Note
Go to the end to download the full example code.
11c. Cross-nested logit with a sparse structureΒΆ
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
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 (
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 b11c_cnl_sparse.py')
Example b11c_cnl_sparse.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 = 'b11c_cnl_sparse'
Biogeme parameters read from biogeme.toml.
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 __b11c_cnl_sparse.iter
Cannot read file __b11c_cnl_sparse.iter. Statement is ignored.
Starting values for the algorithm: {}
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 b_cost alpha_existing existing_nest_p asc_car public_nest_par Function Relgrad Radius Rho
0 -1 -1 -1 0.5 2 -1 2 5.8e+03 0.11 1 0.24 +
1 -0.55 -0.54 -1.2 1 1.9 -0.64 1.9 5.5e+03 0.13 1 0.4 +
2 -0.55 -0.54 -1.2 1 1.9 -0.64 1.9 5.5e+03 0.13 0.5 -0.53 -
3 -0.55 -0.54 -1.2 1 1.9 -0.64 1.9 5.5e+03 0.13 0.25 0.078 -
4 -0.8 -0.79 -1.1 0.75 1.9 -0.39 1.9 5.3e+03 0.059 0.25 0.51 +
5 -0.55 -0.83 -0.94 0.87 1.8 -0.24 1.8 5.2e+03 0.0086 0.25 0.75 +
6 -0.55 -0.83 -0.94 0.87 1.8 -0.24 1.8 5.2e+03 0.0086 0.12 -3.6 -
7 -0.55 -0.83 -0.94 0.87 1.8 -0.24 1.8 5.2e+03 0.0086 0.062 -0.61 -
8 -0.55 -0.83 -0.94 0.87 1.8 -0.24 1.8 5.2e+03 0.0086 0.031 -0.16 -
9 -0.52 -0.86 -0.91 0.87 1.8 -0.27 1.8 5.2e+03 0.01 0.031 0.45 +
10 -0.52 -0.89 -0.94 0.84 1.8 -0.24 1.8 5.2e+03 0.014 0.031 0.3 +
11 -0.49 -0.87 -0.94 0.87 1.9 -0.25 1.8 5.2e+03 0.0073 0.031 0.71 +
12 -0.49 -0.89 -0.93 0.85 1.9 -0.22 1.8 5.2e+03 0.0079 0.031 0.84 +
13 -0.45 -0.89 -0.93 0.85 1.9 -0.22 1.8 5.2e+03 0.0049 0.031 0.81 +
14 -0.44 -0.91 -0.92 0.83 1.9 -0.2 1.8 5.2e+03 0.0066 0.031 0.86 +
15 -0.41 -0.9 -0.92 0.83 1.9 -0.21 1.8 5.2e+03 0.0036 0.031 0.78 +
16 -0.39 -0.91 -0.91 0.8 2 -0.19 1.8 5.2e+03 0.0048 0.031 0.89 +
17 -0.36 -0.91 -0.91 0.79 2 -0.2 1.8 5.2e+03 0.0032 0.031 0.8 +
18 -0.34 -0.91 -0.91 0.76 2 -0.19 1.8 5.2e+03 0.0038 0.031 0.86 +
19 -0.31 -0.91 -0.91 0.75 2 -0.2 1.8 5.2e+03 0.0022 0.031 0.76 +
20 -0.3 -0.91 -0.91 0.72 2.1 -0.2 1.8 5.2e+03 0.0028 0.031 0.82 +
21 -0.28 -0.9 -0.9 0.72 2.1 -0.2 1.8 5.2e+03 0.0017 0.031 0.88 +
22 -0.27 -0.89 -0.9 0.7 2.1 -0.2 1.8 5.2e+03 0.0024 0.031 0.83 +
23 -0.26 -0.89 -0.89 0.7 2.2 -0.21 1.9 5.2e+03 0.0019 0.031 0.82 +
24 -0.26 -0.89 -0.89 0.7 2.2 -0.21 1.9 5.2e+03 0.0019 0.016 -0.51 -
25 -0.25 -0.88 -0.9 0.7 2.2 -0.2 1.9 5.2e+03 0.0015 0.016 0.66 +
26 -0.26 -0.88 -0.89 0.69 2.2 -0.21 1.9 5.2e+03 0.0022 0.016 0.88 +
27 -0.26 -0.88 -0.89 0.69 2.2 -0.21 1.9 5.2e+03 0.0022 0.0078 0.09 -
28 -0.26 -0.88 -0.89 0.69 2.2 -0.21 1.9 5.2e+03 0.0022 0.0039 -0.14 -
29 -0.25 -0.88 -0.9 0.69 2.2 -0.21 1.9 5.2e+03 0.0012 0.0039 0.6 +
30 -0.25 -0.88 -0.9 0.69 2.2 -0.21 1.9 5.2e+03 0.0016 0.0039 0.89 +
31 -0.25 -0.88 -0.89 0.69 2.2 -0.21 1.9 5.2e+03 0.00097 0.039 0.97 ++
32 -0.24 -0.87 -0.89 0.68 2.2 -0.21 1.9 5.2e+03 0.0013 0.39 0.95 ++
33 -0.24 -0.87 -0.89 0.68 2.2 -0.21 1.9 5.2e+03 0.0013 0.2 0.029 -
34 -0.24 -0.87 -0.89 0.68 2.2 -0.21 1.9 5.2e+03 0.0013 0.098 -0.5 -
35 -0.24 -0.87 -0.89 0.68 2.2 -0.21 1.9 5.2e+03 0.0013 0.049 0.058 -
36 -0.23 -0.87 -0.88 0.67 2.3 -0.23 2 5.2e+03 0.0032 0.049 0.6 +
37 -0.21 -0.85 -0.89 0.66 2.3 -0.22 2 5.2e+03 0.0015 0.049 0.87 +
38 -0.19 -0.85 -0.87 0.64 2.3 -0.23 2.1 5.2e+03 0.0023 0.049 0.89 +
39 -0.18 -0.84 -0.88 0.64 2.3 -0.22 2.1 5.2e+03 0.0012 0.49 0.95 ++
40 -0.18 -0.84 -0.88 0.64 2.3 -0.22 2.1 5.2e+03 0.0012 0.24 -0.34 -
41 -0.15 -0.87 -0.9 0.64 2.4 -0.24 2.4 5.2e+03 0.006 0.24 0.15 +
42 -0.15 -0.87 -0.9 0.64 2.4 -0.24 2.4 5.2e+03 0.006 0.12 -0.28 -
43 -0.15 -0.87 -0.9 0.64 2.4 -0.24 2.4 5.2e+03 0.006 0.061 -0.19 -
44 -0.097 -0.82 -0.85 0.59 2.4 -0.24 2.4 5.2e+03 0.0034 0.061 0.83 +
45 -0.097 -0.82 -0.85 0.59 2.4 -0.24 2.4 5.2e+03 0.0034 0.031 -2.7 -
46 -0.097 -0.82 -0.85 0.59 2.4 -0.24 2.4 5.2e+03 0.0034 0.015 -1.3 -
47 -0.097 -0.82 -0.85 0.59 2.4 -0.24 2.4 5.2e+03 0.0034 0.0076 -0.84 -
48 -0.097 -0.82 -0.85 0.59 2.4 -0.24 2.4 5.2e+03 0.0034 0.0038 0.074 -
49 -0.1 -0.82 -0.85 0.58 2.4 -0.24 2.4 5.2e+03 0.0014 0.0038 0.61 +
50 -0.099 -0.82 -0.86 0.59 2.4 -0.24 2.4 5.2e+03 0.0013 0.038 0.94 ++
51 -0.11 -0.82 -0.88 0.59 2.4 -0.23 2.5 5.2e+03 0.002 0.038 0.49 +
52 -0.1 -0.82 -0.86 0.58 2.4 -0.24 2.5 5.2e+03 0.0013 0.038 0.81 +
53 -0.082 -0.82 -0.86 0.57 2.4 -0.23 2.6 5.2e+03 0.0032 0.038 0.66 +
54 -0.075 -0.83 -0.87 0.58 2.4 -0.23 2.6 5.2e+03 0.0011 0.038 0.89 +
55 -0.071 -0.81 -0.86 0.57 2.4 -0.24 2.6 5.2e+03 0.0018 0.038 0.8 +
56 -0.073 -0.82 -0.87 0.57 2.4 -0.24 2.7 5.2e+03 0.001 0.038 0.82 +
57 -0.052 -0.82 -0.85 0.56 2.4 -0.24 2.7 5.2e+03 0.0016 0.038 0.79 +
58 -0.056 -0.82 -0.87 0.56 2.4 -0.24 2.7 5.2e+03 0.00089 0.38 0.95 ++
59 -0.0088 -0.78 -0.84 0.52 2.4 -0.27 3.1 5.2e+03 0.0026 0.38 0.65 +
60 -0.0088 -0.78 -0.84 0.52 2.4 -0.27 3.1 5.2e+03 0.0026 0.19 -4.8 -
61 -0.0088 -0.78 -0.84 0.52 2.4 -0.27 3.1 5.2e+03 0.0026 0.095 -4.7 -
62 -0.0088 -0.78 -0.84 0.52 2.4 -0.27 3.1 5.2e+03 0.0026 0.048 -1.3 -
63 0.0086 -0.81 -0.85 0.55 2.4 -0.22 3.1 5.2e+03 0.0024 0.048 0.39 +
64 0.0086 -0.81 -0.85 0.55 2.4 -0.22 3.1 5.2e+03 0.0024 0.024 -0.68 -
65 0.017 -0.81 -0.85 0.52 2.4 -0.24 3.2 5.2e+03 0.00084 0.024 0.65 +
66 0.017 -0.81 -0.85 0.52 2.4 -0.24 3.2 5.2e+03 0.00084 0.012 0.04 -
67 0.0082 -0.81 -0.85 0.53 2.4 -0.24 3.2 5.2e+03 0.00083 0.012 0.84 +
68 0.01 -0.81 -0.85 0.53 2.4 -0.24 3.2 5.2e+03 0.0013 0.012 0.61 +
69 0.011 -0.8 -0.85 0.53 2.4 -0.24 3.2 5.2e+03 0.00083 0.12 0.94 ++
70 0.011 -0.8 -0.85 0.53 2.4 -0.24 3.2 5.2e+03 0.00083 0.06 -0.59 -
71 0.022 -0.8 -0.83 0.53 2.5 -0.24 3.3 5.2e+03 0.002 0.06 0.27 +
72 0.036 -0.81 -0.85 0.52 2.5 -0.25 3.3 5.2e+03 0.0016 0.06 0.71 +
73 0.036 -0.81 -0.85 0.52 2.5 -0.25 3.3 5.2e+03 0.0016 0.03 -4.2 -
74 0.036 -0.81 -0.85 0.52 2.5 -0.25 3.3 5.2e+03 0.0016 0.015 -1.5 -
75 0.036 -0.81 -0.85 0.52 2.5 -0.25 3.3 5.2e+03 0.0016 0.0075 0.017 -
76 0.03 -0.8 -0.85 0.52 2.5 -0.24 3.3 5.2e+03 0.0012 0.0075 0.56 +
77 0.025 -0.8 -0.84 0.53 2.5 -0.24 3.3 5.2e+03 0.0008 0.0075 0.72 +
78 0.028 -0.8 -0.84 0.52 2.5 -0.24 3.3 5.2e+03 0.0008 0.075 0.94 ++
79 0.033 -0.79 -0.83 0.52 2.5 -0.24 3.4 5.2e+03 0.0018 0.075 0.59 +
80 0.033 -0.79 -0.83 0.52 2.5 -0.24 3.4 5.2e+03 0.0018 0.037 0.067 -
81 0.033 -0.79 -0.84 0.53 2.5 -0.23 3.4 5.2e+03 0.0009 0.037 0.36 +
82 0.059 -0.8 -0.84 0.51 2.5 -0.23 3.5 5.2e+03 0.0013 0.037 0.59 +
83 0.059 -0.8 -0.84 0.51 2.5 -0.23 3.5 5.2e+03 0.0013 0.019 -0.56 -
84 0.04 -0.79 -0.84 0.52 2.5 -0.24 3.5 5.2e+03 0.00077 0.019 0.61 +
85 0.052 -0.79 -0.84 0.51 2.5 -0.24 3.5 5.2e+03 0.00075 0.019 0.66 +
86 0.051 -0.8 -0.83 0.52 2.5 -0.23 3.5 5.2e+03 0.00075 0.019 0.5 +
87 0.044 -0.79 -0.84 0.51 2.5 -0.24 3.5 5.2e+03 0.00074 0.019 0.67 +
88 0.058 -0.79 -0.83 0.51 2.5 -0.24 3.6 5.2e+03 0.00074 0.019 0.82 +
89 0.049 -0.79 -0.83 0.51 2.5 -0.24 3.6 5.2e+03 0.00073 0.019 0.68 +
90 0.055 -0.79 -0.84 0.51 2.5 -0.24 3.6 5.2e+03 0.00072 0.19 0.93 ++
91 0.089 -0.78 -0.82 0.49 2.5 -0.24 3.8 5.2e+03 0.0025 0.19 0.42 +
92 0.08 -0.77 -0.83 0.5 2.5 -0.25 4 5.2e+03 0.0019 0.19 0.54 +
93 0.08 -0.77 -0.83 0.5 2.5 -0.25 4 5.2e+03 0.0019 0.093 -0.87 -
94 0.089 -0.77 -0.82 0.5 2.6 -0.24 4 5.2e+03 0.00057 0.093 0.5 +
95 0.089 -0.77 -0.82 0.5 2.6 -0.24 4 5.2e+03 0.00057 0.041 0.022 -
96 0.085 -0.77 -0.82 0.5 2.5 -0.24 4 5.2e+03 0.00036 0.41 1 ++
97 0.085 -0.77 -0.82 0.5 2.5 -0.24 4 5.2e+03 0.00036 0.04 -0.24 -
98 0.085 -0.77 -0.82 0.5 2.5 -0.24 4 5.2e+03 0.00036 0.02 -1.1 -
99 0.085 -0.77 -0.82 0.5 2.5 -0.24 4 5.2e+03 0.00036 0.01 -0.47 -
100 0.085 -0.77 -0.82 0.5 2.5 -0.24 4 5.2e+03 0.00036 0.005 -0.22 -
101 0.09 -0.78 -0.82 0.5 2.5 -0.24 4 5.2e+03 0.00027 0.005 0.65 +
102 0.091 -0.78 -0.82 0.5 2.5 -0.24 4 5.2e+03 0.00028 0.005 0.4 +
103 0.089 -0.78 -0.82 0.5 2.5 -0.24 4 5.2e+03 0.00017 0.05 0.99 ++
104 0.094 -0.78 -0.82 0.5 2.5 -0.24 4.1 5.2e+03 0.00014 0.5 1.1 ++
105 0.098 -0.78 -0.82 0.5 2.5 -0.24 4.1 5.2e+03 0.0002 5 0.92 ++
106 0.098 -0.78 -0.82 0.49 2.5 -0.24 4.1 5.2e+03 7.3e-05 5 0.58 +
107 0.098 -0.78 -0.82 0.49 2.5 -0.24 4.1 5.2e+03 2.8e-06 5 0.99 +
Optimization algorithm has converged.
Relative gradient: 2.769519681924656e-06
Cause of termination: Relative gradient = 2.8e-06 <= 6.1e-06
Number of function evaluations: 255
Number of gradient evaluations: 147
Number of hessian evaluations: 0
Algorithm: BFGS with trust region for simple bound constraints
Number of iterations: 108
Proportion of Hessian calculation: 0/73 = 0.0%
Optimization time: 0:00:01.589921
Calculate second derivatives and BHHH
File b11c_cnl_sparse.html has been generated.
File b11c_cnl_sparse.yaml has been generated.
print(results.short_summary())
Results for model b11c_cnl_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]
Total running time of the script: (0 minutes 5.545 seconds)