Note
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Mixture with lognormal distribution
Example of a mixture of logit models. The mixing distribution is distributed as a log normal. Compared to Mixture with lognormal distribution, the integration is performed using numerical integration instead of Monte-Carlo approximation.
- author:
Michel Bierlaire, EPFL
- date:
Mon Apr 10 12:13:23 2023
import biogeme.biogeme_logging as blog
import biogeme.biogeme as bio
import biogeme.distributions as dist
from biogeme import models
from biogeme.expressions import (
Beta,
RandomVariable,
exp,
log,
Integrate,
)
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 b17lognormal_mixture_integral.py')
Example b17lognormal_mixture_integral.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_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, -2, 2, 0)
Define a random parameter, log normally distributed, designed to be used for numerical integration.
omega = RandomVariable('omega')
B_TIME_RND = -exp(B_TIME + B_TIME_S * omega)
density = dist.normalpdf(omega)
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 to omega, we have a logit model (called the kernel).
condprob = models.logit(V, av, CHOICE)
We integrate over omega using numerical integration.
logprob = log(Integrate(condprob * density, 'omega'))
Create the Biogeme object.
the_biogeme = bio.BIOGEME(database, logprob)
the_biogeme.modelName = 'b17lognormal_mixture_integral'
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 __b17lognormal_mixture_integral.iter
Cannot read file __b17lognormal_mixture_integral.iter. Statement is ignored.
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 0 0 0 0 1 5.7e+03 0.096 0.5 -0.00035 -
1 0.5 -0.5 -0.5 0.5 1.5 5.4e+03 0.042 0.5 0.39 +
2 0 -0.34 -1 0.69 1.5 5.3e+03 0.04 0.5 0.34 +
3 0 -0.34 -1 0.69 1.5 5.3e+03 0.04 0.25 -0.63 -
4 0.25 -0.39 -1.2 0.44 1.3 5.2e+03 0.017 0.25 0.49 +
5 0.084 -0.38 -1.5 0.61 1.3 5.2e+03 0.0097 0.25 0.25 +
6 0.084 -0.38 -1.5 0.61 1.3 5.2e+03 0.0097 0.12 -2.2 -
7 0.084 -0.38 -1.5 0.61 1.3 5.2e+03 0.0097 0.062 0.093 -
8 0.15 -0.32 -1.4 0.55 1.2 5.2e+03 0.0068 0.062 0.57 +
9 0.17 -0.38 -1.4 0.61 1.2 5.2e+03 0.0065 0.062 0.18 +
10 0.18 -0.34 -1.3 0.56 1.3 5.2e+03 0.0023 0.062 0.44 +
11 0.18 -0.35 -1.4 0.57 1.2 5.2e+03 0.0018 0.062 0.42 +
12 0.18 -0.35 -1.4 0.57 1.2 5.2e+03 0.0018 0.031 -4.6 -
13 0.18 -0.35 -1.4 0.57 1.2 5.2e+03 0.0018 0.016 -1.7 -
14 0.18 -0.35 -1.4 0.57 1.2 5.2e+03 0.0018 0.0078 0.051 -
15 0.17 -0.35 -1.4 0.57 1.2 5.2e+03 0.00057 0.0078 0.55 +
16 0.17 -0.35 -1.4 0.57 1.2 5.2e+03 0.00057 0.0039 0.042 -
17 0.17 -0.35 -1.4 0.57 1.2 5.2e+03 0.0004 0.0039 0.56 +
18 0.17 -0.35 -1.4 0.57 1.2 5.2e+03 0.00014 0.0039 0.74 +
19 0.17 -0.35 -1.4 0.57 1.2 5.2e+03 0.00014 0.002 -0.22 -
20 0.17 -0.35 -1.4 0.57 1.2 5.2e+03 0.00024 0.002 0.2 +
21 0.17 -0.35 -1.4 0.57 1.2 5.2e+03 4.9e-05 0.002 0.77 +
Results saved in file b17lognormal_mixture_integral.html
Results saved in file b17lognormal_mixture_integral.pickle
print(results.short_summary())
Results for model b17lognormal_mixture_integral
Nbr of parameters: 5
Sample size: 6768
Excluded data: 3960
Final log likelihood: -5231.42
Akaike Information Criterion: 10472.84
Bayesian Information Criterion: 10506.94
pandas_results = results.get_estimated_parameters()
pandas_results
Total running time of the script: (0 minutes 11.245 seconds)