Numerical integration

Calculation of a mixtures of logit models where the integral is calculated using numerical integration.

Michel Bierlaire, EPFL Sat Jun 28 2025, 21:09:58

from IPython.core.display_functions import display
from biogeme.biogeme import BIOGEME
from biogeme.expressions import IntegrateNormal, RandomVariable
from biogeme.models import logit

from swissmetro_one 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,
)

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')
b_time_rnd = b_time + b_time_s * omega

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

util = {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}

The choice model is a logit, with availability conditions

integrand = logit(util, av, CHOICE)
numerical_integral = IntegrateNormal(integrand, 'omega')

simulate = {'Numerical': numerical_integral}

biosim = BIOGEME(database, simulate)

results = biosim.simulate(the_beta_values={})
display(results)

   Numerical
0    0.63785

print('Mixture of logit - numerical integration: ', results.iloc[0]['Numerical'])

Mixture of logit - numerical integration:  0.6378498356723438