031 0.031 0.5 1 0.031 0.031 0.031 -0.031 -0.031 1 6.4e+03 4.1 0.00024 -0.68 -
13 -0.031 -0.031 -0.031 -0.031 0.031 0.5 1 0.031 0.031 0.031 -0.031 -0.031 1 6.4e+03 4.1 0.00012 -0.67 -
14 -0.031 -0.031 -0.031 -0.031 0.031 0.5 1 0.031 0.031 0.031 -0.031 -0.031 1 6.4e+03 4.1 6.1e-05 -0.66 -
15 -0.031 -0.031 -0.031 -0.031 0.031 0.5 1 0.031 0.031 0.031 -0.031 -0.031 1 6.4e+03 4.1 3.1e-05 -0.66 -
16 -0.031 -0.031 -0.031 -0.031 0.031 0.5 1 0.031 0.031 0.031 -0.031 -0.031 1 6.4e+03 4.1 1.5e-05 -0.66 -
17 -0.031 -0.031 -0.031 -0.031 0.031 0.5 1 0.031 0.031 0.031 -0.031 -0.031 1 6.4e+03 4.1 7.6e-06 -0.66 -
18 -0.031 -0.031 -0.031 -0.031 0.031 0.5 1 0.031 0.031 0.031 -0.031 -0.031 1 6.4e+03 4.1 3.8e-06 -0.66 -
19 -0.031 -0.031 -0.031 -0.031 0.031 0.5 1 0.031 0.031 0.031 -0.031 -0.031 1 6.4e+03 4.1 1.9e-06 -0.66 -
20 -0.031 -0.031 -0.031 -0.031 0.031 0.5 1 0.031 0.031 0.031 -0.031 -0.031 1 6.4e+03 4.1 9.5e-07 -0.66 -
21 -0.031 -0.031 -0.031 -0.031 0.031 0.5 1 0.031 0.031 0.031 -0.031 -0.031 1 6.4e+03 4.1 4.8e-07 -0.66 -
22 -0.031 -0.031 -0.031 -0.031 0.031 0.5 1 0.031 0.031 0.031 -0.031 -0.031 1 6.4e+03 4.1 2.4e-07 -0.66 -
23 -0.031 -0.031 -0.031 -0.031 0.031 0.5 1 0.031 0.031 0.031 -0.031 -0.031 1 6.4e+03 4.1 1.2e-07 -0.66 -
24 -0.031 -0.031 -0.031 -0.031 0.031 0.5 1 0.031 0.031 0.031 -0.031 -0.031 1 6.4e+03 4.1 6e-08 -0.66 -
25 -0.031 -0.031 -0.031 -0.031 0.031 0.5 1 0.031 0.031 0.031 -0.031 -0.031 1 6.4e+03 4.1 3e-08 -0.66 -
26 -0.031 -0.031 -0.031 -0.031 0.031 0.5 1 0.031 0.031 0.031 -0.031 -0.031 1 6.4e+03 4.1 1.5e-08 -0.66 -
Optimization algorithm has *not* converged.
Algorithm: BFGS with trust region for simple bound constraints
Cause of termination: Trust region is too small: 1.4901161193847656e-08
Number of iterations: 27
Proportion of Hessian calculation: 0/2 = 0.0%
Optimization time: 0:00:04.671759
Calculate second derivatives and BHHH
It seems that the optimization algorithm did not converge. Therefore, the results may not correspond to the maximum likelihood estimator. Check the specification of the model, or the criteria for convergence of the algorithm.
File b11cnl~01.html has been generated.
File b11cnl~01.yaml has been generated.
print(results.short_summary())
Results for model b11cnl
Nbr of parameters: 13
Sample size: 6768
Excluded data: 3960
Final log likelihood: -6433.937
Akaike Information Criterion: 12893.87
Bayesian Information Criterion: 12982.53
pandas_results = get_pandas_estimated_parameters(estimation_results=results)
display(pandas_results)
Name Value ... Robust p-value Active bound
0 asc_train -0.03125 ... 9.223247e-01 False
1 b_time_train -0.03125 ... 7.897335e-01 False
2 b_cost -0.03125 ... 4.847544e-01 False
3 b_headway_train -0.03125 ... 6.834533e-13 False
4 ga_train 0.03125 ... 9.182626e-01 False
5 alpha_existing 0.50000 ... 0.000000e+00 False
6 existing_nest_parameter 1.03125 ... 0.000000e+00 False
7 b_time_swissmetro 0.03125 ... 6.290057e-01 False
8 b_headway_swissmetro 0.03125 ... 6.661338e-16 False
9 ga_swissmetro 0.03125 ... 7.734929e-01 False
10 asc_car -0.03125 ... 7.682654e-01 False
11 b_time_car -0.03125 ... 4.664229e-01 False
12 public_nest_parameter 1.00000 ... 0.000000e+00 True
[13 rows x 6 columns]
Total running time of the script: (0 minutes 25.115 seconds)