Note
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6b. Mixture of logit models with uniform MLHS drawsΒΆ
- Example of a uniform mixture of logit models, using Monte-Carlo
integration. The mixing distribution is uniform. The draws are from the Modified Hypercube Latin Square.
Michel Bierlaire, EPFL Fri Jun 20 2025, 11:24:34
from IPython.core.display_functions import display
import biogeme.biogeme_logging as blog
from biogeme.biogeme import BIOGEME
from biogeme.expressions import Beta, Draws, MonteCarlo, log
from biogeme.models import logit
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 b06b_unif_mixture_MHLS')
Example b06b_unif_mixture_MHLS
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_cost = Beta('b_cost', 0, None, None, 0)
Define a random parameter, normally distributed, designed to be used for Monte-Carlo simulation.
b_time = Beta('b_time', 0, None, None, 0)
It is advised not to use 0 as starting value for the following parameter.
b_time_s = Beta('b_time_s', 1, None, None, 0)
Define a random parameter, uniformly distributed, designed to be used
for Monte-Carlo simulation. The type of draws is set to NORMAL_MLHS.
b_time_rnd = b_time + b_time_s * Draws('b_time_rnd', 'NORMAL_MLHS')
Definition of the utility functions.
v_train = asc_train + b_time_rnd * TRAIN_TT_SCALED + b_cost * TRAIN_COST_SCALED
v_swissmetro = asc_sm + b_time_rnd * SM_TT_SCALED + b_cost * SM_COST_SCALED
v_car = asc_car + b_time_rnd * 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}
Conditional on b_time_rnd, we have a logit model (called the kernel).
conditional_probability = logit(v, av, CHOICE)
We integrate over b_time_rnd using Monte-Carlo
log_probability = log(MonteCarlo(conditional_probability))
Create the Biogeme object.
the_biogeme = BIOGEME(database, log_probability, number_of_draws=10000, seed=1223)
the_biogeme.model_name = 'b06b_unif_mixture_MHLS'
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 __b06b_unif_mixture_MHLS.iter
Cannot read file __b06b_unif_mixture_MHLS.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_time_s b_cost asc_car Function Relgrad Radius Rho
0 -1 -1 2 -1 1 6.1e+03 0.16 1 0.25 +
1 -0.73 -2 3 -0.4 0 5.5e+03 0.049 1 0.36 +
2 -0.95 -2.3 2.6 -1.4 0.51 5.4e+03 0.054 1 0.39 +
3 -0.95 -2.3 2.6 -1.4 0.51 5.4e+03 0.054 0.5 -0.15 -
4 -0.45 -2.8 2.6 -1.1 0.006 5.3e+03 0.03 0.5 0.5 +
5 -0.091 -2.6 2.5 -1.6 0.33 5.3e+03 0.047 0.5 0.13 +
6 -0.091 -2.6 2.5 -1.6 0.33 5.3e+03 0.047 0.25 -0.18 -
7 -0.34 -2.9 2.3 -1.4 0.26 5.2e+03 0.022 0.25 0.65 +
8 -0.29 -2.6 2.2 -1.2 0.22 5.2e+03 0.0084 0.25 0.5 +
9 -0.29 -2.6 2.2 -1.2 0.22 5.2e+03 0.0084 0.12 -2.8 -
10 -0.29 -2.6 2.2 -1.2 0.22 5.2e+03 0.0084 0.062 -0.23 -
11 -0.36 -2.6 2.1 -1.3 0.28 5.2e+03 0.006 0.062 0.46 +
12 -0.29 -2.6 2.1 -1.4 0.22 5.2e+03 0.0065 0.062 0.52 +
13 -0.35 -2.6 2 -1.3 0.21 5.2e+03 0.0092 0.062 0.51 +
14 -0.33 -2.5 2 -1.3 0.21 5.2e+03 0.0034 0.062 0.89 +
15 -0.35 -2.5 1.9 -1.3 0.18 5.2e+03 0.0059 0.062 0.84 +
16 -0.36 -2.4 1.9 -1.3 0.21 5.2e+03 0.0029 0.062 0.62 +
17 -0.37 -2.4 1.8 -1.3 0.16 5.2e+03 0.0018 0.062 0.84 +
18 -0.4 -2.3 1.7 -1.3 0.17 5.2e+03 0.0027 0.062 0.31 +
19 -0.39 -2.3 1.7 -1.3 0.13 5.2e+03 0.002 0.062 0.48 +
20 -0.39 -2.3 1.7 -1.3 0.13 5.2e+03 0.002 0.031 -0.8 -
21 -0.39 -2.3 1.7 -1.3 0.14 5.2e+03 0.0016 0.031 0.28 +
22 -0.39 -2.3 1.7 -1.3 0.14 5.2e+03 0.0016 0.016 0.08 -
23 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 0.0009 0.016 0.79 +
24 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 0.0009 0.0078 -0.47 -
25 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 0.00056 0.0078 0.33 +
26 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 0.00026 0.0078 0.13 +
27 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 0.00026 0.0039 -1.3 -
28 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 0.00026 0.002 0.0035 -
29 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 0.00028 0.002 0.61 +
30 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 0.0001 0.002 0.8 +
31 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 5.8e-05 0.002 0.89 +
32 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 3.3e-05 0.002 0.76 +
33 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 3.2e-05 0.002 0.37 +
34 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 3.2e-05 0.00098 -2.3 -
35 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 3.2e-05 0.00049 0.0087 -
36 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 9.4e-06 0.0049 0.94 ++
37 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 9.4e-06 0.0024 -52 -
38 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 9.4e-06 0.0012 -27 -
39 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 9.4e-06 0.00061 -10 -
40 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 9.4e-06 0.00031 -3.5 -
41 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 9.4e-06 0.00015 -1 -
42 -0.4 -2.3 1.7 -1.3 0.14 5.2e+03 1.6e-06 0.00015 0.86 -
Optimization algorithm has converged.
Relative gradient: 1.632630973351514e-06
Cause of termination: Relative gradient = 1.6e-06 <= 6.1e-06
Number of function evaluations: 98
Number of gradient evaluations: 55
Number of hessian evaluations: 0
Algorithm: BFGS with trust region for simple bound constraints
Number of iterations: 43
Proportion of Hessian calculation: 0/27 = 0.0%
Optimization time: 0:01:37.390006
Calculate second derivatives and BHHH
File b06b_unif_mixture_MHLS.html has been generated.
File b06b_unif_mixture_MHLS.yaml has been generated.
print(results.short_summary())
Results for model b06b_unif_mixture_MHLS
Nbr of parameters: 5
Sample size: 6768
Excluded data: 3960
Final log likelihood: -5214.947
Akaike Information Criterion: 10439.89
Bayesian Information Criterion: 10473.99
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 -0.401859 0.065945 -6.093814 1.102520e-09
1 b_time -2.259754 0.117179 -19.284630 0.000000e+00
2 b_time_s 1.657023 0.132669 12.489949 0.000000e+00
3 b_cost -1.285443 0.086294 -14.896028 0.000000e+00
4 asc_car 0.137021 0.051739 2.648291 8.089997e-03
Total running time of the script: (3 minutes 27.092 seconds)