Note
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9. Nested logit modelΒΆ
Example of a nested logit model.
Michel Bierlaire, EPFL Sat Jun 21 2025, 15:33:00
from IPython.core.display_functions import display
from biogeme import biogeme_logging as blog
from biogeme.biogeme import BIOGEME
from biogeme.expressions import Beta
from biogeme.models import lognested
from biogeme.nests import NestsForNestedLogit, OneNestForNestedLogit
from biogeme.results_processing import (
EstimationResults,
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,
SM_AV,
SM_COST_SCALED,
SM_TT_SCALED,
TRAIN_AV_SP,
TRAIN_COST_SCALED,
TRAIN_TT_SCALED,
database,
)
logger = blog.get_screen_logger(level=blog.INFO)
logger.info('Example b09_nested')
Example b09_nested
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)
nest_parameter = Beta('nest_parameter', 1, 1, 3, 0)
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. 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=nest_parameter, 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.
log_probability = lognested(v, av, nests, CHOICE)
Create the Biogeme object.
the_biogeme = BIOGEME(
database, log_probability, optimization_algorithm='simple_bounds_BFGS'
)
the_biogeme.modelName = "b09_nested"
Biogeme parameters read from biogeme.toml.
/Users/bierlair/MyFiles/github/biogeme/docs/source/examples/swissmetro/plot_b09_nested.py:89: DeprecationWarning: 'modelName' is deprecated. Please use 'model_name' instead.
the_biogeme.modelName = "b09_nested"
Calculate the null log likelihood for reporting.
the_biogeme.calculate_null_loglikelihood(av)
-6964.662979192191
Estimate the parameters.
try:
results = EstimationResults.from_yaml_file(
filename=f'saved_results/{the_biogeme.model_name}.yaml'
)
except FileNotFoundError:
results = the_biogeme.estimate()
*** Initial values of the parameters are obtained from the file __b09_nested.iter
Cannot read file __b09_nested.iter. Statement is ignored.
Starting values for the algorithm: {}
Optimization algorithm: BFGS with simple bounds [simple_bounds_BFGS].
Optimization algorithm: hybrid Newton/BFGS with simple bounds [simple_bounds]
** Optimization: BFGS with trust region for simple bounds
Iter. asc_train b_time b_cost nest_parameter asc_car Function Relgrad Radius Rho
0 -1 -1 -1 2 -1 5.7e+03 0.14 1 0.27 +
1 -1 -1 -1 2 -1 5.7e+03 0.14 0.5 -0.74 -
2 -1.1 -0.5 -1.2 1.9 -0.5 5.3e+03 0.038 0.5 0.6 +
3 -0.7 -0.85 -0.74 1.8 -0.35 5.3e+03 0.031 0.5 0.49 +
4 -0.7 -0.85 -0.74 1.8 -0.35 5.3e+03 0.031 0.25 -0.68 -
5 -0.7 -0.85 -0.74 1.8 -0.35 5.3e+03 0.031 0.12 -0.12 -
6 -0.68 -0.73 -0.87 1.9 -0.23 5.3e+03 0.018 0.12 0.52 +
7 -0.66 -0.85 -0.86 1.9 -0.27 5.2e+03 0.0097 0.12 0.7 +
8 -0.54 -0.89 -0.93 2 -0.15 5.2e+03 0.01 0.12 0.31 +
9 -0.54 -0.89 -0.93 2 -0.15 5.2e+03 0.01 0.062 -1.1 -
10 -0.54 -0.89 -0.93 2 -0.15 5.2e+03 0.01 0.031 -0.21 -
11 -0.51 -0.92 -0.9 2 -0.18 5.2e+03 0.0046 0.031 0.56 +
12 -0.51 -0.92 -0.9 2 -0.18 5.2e+03 0.0046 0.016 -0.85 -
13 -0.52 -0.93 -0.88 2 -0.16 5.2e+03 0.0042 0.016 0.16 +
14 -0.51 -0.92 -0.87 2 -0.16 5.2e+03 0.00098 0.016 0.81 +
15 -0.51 -0.92 -0.87 2 -0.16 5.2e+03 0.00098 0.0078 -0.46 -
16 -0.51 -0.92 -0.87 2 -0.16 5.2e+03 0.00098 0.0039 -0.006 -
17 -0.51 -0.92 -0.87 2 -0.16 5.2e+03 0.00095 0.0039 0.52 +
18 -0.51 -0.91 -0.87 2 -0.16 5.2e+03 0.0011 0.0039 0.39 +
19 -0.51 -0.91 -0.86 2 -0.16 5.2e+03 0.00046 0.0039 0.66 +
20 -0.51 -0.91 -0.86 2 -0.16 5.2e+03 0.00031 0.0039 0.81 +
21 -0.51 -0.91 -0.86 2 -0.16 5.2e+03 0.00035 0.0039 0.78 +
22 -0.51 -0.91 -0.86 2 -0.17 5.2e+03 0.00041 0.0039 0.66 +
23 -0.51 -0.9 -0.86 2 -0.16 5.2e+03 0.00048 0.0039 0.45 +
24 -0.51 -0.91 -0.86 2 -0.17 5.2e+03 0.00049 0.0039 0.27 +
25 -0.51 -0.91 -0.86 2 -0.17 5.2e+03 0.00049 0.002 -0.91 -
26 -0.51 -0.9 -0.86 2 -0.17 5.2e+03 0.00028 0.002 0.6 +
27 -0.51 -0.9 -0.86 2 -0.17 5.2e+03 0.00032 0.002 0.56 +
28 -0.51 -0.9 -0.86 2 -0.17 5.2e+03 0.00025 0.002 0.49 +
29 -0.51 -0.9 -0.86 2 -0.17 5.2e+03 0.00028 0.002 0.48 +
30 -0.51 -0.9 -0.86 2 -0.17 5.2e+03 0.00012 0.002 0.77 +
31 -0.51 -0.9 -0.86 2 -0.17 5.2e+03 0.00015 0.002 0.8 +
32 -0.51 -0.9 -0.86 2 -0.17 5.2e+03 0.00019 0.002 0.25 +
33 -0.51 -0.9 -0.86 2.1 -0.17 5.2e+03 8.2e-05 0.002 0.61 +
34 -0.51 -0.9 -0.86 2.1 -0.17 5.2e+03 8.2e-05 0.00098 -0.36 -
35 -0.51 -0.9 -0.86 2.1 -0.17 5.2e+03 6.7e-05 0.00098 0.56 +
36 -0.51 -0.9 -0.86 2.1 -0.17 5.2e+03 3.2e-05 0.00098 0.77 +
37 -0.51 -0.9 -0.86 2.1 -0.17 5.2e+03 3.2e-05 0.00049 -2.2 -
38 -0.51 -0.9 -0.86 2.1 -0.17 5.2e+03 3.2e-05 0.00024 -0.88 -
39 -0.51 -0.9 -0.86 2.1 -0.17 5.2e+03 3.2e-05 0.00012 -0.33 -
40 -0.51 -0.9 -0.86 2.1 -0.17 5.2e+03 2.4e-05 0.00012 0.52 +
41 -0.51 -0.9 -0.86 2.1 -0.17 5.2e+03 1.5e-05 0.00012 0.69 +
42 -0.51 -0.9 -0.86 2.1 -0.17 5.2e+03 9.6e-06 0.00012 0.9 +
43 -0.51 -0.9 -0.86 2.1 -0.17 5.2e+03 1.2e-05 0.00012 0.82 +
44 -0.51 -0.9 -0.86 2.1 -0.17 5.2e+03 8.5e-06 0.00012 0.5 +
45 -0.51 -0.9 -0.86 2.1 -0.17 5.2e+03 5.6e-06 0.00012 0.95 +
Optimization algorithm has converged.
Relative gradient: 5.5658938507214695e-06
Cause of termination: Relative gradient = 5.6e-06 <= 6.1e-06
Number of function evaluations: 113
Number of gradient evaluations: 67
Number of hessian evaluations: 0
Algorithm: BFGS with trust region for simple bound constraints
Number of iterations: 46
Proportion of Hessian calculation: 0/33 = 0.0%
Optimization time: 0:00:00.524550
Calculate second derivatives and BHHH
File b09_nested.html has been generated.
File b09_nested.yaml has been generated.
print(results.short_summary())
Results for model b09_nested
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 = get_pandas_estimated_parameters(estimation_results=results)
display(pandas_results)
Name Value Robust std err. Robust t-stat. Robust p-value
0 asc_train -0.511953 0.079114 -6.471051 9.732348e-11
1 b_time -0.898716 0.107108 -8.390749 0.000000e+00
2 b_cost -0.856701 0.060033 -14.270452 0.000000e+00
3 nest_parameter 2.053862 0.164154 12.511833 0.000000e+00
4 asc_car -0.167141 0.054528 -3.065218 2.175112e-03
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 1.00000 0.0 0.76294
Swissmetro 0.00000 1.0 0.00000
Car 0.76294 0.0 1.00000
Total running time of the script: (0 minutes 1.930 seconds)