Note
Go to the end to download the full example code.
Mixture of logit models
- 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.
- author:
Michel Bierlaire, EPFL
- date:
Sun Apr 9 17:50:28 2023
import biogeme.biogeme_logging as blog
import biogeme.biogeme as bio
from biogeme import models
from biogeme.expressions import (
Beta,
bioDraws,
exp,
log,
MonteCarlo,
)
from biogeme.parameters import 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 b06unif_mixture_MHLS')
Example b06unif_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 * bioDraws('b_time_rnd', 'NORMAL_MLHS')
Definition of the utility functions.
V1 = ASC_TRAIN + B_TIME_RND * TRAIN_TT_SCALED + B_COST * TRAIN_COST_SCALED
V2 = ASC_SM + B_TIME_RND * SM_TT_SCALED + B_COST * SM_COST_SCALED
V3 = ASC_CAR + B_TIME_RND * CAR_TT_SCALED + B_COST * CAR_CO_SCALED
Associate utility functions with the numbering of alternatives.
V = {1: V1, 2: V2, 3: V3}
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).
prob = exp(models.loglogit(V, av, CHOICE))
We integrate over b_time_rnd using Monte-Carlo
logprob = log(MonteCarlo(prob))
Create the Biogeme object.
the_biogeme = bio.BIOGEME(database, logprob, number_of_draws=100, seed=1223)
the_biogeme.modelName = '06unif_mixture_MHLS'
Biogeme parameters read from biogeme.toml.
Estimate the parameters.
results = the_biogeme.estimate()
As the model is rather complex, we cancel the calculation of second derivatives. If you want to control the parameters, change the name of the algorithm in the TOML file from "automatic" to "simple_bounds"
*** Initial values of the parameters are obtained from the file __06unif_mixture_MHLS.iter
Cannot read file __06unif_mixture_MHLS.iter. Statement is ignored.
The number of draws (100) is low. The results may not be meaningful.
As the model is rather complex, we cancel the calculation of second derivatives. If you want to control the parameters, change the name of the algorithm in the TOML file from "automatic" to "simple_bounds"
Optimization algorithm: hybrid Newton/BFGS with simple bounds [simple_bounds]
** Optimization: BFGS with trust region for simple bounds
Iter. ASC_CAR ASC_TRAIN B_COST B_TIME B_TIME_S Function Relgrad Radius Rho
0 1 -1 -1 -1 2 6.1e+03 0.16 1 0.25 +
1 0 -0.75 -0.34 -2 3 5.5e+03 0.049 1 0.34 +
2 0 -0.75 -0.34 -2 3 5.5e+03 0.049 0.5 -0.039 -
3 0.5 -0.94 -0.84 -2.4 2.6 5.4e+03 0.043 0.5 0.48 +
4 0 -0.44 -1.3 -2.2 2.4 5.3e+03 0.037 0.5 0.45 +
5 0 -0.44 -1.3 -2.2 2.4 5.3e+03 0.037 0.25 -0.035 -
6 0.16 -0.68 -1.4 -2.4 2.2 5.2e+03 0.017 0.25 0.54 +
7 0.084 -0.43 -1.4 -2.5 2.1 5.2e+03 0.0088 0.25 0.73 +
8 0.084 -0.43 -1.4 -2.5 2.1 5.2e+03 0.0088 0.12 -0.66 -
9 0.21 -0.47 -1.4 -2.4 1.9 5.2e+03 0.0088 0.12 0.24 +
10 0.17 -0.34 -1.3 -2.4 1.8 5.2e+03 0.0053 0.12 0.71 +
11 0.17 -0.34 -1.3 -2.4 1.8 5.2e+03 0.0053 0.062 -1.4 -
12 0.17 -0.34 -1.3 -2.4 1.8 5.2e+03 0.0053 0.031 -0.094 -
13 0.2 -0.38 -1.3 -2.4 1.8 5.2e+03 0.0049 0.031 0.2 +
14 0.17 -0.35 -1.3 -2.4 1.8 5.2e+03 0.0025 0.031 0.8 +
15 0.18 -0.37 -1.3 -2.4 1.8 5.2e+03 0.0022 0.031 0.71 +
16 0.15 -0.37 -1.3 -2.3 1.7 5.2e+03 0.0031 0.031 0.76 +
17 0.16 -0.38 -1.3 -2.3 1.7 5.2e+03 0.0016 0.031 0.85 +
18 0.14 -0.39 -1.3 -2.3 1.7 5.2e+03 0.0033 0.031 0.71 +
19 0.15 -0.39 -1.3 -2.3 1.7 5.2e+03 0.0013 0.031 0.84 +
20 0.14 -0.41 -1.3 -2.3 1.6 5.2e+03 0.0026 0.031 0.62 +
21 0.12 -0.4 -1.3 -2.2 1.6 5.2e+03 0.0014 0.031 0.51 +
22 0.12 -0.4 -1.3 -2.2 1.6 5.2e+03 0.0014 0.016 -1.7 -
23 0.14 -0.41 -1.3 -2.2 1.6 5.2e+03 0.001 0.016 0.43 +
24 0.13 -0.41 -1.3 -2.2 1.6 5.2e+03 0.00038 0.16 0.94 ++
25 0.13 -0.41 -1.3 -2.2 1.6 5.2e+03 0.00038 0.078 -5.2 -
26 0.13 -0.41 -1.3 -2.2 1.6 5.2e+03 0.00038 0.039 -1.5 -
27 0.13 -0.41 -1.3 -2.2 1.6 5.2e+03 0.00038 0.02 -0.19 -
28 0.13 -0.41 -1.3 -2.2 1.6 5.2e+03 0.00079 0.02 0.32 +
29 0.13 -0.41 -1.3 -2.2 1.6 5.2e+03 0.00079 0.0098 -0.57 -
30 0.13 -0.41 -1.3 -2.2 1.6 5.2e+03 0.00079 0.0049 0.024 -
31 0.12 -0.41 -1.3 -2.2 1.6 5.2e+03 0.00035 0.0049 0.77 +
32 0.12 -0.41 -1.3 -2.2 1.6 5.2e+03 0.00027 0.0049 0.49 +
33 0.12 -0.41 -1.3 -2.2 1.6 5.2e+03 0.00031 0.0049 0.31 +
34 0.12 -0.41 -1.3 -2.2 1.6 5.2e+03 0.0001 0.0049 0.59 +
Results saved in file 06unif_mixture_MHLS.html
Results saved in file 06unif_mixture_MHLS.pickle
print(results.short_summary())
Results for model 06unif_mixture_MHLS
Nbr of parameters: 5
Sample size: 6768
Excluded data: 3960
Final log likelihood: -5222.242
Akaike Information Criterion: 10454.48
Bayesian Information Criterion: 10488.58
pandas_results = results.get_estimated_parameters()
pandas_results
Total running time of the script: (0 minutes 14.186 seconds)