Note
Go to the end to download the full example code
Numerical integration
Calculation of a mixtures of logit models where the integral is calculated using numerical integration.
- author:
Michel Bierlaire, EPFL
- date:
Thu Apr 13 20:51:32 2023
import biogeme.biogeme as bio
import biogeme.distributions as dist
from biogeme.expressions import RandomVariable, Integrate
from biogeme import models
from swissmetro_one import (
database,
TRAIN_TT_SCALED,
TRAIN_COST_SCALED,
SM_TT_SCALED,
SM_COST_SCALED,
CAR_TT_SCALED,
CAR_CO_SCALED,
TRAIN_AV_SP,
SM_AV,
CAR_AV_SP,
CHOICE,
)
Parameters
ASC_CAR = 0.137
ASC_TRAIN = -0.402
ASC_SM = 0
B_TIME = -2.26
B_TIME_S = 1.66
B_COST = -1.29
Define a random parameter, normally distributed, designed to be used for integration
omega = RandomVariable('omega')
density = dist.normalpdf(omega)
B_TIME_RND = B_TIME + B_TIME_S * 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}
The choice model is a logit, with availability conditions
integrand = models.logit(V, av, CHOICE)
numericalI = Integrate(integrand * density, 'omega')
simulate = {'Numerical': numericalI}
biosim = bio.BIOGEME(database, simulate)
results = biosim.simulate(theBetaValues={})
results
print('Mixture of logit - numerical integration: ', results.iloc[0]['Numerical'])
Mixture of logit - numerical integration: 0.6378498355784457
Total running time of the script: (0 minutes 0.021 seconds)